Exotic $\Omega_c^0$ baryons from meson-baryon scattering

TL;DR

This paper models meson-baryon interactions to explain the observed $ ext{Ω}_c^0$ baryons as molecular states, providing a theoretical framework that matches recent experimental findings from LHCb.

Contribution

It introduces a meson-baryon scattering model using vector meson exchange and unitarization, offering a novel interpretation of $ ext{Ω}_c^0$ states as molecular resonances.

Findings

Identifies two resonances matching observed $ ext{Ω}_c^0$ states.

Suggests these states are molecular with $J^P=1/2^-$.

Provides a theoretical basis for meson-baryon molecular interpretation.

Abstract

A meson-baryon interaction in the charm $+1$, strangeness $-2$ and isospin $0$ sector is built from a t-channel vector meson exchange model employing effective Lagrangians. The implementation of coupled-channel unitarization in the s-wave scattering amplitudes gives rise to two structures that have similar masses and widths to those of the $\Omega_c(3050)^0$ and $\Omega_c(3090)^0$ states recently observed by the LHCb collaboration. A meson-baryon molecular interpretation of these resonances would assign their spin-parity to be $J^P=1/2^-$.

Tables

Table 1. Table 1: The C i ​ j subscript 𝐶 𝑖 𝑗 C_{ij} coefficients for the ( I , S , C ) = ( 0 , − 2 , 1 ) 𝐼 𝑆 𝐶 0 2 1 (I,S,C)=(0,-2,1) sector of the pseudscalar meson-baryon interaction.

\br	$\overline{K}{\Xi }_c$	$\overline{K}{\Xi }_c^{\prime }$	$D\Xi$	$\eta {\Omega }_c^0$	${\eta }^{\prime }{\Omega }_c^0$
\mr$\overline{K}{\Xi }_c$	$1$	$0$	$\sqrt{3/2}{\kappa }_c$	$\text{m}\text{m}0$	$\text{m}\text{m}0$
$\overline{K}{\Xi }_c^{\prime }$		$1$	$\sqrt{1/2}{\kappa }_c$	$-\sqrt{6}$	$\text{m}\text{m}0$
$D\Xi$			$\text{m}2$	$-\sqrt{1/3}{\kappa }_c$	$-\sqrt{2/3}{\kappa }_c$
$\eta {\Omega }_c^0$				$\text{m}\text{m}0$	$\text{m}\text{m}0$
${\eta }^{\prime }{\Omega }_c^0$					$\text{m}\text{m}0$
\br

Table 2. Table 2: The Ω c 0 subscript superscript Ω 0 𝑐 \Omega^{0}_{c} states generated employing vector-type WT zero-range interactions between a pseudoscalar meson and a ground state baryon, within a coupled-channel approach.

\br      $0^{-}\otimes 1/2^{+}$ interaction in the $(I,S,C)=(0,-2,1)$ sector
\mr
$M\left[\mathrm{MeV}\right]$	$3050.3$		$3090.8$
$\Gamma \left[\mathrm{MeV}\right]$	$0.44$		$12$
\mr
	$\left|g_i\right|$	$-g_i^{2}dG/dE$	$\left|g_i\right|$	$-g_i^{2}dG/dE$
$\overline{K}{\Xi }_c\left(2964\right)$	$0.11$	$0.00+i\mathrm{\hspace{0.17em}0.00}$	$0.49$	$-0.02+i\mathrm{\hspace{0.17em}0.01}$
$\overline{K}{\Xi }_c^{\prime }\left(3070\right)$	$1.80$	$0.61+i\mathrm{\hspace{0.17em}0.01}$	$0.35$	$\text{m}0.02-i\mathrm{\hspace{0.17em}0.02}$
$D\Xi \left(3189\right)$	$1.36$	$0.07-i\mathrm{\hspace{0.17em}0.01}$	$4.28$	$\text{m}0.91-i\mathrm{\hspace{0.17em}0.01}$
$\eta {\Omega }_c\left(3246\right)$	$1.63$	$0.14+i\mathrm{\hspace{0.17em}0.00}$	$0.39$	$\text{m}0.01+i\mathrm{\hspace{0.17em}0.01}$
${\eta }^{\prime }{\Omega }_c\left(3656\right)$	$0.06$	$0.00+i\mathrm{\hspace{0.17em}0.00}$	$0.28$	$\text{m}0.00+i\mathrm{\hspace{0.17em}0.00}$
\br

Equations

$$T_{ij}=V_{ij}+V_{il}{G}_l{T}_{lj},$$

$$G_l=i\int \frac{d^{4}q}{(2\pi {)}^4}\frac{2M_l}{(P-q{)}^2-M_l^2+iϵ}\frac{1}{q^2-m_l^2+iϵ},$$

$$V_{ij}(\sqrt{s})=-C_{ij}\frac{1}{4f^2}\left(2\sqrt{s}-M_i-M_j\right)N_i{N}_j$$

$${\mathcal{L}}_{VPP}=ig⟨\left[{\partial }_{\mu }\varphi ,\varphi \right]V^{\mu }⟩,$$

$${\mathcal{L}}_{VBB}=\frac{g}{2}\sum _{i,j,k,l=1}^4{\bar{B}}_{ijk}{\gamma }^{\mu }\left(V_{\mu ,l}^k{B}^{ijl}+2V_{\mu ,l}^j{B}^{ilk}\right),$$

$${\mathcal{L}}_{VVV}=ig⟨\left[V^{\mu },{\partial }_{\nu }{V}_{\mu }\right]V^{\nu }⟩.$$

$$M_1=\mathrm{Re}{z}_1=3051.6\mathrm{MeV},$$

$$M_2=\mathrm{Re}{z}_2=3103.3\mathrm{MeV},$$

$${\Omega }_c(3050{)}^0:$$

$${\Omega }_c(3090{)}^0:$$

Full text

1 Introduction

The recent observation by the LHCb collaboration of five narrow $\Omega_{c}^{0}$ excited resonances [1] has triggered a lot of activity in the field of baryon spectroscopy aiming at understanding their inner structure and possibly establishing their unknown values of spin-parity. Quark models giving a $css$ quark content picture have been revisited [2, 3, 4, 5, 6, 7, 8, 9] and pentaquark interpretations [10, 11, 12] have also been investigated.

An alternative scenario is provided by models that can describe some of these resonances as quasi-bound states of an interacting meson-baryon pair [13, 14, 15], an approach that we have recently re-examined [16] in view of the new experimental data. Similarly to the $P_{c}(4380)$ and $P_{c}(4450)$ excited nucleon resonances, for which a pentaquark structure having a $c\bar{c}$ pair in its composition is more natural rather than being an extremely high energy excitation of the $3q$ system, it is also plausible to expect that some excitations in the $C=1$, $S=-2$ sector can be obtained by adding a $u\bar{u}$ pair to the natural $ssc$ content of the $\Omega_{c}^{0}$. The hadronization of the five quarks could then lead to bound states, generated by the meson-baryon interaction in coupled channels. This possibility is supported by the fact that the $\bar{K}\Xi_{c}$ and $\bar{K}\Xi_{c}^{\prime}$ thresholds are in the energy range of interest, and that the excited $\Omega_{c}^{0}$ baryons under study have been observed from the invariant mass of the spectrum of $K^{-}\Xi^{+}_{c}$ pairs.

2 Formalism

The unitarized scattering amplitude $T_{ij}$ exhibiting dynamically generated resonances is obtained from solving the on-shell Bethe-Salpeter equation in coupled channels,

[TABLE]

which implements the resummation of loop diagrams to infinite order.

The loop function built with the meson and baryon propagators,

[TABLE]

is formally divergent and is regularized by means of the dimensional regularization approach, which introduces the dependence on a subtraction constant $a_{l}(\mu)$ for each intermediate channel $l$ at a given regularization scale $\mu$ (see Eq. (18) in [16]).

The s-wave tree-level amplitude used as the kernel $V_{ij}$ is obtained from the t-channel vector meson exchange [13] that in the $t\ll m_{V}$ limit reduces to a contact Weinberg-Tomozawa (WT) term:

[TABLE]

where $M_{i}$, $M_{j}$ and $E_{i}$, $E_{j}$ are the masses and the energies of the baryons and the normalization factors are ${N=\sqrt{(E+M)/2M}}$.

The coefficients $C_{ij}$ are obtained from the evaluation of the t-channel meson-baryon interaction diagram introducing the effective Lagrangians of the hidden gauge formalism [13]:

[TABLE]

[TABLE]

to describe the vertices coupling the vector meson to pseudoscalars (VPP) and baryons (VBB), respectively, in the pseudoscalar meson-baryon scattering, and assuming $SU(4)$ symmetry.

The interaction of vector mesons with baryons is built in a similar way and involves a three-vector $VVV$ vertex, which is obtained from the effective Lagrangian:

[TABLE]

The resulting interaction is that in Eq. (3), including the product of polarization vectors, $\vec{\epsilon}_{i}\cdot\vec{\epsilon}_{j}$.

Note that, while $SU(4)$ symmetry is encoded in the Lagrangians, the interaction potential is not $SU(4)$ symmetric due to the use of physical masses for the mesons and baryons involved, as well as to a factor $\kappa_{c}=1/4$ accounting for the higher mass of the charmed mesons exchanged in some of the non-diagonal transitions. Actually, the transitions mediated by the exchange of light vector mesons do not make explicit use of $SU(4)$ symmetry. This is for instance the case of the dominant diagonal transitions, which are effectively projected into their $SU(3)$ content.

The available pseudoscalar meson-baryon channels in the $(I,S,C)=(0,-2,1)$ sector are $\bar{K}\Xi_{c}(2964)$, $\bar{K}\Xi^{\prime}_{c}(3070)$, $D\Xi(3189)$, $\eta\Omega_{c}(3246)$, $\eta^{\prime}\Omega_{c}(3656)$, $\bar{D}_{s}\Omega_{cc}(5528)$, and $\eta_{c}\Omega_{c}(5678)$, where the values in parentheses indicate their corresponding threshold. The doubly charmed $\bar{D}_{s}\Omega_{cc}$ and $\eta_{c}\Omega_{c}$ channels will be neglected, as their energy is much larger than that of the other channels. The matrix of $C_{ij}$ coefficients for the resulting 5-channel interaction is given in Table 1.

In the vector meson-baryon case, the allowed states are $D^{*}\Xi(3326)$, $\bar{K}^{*}\Xi_{c}(3363)$, $\bar{K}^{*}\Xi^{\prime}_{c}(3470)$, $\omega\Omega_{c}(3480)$, $\phi\Omega_{c}(3717)$, $\bar{D}_{s}^{*}\Omega_{cc}(5662)$ and $J/\psi\Omega_{c}(5794)$, where, again, we will neglect the doubly charmed states. The coefficients $C_{ij}$ can be straightforwardly obtained from those for the $PB$ interaction in Table 1, by considering the correspondences: $\pi\rightarrow\rho,K\rightarrow K^{\ast},\bar{K}\rightarrow\bar{K}^{\ast},$ $D\rightarrow D^{\ast},\bar{D}\rightarrow\bar{D}^{\ast},{1/\sqrt{3}}\,\eta+\sqrt{2/3}\,\eta^{\prime}\rightarrow\omega$ and $-\sqrt{2/3}\,\eta+{1/\sqrt{3}}\,\eta^{\prime}\rightarrow\phi$.

A resonance generated dynamically from the coupled channel meson-baryon interaction appears as a pole of the scattering amplitude $T$ in the so-called second Riemann sheet of the complex energy plane. The coupling constants $g_{i}$ of the resonance to the various channels are obtained from the residues at the pole position $z_{p}$ while the compositeness, i.e., the amount of $i^{\rm th}$-channel meson-baryon component, is given by the real part of $-g_{i}^{2}(\partial G/\partial\sqrt{s})|_{z_{p}}$.

3 Results and Discussion

The Bethe-Salpeter equation in coupled channels of Eq. (1) has been solved using subtraction constant values, $a_{l}(\mu=1\rm~{}GeV)$, obtained by imposing the loop function to coincide, at the corresponding thresholds, with the loop function regularized with a cut-off $\Lambda=800\rm~{}MeV$. The corresponding pseudoscalar meson-baryon scattering amplitude shows two poles,

[TABLE]

corresponding to resonances with spin-parity $J^{P}=1/2^{-}$. Their energies are very similar to the second and fourth $\Omega_{c}^{0}$ states discovered by LHCb [1], with properties:

[TABLE]

Even if the mass of our heavier state is larger by 10 MeV and its width is about twice the experimental one, our results clearly show the ability of the meson-baryon dynamical models to generate states in the energy range of interest. In an attempt to explore the possibilities of our model, we let the values of the five subtraction constants vary freely within a reasonably constrained range and look for a combination that reproduces the characteristics of the two observed states, $\Omega_{c}(3050)^{0}$ and $\Omega_{c}(3090)^{0}$, within $2\sigma$ of the experimental errors. For a representative set of $a_{l}(\mu=1\rm~{}GeV)$ with equivalent cut-off values in the range $[320-950]\;\rm MeV$ [16], the new properties of the poles are shown in Table 3. We note that the strongest change corresponds to $a_{\bar{K}\Xi_{c}}$, needed to decrease the width of the $\Omega_{c}(3090)^{0}$. Its equivalent cut-off value of 320 MeV is on the low side of the usually employed values but it is still naturally sized.