Speaker
Description
Summary
Inspired by the recently observed $P_c$ and
$P_{cs}$ pentaquarks, we perform a systematic
study on the interactions of the $\Sigma_c^{(*)}D^{(*)}$ systems to
explore the possible $P_{cc}$ states. We include the contact term,
one-pion-exchange, and two-pion-exchange interactions within the
framework of chiral effective field theory.
Due to $G$-parity transformation law, the expressions of the
one-pion-exchange and two-pion-exchange effective potentials of the
$\Sigma_c^{(*)}D^{(*)}$ systems are opposite and identical to those
of the $\Sigma_c^{(*)}\bar{D}^{(*)}$ systems,
respectively.
In principle, the LECs of the $\Sigma_c^{(*)}D^{(*)}$ systems should be fixed from the experimental data or lattice QCD simulations, which are not available at present. Alternatively, we introduce a quark level contact Lagrangian to bridge the LECs determined from the $N\bar{N}$ scattering data to the unknown $\Sigma_c^{(*)}D^{(*)}$ systems.
With the LECs fitted from the $N\bar{N}$ scattering
data, we obtain four sets of ($c_s$, $c_t$) parameters describing
the contributions of the contact terms. We present three cases to
study the binding energies of the $\Sigma_c^{(*)}D^{(*)}$ systems.
The mass spectrum of the $[\Sigma_c^{(*)}D^{(*)}]^{I=1/2}_J$
molecules depend on the values of the LECs. In Case 1, a relatively
small central potential and a large spin-spin interaction are
introduced. The obtained $P_{cc}$ mass spectrum is very similar to
that of the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ systems. However, the
mass spectra obtained in Cases 2 and 3 are different from that of
the Case 1.
In this work, to estimate the binding energies of the $P_{cc}$ pentaquarks, we only consider the $S$-wave interactions between $\Sigma_c^{(*)}$ and $D^{(*)}$. The $S-D$ wave mixing is not included in this work. On the one hand, the LECs introduced from the short-range contact tensor term can not be estimated at present, we only consider the leading order contact terms for cutting down the unknown parameters. Thus, the $S-D$ tensor force from the leading order one-pion-exchange (two-pion-exchange) is neglected for consistency.
From the effective potentials of the $[\Sigma_c^{(*)}D^{(*)}]^{I=1/2}_J$ systems, we find that the attractive force between the $\Sigma_c^{(*)}$ and $D^{(*)}$ arises mainly from the short-range interactions. Although this short-range-interaction-dominant mechanism is consistent with our understanding of the $P_c$, $Z_c$ ($Z_b$), and $X(3872)$ states, these phenomenologically determined LECs still need further support from experimental data or lattice QCD simulations.
We determine the couplings $g$, $g_2$, and $g_4$ by calculating the partial decay widths of the $D^*$, $\Sigma_c$, and $\Sigma_c^{(*)}$ systems, the other axial couplings $g_1$, $g_3$, and $g_5$ can be correspondingly obtained in the framework of the quark model. Thus, the width effects of the $\Sigma_c^{(*)}$ and $D^{(*)}$ are partly encoded in these parameters. However, it is difficult to introduce widths into the Schr\"odinger equation when we solve the binding energies of the $\Sigma_c^{(*)}D^{(*)}$ systems. The present method can only provide rough positions to the considered $P_{cc}$ pentaquark.
We briefly discuss the strong decay behaviors of the $P_{cc}$
pentaquarks. The $(cqq)$-$(c\bar{q})$ and $(ccq)$-$(q\bar{q})$ are
the two types of decay modes. Correspondingly, the $\Lambda_c D$,
$\Lambda_c D \pi$, and $\Xi_{cc}\pi$ are expected to be important
channels to search for these $[\Sigma_c^{(*)}D^{(*)}]^{I=1/2}_J$
molecules.
We also study the interactions of the $\Sigma_c^{(*)}\bar{B}^{(*)}$,
$\Sigma_{b}^{(*)}D^{(*)}$, and $\Sigma_b^{(*)}\bar{B}^{(*)}$ to
search for possible $P_{cb}$, $P_{bc}$, and $P_{bb}$ pentaquarks.
The corresponding systems with $I=1/2$ can also form molecular
states. In addition, among the studied systems, the binding becomes
deeper when the reduced masses of the systems are heavier.
Because the uncertainties from the quark model assumptions cannot be quantified, thus the $\Sigma_c^{(*)}D^{(*)}$ systems still need further study. If lattice QCD calculations are performed to extract physical observable quantities in the future, we can fit the lattice results to extrapolate to the physical pion mass to obtain the LECs for the $\Sigma_c^{(*)}D^{(*)}$ systems. The width effects and the $S-D$ wave mixing effects can also be studied by solving the corresponding Lippmann-Schwinger equations.
