EXCITED $\Omega_{c}$ BARYONS AS 2S STATES

Marek<EMAIL_ADDRESS>and Jonathan L.
<EMAIL_ADDRESS>
a School of Physics and Astronomy

Raymond and Beverly Sackler Faculty of Exact Sciences

Tel Aviv University, Tel Aviv 69978, Israel

b Enrico Fermi Institute and Department of Physics

University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA

ABSTRACT

> The LHCb experiment has recently reported two excited $\Omega_{c}$
> resonances decaying to $\Xi_{c}^{+}K^{-}$, with masses about 3185 and 3327
> MeV. We discuss their assignment to $2S_{1/2}$ and $2S_{3/2}$ states, which
> can be compared with masses based on extrapolation from the observed 1S
> states. The agreement is not perfect, but weighs against an earlier
> alternative assignment.

PACS codes: 12.39.Jh, 13.20.Jf, 13.25.Jx, 14.40.Rt

##  I Introduction

The LHCb experiment has recently reported the discovery of two new
$\Omega_{c}^{0}$ resonances at $3185.1\pm 1.7^{+7.4}_{-0.9}\pm 0.2$ and
$3327.1\pm 1.2^{+0.1}_{-1.3}\pm 0.2$ MeV [1]. Here the errors are statistical,
systematic, and based on the uncertainty of the known $\Xi_{c}^{+}$ mass. Five
previously observed $\Omega_{c}^{0}$ states [2, 3] were confirmed with higher
statistics. These were interpreted as P-wave excitations of a charmed quark
and an $ss$ spin-1 diquark [4]: $J^{P}=1/2^{-}$ for $\Omega_{c}(3000)^{0}$ and
$\Omega_{c}(3050)^{0}$, $3/2^{-}$ for $\Omega_{(}3065)^{0}$ and
$\Omega_{c}(3090)^{0}$, and $5/2^{-}$ for $\Omega_{c}(3119)^{0}$, an
assignment favored by lattice QCD [5]. A less favored picture takes the
$\Omega_{c}(3090)^{0}$ and $\Omega_{c}(3119)^{0}$ as $2S_{1/2}$ and $2S_{3/2}$
[4].

In the present paper we identify the two new resonances as
$\Omega_{c}(3185)^{0}=2S_{1/2}$ and $\Omega_{c}(3327)=2S_{3/2}$, where the
subscript denotes the total spin. The expected 2S–1S splitting is calculated
and compared with experiment in Sec. II, while a similar exercise is performed
for the hyperfine splitting between the 1S and 2S states in Sec. III. The
choice of the favored assignment [4] whereby the five narrow states are all
taken as $1P$ is noted in Sec. IV, while Sec. V concludes.

##  II 2S–1S splitting

We are interested in the difference between 2S and 1S levels after account has
been taken of hyperfine structure. To that end we note that in a system of
spins $s_{1}$ and $s_{2}$ and total spin $S$ the hyperfine interaction for
$s_{1}=s_{2}=1/2$ is proportional to $(1/4,-3/4)$ for $s_{1}=(1,0)$ while for
$s_{1}=1,s_{2}=1/2$ it is proportional to $(1/2,-1)$. Thus in quarkonium
($c\bar{c},b\bar{b}$) systems one is interested in averages
$(1/4)M(J=0)+(3/4)M(J=1)$ while in bound states of a spin-1/2 charmed quark
and a spin-1 $\bar{s}\bar{s}$ antidiquark one is interested in averages
$(1/3)M(J=1/2)+(2/3)M(J=3/2)$. We call these “spin-weighted averages.”

In what follows we treat the $\Omega_{c}^{0}=css$ states as two-body entities
of a charmed quark $c$ with mass $m_{c}=1709$ MeV and a spin-1 $ss$ diquark
with $m_{ss}=1095$ MeV [4]. The corresponding reduced mass,
$\mu_{c,ss}=(m_{c}m_{ss})/(m_{c}+m_{ss})=667$ MeV, is not far from the
charmonium reduced mass $\mu_{c\bar{c}}=m_{c}/2=854.5$ MeV. With the help of
the bottomonium reduced mass $\mu_{b\bar{b}}=m_{b}/2=2521$ MeV and a power-law
extrapolation for the predicted 2S-1S difference

$\Delta=\overline{2S}-\overline{1S}=E_{0}\mu^{p}~{}~{}$ (1)

using the experimental values $\Delta_{c\bar{c}}=605.3\pm 0.3$ MeV,
$\Delta_{b\bar{b}}=572.3\pm 1.2$ MeV, one finds $E_{0}=858.8$ MeV,
$p=-0.0518$, and $\Delta_{c,ss}=613.1$ MeV. Here one has calculated spin-
weighted averages for quarkonia with relative weights (1/4,3/4) for
$J=(1/2,3/2)$.

The observed value of $\Delta$ for the two new resonances, assuming their
assignment to $2S_{J=1/2}$ and $(2S)_{J=3/2}$ states, is based on the masses
in Table I (1S values from Ref. [6]). To eliminate hyperfine contributions in
the $\Omega_{c}$ states listed in Table I we calculate spin-weighted averages
of masses, with weight 1/3 for $J=1/2$ and 2/3 for $J=3/2$. The observed 2S–1S
difference for the spin-weighted $\Omega_{c}$ states is then
$(3279.8^{+2.7}_{-1.4}-2742.3\pm 1.4)$ MeV, or $(537.5^{+3.0}_{-2.0})$ MeV.
This is to be compared with the value of 613 MeV obtained above by power-law
extrapolation from charmonium and bottomonium.

One might suspect a systematic error associated with power-law extrapolation.
Altough it is unlikely to be valid on as large a range, such an estimate gives
$\Delta_{c,ss}=609.0$ MeV, not far from our power-law estimate.

Table I: Masses of 1S and proposed 2S $\Omega_{c}$ resonances, in MeV  | $M(nS_{1/2})$ | $M(nS_{3/2})$ | $\overline{M}(nS)$
---|---|---|---
1S | $2695.2\pm 1.7$ | $2765.9\pm 2.0$ | $2742.3\pm 1.4$
2S | $3185.1\pm 1.7^{+7.4}_{-0.9}\pm 0.2$ | $3327.1\pm 1.2^{+0.1}_{-1.3}\pm 0.2$ | $3279.8^{+2.7}_{-1.4}$

##  III Hyperfine splitting

The hyperfine splitting between the $\Omega_{c}^{0}(1S)_{1/2}$ and
$\Omega_{c}^{0}(1S)_{3/2}$, using Particle Data Group [6] masses, is
$2765.9\pm 2.0-2695.2\pm 1.7=70.7\pm 2.6$ MeV. Normally one would expect it to
be less for the 2S states (see, e.g., [7]) but the value assuming the two new
states are 2S is $3327.1\pm 1.2^{+0.1}_{-1.3}\pm 0.2-[3185.1\pm
1.7^{+7.4}_{-0.9}\pm 0.2]=142.0^{+2.3}_{-7.8}$ MeV. One might ascribe part of
this difference to final-state interactions, as the two new states have widths
$50\pm 7^{+10}_{-20}$ MeV ($J=1/2$ candidate) and $20\pm 5^{+13}_{-1}$ MeV
($J=3/2$ candidate). Mass shifts of the same order as total widths can occur.
The relative widths of the $J=1/2$ and $J=3/2$ 2S candidates are
understandable: the $J=1/2$ state decays to $\Xi_{c}^{+}$ via an S wave, while
the $J=3/2$ state decays to $\Xi_{c}^{+}$ via a more kinematically suppressed
D wave. If the mass shift is greater for the state with the larger total
width, it is natural to ascribe the larger-than-expected 2S hyperfine
splitting mainly to a downward shift of the $J=1/2$ state.

##  IV Favored assignment of five narrow states

In Ref. [4] the favored assignment of the five narrow $\Omega_{c}^{0}$ peaks
was to the five states of a spin-1 $ss$ diquark and a spin-1/2 charmed quark
in a relative P wave. A less likely assignment was to take the two highest
narrow peaks to be 2S, leaving two lower-mass P waves to be found.

With higher statistics, the new LHCb data show no evidence for the lower-mass
P waves. Furthermore, taking $\Omega_{c}^{0}(3090)$ and $\Omega_{c}^{0}(3119)$
to be 2S states would exacerbate the difference between observed and predicted
1S–2S splittings, leaving the two new states without a credible assignment.

A likely solution to both the 2S-1S splitting and the hyperfine problems is to
imagine that final-state interactions mainly lower the mass of the $J=1/2$
state, for which the final-state interactions are indeed greater, while
leaving the $J=3/2$ state mainly unshifted. Significant deviations from naive
quark model predictions due to final-state interactions occur, for example, in
the masses of $\Lambda(1405)$ and $D_{s}^{0}(2317)$ [6].

##  V Conclusions

The two new excited $\Omega_{c}^{0}$ states discovered by LHCb [1], at 3185
and 3327 MeV, have been identified respectively as 2SJ=1/2 and $2S_{J=3/2}$.
The 1S–2S and hyperfine splittings, though smaller and larger, respectivly,
than expected, do not deviate enough from predicted values to jeopardize these
assignments. Confirmation of our methods may be sought in other systems with
no light quarks. The $b\bar{c}$ (1S,2S) system would be ideal except only the
spin-zero $B_{c}(1S,2S)$ masses are known, whereas only the 2S–1S mass
difference is known for the $B_{c}^{*}$ spin-one states [6, 8, 9, 10]. A
useful challenge to resolve this question would be the detection of the soft
photon in $B_{c}^{*+}\to B_{c}^{+}\gamma$.

## Acknowledgments

The work of M.K. was supported in part by NSFC-ISF Grant No. 3423/19.

## References

  * [1] R. Aaij et al. [LHCb Collaboration], “Observation of new $\Omega_{c}^{0}$ states decaying to the $\Xi_{c}^{+}K^{-}$ final state,” [arXiv:2302.04733 [hep-ex]].
  * [2] R. Aaij et al. [LHCb Collaboration], “Observation of five new narrow $\Omega_{c}^{0}$ states decaying to $\Xi_{c}^{+}K^{-}$,” Phys. Rev. Lett. 118, 182001 (2017) [arXiv:1703.04639 [hep-ex]].
  * [3] R. Aaij et al. [LHCb Collaboration], “Observation of excited $\Omega_{c}^{0}$ baryons in $\Omega_{b}^{-}\to\Xi_{c}^{+}K^{-}\pi^{-}$decays,” Phys. Rev. D 104, L091102 (2021) [arXiv:2107.03419 [hep-ex]].
  * [4] M. Karliner and J. L. Rosner, “Very narrow excited $\Omega_{c}$ baryons,” Phys. Rev. D 95, 114012 (2017) [arXiv:1703.07774 [hep-ph]].
  * [5] M. Padmanath and N. Mathur, “Quantum Numbers of Recently Discovered $\Omega^{0}_{c}$ Baryons from Lattice QCD,” Phys. Rev. Lett. 119, 042001 (2017) [arXiv:1704.00259 [hep-ph]].
  * [6] R. L. Workman et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2022, 083C01 (2022).
  * [7] C. Quigg and J. L. Rosner, “Quantum Mechanics with Applications to Quarkonium,” Phys. Rept. 56, 167-235 (1979).
  * [8] G. Aad et al. [ATLAS Collaboration], “Observation of an Excited $B_{c}^{+}$ Meson State with the ATLAS Detector,” Phys. Rev. Lett. 113, 212004 (2014) [arXiv:1407.1032 [hep-ex]].
  * [9] A. M. Sirunyan et al. [CMS Collaboration], “Observation of Two Excited B${}^{+}_{\mathrm{c}}$ States and Measurement of the B${}^{+}_{\mathrm{c}}$(2S) Mass in pp Collisions at $\sqrt{s}=$ 13 TeV,” Phys. Rev. Lett. 122, 132001 (2019) [[arXiv:1902.00571 [hep-ex]].
  * [10] R. Aaij et al. [LHCb Collaboration], “Observation of an excited $B_{c}^{+}$ state,” Phys. Rev. Lett. 122, 232001 (2019) [arXiv:1904.00081 [hep-ex]].