## Higgs Phenomenon

## Higgs Phenomenon

# HIGGS PHENOMENON

In 1934 Enrico Fermi published a descriptive theory of the weak interactions. At the time these were the feeble forces seen at work in nuclear processes. An example is beta decay, in which the neutron decays into a proton, and an electron and a neutrino. This process is slow, and for neutrons trapped in atomic nuclei, it can range from much less than one decay per second to much greater than one decay per many millions of years. Fermi had to introduce a new fundamental constant into physics, later called *G*_{F} (the *F* stands for Fermi), that sets the scale of this process and that controls the overall decay rates in beta decay. This fundamental constant can be mathematically converted into a fundamental unit of mass, which sets the scale of the weak forces, and is approximately 175 GeV. (This equals about 175 times the proton mass; 1 GeV = 1 giga electron volt; energy is used to describe mass because *E* = *mc*^{2}; the proton has a mass of approximately 1 GeV). This is called the mass scale of the weak interactions.

In the intervening years physicists have come to understand a great deal about the weak forces. In the early 1970s the greatest stride along this path occurred when the Standard Model was theoretically and experimentally established. This is a true unified theory of weak, electromagnetic, strong, and gravitational forces under one fundamental symmetry principle, called the gauge principle. Like the discovery of DNA as the basic information carrier of all living things, the gauge principle is the basic underlying defining concept of all known forces in nature. Yet, despite this triumph, the origin of the scale of weak forces as embodied in Fermi's original theory, the 175 GeV, remains a subtle mystery.

The vacuum state in any quantum theory is complicated. Although it is the state of lowest energy, it is not empty and contains vibrational motion of all fundamental particles, known as quantum zero point motion. It is known that the vacuum itself can have bizarre physical properties leading to very dramatic consequences for the observed excited states, which are the particles found in nature. Indeed, nature is mostly controlled by the laws of physics together with the properties of the vacuum.

A superconductor is a block of metal, usually a relatively poor conductor of electricity at room temperatures (such as lead or nickel) that becomes a perfect conductor of electricity when it is cooled to within a few degrees above absolute zero. Superconductors can be readily made in the laboratory (they are used in many commercial devices, such as medical magnetic imaging systems, sensitive magnetometers, etc.). The phenomenon of superconductivity is a quantum effect. At very low temperatures the ground state (vacuum) of the superconductor is rearranged. Electrons become bound together into pairs, known as Cooper pairs, held together by quantum vibrations of the crystal lattice of the material (phonons). Each Cooper pair has an electric charge of -2, and the Cooper pairs act as though they were bosons, particles that can readily occupy the same quantum state (while free electrons are fermions, and no two fermions can occupy the same quantum state). The Cooper pairs form a kind of densely packed "quantum soup" in which every Cooper pair has exactly the same motion as every other. When a low-energy photon, the particle of light, enters the superconducting material, it blends together with the Cooper pair soup and becomes effectively a massive particle. Outside of a superconductor, in free space, the photon is perfectly massless. Hence, it always travels at the speed of light. However, in a superconductor a photon acts as if it were heavy, with a mass of about 1 electron volt, and, in principle, it can be brought to rest. This quantum condensation of the electrically charged Cooper pairs, and the concomitant mass generation for the photon, gives rise to the peculiar features of superconductors, for example, they have absolutely zero electrical resistance to current flow.

In the Standard Model of the electromagnetic and weak interactions (the electroweak theory) there are four gauge particles, including the γ or photon. If the symmetry of the electroweak theory were exact, these four particles would be identically massless. There are symmetry operations that are abstract mathematical "rotations" that allow one to rotate one particle into another in the electroweak theory. These rotations do not occur in ordinary space and time but rather in an abstract mathematical world known as the internal symmetry group of the electroweak theory. The dynamics of these particles, for example, their interactions, masses, etc., is unaffected by these symmetry rotations, just as the color or shape of a chess piece is unchanged when it is rotated in space.

However, at low energies these four particles all behave very differently. The γ, or photon, remains massless in free space in the Standard Model and can be described at low energies by itself in the context of quantum electrodynamics (QED). However, the three other particles, closely related to the photon, are the *W*^{+}, *W*^{-}, and *Z*^{0}. These particles are very heavy in free space: the masses of the *W*^{+} and *W*^{-} (representing the particle and antiparticle and therefore having the same mass) are approximately 80.419 ± 0.056 GeV/*c*^{2}, while the *Z*^{0} is heavier still with a mass of 91.1881 ± 0.0022 GeV/*c*^{2}. The forces that are mediated by the quantum exchange of *W* 's between other particles are exactly the weak forces that Fermi's early theory of beta decay described. Indeed, the weak forces are weak because the *W*^{±} (and *Z* ) are very heavy, and the quantum exchange of heavy particles is a very short-range interaction. The differences between the four particles γ, *W*^{±}, and *Z*^{0} mean that the abstract symmetry interrelating them is broken. The symmetry becomes apparent only at very high energies, energies much higher than the masses of *W*^{±} and *Z*^{0}.

What physical mechanism breaks the symmetry of the electroweak theory at low energies and gives rise to the masses of *W*^{±} and *Z*^{0}? Indeed, it is natural to take a cue from the phenomenon of the superconductor. One conceives of some kind of quantum effect, analogous to what occurs in a superconductor causing the photon to become heavy, but now acting in the vacuum of free space and acting everywhere throughout the universe. This phenomenon must give the *W* 's and *Z* their masses but unlike the superconductor must leave the photon massless. Therefore, whatever undergoes "condensation" in the vacuum must be an electrically neutral particle (unlike the Cooper pairs of the superconductor, which had a net electric charge of -2 and thus affect the photon).

Hence, the question becomes "What condenses in the vacuum to give rise to mass in the Standard Model?" Physicists often build "toy mathematical models" to explain a phenomenon, awaiting additional experimental or theoretical information that will lead to an exact theory of the phenomenon. The toy model is usually incomplete but contains the essence of the gross features of the phenomenon. Indeed, before the correct theory of superconductors (resulting from the work of John Bardeen, Leon Cooper, and John Schreiffer) was constructed, a "toy" model that explained superconductivity was proposed by Vitalii Ginzburg and Lev Landau, building on the earlier ideas of Fritz London. This idea was adapted to particle physics by many authors to give mass to particles such as the *W* and *Z* and has come to become known as the Higgs mechanismm, named after Peter Higgs of the University of Edinburgh, one of its early proponents. Steven Weinberg incorporated the Higgs mechanism into his famous paper "A Model of Leptons," which was one of the earliest works to construct the electroweak Standard Model.

In the Standard Model, to explain the symmetry breaking and masses of *W*^{±} and *Z*^{0}, one introduces a Higgs field. The Higgs field forms what is called a complex doublet and has an electrically neutral component that develops a condensate in the vacuum. The dynamics of the formation of the condensate is largely put in the model "by hand," awaiting a detailed explanation from future experiments. The condensate may be viewed as a nonzero value of the field filling all of space throughout the universe, analogous to an electric or magnetic field filling all of space. The strength of the Higgs field in the vacuum is measured as an energy, and it is postulated to be exactly the Fermi scale, 175 GeV.

The complex doublet Higgs field has four dynamical components (two complex numbers), and three of these components become blended to form the massive *W*^{±} and *Z*^{0}. One remaining component corresponds to small local changes in the vacuum field strength of the condensate. This remaining part of the doublet can show up in the laboratory as a heavy, electrically neutral, spin-0 particle. This particle is often referred to as the Higgs boson although it is really a part of the original Higgs field.

The vacuum condensate is felt by the various particles as they propagate through the vacuum, by their coupling strengths to the Higgs field. This gives rise to their masses. For example, the electron has a coupling strength *g*_{e}. The electron mass is then determined to be *m*_{e} = *g*_{e} (175 GeV). Since *m*_{e} = 0.0005 GeV, *g*_{e} = 0.0005/175 = 0.0000029. This is a very feeble coupling strength, so the electron is a very low mass particle. Other particles, like the top quark that has a mass *m*_{t} ≈ 175 GeV have a coupling strength to the Higgs field that is almost identically equal to 1. Still other particles, like neutrinos, have nearly zero masses and therefore nearly zero coupling strengths.

The Standard Model does not predict in any fundamental way the values of the coupling strengths of quarks and leptons to the Higgs field, that is, these numbers are also put into the theory "by hand." The Standard Model does, however, predict the coupling strength of the *W*^{±} and *Z*^{0} particles to the Higgs, so their masses *M*_{W} and *M*_{Z} are predicted (correctly) by the theory. The couplings of the particles *W*^{±} and *Z*^{0}, called gauge particles, have coupling strengths that are related to known quantities, such as the electric charge *e* and the weak mixing angle *θ*_{W}, which are directly measured in various experiments. Thus, if one measures *e,* and *θ*_{W} and *G*_{F}, the Standard Model correctly and precisely predicts *M*_{W} and *M*_{Z}. Indeed, apart from the properties of the Higgs field and coupling strengths of quarks and leptons to the Higgs field, the Standard Model explains correctly and precisely all the phenomena seen in weak and electromagnetic interactions.

The Standard Model as a quantum theory has been subject to precise tests by experiments at LEP at the European Laboratory for Particle Physics (CERN), Tevatron at Fermilab, and SLC at the Stanford Linear Accelerator Center (SLAC). The Higgs boson, if it exists according to the simple mathematical model, has not yet been seen and is therefore heavier than an experimental lower limit from LEP-II of 115 GeV/*c*^{2}. One can infer an approximate bound on the allowed mass of the Higgs boson from indirect precision measurements of *M*_{Z}, *M*_{W}, and *m*_{top}, and one finds that the Higgs boson should not be heavier than approximately ~200 GeV. This assumes that the Higgs boson is a weakly coupled fundamental particle and that no additional physics is involved in the symmetry-breaking mechanism of the electroweak theory.

So what is the Higgs field in reality? Beyond the simple mathematical model, nothing is certain. However, it is clear there must be something that either really is the Higgs field or that imitates one in a very faithful way. Physicists do know that, with a sufficiently high-energy particle accelerator, they can produce a Higgs-boson-like particle or dynamics in the laboratory.

Two theoretical possibilites have been advanced for the true dynamical origin of the Higgs field. One possibility is that there exists a larger symmetry than the Standard Model structure, known as supersymmetry. Supersymmetry is a very compelling idea for a large number of reasons beyond the scope of this discussion. Supersymmetry is intimately connected with theories of quantum gravity, called superstring theories. In supersymmetry spin-½ particles must be related to spin-0 particles, and hence the Higgs field must be associated with additional, as yet undiscovered, spin-½ particles that would appear as heavy leptons, like the electron and neutrino. Supersymmetry has many desirable theoretical properties, and since it can readily accommodate the Higgs fields, it is perhaps the most popular theory involving the Higgs boson. In supersymmetry there are several Higgs fields, each one of which is a truly fundamental pointlike elementary particle. The lowest-mass Higgs boson could appear in experiments fairly soon with a mass less than of order 140 GeV/*c*^{2} (some evidence for a low-mass Higgs boson may have been observed in 2000 at the end of LEP-II at CERN at a mass scale of 115 GeV/*c*^{2}).

Another possibility is that the Higgs field is composite, associated with new strong dynamics, that is, it is a bound state of other elementary particles, held together by new forces. This idea is closer to the dynamical phenomena that occur within superconductors and has recently been seen to work well with the idea of extra unseen compact dimensions of space at or near the electroweak scale, of order 1 TeV. One possibility is that the strong interactions, described by the theory of gluons and quarks, when extended to extra dimensions can naturally form a bound state of top and bottom quarks and antiquarks (and possibly their excitations in the extra dimensions, known as Kaluza-Klein modes) that has exactly the correct properties to be the Higgs boson. This theory, known as the top quark seesaw model, may ultimately explain why the top quark is much heavier than other quarks and leptons. Although the Higgs boson is characteristically heavy in these schemes, of order 1 TeV, the theories remain consistent with the precision limits because they are nonminimal and contain many additional particles and interactions. In some versions of new strong dynamics, new low-mass spin-0 particles can occur, known as pseudo-Nambu–Goldstone bosons, and these might be confused early on for Higgs bosons.

The Higgs boson, if it is of low mass and in accord with supersymmetry, may be discovered at the current Run-II of the Tevatron (Fermilab). In 2007 the Large Hadron Collider (LHC) at the European Laboratory for Particle Physics (CERN) will begin operations at seven times the energy of the Tevatron, will explore a larger range of Higgs boson masses, and can detect evidence of a new strong dynamics as well as supersymmetry. Beyond these explorations, higher-energy accelerators, such as a Very Large Hadron Collider (VLHC) or an *e*^{+}*e*^{-} Linear Collider, will be required to unravel the details of the true mass-generation mechanism of the Standard Model.

*See also:*Boson, Higgs; Electroweak Symmetry Breaking; Experiment: Search for the Higgs Boson; Particle Physics, Elementary; Standard Model; Supersymmetry; Technicolor

## Bibliography

Bardeen, J.; Cooper, L. N.; and Schrieffer, J. R. "Theory of Superconductivity." *Physical Review***108** , 1175 (1957).

Ellis, J. R. "Supersymmetry for Alp Hikers." <http://arxiv.org/PS_cache/hep-ph/pdf/0203/0203114.pdf>.

Fermi, E. "An Attempt of a Theory of Beta Radiation." *Zeitschrift für Physik***88** , 161 (1934).

Ginzburg, V. L., and Landau, L. D. "On the Theory of Superconductivity." *Journal of Experimental and Theoretical Physics Letters***20** , 1064 (1950).

Hill, C. T., and Simmons, E. H. "Strong Dynamics and Electroweak Symmetry Breaking." http://arxiv.org/PS_ cache/hep-ph/pdf/0203/0203079.pdf

Weinberg, S. "A Model of Leptons." *Physical Review Letters***19** , 1264 (1967).

*Christopher T. Hill*

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