Strange and charm baryon masses with two flavors of dynamical twisted mass fermions

The masses of the low-lying strange and charm baryons are evaluated using two degenerate flavors of twisted mass sea quarks for pion masses in the range of about 260 to 450 MeV. The strange and charm valence quark masses are tuned to reproduce the masses of the kaon and $D$ meson at the physical point. The tree-level Symanzik improved gauge action is employed. We use three values of the lattice spacing, corresponding to $\beta =3.9$, $\beta =4.05$ and $\beta =4.2$, with $r_0/a=5.22(2)$, $r_0/a=6.61(3)$ and $r_0/a=8.31(5)$, respectively. We examine the dependence of the strange and charm baryons on the lattice spacing and the strange and charm quark masses. The pion mass dependence is studied and physical results are obtained using heavy-baryon chiral perturbation theory to extrapolate to the physical point.

Figure 1

Pseudoscalar meson masses for $\beta =4.05$ as a function of the heavy valence quark mass $a{\mu }_2$ in the relevant mass range for the strange quark (left) and charm quark (right). In all the examples shown, the sea quark mass $a{\mu }_{\mathrm{sea}}$ is set equal to the light valence quark mass $a{\mu }_1$.

Figure 2

The dependence of the pseudoscalar masses on the strange quark mass at $\beta =3.9$ (left) and the charm quark mass at $\beta =4.05$ (right). In each plot, the solid line shows the variation of the pseudoscalar mass at the tuned strange (left) and charm (right) quark masses. The dashed vertical line corresponds to the value of $a{\mu }_1$ at which the physical pion mass is recovered. The asterisk denotes the physical value of the kaon or $D$-meson mass in lattice units. The lattice spacing is determined from the nucleon mass.

Figure 3

The low-lying baryons belonging to the octet representation, labeled by value of $I_3$ and hypercharge.

Figure 4

The low-lying baryons belonging to the decuplet representation, labeled by value of $I_3$ and hypercharge.

Figure 5

Representative effective mass plots for $\beta =4.2$ and $a{\mu }_l=0.002$. For the strange baryons (left), we used $a{\mu }_s=0.015$, while for charm baryons (right), we used $a{\mu }_c=0.17$.

Figure 6

Baryon masses versus the strange quark mass for $\beta =3.9$ and $a{\mu }_l=0.0064$ (circles), $\beta =4.05$ and $a{\mu }_l=0.006$ (squares), and $\beta =4.2$ and $a{\mu }_l=0.0065$ (triangles).

Figure 7

Baryon masses versus the charm quark mass for $\beta =3.9$ and $a{\mu }_l=0.004$ (circles), $\beta =4.05$ and $a{\mu }_l=0.003$ (squares), and $\beta =4.2$ and $a{\mu }_l=0.0065$ (triangles).

Figure 8

The masses of ${\Sigma }^{+}$, ${\Sigma }^0$ and ${\Sigma }^{-}$ baryons at $\beta =3.9$, 4.05 and 4.2. Filled symbols at $\beta =3.9$ refer to the larger volume.

Figure 9

Isospin mass splittings in the strange quark sector at the smallest (left) and largest (right) pion masses used in our computation.

Figure 10

Left: The strange octet baryon mass in units of $r_0$ versus $(a/r_0{)}^2$ for several particles in the octet and decuplet at two fixed values of $m_{\pi }{r}_0$. The solid line is a linear fit in $(a/r_0{)}^2$, and the dashed line is a fit to a constant. Right: Chiral extrapolation at fixed strange quark mass. Results at $\beta =3.9$ are shown with open circles for $L=2.1\mathrm{fm}$ and with filled circles for $L=2.7\mathrm{fm}$; results at $\beta =4.05$ are shown with open squares, and results at $\beta =4.2$ with open triangles.

Figure 11

Left: The strange decuplet baryon mass in units of $r_0$ versus $(a/r_0{)}^2$ at two fixed values of $m_{\pi }{r}_0$. The solid line is a linear fit in $(a/r_0{)}^2$, and the dashed line is a fit to a constant. Right: Chiral extrapolation at fixed strange quark mass. In the case of the ${\Sigma }^{*}$ baryon, we show the three charge states ${\Sigma }^{*+}$ (purple), ${\Sigma }^{*0}$ (blue), and ${\Sigma }^{*-}$ (red). The spread seen in the data with the same symbol at a given value of the pion mass indicates the level of isospin breaking for the ${\Sigma }^{*}$ particle. Results at $\beta =3.9$ are shown with open circles for $L=2.1\mathrm{fm}$ and with filled circles for $L=2.7\mathrm{fm}$; results at $\beta =4.05$ are shown with open squares, and results at $\beta =4.2$ with open triangles.

Figure 12

Isospin mass splitting in the charm sector at the smallest and largest values of the pion mass used in this work.

Figure 13

Left: The masses of octet charm baryons in units of $r_0$ versus $(a/r_0{)}^2$ for several particles in the octet and decuplet at two fixed values of the pion mass $m_{\pi }{r}_0$. Right: Chiral fits to the data at $\beta =3.9$ (circles) and at $\beta =4.2$ (triangle) are shown with the green band; fits to all data are shown with the blue band. For the ${\Xi }_{cc}$ baryon we have extrapolated ${\Xi }_{cc}^{+}$, whereas for the ${\Sigma }_c$ baryon we averaged over ${\Sigma }_c^{++}$ (purple), ${\Sigma }_c^{+}$ (blue) and ${\Sigma }_c^0$ (red). The spread seen in the data with the same symbol at a given value of the pion mass indicates the level of isospin breaking for the ${\Sigma }_c$ particle.

Figure 14

Left: The masses of decuplet charm baryons in units of $r_0$ versus $(a/r_0{)}^2$ at two fixed values of $m_{\pi }{r}_0$. Right: Chiral fits to the data at $\beta =3.9$ (circles) and at $\beta =4.2$ (triangle) are shown with the green band; fits to all data are shown with the blue band. For the ${\Xi }_{cc}^{*}$ baryon we averaged over ${\Xi }_{cc}^{*++}$ (blue) and ${\Xi }_{cc}^{*+}$ (red), and for the ${\Sigma }_c^{*}$ baryon we averaged over ${\Sigma }_c^{*++}$ (purple), ${\Sigma }_c^{*+}$ (blue) and ${\Sigma }_c^{*0}$ (red). The spread seen in the data with the same symbol at a given value of the pion mass indicates the level of isospin breaking for the or the ${\Sigma }_c^{*}$ and ${\Xi }_c^{*}$ particles.

Figure 15

The results of this work for the octet strange baryons are shown with filled (red) circles, results by the BMW Collaboration using clover fermions are shown with the open (blue) square [4], results by the PACS-CS Collaboration are shown with filled (magenta) triangles [3], and results by the LHPC using domain-wall valence quarks on a staggered sea are shown with open (green) triangles [5]. The experimental value is shown with an asterisk.

Figure 16

Comparison of the results of this work for the strange decuplet baryons to those obtained using domain wall valence quarks on a staggered sea [5]. The notation is the same as that in Fig. 15.

Figure 17

Lattice data in the strange quark sector extrapolated to the physical pion mass, with the experimental values shown by the horizontal lines. For the twisted mass results of this work, the chiral extrapolation was carried out using NLO $\mathrm{HB}\chi \mathrm{PT}$. We include in addition the results by the QCDSF-UKQCD Collaboration [39] (light blue crosses). The rest of the notation is the same as in Fig. 15.

Figure 18

Masses for charm baryons with spin-$1/2$ (left) and spin-$3/2$ (right) computed within lattice QCD, with the experimental values shown by the horizontal lines. Our results for ${\Lambda }_c$, ${\Sigma }_c$, ${\Xi }_{cc}$, ${\Sigma }_c^{*}$, ${\Xi }_{cc}^{*}$ and ${\Omega }_{ccc}$ baryons are extrapolated to the physical pion mass (shown with filled red circles), whereas for the rest we give the results obtained at $m_{\pi }=260\mathrm{MeV}$ and $\beta =4.2$ (shown with open red circles). We include results obtained using a number of hybrid actions with staggered sea quarks from Refs. [43] (open blue squares), [40, 41] (open green triangles) and [42] (filled magenta triangles).