On the nature of the newly discovered $\Omega_c^{0}$ states

TL;DR

This paper uses QCD sum rules to calculate the properties of excited heavy baryons, specifically the $ ext{Omega}_c^{0}$ and $ ext{Omega}_b^{-}$ states, and compares results with recent LHCb observations.

Contribution

It provides the first detailed QCD sum rule analysis of the radially excited $ ext{Omega}_c^{0}$ and $ ext{Omega}_b^{-}$ baryons, including vacuum condensates up to ten dimensions.

Findings

Calculated masses and residues of excited $ ext{Omega}_c^{0}$ and $ ext{Omega}_b^{-}$ baryons.

Made a comparison with recently observed narrow excited states by LHCb.

Enhanced understanding of heavy baryon spectroscopy.

Abstract

The masses and residues of the radially excited heavy $\Omega_c^{0}$ and $ \Omega_b^{-}$ baryons with spin-parity $J^{P}=\frac{1}{2}^{+}$ and $J^{P}= \frac{3}{2}^{+}$ are calculated by means of QCD two-point sum rule method using the general form of their interpolating currents. In calculations the quark, gluon and mixed vacuum condensates up to ten dimensions are taken into account. In $\Omega_c^{0}$ channel a comparison is made with the narrow excited states recently observed by the LHCb Collaboration.

Tables

Table 1. Table 1: Parameters used in numerical computations.

Parameters	Values
$m_b$	${4.18}_{-0.03}^{+0.04}\mathrm{GeV}$
$m_c$	$\left(1.27\pm 0.03\right)\mathrm{GeV}$
$m_s$	${96}_{-4}^{+8}\mathrm{MeV}$
$⟨\overline{q}q⟩$	$-{\left(0.24\pm 0.01\right)}^3$ ${\mathrm{GeV}}^3$
$⟨\overline{s}s⟩$	$0.8⟨\overline{q}q⟩$
$m_0^2$	$\left(0.8\pm 0.1\right)$ ${\mathrm{GeV}}^2$
$⟨\overline{s}{g}_s\sigma Gs⟩$	$m_0^2⟨\overline{s}s⟩$
$⟨\frac{{\alpha }_s{G}^2}{\pi }⟩$	$\left(0.012\pm 0.004\right)$ ${\mathrm{GeV}}^4$

Table 2. Table 2: The sum rule results for the masses and residues of the 1 ​ S 1 𝑆 1S and 2 ​ S 2 𝑆 2S charmed baryons with the spin-parity 1 / 2 + 1 superscript 2 1/2^{+} and 3 / 2 + 3 superscript 2 3/2^{+} .

$(n,J^P)$	$(1S,{\frac{1}{2}}^{+})$	$(1S,{\frac{3}{2}}^{+})$	$(2S,{\frac{1}{2}}^{+})$	$(2S,{\frac{3}{2}}^{+})$
$M^2(Ge{V}^2$)	$3.5-5.5$	$3.5-5.5$	$3.5-5.5$	$3.5-5.5$
$s_0(Ge{V}^2$)	${3.0}^2-{3.2}^2$	${3.1}^2-{3.3}^2$	${3.5}^2-{3.7}^2$	${3.6}^2-{3.8}^2$
$m_{{\Omega }_c}\left(MeV\right)$	$2685\pm 123$	$2769\pm 89$	$3066\pm 138$	$3119\pm 114$
${\lambda }_{{\Omega }_c}\left(Ge{V}^3\right)$	$\left(6.2\pm 1.8\right)\cdot {10}^{-2}$	$\left(7.1\pm 1.0\right)\cdot {10}^{-2}$	$\left(13.4\pm 1.3\right)\cdot {10}^{-2}$	$\left(17.7\pm 2.4\right)\cdot {10}^{-2}$

Table 3. Table 3: The m Ω b subscript 𝑚 subscript Ω 𝑏 m_{\Omega_{b}} and λ Ω b subscript 𝜆 subscript Ω 𝑏 \lambda_{\Omega_{b}} of the ground-state and radially excited bottom baryons with J P = 1 / 2 + superscript 𝐽 𝑃 1 superscript 2 J^{P}=1/2^{+} and J P = 3 / 2 + superscript 𝐽 𝑃 3 superscript 2 J^{P}=3/2^{+} .

$(n,J^P)$	$(1S,{\frac{1}{2}}^{+})$	$(1S,{\frac{3}{2}}^{+})$	$(2S,{\frac{1}{2}}^{+})$	$(2S,{\frac{3}{2}}^{+})$
$M^2(Ge{V}^2$)	$6.5-9.5$	$6.5-9.5$	$6.5-9.5$	$6.5-9.5$
$s_0(Ge{V}^2$)	${6.3}^2-{6.5}^2$	${6.4}^2-{6.6}^2$	${6.8}^2-{7.0}^2$	${6.9}^2-{7.1}^2$
$m_{{\Omega }_b}\left(MeV\right)$	$6024\pm 65$	$6084\pm 84$	$6325\pm 92$	$6412\pm 108$
${\lambda }_{{\Omega }_b}\left(Ge{V}^3\right)$	$\left(12.1\pm 1.2\right)\cdot {10}^{-2}$	$\left(9.3\pm 1.4\right)\cdot {10}^{-2}$	$\left(20.9\pm 2.5\right)\cdot {10}^{-2}$	$\left(23.6\pm 3.4\right)\cdot {10}^{-2}$

Table 4. Table 4: The theoretical predictions for masses of the ground-state and first radially excited 1 / 2 + 1 superscript 2 1/2^{+} and 3 / 2 + 3 superscript 2 3/2^{+} charmed baryons calculated in the context of the different models and approaches.

${\Omega }_c(1S,{\frac{1}{2}}^{+})$	${\Omega }_c^{\prime }(2S,{\frac{1}{2}}^{+})$	${\Omega }_c^{\star }(1S,{\frac{3}{2}}^{+})$	${\Omega }_c^{\star \prime }(2S,{\frac{3}{2}}^{+})$
($MeV$)	($MeV$)	($MeV$)	($MeV$)
$2685\pm 123$	$3066\pm 138$	$2769\pm 89$	$3119\pm 114$	this work
$2698$	$3088$	$2768$	$3123$	Ref. Ebert:2011kk
$2699$	$3159$	$2767$	$-$	Ref. Valcarce:2008dr
$2718$	$3152$	$2776$	$-$	Ref. Roberts:2007ni
$2720\pm 180$	$-$	$2760\pm 100$	$-$	Refs. Wang:2007sqa ; Wang:2008hz

Table 5. Table 5: The various theoretical predictions for masses of the ground-state and first radially excited 1 / 2 + 1 superscript 2 1/2^{+} and 3 / 2 + 3 superscript 2 3/2^{+} bottom baryons.

${\Omega }_b(1S,{\frac{1}{2}}^{+})$	${\Omega }_b^{\prime }(2S,{\frac{1}{2}}^{+})$	${\Omega }_b^{\star }(1S,{\frac{3}{2}}^{+})$	${\Omega }_b^{\star \prime }(2S,{\frac{3}{2}}^{+})$
($MeV$)	($MeV$)	($MeV$)	($MeV$)
$6024\pm 65$	$6325\pm 92$	$6084\pm 84$	$6412\pm 108$	this work
$6064$	$6450$	$6088$	$6461$	Ref. Ebert:2011kk
$6056$	$6479$	$6079$	$-$	Ref. Valcarce:2008dr
$6081$	$6472$	$6102$	$-$	Ref. Roberts:2007ni
$6130\pm 120$	$-$	$6060\pm 130$	$-$	Refs. Wang:2009cr ; Wang:2008hz

Equations

$$m=2695.2\pm 1.7\mathrm{MeV},m=2765.9\pm 2.0\mathrm{MeV}.$$

$${\Pi }_{(\mu \nu )}(p)=i\int d^{4}x{e}^{ipx}⟨0|\mathcal{T}\{J_{(\mu )}(x){\bar{J}}_{(\nu )}(0)\}|0⟩,$$

$$J$$

$$J_{\mu }$$

$${\Pi }^{\mathrm{Phys}}(p)=\frac{⟨0|J|{\Omega }_Q⟩⟨{\Omega }_Q|\bar{J}|0⟩}{m^2-p^2}+\frac{⟨0|J|{\Omega }_Q^{\prime }⟩⟨{\Omega }_Q^{\prime }|\bar{J}|0⟩}{m^{\prime 2}-p^2}+\dots ,$$

$$⟨0|J|{\Omega }_Q^{(\prime )}(p,s)⟩={\lambda }^{(\prime )}{u}^{(\prime )}(p,s),$$

$$\sum _s{u}^{(\prime )}(p,s){\overline{u}}^{(\prime )}(p,s)=/p+m^{(\prime )},$$

$${\Pi }^{\mathrm{Phys}}(p)=\frac{{\lambda }^2(/p+m)}{m^2-p^2}+\frac{{\lambda }^{\prime 2}(/p+m^{\prime })}{m^{\prime 2}-p^2}+\dots$$

$$\mathcal{B}{\Pi }^{\mathrm{Phys}}(p)={\lambda }^2{e}^{-m^2/M^2}(/p+m)+{\lambda }^{\prime 2}{e}^{-m^{\prime 2}/M^2}(/p+m^{\prime }).$$

$${\lambda }^2{e}^{-m^2/M^2}+{\lambda }^{\prime 2}{e}^{-m^{\prime 2}/M^2}$$

$${\lambda }^{2}m{e}^{-m^2/M^2}+{\lambda }^{\prime 2}{m}^{\prime }{e}^{-m^{\prime 2}/M^2}$$

$$m^{\prime 2}=\frac{d/d(-1/M^2)\left[\mathcal{B}{\Pi }_2^{\mathrm{OPE}}(p)-m\mathcal{B}{\Pi }_1^{\mathrm{OPE}}(p)\right]}{\left[\mathcal{B}{\Pi }_2^{\mathrm{OPE}}(p)-m\mathcal{B}{\Pi }_1^{\mathrm{OPE}}(p)\right]},$$

$${\lambda }^{\prime 2}=\frac{e^{m^{\prime 2}/M^2}}{m^{\prime }-m}\left[\mathcal{B}{\Pi }_2^{\mathrm{OPE}}(p)-m\mathcal{B}{\Pi }_1^{\mathrm{OPE}}(p)\right].$$

$$⟨0|J_{\mu }|{\Omega }_Q^{\ast (\prime )}(p,s)⟩={\lambda }^{(\prime )}{u}_{\mu }^{(\prime )}(p,s),$$

$$\sum _s{u}_{\mu }^{(\prime )}(p,s){\overline{u}}_{\nu }^{(\prime )}(p,s)=-(/p+m^{(\prime )})[g_{\mu \nu }-\frac{1}{3}{\gamma }_{\mu }{\gamma }_{\nu }$$

$$-\frac{2}{3m^{(\prime )2}}{p}_{\mu }{p}_{\nu }+\frac{1}{3m^{(\prime )}}(p_{\mu }{\gamma }_{\nu }-p_{\nu }{\gamma }_{\mu })].$$

$$S_s^{ab}(x)=i{\delta }_{ab}\frac{/x}{2{\pi }^2{x}^4}-{\delta }_{ab}\frac{m_s}{4{\pi }^2{x}^2}-{\delta }_{ab}\frac{⟨\overline{s}s⟩}{12}$$

$$+i{\delta }_{ab}\frac{/x{m}_s⟨\overline{s}s⟩}{48}-{\delta }_{ab}\frac{x^2}{192}⟨\overline{s}{g}_s\sigma Gs⟩+i{\delta }_{ab}\frac{x^2/x{m}_s}{1152}$$

$$\times ⟨\overline{s}{g}_s\sigma Gs⟩-i\frac{g_s{G}_{ab}^{\alpha \beta }}{32{\pi }^2{x}^2}\left[/x{\sigma }_{\alpha \beta }+{\sigma }_{\alpha \beta }/x\right]$$

$$-i{\delta }_{ab}\frac{x^2/x{g}_s^2⟨\overline{s}s{⟩}^2}{7776}-{\delta }_{ab}\frac{x^4⟨\overline{s}s⟩⟨g_s^2{G}^2⟩}{27648}+\dots$$

$$\displaystyle S_{Q}^{ab}(x)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\Bigg{\{}\frac{\delta_{ab}\left({\not{k}}+m_{Q}\right)}{k^{2}-m_{Q}^{2}}$$

$$-\frac{g_s{G}_{ab}^{\alpha \beta }}{4}\frac{{\sigma }_{\alpha \beta }\left(/k+m_Q\right)+\left(/k+m_Q\right){\sigma }_{\alpha \beta }}{(k^2-m_Q^2{)}^2}$$

$$+\frac{g_s^2{G}^2}{12}{\delta }_{ab}{m}_Q\frac{k^2+m_Q/k}{(k^2-m_Q^2{)}^4}+\dots \}.$$

$${\Pi }_{1(2)}^{\mathrm{OPE}}(p)=\underset{(m_Q+2m_s{)}^2}{\overset{s_0}{\int }}\frac{ds{\rho }_{1(2)}^{\mathrm{QCD}}(s)}{s-p^2},$$

$$-0.75\le \cos \theta \le -0.45,0.45\le \cos \theta \le 0.75,$$

Full text

Recently, the LHCb Collaboration reported on discovery of new five narrow states $\Omega_{c}^{0}$ in the $\Xi_{c}^{+}K^{-}$ invariant mass distribution based on the $pp$ collision data at center-of-mass energies $7$, $8$ and $13$ $\mathrm{TeV}$ with an integrated $3.3b^{-1}$ luminosity LHCb . The masses of the $\Omega_{c}^{0}$ states were measured and found to be equal to (in $\mathrm{MeV}$) $M=3000;\,\,\,3050;\,\,\,3066;\,\,\,3090;\,\,\ 3119$. The LHCb determined also their widths through $\Omega_{c}^{0}\rightarrow\Xi_{c}^{+}K^{-}$ decay channels, which did not exceed a few $\mathrm{MeV}$.

Till now the experimental information on the spectrum of the charmed baryons was limited by the $\Omega_{c}^{0}$ and $\Omega_{c}(2770)^{0}$ particles with the masses Olive:2016xmw

[TABLE]

They are presumably the ground states $\Omega_{c}^{0}$ and $\Omega_{c}^{0\star}$ with spin-parity $J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$, respectively .

Theoretical investigations of the charmed (in general, heavy flavored) baryons, on the contrary embrace variety of models and methods. The spectra of the ground and excited states of the charmed baryons were analyzed in the context of the QCD sum rule method Bagan:1991sc ; Bagan:1992tp ; Huang:2000tn ; Wang:2002ts ; Wang:2007sqa ; Wang:2008hz ; Wang:2009cr ; Chen:2015kpa ; Chen:2016phw , different relativistic and non-relativistic quark models Capstick:1986bm ; Ebert:2007nw ; Ebert:2011kk ; Garcilazo:2007eh ; Valcarce:2008dr ; Roberts:2007ni ; Vijande:2012mk ; Yoshida:2015tia ; Shah:2016nxi , the Heavy Quark Effective Theory (HQET) Chiladze:1997ev , and in lattice simulations Padmanath:2013bla . The masses and magnetic moments, radiative decays, various strong couplings and transitions of the heavy flavored baryons were subject of rather intensive theoretical studies, as well Aliev:2008sk ; Aliev:2009jt ; Aliev:2010nh ; Aliev:2010ev ; Aliev:2011kn ; Aliev:2011ufa ; Aliev:2011uf . Some of these theoretical works were carried out using additional assumptions on the structure of the charmed (bottom) baryons. For example, in the relativistic quark model they were considered as the heavy-quark-light-diquark bound states Ebert:2007nw ; Ebert:2011kk . In other papers, QCD sum rule calculations aimed to evaluate spectroscopic parameters of the charmed baryons were performed in the framework of HQET Huang:2000tn ; Wang:2002ts ; Chen:2015kpa ; Chen:2016phw .

New experimental situation emerged due to the discovery of the LHCb Collaboration, necessities a more detailed investigation of charmed (bottom) baryons and their spectra. In the present Letter we will calculate the masses and pole residues of the radially excited $2S$ charmed (bottom) $1/2^{+}$ and $3/2^{+}$ baryons in the framework of the QCD full two-point sum rule approach by employing the most general form of the interpolating currents without any restricting suggestions about their internal organization. We are going to follow a scheme applied to calculate the masses and residues of the radially excited octet and decuplet baryons in Refs. Aliev:2016jnp ; Aliev:2016adl . In these works, the authors get results, which are comparable with existing experimental data on the masses of the radially excited baryons, and demonstrate that the QCD sum rule method can be successfully applied to investigate the radially excited baryons besides their ground-states.

In order to derive the sum rules for the mass and residue of the radially excited $1/2^{+}$ and $3/2^{+}$ baryons we start from the two-point correlation function

[TABLE]

where $J(x)$ is the interpolating current for the $\Omega_{c}^{0}$ ($\Omega_{b}^{-}$) states with $J^{P}=1/2^{+}$. It is given by the expression

[TABLE]

In the case of the $3/2^{+}$ baryons the interpolating current has the form

[TABLE]

In expressions above $C$ is the charge conjugation matrix, and $Q$ is the $c$ or $b$ quark. The current $J$ for the $1/2^{+}$ baryons contains an arbitrary auxiliary parameter $\beta$, where $\beta=-1$ corresponds to the choice of the Ioffe current.

The correlation function has to be calculated using both the physical and quark-gluon degrees of freedom. To this end, we adopt the “ground-state+first excited-state+continuum” scheme and calculate $\Pi^{\mathrm{Phys}}(p)$ in terms of their parameters

[TABLE]

where $\Omega_{Q}$ and $\Omega_{Q}^{\prime}$ are the ground and first radially excited baryons with the masses $m$ and $m^{\prime}$, respectively. The dots stand for contribution of the higher excited states and continuum.

Below we provide some details of calculations for the spin $1/2$ particles omitting, at the same time, corresponding expressions for the spin $3/2$ states. In order to proceed we introduce the matrix elements

[TABLE]

where $\lambda$ and $\lambda^{\prime}$ are the pole residues of the $\Omega_{Q}$ and $\Omega_{Q}^{\prime}$ states, respectively. Employing Eqs. (5) and (6) and performing the summation over the spins of the $1/2^{+}$ baryons

[TABLE]

we get

[TABLE]

The Borel transformation applied to Eq. (8) leads to the result

[TABLE]

As is seen, there are two structures in Eq. (9), namely $\sim\not{p}$ and $\sim I$, both of which can be utilized to derive the required sum rules

[TABLE]

where $B\Pi_{1}^{\mathrm{OPE}}(p)$ and $B\Pi_{2}^{\mathrm{OPE}}(p)$ are the Borel transformation of the corresponding structures in $\Pi(p)$ , but calculated in terms of the quark-gluon degrees of freedom and labeled as $\Pi^{\mathrm{OPE}}(p)$. Here we refrain from writing down rather lengthy expressions of the functions $B\Pi_{1}^{\mathrm{OPE}}(p)$ and $B\Pi_{2}^{\mathrm{OPE}}(p)$, which will be published elsewhere.

After rather simple manipulations we find the desired sum rules for the parameters of the radially excited $\Omega_{Q}^{\prime}$ state

[TABLE]

and

[TABLE]

In the case of the $3/2^{+}$ baryons we use the matrix elements

[TABLE]

where $u_{\mu}(p,s)$ are the Rarita-Schwinger spinors, and perform the summation over the spins $s$ of the baryons by means of the formula

[TABLE]

The interpolating current $J_{\mu}$ couples to the $1/2^{+}$ baryons, which contribute to the sum rules, as well. Their contributions can be separated and removed from the sum rules by special ordering of the Dirac matrices (see, for example Ref. Aliev:2016jnp ). It is not difficult to demonstrate, that the terms, which are formed exclusively due to contribution of the $3/2^{+}$ baryons are proportional to the structures $\sim\not{p}g_{\mu\nu}$ and $\sim g_{\mu\nu}$. Namely, these structures and corresponding invariant amplitudes are employed to get sum rules for the masses and pole residues of the ground-state and radially excited $3/2^{+}$ charmed and bottom baryons.

The correlation function $\Pi^{\mathrm{OPE}}(p)$ should be found using the general expression Eq. (2) and heavy and light quarks’ propagators. In calculations we employ the $s$-quark and heavy quark propagators given by the expressions

[TABLE]

and

[TABLE]

The correlation functions $\Pi_{1(2)}^{\mathrm{OPE}}(p)$ can be written down in the form

[TABLE]

where $\rho_{1(2)}^{\mathrm{QCD}}(s)$ are the corresponding spectral densities, and $s_{0}$ is the continuum threshold parameter. In Eq. (17) contribution of the higher exited states and continuum is subtracted using the quark-hadron duality assumption.

As is seen, the sum rules for the excited states contain the mass of the ground-state particles. We have evaluated the masses and pole residues of the $1/2^{+}$ and $3/2^{+}$ ground-state baryons by employing the two-point sum rule method within the “ground-state + continuum” scheme. In calculations the vacuum condensates up to ten dimensions are taken into account. The masses of the ground-state baryons are used in Eqs. (11) and (12), and in the similar expressions for the $3/2^{+}$ baryons.

The sum rules depend on numerous parameters, which are collected in Table 1. It contains the masses of the bottom, charm and strange quarks, as well as quark, gluon and mixed vacuum condensates. The sum rules also require fixing of the working windows for the Borel parameter $M^{2}$ and continuum threshold $s_{0}$, which are two auxiliary parameters of the calculations. For $1/2^{+}$ particles we have additionally $\beta$ parameter coming from the expression of the interpolating current. The choice of $M^{2}$, $s_{0}$, and $\beta$ is not totally arbitrary, but should satisfy the standard restrictions of the sum rule calculations. Namely, the convergence of the operator product expansion, dominance of the pole contribution, existence of the $(M^{2},s_{0})$ regions, where dependence of the extracted quantities on $M^{2}$ and $s_{0}$ is minimal, have to be obeyed. The same is true for $\beta$: we have to determine a working region for $\beta$ by demanding a weak dependence of our results on its choice.

Predictions obtained in this work for the masses and pole residues of the $1S$ and $2S$ charmed and bottom baryons with spin-parity $1/2^{+}$ and $3/2^{+}$, as well as the working ranges of the parameters $M^{2}$ and $s_{0}$ are shown in Tables 2 and 3, respectively. Results for the $(1S,1/2^{+})$ and $(2S,1/2^{+})$ baryons are obtained by varying the parameter $\beta=\tan\theta$ in Eq. (3) within the limits

[TABLE]

to achieve the stable sum rules’ predictions.

By comparison of our results for the ground state masses $m_{\Omega_{c}}$ and $m_{\Omega_{c}^{\star}}$ with the experimental data given in Eq. (1), we find a nice agreement between them. The same is correct for the bottom $\Omega_{b}^{-}$ baryon with spin-parity $J^{P}=1/2^{+}$ and mass $m=6071\pm 40\,\mathrm{\,MeV}$ (see, Ref. Olive:2016xmw ). Our predictions allow us also to interpret the states $\Omega_{c}(3066)^{0}$ and $\Omega_{c}(3119)^{0}$, recently discovered by the LHCb Collaboration, as the first radially excited $(2S,1/2^{+})$ and $(2S,3/2^{+})$ charmed baryons, respectively.

The spectroscopic parameters of the $\Omega_{c}$ baryons were calculated in the context of different approaches, see Tables 4 and 5. It is interesting to note that our results for the charmed baryons are in accord with the predictions of the relativistic diquark-quark model of Ref. Ebert:2011kk . Other quark models lead almost to same predictions for masses of the $\Omega_{c}^{0}$ and $\Omega_{b}^{-}$ baryons. Within systematic errors of the sum rule computations we agree also with predictions of Refs. Wang:2007sqa ; Wang:2009cr ; Wang:2008hz for the masses of ground-state charmed and bottom baryons.

Investigations performed in the present Letter and results obtained in other works discussed above favor to assign $\Omega_{c}(3066)^{0}$ and $\Omega_{c}(3119)^{0}$ states as the first radially excited $(2S,1/2^{+})$ and $(2S,3/2^{+})$ charmed baryons. Because LHCb did not fix the spin-parity of five $\Omega_{c}^{0}$ states, the remaining three particles, namely $\Omega_{c}(3000)^{0}$, $\Omega_{c}(3050)^{0}$ and $\Omega_{c}(3090)^{0}$ maybe $1P$-wave excitations of the charmed baryons. Our predictions on the ground-state and radially excited b-baryons with $J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$ may help experimentalists in the search for these states.

K. A. thanks Doǧuş University for the partial financial support through the grant BAP 2015-16-D1-B04.