Beyond the God Particle

by Leon M. Lederman

  RUTHERFORD'S RADIOACTIVITY

  We've previously met one of the greatest experimental physicists of all time, the New Zealander Ernest Rutherford. We described his most famous work: the discovery of the atomic nucleus, a discovery that was of paramount importance to the development of the entire quantum theory. However, this work was performed several years after he had already received his Nobel Prize, a rarity in the history physics. So what had Rutherford received his Nobel Prize for if it wasn't for the discovery of the atomic nucleus?

  In 1898 J. J. Thomson, Rutherford's mentor at Cambridge University in England, had arranged an academic post for him at McGill University in Montreal, Canada. Here Rutherford set up his lab and explored the hot new topic of radioactivity. He soon discovered the concept of radioactive “half-life.”4 The main business was understanding all the various “rays” that are emitted from substances displaying radioactivity. The situation was confusing and full of initial mistakes and false hypotheses by all the players—but it was finally sorted out.

  Rutherford's initial hypothesis was that all the radioactivity “rays” were just X-rays. However, by using the much more radioactive elements of polonium and radium, discovered by the Curies, he was able to show that there were two different rays that were deflected in magnetic fields, and therefore that these must be electrically charged particles. One of these was a slow and fairly non-penetrating form, which he called “alpha rays.” The alpha rays required a very strong magnetic field to deflect them, and later Rutherford would prove that alpha rays were actually the nuclei of helium atoms (i.e., helium atoms with no electrons). These are fragments that are produced when a very heavy nucleus falls apart into a lighter one, emitting alpha particles. Some of the other radiation was not bent at all in a magnetic field and was deeply penetrating through matter and must therefore be electrically neutral. Rutherford called these “gamma rays,” which are very energetic photons, of even higher energy than X-rays.

  But Rutherford also found another kind of “ray” that was easily bent in a magnetic field, which he called “beta rays.” Becquerel had also identified the “beta rays,” but Rutherford now found that they were a hundred times more penetrating of matter than the alpha rays. From the deflection of their motion in a magnetic field, the “charge-to-mass ratio” of beta rays could be determined, and it was found to be identical to that of the electron, discovered by J. J. Thompson a few years earlier.

  Usually the emitted beta particles have the negative charge of the electron. But in some rare materials the emitted beta particle has a positive electric charge: that's the antielectron, or positron. So, we'll now take a brief side-excursion into the fascinating phenomenon of antimatter (and by the way, here, too, has emerged another billion-dollar industry—positron-emission tomography—better living through particle physics5).

  ANTIMATTER

  We have all seen and admired the famous equation E = mc2 emblazoned upon T-shirts, opening graphics for TV shows like The Twilight Zone, corporate logos, commercial products, and countless New Yorker cartoons. “E = mc2” has become the universal emblem for “smart” in our culture today.

  But this isn't exactly what Einstein said. What Einstein really said, is that for a particle at rest

  E2 = m2 c4

  To get the energy, E, for the particle, we have to take the square root of both sides of this equation, and sure enough, we'll then get a solution: E=mc2. So what is the difference?

  First, please just bear in mind a simple mathematical fact that you may have forgotten since you took high school algebra: every number has two square roots. For example, the number 4 has the two square roots = 2 and = –2; the latter is “negative 2.” Of course, we all know that 2 × 2 = 4, but we also know that (–2) × (–2) = 4 (two negatives make a positive when you multiply them together). The “other” square root of a positive number is always a negative number (and even negative numbers have square roots, which leads to imaginary numbers, but we digress).

  So, then, here's the puzzle: If the true equation is E2 = m2c4, then how do we know that the energy, E, that we derive from Einstein's formula should be a positive number? Which square root is it? Positive or negative? How does nature know?

  Suppose negative-energy particles exist. These particles would have a negative rest energy of –mc2. If they move, their negative energy would become a still greater negative quantity, that is, they would lose energy as they accelerate, i.e., their energy becomes more and more negative as their velocity increases. In fact, in collisions their energy would become more negative, and after enough collisions, eventually, the negative-energy particles would have an infinitely negative energy. Such particles would continuously accelerate and fall down into an enormous sinkhole of negative infinite energy. The universe would be full of these negative infinite-energy oddball particles, constantly radiating energy as they fell deeper and deeper into the infinite negative-energy abyss.

  In 1926, a young British genius named Paul Dirac sought an equation for the electron, one that would be consistent with Einstein's theory of special relativity, as the equations of the time were based only upon Newton's concepts of space and time and they only worked for slow electrons.6 Dirac found a beautiful equation in the new quantum theory, combined with relativity, and he hoped that it would lead to E = +mc2. But he soon encountered a problem: his equation indeed had the “correct” solutions representing electrons that have spin 1/2 and positive energy, i.e., we do get E = +mc2. But for every positive energy solution there was also a negative energy solution with E = –mc2. The negative-energy electrons should be just as prevalent in nature as the positive-energy ones.

  According to Dirac's equation, the universe should be full of these negative-energy electrons. The universe would become a purgatory, eternally collapsing into a great sinkhole of negative energy. Dirac became frustrated as seemingly nothing could be done about this conundrum—Einstein's theory of relativity combined with quantum physics predicted negative-energy electrons. This would imply that ordinary atoms, even simple hydrogen atoms, indeed, all of ordinary matter, could not possibly be stable. The positive-energy electron with E = mc2 could emit a few photons, adding up to an energy of 2mc2, and could become a negative-energy electron with E = –mc2, and then begin its orbital descent, accelerating and radiating more photons into the abyss of infinite negative energy. The whole universe could not be stable if the negative energy states truly existed. The negative-energy electron solutions of Dirac's equation were now a prime headache for the baby quantum theory.

  However, Dirac soon had a wild idea. Wolfgang Pauli had successfully, and brilliantly, explained the Periodic Table of the Elements with a “rule” that must be obeyed by all electrons, known as the Pauli exclusion principle. This principle says that no two electrons can be put into exactly the same quantum state of motion at the same time. That is, once an electron occupies a given quantum state of motion, like a quantum orbit in an atom—that state is filled. No more electrons can join in. Quantum states are like seats on an airplane—only one passenger per seat is allowed. This is more than a mere “ordinance” or “rule,” and Pauli actually proved it mathematically to apply for all spin-1/2 particles.7

  Dirac's idea was a straightforward extension of Pauli's exclusion principle: He hypothesized that the vacuum itself is completely filled with electrons, occupying all of the negative energy states. And, if all of the negative energy levels in the whole universe are already filled, then positive-energy electrons, such as in atoms, could not drop down into these quantum states—they would be excluded from doing so by the Pauli exclusion principle. The seats in the whole vacuum are all sold out! The vacuum is now stable because it is already filled up with negative-energy electrons.8

  Dirac thought that this was the fix, but he soon realized that it was not the end of the story. It was now theoretically possible to “excite” the vacuum. This means that physicists could arrange a collision in which they could kick a negative-energy electron ou
t of the vacuum, much like a fisherman pulls a deep-sea fish into his boat. Dirac realized that this process would leave behind a hole in the vacuum. The hole, however, would represent the absence of a negative-energy electron. This means that the hole actually would have a positive energy. However, the hole would also represent the absence of a negative electrically charged electron, and hence the hole would be a positively charged particle (see figure 9.33).

  FIGURE 9.32. Dirac Sea. The “Dirac sea,” picture of the vacuum. All of the allowed negative energy levels for fermions, predicted by Dirac's equation where relativity is combined with quantum theory, are filled. The Pauli exclusion principle forbids any more electrons in these levels, so the vacuum is “stable.” The vacuum is like an inert element, e.g., neon, where all orbitals are filled, so neon becomes chemically inactive.

  FIGURE 9.33. Antiparticle as Hole in Dirac Sea. Dirac's sea leads to the prediction that a negative-energy electron can be “ejected out of the vacuum,” e.g., by the nearby collision. The hole left in the vacuum is the absence of a negative-energy, negatively charged electron, and therefore appears as a positive-energy, positively charged particle with identical mass to the electron. Dirac thus predicted the positron, and the phenomenon of “electron–positron pair creation.” The positron was discovered experimentally a few years later by Carl Anderson. The phenomenon of antimatter is now well established and a standard phenomenon in particle physics—the Tevatron discovered the top quark by pair producing top and anti-top quarks in this way.

  Dirac predicted the existence of something bizarre: Antimatter. Every particle species in nature has a corresponding antiparticle. We call the antiparticle of the electron the positron. The positron is the “absence” of a negatively charged electron, a hole, in the vacuum, and is therefore a positively charged particle with positive energy but is otherwise indistinguishable from the electron, with the same mass and the same (though opposite) spin. The laws of special relativity require that the hole in the vacuum, which is the absence of negative energy (note the double negative!), must have a positive energy of exactly E = +mc2, where m is exactly the same as the electron mass. Positrons were predicted by Dirac, and they must exist if both quantum theory and special relativity are true.

  Positrons were subsequently discovered in an experiment in 1933 by Carl Anderson.9 They are the positively charged beta rays that are seen in radioactivity. Antimatter will annihilate matter when the two collide, as the positive-energy electron jumps back into the hole in the vacuum. The annihilation produces a lot of energy (at rest, electron–positron annihilation would release E = 2mc2 by direct conversion of all the rest-mass energy of the two particles into gamma rays). Antimatter can easily be produced by particle accelerators.

  Antimatter is a useful commodity and is already “paying rent.” The positrons naturally generated from radioactive disintegration have found a use in positron-emission tomography (PET) scanners, a form of medical imaging. It is estimated that the cash-flow generated by this one activity, again a by-product of pure and basic research, is larger than the cost of funding all of the science of particle physics today. It is unclear if the future utility of synthesized antimatter will expand to warp-drive starship engines or compact super-energy storage devices, but eventually it will likely find many more practical applications—and yes, we're sure that one day the government will tax it.

  Corresponding to every particle there is an antiparticle in nature. Corresponding to protons we have antiprotons, to neutrons we have antineutrons, to top quarks we have anti-top quarks. When we made top quarks in the good ol’ days at the Fermilab Tevatron—now a staple of the CERN LHC—we made them in pairs: top plus anti-top. We literally go fishing and pull the negative-energy top quark out of the deep depths of the vacuum. This leaves behind a top quark hole (the anti-top quark), and we see the pair, top quark and anti-top quark, produced in our detectors. Particle physicists are simply metaphorical fishermen on the great Dirac sea.

  BETA DECAY: THE SIMPLEST WEAK INTERACTION

  The simplest example of the weak interaction is the beta-decay reaction that occurs with a single neutron, one of the particles found in the atomic nucleus, causing it to decay in about 11 minutes when it is floating about freely in space:

  n0 → p+ + e– + (missing energy)

  Beta decay is observed throughout many atomic nuclei, and it always involves this basic reaction. But beta decay posed a new problem: what is the “missing energy”? From countless observations, the electron and proton energies in the final state of the decay process always added up to something less than the original neutron energy. There thus appeared to be a missing amount of energy in the decay of a neutron. Essentially all beta decays of nuclei are a variation on this process, where the neutron is typically bound within the nucleus, and all revealed the mysterious “missing energy.”10

  Niels Bohr, one of the founding fathers of quantum mechanics, attempted to explain this phenomenon with the radical hypothesis that energy conservation, by which the initial energy is always equal to the final energy in any physical process, has only a limited validity in the world. Bohr proposed that the beta-decay processes were exhibiting, for the first time, a true violation of this time-honored and vaulted conservation law. Bohr, a brilliant and creative thinker, had already seen in the early part of the twentieth century that our detailed understanding of energy was significantly modified by the new rules of quantum mechanics, and he thought that perhaps beta decay was an indicator of deeper novelties and surprises yet to come.

  Wolfgang Pauli, the brash and brilliant theoretical physicist who had developed his exclusion principle to explain how atoms with many electrons are built, could not accept Bohr's idea. The principle of the conservation of energy up to this point had proven valid in all domains of physics. It seemed unnatural to Pauli that the violations would show up only in beta-decay reactions, where it is apparently seen to be a very large effect, and yet it doesn't show up elsewhere. Wouldn't any violation of this fundamental law of physics be universal, felt by all forces in nature, and not just be a property of beta decay? Bohr's proposal made no sense to Pauli.

  In 1930, Pauli therefore did something quite radical by the intellectual standards of his day: he postulated the existence of a new and unseen elementary particle that was also produced, together with the proton and the electron, in the beta-decay reaction. This new particle must carry no electric charge and would therefore escape the decay region totally unobserved, and it would maintain the validity of the conservation law of energy, provided it had a very tiny mass. In other words, physicists could now compute the missing energy required to maintain the conservation law in any beta-decay reaction, and this would be the exact energy carried off by the new particle.

  Pauli announced his new particle in a letter written on December 4, 1930, in a response to an invitation to attend a conference on radioactivity, which he declined.

  Dear Radioactive Ladies and Gentlemen,

  As the bearer of these lines, to whom I graciously ask you to listen, will explain to you in more detail, how because of the “wrong” statistics of the N and Li6 nuclei and the continuous beta spectrum, I have hit upon a desperate remedy to save the…the law of conservation of energy. Namely, the possibility that there could exist…electrically neutral particles, that I wish to call [neutrinos], which have spin 1/2 and obey the exclusion principle…and in any event [have masses] not larger than 0.01 proton masses. The continuous beta spectrum would then become understandable by the assumption that in beta decay a (neutrino) is emitted in addition to the electron such that the sum of the energies of the neutron and the electron is constant…

  I agree that my remedy could seem incredible because one should have seen these (neutrinos) much earlier if they really exist. But only the one who dare can win and the difficult situation, due to the continuous structure of the beta spectrum, is lighted by a remark of my honoured predecessor, Mr Debye, who told me recently in Bruxelles: “Oh, It's well bett
er not to think about this at all, like new taxes.” From now on, every solution to the issue must be discussed. Thus, dear radioactive people, look and judge.

  Unfortunately, I cannot appear in Tubingen personally since I am indispensable here in Zurich because of a ball on the night of 6/7 December. With my best regards to you, and also to Mr Back.

  Your humble servant,

  W. Pauli11

  The process of beta decay with Pauli's neutrino thus looks like this:

  The new particle, n, is the neutrino (with a “bar” over the symbol, it becomes the conventional symbol for antiparticle, or antineutrino).12 Therefore, when the neutron decays in free space, it produces a proton, an electron, and an (anti)neutrino. In our modern parlance, the electron is always produced together with the anti-electron-neutrino in a beta decay. The sums of the final energies of all of the three final particles will be exactly the same as the initial energy (mc2) of the original parent neutron. Notice also that the neutrino, with zero electric charge, allows the beta-decay reaction to satisfy the law of conservation of electric charge. The zero electric charge of the neutrino means that it can't be easily detected—it lacks the “handle” of electric charge that we could otherwise “grab onto” through electromagnetic fields in our particle detectors.

  With the details of beta decay now somewhat better understood, and Pauli's hypothesis of the neutrino, Enrico Fermi was able to write down the first mathematically descriptive quantum theory of the “weak interactions” in 1935. Fermi had to introduce a new fundamental constant into physics to specify the overall strength of the weak interactions, much like Newton had to introduce the “gravitational constant.” In fact, Fermi's constant, called GF, contains a fundamental unit of mass, which sets the scale of the weak forces—about 175 GeV. With Fermi's theory in hand it was now obvious that high-energy particle accelerators would eventually have to take over to study the details of the weak interactions. And not surprisingly, the Higgs boson has shown up with a mass of 126 GeV, not far from Fermi's “weak scale.”13 Pauli's neutrinos have also now been produced and subsequently detected in many experiments. Neutrinos were first directly detected by Clyde Cowan and Frederick Reines in 1956.14 The neutrinos are important to us—they are emitted from the sun as part of the fusion process by which the sun shines. Our very existence depends critically upon the feeble weak interactions in nature.