Light hadron spectroscopy using domain wall valence quarks on an asqtad sea

We calculate the light hadron spectrum in full QCD using two plus one flavor asqtad sea quarks and domain wall valence quarks. Meson and baryon masses are calculated on a lattice of spatial size $L\approx 2.5\mathrm{fm}$, and a lattice spacing of $a\approx 0.124\mathrm{fm}$, for pion masses as light as $m_{\pi }\approx 300\mathrm{MeV}$, and compared with the results by the MILC Collaboration with asqtad valence quarks at the same lattice spacing. Two- and three-flavor chiral extrapolations of the baryon masses are performed using both continuum and mixed action heavy baryon chiral perturbation theory. Both the three-flavor and two-flavor functional forms describe our lattice results, although the low-energy constants from the next-to-leading order $SU(3)$ fits are inconsistent with their phenomenological values. Next-to-next-to-leading order $SU(2)$ continuum formulae provide a good fit to the data and yield an extrapolated nucleon mass consistent with experiment, but the convergence pattern indicates that even our lightest pion mass may be at the upper end of the chiral regime. Surprisingly, our nucleon masses are essentially linear in $m_{\pi }$ over our full range of pion masses, and we show this feature is common to all recent dynamical calculations of the nucleon mass. The origin of this linearity is not presently understood, and lighter pion masses and increased control of systematic errors will be needed to resolve this puzzling behavior.

Figure 1

The left-hand figure shows the residual mass determined from the ratio $R(t)$ in Eq. (9), for the data at $a{m}_{u/d}^{\mathrm{asqtad}}=0.010$; the quoted value of $m_{\mathrm{res}}$ is obtained from a constant fit to the data. Note that the effect of the Dirichlet boundary at $t=22$ relative to the source is apparent as far out as $t=14$. The right-hand figure shows the quark mass dependence of the residual mass. The main reason for the increase in $m_{\mathrm{res}}$ with decreasing $a{m}_{u/d}^{\mathrm{asqtad}}$ is the increased roughening of the gauge field; in typical quenched calculations, a smaller dependence of the residual mass on the quark mass is observed. As a result, chiral symmetry is satisfied to a lesser degree at fixed $L_s$ as the pion mass decreases.

Figure 2

The pion (left) and kaon (right) effective masses at each value of the light quark mass, together with the single-exponential fits to the correlators, as described in the text. The oscillatory terms in the transfer matrix are evident in the effective mass close to the source.

Figure 3

Comparison between pseudoscalar and vector masses with domain wall valence quarks (LHP) and staggered valence quarks (MILC) computed on the coarse ($a\approx 0.125\mathrm{fm}$) MILC ensembles.

Figure 4

Two types of bubble contributions to the scalar meson: the left one is $B_1$ in Eq. (11) while the right panel is $B_2$ in Eq. (11).

Figure 5

The left panel of this figure shows the effective point-point scalar meson correlator plot [as defined in Eq. (15)], along with the mixed action bubble contribution from Eq. (11). Note that the symbols are the same as in Fig. 2. The right-hand panel shows the fits to data from the lightest three ensembles where the largest bubble contributions dominate.

Figure 6

The left- and right-hand figures show the nucleon effective mass at $a{m}_{u/d}^{\mathrm{asqtad}}=0.05$ and $a{m}_{u/d}^{\mathrm{asqtad}}=0.007$, respectively, from the smeared-smeared correlators. The circles denote the lattice data for the effective mass, and the solid line and dash-dotted line denote the single-exponential and oscillating fit, respectively; note that lattice data are offset for clarity.

Figure 7

The left- and right-hand figures show the delta effective mass at $a{m}_{u/d}^{\mathrm{asqtad}}=0.05$ and $a{m}_{u/d}^{\mathrm{asqtad}}=0.007$, respectively, from the smeared-smeared correlators. The circles denote the lattice data for the effective mass, and the solid line and dash-dotted line denote the single-exponential and oscillating fit, respectively; note that lattice data (the circles) are slightly offset for clarity.

Figure 8

The figure shows the nucleon effective mass obtained from a single correlator smeared at both source and sink at each of the values of the light quark masses used in the calculation; the lines show single-exponential fits to the correlator, with the dashed lines showing bootstrap errors corresponding to the 68% confidence levels on the bootstrap distributions.

Figure 9

The figure shows the $\Delta$ effective mass obtained from a single correlator smeared at both source and sink at each of the values of the light quark masses used in the calculation. The lines show single-exponential fits to the correlator, with the dashed lines showing bootstrap errors corresponding to the 68% confidence levels on the bootstrap distributions.

Figure 10

Chiral extrapolations of the nucleon mass, plotted vs the approximate expansion parameter $m_{\pi }/2\sqrt{2}\pi f_0$, with $f_0=121.9\mathrm{MeV}$. For comparison purposes, in all figures, we display the results of fits to the $m007–m040$ mass points denoted by the small black (open) circles with error bars. The black square is the $m050$ mass point not included in any of these fits. The red (solid) circle is the physical nucleon mass, taken to be 939.6 MeV, at a pion mass $m_{\pi }=137\mathrm{MeV}$, which is never included in the minimization. The gray bands represent the 68% confidence interval, and are only determined from the statistical error bar in the lattice results. In Fig. 10a, we plot both the LO and NLO $SU(2)$ $\mathrm{HB}\chi \mathrm{PT}$ results. The light shaded band is from LO and the darker shaded band is NLO. In Fig. 10b, we plot the results of the NNLO $SU(2)$ fit including explicit deltas. In Fig. 10c, we plot the NNLO $SU(2)$ covariant fit without deltas. In comparing Figs. 10b, 10c one needs to note the size of the error band is dictated by the number of free parameters and not by the use of infrared-regularization (covariant expression). In Fig. 10d, we plot the straight-line fit, Eq. (22). All of these fits, except the LO fit of (a), are statistically consistent with our lattice results, as can be seen in Table .

Figure 11

Plot of the contributions to the nucleon mass order-by-order $\delta M_N^{(n)}$ resulting from the NNLO heavy baryon $\chi \mathrm{PT}$ fit including the delta. In (a) we plot each order separately, plotting the magnitude of the (negative) NLO results denoted by a dashed line. In (b) we plot the combined $\mathrm{LO}+\mathrm{NLO}$ and $\mathrm{LO}+\mathrm{NLO}+\mathrm{NNLO}$ results. Plotting any of the NNLO fits of the nucleon mass in this work results in similar values for the $\delta M_N^{(n)}$. The arrows denote the values of the pion mass used in this work. From (a), one notes that the convergence of the resulting NNLO fit may already be breaking down at the second lightest mass point, although the summed contributions in (b) display better convergence.

Figure 12

Nucleon mass results from other lattice groups, the MILC [97], QCDSF/UKQCD [93], RBC/UKQCD [98], and ETM [94] collaborations. We apply our linear fit in $m_{\pi }$ to the lattice results from all groups and plot the corresponding best fit and 68% confidence band. In all cases, the open diamonds correspond to the points included in the analysis. In Fig. 12a, in addition to the coarse MILC ($a\sim 0.124\mathrm{fm}$) nucleon mass results, we also display the same analysis from our lattice results for comparison, as well as the MILC superfine ($a\sim 0.06\mathrm{fm}$) results, which were not used in the analysis. In Fig. 12d, we also display a fit to the ETM lattice data using Eq. (36), which provides a better ${\chi }^2$ description of their nucleon mass results.

Figure 13

Chiral extrapolations of the delta mass. In (a) we display the extrapolation of our lattice data using the continuum infinite volume formula. In (b), we display the extrapolation of our lattice data with the predicted finite volume corrections subtracted from our lattice results. For comparison, both fits are done using all six mass points. The open circles with error bars are our calculational results while the lower open squares are the predicted infinite volume extrapolations. The filled circle with error bar represents our preliminary determination of the delta mass using the $m010$ ensemble on the $L\sim 3.5\mathrm{fm}$ lattices. Note that it is located on the opposite side compared to the predicted infinite volume extrapolation of Ref. 103. The filled (red) circle denotes the pole mass of the delta plotted at $m_{\pi }=137\mathrm{MeV}$. A multiple volume study is required for the extrapolation of the delta mass, which is beyond the scope of this work.

Figure 14

Oscillating fits to the smeared-smeared delta correlation function, Eq. (10) on the $m010$ ${20}^3\times 64$ (a) and the ${28}^3\times 64$ (b) lattices. The delta mass on the larger volume is heavier than on the lighter volume, contrary to expectations.

Figure 15

The Gell-Mann–Okubo (GMO) mass ratio with ${\delta }_{\mathrm{GMO}}=\frac{{\Delta }_{\mathrm{GMO}}}{M_B}$, where ${\Delta }_{\mathrm{GMO}}$ is defined in Eq. (54) and the centroid mass is $M_B=\frac{1}{8}{M}_{\Lambda }+\frac{3}{8}{M}_{\Sigma }+\frac{1}{4}{M}_N+\frac{1}{4}{M}_{\Xi }$. Here we collect the results of this work, as well as those of NPLQCD [14]. We also put the experimental number in the figure, using the charge-neutral baryon masses.

Figure 16

We display the LO $SU(2)$ fits to the mass splittings. The left plot (a) is the result of fitting only to the lightest two mass points plus the 68% confidence bands. The right plot (b) is the result of fitting to the lightest three mass points. The stars represent the physical baryon mass splittings and are not included in the analysis.

Figure 17

We display various chiral extrapolations of the octet baryon masses and mass splittings. In all fits, the (gray) squares are points not included in the analysis, the (colored) open circles with error bars are the points that are included and the (colored) stars are the physical masses/splittings, which are never included in the minimization. The dashed (colored) squares with error bars are the resulting predictions, slightly displaced horizontally for clarity. Figures 17a, 17b are from a combined fit of $M_B-M_N$ and $M_N$, fit to the lightest three quark mass ensembles, using the NLO mixed action extrapolation formulae. Figure 17c is the result of the NLO mixed action fit to the octet baryon masses. Figure 17d is the result of the straight-line analysis. As can be seen, the straight-line fit reproduces the lattice results not included in the minimization but conflicts dramatically with experiment. The error bars/bands represent the 68% confidence interval from the statistical uncertainty.