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Getting Your Quarks in a Row

A tidy lattice is the key to computing with quantum fields

Brian Hayes

Weighing the Quarks

With growing computational resources and algorithmic innovations, QCD finally has the power to make sharp, quantitative predictions. An exemplary case is a recent careful calculation of quark masses.

The importance of these masses is noted in a review article by Andreas S. Kronfeld of the Fermi National Accelerator Laboratory. The two lightest quarks, designated u and d , are the constituents of protons and neutrons. (The proton is uud and the neutron udd .) Patterns among the masses of other quarks suggest that u should weigh more than d . If that were the case, a u could decay into a d . "But then protons would decay into neutrons, positrons and neutrinos.... This universe would consist of neutron stars surrounded by a swarm of photons and neutrinos, and nothing else," Kronfeld says. Since the actual universe exhibits a good deal more variety, we can infer that the d must be heavier than the u . Until recently, however, QCD simulations could not produce reliable or accurate estimates of the u and d masses.

Earlier lattice computations had to ignore a crucial aspect of QCD. Events or pathways in which quark-antiquark pairs are created or annihilated were simply too costly to compute, and so they were suppressed in the simulations. This practice yields acceptable results for some QCD phenomena, but pair-creation events have a major influence on other properties, including estimates of the quark masses.

Algorithmic refinements developed in the past decade have finally allowed quark-antiquark contributions to be included in lattice computations. The new quark-mass estimates based on these methods were made by Quentin Mason, Howard D. Trottier, Ron Horgan, Christine T. H. Davies and Lepage. They derive a u mass of 1.9 MeV (million electron-volts) and a d mass of 4.4 MeV (with estimated systematic and statistical errors of about 8 percent). Thus the uud quarks in a proton weigh about 8 MeV; the mass of the proton itself is 938 MeV.

I am intrigued by this result, and I admire the heroic effort that produced it. On the other hand, I confess to a certain puzzlement that it takes so much effort to pin down a few of the simple numbers that define the universe we live in. Nature, after all, seems to compute these values effortlessly. Why is it such hard work for us?

©Brian Hayes


  • Creutz, Michael. 1983. Quarks, Gluons and Lattices . Cambridge: Cambridge University Press.
  • Creutz, Michael. 2003. The early days of lattice gauge theory. In The Monte Carlo Method in the Physical Sciences , AIP Conference Proceedings 690, pp. 52–60. Preprint:
  • Di Pierro, Massimo. 2007. Visualization for lattice QCD. In Proceedings of Science , XXV International Symposium on Lattice Field Theory,
  • Feynman, Richard. P. 1985. QED: The Strange Theory of Light and Matter . Princeton: Princeton University Press.
  • Kronfeld, Andreas S. 2008. Quantum chromodynamics with advanced computing. Presented at Scientific Discovery with Advanced Computing, July 13–17, Seattle. Preprint:
  • Lepage, G. Peter. 1993. Lattice QCD for small computers. In The Building Blocks of Creation: From Microfermis to Megaparsecs: Proceedings of the 1993 Theoretical Advanced Study Institute in Elementary Particle Physics , University of Colorado at Boulder, 6 June–2 July 1993, Stuart Raby and Terrence Walker, eds., pp. 207–236. River Edge, N.J.: World Scientific.
  • Lepage, G. Peter. 1998. Lattice QCD for novices. In Strong Interactions at Low and Intermediate Energies , edited by J. L. Goity, pp. 49–90. River Edge, N.J.: World Scientific. Preprint:
  • Mason, Quentin, Howard D. Trottier, Ron Horgan, Christine T. H. Davies and G. Peter Lepage. 2006. High-precision determination of the light-quark masses from realistic lattice QCD. Physical Review D 73:114501. Preprint:
  • Rebbi, Claudio. 1983. The lattice theory of quark confinement. Scientific American (February 1983) 248(2):54–65.
  • Wilson, Kenneth G. 1974. Confinement of quarks. Physical Review D 10:2445–2459.
  • Wilson, K. G. 2005. The origins of lattice gauge theory. Nuclear Physics B—Proceedings Supplements 140:3–19. Preprint:

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