First results from $2+1$ dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass

We present the first light-hadron spectroscopy on a set of $N_f=2+1$ dynamical, anisotropic lattices. A convenient set of coordinates that parameterize the two-dimensional plane of light and strange-quark masses is introduced. These coordinates are used to extrapolate data obtained at the simulated values of the quark masses to the physical light and strange-quark point. A measurement of the Sommer scale on these ensembles is made, and the performance of the hybrid Monte Carlo algorithm used for generating the ensembles is estimated.

Figure 1

Temporal (left column) and spatial (right column) plaquette history from the ensemble with ${24}^3\times 128$ volume and $a_t{m}_l=-0.0840$. The x axis is in units of trajectories.

Figure 2

Histogram of the lowest eigenvalues of the Dirac operator for the light and strange quarks from the ensemble with ${24}^3\times 128$ volume and $a_t{m}_l=-0.0840$. The minimum values of the eigenvalues are $a_t\lambda =6.9\times {10}^{-6}$ and $2.0\times {10}^{-4}$ for the light and strange quarks, respectively.

Figure 3

Autocorrelation $\rho (t)$ and integrated autocorrelation length ${\tau }_{\mathrm{int}}$ (in trajectories) for the unsmeared (above) and smeared (below) plaquette involving only spatial links from the ensemble with ${16}^3\times 128$ volume and $a_t{m}_l=-0.0808$.

Figure 4

Autocorrelation $\rho (t)$ and integrated autocorrelation length ${\tau }_{\mathrm{int}}$ (in trajectories) for the up/down (above) and strange (below) quark eigenvalues from the ensemble with ${16}^3\times 128$ volume and $a_t{m}_l=-0.0808$.

Figure 5

Autocorrelation $\rho (t)$ and integrated autocorrelation length ${\tau }_{\mathrm{int}}$ (in trajectories) for the pion (above) and proton (below) correlator at $t=30$ from the ensemble with ${16}^3\times 128$ volume and $a_t{m}_l=-0.0808$.

Figure 6

Pion and rho (above) and proton and Delta (below) fitted masses as functions of $t_{\min }$, where $t_{\mathrm{source}}$ has been shifted to $t=0$. The bands indicated the final fitted masses summarized in Tables 6 and 7.

Figure 7

All measured meson masses as functions of the squared pseudoscalar masses. The diamonds and squares are measured with $m_s=-0.0540$ but with two different volumes, ${12}^3\times 96$ and ${16}^3\times 96$; the upward-pointing triangles are those with $m_s=-0.0618$ and ${12}^3\times 96$ volume; the downward triangles and pentagons are measured with $m_s=-0.0743$ and two different volumes, ${16}^3\times 128$ and ${24}^3\times 128$. The (red) dotted-dashed lines indicate the decay thresholds for the ${12}^3$ (upper) and ${16}^3$ (lower) $m_s=-0.0540$ ensembles, while the (blue) dashed lines are for the ${16}^3$ (upper) and ${24}^3$ (lower) $m_s=-0.0743$. The lowest decay thresholds are $\rho \to \pi (p)+\pi (-p)$, $K^{*}\to K(p)+\pi (-p)$, $\varphi \to K(p)+\overline{K}(-p)$, $a_0\to \pi (0)+\eta (0)$, $a_1\to \pi (0)+\rho (0)$, and $b_1\to \pi (0)+\omega (0)$ (with $\omega$ approximated by $\rho$), where the minimum allowed momentum $p$ on the lattice is $\frac{2\pi }{L_s}$.

Figure 8

All measured baryon masses as functions of the squared pseudoscalar masses. The diamonds and squares are measured with $m_s=-0.0540$ but with two different volumes, ${12}^3\times 96$ and ${16}^3\times 96$; the upward-pointing triangles are those with $m_s=-0.0618$ and ${12}^3\times 96$ volume; the downward triangles and pentagons are measured with $m_s=-0.0743$ and two different volumes, ${16}^3\times 128$ and ${24}^3\times 128$. The (red) dotted-dashed lines indicate the decay thresholds for the ${12}^3$ (upper) and ${16}^3$ (lower) $m_s=-0.0540$ ensembles, while the (blue) dashed lines are for the ${16}^3$ (upper) and ${24}^3$ (lower) $m_s=-0.0743$. The lowest decay thresholds are $\Delta \to N(p)+\pi (-p)$, ${\Sigma }^{*}\to \Lambda (p)+\pi (-p)$, ${\Xi }^{*}\to \Xi (p)+\pi (-p)$, where the minimum allowed momentum $p$ on the lattice is $\frac{2\pi }{L_s}$.

Figure 9

Effective-mass plots using the ground-state principal correlators from the pion and rho-meson masses at four different momenta on the lightest ensemble ($m_s=-0.0743$) with volume ${24}^3\times 128$. The insets show the energy squared in temporal lattice units versus $n^2$, which is related to the momentum by $p^2=\frac{4{\pi }^2}{L_s^2}{n}^2$.

Figure 10

Results for the static-quark potential for the $a_t{m}_l=-0.0808$, $a_t{m}_s=-0.0743$ mass set. The left panel shows the effective energies, $a_t{V}_{\mathrm{eff}}(r)$ for each $r$. The right panel shows the resulting fit to the potential using Eq. (20).

Figure 11

The location of the dynamical ensembles used in this work in the $s_{\Omega }-l_{\Omega }$ plane. The circle (black) indicates the physical point $\{l_{\Omega }^{\mathrm{phys}},s_{\Omega }^{\mathrm{phys}}\}$. The (red) diamonds and squares are generated on ${12}^3\times 96$ and ${16}^3\times 96$ lattices with $a_t{m}_s=-0.0540$; the (green) upper triangle is the ensemble on a ${12}^3\times 96$ lattice with $a_t{m}_s=-0.0618$ and (blue) upside-down triangles and pentagons represent the ensembles on ${16}^3\times 128$ and ${24}^3\times 128$ lattices with $a_t{m}_s=-0.0743$. Detailed parameters can be found in Table 9. The horizontal dashed (pink) line indicates constant $s_{\Omega }$ from the physical point and the diagonal line indicates three-flavor degenerate theories.

Figure 12

Selected meson mass ratios as functions of $l_{\Omega }$ and $s_{\Omega }$. Differently shaded (or colored) points correspond to the $a_t{m}_{l,s}$ combinations in Fig. 15; detailed numbers can be found in Table 6. The smaller-volume ensembles $\{a_t{m}_l,a_t{m}_s\}=\{-0.0826,-0.0540\}$ and $\{-0.0840,-0.0743\}$ are excluded from the fits. The lines indicate the “projected” leading chiral extrapolation fit in $l_{\Omega }$ and $s_{\Omega }$, while keeping the other one fixed. The black (circular) point is the extrapolated point at physical $l_{\Omega }$ and $s_{\Omega }$.

Figure 13

Selected baryon mass ratios as functions of the $l_{\Omega }$ and $s_{\Omega }$. Differently shaded (or colored) points correspond to the $a_t{m}_{l,s}$ combinations in Fig. 15; detailed numbers can be found in Table 7. The smaller-volume ensembles $\{a_t{m}_l,a_t{m}_s\}=\{-0.0826,-0.0540\}$ and $\{-0.0840,-0.0743\}$ are excluded from the fits. The lines indicate the “projected” leading chiral extrapolation fit in $l_{\Omega }$ and $s_{\Omega }$, while keeping the other one fixed. The black (circular) point is the extrapolated point at physical $l_{\Omega }$ and $s_{\Omega }$.

Figure 14

Summary of the extrapolated hadron masses compared with their experimental values.

Figure 15

The assignment of colors from different ensembles in coordinates $a_t\{m_l,m_s\}$. The convention will remain consistent when used again for later extrapolations.

Figure 16

“Slices” of $r_0{m}_{\Omega }/{\xi }_R$ as functions of $l_{\Omega }$ (left) and $s_{\Omega }$ (middle). The straight lines are the fitted functions according to Eq. (25). The assignment of colors is shown in the Fig. 15.

Figure 17

The $s_X-l_X$ plot with $X=\varphi$ (left) $\Xi$ (middle) and the linear combination $2\Xi -\Sigma$ for $N_f=3$ and $N_f=2+1$ at $\beta =1.5$. The circle (black) indicates the physical point $\{l_X^{\mathrm{phys}},s_X^{\mathrm{phys}}\}$. The (red) diamonds and squares are ${12}^3\times 96$ and ${16}^3\times 96$ from $a_t{m}_s=-0.0540$ ensembles; (green) upper triangles are from ${12}^3\times 96$ from $a_t{m}_s=-0.0618$ and (blue) upside-down triangles and pentagons are from ${16}^3\times 128$ and ${24}^3\times 128$ from $a_t{m}_s=-0.0743$ ensembles. Detailed parameters can be found in Table 9. The horizontal dashed (pink) line indicates physical $s_X$, and the straight diagonal lines indicate the SU(3) limit.

Figure 18

$J$-parameter plot for $N_f=2+1$ at $\beta =1.5$. The upper diamond, upside-down triangle, pentagon ($V={12}^3\times 96$) and square ($V={16}^3\times 96$) points are from $m_s=-0.0540$ ensembles and the triangle, lower diamond, upside-down triangle ($V={16}^3\times 128$) and square ($V={24}^3\times 128$) points are from $m_s=-0.0743$; the circle (black) indicates the physical point $\{l_{\Omega }^{\mathrm{phys}},J^{\mathrm{phys}}\}$; the dashed line indicates the physical $J$ value.