Interpretation of the newly observed $\Omega_c^0$ resonances

TL;DR

This paper models the mass spectra and decay behaviors of doubly strange baryons, interpreting newly observed $ ext{Ω}_c^0$ resonances as P-wave excited states, and explains their narrow widths through heavy quark symmetry.

Contribution

It provides a theoretical interpretation of newly observed $ ext{Ω}_c^0$ states as P-wave excitations within a heavy-quark-light-diquark framework, incorporating decay analysis via heavy quark effective theory.

Findings

The $ ext{Ω}_c^0(2695)$ and $ ext{Ω}_c^0(2770)$ fit well as S-wave states.

The five new $ ext{Ω}_c^0$ states are interpreted as P-wave orbitally excited states.

Decay suppression explains the narrow widths of the new $ ext{Ω}_c^0$ states.

Abstract

We study the charmed and bottomed doubly strange baryons within the heavy-quark-light-diquark framework. The two strange quarks are assumed to lie in $S$ wave and thus their total spin is 1. We calculate the mass spectra of the $S$ and $P$ wave orbitally excited states and find the $\Omega_c^0 (2695)$ and $\Omega_c^0 (2770)$ fit well as the $S$ wave states of charmed doubly strange baryons. The five newly $\Omega_c^0(X)$ resonances observed by the LHCb Collaboration, i.e. $\Omega_c^0(3000)$, $\Omega_c^0(3050)$, $\Omega_c^0(3066)$, $\Omega_c^0(3090)$, and $\Omega_c^0(3119)$, can be interpreted as the $P$ wave orbitally excited states. In heavy quark effective theory, we analyze their decays into the $\Xi^+_c K^-$ and $\Xi_c'^+ K^-$, and point out that decays of the five P-wave $\Omega_c^0$ states into the $\Xi_c^+ K^-$ and $\Xi_c'^+K^-$ are suppressed by either heavy quark symmetry or phase space. The narrowness of the five newly observed $\Omega_c^0(X)$ states can then be naturally interpreted with heavy quark symmetry.

Tables

Table 1. Table 1: Masses and widths (MeV) of the Ω c 0 ​ ( X ) superscript subscript Ω 𝑐 0 𝑋 \Omega_{c}^{0}(X) baryons observed by the LHCb Collaboration. The first uncertainty is statistical and the second one is systematic, and the third uncertainty in masses of Ω c 0 ​ ( X ) superscript subscript Ω 𝑐 0 𝑋 \Omega_{c}^{0}(X) baryons is from the Ξ c + superscript subscript Ξ 𝑐 \Xi_{c}^{+} mass.

state	mass	width
${\Omega }_c\left(3000\right)$	$3000.4\pm 0.2\pm {0.1}_{-0.5}^{+0.3}$	$4.5\pm 0.6\pm 0.3$
${\Omega }_c^0\left(3050\right)$	$3050.2\pm 0.1\pm {0.1}_{-0.5}^{+0.3}$	$0.8\pm 0.2\pm 0.1$
${\Omega }_c^0\left(3066\right)$	$3065.6\pm 0.1\pm {0.3}_{-0.5}^{+0.3}$	$3.5\pm 0.4\pm 0.2$
${\Omega }_c^0\left(3090\right)$	$3090.2\pm 0.3\pm {0.5}_{-0.5}^{+0.3}$	$8.7\pm 1.0\pm 0.8$
${\Omega }_c^0\left(3119\right)$	$3119.1\pm 0.3\pm {0.9}_{-0.5}^{+0.3}$	$1.1\pm 0.8\pm 0.4$

Table 2. Table 2: The mass spectra (MeV) of Ω c 0 ​ ( X ) superscript subscript Ω 𝑐 0 𝑋 \Omega_{c}^{0}(X) baryons. The uncertainties of the experimental measurements are squared averages of those from the statistical and systematic, and the Ξ c + superscript subscript Ξ 𝑐 \Xi_{c}^{+} mass.

$n^{2S+1}{L}_J$	This work	Exp. Aaij:2017nav ; Olive:2016xmw  222The following assignment of ${\Omega }_c^0$ baryons to $|n^{2S+1}{L}_J⟩$ is by no means conclusive. For instance, the ${\Omega }_c\left(2695\right)$ has been assigned as the ground state only due to the fact there is no other lower state that has been established on the experimental side.	Shah:2016nxi	Ebert:2011kk	Roberts:2007ni	Yoshida:2015tia	Chen:2015kpa	Valcarce:2008dr	Yamaguchi:2014era	Bali:2015lka
$1^2{S}_{\frac{1}{2}}$	$2695.2\pm 1.7$	$2695.2\pm 1.7$	2695	2698	2718	2731	–	2699	2648	2718
$1^4{S}_{\frac{3}{2}}$	$2773\pm 6$	$2765.9\pm 2.0$	2767	2768	2776	2779	–	2767	–	–
$1^2{P}_{\frac{1}{2}}$	$3068\pm 16$	$3050.2\pm 0.5$	3011	3055	2977	3030	3250	2980	2995	3046
$1^2{P}_{\frac{3}{2}}$	$3095\pm 11$	$3090.2\pm 0.8$	2976	3054	2986	3033	3260	2980	3016	2986
$1^4{P}_{\frac{1}{2}}$	$3017\pm 7$	$3000.4\pm 0.5$	3028	2966	2977	–	–	3035	–	–
$1^4{P}_{\frac{3}{2}}$	$3044\pm 5$	$3065.6\pm 0.6$	2993	3029	2959	–	–	–	–	–
$1^4{P}_{\frac{5}{2}}$	$3140\pm 13$	$3119.1\pm 1.1$	2947	3051	3014	3057	3320	–	–	3014

Equations

$$\Psi (c,s,s)$$

$$\Delta M$$

$$|L=0,{\frac{1}{2}}_J⟩$$

$$|L=0,{\frac{3}{2}}_J⟩$$

$$|L=1,{\frac{1}{2}}_J{⟩}_1$$

$$|L=1,{\frac{1}{2}}_J{⟩}_2$$

$$|L=1,{\frac{3}{2}}_J{⟩}_1$$

$$|L=1,{\frac{3}{2}}_J{⟩}_2$$

$$|L=1,{\frac{5}{2}}_J⟩$$

$$M_{|L=0,{\frac{3}{2}}_J⟩}$$

$$M_{|L=1,{\frac{1}{2}}_J{⟩}_1}$$

$$M_{|L=1,{\frac{1}{2}}_J{⟩}_2}$$

$$M_{|L=1,{\frac{3}{2}}_J{⟩}_1}$$

$$M_{|L=1,{\frac{3}{2}}_J{⟩}_2}$$

$$M_{|L=1,{\frac{5}{2}}_J⟩}$$

$$({\kappa }_{cs}{)}_{\bar{3}}$$

$$K(c,\{u,d\})$$

$$K(c,\{u,s\})$$

$$-2A_c+B$$

$$A_c+B$$

$$M_{{\Omega }_b}(1^4{S}_{\frac{3}{2}})$$

$$M_{{\Omega }_b}(1^2{P}_{\frac{1}{2}})$$

$$M_{{\Omega }_b}(1^2{P}_{\frac{3}{2}})$$

$$M_{{\Omega }_b}(1^4{P}_{\frac{1}{2}})$$

$$M_{{\Omega }_b}(1^4{P}_{\frac{3}{2}})$$

$$M_{{\Omega }_b}(1^4{P}_{\frac{5}{2}})$$

$$|{\frac{1}{2}}_J{⟩}_{1^{\prime }}$$

$$|{\frac{1}{2}}_J{⟩}_{2^{\prime }}$$

$$|{\frac{3}{2}}_J{⟩}_{1^{\prime }}$$

$$|{\frac{3}{2}}_J{⟩}_{2^{\prime }}$$

$$|{\frac{5}{2}}_J⟩$$

$$|{\frac{1}{2}}_J{⟩}_{1^{\prime }}$$

$$|{\frac{1}{2}}_J{⟩}_{2^{\prime }}$$

$$|{\frac{3}{2}}_J{⟩}_{1^{\prime }}=\sqrt{\frac{1}{6}}|L=1,{\frac{3}{2}}_J{⟩}_1+\sqrt{\frac{5}{6}}|L=1,{\frac{3}{2}}_J{⟩}_2,$$

$$|{\frac{3}{2}}_J{⟩}_{2^{\prime }}=\sqrt{\frac{5}{6}}|L=1,{\frac{3}{2}}_J{⟩}_1-\sqrt{\frac{1}{6}}|L=1,{\frac{3}{2}}_J{⟩}_2,$$

$$\mathcal{A}({\Omega }_c(J,J_z)\to {\Xi }_c^{{(}^{\prime })}(J^{\prime },J_z^{\prime })K(L,L_z))$$

$$\Gamma$$

$$S_{\ell }^{\prime }=0,S_{\ell }=(0,1,2),J^{\prime }=\frac{1}{2},J=\left(\frac{1}{2},\frac{3}{2},\frac{5}{2}\right).$$

$$S_{\ell }^{\prime }=1,S_{\ell }=(0,1,2),J^{\prime }=\frac{1}{2},J=(\frac{1}{2},\frac{3}{2},\frac{5}{2}).$$

Full text

Charmed baryons, Bottomed baryons, Diquark

Introduction

The hadron spectroscopy plays an important role in understanding the fundamental theory of strong interactions, i.e. the Quantum Chromodynamics (QCD). In the naive quark model, the mesons are bound states of a quark-antiquark pair while the baryons are composed of three quarks. However, the structure of hadrons is more complicated than the description in the naive quark model. There might be hybrids, glueballs, and multiquark states, which are also allowed under the principle of color confinement. Take the exotic baryon states as example, the LHCb Collaboration have observed two pentaquark candidates $P_{c}(4380)$ and $P_{c}(4450)$ in $\Lambda^{0}_{b}\to J/\psi K^{-}p$ decays Aaij:2015tga , which also have been analyzed in $\Lambda^{0}_{b}\to J/\psi\pi^{-}p$ decays Aaij:2016ymb . The general studies of hadron inner structures will enhance our knowledge on the properties of QCD color confinement.

The charmed doubly strange baryon $\Omega_{c}^{0}(2695)$ with isospin and spin-parity $I(J^{P})=0(\frac{1}{2}^{+})$ was first observed in the hyperon beam experiment WA62 Biagi:1984mu . Later it was confirmed in the electron-positron collider experiment Albrecht:1992xa and the photon beam experiment Frabetti:1992bm . The excited state $\Omega_{c}^{0}(2770)$ with $I(J^{P})=0(\frac{3}{2}^{+})$ was first observed in the radiative decay $\Omega_{c}^{0}(2770)\to\Omega_{c}^{0}(2695)+\gamma$ by the BaBar Collaboration Aubert:2006je , and then confirmed by the Belle Collaboration Solovieva:2008fw .

Using a sample of $pp$ collision data corresponding to an integrated luminosity of 3.3$fb^{-1}$, the LHCb Collaboration has recently observed five new narrow excited $\Omega_{c}^{0}(X)$ states in the $\Xi_{c}^{+}K^{-}$ invariant mass spectrum Aaij:2017nav . They have determined the masses and decay widths of the five new $\Omega_{c}^{0}(X)$ states Aaij:2017nav and the results are collected in Table 1.

After these discoveries, it is natural to ask ourselves three questions: 1) Why are there so small mass differences among these five new states? 2)What are the spin-parities for these five new states? 3)Why are the decay widths so narrow for these five new states?

Investigating their mass spectra and decay properties will answer these questions accordingly. In theoretical aspects, there are already some attempts to interpret of the newly observed $\Omega_{c}^{0}(X)$ resonances. Agaev et al. proposed to assign $\Omega_{c}^{0}(3066)$ and $\Omega_{c}^{0}(3119)$ states as the first radially excited $(2S,\frac{1}{2}^{+})$ and $(2S,\frac{3}{2}^{+})$ charmed baryons in QCD sum rules Agaev:2017jyt . Chen et al. analyzed the newly $\Omega_{c}^{0}(X)$ states with different spins and obtained the related decay widths into $\Xi_{c}^{+}K^{-}$, $\Xi_{c}^{\prime+}K^{-}$ and $\Xi_{c}^{*+}K^{-}$ in QCD sum rules Chen:2017sci . Karliner et al. proposed to assign the newly $\Omega_{c}^{0}(X)$ states as bound states of a charm quark and a $P$ wave $ss$-diquark Karliner:2017kfm . Wang et al. studied the strong and radiative decays of the $\Omega_{c}^{0}(X)$ states in a constituent quark model Wang:2017hej . Besides, Yang et al. proposed to assign some of the newly $\Omega_{c}^{0}(X)$ states as the possible pentaquark states Yang:2017rpg .

In this paper, we will interpret the five new observed $\Omega_{c}^{0}(X)$ states as the $P$ wave orbitally excited states of charmed doubly strange baryons in the heavy-quark-light-diquark picture. The spectra of the bottom partners of $\Omega_{c}^{0}(X)$ states will be also predicted. In the end, the decay properties of charmed doubly strange baryons will be discussed in the heavy quark effective theory.

Interpretation of the newly observed $\Omega_{c}^{0}$ resonances

The notion of diquark is as old as the quark model where Gell-Mann mentioned the possibility of diquarks in the original paper on quarks GellMann:1964nj . According to the color SU(3) group, the color configuration of a diquark can be represented either by an antitriplet or sextet in the decomposition of $\mathbf{3}\otimes\mathbf{3}=\mathbf{\bar{3}}\oplus\mathbf{6}$. The binding of the $q_{1}\bar{q}_{2}$ or $q_{1}q_{2}$ system depends solely on the quadratic Casimir $C_{2}(R)$ of the product color representation R to which the quarks couple according to the discriminator $I=\frac{1}{2}(C_{2}(R)-C_{2}(R_{1})-C_{2}(R_{2}))$, where $R_{i}$ denotes the color representations of two quarks Brodsky:2014xia . The discriminators are then determined as $I=\frac{1}{6}(-8,-4,+2,+1)$ for $R=(\mathbf{1},\mathbf{\bar{3}},\mathbf{6},\mathbf{8})$, respectively. The interaction force becomes attractive when the discriminator is negative, which is somewhat analogous to the Coulomb force in QED. Thus, the only color attractive configuration of $q_{1}\bar{q}_{2}$ is in the color-singlet $\mathbf{1}$, whereas the color attractive configuration of $q_{1}q_{2}$ is in the color antitriplet $\mathbf{\bar{3}}$. The attractive force strength in the color antitriplet diquark is half of that in the color singlet quark-antiquark pair in the one-gluon-exchange model. Thus two quarks in the color antitriplet $\mathbf{\bar{3}}$ have a large possibility to bind into a diquark  Brodsky:2014xia ; Jaffe:2003sg ; Jaffe:2004ph , and thus a baryon can be treated as a quark-diquark system.

In the $css$ system, two strange quarks can form a light diquark system, while the charm and strange quarks may also form a $cs$ diquark. The strength of the attractive force between two quarks is reflected by a coupling constant as given below. A fit of the experimental data have indicated that the coupling constant for the two strange quarks is much larger than that for the $cs$ system, for instance, $\kappa_{ss}=72$MeV and $\kappa_{cs}\simeq(24-25)$MeV Maiani:2004vq ; Ali:2011ug ; Hambrock:2011qga . Following this scheme, we will treat the charmed doubly strange baryons as heavy-quark-light-diquark bound states in order to explain the newly observed five narrow $\Omega_{c}^{0}(X)$ states.

The wave function of the charmed doubly strange baryon is composed of four parts, coordinate-space, color, flavor, and spin subspaces He:2016xvd

[TABLE]

where we denote numbers 1, 2, 3 to charm and two strange quarks respectively; $\psi(x_{i})$, $\chi$, $f$, and $s_{i}$ denote the coordinate-space, color, flavor, and spin wave functions, respectively. The total wave function should satisfy the Pauli exclusion principle when we interchange the two strange quarks. We will restrict ourselves to the ground state of the diquark, namely the coordinate-space wave function is in the $S$-wave with $L=0$, and thus symmetric. The color wave function is anti-symmetrical because the baryon system is in the color singlet. The flavor wave function is also symmetrical to the interchange of the two strange quarks. Thus the spin wave function should be also symmetrical, i.e. the spin of two strange quarks should be 1 in the charmed doubly strange baryon.

The charmed doubly strange baryons are composed of a charm quark and two strange quarks. We assume the two strange quarks form a diquark $\delta=ss$, which along with the charm quark make it true for the stable spectra of the $\Omega_{c}^{0}(X)$ system. The baryons mass splitting $\Delta M$ can be estimated as Jaffe:2003sg ; Maiani:2004vq

[TABLE]

where the first two terms are spin-spin interaction between the diquark and charm quarks and inside the diquark. The third and fourth terms are the spin-orbital interactions. The fifth term is the pure orbital interactions. The $\mathbf{S}_{\delta}$ corresponds to the spin operator of diquark. The spin operators of strange quark and charm quark are given by $\mathbf{S}_{s}$ and $\mathbf{S}_{c}$, respectively. The coefficients $(\kappa_{q_{1}q_{2}})_{\bar{3}}$ are the spin-spin couplings for two quarks in color antitriplet, respectively.

Unlike the case in the $\Omega^{-}$ where the total angular momentum $J$ is $3/2$ with $L=0$, the $S$ wave states of the $\Omega_{c}^{0}$ system have two states where the total angular momentum $J$ can be either $1/2$ or $3/2$.

[TABLE]

where $|S_{c},S_{\delta};S_{c\delta};L=0;N_{J}\rangle$ stands for the baryon; the $S_{\delta}$ and $S_{c}$ denote the spin of the diquark $[ss]$ and the charm quark, respectively, and the $N_{J}$ denotes the total angular momentum of the baryon.

There are five $P$ wave states of $\Omega_{c}^{0}$ system with $L=1$ and negative parity

[TABLE]

[TABLE]

There are some simple relations among the $S$ and $P$ wave states of $\Omega_{c}^{0}$ system when using the mass splitting formulae. Their relations are

[TABLE]

where we simply assume $A_{\delta}=A_{c}$.

For convenience, we write the possible states into the corresponding form $|n^{2S+1}L_{J}\rangle$, i.e. $|1^{2}S_{\frac{1}{2}}\rangle=|L=0,\frac{1}{2}_{J}\rangle$, $|1^{4}S_{\frac{3}{2}}\rangle=|L=0,\frac{3}{2}_{J}\rangle$, $|1^{2}P_{\frac{1}{2}}\rangle=|L=1,\frac{1}{2}_{J}\rangle_{1}$, $|1^{4}P_{\frac{1}{2}}\rangle=|L=1,\frac{1}{2}_{J}\rangle_{2}$, $|1^{2}P_{\frac{3}{2}}\rangle=|L=1,\frac{3}{2}_{J}\rangle_{1}$, $|1^{4}P_{\frac{3}{2}}\rangle=|L=1,\frac{3}{2}_{J}\rangle_{2}$, and $|1^{4}P_{\frac{5}{2}}\rangle=|L=1,\frac{5}{2}_{J}\rangle$. Assuming the $\Omega_{c}^{0}(2695)$ is the ground state with $1^{2}S_{\frac{1}{2}}$ and then the $\Omega_{c}^{0}(2695)$ is the lightest state, the mass spectra of the $S$ and $P$ wave states of $\Omega_{c}^{0}(X)$ baryons can be obtained from the relations in Eqs. (10-15).

The coupling constants in Eq. 2 are described in detail in Refs. Jaffe:2004ph ; Maiani:2004vq ; Ali:2009es ; Zhu:2015bba ; Wang:2016tsi ; Zhu:2016arf . In order to give more information of the coupling constants, we extract the coupling constants from the baryon mass relations  Jaffe:2004ph

[TABLE]

Inputting the related charmed baryon masses Olive:2016xmw , i.e. $m_{\Xi_{c}^{*0}}=(2645.9\pm 0.5)$MeV, $m_{\Omega_{c}^{0}}=(2695.2\pm 1.7)$MeV, $m_{\Sigma_{c}^{+}}=(2452.9\pm 0.4)$MeV, and $m_{\Sigma_{c}^{*+}}=(2517.5\pm 2.3)$MeV, the value of the coupling constant $(\kappa_{cs})_{\bar{3}}$ can be extracted as $(\kappa_{cs})_{\bar{3}}=(26\pm 1.5)$MeV.

The parameters $A_{c}$ and $B$ which describe the orbital couplings of the excited states can be estimated by the comparison with the observed spin-orbitally splitting in the $\Xi_{c}^{0}(X)$ states. We have the estimation

[TABLE]

Inputting the related charmed baryon masses Olive:2016xmw , i.e. $m_{\Xi_{c}^{0}(\frac{1}{2}^{+})}=(2470.85^{+0.28}_{-0.40})$MeV, $m_{\Xi_{c}^{0}(\frac{1}{2}^{-})}=(2791.9\pm 3.3)$MeV, and $m_{\Xi_{c}^{0}(\frac{3}{2}^{-})}=(2819.6\pm 1.2)$MeV, the value of the coupling constants can be extracted as $A_{c}(\Omega_{c})=(9\pm 1.5)$MeV and $B(\Omega_{c})=(340\pm 2)$MeV.

Considering the uncertainties of the inputting parameters, the mass spectra of the $S$ and $P$ wave states of $\Omega_{c}^{0}(X)$ baryons are given in Tab. 2. In this table, the assignment of $\Omega_{c}^{0}$ baryons to $|n^{2S+1}L_{J}\rangle$ is by no means conclusive. For instance, the $\Omega_{c}(2695)$ has been assigned as the ground state only due to the fact there is no other lower state that has been established on the experimental side. In Tab. 2 we also list the experimental data and other theoretical predictions. Most of them are based on the potential model, QCD sum rules, and Lattice QCD simulation. Besides, some excited states of $\Omega_{c}^{0}(X)$ baryons are also predicted from meson-baryon unitarization starting from a lowest order potential in Refs. JimenezTejero:2009vq ; Romanets:2012hm , where the existence of a bound state at 2959 MeV, near the lowest threshold, and two resonances placed at 2966 and 3117 MeV are predicted in this scheme. The widths of the two resonances are calculated as $\Gamma(2966)=1.1$ MeV and $\Gamma(3117)=16$ MeV.

The bottom partners of the $\Omega_{c}^{0}(X)$ baryons can also be predicted. Assuming the $\Omega_{b}^{-}(6046)$ with the mass $(6046\pm 1.9)\mathrm{MeV}$ is the lightest state with $1^{2}S_{\frac{1}{2}}$, the spectra of $\Omega_{b}^{-}(X)$ baryons are very similar to that of $\Omega_{c}^{0}(X)$ baryons. Their masses and spin-parities are estimated as

[TABLE]

where the parameters are adopted as $(\kappa_{bs})_{\bar{3}}=25\pm 2$MeV, $A_{b}(\Omega_{b})=5\pm 2$MeV, and $B(\Omega_{b})=408\pm 4$MeV Ali:2011ug ; Hambrock:2011qga ; Ali:2009es . Since the observed spin-orbitally splitting in the $\Xi_{b}^{-}(X)$ states is limited, we only give the approximate error and will discuss the uncertainties of the coupling constants in future works. The mass splitting for the P-wave orbitally excited states is very small. Currently, only the ${\Omega_{b}}(1^{2}S_{\frac{1}{2}})$ has been observed Olive:2016xmw . The S-wave orbitally excited state ${\Omega_{b}}(1^{4}S_{\frac{3}{2}})$ and the five P-wave orbitally excited states can be also reconstructed by the electro-weak decay channel $\Omega^{-}_{b}(X)\to J/\psi+\Omega^{-}$ with the sub-decays $J/\psi\to\mu^{+}\mu^{-}(e^{+}e^{-})$ and $\Omega^{-}\to\Lambda K^{-}(\Xi^{0}\pi^{-})\to p\pi^{-}K^{-}(p\pi^{-}\pi^{0}\pi^{-})$. This can be examined in future.

Decays into $\Xi_{c}K$ and $\Xi_{c}^{{}^{\prime}}K$

In the heavy quark limit, the static heavy quark can only interact with gluons via its chromoelectric charge, which leads to the heavy quark spin symmetry. In this heavy quark limit, the spin of the heavy quark and the light degrees of freedom $\mathbf{S}_{\ell}=\mathbf{J}-\mathbf{S}_{Q}$ with $Q=c,b$ is conserved, respectively. Thus some relations for the strong decays can be obtained.

In the heavy quark limit, the five P-wave baryonic states are given as

[TABLE]

[TABLE]

Apparently, the spin-$5/2$ baryonic state is the same with the one in Eq. (9), while the two spin-$1/2$ and $3/2$ states will mix with each other respectively. The mixing matrix is given as

[TABLE]

for the two spin-1/2 states and

[TABLE]

for the spin-$3/2$ baryons.

In the heavy quark limit, the amplitudes of $\Omega_{c}^{0}(X)\to\Xi_{c}^{+}(\Xi_{c}^{{}^{\prime}+})K^{-}$ can be expressed as

[TABLE]

where the quantum numbers $S_{\ell}$ and $S^{\prime}_{\ell}$ are the spin of the light degrees of freedom in $\Omega_{c}^{0}(X)$ and $\Xi_{c}^{+}(\Xi_{c}^{{}^{\prime}+})$ respectively, the quantum numbers $J$ and $J^{\prime}$ are the total angular momentum of $\Omega_{c}^{0}(X)$ and $\Xi_{c}^{+}(\Xi_{c}^{{}^{\prime}+})$ respectively.

The decay widths of $\Omega_{c}^{0}(X)\to\Xi_{c}^{+}(\Xi_{c}^{{}^{\prime}+})K^{-}$ are proportional to Clebsch-Gordan coefficients

[TABLE]

where the product of Clebsch-Gordan coefficients are in terms of 6j symbols.

For $\Omega_{c}^{0}(X)\to\Xi_{c}^{+}K^{-}$, the quantum numbers are

[TABLE]

We find the following results:

• •

Due to the parity conservation, the decays can proceed through S-wave or D-wave.

• •

Only the lowest-lying state, $|\frac{1}{2}\rangle_{S_{\ell}=0}$ can decays into the $\Xi_{c}K$ in S-wave. The $|\frac{1}{2}\rangle_{S_{\ell}=0}$ may mix with $|\frac{1}{2}\rangle_{S_{\ell}=1}$ in QCD. However we expect that their low masses do not allow a large phase space. So the $1/2$ states will have not large decay widths.

• •

The $|\frac{3}{2}\rangle_{S_{\ell}=2}$ and $|\frac{5}{2}\rangle_{S_{\ell}=2}$ can decays into the $\Xi_{c}K$ through D-wave. For the $|\frac{5}{2}\rangle_{S_{\ell}=2}$, this is guaranteed by the angular momentum conservation, and while the heavy quark symmetry relates the decays of $|\frac{3}{2}\rangle_{S_{\ell}=2}$. Such amplitudes are also suppressed due to the phase space. Thus the total widths are expected to be small again.

• •

The breaking of heavy quark symmetry may induce small contributions to decay widths.

For the channel $\Omega_{c}^{0}(X)\to\Xi_{c}^{{}^{\prime}+}K^{-}$, the related quantum numbers of the initial and final states are

[TABLE]

The following remarks are given in order.

• •

The threshold of $\Xi_{c}^{{}^{\prime}+}K^{-}$ is about 3069 MeV, which prohibits decays of the lower three baryons.

• •

Decays of $\Omega(3090)$ and $\Omega(3119)$ into $\Xi_{c}^{{}^{\prime}+}K^{-}$ have some phase space.

• •

From the 6j symbol, we find the S-wave decay is through $|1/2\rangle_{S_{\ell}=1}\to\Xi_{c}^{\prime}K$. But considering the threshold of $\Xi_{c}^{{}^{\prime}+}K^{-}$ is about 3069 MeV, this will not be kinematically allowed.

• •

There are D-wave decay amplitudes for $|1/2\rangle_{S_{\ell}=1}\to\Xi_{c}^{\prime}K$, $|3/2\rangle_{S_{\ell}=2}\to\Xi_{c}^{\prime}K$, $|5/2\rangle_{S_{\ell}=2}\to\Xi_{c}^{\prime}K$. However these contributions are not big since the phase space is limited.

Since both decays into $\Xi_{c}^{+}K^{-}$ and $\Xi_{c}^{{}^{\prime}+}K^{-}$ are suppressed, the narrowness of the five newly observed $\Omega_{c}$ states can be understood using heavy quark symmetry.

In the heavy-quark-light-diquark model, the decay of $\Omega_{c}$ into $\Xi_{c}^{+}K^{-}$ requests to tear the $ss$ diquark apart, and thus the calculation of the width decay into $\Xi_{c}K$ is beyond the quark-diquark scheme mainly used in this work. A tool to estimate the decay width might be using the flavor SU(3) symmetry to relate to other charmed baryons, for instance $\Gamma(\Lambda_{c}(2595))=(2.6\pm 0.6)$ MeV, $\Gamma(\Lambda_{c}(2625))<0.97$ MeV Olive:2016xmw , $\Xi_{c}^{+}(2645)=(2.1\pm 0.2)$ MeV, $\Xi_{c}^{+}(2790)=(8.9\pm 1.0)$ Yelton:2016fqw . This can give us a hint that the corresponding $\Omega_{c}$ states might be narrow. However a conclusive result requests the classification of the $\Lambda_{c}$ and $\Xi_{c}$ baryons and a more comprehensive analysis to be published in future.

Conclusion

In this work, we have studied the charmed and bottomed baryons with two strange quarks in a quark-diquark model. The two strange quarks lie in S wave and thus their total spin is 1. Within the heavy-quark-light-diquark framework, we calculate the mass spectra of the $S$ and $P$ wave orbitally excited states. We find the $\Omega_{c}^{0}(2695)$ and $\Omega_{c}^{0}(2770)$ fit well as the $S$ wave states of charmed doubly strange baryons. There are five P-wave states. The five newly $\Omega_{c}^{0}$ resonances observed by the LHCb Collaboration, i.e. $\Omega_{c}^{0}(3000)$, $\Omega_{c}^{0}(3050)$, $\Omega_{c}^{0}(3066)$, $\Omega_{c}^{0}(3090)$, and $\Omega_{c}^{0}(3119)$, can be interpreted as the $P$ wave orbitally excited states of charmed doubly strange baryons. We have analyzed their decays into the $\Xi_{c}K$ and $\Xi_{c}^{\prime}K$ in the heavy quark effective theory. We find decays of the five new $\Omega_{c}$ states into the $\Xi_{c}K$ and $\Xi_{c}^{\prime}K$ are suppressed by the heavy quark symmetry or the phase space. The narrowness of the five newly observed $\Omega_{c}$ states can be understood using heavy quark symmetry.

[TABLE]

While this paper was submitted, there are studies of the masses or (and) decay properties of the newly observed $\Omega_{c}^{0}(X)$ states using different approaches: the QCD sum rules Wang:2017zjw ; Zhao:2017fov ; Chen:2017gnu ; Aliev:2017led ; Agaev:2017lip , heavy hadron chiral perturbation theory Cheng:2017ove , the chiral quark-soliton model Kim:2017jpx , and lattice QCD Padmanath:2017lng , the constituent quark models and treatment as pentaquarks Huang:2017dwn ; An:2017lwg .

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant No. 11575110, 11647163, 11655002, and by the Research Start-up Funding (R.L. Zhu) of Nanjing Normal University, by Natural Science Foundation of Shanghai under Grant No. 15ZR1423100 and No. 15DZ2272100, by the Young Thousand Talents Plan, and by Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education.