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NERSC 3 Greenbook

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Lattice Field Theory

Doug Toussaint, Physics Department, University of Arizona

Elementary particle physics is concerned with the fundamental questions ``what are things made of'' and ``what holds them together''. Current theories of fundamental forces are theories of fields, objects which are functions of space and time. Quantum field theories of the strong, weak and electromagnetic interaction, namely quantum chromodynamics (QCD), and the Weinberg-Salam model, are generally accepted, while quantum theory of gravity is at best in its infancy. All of these theories involve local invariance principles, or ``gauge symmetries''.

Quantum chromodynamics describes the strong interactions between quarks; these interactions bind quarks together to form protons, neutrons, and other particles known collectively as hadrons. The strong nuclear force is believed to be a consequence of the much stronger forces of QCD. The other half of the Standard Model, the Weinberg-Salam model, unifies the weak force with electromagnetism. The Weinberg-Salam model predicts the existence of the W, the Z, and the Higgs particle. Although very successful, the Standard Model is incomplete: a true ``Theory of Everything'' still requires much theoretical and experimental work. Furthermore, many predictions of the Standard Model will be extracted only with considerable effort.

An important tool for this work is lattice gauge theory, introduced by Kenneth Wilson in 1975, in which continuous space-time is replaced by a four dimensional lattice. Wilson's work made possible the use of numerical simulation in the study of the Standard Model. The introduction of a space-time lattice is an approximation; to represent reality accurately, the lattice spacing must be small and the overall lattice size large compared to the scale of the problem studied.

Numerical simulation has been used extensively in the study of Quantum Chromodynamics. QCD is a very successful theory, explaining many results from high energy scattering experiments. However, QCD is also a very complicated theory and defies analytic solution for the fundamental properties of strongly interacting particles, such as their masses. The basic problem is the fact that the strong interactions, which are one of four basic forces of nature, are ``strong''. This means that they entail highly non-perturbative and non-linear interactions that are not amenable to analytical methods. Over the past decade, computer simulations have proven to be valuable theoretical tools for the study of QCD and other quantum field theories. Many properties of strongly interacting particles have been studied, including the following:

• Particle Masses: Hundreds of strongly interacting particles and their masses are, in principle, completely determined by the handful of parameters that define QCD. The explanation of these masses is the first important step in a successful first-principles analysis of the properties of QCD. There is now strong evidence that lattice calculations in QCD give the correct hadron spectrum when the lattice spacing and quark masses are made small enough.
• Glueball Spectroscopy: QCD firmly suggests the existence of strongly interacting particles that are not made of quarks. These particles, known as glueballs, are bound states of the gluons that bind quarks. The experimental situation is confused, with several glueball candidates but no definitive evidence. Lattice studies of QCD give important information on glueball properties.
• Heavy Quark Potential: The potential between heavy quark-antiquark pairs is important for understanding the spectroscopy of heavy quarks such as the bottom (``b'') and top quark.
• QCD Thermodynamics: QCD appears to exhibit a phase transition at high temperature to a new form of matter in which quarks are no longer confined inside the proton and the neutron. Such a phase was presumably present during the very early universe. More immediately, experiments at RHIC, the ``Relativistic Heavy Ion Collider'' under construction at Brookhaven National Laboratory, and at CERN are expected to create short lived ``fireballs'' of matter in this new phase. Theoretical calculations are important to understand the properties of this phase, and to help interpret these experiments.
• Hadron Form Factors: Recent simulations have measured the electromagnetic structure functions of some strongly interacting particles. These results are consistent with experiments, within acceptable error limits.
• Weak interaction matrix elements: The past several years have seen slow but steady progress in the calculation of weak interaction matrix elements, which determine the effect of the strong interaction on the weak interaction. These calculations aim, for example, to explain the $\Delta I = 1/2$ rule in Kaon decay and to predict the charge-parity (CP) violation parameters. The purely theoretical background needed for these calculation is formidable, as are the simulations themselves and the data analysis.
• B meson properties: A specific set of weak matrix elements which have attracted a lot of attention during the last few years are those involving mesons containing a b quark, the second heaviest of the six known quarks. These mesons are quite massive ($\sim5$ GeV) and are considered to be the best candidates for elucidating the extremely important phenomena of CP-violation. The phenomena of CP violation is intimately related to the observed baryon asymmetry in the universe. The B-factories being constructed at SLAC and at KEK (Japan) will produce a large number of B-mesons, about 10⁸/year, which will be used to study the properties of B-meson especially to view the CP violation phenomena. Interpretation of these experiments will require the input from computer calculations that we are doing. The methodology that we have developed allows us to calculate many quantities with a precision of about 25%-30%. Cost-effective use of the forthcoming results from the B-factory experiments demands that computational methods such as ours must continue to further improve the accuracy in our calculations. With increased computing power and improvements that are now underway, we believe that we can increase our accuracy for calculating many of the important entities to a precision of about 10% in the next few years. This should be very helpful in a confrontation of the predictions of the Standard Model with results from experiments at the B-factories. Amarjit Soni, working at NERSC, and Claude Bernard and the MILC collaboration, working on the Paragon at ORNL, have been pursuing these studies on the DOE supercomputers.

In addition to studies of QCD, supercomputers are essential to the study of other theories. The simulation of the Weinberg-Salam model is of special interest, notably the simulation of the Higgs particle. Numerical simulation of the Higgs sector have been used to bound the energy region where experimentalists must look to find the Higgs particle. Also, simulations allow us to increase our understanding and test our ideas by simulating models other than the real world. We may wish to understand the effects of changing the number of kinds of quarks, or to isolate a phenomenon such as the breaking of chiral symmetry by studying simpler models, often in fewer than four space-time dimensions. For example, John Kogut and his collaborators have used DOE supercomputer facilities to study chiral symmetry breaking using a simpler interaction than the eight kinds of gluons required in QCD.

Lattice QCD simulations make large computational demands. The necessary computational power and memory are available only on massively parallel or vector supercomputers. Lattice theorists have made aggressive use of the newest parallel machines and even built specially designed machines, to do their calculations.

 

Figure 1: Lattice gauge theory derives its name from the four-dimensional space-time lattice used to simulate the behavior of elementary particles. This drawing illustrates the need for finer lattice spacing in resolving the quark structure inside the proton. Listed below each lattice are the times required for a single step of the computer code on a modern parallel supercomputer (e.g., the CRI T3D). Thousands of steps are required for a complete simulation.

NERSC 3 Greenbook

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