A serious problem arises immediately with the Yang–Mills theory and renormalizability: not all force-carrying bosons are massless! What does this do for the essential feature of gauge invariance in quantum field theory? The immediate answer is that it destroys the gauge invariance, and therefore the theory is no longer renormalizable and no longer conserves scattering probabilities. This problem was recognized already by Yang and Mills, and has been a topic of research since the 1950s, starting with Schwinger and Lee and Yang, and continues to be pursued by many physicists up to the present day. One possible resolution of this problem has been the development of the standard-model electroweak interactions involving spontaneous symmetry breaking and a Higgs particle. The idea of a Higgs boson whose lowest energy state breaks spontaneously the basic group symmetry of SU(2) × U(1) and allows for different masses of the W and Z bosons to emerge within the standard-model electroweak theory, while keeping the photon massless, has been the most popular resolution of this problem.
Fortunately, QCD, the theory of the strong interactions of particles, developed in 1973 by Gell-Mann, Fritzsch, and Leutwyler, using nonabelian SU(3) quantum field theory with eight colored gluons, does not suffer from the consequences of having massive force-carrying bosons because the eight colored gluons are massless. It can be demonstrated that this strong-interaction theory has an extended form of gauge invariance, and the Ward identities in it have been generalized by Yasushi Takahashi, Andrei A. Slavnov, and John C. Taylor. The technical issues involved in proving the renormalizability of the nonabelian gauge theory for strong interactions are somewhat formidable. Feynman, in his research on quantum gravity, which is a form of nonabelian gauge theory, with gravitons being the massless force carriers of gravity, discovered that certain “ghost” fields had to be included in the calculations of QCD to guarantee renormalizability. Russian theoretical physicists Ludvig Faddeev and Victor Popov played an important role in developing the mathematics of these ghost fields, which actually do not appear as physical particles in calculations of the scattering amplitudes. A new gauge symmetry was discovered by Carlo Maria Becchi, Alain Rouet, and Raymond Stora, and independently by Igor Tyutin (called BRST) that clarifies the deeper meaning of how these ghost fields enter into the renormalizable nonabelian calculations.
The problem of the impossibility of having a renormalizable quantum field theory of weak interactions became critical during the late 1950s and early 1960s. A massive intermediate charged vector boson, W, which was necessary to make the weak interaction a short-range force, destroyed any possibility of having a renormalizable and finite theory of weak interactions. Because the W was massive, the basic gauge invariance of the theory was lost. In a seminal paper published by Sheldon Glashow in 1961,4 in which he introduced the need for electrically neutral currents in weak interactions, mediated by the exchange of a neutral vector particle called Z, he put the masses of the W and the Z “by hand” into the weak-interaction model. He was aware that this would ruin the possibility of getting finite scattering amplitudes involving the W and the Z particles, and the leptons such as electrons and muons. If one entertained the idea that weak interactions began in some phase in the early universe with massless particles, then this initial phase of the theory could be gauge invariant, renormalizable, and finite, just like QED with the massless photon. But, then, where would the masses of the W and the Z and the fermions come from?
SOLID-STATE PHYSICS TO PARTICLE PHYSICS
During the early 1960s, more theoretical physicists began to speculate on the nature of symmetry breaking in quantum theory. Previously, Werner Heisenberg in Germany, among others, had found an interesting new way of explaining ferromagnetism. He viewed atoms in a metal, such as iron, as little bar magnets with a north pole and a south pole. When the temperature of the metal was above a critical point called the Curie temperature, the little bar magnets were oriented randomly and the system was rotationally invariant under rotations of the group O(3). Below the critical Curie temperature, the little bar magnets aligned themselves in a certain direction, and the rotational symmetry of O(3) was broken spontaneously.
Jeffrey Goldstone, a theorist who was once a colleague of mine at Trinity College Cambridge, theorized that this phenomenon of spontaneous symmetry breaking in solid-state physics could be imported into relativistic quantum field theory. He published a paper in 19615 in which he demonstrated that, within a certain quantum field theory model involving a scalar spin-0 field like Schwinger’s sigma field, you could invoke spontaneous symmetry breaking of a mathematical group. This symmetry breaking always predicted a massless, scalar spin-0 particle. A similar result had been obtained by Yoichiro Nambu in 1960, and by Nambu and Giovanni Jonah-Lasinio in 1961, when they investigated the nature of superconductors, which are metals cooled to such a low temperature that the electrical resistance of the electrons moving through the metal approaches zero, meaning that the electrons move freely through the metal. Nambu discovered that if the superconducting nature of the material was caused by spontaneous symmetry breaking of a symmetry at higher temperatures, then this predicted the existence of a massless spin-0 particle. These ideas became known as Goldstone’s theorem or, later, as the Nambu-Goldstone theorem.
However, if you invoke this phenomenon in quantum field theory and try to relate it to reality, then you run into the problem of predicting the existence of a massless particle that had never been observed. The only massless particles known at the time were the photon, which has spin 1, not spin 0, and the neutrino, with spin ½. This situation led to an impasse in theoretical particle physics that lasted about three years. Salam and Weinberg, who met with Goldstone when he visited Harvard, investigated this problem and tried to resolve it, but were unable to do so successfully.6 Another theorist, my former colleague at Trinity College Cambridge, Walter Gilbert, published a paper claiming that spontaneous symmetry breaking in particle physics could not be considered physical.7 With this impasse, it was not possible even to begin to attempt to explain how the W and Z bosons in weak interactions got their masses, because any such mechanism would be accompanied by these nonexistent, massless scalar particles.
Meanwhile, back in solid-state physics, Philip Anderson, a theorist at Princeton, discovered that when you invoke spontaneous symmetry breaking to explain superconductivity in materials, the photon becomes massive. The spontaneous symmetry-breaking mechanism had generated an “effective mass” for the photon without producing the pesky massless scalar particles suggested by the Nambu-Goldstone theorem. How had this come about?
In 1963, Anderson published a paper8 explaining that the reason this happened was because the spontaneous breaking of the U(1) abelian symmetry occurred in such a way that the massless photon field “ate” the Nambu-Goldstone scalar field boson, thereby putting on weight and acquiring mass. In Anderson’s paper, he refers to a seminal paper titled “Gauge Invariance and Mass,” published by Julian Schwinger in Physical Review in 1962,9 in which Schwinger discusses the significance of gauge invariance and the origin of masses of the elementary particles. It is noteworthy that at the end of Anderson’s paper, he discusses relativistic Yang–Mills gauge bosons and the Goldstone bosons in a relativistic context. He ends the article with the prophetic statement, “We conclude, then, that the Goldstone zero-mass difficulty is not a serious one, because we can probably cancel it off against an equal Yang–Mills zero-mass problem.” We see that Anderson uses the language of relativistic field theory. This indicates that Anderson comprehended the importance of a so-called Higgs mechanism a year before the publication of the papers in Physical Review Letters by Brout and Englert,10 Higgs,11 Guralnik, Hagen, and Kibble12 in 1964. Therefore, it is perhaps justified to include Philip Anderson’s name among those who discovered the Higgs mechanism. However, Anderson’s work was in the context of nonrelativistic solid-state physics. Particle physicists generally ignored solid-state physics. In fact, Gell-Mann was known to call it “squalid-state physics.”
How
physicists realized that the photon inside the superconductor actually became a heavy photon is a long story. The phenomenon of superconductivity was first discovered in 1908 by Dutch physicist Heike Kamerlingh Onnes. In contrast to normal electrical conducting materials, which have some resistance to electrical currents, when “superconductors” are cooled close to 0 K—for example, liquid nitrogen at temperatures of −321°F, or 77 K—they have no resistance to electrical currents. At very low temperatures, the material reaches a critical temperature, which varies with each superconducting material, at which the electrons can travel through the material without losing heat or energy. The atoms in these supercooled materials assume the configuration of a lattice, with the electrons moving freely among the atoms within the lattice. This is how metals conduct heat and electricity. As the metal cools down, the repulsion between two electrons is reduced, and they actually bind together to form what is called Cooper pairs, named after theoretical physicist Leon Cooper, who did a theoretical investigation of superconductivity. In 1957, John Bardeen, Cooper, and John Robert Schrieffer published a paper13 in which they explained the origin of superconductivity in metals by means of the Cooper pairs of electrons. It is the Cooper pairs, which form a composite “condensate” of pairs of electrons, that enable the electrons to move freely through the metal without resistance.
Another significant physical phenomenon associated with superconductors is the Meissner effect, which was discovered by German physicists Walther Meissner and Robert Ochsenfeld14 in 1933 by measuring the magnetic field outside superconducting tin and lead samples. They discovered that the superconducting metals expelled the magnetic fields—actually pushed the fields away in space. These experiments demonstrated that superconductors have a unique physical property beyond reducing electrical resistance to zero.
The history of the theory of superconductors is a long one. Original work done by Vitaly Ginzberg and Lev Landau, first published in 1950, seven years before the publication of the paper by Bardeen, Cooper, and Schrieffer, provided an effective theory of how superconductors work.15 In the Ginzberg–Landau theory, an “order parameter” measured the degree of transition the metal attained when it went through a critical temperature threshold, giving rise to what is called a phase transition. When the transition temperature was reached, and the system was at its lowest energy state—the ground state—then electron (Cooper) condensates formed and a superfluid was created. The Ginzberg–Landau phenomenon was described by a wave function in the presence of an electromagnetic field. The wave function has the characteristic features associated with the gauge invariance of Maxwell’s equations for the electromagnetic field.
Now the Meissner effect comes into play. At a certain length from the surface of the superconductor, called the screening length, an external magnetic field cannot penetrate the metal, and the interior magnetic field is expelled macroscopically from the interior of the superconductor. This effect can be explained in terms of an “effective” nonzero photon mass. Indeed, there is a connection between the screening length and the effective mass of the photon, which carries the electromagnetic force. In certain physical units, the mass of the photon is inversely proportional to the screening length; when the photon mass increases, the screening length gets smaller. The physical origin of the screening length can be explained by observing what happens when a magnetic field is applied to a field of charged particles. The magnetic field accelerates the charged particles, resulting in currents that tend to cancel or screen the applied magnetic field.
Figure 5.1 The Meissner effect. This diagram shows how the magnetic field is screened as a result of the Meissner effect. © Dwi Prananto, Department of Physics, Tohoku University.
SOURCE: Simpliphy.wordpress.com
This is the Meissner effect; it pushes out the magnetic flux from the interior of the superconductor (Figure 5.1). Screening currents are set up within the superconductor over distances about as long as the screening length from the exterior boundary of the material. These currents cancel exactly the applied magnetic flux in the interior of the superconductor, and the size of the screening length can be measured to be approximately 10−8 of a meter.
The Meissner effect illustrates much of the essential physics involved in the generation of a massive photon inside a superconductor. The Ginzberg–Landau wave field plays an important role in addition to the electromagnetic field. It explains the “discontinuous” transition between the massless photon and the massive one—namely, going from two degrees of freedom to three degrees of freedom. The degrees of freedom are the two transverse and one longitudinal degree of freedom. A massless photon is described by electromagnetic waves that propagate only in transverse directions along the photon’s axis of propagation. A massive photon would have an additional degree of freedom—namely, the longitudinal one directed along its axis of propagation. We are now close to understanding why the spontaneous symmetry-breaking Higgs mechanism was “borrowed” from the condensed-matter physics of superconductors.
INVENTING THE HIGGS BOSON IN THE STANDARD MODEL
In 1964, six physicists considered whether the phenomenon of superconductivity and ferromagnetism in solid-state physics could apply to relativistic particle physics. They were still trying to resolve the problem of the massless Goldstone boson, and how to give mass to the vector bosons of weak interactions, the W and the Z particles. Robert Brout and Françoise Englert in Belgium, Peter Higgs in the United Kingdom, and Carl Hagen, Gerald Guralnik, and Tom Kibble also in the United Kingdom considered the interaction of a massless photon gauge field, studied by Philip Anderson in superconducting materials, with a scalar field in the context of a relativistic field theory. Through the interaction of this particle physics gauge field with the scalar field, the irritating massless Goldstone bosons were “eaten” by the vector gauge particle, producing a third degree of freedom, which allowed the gauge field particle to become massive. This compares directly with the phenomena of the superconductors and the Meissner effect, in which the massless photon acquires an effective mass when it interacts with the matter field or Ginzberg–Landau wave function.
Peter Higgs submitted a paper to a journal in Europe, and the editor of the journal at CERN rejected the paper because he did not consider it was of physical significance for particle physics; it was just a mathematical speculation. In his paper, Higgs had offered a simple, elegant model in which a photon with its U(1) electromagnetic gauge field interacted with a scalar field. This broke spontaneously the gauge symmetry of the vacuum associated with this physical system, generating a mass for the U(1) gauge field—that is, the photon.
After his paper was rejected, Higgs included an addendum to his manuscript proposing that the mechanism he discovered required the existence of a new particle, with spin 0, associated with a scalar field. Now the reviewing editor of Physics Letters saw the physical significance of this work, for accelerators could hunt for this predicted particle.16 The other five physicists discussed the same spontaneous breaking of gauge invariance as did Higgs; however, they suggested only indirectly that a new particle had to accompany the mechanism to produce masses for the gauge bosons. Then Higgs, as well as Brout, and Englert, and the trio Hagen, Guralnik, and Kibble published letters in Physical Review Letters in the United States within weeks of one another. (See citations in footnotes 10–12.) The letter by Hagen, Guralnik, and Kibble did cite the already published papers of Brout and Englert, and of Higgs. In recent interviews, Guralnik claimed that it was a mistake that he and his collaborators had cited the other papers because it implied that they had come to the subject later, whereas in fact they had come up with these ideas independently and only learned about the other papers after they had drafted theirs. Because Higgs actually predicted the existence of a particle that was responsible for providing gauge bosons with mass, this predicted boson subsequently became known as the Higgs particle (boson), and only recently has the term Higgs mechanism been replaced by the incredibly cumbersome but more his
torically accurate term, the Brout–Englert–Higgs–Hagen–Guralnik–Kibble (BEHHGK; pronounced “beck”) mechanism. To do justice to the seminal published paper by Anderson, this mechanism is even sometimes dubbed the ABEHHGK mechanism!
In 1966, Higgs was visiting the States and was invited to Harvard to give a talk on his mechanism for producing mass for gauge particles through spontaneous symmetry breaking. It appears that some members of the audience, including Sydney Coleman, dismissed the whole idea. However, Steven Weinberg, who was in the audience, was at the time engaged actively in trying to understand how to unify electromagnetism and the weak interactions, and fit in the W and Z bosons of Glashow’s 1961 model. It occurred to him, apparently while driving to his Harvard office, as he later explained, that he could incorporate the spontaneous symmetry-breaking mechanism explained in Higgs’s lecture into a unification scheme of the weak force and electromagnetism involving leptons. He came up with a model in which the nonzero vacuum expectation value of a scalar field interacting with leptons was able to break spontaneously the gauge symmetry proposed originally by Glashow in such a way that the W and Z bosons would acquire a mass whereas the photon remained massless.
Cracking the Particle Code of the Universe Page 12
