Nonperturbative QCD simulations with $2+1$ flavors of improved staggered quarks

Dramatic progress has been made over the last decade in the numerical study of quantum chromodynamics (QCD) through the use of improved formulations of QCD on the lattice (improved actions), the development of new algorithms, and the rapid increase in computing power available to lattice gauge theorists. In this article simulations of full QCD are described using the improved staggered quark formalism, “asqtad” fermions. These simulations were carried out with two degenerate flavors of light quarks (up and down) and with one heavier flavor, the strange quark. Several light quark masses, down to about three times the physical light quark mass, and six lattice spacings have been used. These enable controlled continuum and chiral extrapolations of many low energy QCD observables. The improved staggered formalism is reviewed, emphasizing both advantages and drawbacks. In particular, the procedure for removing unwanted staggered species in the continuum limit is reviewed. Then the asqtad lattice ensembles created by the MILC Collaboration are described. All MILC lattice ensembles are publicly available, and they have been used extensively by a number of lattice gauge theory groups. The physics results obtained with them are reviewed, and the impact of these results on phenomenology is discussed. Topics include the heavy quark potential, spectrum of light hadrons, quark masses, decay constants of light and heavy-light pseudoscalar mesons, semileptonic form factors, nucleon structure, scattering lengths, and more.

Figure 1

(Color online) Comparison of the ratio of lattice QCD and experimental values for several observables, where the lattice QCD calculations are done in the quenched approximation (left) and with $2+1$ flavors of asqtad sea quarks (right). Adapted from 156.

Figure 2

Lüscher-Weisz action Wilson loops: (a) standard plaquette, (b) $2\times 1$ rectangle, and (c) $1\times 1\times 1$ parallelogram.

Figure 3

(Color online) Illustration of taste violations for staggered fermion actions with various link fattenings. The valence quark masses were adjusted to give the same $m_{{\pi }_5}∕m_{\rho }=0.55$ for all fermion actions. The results are for quenched gauge field configurations with a Symanzik improved gauge action using $\beta =7.30$. The staggered fermion actions are standard, or one-link (OL), $\mathrm{fat}3+\text{Naik}$ (OFN), fat5, and asqtad. The pions are labeled by the taste structure, with the taste singlet the heaviest, and the taste pseudoscalar $\left({\pi }_5\right)$, the pseudo-Goldstone boson, the lightest. For more comparisons see 330.

Figure 4

(Color online) The pressure (left) per fermion degree of freedom for free Kogut-Susskind, Naik, Wilson, and p4 (247) fermions as a function of $N_T=1∕aT$. The continuum value is shown as the horizontal solid line. From 80; an earlier version appeared in 71. The “speed of light squared” (right) calculated from the pion dispersion relation, for Naik and K-S pions. From 71.

Figure 5

(Color online) Rho masses (left) and nucleon masses (right) in units of $r_1\approx 0.32\mathrm{fm}$. Octagons are unimproved staggered fermions with Wilson gauge action, diamonds are unimproved staggered fermions with Symanzik improved gauge action, crosses are Naik fermions, and squares are asqtad fermions, both with Symanzik improved gauge action. For comparison we also show tadpole clover improved Wilson fermions with Wilson gauge action (112) (fancy squares) and with Symanzik improved gauge action (139) (fancy diamonds). Adapted from 72.

Figure 6

(Color online) Squared charged pion masses, in units of $r_1$, as function of quark mass (left). From 83, 86. A previous version appeared in 74. The splittings appear to be independent of the quark mass. The taste splittings as function of ${\alpha }^2{a}^2$ (right) in a log-log plot, showing the expected behavior, indicated by the diagonal straight line. Adapted from 84.

Figure 7

(Color online) Relative weights (shown at the right of each line) of two-particle intermediate states in the scalar, taste-singlet correlator in the one-flavor case. The two-${\eta }_S^{\prime }$ state ($S$ indicates taste singlet) is shown at top; while the various two-pion states below are labeled by the pion taste $(S,V,T,A,P)$. The height of each line represents, qualitatively, the relative mass of the state.

Figure 8

(Color online) Lattice spacings and quark masses used. The octagons indicate ensembles with the strange quark near its physical value, while the crosses indicate those with an unphysically light strange quark. The burst at lower left shows the physical light quark mass. Here the quark masses are in units of MeV, but using the asqtad action lattice regularization.

Figure 9

(Color online) The plaquette as a function of integration step size squared for ${20}^3\times 64$ lattices with $\beta =6.76$ and $a{m}_q=0.01∕0.05$. The point at ${\epsilon }^2=0$ is from the RHMC algorithm, and the point indicated by $R$ is the value used in the R algorithm production runs. The remaining two points are from short test runs described by 33.

Figure 10

The static quark potential for the ensemble with $a\approx 0.09\mathrm{fm}$ and $m_l=0.2m_s$. This was obtained from time range five to six. The inset magnifies the short distance part, showing a lattice artifact which is discussed in the text.

Figure 11

The static quark potential and first excited state potential for the ensemble with $a\approx 0.06\mathrm{fm}$ and $m_l=0.1m_s$. This was obtained from time range three to twenty, using the APE smeared time links discussed in the text.

Figure 12

(Color online) The static quark potential in units of $r_1$ for five different lattice spacings. In all cases, these are for light quark mass of two-tenths the simulation strange quark mass. For each lattice spacing, a constant has been subtracted to set $V\left(r_1\right)=0$. The ruler near the bottom of the plot shows distance in units of fm, using $r_1=0.318\mathrm{fm}$. The multiple rulers in the upper half of the plot show distance in units of the lattice spacings for the different ensembles.

Figure 13

(Color online) Points used to compute $⟨q\left(r\right)q\left(0\right)⟩$. Measured points (open symbols) are used for $r⩽r_c\sim 9a$. For $r>r_c$ the fit function (solid curve) is used in Eq. (118). From 86.

Figure 14

(Color online) Topological susceptibility data, and its continuum extrapolation, compared with the prediction of Eq. (117). Adapted from 86.

Figure 15

(Color online) The “big picture”—comparison of masses calculated on the asqtad ensembles with experimental values. For the light quark hadrons we plot the hadron mass, and for the $\overline{c}c$ and $\overline{b}b$ masses the difference from the ground state (1S) mass. The continuum and chiral extrapolations of the pion and kaon masses are described in Sec. 6, and most other meson masses were extrapolated to the continuum and physical light quark masses using simple polynomials. Masses of hadrons containing strange quarks were adjusted for the difference in the strange quark mass used in generating the ensembles from the correct value. The nucleon mass extrapolation, described by 87, used a one-loop chiral perturbation theory form. The charmonium mass splitting is from 193 and the $\overline{b}b$ splittings from 225, 226 and 381. Experimental values are from 14. The ${\rm Y}$ 2S-1S splitting and the $\pi$ and $K$ masses are shown with a different symbol since these quantities were used to fix $r_1$ in physical units and the light and strange quark masses. Earlier versions of the plot appeared in 33 and the PDG “Review of Particle Physics” (14).

Figure 16

Pion and nucleon correlators plotted vs the distance from the source. These correlators are from the $\beta =6.76$, $a{m}_l∕a{m}_s=0.007∕0.05$ ensemble. The small symbols in the center of the octagons in the pion correlator are error bars. Note the increasing fractional errors with distance in the nucleon correlator, and the constant fractional errors in the pion correlator.

Figure 17

(Color online) Result of fitting the correlators in Fig. 16 from a minimum distance to the center of the lattice (for the pion) or distance at which the correlator loses statistical significance (for the nucleon). For the pion correlator (left panel), octagons correspond to single-particle fits and squares to two-particle fits. The diamonds are from single-particle fits ignoring correlations among the data points. For the nucleon fits (right panel), all fits use two particles, one of each parity. Octagons are correlated fits, and diamonds are fits ignoring the correlations. The sizes of the symbols are proportional to the confidence level of the fits, with the symbol size in the legends corresponding to 50% confidence.

Figure 18

(Color online) The $\rho$ mass in units of $r_1$, plotted vs the squared pion mass. Since $m_{\pi }^2\propto m_q$, this is effectively a plot vs light quark mass. The octagons are from ensembles with $a\approx 0.12\mathrm{fm}$, the squares from ensembles with $a\approx 0.09\mathrm{fm}$, and the bursts from ensembles with $a\approx 0.06\mathrm{fm}$. The decorated plus at the left is the physical $\rho$ mass, with the error on this point coming from the error in $r_1$. For reference, the upward arrow indicates approximately where the quark mass equals the strange quark mass.

Figure 19

(Color online) The nucleon and a chiral fit. Nucleon masses are shown for different light quark masses at three lattice spacings. The cross at the left is the experimental value. The slightly curved line and the diamond at the physical quark mass are a continuum and chiral extrapolation. Lattice spacing errors are assumed to be linear in $a^2{\alpha }_s$. The particular chiral form used here is a one-loop calculation with $\pi -N$ and $\pi -\Delta$ intermediate states (254; 91). Adapted and updated from 87.

Figure 20

(Color online) The ${\Omega }^{-}$ mass. Results are shown for three different lattice spacings. The points with $a\approx 0.09$ and $\approx 0.06\mathrm{fm}$ were fit to the form $M_{\Omega }{r}_1=A+B{a}^2{\alpha }_s+C{\left(m_{\pi }{r}_1\right)}^2$. The sloping lines show this fit form evaluated at the values of $a^2{\alpha }_s$ for these lattice spacings, and at $a=0$. Finally, the fancy cross with error bars is the fit form evaluated at the physical pion mass, and the small diamond is the experimental value. Note that in this case the vertical axis does not begin at zero. Adapted and updated from 365 and 87.

Figure 21

(Color online) Best fit to the $a_0$ correlator (left panel) for five momenta and the $f_0$ correlator (right panel) for four momenta. The fitting range is indicated by darker points and lines. Occasional points with negative central values are not plotted. Data are determined from the $a\approx 0.12\mathrm{fm}$ (coarse) ensemble with $a{m}_l=0.005$ and $a{m}_s=0.05$. From 63.

Figure 22

(Color online) The isovector scalar $\left(a_0\right)$ correlator on the MILC coarse $a{m}_l∕a{m}_s=0.007∕0.05$ ensemble with three different domain-wall valence masses. Overlaid on the data are the predicted bubble contributions, which should dominate over the exponentially decaying contributions at sufficiently large times. From 30.

Figure 23

(Color online) NNNLO fit to partially quenched squared meson masses. Only the lightest sea-quark ensemble for each lattice spacing is shown. The data fit includes the results for decay constants and is reflected in the number of degrees of freedom. From 85.

Figure 24

(Color online) The meson decay constants are plotted along with the NNNLO fit that was shown for the masses in Fig. 23. The left plot shows partially quenched data from more ensembles than in Fig. 23, but still only a fraction of the data fit. On the right, still more ensembles are included, but only full QCD data points are plotted. From 85.

Figure 25

(Color online) The ratio of light decay constants $f_K∕f_{\pi }$ from six calculations. The top four use MILC asqtad configurations and the lower two use other types of quarks. Details and references can be found in the text.

Figure 26

(Color online) Chiral extrapolation for ${\Phi }_D$ (octagons) and ${\Phi }_{D_s}$ (crosses or diamonds) by the Fermilab/MILC Collaboration (90). Solid lines are the $\mathrm{HMS}\chi \mathrm{PT}$ fit to ${\Phi }_D$; dotted lines, to ${\Phi }_{D_s}$. Although the full set of partially quenched data is included in the fit, for ${\Phi }_D$ the plot shows only those full QCD points for which the light valence and sea masses are equal to $m_x$, the mass on the abscissa. For ${\Phi }_{D_s}$, only points with the strange valence mass $\left(m_{sv}\right)$ equal to the strange sea mass are shown, plotted either as a function of $m_l$ (crosses), or at $m_{sv}$ (diamonds). The dashed lines show the fit after removal of light-quark discretization errors, with the fancy plus signs giving the chirally extrapolated results with statistical errors.

Figure 27

(Color online) Comparison of lattice QCD and experimental results for $f_D$ and $f_{D_s}$ (left panel) and of lattice QCD results for $f_B$ and $f_{B_s}$ (right panel).

Figure 28

(Color online) Comparison of the Fermilab/MILC/HPQCD lattice prediction for the normalized $D\to K\mathcal{l}\nu$ form factor (bands) with the subsequent results from Belle (diamonds), BABAR (squares), and CLEO (triangles). The dark gray band is the 1$\sigma$ error band from statistics, and the light gray band is the 1$\sigma$ band for all errors added in quadrature. Adapted from 89. An earlier version appeared in 267.

Figure 29

(Color online) Results for the normalized $B\to \pi \mathcal{l}\nu$ form factor $P_{+}{\varphi }_{+}{f}_{+}$ from the Fermilab/MILC lattice calculations (circles) and BABAR (stars). The solid line is the results of a fully correlated simultaneous fit. Requiring that lattice and experiment have the same normalization yields $\left|V_{ub}\right|$. From 38.

Figure 30

(Color online) Values of $\left|V_{ub}\right|$ obtained from averaging the exclusive determinations compared with inclusive determinations using different theoretical inputs.

Figure 31

(Color online) Values of $\left|V_{cb}\right|$ from the exclusive decays $B\to D\mathcal{l}\nu$, $B\to D^{*}\mathcal{l}\nu$, and the inclusive determination.

Figure 32

(Color online) Summary of determinations of the strong coupling constant ${\alpha }_s\left(M_Z\right)$. The lattice QCD determination is the most precise one. From 14.

Figure 33

(Color online) The ${\rm Y}$ spectrum. Left: spin-averaged radial and orbital levels in GeV. Closed and open symbols are from coarse and fine lattices, respectively. Squares and triangles denote unquenched and quenched results, respectively. Lines represent experiment. Right: Spin-averaged mass differences from the same data divided by experiment, in the quenched approximation (left narrow figure) and unquenched (right narrow figure). From 226.

Figure 34

(Color online) Summary of the charmonium (left) and bottomonium (right) spectra. The fine ensemble results are in fancy squares, the coarse in circles, the medium coarse are in diamonds, and the extracoarse results are in squares. Included in the error budget is an estimated systematic uncertainty from setting the heavy quark masses.

Figure 35

(Color online) Independent mass differences of $J^p={\frac{1}{2}}^{+}$ singly charmed baryons (a) and singly bottom baryons (b). From 316.

Figure 36

(Color online) The mass spectrum of doubly charmed (a) and bottom (b) baryons. The error bars are statistical only. From 316.

Figure 37

(Color online) The ratio ${\xi }^{\prime }=\xi \sqrt{M_{B_s}∕M_{B_d}}=f_{B_s}\sqrt{M_{B_s}{B}_{B_s}}∕f_{B_d}\sqrt{M_{B_d}{B}_{B_d}}$ as a function of the light valence quark mass together with $\mathrm{rS}\chi \mathrm{PT}$ fits and the chiral and continuum extrapolation. The left panel is from the HPQCD Collaboration (204) and the right panel from the Fermilab/MILC Collaboration (189).

Figure 38

(Color online) The lowest-order diagram for the QCD correction to the muon anomalous magnetic moment at $\mathcal{O}\left({\alpha }^2\right)$. The bubble represents all possible hadronic states. From 27.

Figure 39

(Color online) Two different $\mathrm{rS}\chi \mathrm{PT}$ fits to $\Pi \left(q^2\right)$ for three light masses: $a{m}_l=0.0031$ (diamonds), 0.0062 (squares), and 0.0124 (circles) with $a{m}_s=0.031$. From 27, which contains the details.

Figure 40

(Color online) The gluon dressing function $q^{2}D\left(q^2\right)$ for quenched and dynamical configurations with $a\approx 0.09\mathrm{fm}$, from 116 (left), and the quark mass function for light sea-quark mass in full QCD at $a\approx 0.12$ and $0.09\mathrm{fm}$, from 334 (right).

Figure 41

(Color online) Correction to Dashen’s theorem, as a function of the LO $\pi$ mass squared (equivalent to the pion mass squared with $e^2=0$). From 47.

Figure 42

(Color online) The taste splittings as function of ${\alpha }^2{a}^2$ for the asqtad and HISQ actions (with the latter indicated by dashed boxes).