# Spectroscopy of light baryons with strangeness -1, -2, and -3

Juhi Oudichhya  Keval Gandhi  Ajay Kumar Rai  Department of Physics, Sardar
Vallabhbhai National Institute of Technology, Surat, Gujarat-395007, India

###### Abstract

The present article contains the descriptive study of light strange baryons
$\Lambda$, $\Sigma$, $\Xi$, and $\Omega$. The Regge phenomenology with linear
Regge trajectories has been employed and the relations between Regge slopes,
intercepts, and baryon masses have been extracted. With the aid of these
relations, ground state masses are obtained for $\Xi$ and $\Omega$ baryons.
Regge parameters have also been estimated to calculate the excited state
masses for $\Lambda$, $\Sigma$, $\Xi$, and $\Omega$ baryons in both the
$(J,M^{2})$ and $(n,M^{2})$ planes. The obtained results are compared with
available experimental data and various theoretical approaches. We predict the
spin-parity of recently observed light baryons and our predictions may be
useful in future experimental searches of these baryons and their $J^{P}$
assignments.

††preprint: APS/123-QED

##  I Introduction

Hadron spectroscopy is a tool for understanding the dynamics of quark
interactions in composite systems like mesons, baryons, and exotics. The
spectroscopic study of baryon with a strange quark(s) will have a great
interest because the strange quark would be slightly heavier than up and down
quarks and considerably lighter than charm and bottom quarks. In this paper we
present a description of hyperons which are $\Lambda$, $\Sigma$ baryons with
strangeness -1, $\Xi$ and $\Omega$ baryons with strangeness -2 and -3,
respectively. Many experimental facilities around the world have been seeking
to study the strange baryons. Recently the ALICE Collaboration observed an
attractive interaction between protons and $\Xi^{-}$ baryon [1]. In 2019 the
BESIII Collaboration observed $\Xi(1530)$ in a baryon-antibaryon pair from
charmonium decay [2]. At Jefferson Lab, the photoproduction of $\Xi$ has been
observed using the CLAS detector [3]. Furthermore, JLab has proposed to
investigate the strange hyperon spectroscopy using a secondary KL beam and a
GlueX experiment and the results are expected to provide more knowledge of
strange hyperons $\Lambda$, $\Sigma$, and $\Xi$ [4]. Extensive research have
also been conducted by the BABAR Collaboration [5]. Recently in 2018, the
Belle Collaboration observed new excited $\Omega^{-}$ state as $\Omega(2012)$
through $e^{-}e^{+}$ annihilations decaying into $\Xi^{0}K^{-}$ and
$\Xi^{-}\bar{K}^{0}$ channels [6].

By studying the properties these, so-called, strange baryons or hyperons,
hadron physicists will be able to make progress in answering various critical
questions, namely; what is the intrinsic structure of these baryons? what are
the important degrees of freedom in a baryon? and, are there exotic forms of
baryon-like matter?. Eventually, addressing these questions helps in shedding
light on the deeper fundamental question, namely “how to understand the
underlying confinement mechanism?”. The anticipated multiplet structure of the
baryons must be established experimentally to understand the symmetries and
dynamics of the strong interaction, and the details of their excitation
spectrum are vital for that. However, experimental information is presently
very limited, in particular for $\Xi$ and $\Omega$ baryons with strangeness -2
and -3, respectively. For a large part, the lack of experimental knowledge can
be understood by the fact that with the widely usage of electromagnetic probes
the production cross section of strange baryons is very limited making it
difficult to generate sufficient statistics. The $\Xi$ baryons are produced
only at the final state of a decay process and have small cross-sections
(typically a few $\mu b$) [7]. Somewhat this comment is also valid for
$\Omega^{-}$ baryons. Also, $\Omega^{-}$ baryons have zero isospins, which
means that $\Omega^{*-}\rightarrow\Omega^{-}\pi^{0}$ decays are highly
suppressed, and this restricts the possible decays of excited states.
Therefore, $\Xi K$ being the excepted decay mode for the low-lying
$\Omega^{-}$ states [6]. These decays are similar to the
$\Omega_{c}^{0}\rightarrow\Xi_{c}^{+}K^{-}$ decays discovered by the LHCb [8]
Collaboration and confirmed by the Belle [9] Collaboration shortly after.

$\overline{\mbox{P}}$ANDA (antiProton ANnihilations at DArmstadt), the
upcoming experimental facility at FAIR (Facility for AntiProton and Ion
Research), will have a task to establish the whole spectrum of hyperons
through antiproton-proton annihilation [14, 10, 11, 12, 13, 16]. At the
latest, a member of the $\overline{\mbox{P}}$ANDA group studied the
feasibility of the reaction $\bar{p}p\rightarrow\bar{\Xi}^{+}\Xi^{*-}$ and its
charge conjugate channel, where the $\Xi^{*}$ denotes the following
intermediate resonance states: $\Xi$ (1530), $\Xi$ (1690), and $\Xi$ (1820)
[15]. A major goal of the $\Xi$ spectroscopy program at
$\overline{\mbox{P}}$ANDA is the determination of the spin and parity quantum
numbers of the $\Xi$ states [17]. Unlike $\Xi$ and $\Omega$, $\Lambda$ and
$\Sigma$ baryons have quite a number of experimentally established states.
Table 1 represents the excited $\Xi$ and $\Omega$ baryons listed by the latest
version of the Particle Data Group (PDG) [18]. Spin and parity quantum numbers
of newly observed $\Xi$ and $\Omega$ states are not yet confirmed and the PDG
needs more confirmations. It is crucial to assign the spin-parity ($J^{P}$) of
hadrons which facilitate the determination of properties such as decay width,
branching fraction, isospin mass splitting, polarization amplitude, etc.

Since, we have studied the light baryons which are made of the quarks $u$,
$d$, and $s$ only. The SU(3) flavour symmetry can be only approximate because
the mass of the strange quark is about 0.1 GeV greater than the masses of the
up and down quarks, although this mass difference is relatively small compared
to the typical QCD binding energy which is of order 1 GeV. The eightfold way
and the standard SU(3) Gell-Mann–Okubo (GMO) formula [19] have played an
important role in particle physics. However, the direct generalization of the
GMO formula to the charmed and bottom hadrons cannot agree well with
experimental data due to higher-order breaking effects. Hence for light
baryons the breaking of SU(3) symmetry is minimum.

Several phenomenological and theoretical model have been employed to study the
properties of light baryons. The authors of Ref.[20] estimated the mass
spectra of strange baryons using the relativistic quark model based on the
quasipotential approach and baryons are treated as relativistic quark-diquark
bound systems. In the recent study [21, 22], the authors employed hypercentral
Constituent Quark Model with linear confining potential along with first order
correction term to obtain the mass spectra of light strange baryons with
strangeness -1,-2, and -3. Regge trajectories are explored for the linearity
of the calculated masses for $(J,M^{2})$ and $(n,M^{2})$. Ref. [23] present
the complete excited $\Lambda$, $\Sigma$, $\Xi$, and $\Omega$ spectrum in a
relativistic quark model based on the three quark Bethe-Salpeter equation with
instantaneous two and three body forces. The investigation of instanton-
induced effects in the baryon mass spectrum plays a central role in this work.
The authors of Ref. [24] present a systematic analysis of spectra and
transition rates of strange baryons within the framework of a collective
string-like $qqq$ model in which the orbital excitations are treated as
rotations and vibrations of the strings.

Table 1:  Masses and $J^{P}$ values of $\Xi$ and $\Omega$ baryons are listed in PDG [18]. The status is given as poor(*), fair(**), very likely(***), and certain(****).  Resonance | Mass (MeV) | $J^{P}$ | Status
---|---|---|---
$\Xi^{0}$ | 1314.86 $\pm$ 0.20 | $\frac{1}{2}^{+}$ | ****
$\Xi^{-}$ | 1321.71 $\pm$ 0.07 | $\frac{1}{2}^{+}$ | ****
$\Xi(1530)^{0}$ | 1531.80 $\pm$ 0.32 | $\frac{3}{2}^{+}$ | ****
$\Xi(1530)^{-}$ | 1535.0 $\pm$ 0.6 | $\frac{3}{2}^{+}$ | ****
$\Xi(1620)$ | 1620 |  | *
$\Xi(1690)$ | 1690 $\pm$ 10 |  | ***
$\Xi(1820)$ | 1823 $\pm$ 5 | $\frac{3}{2}^{-}$ | ***
$\Xi(1950)$ | 1950 $\pm$ 15 |  | ***
$\Xi(2030)$ | 2025 $\pm$ 5 | $\frac{5}{2}^{?}$ | ***
$\Xi(2120)$ | 2120 |  | *
$\Xi(2250)$ | 2250 |  | **
$\Xi(2370)$ | 2370 |  | **
$\Xi(2500)$ | 2500 |  | *
$\Omega^{-}$ | 1672.45 $\pm$ 0.29 | $\frac{3}{2}^{+}$ | ****
$\Omega(2012)^{-}$ | 2012.4 $\pm$ 0.9 |  | ***
$\Omega(2250)^{-}$ | 2252 $\pm$ 9 |  | ***
$\Omega(2380)^{-}$ | 2380 |  | **
$\Omega(2470)^{-}$ | 2474 $\pm$ 12 |  | **

In the present article, we give a systematic study of strange baryons
$\Lambda$, $\Sigma$, $\Xi$, and $\Omega$. Here we employ the Regge
phenomenology with the assumption of linear Regge trajectories. We find the
relations between the intercept, slope ratios, and baryon masses in both the
$(J,M^{2})$ and $(n,M^{2})$ planes. With the help of these relations, we
determine the ground state masses of $\Xi$ and $\Omega$ baryons. We extract
the Regge parameters (a(0)and $\alpha^{,}$) to determine the mass spectra of
all light strange baryons in both the $(J,M^{2})$ and $(n,M^{2})$ planes. It
is evident that the ground and low-lying resonance states are within a
reasonable range for almost all of the models and approaches. However, the
higher excited states exhibit huge variations in their mass predictions, which
motivated us to study the experimental determination of the spin and parity
quantum numbers of these light strange baryons.

The remainder of this paper is organized as follows. After briefing various
experimental and theoretical approaches, in Sec. II we describe Regge theory
and extract the mass relations. By using these relations we obtain the masses
for $\Lambda$, $\Sigma$, $\Xi$, and $\Omega$ baryons in the $(J,M^{2})$ plane
for natural and unnatural parity states. Further we obtain the Regge
parameters for each Regge line and calculate the radial and orbital excited
states of these baryons in the $(n,M^{2})$ plane. We extend this model and try
to determine the remaining states other than natural and unnatural parity
states in the $(J,M^{2})$ plane. The detailed description of our results
obtained is discussed in Sec. III. Finally we concluded our study in Sec. IV.

##  II Theoretical Framework

To study the hadron spectroscopy, the linear Regge trajectory is one of the
most effective and widely used phenomenological approach. The plots of Regge
trajectories of hadrons in the $(J,M^{2})$ plane are usually called Chew-
frautschi plots [25]. They used the theory to study the strong quark gluon
interaction and observed that the excited states of experimentally missing
mesons and baryons exists on the linear trajectories in the $(J,M^{2})$ plane.
The trajectory of a particular pole is characterized by a set of internal
quantum numbers and the hadrons lying on the particular Regge line have the
same internal quantum numbers. The most general form of linear Regge
trajectories can be expressed as [26, 27, 28, 29],

$J=\alpha(M)=a(0)+\alpha^{{}^{\prime}}M^{2},$ (1)

where $a(0)$ and $\alpha^{{}^{\prime}}$ are, respectively, the slope and
intercept of the Regge trajectory. Regge intercepts and Regge slopes for
different flavors of a baryon multiplet can be related by the following
relations [30, 31, 32, 33, 34],

$a_{iiq}(0)+a_{jjq}(0)=2a_{ijq}(0),$ (2)

$\dfrac{1}{{\alpha^{{}^{\prime}}}_{iiq}}+\dfrac{1}{{\alpha^{{}^{\prime}}}_{jjq}}=\dfrac{2}{{\alpha^{{}^{\prime}}}_{ijq}},$
(3)

where $i,j,q$ represent the quark flavors. Using Eqs. (1) and (2) we obtain,

$\alpha^{{}^{\prime}}_{iiq}M^{2}_{iiq}+\alpha^{{}^{\prime}}_{jjq}M^{2}_{jjq}=2\alpha^{{}^{\prime}}_{ijq}M^{2}_{ijq}.$
(4)

We get two pairs of solutions after combining the relations (3) and (4) which
are expressed as,

$\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{iiq}}=\dfrac{1}{2M^{2}_{jjq}}\times[(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})\pm\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^{2}}-4M^{2}_{iiq}M^{2}_{jjq}}],$
(5)

and,

$\dfrac{\alpha^{{}^{\prime}}_{ijq}}{\alpha^{{}^{\prime}}_{iiq}}=\dfrac{1}{4M^{2}_{ijq}}\times[(4M^{2}_{ijq}+M^{2}_{iiq}-M^{2}_{jjq})\pm\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^{2}}-4M^{2}_{iiq}M^{2}_{jjq}}].$
(6)

These are the significant relationships that we have obtained between slope
ratios and baryon masses.

Now the above obtained Eq. (5) can also be expressed as,

$\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{iiq}}=\dfrac{\alpha^{{}^{\prime}}_{kkq}}{\alpha^{{}^{\prime}}_{iiq}}\times\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{kkq}},$
(7)

here $k$ can be any quark flavor. Thus we have,

$\begin{split}\dfrac{[(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})+\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^{2}}-4M^{2}_{iiq}M^{2}_{jjq}}]}{2M^{2}_{jjq}}\\\
=\dfrac{[(4M^{2}_{ikq}-M^{2}_{iiq}-M^{2}_{kkq})+\sqrt{{{(4M^{2}_{ikq}-M^{2}_{iiq}-M^{2}_{kkq}})^{2}}-4M^{2}_{iiq}M^{2}_{kkq}}]/2M^{2}_{kkq}}{[(4M^{2}_{jkq}-M^{2}_{jjq}-M^{2}_{kkq})+\sqrt{{{(4M^{2}_{jkq}-M^{2}_{jjq}-M^{2}_{kkq}})^{2}}-4M^{2}_{jjq}M^{2}_{kkq}}]/2M^{2}_{kkq}}.\end{split}$
(8)

This is the general relation we have derived in terms of baryon masses which
can be used to estimate the mass of any baryon state if all other masses are
known. In the present work, the ground-state ($J^{P}=\frac{1}{2}^{+}$ and
$\frac{3}{2}^{+}$) masses of the light strange baryons having strangeness -2
and -3 are evaluated using the above equation.

###  II.1 Masses in the $(J,M^{2})$ plane

In this section using the relations we have extracted above, we determine the
ground-state masses of $\Xi$ and $\Omega$ baryons, as well as the orbitally
excited state masses of all the light strange baryons $\Lambda$, $\Sigma$,
$\Xi$, and $\Omega$ for natural ($P=(-1)^{J-\frac{1}{2}}$) and unnatural
($P=(-1)^{J+\frac{1}{2}}$) parities in the $(J,M^{2})$ plane. The quark
combination of $\Xi^{0}$ baryon is $uss$, hence we put $i=d$, $j=s$, $q=u$,
and $k=d$ in Eq. (8). We obtain the mass expression for $\Xi^{0}$ as a
function of squared masses of neutron ($n$) and $\Lambda^{0}$ baryons, which
is expressed as

Table 2:  Regge slopes of $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ trajectories of the light strange baryons. (in GeV-2) .  | $\Lambda$ | $\Sigma$ | $\Xi$ | $\Omega$
---|---|---|---|---
$\alpha^{{}^{\prime}}$ (S = 1/2) | 0.8597 | 0.8597 | 0.7420 | -
$\alpha^{{}^{\prime}*}$ (S = 3/2) | - | 0.8051 | 0.7250 | 0.6585
Table 3:  Masses of excited states of $\Lambda$ baryon in the $(J,M^{2})$ plane for natural parities. The numbers in the boldface are the experimental values taken as the input [18] (in MeV).  States | Present |  |  | Others |  |  |
---|---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ | $\Lambda^{0}$ | PDG [18] | [20] | [35] | [21] | [23] | [36]
$1^{2}S_{\frac{1}{2}}$ | 1116 | 1116 | 1115 | 1115 | 1115 | 1108 | 1116
$1^{2}P_{\frac{3}{2}}$ | 1552 | 1519 | 1549 | 1545 | 1534 | 1508 | 1650
$1^{2}D_{\frac{5}{2}}$ | 1890 | 1820 | 1825 | 1890 | 1746 | 1834 | 1896
$1^{2}F_{\frac{7}{2}}$ | 2176 | 2100 | 2097 | 2150 | 1970 | 2090 |
$1^{2}G_{\frac{9}{2}}$ | 2429 | 2350 | 2360 |  | 2204 | 2340 |
$1^{2}H_{\frac{11}{2}}$ | 2658 |  | 2605 |  |  |  |
Table 4:  Masses of excited states of $\Sigma$ baryon in the $(J,M^{2})$ plane with natural and unnatural parities. The numbers in the boldface are the experimental values taken as the inputs [18] (in MeV).  States | Present |  |  | Others |  |
---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Sigma^{-}$ | PDG [18] | [20] | [35] | [21] | [23]
$1^{2}S_{\frac{1}{2}}$ | 1189 | 1193 | 1197 |  | 1187 | 1190 | 1193 | 1190
$1^{2}P_{\frac{3}{2}}$ | 1605 | 1608 | 1611 | 1670 | 1706 | 1655 | 1698 | 1669
$1^{2}D_{\frac{5}{2}}$ | 1934 | 1936 | 1939 | 1915 | 1991 | 1995 | 1998 | 1956
$1^{2}F_{\frac{7}{2}}$ | 2214 | 2216 | 2219 |  | 2259 | 2245 | 2318 | 2236
$1^{2}G_{\frac{9}{2}}$ | 2463 | 2464 | 2467 |  | 2548 |  |  |
$1^{2}H_{\frac{11}{2}}$ | 2689 | 2690 | 2692 |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 1383 | 1384 | 1387 |  | 1381 | 1370 | 1384 | 1411
$1^{4}P_{\frac{5}{2}}$ | 1776 | 1777 | 1779 | 1775 | 1757 | 1755 | 1680 | 1770
$1^{4}D_{\frac{7}{2}}$ | 2097 | 2098 | 2099 | 2030 | 2033 | 2060 | 1962 | 2070
$1^{4}F_{\frac{9}{2}}$ | 2375 | 2376 | 2377 |  | 2289 |  | 2257 | 2325
$1^{4}G_{\frac{11}{2}}$ | 2623 | 2624 | 2625 | 2620 | 2529 |  |  |
$1^{4}H_{\frac{13}{2}}$ | 2850 | 2851 | 2852 |  |  |  |  |
Table 5:  Masses of excited states of $\Xi$ baryon in the $(J,M^{2})$ plane with natural and unnatural parities. (in MeV).  States | Present |  |  | Others |  |  |  |
---|---|---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ | $\Xi^{0}$ | $\Xi^{-}$ | PDG [18] | [20] | [35] | [23] | [21] | [37] |
$1^{2}S_{\frac{1}{2}}$ | 1293 | 1294 | 1315 ($\Xi^{0}$) | 1330 | 1305 | 1310 | 1322 | 1317 |
|  |  | 1322 ($\Xi^{-}$) |  |  |  |  |  |
$1^{2}P_{\frac{3}{2}}$ | 1738 | 1738 |  | 1764 | 1785 | 1780 | 1871 | 1801 |
$1^{2}D_{\frac{5}{2}}$ | 2090 | 2090 | 2025 | 2108 | 2045 | 2013 | 2234 | 1959 |
$1^{2}F_{\frac{7}{2}}$ | 2391 | 2391 | 2370 | 2460 | 2355 | 2320 | 2647 |  |
$1^{2}G_{\frac{9}{2}}$ | 2658 | 2658 |  |  |  |  |  |  |
$1^{2}H_{\frac{11}{2}}$ | 2900 | 2900 |  |  |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 1537 | 1533 | 1532 ($\Xi^{*0}$) | 1518 | 1505 | 1539 | 1531 | 1526 |
|  |  | 1535 ($\Xi^{*-}$) |  |  |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 1934 | 1931 | 1950 | 1853 | 1900 | 1955 | 1859 | 1917 |
$1^{4}D_{\frac{7}{2}}$ | 2263 | 2260 | 2250 | 2189 | 2180 | 2169 | 2203 | 2074 |
$1^{4}F_{\frac{9}{2}}$ | 2550 | 2547 |  | 2502 |  | 2505 | 2588 |  |
$1^{4}G_{\frac{11}{2}}$ | 2807 | 2805 |  |  |  |  |  |  |
$1^{4}H_{\frac{13}{2}}$ | 3043 | 3041 |  |  |  |  |  |  |
Table 6:  Masses of excited states of $\Omega$ baryon in the $(J,M^{2})$ plane with unnatural parities (in MeV).  States | Present | PDG [18] | [20] | [35] | [23] | [22]
---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ |  |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 1691 | 1672 | 1678 | 1635 |  | 1672
$1^{4}P_{\frac{5}{2}}$ | 2092 |  | 2653 | 2490 | 2528 | 1970
$1^{4}D_{\frac{7}{2}}$ | 2428 | 2380 | 2369 | 2295 | 2292 | 2233
$1^{4}F_{\frac{9}{2}}$ | 2723 |  | 2649 |  | 2606 | 2521
$1^{4}G_{\frac{11}{2}}$ | 2989 |  |  |  |  |
$1^{4}H_{\frac{13}{2}}$ | 3233 |  |  |  |  |

$\begin{split}\left[(M_{n}+M_{\Xi^{0}})^{2}-4M^{2}_{\Lambda^{0}}\right]\\\
=\sqrt{(4M^{2}_{\Lambda^{0}}-M^{2}_{n}-M^{2}_{\Xi^{0}})^{2}-4M^{2}_{n}M^{2}_{\Xi^{0}}};\end{split}$
(9)

substituting the masses of $n$ and $\Lambda^{0}$ into Eq. (9) we get the
ground-state mass of $\Xi^{0}$ as 1293 MeV for $J^{P}=\frac{1}{2}^{+}$.
Similarly we can get 1537 MeV for $J^{P}=\frac{3}{2}^{+}$. In the same manner
we can calculate for $\Xi^{-}$ baryon also (see Table 5). Similarly to
evaluate the ground state ($J^{P}=\frac{3}{2}^{+}$) mass of $\Omega^{-}$
baryon, composed of three strange quarks ($sss$), we put $i=u$, $j=s$, $q=s$,
and $k=u$ in Eq. (8) we get,

$\begin{split}\left[(M_{\Sigma^{*+}}+M_{\Omega^{-}})^{2}-4M^{2}_{\Xi^{*0}}\right]\\\
=\sqrt{(4M^{2}_{\Xi^{*0}}-M^{2}_{\Sigma^{*+}}-M^{2}_{\Omega^{-}})^{2}-4M^{2}_{\Sigma^{*+}}M^{2}_{\Omega^{-}}};\end{split}$
(10)

again substituting the masses of $\Sigma^{*+}$ [18] and $\Xi^{*0}$ (calculated
above) into the above equation, we obtain the ground-state mass of
$\Omega^{-}$ baryon as 1691 MeV.

After evaluating the ground-state masses, the orbitally excited states of
light strange baryons are calculated lying on $\frac{1}{2}^{+}$ and
$\frac{3}{2}^{+}$ trajectories by obtaining the Regge slopes
$\alpha^{{}^{\prime}}$. For instance using Eq. (5),

$\begin{split}\dfrac{\alpha^{{}^{\prime}}_{\Xi^{0}}}{\alpha^{{}^{\prime}}_{N}}=\dfrac{1}{2M^{2}_{\Xi^{0}}}\times[(4M^{2}_{\Lambda^{0}}-M^{2}_{n}-M^{2}_{\Xi^{0}})\\\
+\sqrt{{{(4M^{2}_{\Lambda^{0}}-M^{2}_{n}-M^{2}_{\Xi^{0}}})^{2}}-4M^{2}_{n}M^{2}_{\Xi^{0}}}],\end{split}$
(11)

inserting the masses of $\Xi^{0}$, $n$, and $\Lambda^{0}$ baryons in the above
equation we have $\alpha^{{}^{\prime}}_{\Xi^{0}}/\alpha^{{}^{\prime}}_{N}$.
Since from Eq. (1) we can have
$\alpha^{{}^{\prime}}=2/(M^{2}_{J+2}-M^{2}_{J})$, hence we get
$\alpha^{{}^{\prime}}_{N}$ = 1.0217 GeV-2. So we can calculate
$\alpha^{{}^{\prime}}_{\Xi^{0}}$ = 0.7420 GeV-2 for $\frac{1}{2}^{+}$
trajectory .Similarly with the help of Eqs. (5) and (6) we can determine the
Regge slopes ($\alpha^{{}^{\prime}}$) of $\Lambda$, $\Sigma$, and $\Omega$
baryons for both $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ trajectories.
According to the Ref [38]. $\alpha^{{}^{\prime}}_{\Lambda}$ $\simeq$
$\alpha^{{}^{\prime}}_{\Sigma}$, in this work we take this approximation.
Table 2 shows the estimated values of $\alpha^{{}^{\prime}}$ and
$\alpha^{{}^{\prime}*}$ of the light strange baryons. Now from Eq. (1) one can
write,

$M_{J+1}=\sqrt{M_{J}^{2}+\dfrac{1}{\alpha^{{}^{\prime}}}},$ (12)

With the help of values $\alpha^{{}^{\prime}}$ extracted for light-strange
baryons, from Eq.(12), the masses of orbitally excited states with
$J^{P}=\frac{3}{2}^{-},\frac{5}{2}^{+},\frac{7}{2}^{-}$…. and
$J^{P}=\frac{5}{2}^{-},\frac{7}{2}^{+},\frac{9}{2}^{-}$…. for natural and
unnatural parities respectively, in the $(J,M^{2})$ plane can be evaluated.
The numerical results are shown in Tables 3-6. Here the spectroscopic
notations $N^{2S+1}L_{J}$ are used to represent the state of the particles,
where $N$, $L$, $S$ denote the radial excited quantum number, orbital quantum
number, and intrinsic spin, respectively.

Table 7:  The values for Regge intercepts ($\beta_{0}$) and Regge slopes ($\beta$) (in GeV-2) of $\Xi^{0}$ baryon.  | Spin (S) | $\beta_{0}$ | $\beta$
---|---|---|---
S-states | 1/2 | 0.11315 | 0.53046
| 3/2 | -0.57199 | 0.66543
P-states | 1/2 | -0.49582 | 0.49520
| 3/2 | -1.19693 | 0.58736
D-states | 1/2 | -0.80656 | 0.41358
| 3/2 | -1.44628 | 0.47768
Table 8:  Masses of excited states of $\Lambda$ baryon in $(n,M^{2})$ plane. The masses in the boldface are taken as input from Ref. [18] (in MeV). | $N^{2S+1}L_{J}$ | Present | [21] | [23] | [35] | [37] | [36]
---|---|---|---|---|---|---|---
(S=1/2) | $1^{2}S_{1/2}$ | 1116 | 1115 | 1108 | 1115 | 1113 | 1116
| $2^{2}S_{1/2}$ | 1600 | 1592 | 1677 | 1680 | 1606 | 1518
| $3^{2}S_{1/2}$ | 1968 | 1885 | 1747 | 1830 | 1880 | 1955
| $4^{2}S_{1/2}$ | 2278 | 2202 | 2077 | 2010 | 2173 |
| $5^{2}S_{1/2}$ | 2550 | 2540 | 2132 | 2120 |  |
| $6^{2}S_{1/2}$ | 2796 |  |  |  |  |
(S=1/2) | $1^{2}P_{3/2}$ | 1552 | 1534 | 1508 | 1545 | 1560 | 1650
| $2^{2}P_{3/2}$ | 1690 | 1819 | 1775 | 1770 | 1859 | 1854
| $3^{2}P_{3/2}$ | 1818 | 2131 | 2147 | 2185 |  |
| $4^{2}P_{3/2}$ | 1937 | 2464 | 2313 |  |  |
| $5^{2}P_{3/2}$ | 2049 |  |  |  |  |
(S=1/2) | $1^{2}D_{5/2}$ | 1890 | 1746 | 1834 | 1890 | 1839 | 1896
| $2^{2}D_{5/2}$ | 2090 | 2051 | 2078 | 2115 | 2103 |
| $3^{2}D_{5/2}$ | 2272 | 2378 |  |  |  |
| $4^{2}D_{5/2}$ | 2441 |  |  |  |  |
| $5^{2}D_{5/2}$ | 2599 |  |  |  |  |
Table 9:  Masses of excited states of $\Sigma$ baryon in $(n,M^{2})$ plane. The masses in the boldface are taken as input from Ref. [20] (in MeV). | $N^{2S+1}L_{J}$ | Present |  | Others |  |
---|---|---|---|---|---|---
|  | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Sigma^{-}$ | [21] | [23] | [35] | [37]
(S=1/2) | $1^{2}S_{1/2}$ | 1189 | 1193 | 1197 | 1193 | 1190 | 1190 | 1192
| $2^{2}S_{1/2}$ | 1711 | 1711 | 1711 | 1643 | 1760 | 1720 | 1664
| $3^{2}S_{1/2}$ | 2107 | 2105 | 2103 | 2083 | 1947 | 1915 | 1924
| $4^{2}S_{1/2}$ | 2440 | 2436 | 2432 | 2560 | 2098 | 2030 | 2069
| $5^{2}S_{1/2}$ | 2733 | 2728 | 2722 | 3067 |  |  |
| $6^{2}S_{1/2}$ | 2997 | 2991 | 2984 |  |  |  |
(S=3/2) | $1^{4}S_{3/2}$ | 1383 | 1384 | 1387 | 1384 | 1411 | 1370 | 1383
| $2^{4}S_{3/2}$ | 1862 | 1862 | 1862 | 1827 | 1896 | 1920 | 1868
| $3^{4}S_{3/2}$ | 2241 | 2240 | 2238 | 2229 | 2044 | 2030 | 2039
| $4^{4}S_{3/2}$ | 2564 | 2563 | 2560 | 2675 | 2112 | 2115 | 2122
| $5^{4}S_{3/2}$ | 2851 | 2850 | 2845 | 3159 |  |  |
| $6^{4}S_{3/2}$ | 3112 | 3110 | 3105 |  |  |  |
(S=1/2) | $1^{2}P_{3/2}$ | 1605 | 1608 | 1611 | 1698 | 1669 | 1655 | 1698
| $2^{2}P_{3/2}$ | 2175 | 2175 | 2175 | 2099 | 1781 | 1755 | 1802
| $3^{2}P_{3/2}$ | 2624 | 2622 | 2620 | 2545 | 2203 | 2200 |
| $4^{2}P_{3/2}$ | 3007 | 3003 | 3000 | 3027 |  |  |
| $5^{2}P_{3/2}$ | 3346 | 3341 | 3337 | 3541 |  |  |
(S=3/2) | $1^{4}P_{5/2}$ | 1776 | 1777 | 1779 | 1680 | 1770 | 1755 | 1743
| $2^{4}P_{5/2}$ | 2214 | 2214 | 2214 | 2076 | 2174 | 2205 |
| $3^{4}P_{5/2}$ | 2579 | 2578 | 2577 | 2516 |  |  |
| $4^{4}P_{5/2}$ | 2898 | 2896 | 2894 | 2994 |  |  |
| $5^{4}P_{5/2}$ | 3185 | 3183 | 3180 | 3502 |  |  |
(S=1/2) | $1^{2}D_{5/2}$ | 1934 | 1936 | 1939 | 1998 | 1956 | 1995 | 1949
| $2^{2}D_{5/2}$ | 2459 | 2459 | 2459 | 2432 | 2071 | 2095 | 2062
| $3^{2}D_{5/2}$ | 2890 | 2889 | 2887 | 2905 |  |  | 2154
| $4^{2}D_{5/2}$ | 3265 | 3262 | 3259 | 3410 |  |  |
| $5^{2}D_{5/2}$ | 3601 | 3598 | 3593 |  |  |  |
(S=3/2) | $1^{4}D_{7/2}$ | 2097 | 2098 | 2099 | 1962 | 2070 | 2060 | 2002
| $2^{4}D_{7/2}$ | 2470 | 2470 | 2470 | 2390 | 2161 | 2125 | 2106
| $3^{4}D_{7/2}$ | 2794 | 2793 | 2792 | 2855 |  |  |
| $4^{4}D_{7/2}$ | 3083 | 3082 | 3081 | 3353 |  |  |
| $5^{4}D_{7/2}$ | 3348 | 3346 | 3344 |  |  |  |
Table 10:  Masses of excited states of $\Xi$ baryon in $(n,M^{2})$ plane. The masses in the boldface are taken as input from Ref. [20] (in MeV). | $N^{2S+1}L_{J}$ | Present |  | Others |  |
---|---|---|---|---|---|---
|  | $\Xi^{0}$ | $\Xi^{-}$ | [21] | [23] | [35] | [37]
(S=1/2) | $1^{2}S_{1/2}$ | 1293 | 1294 | 1322 | 1310 | 1305 | 1317
| $2^{2}S_{1/2}$ | 1886 | 1886 | 1884 | 1876 | 1840 | 1750
| $3^{2}S_{1/2}$ | 2333 | 2332 | 2361 | 2131 | 2100 | 2054
| $4^{2}S_{1/2}$ | 2707 | 2706 | 2935 | 2215 | 2150 | 2149
| $5^{2}S_{1/2}$ | 3035 | 3034 | 3591 |  | 2345 |
| $6^{2}S_{1/2}$ | 3331 | 3330 |  |  |  |
(S=3/2) | $1^{4}S_{3/2}$ | 1537 | 1533 | 1531 | 1539 | 1505 | 1526
| $2^{4}S_{3/2}$ | 1966 | 1966 | 1971 | 1988 | 2045 | 1952
| $3^{4}S_{3/2}$ | 2317 | 2319 | 2457 | 2170 | 2165 | 2114
| $4^{4}S_{3/2}$ | 2621 | 2626 | 3029 | 2257 | 2230 | 2218
| $5^{4}S_{3/2}$ | 2894 | 2900 | 3679 |  |  |
| $6^{4}S_{3/2}$ | 3143 | 3150 |  |  |  |
(S=1/2) | $1^{2}P_{3/2}$ | 1738 | 1738 | 1871 | 1780 | 1785 | 1801
| $2^{2}P_{3/2}$ | 2245 | 2245 | 2337 | 1924 | 1895 | 1976
| $3^{2}P_{3/2}$ | 2657 | 2657 | 2894 | 2353 | 2330 |
| $4^{2}P_{3/2}$ | 3013 | 3013 | 3532 |  |  |
| $5^{2}P_{3/2}$ | 3331 | 3331 |  |  |  |
(S=3/2) | $1^{4}P_{5/2}$ | 1934 | 1931 | 1859 | 1955 | 1900 | 1917
| $2^{4}P_{5/2}$ | 2333 | 2333 | 2318 | 2292 | 2345 |
| $3^{4}P_{5/2}$ | 2673 | 2675 | 2865 | 2438 |  |
| $4^{4}P_{5/2}$ | 2974 | 2978 | 3494 |  |  |
| $5^{4}P_{5/2}$ | 3248 | 3253 |  |  |  |
(S=1/2) | $1^{2}D_{5/2}$ | 2090 | 2090 | 2234 | 2013 | 2045 | 1959
| $2^{2}D_{5/2}$ | 2605 | 2605 | 2771 | 2197 | 2230 | 2170
| $3^{2}D_{5/2}$ | 3034 | 3034 | 3391 | 2279 | 2240 | 2239
| $4^{2}D_{5/2}$ | 3409 | 3409 |  |  |  |
| $5^{2}D_{5/2}$ | 3747 | 3747 |  |  |  |
(S=3/2) | $1^{4}D_{7/2}$ | 2263 | 2260 | 2203 | 2169 | 2180 | 2074
| $2^{4}D_{7/2}$ | 2686 | 2686 | 2729 | 2289 | 2240 | 2189
| $3^{4}D_{7/2}$ | 3051 | 3053 | 3336 |  |  |
| $4^{4}D_{7/2}$ | 3377 | 3381 |  |  |  |
| $5^{4}D_{7/2}$ | 3673 | 3679 |  |  |  |
Table 11:  Masses of excited states of $\Omega$ baryon in $(n,M^{2})$ plane. The masses in the boldface are taken as input from Ref [22] (in MeV).  | $N^{2S+1}L_{J}$ | Present | [20] | [37] | [23] | [35]
---|---|---|---|---|---|---
(S=3/2) | $1^{4}S_{3/2}$ | 1691 | 1678 | 1673 |  | 1635
| $2^{4}S_{3/2}$ | 2057 | 2173 | 2078 | 2177 | 2165
| $3^{4}S_{3/2}$ | 2367 |  |  |  |
| $4^{4}S_{3/2}$ | 2641 |  |  |  |
| $5^{4}S_{3/2}$ | 2889 |  |  |  |
| $6^{4}S_{3/2}$ | 3117 |  |  |  |
(S=3/2) | $1^{4}P_{5/2}$ | 2092 | 2653 |  | 2528 | 2490
| $2^{4}P_{5/2}$ | 2321 |  |  | 2534 |
| $3^{4}P_{5/2}$ | 2529 |  |  |  |
| $4^{4}P_{5/2}$ | 2722 |  |  |  |
| $5^{4}P_{5/2}$ | 2902 |  |  |  |
(S=3/2) | $1^{4}D_{7/2}$ | 2428 | 2369 | 2205 | 2292 | 2295
| $2^{4}D_{7/2}$ | 2623 |  |  |  |
| $3^{4}D_{7/2}$ | 2804 |  |  |  |
| $4^{4}D_{7/2}$ | 2975 |  |  |  |
| $5^{4}D_{7/2}$ | 3136 |  |  |  |
Table 12:  Masses of other excited states of $\Lambda$ baryon in the $(J,M^{2})$ plane. The numbers in the boldface are the experimental values taken as the input [18] (in MeV).  $N^{2S+1}L_{J}$ | Present | PDG [18] | [21] | [20] | [35] | [37]
---|---|---|---|---|---|---
$1^{2}P_{\frac{1}{2}}$ | 1670 | 1670 | 1546 | 1406 | 1550 | 1559
$1^{2}D_{\frac{3}{2}}$ | 1861 | 1890 | 1769 | 1854 | 1900 | 1836
$1^{2}F_{\frac{5}{2}}$ | 2034 | 2082 | 2005 | 2136 | 2180 |
$1^{2}G_{\frac{7}{2}}$ | 2193 |  | 2253 |  |  |
$1^{2}H_{\frac{9}{2}}$ | 2342 |  |  |  |  |
Table 13:  Masses of other excited states of $\Sigma$ baryon in the $(J,M^{2})$ plane. The numbers in the boldface are taken as the input (in MeV).  $N^{2S+1}L_{J}$ | Present | PDG[18] | [20] | [35] | [37] | [23]
---|---|---|---|---|---|---
$1^{2}P_{\frac{1}{2}}$ |  1620 [18] |  | 1620 | 1630 | 1657 | 1628
$1^{4}P_{\frac{1}{2}}$ |  1750 [18] |  | 1693 | 1675 | 1746 | 1771
$1^{4}P_{\frac{3}{2}}$ |  1709 [21] |  | 1731 | 1750 | 1790 | 1728
$1^{2}D_{\frac{3}{2}}$ | 1816 |  | 2025 | 1970 | 1947 | 1961
$1^{4}D_{\frac{3}{2}}$ | 2062 | 2080 | 2076 | 2010 | 1993 | 2011
$1^{4}D_{\frac{5}{2}}$ | 1997 |  | 2062 | 2030 | 2028 | 2027
$1^{2}F_{\frac{5}{2}}$ | 1993 |  | 2347 | 2250 |  | 2226
$1^{4}F_{\frac{5}{2}}$ | 2333 |  |  |  |  |
$1^{4}F_{\frac{7}{2}}$ | 2249 | 2250 | 2349 |  |  | 2285
$1^{2}G_{\frac{7}{2}}$ | 2155 |  |  |  |  |
$1^{4}G_{\frac{7}{2}}$ | 2575 |  |  |  |  |
$1^{4}G_{\frac{9}{2}}$ | 2475 | 2455 |  |  |  |
Table 14:  Masses of other excited states of $\Xi$ baryon in the $(J,M^{2})$ plane. (in MeV).  $N^{2S+1}L_{J}$ | Present | PDG [18] | [20] | [35] | [23] | [37]
---|---|---|---|---|---|---
$1^{2}P_{\frac{1}{2}}$ | 1810 |  | 1682 | 1755 | 1770 | 1772
$1^{4}P_{\frac{1}{2}}$ | 1890 |  | 1758 | 1810 | 1922 | 1894
$1^{4}P_{\frac{3}{2}}$ | 1825 | 1823 | 1798 | 1880 | 1873 | 1918
$1^{2}D_{\frac{3}{2}}$ | 2155 |  | 2100 | 2065 | 2076 | 1970
$1^{4}D_{\frac{3}{2}}$ | 2204 |  | 2121 | 2115 | 2128 | 2065
$1^{4}D_{\frac{5}{2}}$ | 2115 | 2120 | 2147 | 2165 | 2141 | 2102
$1^{2}F_{\frac{5}{2}}$ | 2318 |  | 2411 | 2350 | 2409 |
$1^{4}F_{\frac{5}{2}}$ | 2478 |  |  | 2385 | 2425 |
$1^{4}F_{\frac{7}{2}}$ | 2370 | 2370 | 2474 | 2425 |  |
$1^{2}G_{\frac{7}{2}}$ | 2470 |  |  |  |  |
$1^{4}G_{\frac{7}{2}}$ | 2725 |  |  |  |  |
$1^{4}G_{\frac{9}{2}}$ | 2600 |  |  |  |  |
Table 15:  Masses of other excited states of $\Omega$ baryon in the $(J,M^{2})$ plane (MeV).  $N^{2S+1}L_{J}$ | Present | PDG [18] | [20] | [35] | [23] | [37]
---|---|---|---|---|---|---
$1^{4}P_{\frac{1}{2}}$ | 2030 | 2012 | 2463 | 2410 | 2456 |
$1^{4}P_{\frac{3}{2}}$ | 1941 |  | 2537 | 2440 | 2446 |
$1^{4}D_{\frac{3}{2}}$ | 2338 | 2380 | 2332 | 2345 | 2287 | 2263
$1^{4}D_{\frac{5}{2}}$ | 2249 | 2252 |  | 2345 | 2312 | 2260
$1^{4}F_{\frac{5}{2}}$ | 2610 |  |  |  | 2617 |
$1^{4}F_{\frac{7}{2}}$ | 2519 |  | 2599 |  | 2531 |
$1^{4}G_{\frac{7}{2}}$ | 2856 |  |  |  |  |
$1^{4}G_{\frac{9}{2}}$ | 2763 |  |  |  |  |
Table 16:  Masses of excited states of $\Xi$ baryon in the $(J,M^{2})$ plane including natural, unnatural, and other states (in MeV).  $N^{2S+1}L_{J}$ | Present | PDG [18] | [20] | [35] | [23] | [37]
---|---|---|---|---|---|---
$1^{2}S_{\frac{1}{2}}$ | 1293 | 1315 | 1330 | 1305 | 1310 | 1317
$1^{4}S_{\frac{3}{2}}$ | 1537 | 1532 | 1518 | 1505 | 1539 | 1526
$1^{2}P_{\frac{1}{2}}$ | 1810 |  | 1682 | 1755 | 1770 | 1772
$1^{2}P_{\frac{3}{2}}$ | 1738 |  | 1764 | 1785 | 1780 | 1801
$1^{4}P_{\frac{1}{2}}$ | 1890 |  | 1758 | 1810 | 1922 | 1894
$1^{4}P_{\frac{3}{2}}$ | 1825 | 1823 | 1798 | 1880 | 1873 | 1918
$1^{4}P_{\frac{5}{2}}$ | 1934 | 1950 | 1853 | 1900 | 1955 | 1917
$1^{2}D_{\frac{3}{2}}$ | 2155 |  | 2100 | 2065 | 2076 | 1970
$1^{2}D_{\frac{5}{2}}$ | 2090 | 2025 | 2108 | 2042 | 2013 | 1959
$1^{4}D_{\frac{3}{2}}$ | 2204 |  | 2121 | 2115 | 2128 | 2065
$1^{4}D_{\frac{5}{2}}$ | 2115 | 2120 | 2147 | 2165 | 2141 | 2102
$1^{4}D_{\frac{7}{2}}$ | 2263 | 2250 | 2189 | 2180 | 2169 | 2074
$1^{2}F_{\frac{5}{2}}$ | 2318 |  | 2411 | 2350 | 2409 |
$1^{2}F_{\frac{7}{2}}$ | 2391 | 2370 | 2460 | 2355 | 2320 |
$1^{4}F_{\frac{5}{2}}$ | 2478 |  |  | 2385 | 2425 |
$1^{4}F_{\frac{7}{2}}$ | 2370 | 2370 | 2474 | 2425 |  |
$1^{4}F_{\frac{9}{2}}$ | 2550 |  | 2502 |  | 2505 |
$1^{2}G_{\frac{7}{2}}$ | 2470 |  |  |  |  |
$1^{2}G_{\frac{9}{2}}$ | 2658 |  |  |  |  |
$1^{4}G_{\frac{7}{2}}$ | 2725 |  |  |  |  |
$1^{4}G_{\frac{9}{2}}$ | 2600 |  |  |  |  |
$1^{4}G_{\frac{11}{2}}$ | 2807 |  |  |  |  |
$1^{2}H_{\frac{11}{2}}$ | 2900 |  |  |  |  |
$1^{4}H_{\frac{13}{2}}$ | 3043 |  |  |  |  |

###  II.2 Masses in the $(n,M^{2})$ plane

In this section the masses for radial and orbital excited states from
$1^{2}S_{\frac{1}{2}}-6^{2}S_{\frac{1}{2}},1^{4}S_{\frac{3}{2}}-6^{4}S_{\frac{3}{2}},1^{2}P_{\frac{3}{2}}-5^{2}P_{\frac{3}{2}},1^{4}P_{\frac{5}{2}}-5^{4}P_{\frac{5}{2}},1^{2}D_{\frac{5}{2}}-5^{2}D_{\frac{5}{2}}$
and $1^{4}D_{\frac{7}{2}}-5^{4}D_{\frac{7}{2}}$ are estimated in the
$(n,M^{2})$ plane. The general equation for linear Regge trajectories in the
($n,M^{2}$) plane can be expressed as,

$n=\beta_{0}+\beta M^{2},$ (13)

where $n$ = 1, 2, 3…. is the radial principal quantum number, $\beta_{0}$, and
$\beta$ are the Regge intercept and slope of the trajectories. These
parameters are extracted for each Regge line for $\Lambda$, $\Sigma$, $\Xi$,
and $\Omega$ baryons. Since, the baryon multiplets lying on the single Regge
line have the same Regge slope ($\beta$) and Regge intercept ($\beta_{0}$).
Using relation (13) and the values for $\beta_{0}$ and $\beta$ obtained, we
estimated the excited state masses of light strange baryons ($\Lambda$,
$\Sigma$, $\Xi$, and $\Omega$) lying on each Regge lines for natural and
unnatural parity states.

For instance, using the slope equation, we have
$\beta_{(S)}=1/(M^{2}_{\Xi(2S)}-M^{2}_{\Xi(1S)})$ for $\Xi^{0}$ baryon, where
$M_{\Xi(1S)}$ = 1293 MeV (calculated above) and taking $M_{\Xi(2S)}$ = 1886
MeV from [20] for the $1/2^{+}$ trajectory, we get $\beta_{(S)}$ = 0.53046
GeV-2. From Eq. (13) we can write,

$\begin{split}1=\beta_{0(S)}+\beta_{(S)}M^{2}_{\Xi(1S)},\\\
2=\beta_{0(S)}+\beta_{(S)}M^{2}_{\Xi(2S)},\end{split}$ (14)

using the above relations, we get $\beta_{0(S)}$ = 0.11315. With the help of
$\beta_{(S)}$ and $\beta_{0(S)}$, we calculate the masses of the excited
$\Xi^{0}$ baryon for $n$ = 3, 4, 5… Similarly, we can express these relations
for $P$ and $D$-wave as,

$\begin{split}1=\beta_{0(P)}+\beta_{(P)}M^{2}_{\Xi(1P)},\\\
2=\beta_{0(P)}+\beta_{(P)}M^{2}_{\Xi(2P)},\\\
1=\beta_{0(D)}+\beta_{(D)}M^{2}_{\Xi(1D)},\\\
2=\beta_{0(D)}+\beta_{(D)}M^{2}_{\Xi(2D)},\end{split}$ (15)

Table 7 shows the values of $\beta_{0}$ and $\beta$ for the Regge trajectories
of S, P, and D states for spin S = 1/2 and 3/2. In the same manner, we
estimated the radial and orbital excited states of other light strange baryons
for natural and unnatural parity states. The calculated results are summarized
in tables 8-11.

###  II.3 Other states in the ($J,M^{2}$) plane

So far, we have used conventional formulae to compute the masses of light
strange baryons for natural and unnatural parity states. After the successful
implementation of this model, now in this section, we try to obtain the
remaining other states in the ($J,M^{2}$) plane by using the same method.
Since we have calculated $1^{2}P_{\frac{3}{2}}$ and $1^{4}P_{\frac{5}{2}}$
states earlier, now here we firstly calculate the other three $1P$ states
i.e., $1^{2}P_{\frac{1}{2}}$, $1^{4}P_{\frac{1}{2}}$, and
$1^{4}P_{\frac{3}{2}}$ by using the Eq. (8). For $\Xi$ baryon we put $i=d$,
$j=s$, $q=u$, and $k=d$ in Eq. (8) we have,

$\begin{split}\left[(M_{n}+M_{\Xi})^{2}-4M^{2}_{\Lambda}\right]\\\
=\sqrt{(4M^{2}_{\Lambda}-M^{2}_{n}-M^{2}_{\Xi})^{2}-4M^{2}_{n}M^{2}_{\Xi}};\end{split}$
(16)

After putting the masses in above equation we get masses for
$1^{2}P_{\frac{1}{2}}$, $1^{4}P_{\frac{1}{2}}$, and $1^{4}P_{\frac{3}{2}}$
states. Similarly we can determine the other $1P$ states masses for $\Omega$
baryons. Once we have calculated the $1P$ states, we extract the Regge slopes
for all the light strange baryons in the $(J,M^{2})$ plane as we have done in
previous section. Again using Eq. (1) we have,

$M_{J+1}=\sqrt{M_{J}^{2}+\dfrac{1}{\alpha^{{}^{\prime}}}}.$ (17)

Excited state masses can be obtained by using the above relation. Tables 12 \-
15 shows our calculated results for the remaining other states for light
strange baryons. We compared our estimated masses with experimental data and
other theoretical studies.

##  III Results and Discussion

In the present work, an attempt has been made to obtain the mass spectra of
hyperons under the methodology of Regge Phenomenology. Regge slopes were
calculated in the $(J,M^{2})$ plane. With the aid of these Regge slopes, the
masses of the orbitally excited baryons were estimated for both natural and
unnatural parity states. After that, the Regge slopes and intercepts were
extracted for each Regge line in the $(n,M^{2})$ plane, and with the help of
these parameters mass spectra of light baryons were obtained successfully.
Tables II-V and VI-IX summarizes the calculated masses in the $(J,M^{2})$ and
$(n,M^{2})$ planes respectively, for natural and unnatural parity states along
with the other theoretical outcomes and the experimental observations where
available. Also, our estimated masses for remaining other states in the
$(J,M^{2})$ plane are shown in tables X-XIII.

  1. 1.

For $\Lambda$ baryon the four star and three star status states
$\Lambda(1520)$, $\Lambda(1820)$, $\Lambda(1890)$, $\Lambda(2080)$,
$\Lambda(2100)$, and $\Lambda(2350)$ in the PDG [18] are in good agreement
with our calculated results with a mass difference of 33-80 MeV. We confirmed
the $J^{P}$ values of these states in the present work (see tables 3 and X).
We compared our results with the predictions of other theoretical models also
and our estimated masses are consistent with the results of Refs. [20, 35, 23]
with a mass difference of few MeV as shown in Tables 3, 8, and 12.

  2. 2.

The calculated results in the $(J,M^{2})$ and $(n,M^{2})$ plane for $\Sigma$
baryon are summerized in Tables 4, 9, and 13. For the case of $\Sigma$ the
masses of well established states; $\Sigma(1670)$, $\Sigma(1775)$,
$\Sigma(1915)$, $\Sigma(2030)$, and $\Sigma(2080)$ closely matches with the
masses of our estimated results, and our model confirmed the spin parity of
these states. The PDG states $\Sigma(2455)$, $\Sigma(2620)$, and
$\Sigma(2250)$ are two and three starred, their $J^{P}$ values are still
unknown. The $\Sigma(2250)$ is very close to our predicted mass 2249 MeV
having a mass difference of 1 MeV only (see table 13). So we predicted this
state to be $1^{4}F_{\frac{7}{2}}$ having spin parity $\frac{7}{2}^{-}$. The
states $\Sigma(2455)$ and $\Sigma(2620)$ are fairly close to our estimated
masses 2475 MeV and 2624 MeV respectively. So we can say that these two states
may belong to $1G$ state having $J^{P}=\frac{9}{2}^{+}$ and $\frac{11}{2}^{+}$
(see tables 4 and 13). The calculated masses for low lying resonance fairly
matches with the results of other theoretical predictions [20, 35, 23], but a
wide range of mass difference is shown for higher excited states.

  3. 3.

Our calculated ground-state masses of $\Xi$ baryon 1293 MeV ($\Xi^{0}$), 1294
MeV ($\Xi^{-}$) with $J^{P}=\frac{1}{2}^{+}$ and 1537 MeV ($\Xi^{0*}$), 1533
MeV ($\Xi^{-*}$) with $J^{P}=\frac{3}{2}^{+}$ are found to be very close to
the experimentally well established masses with a few difference of MeV and
also matches very well with the predictions of other theoretical models (see
table 5). Very few states are confirmed with spin-parity in the $\Xi$ family.
The $\Xi(1820)$ is the only negative parity state assigned with
$J^{P}=\frac{3}{2}^{-}$ in PDG having mass 1823 MeV which is some what close
to our predicted mass 1783 having mass difference of 85 MeV and very close
1825 MeV with a slight difference of 2 MeV only for $1^{2}P_{\frac{3}{2}}$ and
$1^{4}P_{\frac{3}{2}}$ respectively. The $\Xi(2030)$ is assigned with angular
momentum having value $\frac{5}{2}$, parity of this state is not confirmed
yet. Here we predicted this state with positive parity having mass 2090 MeV
belongs to $1D$ state having $J^{P}=\frac{5}{2}^{+}$. The $J^{P}$ value of
three stared $\Xi(1950)$ state is still not confirmed in PDG. Our predicted
mass 1934 MeV for $1^{4}P_{\frac{5}{2}}$ state is near to 1950 Mev with a mass
difference of 16 MeV. So, we predicted the spin-parity of this state to be
$\frac{5}{2}^{-}$ for S = 3/2. The two starred state $\Xi(2250)$ is close to
our estimated mass 2263 MeV with a mass difference of 13 MeV, so we predicted
this state as $1D$ state with $J^{P}=\frac{7}{2}^{+}$ for $S=3/2$. The
$\Xi(2370)$ state with two star matches exactly with our predicted mass 2370
MeV (see table 14) and also it is reasonably close to 2391 MeV (see table 5).
So this state may belongs to either $1^{2}F_{\frac{7}{2}}$ or
$1^{4}F_{\frac{7}{2}}$ having $J^{P}=\frac{7}{2}^{-}$. Also, one more state
$\Xi(2120)$ with one star is found to be very close to our mass 2115 MeV with
a mass difference of 5 MeV (see table 14). Hence we can say that, this may
belongs to $1D$ state with $J^{P}=\frac{5}{2}^{+}$ for $S=3/2$. We compared
our calculated results with various theoretical approaches and they are
consistent with the prediction of Refs. [20, 35, 23]. For higher excited
states variation in mass differences observed with other theoretical results.

  4. 4.

The estimated results for $\Omega$ baryon in the $(J,M^{2})$ and $(n,M^{2})$
planes are summerized in tables 6, 11, and 15. Experimentally observed ground
state of $\Omega$ baryon with known spin parity $\frac{3}{2}^{+}$ having mass
1672 MeV is found to be very close to our estimated mass 1691 MeV with slight
mass difference of 19 Mev (see table 6). Four new resonances states of
$\Omega$ baryon; $\Omega(2012)$, $\Omega(2250)$, $\Omega(2380)$, and
$\Omega(2470)$ are experimentally observed. Their spin-parity are still not
confirmed. The three-star state $\Omega(2012)$ is still a mystery. Because it
is so close to the $\Xi(1530)K$ threshold, a recent study suggested that this
state is a molecular one [39]. Other researches [40, 41] indicate that the
current information is insufficient to consider it as a molecular state.
However in the present study this state is close to the predicted mass 2030
MeV vary by 18 MeV (see table 15), so this state may belongs to
$1^{4}P_{\frac{1}{2}}$ having $J^{P}=\frac{1}{2}^{-}$. The $\Omega(2250)$
state having mass 2252 MeV is very close to our calculated mass 2249 MeV with
a slight mass difference of 3 MeV, hence we can say that this state can be a
good candidate of $D$-wave and identified with spin parity
$J^{P}=\frac{5}{2}^{+}$ for S=3/2. The two starred state $\Omega(2380)$ in PDG
is close to both 2338 MeV and 2428 MeV with a mass difference of 42 MeV and 48
MeV respectively. So this state belongs to $D$-wave and may have spin-parity
$\frac{3}{2}^{+}$ or $\frac{7}{2}^{+}$. The $\Omega(2470)$ state is not
identified in this work. We compared our results with other theoretical
predictions as well and they are in good agreement with the assumptions of
Refs. [20, 35, 23, 37]. Also we observed a wide range of mass difference for
$\Omega(2012)$ state with other theoretical results.

##  IV Conclusion

Regge phenomenology has been effectively implemented to study the mass-spectra
of light baryons with strangeness -1,-2, and -3. Here our aim of determining
the spin parity of experimentally observed states : $\Sigma(2250)$,
$\Sigma(2455)$, $\Sigma(2620)$,$\Xi(1950)$, $\Xi(2030)$, $\Xi(2120)$,
$\Xi(2250)$, $\Xi(2370)$, $\Omega(2012)$, $\Omega(2250)$, and $\Omega(2380)$
is accomplished. Also we confirmed the spin parity of many experimentally well
established states of $\Lambda$ and $\Sigma$ baryons. It is evident that the
low-lying resonance masses are in good agreement with other theoretical
predictions but for higher excited states we see a wide range of mass
difference. This possibly because of the fact that not a single model exactly
predicts the spin-parity assignments, and also there is no experimental
evidence for the states. As a result, our work with a huge number of spin-
parity predictions could be useful for future facilities like
$\overline{\mbox{P}}$ANDA, which is intended to explore light strange baryons
in depth.

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