Quantum chromodynamics with advanced computing

arXiv:0807.2220v1 [physics.comp-ph] 14 Jul 2008 Quantum chromodynamics with advanced computing Andreas S Kronfeld Fermi National Accelerator Laboratory, Batavia, IL 60510, USA (for the USQCD Collaboration) E-mail: ask@fnal.gov Abstract. We survey results in lattice quantum chromodynamics from groups in the USQCD Collaboration. The main focus is on physics, but many aspects of the discussion are aimed at an audience of computational physicists. 1. Introduction and background Quantum chromodynamics (QCD) is the modern theory of the strong nuclear force. The physical degrees of freedom are quarks and gluons. The latter are the quanta of gauge fields, in some ways analogous to photons. The crucial difference is that gluons couple directly to each other, whereas photons do not. A consequence of the self-coupling is that the force between quarks does not vanish at large distances (as the Coulomb force e2/(4πr2) does) but becomes a constant FQCD ≈800 MeV fm−1 ≈15, 000 N. It would, thus, require a vast amount energy to separate quarks out to a macroscopic distance. Indeed, long before the separation becomes large enough to measure, the energy stored in the gluon field “sparks” into quark-antiquark pairs. This phenomenon of QCD explains why freely propagating quarks are not observed in nature, and it is called confinement. QCD is a quantum field theory, which means that from the outset one must deal with mathematical objects that are unfamiliar even to many physicists. The most widely used theoretical tool for quantum field theories is relativistic perturbation theory, as developed for quantum electrodynamics (QED) by Feynman, Schwinger, Tomonaga and others sixty years ago (1). Perturbation theory provides the key connection between the mathematical theory with experiments in QED (2) and the Glashow-Weinberg-Salam theory merging QED with the weak nuclear force (3). A textbook example of perturbative quantum field theory is to calculate how virtual pairs induce a distance dependence on the coupling in quantum gauge theories. In QED, one considers the fine structure constant, α = e2 4π¯hc, (1) where −e is the charge of the electron. In QCD, one has the strong coupling αs, related to the gauge coupling g by analogy with equation (1). When virtual pairs are taken into account, α and αs are not constant but depend on distance; they are said to “run.” The running, depicted in figure 1, is completely different in the two theories. In QED α becomes constant at distances r > ¯h/mec (the electron’s Compton wavelength), but at short distances it grows. In QCD αs grows at long distance; perturbative running no longer makes sense once r is in the confining regime. At short distances—or, equivalently, high energies—the QCD coupling is small enough that perturbation theory can again be used. For example, the cross section for an electron- positron pair to annihilate and produce a quark-antiquark pair can be reliably computed as a power series in αs. The quark and antiquark each fragment into a jet of hadrons—mostly pions, but some protons and neutrons, too—and general properties such as the energy flow and angular distributions of these jets can be traced back to the quark and antiquark. In this way, the experiments can measure quark properties, and these are found to agree with theoretical calculations (4). QCD and the electroweak theory form the foundation of the Standard Model of elementary particles. They are so well established that many particle physicists consider the chromodynamic and electroweak gauge symmetries, and the associated quantum-number assignments of the quarks and leptons, to be laws of nature. We cannot conceive of a more comprehensive theory that does not encompass them. But there is more to the Standard Model. We know that something must spontaneously break the electroweak symmetry and that something (perhaps the same thing) must generate masses for the quarks and leptons. The Standard Model contains interactions that model these phenomena. The model interactions are consistent with observations yet also incomplete. It is hoped that insights obtained with the Large Hadron Collider (LHC), to commence operations at CERN later this year, will enable particle physicists to solve many of the puzzles raised by the Standard Model. The purpose of this paper is to cover QCD, particularly in the strong-coupling regime where perturbation theory is insufficient. It is worth bearing in mind, however, that there is a close connection between many of the results discussed below and some of the big questions in particle physics. A simple example is the masses of the quarks. To determine the quark masses (which because of confinement cannot be measured), we have to calculate the relation between quark masses and measurable quantities, such as hadron masses. Quark masses are interesting for many reasons, the most unsettling of which is as follows. The mass of the top quark is larger than that of the bottom quark; see figure 2. Similarly,

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