Axial-vector tetraquark with $S=+2$

The possibility of an axial-vector isoscalar tetraquark with $ud\overline{s}\overline{s}$ is discussed. If the pentaquark ${\Theta }^{+}(1540)$ has the $(qq{)}_{\overline{3}}(qq{)}_{\overline{3}}\overline{q}$ configuration, the isoscalar $ud\overline{s}\overline{s}$ (${\vartheta }^{+}$-meson) state with $J^P=1^{+}$ is expected to exist in the mass region lower than, or close to, the mass of ${\Theta }^{+}(1540)$. Within a flux-tube quark model, a possible resonant state of $ud\overline{s}\overline{s}(J^P=1^{+})$ is suggested to appear around 1.4 GeV with the width $\mathcal{O}(20\sim 80)\mathrm{MeV}$. We propose that the ${\vartheta }^{+}$-meson is a good candidate for the tetraquark search, which would be observed in the $K^{+}{K}^{+}{\pi }^{-}$ decay channel.

Figure 1

Flux-tube configurations for $q\overline{q}$ meson (a), $q^3$ baryon (b), $q^2{\overline{q}}^2$ states (c),(d), and $q^4\overline{q}$ states (e),(f). For the $q^2{\overline{q}}^2$ states, the exotic tube configuration (c) correspond, to the $(qq{)}_{\overline{3}}(\overline{q}\overline{q}{)}_3$, and the disconnected tube (d) represents the meson-meson state, $(q\overline{q}{)}_1(q\overline{q}{)}_1$. The configurations $(qq{)}_{\overline{3}}(qq{)}_{\overline{3}}\overline{q}$ and $(qqq{)}_1(q\overline{q}{)}_1$ for the $q^4\overline{q}$ system are illustrated in (e) and (f), respectively.

Figure 2

Mass spectra for $q\overline{q}$ mesons and $3q$ baryons. The experimental and calculated masses are shown by dashed and solid lines, respectively. We adjust the zero-point energy of the string potential for the $q\overline{q}$ system as $M_{q\overline{q}}^0=584\mathrm{MeV}$ to fit the experimental $\rho$-meson mass. For the $3q$ system, $M_{q^3}^0$ is chosen to be $M_{q^3}^0=972\mathrm{MeV}$ to reproduce the nucleon masses.

Figure 3

Adopted zero-point energy ($M^0$) of the string potential as a function of the quark number $N_q$. $M_{q\overline{q}}^0$, $M_{q^3}^0$, and $M_{[qq][qq]\overline{q}}^0$ are adjusted to reproduce the $\rho$-meson, nucleon, and ${\Theta }^{+}$ masses, respectively. The $M_{[qq][\overline{q}\overline{q}]}^0$ for the $(qq{)}_{\overline{3}}(\overline{q}\overline{q}{)}_3$ is deduced by assuming the linear function $M^0(N_q)=a_0{N}_q+b_0$, where the parameters $a_0$ and $b_0$ are determined by fitting (i) $M^0(N_q=2)=M_{q\overline{q}}^0=584\mathrm{MeV}$ and $M^0(N_q=5)=M_{[qq][qq]\overline{q}}^0=2375\mathrm{MeV}$, or (ii) $M^0(N_q=3)=M_{q^3}^0=972\mathrm{MeV}$ and $M^0(N_q=5)=M_{[qq][qq]\overline{q}}^0=2375\mathrm{MeV}$. The circle and triangle indicate the $M_{[qq][\overline{q}\overline{q}]}^0$ values obtained by the former fitting (i) and the latter one (ii), respectively.

Figure 4

Masses of the ${\vartheta }^{+}(J^{PC}=1^{+})$-meson and $f_1$-mesons. The theoretical values (solid lines) shown in the left and right panels were obtained by using (i) $M_{[qq][\overline{q}\overline{q}]}^0=1785\mathrm{MeV}$ and (ii) $M_{[qq][\overline{q}\overline{q}]}^0=1679\mathrm{MeV}$, respectively. The mass of the $f_1$ with the $4q$ component was obtained by calculations of the $us\overline{u}\overline{s}$ with $J^{PC}=1^{++}$. The dashed lines are experimental masses of the $f_1$-mesons. The experimental $K{K}^{*}$ threshold energy is also displayed by dotted lines.

Figure 5

Energy dependence of the two-meson $K{K}^{*}(892)$ decay width $\Gamma$. The coupling is chosen to fit the full-width $\Gamma =55\mathrm{MeV}$ of the $f_1(1420)$ at $m_0=1.42\mathrm{GeV}$, and is assumed to be energy independent.