Quantum Chromodynamics

Too Strong to Work Out on Paper

Physicists usually get numbers out of a theory by adding up corrections. For electricity and magnetism they work out the simplest version of an event, then add a correction for one more brief stir of the electromagnetic field, then for two. Each correction is hundreds of times smaller than the one before, because the electric force has a strength of about one in 137. A handful of terms predicts how strong a magnet the electron is, and the prediction matches the measurement to about one part in a trillion. It also needs one more measured number, the strength of the electric force, whose two best measurements disagree with each other, so for now the check can go no closer.

The same method works for the strong force only where the force is weak: between quarks very close together, in the debris of the most violent collisions, which is where that falling strength has been tested. Probed at a tenth of a proton’s width, the strength is already about one half, and across the whole proton it grows larger still. Every correction is then nearly as big as the one before it, and adding them up gets nowhere. That size is not an accident. The strength has no units, and apart from the quarks’ masses nothing with a size goes into the theory, yet a size comes out: the distance at which the strength grows large, about a femtometer. That distance sets the size of the proton, and with it most of its mass. For decades, the theory of the strong force could not give the mass of the proton, the particle it was written to describe.

Space as a Grid

There is a way out. Replace smooth space and time with a grid of points. Put the quarks on the points and the gluon field on the short links between them, and the endless ways the gluon field can vary become a list of numbers: very long, but finite. Quantum physics says that every arrangement of the fields takes part in what happens, each with a weight of its own. Those weights are not chances: they swing like the hands of a clock and cancel one another. The quarks’ share can be worked out exactly for each arrangement of the gluon field and folded into its weight. Then a mathematical step that trades time for a fourth direction of space, called imaginary time, turns the weights into chances, numbers between nothing and one. A computer cannot try every arrangement of the gluon field, but it can then draw them at random, each as often as its chance says, the way a poll samples voters instead of asking everyone. One drawn field is pure noise. The average of thousands is the answer. Below, one slice of the grid is drawn as a sheet whose height at each point is the field’s strength there, with a quark glowing red and its antiquark cyan, the color that cancels red.

The first of these calculations, on a simpler two-color cousin of the theory, found that the pull between a quark and an antiquark stopped weakening with distance. Confinement showed up in the numbers with nothing put in by hand, and later grids found it with three colors too. Since then the grids have grown finer and wider. Each answer is followed to the limit where the spacing between points would shrink to nothing, because the grid is only scaffolding and the answer must not depend on it. The largest grids today have more than a hundred points along each side, as little as three to four hundredths of a femtometer apart, and keep the biggest supercomputers busy for years.

What the Grid Weighed

The first calculation that settled it fed in three measured masses: one to fix how heavy the up and down quarks are, taken as equal, one for the strange quark, and one to set the scale. Out came the masses of the other light particles built of quarks, the proton and neutron among them, as one number for the two. Every one matched experiment within its uncertainty, which for the proton was about four percent. Calculations since have brought that down to about one percent.

i The grids run for years on machines like these ×

Grids have since computed something far finer: that the neutron outweighs the proton, by about a seventh of a percent. Two small effects nearly cancel there. The neutron’s extra down quark is a little heavier than the up quark it replaces, while the proton’s electric charge adds some energy of its own. The result matters more than its size suggests. If the neutron were not heavier than the proton by more than an electron’s mass, a hydrogen atom’s electron would merge with its proton into a neutron. Hydrogen would not last, and there would be no water and no stars like the Sun.

The same grids look inside the proton. The masses of its three quarks add up to about one percent of it. Counting the extra pairs of quarks and antiquarks its field also holds, quark masses give about a tenth. The rest is the energy of the gluon field and of the quarks’ motion in it. The quarks’ spins, which textbooks long said made the proton’s whole spin, supply only about a third of it, as experiments have shown. The gluons and the motion of everything inside supply the rest, and grid calculations now split the total between them.

They also show why the pion, the particle whose field holds nucleons together, is so light, about a seventh of a proton. Empty space in this theory is not empty, in one precise sense: its quark field holds a steady value everywhere, as if quarks and antiquarks were paired throughout. It is not a crowd of particles, and nothing in it can be caught. A quark moving through it acts as if it weighed about a third of a proton. The pions are ripples in that steady value itself. They would weigh nothing at all if the up and down quarks had no mass of their own, and that lightness is why their field reaches as far as it does.

Recently the grid passed a test with high stakes. The muon, the electron’s heavier cousin, is a tiny magnet, and its strength depends on every field it stirs around it, the quark and gluon fields included. For two decades the measured strength disagreed with the predicted one, a hint, many hoped, of particles no one had found. The prediction had taken its strong-force part from other experiments. Worked out on grids instead, that part shifts, and the prediction now agrees with the final measurement. Why the older estimate came out different is still argued over. With the grid’s answer, the hint of new particles has faded; it would return only if the older estimate proved right.

Particles Nobody Ordered

Confinement asks only that the colors cancel. Three quarks of three colors do it, and so do a quark and an antiquark, which between them make every particle the strong force needs for nuclei. But four quarks would do it too, or five, or gluons with no quarks at all, since gluons carry color. For decades these were only allowed. Then they began to turn up, first of four quarks, then of five, and by now more than twenty such particles at the Large Hadron Collider alone. Whether each one is a single tight knot or two ordinary particles loosely stuck together is argued case by case.

i Two quarks, four, five, or glue alone ×

The particle made of gluons alone, the glueball, is the one the grids predict most firmly and the hardest to pin down. The lightest should weigh nearly twice as much as a proton, but a glueball mixes with ordinary particles of similar mass and looks like them. A strong recent candidate is for a heavier kind, which a mirror treats differently from the lightest. It is some two and a half times a proton’s mass, and its mass, its spin and the way a mirror treats it match what the grids predict for a glueball of that kind. It turned up in a decay rich in gluons, the kind where glueballs should be made most often. It also skips a way of breaking up that most particles built of quarks take, a sign that it treats every light quark alike, as glue does. Whether that settles it is still argued.

A Dial Set to Zero

The one line has room for one more piece, which none of the strong force’s behavior needs. It comes with an angle, called theta, that could take any value round a circle. If it were anything but zero, the strong force would not look the same in a mirror, nor in the mirror that also swaps matter for antimatter. It would also make every neutron lopsided. A neutron has no charge overall, but its quarks do: one up quark with two thirds of a proton’s charge and two down quarks with minus a third each. Theta would shift the positive charge a little toward one end of the neutron’s spin and the negative charge toward the other.

A lopsided neutron can be caught in the act. Kept in a chamber with a magnetic field, a neutron’s spin turns round the field like the hand of a clock. Add a strong electric field along the magnetic one, and a lopsided neutron would turn a little faster with the electric field one way than the other. The newest experiment, which recently began collecting data, stacks two chambers with the electric field up in one and down in the other, so any difference would show as one chamber’s neutrons pulling ahead, and the lead would change sides each time the field is reversed.

No neutron has ever been caught lopsided. The best measurement so far means that if a neutron were blown up to the size of the Earth, its charges could sit no farther apart than the width of a bacterium. That holds the dial within about a ten-billionth of zero. Nothing in the theory asks for that. Of all the numbers the theory leaves open, this one alone sits on a special value, like a pointer spun at random that stops within a hair of the mark. This is the strong CP problem, named for the combined mirror it would break, CP, which swaps matter for antimatter (C) and reflects space (P) at the same time.

The leading explanation is that the dial is not fixed, but a field that rolls to zero by itself, the way a marble settles at the bottom of a bowl. The field’s ripples would be a new, light particle, named the axion after a laundry detergent, because it would clean up the problem. Axions would also be among the best candidates for the dark matter that holds galaxies together. Experiments that listen for axions turning into faint radio waves inside strong magnets have found none yet, across the range of masses they have covered.

Still Out of Reach

The grid that weighed the proton has a hard limit. Drawing fields at random works only if each one’s weight, in imaginary time, behaves like a chance. Pack quarks densely, as in the core of a neutron star, and the weights turn into numbers that cancel one another. The sampling stops settling down, and no computer is big enough to repair it. This is the sign problem. It is why no one can yet compute from the one line what matter is like at the center of a neutron star. Nor can the grid say whether the change from ordinary matter to quark-gluon plasma, smooth at the low densities it can reach, turns sharp at high density, with a critical point where the two behaviors meet. Recently an experiment smashing gold nuclei at lower energies, to make denser drops, reported a dip near where such a point might show, in the shape of the spread of proton numbers from one collision to the next (how often a collision makes far more or far fewer protons than usual), without claiming one.

i What fills a neutron star’s core is beyond today’s grids ×

The other gap is a proof. Every experiment and every grid calculation shows confinement, yet it has not been derived from the one line. A related statement is one of the seven Millennium Prize Problems, with a million dollars offered for a proof: that a world of gluons alone can be built in exact mathematics and has a mass gap, with no ripple of its field lighter than some smallest mass. It is still open. Physicists do not doubt the answers. What is missing is an understanding of why a theory this small has to give them.