Supersymmetry at Work for Quantum Chromodynamics
Quantum Chromodynamics (QCD) is the theory describing the interactions of quarks and gluons. One of the continuing mysteries of QCD is that quarks and gluons never appear in isolation. Instead, they are always confined into protons, pions, and other strongly interacting particles. Experiments here at SLAC and elsewhere have shown conclusively that particles with the properties of quarks are found inside the proton. We know this because we can compute the interactions of quarks at short distances using Feynman diagrams. But we do not have good mathematical tools to understand the long-distance forces that keep quarks confined.
There are other theories in which we can compute some aspects of the strong interaction responsible for quark confinement. These are theories with a special property called supersymmetry. Supersymmetry has been proposed as a part of the ultimate theory of nature. But even if it is not found in the real world, this symmetry has amazing properties that make some quantities of strongly coupled models with supersymmetry exactly calculable. It would be wonderful if we were able to benefit from this knowledge and translate some part of it to an improved understanding of QCD.
To realize this goal, many theorists are trying to construct precise paths that lead from QCD to a supersymmetric theory. Mikhail Shifman of the University of Minnesota and his collaborators gave a remarkable example: An important property of QCD in the real world is that quarks come in three colors and change their color by emission or absorption of a gluon. So, take QCD with gluons only, make the theory supersymmetric, and let the number of colors in the theory become large. Take QCD, without supersymmetry, with one flavor of quark, and let the number of colors become large. When this is done in a certain way, Shifman suggested, the two theories should be equivalent.
Working with Larry Yaffe from the Unversity of Washington and Pavel Kovtun from U C Santa Barbara, I have been trying to pin down the conditions under which this and similar equivalences are valid. To test the equivalences, we have subjected QCD and the supersymmetric theories to drastic modifications that make them easier to solve. For example, we consider one dimension of space, or all three, wrapped up so that they are very small.
Some of these modified versions of QCD have very weird behavior. The QCD that we see in the real world has particle-antiparticle symmetry, also called charge conjugation or C invariance. When we take one dimension to be small, we findfor QCD but not for its supersymmetric analogue—a system in which C is spontaneously broken. Tom DeGrand and Roland Hoffmann at the University of Colorado constructed a lattice model of QCD in which our results could be tested by numerical simulation. They found that there is a critical size below which C is broken. For large sizes, C is a good symmetry and the equivalence seems to hold.
We do not know yet where the insights from these modifications of QCD will lead. An exact equivalence between QCD and a supersymmetric theory might allow us to compute many properties of the real strong interactions. On the other hand, strong-interaction breaking of discrete symmetries, as we see in some of our weirder realizations of QCD, might be applied to build new models of observed breaking of CP and flavor symmetries by the weak interactions. One way or another, we will deepen our understanding of QCD and the family of theories that it belongs to.