Physics / Critical Essay

Vol. 4, NO. 1 / May 2018

The Standard Model

Sheldon Lee Glashow

This is the third and final essay in a three-part series. Parts one and two told of the development of the physical sciences from ancient times to the discovery of relativity and quantum mechanics in the first quarter of the twentieth century. This part begins with the challenges posed by the atomic nucleus. Two forces of nature had been recognized, gravity and electromagnetism, but two more would needed to understand nuclear phenomena: a strong force to bind nucleons into nuclei, and a weak force to explain how they decay.

In this part, I describe the Standard Model of particle physics, which encompasses three of the four forces of nature. Gravity seems to play no role in the subatomic world. The strong force results from a gauge theory based on an unbroken SU(3) symmetry called quantum chromodynamics, the weak and electromagnetic forces from a broken SU(2) × U(1) symmetry. Together they form the Standard Model of particle physics, offering a complete, correct, and consistent description of all known elementary particle phenomena. Its formulation, unlike the theories of Isaac Newton and Albert Einstein, depended on the work of thousands of physicists and engineers and the generosity of many governments. Triumph though it is, for reasons both experimental and theoretical, the Standard Model is known to be incomplete.

Much more remains to be understood.

In the beginning, the elementary particles comprised protons, photons, and electrons. In 1932, the American physicist Carl Anderson found positively charged electron-like particles among cosmic rays. He had discovered positrons, antiparticles of electrons and the first known form of antimatter.1 Neutrons, discovered in the same year by the British physicist James Chadwick, revealed that atomic nuclei, once regarded as composites of protons and electrons, were comprised of protons and their electrically neutral counterparts, neutrons. These two particles have about the same mass and are collectively called nucleons. A nucleus containing A nucleons and Z protons is designated ANZ, where N is the chemical symbol for the element. 3H1, or tritium, is the radioactive hydrogen isotope made of one proton and two neutrons,2 and 235U92 is the fissile isotope of uranium containing 143 neutrons. In general, A is the nearest integer to the atomic weight of an isotope; Z is its atomic number or place in the periodic table.

Soon after radioactivity was found, Ernest Rutherford and the English radiochemist Frederick Soddy demonstrated that each radioactive species has a characteristic half-life. In a one-microgram sample of 238U92, an average of one atom decays per minute. Just which atom or atoms, if any, decay in that minute, and which survive for billions of years, no one can tell. Two atoms of the same nuclear species are absolutely identical. There is no subtle difference between them that explains which will live another billion years and which will decay. In quantum mechanics, certainty is replaced by chance, precise trajectories by probable paths.

Rutherford identified three distinct varieties of radioactive emanations and called them alpha, beta, and gamma rays. Each is emitted from a radioactive atomic nucleus by a different mechanism of radioactive decay. In the process of alpha decay, an energetic alpha particle, soon identified as the nucleus of a helium atom, is expelled from an atomic nucleus:

ANZ → A–4NZ–2 + 4He2.

Decay proceeds if, and only if, the mass of the parent nucleus exceeds the sum of the masses of the daughter and alpha particle. The mass difference is carried off as kinetic energy. The α particle emerges with a well-defined energy.

In β– decay, the most common form of beta decay, a neutron in the parent nucleus becomes a proton as an energetic electron is emitted, but the energy of the electron is not always the same. What happens to the energy? This problem proved so intractable that both Niels Bohr and Einstein were prepared to abandon the law of energy conservation.3 Wolfgang Pauli solved the problem, and saved energy conservation, with a daring suggestion. The electron is emitted along with a neutral and almost massless particle now called an antineutrino. The two light particles share the energy released by the decay:

ANZ → ANZ+1 + e– + v¯.

The electron is denoted by e–, the antineutrino by v¯.4 As with alpha decay, beta decay takes place whenever the mass of the parent exceeds the total mass of the potential products. In beta decay, the parent atom moves one step up the periodic table, while in alpha decay it steps down by two.

Rutherford’s third form of radioactivity is the emission of an energetic photon by a nucleus in an excited state. The process is analogous to the emission of a photon by an excited atom. The difference is that gamma rays, as well alpha and beta rays, are millions of times more energetic than those involved in atomic phenomena, and energies associated with subnuclear phenomena are a million times greater still. These vast energy gaps explain why most chemists and atomic physicists need not know much nuclear physics, just as nuclear physicists need not know much high-energy physics.

Although Frédéric and Irène Joliot-Curie failed to discover neutrons, in 1933 they did discover β+ decay, where an atom descends one step of the periodic table as it emits a positron and a neutrino:

ANZ → ANZ–1 + e+ + v.

In this equation, e+ denotes the positron and v the neutrino. The Joliot-Curies received the 1935 Nobel Chemistry Prize for their discovery. Decades later their work enabled positron–electron tomography, thereby contributing to the science of nuclear medicine that Irène’s mother had pioneered. According to Enrico Fermi’s important work of 1933,5 and in the emerging language of quantum field theory, both β– and β+ decay involve the annihilation of one nucleon and the creation of a different one within the nucleus, with the simultaneous creation and ejection of a charged lepton-antilepton pair.6

Zero-Point Energy

Light propagates as waves, but is emitted or absorbed as particles. In quantum electrodynamics, the electromagnetic field acts as an operator describing the creation and annihilation of photons. As Max Planck had originally recognized, a field behaves as an infinite system of harmonic oscillators with all possible frequencies. Quantum mechanics requires each oscillator to have a non-vanishing zero-point energy of hf/2 in its ground state, which results in an infinite vacuum energy. If this energy were real, it would gravitate, which is contrary to experiment. Most physicists, quite properly, simply define it away.7

In 1948, Hendrik Casimir showed how constraints placed on zero-point electromagnetic oscillations by two closely-spaced conducting plates produced a force between them. The Casimir force has been measured most recently by Steve Lamoreaux to a precision of one percent, in perfect agreement with theoretical predictions. Does this mean that the zero-point energy of the vacuum exists? Not according to Robert Jaffe. “Casimir forces can be computed,” Jaffe remarked, “without reference to zero-point energies. They are relativistic, quantum forces between charges and currents … [N]o known phenomenon, including the Casimir effect, demonstrates that zero-point energies are real.”8

Other contributions to the energy of the vacuum are not so easily dealt with.

The Dirac Equation

Schrödinger’s equation was devised to explain the behavior of electrons, especially those in atoms. The special theory of relativity demands that space and time must be treated mathematically in the same way. Schrödinger’s equation is second order in its space derivative, but first order in its time derivative. It is not invariant under Lorentz transformations, and consequently cannot be relativistic.9 Schrödinger’s equation works well for electrons moving much more slowly than the speed of light, such as those in hydrogen atoms or cathode ray tubes; it does not work at all for electrons at close to the speed of light, such as beta rays.

In 1926, Oskar Klein and Walter Gordon devised the eponymous relativistic equation that describes the propagation of particles without spin, such as pions and other mesons. Spin is a measure of the intrinsic angular momentum of particles, nuclei, or atoms. Systems with integer spin, like photons and hydrogen atoms, are called bosons. Many bosons can cluster in the same quantum state, like photons in a laser or atoms in a Bose–Einstein condensate.

Leptons and nucleons carry half a unit of spin, the least non-vanishing possibility. Each such particle has two possible spin states, often called “up” and “down.” Nuclei can have larger spins, both integers and half-integers. Atoms, nuclei or particles with half-integer spin are called fermions. They satisfy the Pauli exclusion principle: only one fermion at a time can occupy the same quantum state. This property of electrons underlies the shell structure of atoms and the inertness of atoms with completed shells, such as helium, neon and argon.10 In 1927, Pauli devised a two-component form of Schrödinger’s equation to describe the electromagnetic interactions of electrons with spin. It too was not relativistic. The Klein–Gordon equation is relativistic, but it does not describe spin.

The time had come for Paul Dirac to find the equation that could consummate the marriage between quantum mechanics and special relativity,

γμ(pμ – eAμ)ψ = mcψ,

with summation implied over the space-time indices μ = 1…4. In this equation, ψ is the electron’s wave function, m its mass, Aμ is the electromagnetic 4-vector potential, and γμ are the four 4 × 4 purely numerical Dirac matrices. The electron’s 4-momenta pμ correspond to the differential operators iℏ∂μ with ℏ = h/2π.

Dirac’s equation predicted the value of the electron’s magnetic moment, enabled precise calculations of the fine structure of atomic spectra, and described the scattering of photons, electrons, or positrons by electrons. This was remarkable, but as Dirac was only too well aware, behind each of its positive energy solutions there lurked an unphysical solution with negative energy. At first, Dirac thought his equation might describe both electrons and protons. In 1929, he thought better. “One cannot, however, simply assert,” he wrote, “that a negative-energy electron is a proton [emphasis original] … which would violate the law of conservation of electric charge.”11 Instead, Dirac continued, “Let us assume … that all the states of negative energy are occupied except perhaps a few of small velocity [emphasis original].” The vacuum consists of a sea of electrons with infinite negative energy and charge. If one negative electron is plucked from the sea, the empty site behaves like a particle of positive energy and charge. “We are therefore led to the assumption,” Dirac concluded, “that the holes in the distribution of negative energy electrons are the protons.”12 But as Dirac once told me while we were sunbathing at the beach by Sant Vito Lo Capo, “my equation was smarter than I was.” By May 1931, Dirac realized his equation described both electrons and their antiparticles, positrons. “A hole, if there were one,” Dirac wrote, “would be a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron.”13 A year later, experimental physicists discovered the positron.

The Dirac sea of negative-energy electrons evaporated, only to be replaced by the unexpected and uncontrollable divergent integrals of quantum field theory.

Mastering Divergences

Quantum electrodynamics (QED) emerged in the late 1940s from the work of Julian Schwinger, Shin’ichirō Tomonaga, and Richard Feynman.14 Schwinger and Tomonaga both approached QED by means of variational differentiation while Feynman relied on functional integration. Freeman Dyson proved that these techniques were equivalent.15 Within QED, infinite integrals were controlled by renormalizing the mass and charge of the electron. No infinities appear in calculations of observable quantities for QED, or any other renormalizable quantum field theory.

Schwinger’s methods worked well for Schwinger, but were often opaque to others—I should know, having been his graduate student. It was Feynman who invented a diagrammatic scheme that made quantum field theory accessible. Imagine time proceeding from left to right. Propagating electrons are depicted as rightward arrows, positrons by leftward arrows, and photons by undirected wavy lines.16 Electromagnetic interactions are depicted by vertices from which a wavy line emerges from a directed arrow. This fundamental act of becoming in QED can signify six different processes. Four are the emission or absorption of a photon by an electron or positron; the others, a photon creating an electron-positron pair, or a pair annihilating into a photon. None of these processes can take place: they are forbidden by the laws of energy and momentum conservation.

Physically permitted processes involve two or more acts. Simple two-act Feynman diagrams represent the lowest-order contributions to the effect of one charge on another. For example, positron-electron scattering is described by two such diagrams. Feynman’s rules convert each diagram into a mathematical expression. The results are added together, and then squared, to obtain the formula describing the process—its differential scattering cross-section. Photon exchange is the mechanism underlying the electrical force between any two charged bodies.

Exact calculations cannot be carried out in QED. Results are obtained as a power series in the fine structure constant, a dimensionless number α ≈ 1/137. The lowest order diagram for any process is always a loopless tree. Higher order corrections are described by Feynman diagrams with one or more closed loops, such as the order α2 contribution to positron-electron scattering. In 1947, Schwinger was the first to compute the leading (one loop) correction to Dirac’s predicted magnetic moment of the electron. The next and much tinier correction (two loops) was first worked out a decade later by my classmate Charles Sommerfield. Today, the theoretical result is known to nine decimal places and is in virtually perfect agreement with observations.17 Other predictions of the theory are equally impressive.

QED has been a spectacular success.

Meson Theory

With the photon accepted as the mediator of the electromagnetic force, Hideki Yukawa applied analogous reasoning to nuclear forces. In 1934, Yukawa argued for the existence of charged bosons with masses lying between those of electrons and nucleons. His conjectured particles were strongly coupled to nucleons so that their exchange could produce the attractive force holding atomic nuclei together. It was a simple but radical hypothesis.18 Yukawa’s particles were also weakly coupled to leptons so that they might describe beta decay. But his ambitious dream of a unified theory of the weak and strong nuclear forces was not fulfilled.

One year later, charged particles with masses about a tenth of the nucleon mass were observed in cosmic rays.19 At first, physicists believed these particles to be those Yukawa had predicted. As it turned out, the particles did not interact strongly with nucleons—they were, in fact, fermions and not bosons. Now called muons, these particles have nothing to do with nuclear forces. Negatively charged muons are leptons, the heavier and short-lived cousins of electrons. Their positively charged antiparticles are cousins of positrons.

When Isidor Rabi learned about muons, he asked: “Who ordered that?20

Many nuclear physicists were convinced that, muons notwithstanding, something like Yukawa’s particles must exist to mediate the nuclear forces. In 1940, Hans Bethe fleshed out Yukawa’s theory in two historic papers.21 “It seems that one type of mesons,” Bethe wrote, “is entirely sufficient to account qualitatively for all properties of nuclei, i.e., the forces between the nuclear particles as well as the β-transformation.”22 Yukawa’s predicted particles were finally found in 1947 by a British collaboration studying high-altitude cosmic rays. Called π-mesons or pions, these particles comprise a triplet (π+, π0 and π–) of spinless bosons, all with approximately the same mass.23 Yukawa became Japan’s first Nobel laureate in 1949, “for his prediction of the existence of mesons on the basis of theoretical work on nuclear forces.”24

For all that, Yukawa’s meson could not explain beta decay, nor provide a quantitative description of the nucleon–nucleon force. In their classic textbook, Theoretical Nuclear Physics, published in 1952, John Blatt and Victor Weisskopf observed that the “numerous attempts to predict nuclear forces on the basis of meson fields … have failed to reproduce quantitatively the observed forces between nuclear particles.”25

In addition to the pions, a number of strange particles were found in cosmic rays. These particles are strange because they are produced by the strong force, but decay by the weak. They include charged and neutral spinless K-mesons, or kaons, and five strange spin-1/2 baryons. The lightest of the strange baryons, Λ0, is only ten percent heavier than either nucleon.26

Quarks with Flavor

Shoichi Sakata argued in 1956 that the “curious properties of the new particles could be reduced to those of Λ0, just like the mysterious properties of the atomic nuclei were reduced to those of [the] neutron.”27 Sakata proposed that only three strongly-interacting particles are truly elementary: neutrons, protons, and Λ0s. All other hadrons—pions, kaons, and strange baryons—were built from these particles and their antiparticles.

Sakata’s theory is wrong, but it contains an element of the truth.

In 1961, Murray Gell-Mann and Yuval Ne’eman, working independently, devised the unitary symmetry model of elementary particles.28 Gell-Mann used an abstract Sakata model as the framework of a theory in which all eight spin-1/2 baryons were regarded as equally elementary.29 According to unitary symmetry, all hadrons occur as multiplets corresponding to representations of the group SU(3). The spinless mesons and spin-1/2 baryons form octuplets, which is why the scheme became known as the eightfold way. The predicted eighth meson was discovered while Gell-Mann was completing his paper.

Following the deployment of powerful particle accelerators in the 1950s and 1960s, new particles were discovered at a remarkable rate, including octuplets and singlets of spin-1 and spin-2, and many new baryon states. An SU(3) decuplet of spin-3/2 baryons was completed in 1964 with the discovery of the triply-strange Ω– with just the properties Gell-Mann had predicted. Some physicists, like Geoffrey Chew, invoked a nuclear democracy in which all particles are equally elementary. Chew believed that principles of mathematical consistency could themselves permit the derivation of Werner Heisenberg’s S-matrix, which describes the scattering of any two particles.

Others returned to Sakata’s vision of a small number of building blocks from which all hadrons could built, a stepping-stone toward the notion of fractionally charged nuclear constituents.30 Gell-Mann referred to his three hypothetical entities as quarks. Three quarks for Muster Mark! Gell-Mann’s quarks came in three varieties, or flavors: u for up quarks, d for down quarks, and s for strange quarks. Their strangeness is a simple consequence of the relatively large mass of the strange quark. Today we know of three more quark flavors. Charmed quarks, bottom quarks and top quarks are all much more massive than strange quarks.

Quarks with Color

To make his model work, Gell-Mann imposed on quarks an arbitrary rule of quark confinement. Quarks are forever confined to the hadrons of which they form parts as three-quark baryons, three-antiquark antibaryons, or quark-antiquark mesons. No one has ever observed an isolated quark, or any other fractionally charged particle. What could give rise to the force that holds quarks together to form hadrons? Part of the answer was provided by Oscar Greenberg.31 The original quark model assumed that quarks within a baryon had to act as if they were bosons, rather than fermions. Greenberg proposed that quarks of each flavor—up, down, and strange—come in three different colors. The antisymmetry of the quark wavefunction in a baryon is guaranteed by the constraint that hadrons must be colorless.

Quark color is the arena in which the strong force operates. Quark flavor, on the other hand, is reserved for the electroweak force. Quantum chromodynamics (QCD) is a non-abelian gauge theory based on the color charge, just as QED is an abelian gauge theory based upon electric charge. The successes of the eightfold way result from a flavor-SU(3) group whose symmetries are violated by quark masses. QCD is a gauge theory based upon an exact color-SU(3) gauge group. The theory was first proposed in 1973 by Harald Fritzsch and Heinrich Leutwyler, and separately by Gell-Mann.32 The force between colored quarks is mediated by the exchange of massless gluons, just as the force between charged particles is mediated by the exchange of massless photons. Gluons, like quarks, bear the color charge. Unlike photons, they cannot propagate as free particles.

QCD explains how quarks are permanently confined within hadrons but behave like freely moving particles when studied at high energy. Often referred to as asymptotic freedom and infrared slavery, these properties were discovered shortly after the theory was formulated.33 The nuclear force between two colorless nucleons is a secondary effect of the chromodynamic attraction of the colored quarks within them just as the chemical force between two electrically neutral atoms is an effect of the electric force among the charged constituents within them. The theory’s elegance and quantitative successes have convinced most physicists that QCD is the correct explanation of the strong force.

Electroweak Thoughts

Schwinger suggested the possibility of a unified theory of weak and electromagnetic interactions in 1956.34 He assigned its development to me as my thesis subject. Schwinger directed me toward non-Abelian gauge theories; he thought that they would assure the vectorial nature and universality of an electroweak theory.35 I found no three-parameter gauge theory that could yield a phenomenologically sound electroweak theory. One problem lay with the then-recent discovery that weak interactions do not share the mirror symmetry possessed by strong and electromagnetic interactions. How can an electroweak theory link the parity-conserving photon A with the parity-violating charged weak mediators W±? A second problem had to do with mass. How can a proposed electroweak symmetry link the massless photon to a massive, electrically charged intermediary? Having completed my thesis in 1958, I remained convinced that a unified electroweak theory must exist.

At the end of a two-year postdoctoral fellowship in Copenhagen, I found the answer to the first question. A suitable algebraic structure of the electroweak theory required the four-parameter gauge group SU(2) × U(1). Four intermediate bosons had to exist: A, W±, and the heavy Z0. The neutral intermediary would mediate neutral current interactions, for which there was no evidence. I had no answer to the second question.

How does the electroweak symmetry break down, leaving the photon massless, but endowing the remaining three gauge bosons with large masses?

Spontaneous Symmetry Breaking

The answer involves the phenomenon of spontaneous symmetry breaking. Jeffrey Goldstone provided a hint just a week before I submitted my paper on the partial symmetries of weak interactions to Nuclear Physics in 1960.36 Goldstone showed that massless bosons arise whenever a continuous symmetry of a system of spinless mesons spontaneously breaks. Begin with the assumption of a U(1)-invariant Lagrangian of a complex scalar field. Its non-negative quartic potential is chosen to be shaped like a sombrero. It vanishes on a circle in the complex plane, thereby defining the degenerate ground states of the system. In expanding about any of these ground states, one finds the U(1) symmetry is broken. One of the two bosons has become massless. For other Lie groups, one Goldstone boson is produced for each broken symmetry, but at least one massive boson must survive.

Although Goldstone’s work was very important, nobody has ever seen his massless bosons and none is likely to exist.

In three separate issues of the Physical Review Letters for 1964, three papers by six authors were devoted to the application of spontaneous symmetry breaking to gauge theories.37 In such systems, the massless spin-1 gauge bosons are coupled to a system of spinless bosons. When the symmetry spontaneously breaks, no massless Goldstone bosons appear, but some of the initially massless gauge bosons acquire mass—one for each broken symmetry. It is as if the potentially massless bosons were consumed by the gauge bosons. At first no one found the work relevant to the electroweak theory. It would soon become famous as the Higgs mechanism, in honor of my dear friend Peter Higgs, one of its five ingenious creators.

Three years later, Steven Weinberg brilliantly invoked the Higgs mechanism. In order to give mass to the three weak interaction intermediaries in SU(2) × U(1), Weinberg coupled a complex doublet of scalar bosons to the electroweak gauge bosons. In the course of spontaneous symmetry breaking, three of the four scalar bosons sacrificed themselves to give mass to the W and Z bosons, while the surviving spinless particle became known as the Higgs boson.

How does the electroweak symmetry break down, leaving the photon massless, but endowing the remaining three gauge bosons with large masses?

Weinberg provided the answer.38

The Charmed Quark

The brilliant work of Weinberg and Abdus Salam promoted my naive model into a theory of the weak and electromagnetic interactions of leptons. Nonetheless, a complete electroweak theory must include the interactions of nuclear particles as well as leptons. If applied directly to the original flavor triplet of quarks, the theory leads to unacceptable strangeness-changing neutral currents. These are weak-interaction phenomena that are not found in nature. An example? The decay of a charged kaon into a charged pion and an electrically neutral lepton-antilepton pair. This was the third problem faced by the electroweak model. In 1964, James Bjorken and I introduced a fourth quark on essentially aesthetic grounds.39 How could there be four leptons and only three quarks? Surely there had to exist two doublets of quarks, just as there were two doublets of leptons? Our charmed quark, together with Gell-Mann’s strange quark, formed the second quark doublet. It was another six years before John Iliopoulos, Luciano Maiani, and I realized that charm could justify its meaning as a talisman worn to avert the evil eye. Inserted into the electroweak model, the charmed quark yielded the Glashow–Iliopoulos–Maiani mechanism, eliminating the theory-killing threat of strangeness changing currents, both at tree and higher order levels.40

Confirmation

Gerard ’t Hooft, guided by his supervisor Martinus Veltman, proved in 1971 that spontaneously broken gauge interactions were renormalizable.41 The electroweak theory, along with QCD, formed the first version of the Standard Model. The first family of quarks and leptons is essential. Up and down quarks, along with electrons and their neutrinos, are required for the sun to shine, but what conceivable purpose could the second family serve? Who needs muons and quarks, both strange and charmed? A few years later we would learn that a third family of quarks and leptons was awaiting discovery.

Two critical experimental developments took place between 1970 and 1974. One pertained to the weak force and the other to the strong. Experimenters at CERN and at Fermilab competed to detect the neutral currents predicted by the electroweak model. Inelastic collisions between neutrinos and nucleons, the model suggested, should allow neutrinos to emerge unchanged.

They did.

At the same time, strong evidence appeared for the existence of quarks. Inelastic electron scattering experiments performed at the Stanford Linear Accelerator (SLAC) revealed point-like constituents within protons. Feynman called them partons, but further evidence from the study of neutrino collisions at CERN convinced physicists that partons were, in fact, quarks.42

The November Revolution

On the morning of November 14, 1974, I was awakened by my friend, the MIT experimental physicist, Samuel Ting. His group had discovered a mystifying new particle. Produced by energetic proton collisions, the J particle decays into an electron-positron pair. On the same day, Burton Richter’s SLAC group announced the discovery of its own mystifying particle. Produced via electron–positron annihilation, the Ψ particle had the same mass as the J particle. The same particle had been discovered at the same time on opposite American coasts and with very different experimental techniques. It is now known by a double name: J/Ψ.43

Many interpretations of the new particle were proposed. Schwinger thought it confirmed his own dyon theory; Maiani imagined it to be the weak intermediary Z0. J/Ψ comprises, in fact, a charmed quark bound to its antiquark. It is now called charmonium, by way of an analogy to positronium. Particles containing a single charmed quark were observed soon afterward—the first charmed baryon in 1975, charmed mesons a year later.

Physicists had shown the electroweak theory to be both mathematically consistent and phenomenologically correct. The obligatory fourth quark had been discovered and strangeness-conserving neutral currents seen. The theory provided a correct description of all observed weak interactions. By 1979 it was generally accepted that Makoto Kobayashi and Toshihide Maskawa’s extra quark flavors could and most likely did explain observed charge conjugation and parity (CP) violation and it was generally believed that the top quark would be found as soon as sufficient accelerator energy became available. The 1995 discovery of the top quark was the culmination of a long effort, but it was in no way surprising because other data indicated that the top quark mass had to be particularly high.

The CP operation combines the discrete operations of space reflection and charge conjugation. Experimenters in the 1950s established that neither is a symmetry of nature. Both are violated by the weak interactions. In 1964, Val Fitch and James Cronin discovered a small and entirely unexpected violation of CP invariance in the decay of neutral kaons.44 The effect could not be explained by the original two-family Standard Model. Kobayashi and Maskawa pointed out in 1973 that schemes involving more than four quark flavors could incorporate CP violation.45 Their idea would soon bear fruit.

The discovery of a third charged lepton in 1975 by Martin Perl was almost entirely unexpected. The tau lepton, or tauon, is about seventeen times heavier than the muon. It was the first member of a third family of quarks and leptons that included the neutrino and another doublet of quarks, rather tamely dubbed “top” and “bottom.” The bottom quark was found at Fermilab in 1977, but the heavy top quark was not produced and detected until 1995. The existence of a third family of quarks and leptons enabled the Standard Model to provide a correct and quantitative account of the many instances of CP violation that experimenters have since discovered.

Beyond the Standard Model

The Higgs boson is the particle responsible for the masses of the W and Z bosons, as well as most other particles. Nucleons obtain most of their mass from another well-understood mechanism. The search for the Higgs, because it was difficult, was also protracted. It was on July 4, 2012 that its discovery was announced by the two teams that had been working at CERN’s Large Hadron Collider.

We have reached our goal. The Standard Model is a successful theory. It offers a complete, correct, consistent, and elegant description of the known elementary particles and their interactions. It explains the weak, strong and electromagnetic forces in terms of gauge interactions. Formulated almost half a century ago, the Standard Model has been confirmed by many experiments. If many experiments confirm its predictions, none contradicts them.

And that is our problem. No flaws have appeared in the Standard Model, but many questions remain unanswered. Why are top quarks roughly a trillion times heavier than neutrinos? Why is the mass of the neutrino comparable to the cosmological constant? What stabilizes the mass of the Higgs boson? What explains the tripartite group structure of the Standard Model? Why are there three fermion families, and not four or seven? Can the numbers describing particle masses, mixings, and coupling strengths be related to one another?

Three decades ago, string theorists were optimistic that they had answers in hand. David Gross boasted that, “there seem to be no insuperable obstacles to deriving all of known physics from the E8 × E8 heterotic string.”46 John Schwartz remarked of the theory he helped create that it “could be a ‘theory of everything.’”47 Weinberg was more guarded, going no further than to remark that string theory has the kind of rigidity that may “in the end turn out to have something to do with the real world.”48

“I think all this superstring stuff is crazy,” Feynman remarked.49

The latest edition of string theory addresses none of our questions, makes no predictions, and cannot be falsified. “If one’s theory can’t predict anything,” Peter Woit observed with some asperity, “it is just wrong and one should try something else.”50

Things are as they are, but why they are as they are, the theory does not say.

Endmark
DOI: 10.37282/991819.18.15
  1. Every particle has an antiparticle with the same mass but opposite quantum numbers. Some neutral particles—like photons, but not neutrons—are identical to their antiparticles. Antiprotons were first produced and detected in 1955 at the Berkeley Bevatron, then the most powerful particle accelerator; antineutrons were observed soon thereafter. Neutrons and protons are each assigned baryon number +1, their antiparticles –1. No process has yet been observed that violates the conservation of baryon number, but the search continues. ↩
  2. Every chemical element has several different isotopes. Their nuclei have the same Z, but different A. Hydrogen has three isotopes with A = 1, 2, and 3. Deuterium was discovered by the American chemist Harold Urey in 1931, and identified a year later as a bound system of a proton and neutron. ↩
  3. Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World (Oxford: Clarendon Press, 1986), 311–12, 105–12. Marie Curie and William Thompson, Lord Kelvin also invoked energy nonconservation to explain the large and seemingly eternal energy release by radioactivity. ↩
  4. Kan Chang Wang first proposed a method to detect neutrinos, but did not attempt it. See Kan Chang Wang, “A Suggestion on the Detection of the Neutrino,” Physical Review 61 (1942): 97. Antineutrinos were first detected in 1956. See Clyde Cowan et al., “Detection of the Free Neutrino: A Confirmation,” Science 124, no. 3,212 (1956): 103–104. Both neutrinos and antineutrinos are now routinely studied at power reactors, particle accelerators, and from radioactivity, cosmic rays, the earth’s interior, the sun, and distant supernovae. We know of three different neutrino species, each with its own antiparticle. ↩
  5. His paper on beta decay was submitted to Nature, which rejected it. It first appeared in Italian. See Enrico Fermi, “Tentativo di una teoria dei raggi β,” (Tentative Theory of Beta Rays), La Ricerca Scientifica 2, no. 12 (1933). An English version was published decades later. See Fred Wilson, “Fermi’s Theory of Beta Decay,” American Journal of Physics 36 (1968): 1,150–60. ↩
  6. Leptons are particles with only weak and electromagnetic interactions. Today we know of three different charged leptons: electrons, muons, and tauons. Each charged lepton is associated with its own neutrino. Leptons carry lepton number +1 while antileptons carry –1. Every process yet observed conserves lepton number. The search for its anticipated violation continues. ↩
  7. This is not as silly as it seems. The negative gravitational energy of a one-kilogram brick on the surface of the earth is fifty megajoules, and can be ignored unless one wants to launch the brick into orbit. ↩
  8. Robert Jaffe, “The Casimir Effect and the Quantum Vacuum,” Physical Review D 72, no. 2 (2005). See also Hendrik Casimir, “On the Attraction Between Two Perfectly Conducting Plates,” Koninklijke Nederlandse Akademie Van Wetenschappen Proceedings 51 (1948): 793; and Steve Lamoreaux, “Demonstration of the Casimir Force in the 0.6 to 6μm Range,” Physical Review Letters 78, no. 5 (1997). ↩
  9. It is a first-order differential equation in time, but second order in space. ↩
  10. The Pauli exclusion principle leads to the shell model of atomic nuclei. The alpha particle is exceptionally stable because it contains a complete shell of two protons and another complete shell of two neutrons. ↩
  11. Paul Dirac, “A Theory of Electrons and Protons,” Proceedings of the Royal Society A 126, no. 801 (1930): 361–62. ↩
  12. Paul Dirac, “A Theory of Electrons and Protons,” Proceedings of the Royal Society A 126, no. 801 (1930): 363. ↩
  13. Paul Dirac, “Quantised Singularities in the Electromagnetic Field,” Proceedings of the Royal Society A 133, no. 821 (1931): 61. ↩
  14. Shin’ichirō Tomonaga, Julian Schwinger, and Richard Feynman shared the 1965 Nobel Prize in Physics for “their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.” See Nobelprize.org, “The Nobel Prize in Physics 1965.” ↩
  15. Freeman Dyson, “The Radiation Theories of Tomonaga, Schwinger, and Feynman,” Physical Review 75, no. 3 (1949): 486–502, 1,736–55. ↩
  16. Positrons were once thought of as electrons moving backwards in time. In 1940, John Wheeler imagined a one-electron universe in which electrons and positrons everywhere were actually the same one particle shuttling back and forth in time. See Richard Feynman, “The Developments of the Space-Time View of Quantum Electrodynamics,” Nobel Lecture (1965). ↩
  17. A small three-sigma discrepancy persists. The threshold for discovery in particle physics is five sigmas. Ongoing experiments will soon tell us whether this tantalizing discrepancy is due to physics beyond the Standard Model. ↩
  18. Hideki Yukawa, “On the Interaction of Elementary Particles. I,” Proceedings of the Physico-Mathematical Society of Japan 17 (1935): 48–57. ↩
  19. Carl Anderson and Seth Neddermeyer, “Cloud Chamber Observations of Cosmic Rays at 4,300 Meters Elevation and Near Sea-Level,” Physical Review 50, no. 4 (1936): 263–71. ↩
  20. Several decades later, Rabi confirmed the story to me, adding that the event occurred over lunch at a Chinese restaurant. As Schwinger’s thesis advisor, Rabi was my professional grandfather. ↩
  21. Hans Bethe, “The Meson Theory of Nuclear Forces. I. General Theory,” Physical Review 57, no. 4 (1940): 260–72; Hans Bethe, “The Meson Theory of Nuclear Forces. II. Theory of the Deuteron,” Physical Review 57, no. 5 (1940): 390–413. ↩
  22. Hans Bethe, “The Meson Theory of Nuclear Forces. II. Theory of the Deuteron,” Physical Review 57, no. 5 (1940): 412. ↩
  23. Charged pions decay into a lepton–antilepton pair, with a halflife of about twenty nanoseconds: π+ → μ+ + v. Neutral pions decay into two photons even more rapidly. ↩
  24. Nobelprize.org, “The Nobel Prize in Physics 1949.” ↩
  25. John Blatt and Victor Weisskopf, Theoretical Nuclear Physics (New York: Wiley & Sons, 1952): vi. ↩
  26. Baryons, like nucleons, are fermions carrying one unit of baryon number. Strongly interacting particles, whether mesons, baryons, or antibaryons are hadrons, i.e., particles made from quarks. ↩
  27. Soichoi Sakata, “On a Composite Model for the New Particles,” Progress of Theoretical Physics 16, no. 6 (1956): 688. See also Lev Okun, “The Impact of the Sakata Model,” Progress of Theoretical Physics, supplement 167 (2007): 163–74. ↩
  28. Murray Gell-Mann, “The Eightfold Way: A Theory of Strong Interaction Symmetry,” California Institute of Technology Synchrotron Laboratory Report TID-12608, CTSL-20 (Pasadena, March 1961); Yuval Ne’eman, “Derivation of Strong Interactions from a Gauge Invariance,” Nuclear Physics 26, no. 2 (1961): 222–29. ↩
  29. More precisely, Sakata assigned his baryon triplet to the fundamental representation of the group SU(3), whereas in the unitary symmetry scheme all eight baryons are placed in its 8-dimensional adjoint representation. That is, Sakata found the correct higher symmetry group, while Gell-Mann and Ne’eman found its relevant representations. ↩
  30. Murray Gell-Mann, “A Schematic Model of Baryons and Mesons,” Physics Letters 8, no. 3 (1964): 214–15; André Petermann, “Propriétés de l’étrangeté et une formule de masse pour les mésons vectoriels,” (Strangeness Properties and a Mass Formula for Vector Mesons), Nuclear Physics 63, no. 2 (1965): 349–52; George Zweig, “An SU3 Model for Strong Interaction Symmetry and Its Breaking,” CERN Report TH-401 (January 1964). ↩
  31. Oscar Greenberg, “Spin and Unitary-Spin Independence in a Paraquark Model of Baryons and Mesons,” Physical Review Letters 13, no. 20 (1964): 598. ↩
  32. Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler, “Advantages of the Color Octet Gluon Picture,” Physics Letters B 47, no. 4 (1973): 365–68. ↩
  33. David Gross and Frank Wilczek, “Ultraviolet Behavior of Non-Abelian Gauge Theories,” Physical Review Letters 30, no. 26 (1973): 1,348; Hugh David Politzer, “Reliable Perturbative Results for Strong Interactions?” Physical Review Letters 30, no. 26 (1973): 1,346. ↩
  34. Julian Schwinger, “A Theory of the Fundamental Interactions,” Annals of Physics 2 (1956): 407–34. ↩
  35. Many theorists besides Yukawa had proposed mesons to mediate the weak force, among them Shuzo Ogawa and Yasutaka Tanikawa in Japan, Semjon Gershtein and Yakov Zeldovich in the USSR, and Feynman and Gell-Mann in the US. Attempts to create an electroweak model were published by Abdus Salam and John Ward in 1959, 1962, and 1964, but none had the correct gauge group, none agreed with experiment, and none predicted neutral currents. ↩
  36. Sheldon Lee Glashow, “Partial-Symmetries of Weak Interactions,” Nuclear Physics 22, no. 4 (1961): 579–88; Jeffrey Goldstone, “Field Theories with ‘Superconductor’ Solutions,” Il Nuovo Cimento 19, no. 1 (1961): 154–64. ↩
  37. François Englert and Robert Brout, “Broken Symmetry and the Mass of Gauge Vector Mesons,” Physical Review Letters 13, no. 9 (1964): 321; Peter Higgs, “Broken Symmetries and the Masses of Gauge Bosons,” Physical Review Letters 13, no. 16 (1964): 508; Gerald Guralnik, Carl Hagen, and Thomas Kibble, “Global Conservation Laws and Massless Particles,” Physical Review Letters 13, no. 20 (1964): 585. ↩
  38. Steven Weinberg, “A Model of Leptons,” Physical Review Letters 19, no. 21 (1967): 1,264–66; Abdus Salam, “Weak and Electromagnetic Interactions,” in Elementary Particle Theory: Relativistic Groups and Analyticity. Proceedings of the Eighth Nobel Symposium, ed. Nils Svartholm (Stockholm: Almquist & Wiksell, 1968), 367–77. Salam cites Weinberg’s earlier work.  ↩
  39. James Bjørken and Sheldon Lee Glashow, “Elementary Particles and SU(4),” Physics Letters 11, no. 3 (1964): 255–7. 1964 was an amazing year for physics: quarks with color and charm were first proposed, along with the Higgs mechanism, as well as the discoveries of the Gell-Mann’s predicted Ω– baryon, CP violation, quasars, the cosmic background radiation, and John Bell’s inequality! ↩
  40. Sheldon Lee Glashow, John Iliopoulos, and Luciano Maiani, “Weak Interactions with Lepton-Hadron Symmetry,” Physical Review D 2 (1970): 1,285. ↩
  41. Gerard ’t Hooft, “Renormalization of Massless Yang–Mills Fields,” Nuclear Physics 33 (1971): 173–99. ↩
  42. Richard Feynman, “The Behavior of Hadron Collisions at Extreme Energies,” in Proceedings of the Third International Conference on High Energy Collisions at Stony Brook, ed. Chen-Ning Yang (New York: Gordon & Breach, 1969), 237–58; James Bjorken, “The November Revolution: A Theorist Reminisces,” SLAC Beam Line 8 (July 1985): 1–6. ↩
  43. Jean-Jacques Aubert et al., “Experimental Observation of a Heavy Particle J,” Physical Review Letters 33, no. 23 (1974): 1,404; Jean-Eudes Augustin et al., “Discovery of a Narrow Resonance in e+e– Annihilation,” Physical Review Letters 33, no. 23 (1974): 1,406. ↩
  44. James Christenson et al., “Evidence for the 2π Decay of the K02 Meson,” Physical Review Letters 13, no. 4 (1964): 138. ↩
  45. Makoto Kobayashi and Toshihide Maskawa, “CP Violation in the Renormalizable Theory of Weak Interaction,” Progress of Theoretical Physics 49, no. 2 (1973): 652–57. ↩
  46. David Gross quoted in Peter Galison, “Theory Bound and Unbound: Superstrings and Experiments,” in Laws of Nature: Essays on the Philosophical, Scientific and Historical Dimensions, ed. Friedel Weinert (Berlin: Walter de Gruyter, 1995), 383. ↩
  47. John Schwartz quoted in Peter Galison, “Theory Bound and Unbound: Superstrings and Experiments,” in Laws of Nature: Essays on the Philosophical, Scientific and Historical Dimensions, ed. Friedel Weinert (Berlin: Walter de Gruyter, 1995), 383. ↩
  48. Steven Weinberg quoted in Peter Galison, “Theory Bound and Unbound: Superstrings and Experiments,” in Laws of Nature: Essays on the Philosophical, Scientific and Historical Dimensions, ed. Friedel Weinert (Berlin: Walter de Gruyter, 1995), 386. ↩
  49. Interview of Richard Feynman in Superstrings: A Theory of Everything? eds. Paul Davies and Julian Brown (Cambridge: Cambridge University Press, 1988): 194. ↩
  50. Peter Woit, Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law (New York: Basic Books, 2007), xii. ↩

Sheldon Lee Glashow is a Nobel Laureate, Higgins Professor of Physics, emeritus, at Harvard University, and University Professor, emeritus, at Boston University.

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