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§ Phase xiii · Feynman Diagrams · Chapter 03

Beta decay, diagrammed

A neutron in free space lives, on average, about fifteen minutes before it falls apart into a proton, an electron, and an unseeable companion. Enrico Fermi wrote the first equation for that process in 1934 on the same trip where he was writing about slow neutrons, and his hand-drawn picture of it (four lines meeting at one point) became the seed of every Feynman diagram drawn since. Inside that knot of lines lives the second force of nature: the weak interaction, the one that powers the Sun, dates the Shroud of Turin, and prefers its electrons left-handed.

The first time anyone wrote down a Feynman diagram, it was not by Feynman, it was not called a diagram, and it had no propagator on it. It was Enrico Fermi’s 1934 sketch of beta decay, and it was a single dot where four lines met: a neutron coming in, and a proton, an electron, and a neutrino going out. The picture was a stand-in for an equation, the equation was a guess at the simplest thing that could possibly work, and the guess turned out to be the entire low-energy structure of the weak force, dressed up in the smallest number of symbols a theorist could get away with.

To see why Fermi was driven to a guess at all, back up to the 1920s. Beta decay had been a known phenomenon since Rutherford had named it in 1899. By the late 1920s, three things about it were puzzling. First, the beta particle (which by then everyone agreed was an electron) came out with a continuous range of energies, instead of the single sharp line you would expect if a parent nucleus simply broke into a daughter nucleus and a single electron. Second, the parent and daughter nuclei often had different spins, and the spin of one electron could not, on its own, balance the books. Third, calorimetry experiments in 1929 by Lise Meitner and Wilhelm Orthmann had pinned down that energy was being quietly lost, not just smeared around. Something was leaving the apparatus carrying angular momentum and energy and refusing to deposit either anywhere measurable.

The community split over what to do. Niels Bohr, of all people, suggested that perhaps energy was not exactly conserved in single nuclear events, only on average. He was prepared to give up the conservation laws sooner than invent a new particle. Wolfgang Pauli took the opposite tack. In a now-famous letter dated December 4, 1930, addressed playfully to a gathering of nuclear physicists in Tübingen as Liebe Radioaktive Damen und Herren, he proposed what he called a desperate remedy: a third particle, neutral, very light, escaping every detector then known, slipping out alongside the electron and carrying away the missing books. He apologised in the same letter for not coming to the meeting in person; he was at a ball in Zurich.

Fermi liked the new particle. He gave it the name neutrino, the little neutral one, to distinguish it from the heavy neutral particle Chadwick had just discovered (the neutron) and to flatter Pauli with the diminutive. Then, in the autumn of 1933, Fermi sat down at his desk in Rome and asked the practical question: granted that the neutrino exists, what is the simplest mathematical interaction that turns a neutron into a proton plus an electron plus an antineutrino, in a way that fits everything we have measured?

He wrote down four operators in a row. One destroyed a neutron at a point. One created a proton at the same point. One created an electron at the same point. One created an antineutrino at the same point. Multiply them, integrate over all space and time, and you have an interaction. There was no exchange particle in the formula, no propagator, no internal lines. Just four fields meeting at a single spacetime vertex, with one new constant of nature out front to set the strength. Fermi called it G. We now call it the Fermi constant, G_F. Its measured value, in modern units, is about 1.166 × 10⁻⁵ GeV⁻², a number that comes back in every weak-interaction calculation done since.

The paper landed at Nature, which rejected it as too speculative. Fermi sent it instead to the Italian journal Ricerca Scientifica, where it appeared at the end of 1933, and later to Zeitschrift für Physik. It is now widely treated as the founding document of weak interactions. From the four-field vertex Fermi predicted the shape of the beta spectrum (the famous Kurie plot, falling smoothly to zero at the endpoint energy if the neutrino is massless) and the half-lives of neutron and tritium decay within an order of magnitude. He published it before Yukawa wrote anything. The picture of physics it carved out (interactions happen at a point, mediated by a coupling constant, drawn as a dot where lines meet) is what Feynman would formalise a decade later. The original Feynman diagram was Fermi’s.

For Fermi, the four-field theory was a phenomenology, not a final answer. It worked at low energies, the only energies anyone could probe in the 1930s. It said nothing about what was really going on inside the vertex. Through the 1940s and 1950s, as the calculation tools got better, theorists like Julian Schwinger started to wonder whether the same trick that had worked for electromagnetism (a long-range force from the exchange of a massless photon) could be modified to produce a short-range force by exchanging a massive particle.

For the weak interaction the same logic suggested a charged exchange particle, very heavy, with a name that has stayed: the W boson, for weak. Apply Yukawa’s recipe to the beta-decay vertex and the four-fermion interaction splits in two. Instead of (neutron, proton, electron, antineutrino) all touching at one point, you have two vertices joined by a W propagator. At one vertex a d quark inside the neutron emits a W⁻ and turns into a u quark, which is what changes the neutron into a proton at the nucleon level. The W⁻ then propagates a tiny distance and decays at a second vertex into the electron and the antineutrino. The Fermi constant becomes a derived quantity: G_F is roughly g²/M_W², where g is the W’s coupling to fermions and M_W is its mass. The mass of the W came out, when it was finally measured at CERN in 1983, at about 80.4 GeV. The corresponding force range is about 10⁻¹⁸ metres, far smaller than a proton. At any energy much lower than 80 GeV the W propagator is effectively a constant, and the whole picture collapses back to Fermi’s single point. Fermi was not wrong; he was working at low energy.

The exchange picture explains something the original Fermi theory could not: parity violation. The story is one of the most dramatic plot twists in twentieth-century physics. By the mid 1950s a puzzle had appeared in the cosmic-ray data. Two particles, called the tau and the theta, were observed decaying through the weak force into different numbers of pions. The decay products of one had the opposite spatial-mirror behaviour from the other, yet the tau and theta otherwise looked identical: same mass, same lifetime, same charge. Either they were two distinct particles that coincidentally shared every measurable property, or they were the same particle decaying through a force that did not respect mirror symmetry.

In June 1956 Tsung-Dao Lee and Chen-Ning Yang published a careful theoretical paper pointing out that no experiment had ever tested whether the weak force conserved parity. The strong force did. Electromagnetism did. Gravity did. Everyone had simply assumed the weak force would behave the same way. Lee and Yang sketched a handful of experiments that could check. Chien-Shiung Wu, an experimental physicist at Columbia who had built her career on precision beta-decay measurements, picked one of them up that summer. By December her team had aligned the spins of cobalt-60 nuclei in a magnetic field at 0.01 kelvin and started counting the directions of the emitted electrons. The result, published in January 1957, was as clean a violation as nature ever delivers: electrons came out preferentially in the direction opposite to the nuclear spin. The mirror image of the experiment, in which all spins are reversed, would have given electrons coming out parallel to the spin. The two pictures are physically distinct. The weak force can tell its left hand from its right.

D_W(q) = 1 / ( q² − M_W² )

D_W(q) ≈ −1 / M_W²

( g / √8 )² · ( −1 / M_W² )  =  − g² / ( 8 M_W² )

G_F / √2  =  g² / ( 8 M_W² )

The story of beta decay does not end with diagrams. The weak force has fingerprints all over the universe, most of them familiar in shape but quietly weak-mediated under the hood. The Sun shines because, at its core, two protons fuse into deuterium via a weak conversion: one of the two protons turns into a neutron, emitting a positron and a neutrino. Without parity-violating weak interactions, hydrogen would not burn. Stellar nucleosynthesis grinds to a halt. Carbon-14 dating, the technique that puts the Shroud of Turin at the medieval period and gave us the chronology of the cave paintings at Lascaux, depends on the steady beta decay of carbon-14 to nitrogen-14 with a 5,730-year half-life. Every time a smoke detector buzzes, the chip inside is registering alpha particles from a small chip of americium, but the chain of nuclear reactions that built that americium in a reactor ran through a half-dozen beta decays. Supernova neutrinos, the burst that arrives a few hours before the visible flash, are weak-force fingerprints from the collapse of a dying star’s iron core. The 1987A burst registered in the Kamiokande detector was the first time we ever heard a star’s last breath in real time.

Pull on any of those threads and you arrive back at the same diagram. A neutron sits there. One of its down quarks emits a W⁻ boson. The W⁻ flicks out of existence and leaves an electron and an antineutrino on the table. Three lines come in, three lines go out, a fourth line (the propagator) connects two vertices. The propagator is short and stubby because the W is heavy. The two vertices each carry a coupling that, when squared and divided by the W mass squared, reproduces the constant G_F that Fermi guessed in 1934. The diagram and the equation are the same object, written in two languages.

For the last detail, glance back at the figure. The neutrino line is drawn dashed in both panels. That is not a stylistic decision; it is a convention to mark a particle that almost never interacts. It travels through the wall of the laboratory, through the Earth, through three light-years of lead without leaving a track. Pauli, in his 1930 letter, called it the desperate remedy. Twenty-six years later, Cowan and Reines pulled the first neutrinos out of the antineutrino flux from a reactor at Savannah River. They sent a telegram to Pauli on June 14, 1956: We are happy to inform you that we have definitely detected neutrinos. Pauli, in Zurich, opened a bottle of champagne. The bet was lost. The particle was real. The diagram was right.