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Learning about QCD from String Theory

Quantum Chromodynamics (QCD) is a very successful theory of the strong nuclear interactions, the interactions that bind together quarks to form protons. The theory is well tested when applied to high-energy reactions, including the processes of inelastic electron-proton scattering and quark production from electron-positron annihilation studied here at SLAC, and agrees excellently with the data. QCD has the property, called "asymptotic freedom," that its strong interactions become weak at high energies, allowing its predictions to be worked out systematically. At low energies, the QCD interactions become very strong, and it becomes difficult for theorists to make precise predictions. So far, most of our knowledge about QCD at low energies has come from simulating the theory on powerful computers. However, it seems that string theory can give us insight into strongly coupled low-energy QCD. During my visit to SLAC and Stanford over the past year, I have been using string theory to work out the properties of strongly-coupled QCD and other similar theories.

QCD involves quarks of three different types, called "colors." Hadrons—protons, pions, and other strongly interacting particles—are bound states of quarks with no visible color. The most basic properties of QCD would be similar if we change the number of colors, and it is interesting to think about this generalization, even considering the case in which the number of colors is very large. Many years ago, Gerard 't Hooft (of the University of Utrecht) showed that this limit of QCD with a large number of colors gives a string theory. The confinement of quarks into hadrons becomes obvious in this picture; every quark is connected by a string to an antiquark of the opposite color. It seems that the string theory of QCD is similar to the string theories that were later proposed to give "theories of everything." Many solutions of the equations of string theory are now known, and there are more to be discovered. It is not yet known which solution of string theory describes QCD with a large number of colors.

About ten years ago, Juan Maldacena (of the Institute for Advanced Study) discovered a direct connection between a field theory closely related to QCD and a specific string theory in a curved 10-dimensional space. Following Maldacena's route, we have been able to construct new string theories equivalent to field theories that come increasingly close to QCD with a large number of colors. These relatives of QCD that we can represent by string theories involve additional unwanted matter fields. There is a limit in which these additional fields are made very heavy and become irrelevant, so that the string theories become the string theory for QCD. This limit is very hard to analyze. However, the theories with extra matter fields can be easily analyzed by standard string theory tools. This limit is not quantitatively the same as QCD, but we might hope for that these models are qualitatively similar to QCD.

In fact, these theories have been proven to have many of the characteristic properties of low-energy QCD. Quarks are confined. Quarks acquire mass through their strong interactions. The pi mesons are present as the lightest hadrons, as in QCD, and their properties are the same as those observed. The properties of pions are related to the spontaneous breaking of a symmetry of QCD called chiral symmetry, and we can prove that this same symmetry is broken in the string theories. We can compute the spectrum of mesons, and, with some more work, the spectrum of baryons. Of course, the string theory contains extra states. Many of these extra states are related to the fact that the strings actually move in a ten-dimensional space-time. The six extra dimensions, which we might like to be very small, are taken to have the size of hadrons in order to make the theory easier to solve.

Since these string theories are solvable and we believe that they are similar to QCD, we can use them as models to explore aspects of QCD that are more difficult to learn about directly. For example, we can ask how these theories behave when we heat them. At high temperature, QCD is expected to make a transition to a new phase in which chiral symmetry is unbroken and quarks are not confined. We would like to know whether these two effects, which are in principle distinct, set in at the same temperature or at different temperatures. Computer simulations of QCD indicate that the two transitions happen together, but it is not clear whether the two phenomena of confinement and chiral symmetry breaking are intrinsically connected. In work with Jacob Sonnenschein and Shimon Yankielowicz of Tel-Aviv University, I studied the high-temperature transitions of string models related to QCD. We showed there is no strict connection between quark confinement and chiral symmetry breaking. In some range of parameters the two phase transitions happen at the same temperature, while for another range of parameters they are separate. This conclusion is important guidance in how to think about QCD. It might also be relevant to the early universe, if a phase transition like that in QCD is responsible for the spontaneous breaking of electroweak symmetry.

We continue to work on trying to find the string theory equivalent to QCD, and to find additional qualitative features of QCD that we can study from the string theory viewpoint. It would be remarkable if the first practical applications and experimental tests of string theory would come from QCD rather than from unification models at very high energy.

—Ofer Aharony, SLAC Today, August 2, 2007
Aharony is a Professor of Physics at the Weizmann Institute, and has spent the past 7 months on sabbatical at SLAC and Stanford.