Analysis of the $\Omega_c(3000)$, $\Omega_c(3050)$, $\Omega_c(3066)$, $\Omega_c(3090)$ and $\Omega_c(3119)$ with QCD sum rules

TL;DR

This paper uses QCD sum rules to analyze the quantum states of several Omega_c baryons, assigning their internal quark configurations and quantum numbers based on theoretical predictions.

Contribution

It introduces a method to assign quantum numbers and internal quark arrangements to Omega_c states using explicit P-wave considerations in QCD sum rules.

Findings

Supports assigning Omega_c(3050), Omega_c(3066), Omega_c(3090), Omega_c(3119) as P-wave states with specific J^P values.

Omega_c(3000) is assigned as a P-wave state with J^P=1/2^- and S-wave between s quarks.

Provides theoretical support for the internal structure of these baryons.

Abstract

In this article, we assign the $\Omega_c(3000)$, $\Omega_c(3050)$, $\Omega_c(3066)$, $\Omega_c(3090)$ and $\Omega_c(3119)$ to be the P-wave baryon states with $J^P={\frac{1}{2}}^-$, ${\frac{1}{2}}^-$, ${\frac{3}{2}}^-$, ${\frac{3}{2}}^-$ and ${\frac{5}{2}}^-$, respectively, and study them with the QCD sum rules by introducing an explicit relative P-wave between the two $s$ quarks. The predictions support assigning the $\Omega_c(3050)$, $\Omega_c(3066)$, $\Omega_c(3090)$ and $\Omega_c(3119)$ to be the P-wave baryon states with $J^P={\frac{1}{2}}^-$, ${\frac{3}{2}}^-$, ${\frac{3}{2}}^-$ and ${\frac{5}{2}}^-$, respectively, where the two $s$ quarks are in relative P-wave; while the $\Omega_c(3000)$ can be assigned to the P-wave baryon state with $J^{P}={\frac{1}{2}}^-$, where the two $s$ quarks are in relative S-wave.

Tables

Table 1. Table 1: The Borel parameters T 2 superscript 𝑇 2 T^{2} , continuum threshold parameters s 0 subscript 𝑠 0 s_{0} , pole contributions (pole) and perturbative contributions (perturbative).

currents	$T^2\left({\mathrm{GeV}}^2\right)$	$\sqrt{s_0}\left(\mathrm{GeV}\right)$	pole	perturbative
$J^1$	$1.8-2.2$	$3.6\pm 0.1$	$\left(43-73\right)\%$	$\left(95-99\right)\%$
$J_{\mu }^1$	$1.8-2.2$	$3.6\pm 0.1$	$\left(42-73\right)\%$	$\left(100-104\right)\%$
$J_{\mu }^2$	$1.9-2.3$	$3.6\pm 0.1$	$\left(43-72\right)\%$	$\left(100-102\right)\%$
$J_{\mu \nu }$	$2.4-2.8$	$3.7\pm 0.1$	$\left(42-66\right)\%$	$\left(94-96\right)\%$

Table 2. Table 2: The masses M 𝑀 M , pole residues λ 𝜆 \lambda and possible assignments of the charmed baryon states, where the j l subscript 𝑗 𝑙 j_{l} denotes the total angular momentum of the light degree of freedom.

currents	$J_{j_l}^P$	$M\left(\mathrm{GeV}\right)$	$\lambda \left({10}^{-1}{\mathrm{GeV}}^4\right)$	assignments
$J^1$	${\frac{1}{2}}_0^{-}$	$3.05\pm 0.11$	$2.34\pm 0.50$	${\Omega }_c\left(3050\right)$
$J_{\mu }^1$	${\frac{3}{2}}_1^{-}$	$3.06\pm 0.11$	$1.03\pm 0.23$	${\Omega }_c\left(3066/3090\right)$
$J_{\mu }^2$	${\frac{3}{2}}_2^{-}$	$3.06\pm 0.10$	$2.47\pm 0.47$	${\Omega }_c\left(3066/3090\right)$
$J_{\mu \nu }$	${\frac{5}{2}}_2^{-}$	$3.11\pm 0.10$	$1.07\pm 0.17$	${\Omega }_c\left(3119\right)$

Table 3. Table 3: The masses of the P-wave Ω c subscript Ω 𝑐 \Omega_{c} baryon states, where the unit is GeV, the j l subscript 𝑗 𝑙 j_{l} denotes the total angular momentum of the light degree of freedom. We neglect the mixing effects of the 1 2 0 − − 1 2 1 − subscript superscript 1 2 0 subscript superscript 1 2 1 {\frac{1}{2}}^{-}_{0}-{\frac{1}{2}}^{-}_{1} and 3 2 1 − − 3 2 2 − subscript superscript 3 2 1 subscript superscript 3 2 2 {\frac{3}{2}}^{-}_{1}-{\frac{3}{2}}^{-}_{2} in the potential quark models for simplicity.

$J_{j_l}^P$	This Work	[4]	[26]	[27]	[28]	[29]	[30]	[31]
${\frac{1}{2}}_0^{-}$	$3.05\pm 0.11$				3.011	2.977	3.035	3.055
${\frac{1}{2}}_1^{-}$		$2.98\pm 0.16$	$2.990\pm 0.129$		3.028	2.990	3.125	2.966
${\frac{3}{2}}_1^{-}$	$3.06\pm 0.11$	$2.98\pm 0.16$	$3.056\pm 0.103$	$3.08\pm 0.12$	2.976	2.986		3.054
${\frac{3}{2}}_2^{-}$	$3.06\pm 0.10$				2.993	2.994		3.029
${\frac{5}{2}}_2^{-}$	$3.11\pm 0.10$				2.947	3.014		3.051

Equations

$${\Omega }_c(3000):M=3000.4\pm 0.2\pm 0.1\text{mbox}MeV,\Gamma =4.5\pm 0.6\pm 0.3\text{mbox}MeV,$$

$${\Omega }_c(3050):M=3050.2\pm 0.1\pm 0.1\text{mbox}MeV,\Gamma =0.8\pm 0.2\pm 0.1\text{mbox}MeV,$$

$${\Omega }_c(3066):M=3065.6\pm 0.1\pm 0.3\text{mbox}MeV,\Gamma =3.5\pm 0.4\pm 0.2\text{mbox}MeV,$$

$${\Omega }_c(3090):M=3090.2\pm 0.3\pm 0.5\text{mbox}MeV,\Gamma =8.7\pm 1.0\pm 0.8\text{mbox}MeV,$$

$${\Omega }_c(3119):M=3119.1\pm 0.3\pm 0.9\text{mbox}MeV,\Gamma =1.1\pm 0.8\pm 0.4\text{mbox}MeV.$$

$$\Pi (p)$$

$${\Pi }_{\mu \nu }(p)$$

$${\Pi }_{\mu \nu \alpha \beta }(p)$$

$$J^1(x)$$

$$J^2(x)$$

$$J_{\mu }^1(x)$$

$$J_{\mu }^2(x)$$

$$J_{\mu \nu }(x)$$

$$⟨0|J(0)|B_{\frac{1}{2}}^{-}(p)⟩$$

$$⟨0|J_{\mu }(0)|B_{\frac{1}{2}}^{+}(p)⟩$$

$$⟨0|J_{\mu }(0)|B_{\frac{3}{2}}^{-}(p)⟩$$

$$⟨0|J_{\mu \nu }(0)|B_{\frac{1}{2}}^{-}(p)⟩$$

$$⟨0|J_{\mu \nu }(0)|B_{\frac{3}{2}}^{+}(p)⟩$$

$$⟨0|J_{\mu \nu }(0)|B_{\frac{5}{2}}^{-}(p)⟩$$

$$⟨0|J(0)|B_{\frac{1}{2}}^{+}(p)⟩$$

$$⟨0|J_{\mu }(0)|B_{\frac{1}{2}}^{-}(p)⟩$$

$$⟨0|J_{\mu }(0)|B_{\frac{3}{2}}^{+}(p)⟩$$

$$⟨0|J_{\mu \nu }(0)|B_{\frac{1}{2}}^{+}(p)⟩$$

$$⟨0|J_{\mu \nu }(0)|B_{\frac{3}{2}}^{-}(p)⟩$$

$$⟨0|J_{\mu \nu }(0)|B_{\frac{5}{2}}^{+}(p)⟩$$

$$\Pi (p)$$

$${\Pi }_{\mu \nu }(p)$$

$${\Pi }_{\mu \nu \alpha \beta }(p)$$

$$\sum _{s}U\overline{U}$$

$$\sum _s{U}_{\mu }{\overline{U}}_{\nu }$$

$$\sum _s{U}_{\mu \nu }{\overline{U}}_{\alpha \beta }$$

$$\Pi (p)$$

$${\Pi }_{\mu \nu }(p)$$

$${\Pi }_{\mu \nu \alpha \beta }(p)$$

$$\frac{\mathrm{Im}{\Pi }_j(s)}{\pi }$$

$${\int }_{m_c^2}^{s_0}ds\left[\sqrt{s}{\rho }_{j,H}^1(s)+{\rho }_{j,H}^0(s)\right]\exp \left(-\frac{s}{T^2}\right)$$

$$\frac{\mathrm{Im}{\Pi }_j(s)}{\pi }$$

$$2M_{-}{{\lambda }_j^{-}}^2\exp \left(-\frac{M_{-}^2}{T^2}\right)$$

$$M_{-}^2$$

$$m_c(\mu )$$

$$m_s(\mu )$$

$${\alpha }_s(\mu )$$

$$\mathrm{per}$$

Full text

PACS number: 14.20.Lq

Key words: Charmed baryon states, QCD sum rules

1 Introduction

In the past years, several new charmed baryon states have been observed, and the spectroscopy of the charmed baryon states have re-attracted much attentions [1], the QCD sum rules plays an important roles in assigning those new baryon states. The masses of the heavy baryon states with $J^{P}={\frac{1}{2}}^{\pm}$, ${\frac{3}{2}}^{\pm}$, ${\frac{5}{2}}^{\pm}$ have been studied with the full QCD sum rules [2, 3, 4, 5] or the QCD sum rules combined with the heavy quark effective theory [6].

Recently, the LHCb collaboration studied the $\Xi_{c}^{+}K^{-}$ mass spectrum with a sample of $pp$ collision data corresponding to an integrated luminosity of $3.3\rm{fb}^{-1}$ collected by the LHCb experiment, and observed five new narrow excited $\Omega_{c}^{0}$ states, $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$, $\Omega_{c}(3119)$ [7]. The measured masses and widths are

[TABLE]

There have been several assignments for those new charmed states. In Ref.[8], the $\Omega_{c}(3066)$ and $\Omega_{c}(3119)$ are assigned to be the 2S $\Omega_{c}^{0}$ states with $J^{P}={\frac{1}{2}}^{+}$ and ${\frac{3}{2}}^{+}$, respectively. In Ref.[9], possible assignments of those $\Omega_{c}^{0}$ states to be the P-wave baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$ are discussed. In Refs.[10, 11, 12], the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ are assigned to be the P-wave baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$, respectively. In Ref.[13], those $\Omega_{c}^{0}$ states are assigned to be the pentaquark states or molecular pentaquark states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$ or ${\frac{5}{2}}^{-}$. In Ref.[14], the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$ and $\Omega_{c}(3090)$ are assigned to be the P-wave baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{1}{2}}^{-}$, respectively. In Ref.[15], the $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ are assigned to be the 2S $\Omega_{c}^{0}$ states with $J^{P}={\frac{1}{2}}^{+}$ and ${\frac{3}{2}}^{+}$, respectively, while the $\Omega_{c}(3000)$, $\Omega_{c}(3066)$ and $\Omega_{c}(3050)$ are assigned to be the P-wave baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$, respectively.

In this article, we tentatively assign the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ to be the P-wave baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$, respectively, and study their masses and pole residues with the QCD sum rules in details.

The ground state quarks have the spin-parity ${1\over 2}^{+}$, two quarks can form a scalar diquark or an axialvector diquark with the spin-parity $0^{+}$ or $1^{+}$, the diquark then combines with a third quark to form a positive parity baryon with spin ${1\over 2}$ or ${3\over 2}$. We can construct the baryon currents $\eta$ and $\eta_{\mu}$ with positive parity without introducing additional P-wave. As multiplying $i\gamma_{5}$ to the baryon currents changes their parity, the currents $i\gamma_{5}\eta$ and $i\gamma_{5}\eta_{\mu}$ couple potentially to the negative parity heavy baryon states. In Refs.[4, 5], we construct the currents without introducing relative P-wave to study the negative parity heavy, doubly-heavy and triply-heavy baryon states, and obtain satisfactory results. The predictions $M=2.98\pm 0.16\,\rm{GeV}$ for the $\Omega_{c}^{0}$ states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$ are consistent with the masses of the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ from the LHCb collaboration [4].

In Ref.[16], we construct the interpolating currents by introducing the relative P-wave explicitly, study the negative parity charmed baryon states $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ with the full QCD sum rules, and reproduce the experimental values of the masses. In this article, we extend our previous work to study the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ with QCD sum rules by introducing the relative P-wave explicitly.

The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the $\Omega_{c}^{0}$ states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusions.

2 QCD sum rules for the $\Omega_{c}^{0}$ states

In the following, we write down the two-point correlation functions $\Pi(p)$, $\Pi_{\mu\nu}(p)$, $\Pi_{\mu\nu\alpha\beta}(p)$ in the QCD sum rules,

[TABLE]

where $J(x)=J^{1}(x),\,J^{2}(x)$, $J_{\mu}(x)=J^{1}_{\mu}(x),\,J^{2}_{\mu}(x)$,

[TABLE]

$\widetilde{g}_{\mu\nu}=g_{\mu\nu}-\frac{1}{4}\gamma_{\mu}\gamma_{\nu}$, the $i$, $j$, $k$ are color indices, the $C$ is the charge conjugation matrix. We construct the currents with the light diquarks $S^{i}_{\mu\nu}=\varepsilon^{ijk}\left[\partial_{\mu}s^{T}_{i}C\gamma_{\nu}s_{j}+s^{T}_{i}C\gamma_{\nu}\partial_{\mu}s_{j}\right]$. The $S^{i}_{\mu\nu}$ have two Lorentz indices $\mu$ and $\nu$, but they are neither symmetric nor anti-symmetric when interchanging the indices $\mu$ and $\nu$. The scalar components $S^{i}_{\mu\nu}g^{\mu\nu}$ and $S^{i}_{\mu\nu}\sigma^{\mu\nu}$ couple potentially to the spin-0 diquarks. The Dirac matrixes $\widetilde{g}^{\alpha\mu}\gamma^{\nu}-\widetilde{g}^{\alpha\nu}\gamma^{\mu}$ and $g^{\alpha\mu}\gamma^{\nu}+g^{\alpha\nu}\gamma^{\mu}-\frac{1}{2}g^{\mu\nu}\gamma^{\alpha}$ are anti-symmetric and symmetric respectively when interchanging the indices $\mu$ and $\nu$, the vector components $S^{i}_{\mu\nu}\left(\widetilde{g}^{\alpha\mu}\gamma^{\nu}-\widetilde{g}^{\alpha\nu}\gamma^{\mu}\right)$ and $S^{i}_{\mu\nu}\left(g^{\alpha\mu}\gamma^{\nu}+g^{\alpha\nu}\gamma^{\mu}-\frac{1}{2}g^{\mu\nu}\gamma^{\alpha}\right)$ couple potentially to the spin-1 diquarks. The symmetric components $S^{i}_{\mu\nu}+S^{i}_{\nu\mu}$ couple potentially to the spin-0 and 2 diquarks. So we choose the currents $J(x)$, $J_{\mu}(x)$ and $J_{\mu\nu}(x)$ to study the spin-$\frac{1}{2}$, $\frac{3}{2}$ and $\frac{5}{2}$ baryon states, respectively.

The currents $J(0)$, $J_{\mu}(0)$ and $J_{\mu\nu}(0)$ couple potentially to the ${\frac{1}{2}}^{-}$, ${\frac{1}{2}}^{+}$, ${\frac{3}{2}}^{-}$ and ${\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{+}$, ${\frac{5}{2}}^{-}$ charmed baryon states $B_{\frac{1}{2}}^{-}$, $B_{\frac{1}{2}}^{+}$, $B_{\frac{3}{2}}^{-}$ and $B_{\frac{1}{2}}^{-}$, $B_{\frac{3}{2}}^{+}$, $B_{\frac{5}{2}}^{-}$, respectively,

[TABLE]

On the other hand, the currents $J(0)$, $J_{\mu}(0)$ and $J_{\mu\nu}(0)$ couple potentially to the ${\frac{1}{2}}^{+}$, ${\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{+}$ and ${\frac{1}{2}}^{+}$, ${\frac{3}{2}}^{-}$, ${\frac{5}{2}}^{+}$ charmed baryon states $B_{\frac{1}{2}}^{+}$, $B_{\frac{1}{2}}^{-}$, $B_{\frac{3}{2}}^{+}$ and $B_{\frac{1}{2}}^{+}$, $B_{\frac{3}{2}}^{-}$, $B_{\frac{5}{2}}^{+}$, respectively [19, 20],

[TABLE]

The spinors $U^{\pm}(p,s)$ satisfy the Dirac equations $(\not\!\!p-M_{\pm})U^{\pm}(p)=0$, while the spinors $U^{\pm}_{\mu}(p,s)$ and $U^{\pm}_{\mu\nu}(p,s)$ satisfy the Rarita-Schwinger equations $(\not\!\!p-M_{\pm})U^{\pm}_{\mu}(p)=0$ and $(\not\!\!p-M_{\pm})U^{\pm}_{\mu\nu}(p)=0$, and the relations $\gamma^{\mu}U^{\pm}_{\mu}(p,s)=0$, $p^{\mu}U^{\pm}_{\mu}(p,s)=0$, $\gamma^{\mu}U^{\pm}_{\mu\nu}(p,s)=0$, $p^{\mu}U^{\pm}_{\mu\nu}(p,s)=0$, $U^{\pm}_{\mu\nu}(p,s)=U^{\pm}_{\nu\mu}(p,s)$. The $\lambda^{\pm}_{\frac{1}{2}/\frac{3}{2}/\frac{5}{2}}$, $f^{\pm}_{\frac{1}{2}/\frac{3}{2}}$ and $g^{\pm}_{\frac{1}{2}}$ are the pole residues or current-baryon coupling constants.

At the phenomenological side, we insert a complete set of intermediate charmed baryon states with the same quantum numbers as the current operators $J(x)$, $i\gamma_{5}J(x)$, $J_{\mu}(x)$, $i\gamma_{5}J_{\mu}(x)$, $J_{\mu\nu}(x)$ and $i\gamma_{5}J_{\mu\nu}(x)$ into the correlation functions $\Pi(p)$, $\Pi_{\mu\nu}(p)$ and $\Pi_{\mu\nu\alpha\beta}(p)$ to obtain the hadronic representation [17, 18]. After isolating the pole terms of the lowest states of the charmed baryon states, we obtain the following results:

[TABLE]

[TABLE]

where $\widetilde{g}_{\mu\nu}=g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}$. In calculations, we have used the following summations [21],

[TABLE]

and $p^{2}=M^{2}_{\pm}$ on the mass-shell.

We can rewrite the correlation functions $\Pi(p)$, $\Pi_{\mu\nu}(p)$ and $\Pi_{\mu\nu\alpha\beta}(p)$ into the following form according to Lorentz covariance,

[TABLE]

In this article, we choose the tensor structures $g_{\mu\nu}$ and $g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha}$ for analysis, and separate the contributions of the ${\frac{3}{2}}^{\pm}$ and ${\frac{5}{2}}^{\pm}$ charmed baryon states unambiguously. For detailed discussions on this subject, one can consult Ref.[20].

We obtain the hadronic spectral densities at phenomenological side through the dispersion relation,

[TABLE]

where $j=\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$, the subscript $H$ denotes the hadron side, then we introduce the weight function $\exp\left(-\frac{s}{T^{2}}\right)$ to obtain the QCD sum rules at the phenomenological side,

[TABLE]

where the $s_{0}$ are the continuum thresholds and the $T^{2}$ are the Borel parameters [20].

At the QCD side, we calculate the light quark parts of the correlation functions $\Pi(p)$, $\Pi_{\mu\nu}(p)$, $\Pi_{\mu\nu\alpha\beta}(p)$ with the full light quark propagators in the coordinate space and take the momentum space expression for the full $c$-quark propagator. It is straightforward but tedious to compute the integrals both in the coordinate and momentum spaces to obtain the correlation functions $\Pi_{j}(p^{2})$, therefore the QCD spectral densities through the dispersion relation,

[TABLE]

where $j=\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$, the explicit expressions of the QCD spectral densities $\rho^{1}_{j,QCD}(s)$ and $\rho^{0}_{j,QCD}(s)$ are neglected for simplicity. In this article, we carry out the operator product expansion up to the vacuum condensates of dimension 10 and take into account the vacuum condensates $\langle\bar{s}s\rangle$, $\langle\frac{\alpha_{s}GG}{\pi}\rangle$, $\langle\bar{s}g_{s}\sigma Gs\rangle$, $\langle\bar{s}s\rangle\langle\bar{s}g_{s}\sigma Gs\rangle$, $\langle\bar{s}g_{s}\sigma Gs\rangle^{2}$.

Once the analytical QCD spectral densities $\rho^{1}_{j,QCD}(s)$ and $\rho^{0}_{j,QCD}(s)$ are obtained, we can take the quark-hadron duality below the continuum thresholds $s_{0}$ and introduce the weight function $\exp\left(-\frac{s}{T^{2}}\right)$ to obtain the QCD sum rules:

[TABLE]

We derive Eq.(22) with respect to $\frac{1}{T^{2}}$, then eliminate the pole residues $\lambda^{-}_{j}$ and obtain the QCD sum rules for the masses of the charmed baryon states,

[TABLE]

3 Numerical results and discussions

The vacuum condensates are taken to be the standard values $\langle\bar{q}q\rangle=-(0.24\pm 0.01\,\rm{GeV})^{3}$, $\langle\bar{s}s\rangle=(0.8\pm 0.1)\langle\bar{q}q\rangle$, $\langle\bar{s}g_{s}\sigma Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle$, $m_{0}^{2}=(0.8\pm 0.1)\,\rm{GeV}^{2}$, $\langle\frac{\alpha_{s}GG}{\pi}\rangle=(0.33\,\rm{GeV})^{4}$ at the energy scale $\mu=1\,\rm{GeV}$ [17, 18, 22]. The quark condensate and mixed quark condensate evolve with the renormalization group equation, $\langle\bar{s}s\rangle(\mu)=\langle\bar{s}s\rangle(Q)\left[\frac{\alpha_{s}(Q)}{\alpha_{s}(\mu)}\right]^{\frac{4}{9}}$, $\langle\bar{s}g_{s}\sigma Gs\rangle(\mu)=\langle\bar{s}g_{s}\sigma Gs\rangle(Q)\left[\frac{\alpha_{s}(Q)}{\alpha_{s}(\mu)}\right]^{\frac{2}{27}}$.

In the article, we take the $\overline{MS}$ masses $m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ and $m_{s}(\mu=2\,\rm{GeV})=(0.095\pm 0.005)\,\rm{GeV}$ from the particle data group [1], and take into account the energy-scale dependence of the $\overline{MS}$ masses from the renormalization group equation,

[TABLE]

where $t=\log\frac{\mu^{2}}{\Lambda^{2}}$, $b_{0}=\frac{33-2n_{f}}{12\pi}$, $b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$, $b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$, $\Lambda=213\,\rm{MeV}$, $296\,\rm{MeV}$ and $339\,\rm{MeV}$ for the flavors $n_{f}=5$, $4$ and $3$, respectively [1].

In Refs.[23, 24], we study the acceptable energy scales of the QCD spectral densities for the hidden-charm (bottom) tetraquark states and molecular states in the QCD sum rules for the first time, and suggest an empirical formula $\mu=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{Q})^{2}}$ to determine the optimal energy scales, where the $X$, $Y$, $Z$ denote the four-quark states, and the ${\mathbb{M}}_{Q}$ is the effective heavy quark mass. The empirical energy scale formula also works well in studying the hidden-charm pentaquark states [20]. In Ref.[16], we use the diquark-quark model to construct the interpolating currents, and take the analogous formula $\mu=\sqrt{M_{\Lambda_{c}/\Xi_{c}}^{2}-{\mathbb{M}}_{c}^{2}}$ to determine the energy scales of the QCD spectral densities of the QCD sum rules for the charmed baryon states $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$, and obtain satisfactory results. In this article, we use the formula $\mu=\sqrt{M_{\Omega_{c}}^{2}-{\mathbb{M}}_{c}^{2}}$ to determine the energy scales of the QCD spectral densities. If we take the updated value ${\mathbb{M}}_{c}=1.82\,\rm{GeV}$ [25], then $\mu\approx 2.5\,\rm{GeV}$. In calculations, we set the energy scales of the QCD spectral densities to be $\mu=2.5\,\rm{GeV}$.

Now we search for the Borel parameters $T^{2}$ and continuum threshold parameters $s_{0}$ to satisfy the following three criteria:

$\bf{1_{\cdot}}$ Pole dominance at the phenomenological side;

$\bf{2_{\cdot}}$ Convergence of the operator product expansion;

$\bf{3_{\cdot}}$ Appearance of the Borel platforms.

In calculations, we observe that no stable QCD sum rules can be obtained for the current $J^{2}(x)$. The resulting Borel parameters $T^{2}$, continuum threshold parameters $s_{0}$, pole contributions and perturbative contributions (per) are shown explicitly in Table 1, where the perturbative contributions are defined by

[TABLE]

the $\rho_{per}(s)$ and $\rho_{tot}(s)$ denote the perturbative and total QCD spectral densities, respectively. From the Table, we can see that the criteria $\bf 1$ and $\bf 2$ can be satisfied.

We take into account all uncertainties of the relevant parameters, and obtain the values of the masses and pole residues of the $\Omega_{c}^{0}$ baryon states, which are shown in Figs.1-2 and Table 2. In Figs.1-2, we plot the masses and pole residues with variations of the Borel parameters at much larger intervals than the Borel windows shown in Table 1. In the Borel windows, the uncertainties originate from the Borel parameters in the Borel windows are very small, $\delta M_{\Omega_{c}}/M_{\Omega_{c}}=(1.2-1.6)\%$, the criterion $\bf 3$ is also satisfied. The three criteria are all satisfied, we expect to make reliable predictions. In Figs.1-2 and Table 2, we also present the possible assignments of the $\Omega_{c}^{0}$ states according to the masses.

In Ref.[4], we choose the currents without introducing relative P-wave to study the negative parity heavy and doubly-heavy baryon states, and obtain the predictions $M=2.98\pm 0.16\,\rm{GeV}$ for the $\Omega_{c}^{0}$ states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, where the diquark constituent $\varepsilon^{ijk}s^{T}_{j}C\gamma_{\mu}s_{k}$ is taken to construct the currents. Multiplying $i\gamma_{5}$ to the baryon currents changes their parity, we can choose currents without introducing relative P-wave to study the P-wave baryon states. The current $\varepsilon^{ijk}s^{T}_{j}C\gamma_{\mu}s_{k}\gamma^{\mu}c_{k}$ couples potentially to the $\Omega_{c}^{0}$ state with $J^{P}={\frac{1}{2}}^{-}$ [4], the mass of the $\Omega_{c}(3000)$ is in excellent agreement with the prediction $M=2.98\pm 0.16\,\rm{GeV}$ [4] or the prediction $M=2.990\pm 0.129\,\rm{GeV}$ based on a more general interpolating current with additional parameter [26], the $\Omega_{c}(3000)$ can be assigned to the P-wave charmed baryon state with $J^{P}={\frac{1}{2}}^{-}$, where two $s$ quarks are in relative S-wave. In Table 3, we present some predictions for the masses of the P-wave $\Omega_{c}^{0}$ baryon states from the full QCD sum rules [4, 26, 27] and potential quark models [28, 29, 30, 31]. We cannot identify a baryon state unambiguously with the mass alone, it is necessary to study the decay widths of those P-wave baryon states with the QCD sum rules. In Ref.[26], Agaev, Azizi and Sundu study the masses and widths of the 1P ${\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and 2S ${\frac{1}{2}}^{+}$, ${\frac{3}{2}}^{+}$ $\Omega_{c}^{0}$ baryon states with the full QCD sum rules, and assign the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$ and $\Omega_{c}(3119)$ to be the $\Omega_{c}^{0}$ baryon states with the quantum numbers $({\rm 1P},\,\frac{1}{2}^{-})$, $({\rm 1P},\,\frac{3}{2}^{-})$ and $({\rm 2S},\,\frac{3}{2}^{+})$, respectively, and assign the $\Omega_{c}(3066)$ or $\Omega_{c}(3090)$ to be the $\Omega_{c}^{0}$ baryon state with the quantum numbers $({\rm 2S},\,\frac{1}{2}^{+})$.

4 Conclusion

In this article, we assign the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ to be the P-wave charmed baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$, respectively, and study their masses and pole residues with the QCD sum rules in details by introducing an explicit relative P-wave between the two constituents of the light diquarks. The predictions support assigning the $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ to be the P-wave baryon states with $J^{P}={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$, respectively, where the two constituents of the light diquark are in relative P-wave; while the $\Omega_{c}(3000)$ can be assigned to the P-wave charmed baryon state with $J^{P}={\frac{1}{2}}^{-}$, where the two constituents of the light diquark are in relative S-wave.

Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number 11375063.