The Higgs Boson: Searching for the God Particle

by Scientific American Editors

  As in other instances where a global symmetry is converted into a local one, the invariance can be maintained only if something more is added to the theory. Because the Yang-Mills theory is more complicated than earlier gauge theories it turns out that quite a lot more must be added. When isotopic-spin rotations are made arbitrarily from place to place, the laws of physics remain invariant only if six new fields are introduced. They are all vector fields, and they all have infinite range.

  The Yang-Mills fields are constructed on the model of electromagnetism, and indeed two of them can be identified with the ordinary electric and magnetic fields. In other words, they describe the field of the photon. The remaining Yang-Mills fields can also be taken in pairs and interpreted as electric and magnetic fields, but the photons they describe differ in a crucial respect from the known properties of the photon: they are still massless spin-one particles, but they carry an electric charge. One photon is negative and one is positive.

  The imposition of an electric charge on a photon has remarkable consequences. The photon is defined as the field quantum that conveys electromagnetic forces from one charged particle to another. If the photon itself has a charge, there can be direct electromagnetic interactions among the photons. To cite just one example, two photons with opposite charges might bind together to form an "atom" of light. The familiar neutral photon never interacts with itself in this way.

  The surprising effects of charged photons become most apparent when a local symmetry transformation is applied more than once to the same particle. In quantum electrodynamics, as was pointed out above, the symmetry operation is a local change in the phase of the electron field, each such phase shift being accompanied by an interaction with the electromagnetic field. It is easy to imagine an electron undergoing two phase shifts in succession, say by emitting a photon and later absorbing one. Intuition suggests that if the sequence of the phase shifts were reversed, so that first a photon was absorbed and later one was emitted, the end result would be the same. This is indeed the case. An unlimited series of phase shifts can be made, and the final result will be simply the algebraic sum of all the shifts no matter what their sequence.

  In the Yang-Mills theory, where the symmetry operation is a local rotation of the isotopic-spin arrow, the result of m ultiple transformations can be quite different. Suppose a hadron is subjected to a gauge transformation, A, followed soon after by a second transformation, B; at the end of this sequence the isotopic-spin arrow is found in the orientation that corresponds to a proton. N ow suppose the same transformations were applied to the same hadron but in the reverse sequence: B followed by A. In general the final state will not be the same; the particle may be a neutron instead of a proton. The net effect of the two transformations depends explicitly on the sequence in which they are applied.

  Because of this distinction quantum electrodynamics is called an Abelian theory and the Yang-Mills theory is called a non-Abelian one. The terms are borrowed from the mathematical theory of groups and honor Niels Henrik Abel, a Norwegian mathematician who lived in the early years of the 19th century. Abelian groups are made up of transformations that, when they are applied one after another, have the commutative property; non-Abelian groups are not commutative.

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  EFFECTS OF REPEATED TRANSFORMATIONS distinguish quantum electrodynamics, which is an Abelian theory, from the Yang-Mills theory, which is non-Abelian. An Abelian transformation is commutative: if two transformations are applied in succession, the outcome is the same no matter which sequence is chosen. An exmple is rotation in two dimensions. Non-Abelian transformations are not commutative, so that two transformations will generally yield different results if their sequence is reversed. Rotations in three dimensions exhibit this dependence on sequence. Quantum electrodynamics is Abelian in that successive phase shifts can be applied to an electron field without regard to the sequence. The Yang-Mills theory is non-Abelian because the net effect of two isotopic-spin rotations is generally different if the sequence of rotations is reversed. One sequence might yield a proton and the opposite sequence a neutron.

  Illustration by Allen Beechel

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  Commutation is familiar in arithmetic as a property of addition and multip lication, where for any numbers A and B it can be stated that A + B = B + A and A X B = B X A. How the principle can be applied to a group of transformations can be ill ustrated with a familiar example: the group of rotations. All possible rotations of a two-dimensional object are commutative, and so the group of such rotations is Abelian. For instance, rotations of + 60 degrees and -90 degrees yield a net rotation of -30 degrees no matter which is applied first. For a three-dimensional object free to rotate about three axes the commutative law does not hold, and the group of three-dimensional rotations is nonAbelian. As an example, consider an airplane heading due north in level flight. A 90-degree yaw to the left followed by a 90-degree roll to the left leaves the airplane heading west with its left wing tip pointing straight down. Reversing the sequence of transformations, so that a 90-degree roll to the left is followed by a 90-degree left yaw, puts the airplane in a nose dive with the wings aligned on the north-south axis.

  Like the Yang-Mills theory, the general theory of relativity is non-Abelian: in making two succe ssive coordinate transformations, the order in which the y are made usually has an e ffect on the outcome. In the past 10 years or so several more non-Abelian theories have been devised, and even the electromagnetic interactions have been incorporated into a larger theory that is non-Abelian. For now, at least, it seems all the forces of nature are governed by non-Abelian gauge the ories.

  The Yang-Mills theory has proved to be of monumental importance, but as it was originally formulated it was totally unfit to describe the real world. A first objection to it is that isotopic-spin symmetry becomes exact, with the result that protons and neutrons are indistinguishable; this situation is obviously contrary to fact. Even more troubling is the prediction of electrically charged photons. The photon is necessarily massless because it must have an infinite range. The existence of any electrically charged particle lighter than the electron would alter the world beyond recognition. Of course, no such particle has been observed. In spite of these difficulties the theory has great beauty and philosophical appeal. One strategy adopted in an attempt to fix its defects was to artificially endow the charged field quanta with a mass greater than zero.

  Imposing a mass on the quanta of the charged fields does not make the fields disappear, but it does confine them to a finite range. If the mass is large enough, the range can be made as small as is wished. As the long-range effects are removed the existence of the fields can be reconciled with experimental observations. Moreover, the selection of the neutral Yang-Mills field as the only real long-range one automatically distinguishes protons from neutrons. Since this field is simply the electromagnetic field, the proton and the neutron can be distinguished by their differing interactions with it, or in other words by their differing electric charges.

  With this modification the local symmetry of the Yang-Mills theory would no longer be exact but approximate, since rotation of the isotopic-spin arrow would now have observable consequences. That is not a fundamental objection: approximate symmetries are quite commonplace in nature. (The bilateral symmetry of the human body is only approximate.) Moreover, at distance scales much smaller than the range of the massive components of the Yang-Mills field, the local symmetry becomes better and better. Thus in a sense the microscopic structure of the theory could remain locally symmetric, but not its predictions of macroscopic, observable events.

  The modified Yang-Mills theory was easier to understand, but the theory still had to be given a quantum-mechanicalinterpretation. The problem of infinities turned out to be severer than it had been in quantum electrodynamics, and the standard recipe for renormalization would not solve it. New techniques had to be devised.

  An important idea was introduced in 1963 by Feynman: it is the notion of a "
ghost" particle, a particle added to a theory in the course of a calculation that vanishes when the calculation is finished. It is known from the outset that the ghost particle is fictitious, but its use can be justified if it never appears in the final state. This can be ensured by making certain the total probability of producing a ghost particle is always zero.

  Among theoretical groups that continued work on the Yang-Mills theory the ghost-particle method was taken seriously only at the University of Utrecht, where I was then a student. Martin J. G. Veltman, my thesis adviser, together with John S. Bell of the E uropean Organization for Nuclear Research (CERN) in Geneva, was led to the conclusion that the weak interactions might be described by some form of the Yang-Mills theory. He undertook a systematic analysis of the renormalization problem in the modified Yang-Mills model (with massive charged fields), examining each class of Feynman diagrams in turn. The diagrams having no closed loops were readily shown to make only finite contributions to the total interaction probability. The diagrams with one loop do include infinite terms, but by exploiting the properties of the ghost particles it was possible to make the positive infinities and the negative ones cancel exactly.

  As the number of loops increases, the number of diagrams rises steeply; moreover, the calculations required for each diagram become more intricate. To assist in the enormous task of checking all the two-loop diagrams a computer program was written to handle the algebraic manipulation of the probabilities. The output of the program is a list of the coefficients of the infinite quantities remaining after the contributions of all the diagrams have been summed. If the infinities are to be expunged from the theory, the coefficients must without exception be zero. By 1970 the results were known and the possibility of error had been excluded; some infinities remained.

  The failure of the modified Yang-Mills theory was to be blamed not on any defect in the Yang-Mills formulation itself but rather on the modifications. The masses of the charged fields had to be put in "by hand" and as a result the invariance with respect to local isotopic-spin rotations was not quite perfect. It was suggested at the time by the Russian investigators L. D. Faddeev, V. N. Popov, E. S. Fradkin and I. V. Tyutin that the p ure Yang-Mills theory, with only massless fields, could indeed be renormalized. The trouble with this theory is that it not only is unrealistic but also has long-range fields that are difficult to work with.

  In the meantime another new ingredient for the formulation of gauge theories had been introduced by F. Englert and Robert H. Brout of the University of Brussels and by Peter Higgs of the University of Edinburgh. They found a way to endow some of the Yang-Mills fields with mass while retaining exact gauge symmetry. The technique is now called the Higgs mechanism.

  The fundamental idea of the Higgs mechanism is to include in the theory an extra field, one having the peculiar property that it does not vanish in the vacuum. One usually thinks of a vacuum as a space with nothing in it, but in physics the vacuum is defined more precisely as the state in which all fields have their lowest possible energy. For most fields the energy is minimized when the value of the field is zero everywhere, or in other words when the field is "turned off. " An electron field, for example, has its minimum energy when there are no electrons. The Higgs field is unusual in this respect. Reducing it to zero costs energy; the energy of the field is smallest when the field has some uniform value greater than zero.

  The effect of the Higgs field is to provide a frame of reference in which the orientation of the isotopic-spin arrow can be determined. The Higgs field can be represented as an arrow superposed on the other isotopic-spin indicators in the imaginary internal space of a hadron. What distinguishes the arrow of the Higgs field is that it has a fixed length, established by the vacuum value of the field. The orientation of the other isotopic-spin arrows can then be measured with respect to the axis defined by the Higgs field. In this way a proton can be distinguished from a neutron.

  It might seem that the introduction of the Higgs field would spoil the gauge symmetry of the theory and thereby lead again to insoluble infinities. In actuality, however, the gauge symmetry is not destroyed but merely concealed. The symmetry specifies that all the laws of physics must remain invariant when the isotopic-spin arrow is rotated in an arbitrary way from place to place. This implies that the absolute orientation of the arrow cannot be determined, since any experiment for measuring the orientation would have to detect some variation in a physical quantity when the arrow was rotated. With the inclusion of the Higgs field the absolute orientation of the arrow still cannot be determined because the arrow representing the Higgs field also rotates during a gauge transformation. All that can be measured is the angle between the arrow of the Higgs field and the other isotopic-spin arrows, or in other words their relative orientations.

  The Higgs mechanism is an example of the process called spontaneous symmetry breaking, which was already well established in other areas of physics. The concept was first put forward by Werner Heisenberg in his description of ferromagnetic materials. Heisenberg pointed out that the theory describing a ferromagnet has perfect geometric symmetry in that it gives no special distinction to any one direction in space. When the material becomes magnetized, however, there is one axis–the direction of magnetization–that can be distinguished from all other axes. The theory is symmetrical but the object it describes is not. Similarly, the Yang-Mills theory retains its gauge symmetry with respect to rotations of the isotopic-spin arrow, but the objects described–protons and neutrons–do not express the symmetry.

  How does the Higgs mechanism lend mass to the quanta of the Yang-Mills field? The process can be explained as follows. The Higgs field is a scalar quantity, having only a magnitude, and so the quantum of the field must have a spin of zero. The Yang-Mills fields are vectors, like the electromagnetic field, and are represented by spin-one quanta. Ordinarily a particle with a spin of one unit has three spin states (oriented parallel, antiparallel and transverse to its direction of motion), but because the Yang-Mills particles are massless and move with the speed of light they are a special case; their transverse states are missing. If the particles were to acquire a mass, they would lose this special status and all three spin states would have to be observable. In quantum mechanics the accounting of spin states is strict and the extra state must come from somewhere; it comes from the Higgs field. Each Yang-Mills quantum coalesces with one Higgs particle; as a result the Yang-Mills particle gains mass and a spin state, whereas the Higgs particle disappears. A picturesque description of this process has been suggested by Abdus Salam of the International Center for Theoretical Physics in Trieste: the massless Yang-M ills particles "eat" the Higgs particles in order to gain weight, and the swallowed Higgs particles become ghosts.

  In 1971, Veltman suggested that I investigate the renormalization of the pure Yang-Mills theory. The rules for constructing the needed Feynman diagrams had already been formulated by Faddeev, Popov, Fradkin and Tyutin, and independently by Bryce S. DeWitt of the University of Texas at Austin and Stanley Mandelstam of the University of California at Berkeley. I could adapt to the task the powerful methods for renormalization studies that had been developed by Veltman.

  Formally the results were encouraging, but if the theory was to be a realistic one, some means had to be found to confine the Yang-Mills fields to a finite range. I had just learned at a summer school how Kurt Symanzik of the German Electron Synchrotron and Benjamin W. Lee of the Fermi National Accelerator Laboratory had successfully hand led the renormalization of a theoretical model in which a global symmetry is spontaneously broken. It therefore seemed natural to try the Higgs mechanism in the Yang-Mills theory, where the broken symmetry is a local one.

  A few simple models gave encouraging results: in these selected instances all infinities canceled no matter how many gauge particles were exchanged and no matter how many loops were included in the Feynman diagrams. The decisive test would come when the theory was checked by the comp uter program for infinities in all p
ossible diagrams with two loops. The results of that test were available by July, 1971; the output of the program was an uninterrupted string of zeros. Every infinity canceled exactly. Subsequent checks showed that infinities were also absent even in extremely complicated Feynman diagrams. My results were soon confirmed by others, notably by Lee and by Jean Zinn-Justin of the Saclay Nuclear Research Center near Paris.

  The Yang-Mills theory had begun as a model of the strong interactions, but by the time it had been renormalized interest in it centered on applications to the weak interactions. In 1967 Steven Weinberg of Harvard University and independently (but later) Salam and John C. Ward of Johns Hopkins University had proposed a model of the weak interactions based on a version of the Yang-Mills theory in which the gauge quanta take on mass through the Higgs mechanism. They speculated that it might be possible to renormalize the theory, but they did not demonstrate it. Their ideas therefore joined many other untested conjectures until some four years later, when my own results showed it was just that subclass of Yang-Mills theories incorporating the Higgs mechanism that can be renormalized.

  The most conspicuous trait of the weak force is its short range: it has a significant influence only to a distance of 10-15 centimeter, or roughly a hundredth the radius of a proton. The force is weak largely because its range is so short: particles are unlikely to approach each other closely enough to interact. The short range implies that the virtual particles exchanged in weak interactions must be very massive. Present estimates run to between 80 and 100 times the mass of the proton.

  The Weinberg-Salam-Ward model actually embraces both the weak force and electromagnetism. The conjecture on which the model is ultimately founded is a postulate of local invariance with respect to isotopic spin; in order to preserve that invariance four photon-like fields are introduced, rather than the three of the original Yang-Mills theory. The fourth photon could be identified with some primordial form of electromagnetism. It corresponds to a separate force, which had to be added to the theory without explanation. For this reason the model should not be called a unified field theory. The forces remain distinct; it is their intertwining that makes the model so peculiar.