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An Elegant Approximation

SLAC theoretical physicist Stan Brodsky. (Photo by Lauren Schenkman.)

SLAC theoretical physicist Stan Brodsky and University of Costa Rica physicist Guy de Téramond have found a simple equation describing the behavior of the subatomic particles within the proton. Their paper, published in the February 27 issue of Physical Review Letters, is an important step in SLAC's long history of investigations into the quantum mechanical world of the proton.

"A better understanding of the structure of the proton has traditionally been one of the main goals of SLAC physics," Brodsky said. "How is it made up, at the fundamental level? That's one of our main driving points at SLAC, and, in fact, in the whole field of high energy physics and nuclear physics."

The SLAC tradition dates back to the 1960s, when Richard Taylor, Henry Kendall and Jerome Friedman used SLAC's linear accelerator to shoot high-energy electrons at the protons and neutrons in a liquid hydrogen target. The experiments confirmed a prediction of a young SLAC theorist named James Bjorken—the apparently fundamental protons and neutrons were made up of smaller particles called quarks. The discovery earned Taylor and his collaborators the 1990 Nobel Prize in physics.

Since that time, physicists have been wrestling with the mathematical description of the proton structure, a theory known as quantum chromodynamics, or QCD. The "chromo" refers to "color," a quality quarks have, similar to the way electrons have charge. Unfortunately, due to the complex nature of the fundamental force that binds quarks together, QCD so far has been extremely difficult to work with. In the language of QCD, asking the question, "Where are the quarks inside the proton, and how are they moving?" ties variables like color, spin, position and number of particles into a nightmarish Gordian knot.

Physicists have tried to simplify the problem by simulating space as a giant grid or "lattice," with quarks living only on the vertices of the squares and gluons, the force-carrying particles within the proton, traveling along the lines. Lattice gauge theorists construct the question of proton structure in this simplified universe, then feed it into a computer. By shrinking the squares of the lattice, theorists can make lattice gauge theory more accurate, but doing so increases the computing power required.

"I like to joke that the lattice people have half the computing power on earth," Brodsky said. "The largest computers in the world work on lattice gauge theory. But the theory only gives an indirect picture."

While lattice gauge theorists have made great progress, Brodsky said, they long for the simplicity of the equation Austrian physicist Erwin Schrödinger published in 1926, describing the force binding an electron to the nucleus in an atom. With a gentle shake, two extremely important pieces of information fall out of this equation—the electron's energy and its behavior, called the wavefunction.

What physicists would really like is a sort of Schrödinger equation for the proton—simple, straightforward, and solvable analytically, that is, with pen, paper and a little brainpower. Brodsky and de Téramond have discovered just such an equation. Although it doesn't provide an exact description, the equation promises to be a powerful first approximation to QCD, a more direct picture of the particle's behavior than is provided by the lattice approach. Most importantly, Brodsky said, "it can be systematically improved."

Although Brodsky is a theorist, his tools are in some ways as spectacular as the linear accelerator of Taylor et al.'s experiments. Brodsky and de Téramond borrowed a trick from string theory, a five-dimensional space called "Anti-de Sitter space," and changed ordinary time to the time seen by the front of a traveling light wave, called light-front time. By looking at QCD from this new perspective, called Light Front Holography, the nasty Gordian knot of many variables shrank down to a simple shoelace of only one variable.

"This is very analytic and very simple," Brodsky said. "It requires no computers and no crunching."

Like the Schrödinger equation, Brodsky and de Téramond's equation yields two pieces of information. The first is the mass of the proton—or other type of hadron, a conglomeration of quarks and gluons. When Brodsky and de Téramond write their equation for specific hadrons, the masses that emerge are very close to the particles' observed masses. The second piece of information is the wavefunction—an elegant mathematical picture of the quarks and gluons at a given light-front time.

"This is what excites me most," Brodsky said. "We've never known the wavefunction before, only approximate features, and there's so much that we need to calculate that requires the wavefunction."

One of those things is the "form factor," the mathematical function that describes the probability that a hadron will not fall apart when hit by an electron. Brodsky and de Téramond used the emergent wavefunctions to calculate this form factor as a second test of their equation's viability, and it appeared very successful.

"If we use these wavefunctions, we get a good representation of the experimental data," Brodsky said. "The form factor looks just like what Robert Hofstadter measured at Stanford, and what Taylor and his collaborators continued to measure here at SLAC."

Brodsky is most interested in using these wavefunctions to understand the conundrum of "color confinement," or why the quarks and gluons are always stuck together in a hadron, such as a proton or neutron.

"These wavefunctions are exactly what you need to convert the quarks into the hadrons you see at the final state," Brodsky said. "For the first time, people can now attempt to really understand the transition from quarks and gluons into hadrons."

In addition to his collaboration with de Téramond in Costa Rica, Brodsky is moving forward in this line of inquiry with groups as far flung as Iowa and Chile. He hopes to see others using his equation as a springboard for inquiries into confinement. "This is just the first step," Brodsky said.

Brodsky and de Téramond offer a more in-depth technical description of their recent work on the SLAC Document Server.

—Lauren Schenkman
SLAC Today, March 5