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§ Phase xii · Quarks & Hadrons · Chapter 03

The proton, up close

Open a chemistry textbook and the proton is a small red ball labeled p+. Open a quark-model textbook and it is three colored marbles, two up and one down. Open a high-energy physics journal and it is a roaring swarm of quarks, antiquarks, and gluons that no painter has ever drawn faithfully. This chapter is about how the simple picture broke open.

If you have read this far in the book, the proton already has a face. In the periodic-table chapter it was the dot at the center of hydrogen. In the nuclear chapters it was the unit of positive charge that made fluorine fluorine and gold gold. By the time you reached the quark model two chapters ago, the proton had grown a little more complicated. It was three quarks in a bag. Two ups, one down, charges summing tidily to plus one, spins arranging themselves to make a half. A picture you could draw in three seconds. A picture that, for most chemistry and most nuclear physics, is all you ever need.

But three quarks in a bag was an idea waiting to be tested. By the mid 1960s the quark model existed only on paper. Murray Gell-Mann had proposed it, Yuval Ne’eman had organized the hadron families into the same rosettes, and George Zweig (independently, at CERN) had drawn the same picture with different names. None of them had seen a quark. Worse, no one ever would. Free quarks did not turn up in cosmic-ray plates, in bubble chambers, in oyster shells (Murray Gell-Mann actually suggested checking oyster shells, on the off chance that fractional charges had settled to the seafloor). Quarks remained, in 1967, hypothetical bookkeeping objects. The proton remained, on the page, a featureless ball of charge with three labels stamped on it.

The thing that changed this was an enormous machine in a brown field south of San Francisco. The Stanford Linear Accelerator, two miles of klystron-driven copper sections running due west under Highway 280, could push electrons to twenty billion electron volts. By 1968 a team of experimentalists led by Jerome Friedman, Henry Kendall, and Richard Taylor was firing this beam at a small flask of liquid hydrogen and watching, with a spectrometer the size of a house, what came out the other side. They were doing for the proton what Rutherford had done for the atom fifty-seven years earlier. They were looking inside it.

What they expected was boredom. Standard form-factor theory said that at high momentum transfer the elastic scattering rate (electron in, electron out, proton recoils intact) should fall off catastrophically. The proton is fuzzy. A hard punch should glance off it the way a billiard ball glances off a pillow. So the elastic rate would die. What was left over was the inelastic channel: the proton, struck hard, would shatter into a spray of pions and other debris, and only the scattered electron would be measured. This was called deep inelastic scattering, or DIS. The community’s prevailing guess, in 1967, was that the DIS rate would also fall fast, because in a fuzzy extended object every part of the charge participates and high-momentum kicks have to be shared. The spectrum would be smooth, featureless, and uninteresting.

The data refused to be uninteresting. The DIS rate did not fall fast. It fell hardly at all. Above a certain momentum transfer the electrons came back as if they had bounced off a hard pebble, not a pillow. Worse, the rate had a specific shape, controlled by a single dimensionless ratio that James Bjorken at SLAC had predicted on theoretical grounds the year before. Bjorken called the variable x. Its value was the fraction of the proton’s momentum carried by whatever the electron had hit. The scattering rate depended only on x and not on the energy of the beam, a phenomenon Bjorken called scaling. Scaling is what point-like targets do. It is the signature of Rutherford’s gold-foil experiment translated into the language of fields. Scaling said that whatever was inside the proton, it was not a smear of charge. It was a collection of hard, small, point-like things.

Richard Feynman, on a visit to SLAC in 1968, drew the picture that nailed the interpretation. He drew the proton as a bag full of free particles, all moving forward at nearly the speed of light, and he gave them a name that was deliberately agnostic about what they were. He called them partons. The DIS cross section was, in his picture, just the incoherent sum of elastic scatterings of the electron off individual partons. Feynman did not, in his 1969 paper, commit himself to identifying partons with Gell-Mann’s quarks. He kept the language general. But the rest of the community made the connection within a year. The fractional electric charges of the partons, extracted from the relative DIS rates on protons and neutrons, came out to plus two thirds and minus one third. Quarks had been spotted, indirectly but unmistakably, by the bruises they left on a beam of electrons.

So the modern picture of the proton is not three quarks in a bag. It is three quarks in a bag plus everything else the strong force is allowed to put there. The strong force is mediated by gluons, eight of them, and gluons can split into pairs of quarks and antiquarks that live for a moment and then recombine. They can also split into more gluons. At any given instant a high-energy proton is teeming with these short-lived bystanders, which collectively are called the sea. The two ups and one down that you can never get rid of, the ones whose quantum numbers determine that the proton is a proton, are now called the valence quarks. The valence quarks are the permanent residents. The sea is the rotating cast of guests.

How many guests, on average? It depends on the energy at which you ask. This is one of the most counterintuitive facts in particle physics. The proton, viewed at low resolution, looks like three quarks. The proton, viewed at high resolution, looks like a swarm of hundreds of partons. There is no contradiction. When you fire a harder probe you resolve shorter time scales, and on a shorter time scale you see more of the virtual fluctuations that quantum field theory allows. The same proton, examined at different shutter speeds, has a different inventory. Quantum chromodynamics (QCD), the theory that handles all of this, predicts the shutter-speed dependence quantitatively. The predictions have been confirmed to fractions of a percent at HERA, at the LHC, and at Jefferson Lab. The proton is not an object with a fixed contents. It is a process with a fixed identity.

Now to the most spectacular consequence of this picture. The proton weighs 938 mega-electron-volts. The up quark, measured by itself, weighs about 2.2. The down quark weighs about 4.7. Add them with the standard recipe (up, up, down) and you get something like 9 MeV. That is one percent of the proton mass. The other ninety-nine percent is not there. It does not exist as rest mass of any constituent. It is, almost entirely, the kinetic and field energy of the quarks and gluons sloshing around inside. Einstein’s E = mc² is usually quoted to explain mass-to-energy conversion in nuclear reactions. The proton runs the same equation in reverse. Energy of a confined relativistic flow, divided by c², shows up on a balance as mass. Yours, mine, and every atom of the visible universe is mostly that. Strip out the contribution of confined gluon energy and the world would weigh about a percent of what it does.

The same partonic picture also exposes a problem that has been embarrassing the proton community for forty years. The proton’s spin is exactly one half. In the naive three-quark model that one half is easy to assemble: two ups paired antisymmetrically with one spin up and one spin down, the third quark spinning either way, and the totals come out right. Done in a paragraph. But when experimentalists at the European Muon Collaboration measured how much of the proton’s spin actually sits in the spins of its quarks (the experiment, run at CERN starting in 1987, is known as the EMC spin crisis), the answer was about thirty percent. Not one hundred. Not a number you can round to one. Thirty. The quark spins by themselves account for less than a third of the proton’s spin. The rest is split among gluon spin, gluon orbital angular momentum, and quark orbital angular momentum, in proportions that experimentalists at RHIC and Jefferson Lab are still nailing down. The story is unfinished, and an entire planned facility (the Electron-Ion Collider at Brookhaven, scheduled for the 2030s) is being built largely to settle it.

Q² = -q²       (positive, by convention)
ν   = q·P / M  (energy transfer in the proton rest frame)

x = Q² / (2 M ν)

F₂(x, Q²) = Σᵢ eᵢ² · x · fᵢ(x, Q²)

Step back from the swarm for a moment. The proton, as a chemistry textbook draws it, is correct. It really does carry charge +1, it really does have spin one half, and it really does combine with electrons to make hydrogen in exactly the way the periodic table promises. The naive picture is not wrong. It is coarse-grained. It is what you see when you do not look hard. The partonic picture is not a correction to the naive picture; it is the same picture at higher resolution. When you fire a soft probe at a proton (the photon emitted by an orbiting electron, for example) you see three quarks in a bag. When you fire a hard probe (a 20-GeV electron, or anything in a collider) you see the seething ensemble that QCD allows. Both views are accurate. They are answers to different questions.

This is, in fact, the lesson the proton teaches the entire Standard Model. Particles are not little balls. They are configurations of fields whose effective contents depend on the energy scale at which you interrogate them. The electron looks point-like at every scale we can probe (down to 10⁻¹⁸ meters), and so we call it elementary. The proton fails this test catastrophically, and so we call it composite. The line between the two is a sliding line, fixed only by the experimental energies you have available. And the rules that govern the proton’s inner life (color charge, gluon exchange, asymptotic freedom, confinement) are universal. They will appear again, in different colors, when we draw the diagrams of the strong interaction itself. Vertices and propagators are the language of those diagrams, and that is where we go next.