Electroweak Symmetry Breaking

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ELECTROWEAK SYMMETRY BREAKING

The electroweak theory proposed by Sheldon Glashow, Steven Weinberg, and Abdus Salam provides a unified description of the electromagnetic and weak forces. At first glance, such a unification hardly seems possible since these two forces mediate very different phenomena. Electromagnetism, for example, is responsible for binding electrons to nuclei in atoms and for binding atoms into molecules. The weak interactions, on the other hand, mediate the transmutation of neutrons into protons via reactions involving electrons and neutrinos, are responsible for the reactions that produce energy in stars, and cause the radioactive decay of unstable nuclei. As will be discussed, the differences between the everyday manifestations of electromagnetism and the weak force arise because the unified electroweak force exists in a broken phase.

Key differences between electromagnetism and the weak interactions are manifest in the properties of the force-carrying particles associated with the interactions. Electromagnetism results from the exchange of photons, quanta of light, whereas the weak interactions result from the exchange of three particles: the W⁺, the W^-, and an electrically neutral Z . All four force carriers have an intrinsic spin angular momentum of ℏ and are therefore referred to as spin-1 particles. The photon is a massless particle (which can never be at rest and travels at the "speed of light"), whereas the W and Z particles have masses of roughly eighty-five and a hundred times the mass of a proton (938 GeV/c²), respectively. The mass of a force carrier restricts the range over which the corresponding force can act. The massless photon gives rise to the long-range Coulomb potential, whereas the masses of the W and Z particles restrict the distance scale over which the weak interaction can be felt to of order 10^-16 cm (one one-thousandth the diameter of a proton).

Another, more subtle, distinction between electromagnetism and the weak force is that the weak interactions do not respect the symmetry of parity, whereas electromagnetic interactions do. Parity violation in the weak interactions was initially proposed by Tsung-Dao Lee and Chen Ning Yang in 1956 to explain some properties of the decay of particles that contain strange quarks. Murray Gell-Mann and Richard Feynman showed that parity violation arises because the charged weak force only affects particles whose spin points antiparallel to their direction of motion. In modern terms, this would be stated as follows: only quarks and leptons that are left-handed react via the exchange of W particles. The symmetry of parity involves changing the sign of all the spatial coordinates, from (x , y , z ) to (-x , -y , -z ), and under parity a left-handed particle would be exchanged for its right-handed complement. The mystery of parity violation in the weak interactions is compounded by the fact that the ordinary quarks and (charged) leptons have mass and may therefore be brought to rest—in which case, the distinction between left-handed and right-handed is meaningless!

Despite these physical differences between the two forces, Glashow, as well as Salam and John Ward, proposed that the electromagnetic and weak interactions were governed by one underlying gauge interaction. Electromagnetism was already known to display the property called gauge symmetry, arising from the fact that the photon is a spin-1 particle. In fact, gauge symmetry ensures the mathematical consistency of the quantum theory of electromagnetism, quantum electrodynamics. Such consistency was, apparently, only possible for massless spin-1 force-carrying particles like the photon. This was a potential stumbling block to electroweak unification: observed phenomena associated with the weak interactions implied that theW and Z particles must be massive. Moreover, as mentioned above, since theW particles only couple to left-handed matter particles, the mathematical consistency of the theory seemed to imply that all quarks and leptons must be massless, again in contradiction with observation.

Salam and Weinberg realized that an additional theoretical ingredient would enable the fundamental interactions to respect a combined electroweak gauge symmetry, while giving rise to massive W , Z , and matter particles. The key was gauge symmetry breaking, as described by Peter Higgs (and by P. Anderson in the context of superconductivity, and independently discovered by F. Englert and R. Brout, as well as G. Guralnik, C. Hagan, and T. Kibble). Building on the ideas of Jeffrey Goldstone and Yoichiro Nambu, Higgs realized that it is possible for the interactions to respect a gauge symmetry even though the ground state of the system governed by the interactions does not. Consider an ant moving on a surface shaped like a sombrero or the bottom of a wine bottle: perched on the hill at the center, the ant appreciates the rotational symmetry; standing at any point in the circular trough, the symmetry though present is not manifest. Salam and Weinberg realized that the Higgs mechanism of gauge symmetry breaking could be used to break an electroweak gauge symmetry. The fundamental forces would respect the unified symmetry, but this would not be manifest from the vantage point of the ground state in which one lives.

In particular, Salam and Weinberg proposed adding additional scalar fields (fields with spin 0 and therefore no spin angular momentum) to the gauge theory proposed by Glashow, in such a way that the lowest-energy state(s) would not manifest the electroweak gauge symmetry. The theory maintains that the direction in gauge field space picked out by the scalar field leaves one manifest symmetry, that associated with electromagnetism. Some of the additional states of the scalar field can be interpreted as the extra longitudinal polarization state that the massive W and Z particles possess, but the massless photon does not. Salam and Weinberg also showed that the scalar fields could couple to matter particles in such a way as to provide mass to the quarks and leptons as well.

An important prediction of the model proposed by Salam and Weinberg was that one of the additional scalar states should be a new, physical Higgs particle. The Higgs particle has the following property: it couples to pairs of W or Z bosons, quarks, or leptons with a coupling proportional to the masses of these particles. As shown by Gerard 't Hooft, the existence of this Higgs particle enables the electroweak theory to be renormalizable. In other words, the theory is mathematically consistent and calculations of physically observable quantities yield finite answers. Dozens of the predictions of the renormalizable electroweak theory about the properties of the W and Z bosons have been experimentally tested to a fraction of a percent, and thus far, all measurements are in accord with the theory.

The existence of the Higgs particle itself, however, has not yet been directly experimentally confirmed. Unfortunately, although its couplings to other particles are completely determined by the measured masses of those states, the mass of the Higgs particle is related to an arbitrary self-coupling constant and is essentially undetermined. Experimental searches currently (2002) restrict the mass of the Higgs to be higher than approximately 100 GeV/c² (slightly higher than the mass of the Z particle). The theory becomes inconsistent if the coupling constant determining the Higgs mass is too large, leading to an upper bound of order 800 GeV/c² on its mass.

Although a standard Higgs theory can accommodate electroweak symmetry breaking in a manner consistent with experimental data, such a theory does not explain the necessity for electroweak symmetry breaking or why it should occur at an energy scale that produces masses of order 100 GeV/c² for the W and Z bosons. Moreover, if one extends the theory to encompass additional physical dynamics at higher-energy scales that does address this issue, one finds that quantum mechanical corrections in the theory tend to force electroweak symmetry breaking to occur at the highest-energy scale pertinent to the theory. This instability of the symmetry-breaking scale is known as the gauge hierarchy problem. Two theoretical approaches have been taken in constructing a theory of electroweak symmetry breaking that does not suffer from the hierarchy problem: supersymmetry and strongly interacting symmetry breaking.

In a supersymmetric theory, one introduces a symmetry that relates matter (particles with spin ℏ/2, known as fermions) and forces (mediated by spin-0 or spin-1 particles, known collectively as bosons). For every fermion, quark, or lepton, in the Standard Model one introduces a boson, scalar squark, or slepton. For every gauge boson, one introduces a fermionic gauge particle, called a gaugino. For the scalar bosons associated with electroweak symmetry breaking (twice as many are required as in the model of Salam and Weinberg), one must add fermionic partners. The couplings of these new particles to each other and to the Standard Model particles are fixed by supersymmetry.

Once the superpartners are included in the calculation of the quantum corrections to the Higgs boson mass, it is found that contributions from bosonic and fermionic species cancel. More precisely, the gauge hierarchy is stabilized so long as the mass splittings between the ordinary particles and their superpartners are of order of the electroweak scale. Consequently, if supersymmetry is relevant to the hierarchy problem, the masses of the supersymmetric partners must lie below about 1,000 GeV/c². Moreover, in supersymmetric theories, the masses of the Higgs bosons are related to the gauge couplings (and possibly to other couplings as well). In the simplest models, the lightest Higgs boson must have a mass of less than approximately 130 GeV/c².

In a strongly interacting symmetry-breaking theory one introduces a new gauge interaction that becomes strong at an energy scale of order 1,000 GeV, as well as new, massless, weakly charged fermions that experience this new force. The simplest forms of this new interaction are modeled after quantum chromodynamics, the modern theory of the strong nuclear force. When the new force becomes sufficiently strongly interacting, it binds the new fermions together in a way that spontaneously breaks the electroweak symmetry. As a side effect, the binding can also produce additional particles (called technipions, by analogy with the pions of quantum chromodynamics) with masses in the range of a few hundred GeV/c². More generally, the new strong dynamics predicts an enhancement over the Standard Model value of the scattering cross section for W and Z bosons, and also the presence of numerous resonances in W and Z boson scattering with masses in the range of a few hundred to a few thousand GeV/c². The gauge hierarchy is rendered natural in this kind of theory by the fact that the new gauge coupling must evolve over many decades of energy scale before it becomes strong enough to break the electroweak symmetry.

Experiments planned for the Large Hadron Collider (LHC) currently under construction at the European Center for Particle Physics (CERN) are designed to shed light on the mechanism of electroweak symmetry breaking. If a Higgs boson exists, it will be found; if supersymmetry or new strong dynamics is related to electroweak symmetry breaking, the signatures of this new physics will be discovered. Most likely, uncovering the agent responsible for electroweak symmetry breaking will generate a new set of questions about the fundamental properties of nature.