Physics Overview

The dominant cost of lattice QCD simulations comes from including the effects of quark vacuum polarization (sea quarks). Until recently most simulations were done in the so-called quenched approximation, in which these effects are simply excluded from the QCD path integral, which induces a large and uncontrolled systematic error. The development of the improved staggered discretization scheme for light quarks, however, dramatically improves the efficiency of unquenched simulations. Working together with the MILC Collaboration and the Fermilab Lattice QCD Collaboration, we have demonstrated that high-precision results are feasible using the improved staggered quark action for light sea quarks. This dramatic improvement is illustrated in Fig. 1 below, which compares experimental data for ten different quantities with lattice QCD results, from both quenched simulations and from simulations of QCD including the full effect of u, d and s sea quarks.

Fig. 1: Lattice QCD results divided by experimental measurements for π and K decay constants, and for mass splittings in light meson, light baryon, charmonium, and bottomonium systems. These figure shows that including the effect of light sea quarks is necessary to achieve high-precision results.

Another key ingredient in high-precision lattice QCD is perturbation theory, which connects lattice simulation results to continuum quantities. The perturbative matching is the largest source of systematic error which remains once sea quarks are included. These perturbative calculations must be done through two-loop order so as to reduce the matching error to an acceptable level of a few percent. Analytic perturbation theory is extremely difficult however, because the lattice formulation of QCD leads to a proliferation of diagrams that are not present in the continuum. The HPQCD Collaboration has developed computer algorithms to automate the generation of the lattice Feynman rules, for arbitrarily complicated actions. The computer program has recently been applied to obtain a three-loop determination of the strong coupling constant (Fig. 2), which gives α_s(M_z,MSbar) = 0.1170(12), which agrees with (and is more precise than) α_s(M_z,MSbar) = 0.1187(20) the accepted world average given by the Particle Data Group. The quark masses have also been determined from these lattice calculations and converted to a continuum quark mass using lattice perturbation theory. We recently reported a two-loop determination of the light quark (up, down and strange) masses and a one-loop determination of the charm and bottom quark masses. These results are summarized in Fig. 3.

Fig. 2: A three-loop determination of the strong coupling constant from lattice QCD. Twenty eight different quantities were used to extract the coupling. The left panel shows α_s at different scales, which demonstrates the evolution of the coupling constant. The right panel shows α_s from the different quantities using a standard renormalisation scheme and scale for comparison to that quoted by the particle data tables.

Fig. 3: Quark masses determined using lattice QCD (circles) compared with the existing world averages from the particle data group (lines).