Scalar-quark systems and chimera hadrons in $\mathrm{SU}(3{)}_c$ lattice QCD

In terms of mass generation in the strong interaction without chiral symmetry breaking, we perform the first study for light scalar-quarks $\varphi$ (colored scalar particles with ${\mathbf{3}}_c$ or idealized diquarks) and their color-singlet hadronic states using quenched $\mathrm{SU}(3{)}_c$ lattice QCD with $\beta =5.70$ (i.e., $a\simeq 0.18\mathrm{fm}$) and lattice size ${16}^3\times 32$. We investigate “scalar-quark mesons” ${\varphi }^{†}\varphi$ and “scalar-quark baryons” $\varphi \varphi \varphi$ as the bound states of scalar-quarks $\varphi$. We also investigate the color-singlet bound states of scalar-quarks $\varphi$ and quarks $\psi$, i.e., ${\varphi }^{†}\psi$, $\psi \psi \varphi$, and $\varphi \varphi \psi$, which we name “chimera hadrons.” All the new-type hadrons including $\varphi$ are found to have a large mass even for zero bare scalar-quark mass $m_{\varphi }=0$ at $a^{-1}\simeq 1\mathrm{GeV}$. We find a “constituent scalar-quark/quark picture” for both scalar-quark hadrons and chimera hadrons. Namely, the mass of the new-type hadron composed of $m$ $\varphi$’s and $n$ $\psi$’s, $M_{m\varphi +n\psi }$, approximately satisfies $M_{m\varphi +n\psi }\simeq m{M}_{\varphi }+n{M}_{\psi }$, where $M_{\varphi }$ and $M_{\psi }$ are the constituent scalar-quark and quark masses, respectively. We estimate the constituent scalar-quark mass $M_{\varphi }$ for $m_{\varphi }=0$ at $a^{-1}\simeq 1\mathrm{GeV}$ as $M_{\varphi }\simeq 1.5–1.6\mathrm{GeV}$, which is much larger than the constituent quark mass $M_{\psi }\simeq 400\mathrm{MeV}$ in the chiral limit. Thus, scalar quarks acquire a large mass due to large quantum corrections by gluons in the systems including scalar quarks. Together with other evidences of mass generation of glueballs and charmonia, we conjecture that all colored particles generally acquire a large effective mass due to dressed gluon effects. In addition, the large mass generation of pointlike colored scalar particles indicates that plausible diquarks used in effective hadron models cannot be described as the pointlike particles and should have a much larger size than $a\simeq 0.2\mathrm{fm}$.

Figure 1

Diagrams for scalar-quark hadrons, corresponding to (a) the two-point correlator of scalar-quark mesons (${\varphi }^{†}\varphi$) and (b) that of scalar-quark baryons ($\varphi \varphi \varphi$). The dotted line denotes the scalar-quark $\varphi$, and the arrow denotes the current of the color in fundamental representation. As for the scalar-quark meson, we only consider the connected diagram, i.e., nonsinglet states of the scalar-quark flavor. For scalar-quark baryons, no disconnected diagram appears because of the color conservation.

Figure 2

Diagrams for chimera hadrons corresponding to (a) the two-point correlator of chimera mesons (${\varphi }^{†}\psi$) and (b) that of chimera baryons ($\psi \psi \varphi$ and $\varphi \varphi \psi$). The dotted line corresponds to the scalar-quark $\varphi$ and the solid line corresponds to the (ordinary) quark $\psi$. For chimera mesons and chimera baryons, no disconnected diagram appears because of the color and the quark-number conservation.

Figure 3

Typical examples of effective mass plots for (a) scalar-quark mesons ${\varphi }^{†}\varphi$, (b) scalar-quark baryons $\varphi \varphi \varphi$, (c) chimera mesons ${\varphi }^{†}\psi$, (d) chimera baryons $\psi \psi \varphi$, and (e) chimera baryons $\varphi \varphi \psi$. In the figure, the bare scalar-quark mass $m_{\varphi }$ for scalar-quark hadrons and chimera hadrons is $m_{\varphi }=0$ and the pion mass $M_{\pi }$ for chimera hadrons is $M_{\pi }=0.75\mathrm{GeV}$. The unit of the horizontal axis is lattice spacing $a$ and that of the vertical axis is $a^{-1}$. For large $t$, there is a plateau region and the correlator $G(t)$ for the scalar-quark meson is fitted in the corresponding region by a single exponential function to extract the lowest-state mass. The vertical dashed lines denote the fit ranges of corresponding correlators.

Figure 4

(a) The scalar-quark meson mass $M_{{\varphi }^{†}\varphi }$ and (b) the scalar-quark baryon mass $M_{\varphi \varphi \varphi }$ plotted against the bare scalar-quark mass $m_{\varphi }$. Even at $m_{\varphi }=0$, large masses of about 3 and 4.7 GeV are observed for the scalar-quark meson and the scalar-quark baryon, respectively. (c) The scalar-meson mass squared $M_{{\varphi }^{†}\varphi }^2$ and (d) the scalar-quark baryon mass squared $M_{\varphi \varphi \varphi }^2$ are plotted against $m_{\varphi }^2$ in both cases of $m_{\varphi }^2\ge 0$ and $m_{\varphi }^2<0$. Calculations can be performed even with $m_{\varphi }^2\le 0$. Almost linear $m_{\varphi }^2$-dependence is observed for both scalar-quark mesons and scalar-quark baryons.

Figure 5

The lattice results for scalar-quark hadrons for the negative region of bare scalar-quark mass squared $m_{\varphi }^2$: (a) scalar-quark meson mass squared $M_{{\varphi }^{†}\varphi }^2$ vs $m_{\varphi }^2$, (b) scalar-quark baryon mass squared $M_{\varphi \varphi \varphi }^2$ vs $m_{\varphi }^2$, and (c) the ratio $M_{\varphi \varphi \varphi }/M_{{\varphi }^{†}\varphi }$ vs $m_{\varphi }^2$. Both the scalar-quark meson mass and the scalar-quark baryon mass become almost zero at about $m_{\varphi }^2\simeq -(1.2\mathrm{GeV}{)}^2$. Except for the large negative region of $m_{\varphi }^2$, one observes the relation of the constituent scalar-quark picture as $M_{\varphi \varphi \varphi }/M_{{\varphi }^{†}\varphi }\simeq 1.5$ denoted by the dotted line.

Figure 6

The mass of chimera mesons, $M_{{\varphi }^{†}\psi }$, plotted against pion mass squared $M_{\pi }^2$ in the unit of ${\mathrm{GeV}}^2$. (a), (b), (c), and (d) are $M_{{\varphi }^{†}\psi }$ at $m_{\varphi }=0$, 110, 220, and 330 MeV, respectively, plotted against $M_{\pi }^2$. The data at $M_{\pi }^2=0$ is obtained from linear extrapolation, and the solid line denotes the best-fit linear function of the data. In (e), the central values of (a), (b), (c), and (d) are plotted all together. Different symbols correspond to the different bare scalar-quark mass $m_{\varphi }$ (circle: $m_{\varphi }=0$; square: $m_{\varphi }=0.11\mathrm{GeV}$; triangle: $m_{\varphi }=0.22\mathrm{GeV}$; diamond: $m_{\varphi }=0.33\mathrm{GeV}$).

Figure 7

The mass of chimera baryons, $M_{\psi \psi \varphi }$, plotted against pion mass squared $M_{\pi }^2$ in the unit of ${\mathrm{GeV}}^2$. The notation, unit, and notices are the same as those of Fig. 6.

Figure 8

The mass of chimera baryons, $M_{\varphi \varphi \psi }$, plotted against pion mass squared $M_{\pi }^2$ in the unit of ${\mathrm{GeV}}^2$. The notation, unit, and notices are same as those of Fig. 6.

Figure 9

Chimera meson mass $M_{{\varphi }^{†}\psi }$, chimera baryon mass $M_{\psi \psi \varphi }$, and chimera baryon mass $M_{\varphi \varphi \psi }$ plotted against bare scalar-quark mass squared $m_{\varphi }^2$ in unit of GeV. The pion mass of the figure data is $M_{\pi }\simeq 0.49\mathrm{GeV}$. Linear dependence of these masses for $m_{\varphi }^2$ can be seen.

Figure 10

$\Delta M_{{\varphi }^{†}\psi }\equiv M_{{\varphi }^{†}\psi }(m_{\varphi }^2,M_{\pi }^2)-M_{{\varphi }^{†}\psi }(m_{\varphi }^2,M_{\pi }^2=0)$, $\Delta M_{\psi \psi \varphi }/2\equiv (M_{\psi \psi \varphi }(m_{\varphi }^2,M_{\pi }^2)-M_{\psi \psi \varphi }(m_{\varphi }^2,M_{\pi }^2=0))/2$, and $\Delta M_{\varphi \varphi \psi }\equiv M_{\varphi \varphi \psi }(m_{\varphi }^2,M_{\pi }^2)-M_{\varphi \varphi \psi }(m_{\varphi }^2,M_{\pi }^2=0)$ plotted against $M_{\pi }^2$ at $m_{\varphi }=0$, 0.11, 0.22, and 0.33 GeV. Different symbols correspond to the different types of the chimera hadrons (triangle: chimera mesons ${\varphi }^{†}\psi$; circle: chimera baryons $\varphi \varphi \psi$; square: chimera baryons $\psi \psi \varphi$). The four data at $m_{\varphi }=0$, 0.11, 0.22, and 0.33 almost coincide at each case, indicating $m_{\varphi }$-independence for these mass differences. The difference between $\Delta M_{{\varphi }^{†}\psi }$ and $\Delta M_{\varphi \varphi \psi }$ is only several MeV.

Figure 11

Structure of chimera mesons ${\varphi }^{†}\psi$ and chimera baryons $\varphi \varphi \psi$. The wave function of the quark $\psi$ in the chimera meson distributes around the heavy scalar-quark ${\varphi }^{†}$. Similar to chimera mesons, the wave function of $\psi$ in chimera baryons distributes around the two $\varphi$’s localized near the center of mass.

Figure 12

Schematic figure for dynamical mass generation of colored particles. Even without chiral symmetry breaking, colored particles generally acquire a large effective mass due to dressed gluons.

Figure 13

The lattice results for scalar-quark mesons ${\varphi }^{†}\varphi$ at $\beta =6.10$: (a) The scalar-quark meson mass $M_{{\varphi }^{†}\varphi }$ plotted against the bare scalar-quark mass $m_{\varphi }$ and (b) $M_{{\varphi }^{†}\varphi }^2$ against $m_{\varphi }^2$. A large mass of scalar-quark mesons about 5.5 GeV even at $m_{\varphi }=0$ and approximate linear $m_{\varphi }^2$-dependence of $M_{{\varphi }^{†}\varphi }^2$ are observed.