00footnotetext: Received 31 June 2015

# Spectroscopic Investigation of Light Strange S=-1 $\Lambda$, $\Sigma$ and
S=-2 $\Xi$ Baryons

Chandni Menapara  and Ajay Kumar Rai<EMAIL_ADDRESS>Department of
Physics, Sardar Vallabhbhai National Institute of Technology, Surat-395007,
Gujarat, India

###### Abstract

The present study is dedicated to light-strange $\Lambda$ with strangeness
S=-1, isospin I=0; $\Sigma$ with S=-1, I=1 and $\Xi$ baryon with S=-2 and
$I=\frac{1}{2}$. In this article, hypercentral Constituent Quark Model with
linear confining potential has been employed along with first order correction
term to obtain the resonance masses for nearly upto 4 GeV. The calculated
states include 1S-5S, 1P-4P, 1D-3D, 1F-2F and 1G (in few case) along with all
the possible spin-parity assignments. Regge Trajectories have been explored
for the linearly of the calculated masses for $(n,M^{2})$ and $(J,M^{2})$
respectively. Magnetic moments have been intensively studied for ground state
spin $\frac{1}{2}$ and $\frac{3}{2}$, in addition to the configuration mixing
for first negative parity state for $\Xi$. Lastly, transition magnetic moment
and radiative decay width have been presented.

###### keywords:

Mass spectra, Strange baryon, Regge trajectory, Magnetic moment

###### pacs:

1

– 3 PACS codes (Physics and Astronomy Classification Scheme,
https://publishing.aip.org/publishing/pacs/pacs-2010-regular-edition/)

00footnotetext: $\scriptstyle\copyright$2013 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd

##  1 Introduction

The objective for the study of hadrons is to reveal the possible degrees of
freedom responsible for the way a given system appears. The quark confinement
and asymptotic freedom has been the starting point of any theoretical and
phenomenological study to understand quark dynamics. Hadron spectroscopy
attempts to explore the excited mass spectrum along with the multiplet
structure as well as spin-parity assignments. The light quark baryons form the
basis of octet and decuplet ranging from strangeness S=0 to S=-3 based on the
symmetric, asymmetric and mixed-symmetric flavor-spin combinations.

$3\otimes 3\otimes 3=1^{A}\oplus 8^{M}\oplus 8^{M}\oplus 10^{S}$

The presence of strange quark in a baryon draws attention because of the fact
that it is a bit heavier compared to u and d quarks whereas considerably light
compared to c and b quarks. Particularly the strangeness S=-2 $\Xi$ baryons
have not been observed experimentally like other light sector baryons [2] as
depicted in table 1. The limited observations in the $\Xi$ baryon group owes
to the fact that they are produced only as the final state in a process in
addition to considerably small cross-sections [3]. Unlike the $\Xi$ baryons,
$\Sigma$ and $\Lambda$ with S=-1 have quite a number of experimentally
established states.

The present article is dedicated to the study of $\Lambda$, $\Sigma$ and $\Xi$
baryons. Cascade baryons appear with isospin $I=\frac{1}{2}$ in the octet
($J=\frac{1}{2}$) and decuplet ($J=\frac{3}{2}$) as $\Xi$ and $\Xi^{*}$
respectively. The quark combination is uss for $\Xi^{0}$ and dss for
$\Xi^{-}$.
For mixed symmetry flavour wave-function in octet group,

$\Xi^{0}\rightarrow\quad\frac{1}{\sqrt{6}}(sus+uss-2ssu)$
$\Xi^{-}\rightarrow\quad\frac{1}{\sqrt{6}}(dss+sds-2ssd)$
$\Xi^{0}\rightarrow\quad\frac{1}{\sqrt{2}}(uss-sus)$
$\Xi^{-}\rightarrow\quad\frac{1}{\sqrt{2}}(dss-sds)$

For symmetric flavour wave-function in decuplet group,

$\Xi^{*0}\rightarrow\quad\frac{1}{\sqrt{3}}(uss+sus+ssu)$
$\Xi^{*-}\rightarrow\quad\frac{1}{\sqrt{3}}(dss+sds+ssd)$

Similarly the $\Sigma$ baryon with u, d and s constituent quarks have place in
octet and decuplet with three possible combinations as uus, uds and dds
respectively. The $\Lambda$ baryon appearing in octet as I=0 has uds quark
content however, it differs from $\Sigma^{0}$ based on the wave-function.

Table 1: The $\Xi$ listed by Particle Data Group (PDG[2]) State | $J^{P}$ | Status
---|---|---
$\Xi^{0}$(1314) | $\frac{1}{2}^{+}$ | ****
$\Xi^{-}$(1321) | $\frac{1}{2}^{+}$ | ****
$\Xi$(1530) | $\frac{3}{2}^{+}$ | ****
$\Xi$(1620) |  | *
$\Xi$(1690) |  | ***
$\Xi$(1820) | $\frac{3}{2}^{-}$ | ***
$\Xi$(1950) |  | ***
$\Xi$(2030) | $\frac{5}{2}^{?}$ | ***
$\Xi$(2120) |  | *
$\Xi$(2250) |  | **
$\Xi$(2370) |  | **
$\Xi$(2500) |  | *

Experimental facilities across the world have been striving towards the study
of strange hyperons. A recent study at CERN by ALICE Collaboration has
established an attractive interaction of proton and $\Xi^{-}$ [4]. Activities
in measuring weak decays of $\Xi$ hyperon were reported by the KTeV
Collaboration [5] and by the NA48/1 Collaboration [6] as well as BABAR
Collaboration has been carrying out extensive studies [7]. The photoproduction
of $\Xi$ has been observed by CLAS detector at Jefferson Lab [8]. ALso,
recently JLab has proposed to explore the strange hyperon spectroscopy through
secondary KL beam alongwith GlueX experiment [9] and the findings shall
expectantly give new directions and understanding of strange hyperons
$\Sigma$, $\Lambda$ and $\Xi$. The BESIII Collaboration observed $\Xi$(1530)
in baryon-antibaryon pair from charmonium decay [10]. The upcoming
experimental facility PANDA at FAIR-GSI has been highly expected to establish
the whole spectrum of hyperons through proton-antiproton collisions [11]. A
$\Xi$ dedicated study has been undertaken for mass, widths and decay modes by
one of the PANDA group [12, 13].

In case of $\Sigma$ and $\Lambda$ baryons, all the properties are not known
completely. Most of the data for strange baryons have been based from earlier
studies from bubble chamber for $K^{-}$ reactions. The $\Lambda$(1405) with
$J^{P}=\frac{1}{2}^{-}$ is still a mysterious state in the lambda spectrum.
This state is lower than the non-strange counterpart $N^{*}$(1535). Recent
studies have attempted to understand this state as some hadronic molecular
[14, 15]. JPAC and Osaka-ANL group have applied coupled channel approach to
study this dynamics [16, 17]. Also, two pole structure of $\Lambda$(1405) is
analyzed using chiral effective field theory [18]. E. Klempt et al. has
extensively reviewed the $\Lambda$ and $\Sigma$ hyperon spectrum based on
experimental and theoretical studies focusing on all the known states of the
spectrum [19]. These spectrums are studied through photoproduction off the
proton in ref [20]. The study of strangeness S=-1, -2 becomes more interesting
not only in high energy arena but also in astrophysical bodies as in neutron
stars [21].

The excited states of hyperons have been investigated using phenomenological
as well as theoretical approaches. Various models have attempted to reproduce
the octet and decuplet ground states and a range of excited states. A recent
review has summarized in details few of the states of strange baryons spectrum
with theoretical and experimental aspects [22]. L. Xiao et al. has intensively
studied the strong decays of $\Xi$ under Chiral quark model which may assist
in determining possible spin-parity of a given state in strong decay [23].
Some of the models have been summarized briefly in Section 3 which ranges from
relativistic appraoch [30], instanton induced model [31], CI model [32],
algebraic model [33], Skyrme model [36], etc. The present work has been lead
on the phenomenological non-relativistic hypercentral Constituent Quark Model
[37]. Similar studies have been carried out for nucleons and delta baryons
earlier which serve as a key to proceed towards exploring the strange baryons
[45, 46].

This paper is organized as follows: after the briefing of hadron spectroscopy
and various experimental and theoretical approaches, section 2 deals with the
background of the model used. Section 3 describes the results for resonance
masses alongwith the detailed discussion about the data presented in
respective tables. The later section 4 represents the Regge trajectories and
deductions based on them. Section 5 is dedicated to magnetic moments of
isospin states of $\Xi$, $\Sigma$ and $\Lambda$ based on their effective
masses. Section 6 focuses on the transition magnetic moment and radiative
decay widths and lastly the conclusion.

##  2 Theoretical Background

The Constituent Quark Model (CQM) is based on a simple assumption of baryon as
a system of three quarks (or anti-quarks)interacting by some potential which
ultimately may provide some quantitative description of baryonic properties.
It is obvious that the QCD quark masses are considerably smaller than
constituent quark masses however this large mass parametrizes all the other
effects within a baryon. Thus, the CQMs have been employed in various studies
through various modifications in non-relativistic or semi-relativistic
approaches.

The hypercentral Constituent Quark Model (hCQM) has been employed for the
present study which is a non relativistic approach [24]. It undertakes the
baryon as a confined system of three quarks wherein the potential is
hypercentral one. The dynamics of three body system are taken care of using
the Jacobi coordinates introduced as $\rho$ and $\lambda$ reducing it to two
body parameters [25].

$\displaystyle{\bf\rho}=\frac{1}{\sqrt{2}}({\bf r_{1}}-{\bf r_{2}})$ (1a)
$\displaystyle{\bf\lambda}=\frac{(m_{1}{\bf r_{1}}+m_{2}{\bf
r_{2}}-(m_{1}+m_{2}){\bf
r_{3}})}{\sqrt{m_{1}^{2}+m_{2}^{2}+(m_{1}+m_{2})^{2}}}$ (1b)

The hyperradius and hyperangle are defined as

$x=\sqrt{{\bf\rho^{2}}+{\bf\lambda^{2}}};\;\;\xi=arctan(\frac{\rho}{\lambda})$
(2)

The the Hamiltonian of the system is written with potential term solely
depending on hyperradius x of three body systems

$H=\frac{P^{2}}{2m}+V^{0}(x)+V_{SD}(x)$ (3)

where $m=\frac{2m_{\rho}m_{\lambda}}{m_{\rho}+m_{\lambda}}$ is the reduced
mass. Thus the hyper-radial part of the wave-function as determined by
hypercentral Schrodinger equation is [37]

$\left[\frac{d^{2}}{dx^{2}}+\frac{5}{x}\frac{d}{dx}-\frac{\gamma(\gamma+4)}{x^{2}}\right]\psi(x)=-2m[E-V(x)]\psi(x)$
(4)

Here $\gamma$ replaces the angular momentum quantum number by the relation as
$l(l+1)\rightarrow\frac{15}{4}+\gamma(\gamma+4)$. The choice of hypercentral
potential narrows down to hyperCoulomb one i.e. $-\frac{\tau}{x}$. The
confinement term here is chosen to be of linear nature.

$V^{0}(x)=-\frac{\tau}{x}+\alpha x$ (5)

Here, $\tau=\frac{2}{3}\alpha_{s}$ with $\alpha_{s}$ being running coupling
constant and $\alpha$ is a parameter based on the fitting of ground state for
a given system.

The $V_{SD}(x)$ accounts for the spin-dependent terms leading to hyperfine
interactions.

$\begin{split}V_{SD}(x)=V_{SS}(x)({\bf S_{\rho}\cdot S_{\lambda}})+V_{\gamma
S}(x)({\bf\gamma\cdot S})\\\ +V_{T}\times[S^{2}-\frac{3({\bf S\cdot x})({\bf
S\cdot x})}{x^{2}}]\end{split}$ (6)

where $V_{SS}(x)$, $V_{\gamma S}(x)$ and $V_{T}(x)$ are spin-spin, spin-orbit
and tensor terms respectively [27]. In the present study, first order
correction term to potential with $\frac{1}{m}$ dependence has also been added
as $\frac{1}{m}V^{1}(x)$ [28].

$V^{1}(x)=-C_{F}C_{A}\frac{\alpha_{s}^{2}}{4x^{2}}$ (7)

where $C_{F}$ and $C_{A}$ are Casimir elements of fundamental and adjoint
representation.
The resonance masses have been obtained with and without the firt order
correction term. The constituent quark masses have been taken to be
$m_{u}=m_{d}=0.290$ MeV and $m_{s}=0.500$ MeV. Mathematica Notebook has been
employed for numerical solutions [29].

##  3 Results and Discussion for the Resonance Mass Spectra

In the present work, the resonance masses are calculated for radial and
orbital states from 1S-5S, 1P-4P, 1D-3D and 1F-2F. Also, all the possible
spin-parity configuration for each state with $S=\frac{1}{2}$ and
$S=\frac{3}{2}$ have been considered and calculated the respective
contribution based on the model discussed in section 2 including with and
without first order corrections. For S-state, possible total angular momentum
and parity are $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$, for P-state the range
goes from $\frac{1}{2}^{-}$ to $\frac{5}{2}^{-}$, for D-state the range is
$\frac{3}{2}^{+}$ to $\frac{7}{2}^{+}$ and for F-state, it is
$\frac{3}{2}^{-}$ to $\frac{9}{2}^{-}$. Only few experimentally established
states with four star status available are mentioned in respective tables. In
the tables, $Mass_{cal}1$ and $Mass_{cal}2$ represents resonance masses
without and with first order correction respectively in the units of MeV.

Table 2: S-wave of $\Xi$ baryon (in MeV) State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$ [2]
---|---|---|---|---
1S | $\frac{1}{2}^{+}$ | 1322 | 1321 | 1321
| $\frac{3}{2}^{+}$ | 1531 | 1524 | 1532
2S | $\frac{1}{2}^{+}$ | 1884 | 1891 |
| $\frac{3}{2}^{+}$ | 1971 | 1964 |
3S | $\frac{1}{2}^{+}$ | 2361 | 2372 |
| $\frac{3}{2}^{+}$ | 2457 | 2459 |
4S | $\frac{1}{2}^{+}$ | 2935 | 2954 |
| $\frac{3}{2}^{+}$ | 3029 | 3041 |
5S | $\frac{1}{2}^{+}$ | 3591 | 3620 |
| $\frac{3}{2}^{+}$ | 3679 | 3702 |
Table 3: P-wave of $\Xi$ baryon (in MeV) State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1886 | 1889 |
$1^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1871 | 1873 | 1823
$1^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1894 | 1897 |
$1^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1879 | 1881 | 1823
$1^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1859 | 1859 |
$2^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2361 | 2373 |
$2^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2337 | 2347 |
$2^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2373 | 2386 |
$2^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2349 | 2360 |
$2^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2318 | 2325 |
$3^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2929 | 2948 |
$3^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2894 | 2913 |
$3^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2946 | 2966 |
$3^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2912 | 2931 |
$3^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2865 | 2884 |
$4^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 3577 | 3609 |
$4^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 3532 | 3563 |
$4^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 3599 | 3632 |
$4^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 3554 | 3586 |
$4^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 3494 | 3524 |
Table 4: D-wave of $\Xi$ baryon (in MeV) State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2270 | 2281 |
$1^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2234 | 2244 |
$1^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2310 | 2322 |
$1^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2283 | 2295 |
$1^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2247 | 2257 |
$1^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2203 | 2211 |
$2^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2819 | 2842 |
$2^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2771 | 2791 |
$2^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2874 | 2899 |
$2^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2838 | 2861 |
$2^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2790 | 2810 |
$2^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2729 | 2747 |
$3^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 3455 | 3489 |
$3^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 3391 | 3423 |
$3^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 3527 | 3562 |
$3^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 3479 | 3513 |
$3^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 3415 | 3448 |
$3^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 3336 | 3366 |
Table 5: F-wave of $\Xi$ baryon (in MeV) State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2713 | 2736 |
$1^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2647 | 2666 |
$1^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2786 | 2813 |
$1^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2733 | 2757 |
$1^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2667 | 2687 |
$1^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2588 | 2603 |
$2^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 3333 | 3368 |
$2^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 3249 | 3280 |
$2^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 3426 | 3465 |
$2^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 3358 | 3394 |
$2^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 3274 | 3306 |
$2^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 3173 | 3201 |
Table 6: Comparison of masses with other predictions based on $J^{P}$ value for $\Xi$ baryon (in MeV) $J^{P}$ | $Mass_{cal}1$ | $Mass_{cal}2$ | [30] | [31] | [32] | [33] | [34] | [35] | [36] | [3] | [43] | [44]
---|---|---|---|---|---|---|---|---|---|---|---|---
$\frac{1}{2}^{+}$ | 1322 | 1321 | 1330 | 1310 | 1305 | 1334 | 1348 | 1317 | 1318 | 1325 | 1317 | 1303 $\pm$ 13
| 1884 | 1891 | 1886 | 1876 | 1840 | 1727 | 1805 | 1772 | 1932 | 1891 | 1750 | 2178 $\pm$ 48
| 2310 | 2322 | 1993 | 2062 | 2040 | 1932 | 1868 |  |  |  | 1980 | 2231 $\pm$ 44
| 2361 | 2372 | 2012 | 2131 | 2100 |  |  | 1874 |  |  | 2054 | 2408 $\pm$ 45
| 2874 | 2899 | 2091 | 2176 | 2130 |  |  |  |  |  | 2107 |
| 2935 | 2954 | 2142 | 2215 | 2150 |  |  |  |  |  | 2149 |
| 3527 | 3562 | 2367 | 2249 | 2230 |  |  |  |  |  | 2254 |
| 3591 | 3620 |  |  | 2345 |  |  |  |  |  |  |
$\frac{3}{2}^{+}$ | 1531 | 1524 | 1518 | 1539 | 1505 | 1524 | 1528 | 1552 | 1539 | 1520 | 1526 | 1553 $\pm$ 18
| 1971 | 1964 | 1966 | 1988 | 2045 | 1878 |  | 1653 | 2120 | 1934 | 1952 | 2228 $\pm$ 40
| 2270 | 2281 | 2100 | 2076 | 2065 | 1979 |  |  |  |  | 1970 | 2398 $\pm$ 52
| 2283 | 2295 | 2121 | 2128 | 2115 |  |  |  |  |  | 2065 | 2574 $\pm$ 52
| 2457 | 2459 | 2122 | 2170 | 2165 |  |  |  |  |  | 2114 |
| 2819 | 2842 | 2144 | 2175 | 2170 |  |  |  |  |  | 2174 |
| 2838 | 2861 | 2149 | 2219 | 2210 |  |  |  |  |  | 2184 |
| 3029 | 3041 | 2421 | 2257 | 2230 |  |  |  |  |  | 2218 |
| 3455 | 3489 |  | 2279 | 2275 |  |  |  |  |  | 2252 |
| 3479 | 3513 |  |  |  |  |  |  |  |  |  |
| 3679 | 3702 |  |  |  |  |  |  |  |  |  |
$\frac{5}{2}^{+}$ | 2234 | 2242 | 2108 | 2013 | 2045 |  |  |  |  | 1936 | 1959 |
| 2247 | 2295 | 2147 | 2141 | 2165 |  |  |  |  | 2025 | 2102 |
| 2771 | 2791 | 2213 | 2197 | 2230 |  |  |  |  |  | 2170 |
| 2790 | 2810 |  | 2231 | 2230 |  |  |  |  |  | 2205 |
| 3391 | 3423 |  | 2279 | 2240 |  |  |  |  |  | 2239 |
| 3415 | 3448 |  |  |  |  |  |  |  |  |  |
$\frac{7}{2}^{+}$ | 2203 | 2211 | 2189 | 2169 | 2180 |  |  |  |  | 2035 | 2074 |
| 2729 | 2747 |  | 2289 | 2240 |  |  |  |  |  | 2189 |
| 3336 | 3366 |  |  |  |  |  |  |  |  |  |
Table 7: Comparison of masses with other predictions based on $J^{P}$ value $\Xi$ baryon (in MeV) $J^{P}$ | $Mass_{cal}1$ | $Mass_{cal}2$ | [30] | [31] | [32] | [33] | [34] | [35] | [36] | [3] | [43] | [44] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\frac{1}{2}^{-}$ | 1886 | 1889 | 1682 | 1770 | 1755 | 1869 |  |  | 1658 | 1725 | 1772 | 1716 $\pm$ 43 |
| 1894 | 1897 | 1758 | 1922 | 1810 | 1932 |  |  |  | 1811 | 1894 | 1837 $\pm$ 28 |
| 2361 | 2373 | 1839 | 1938 | 1835 | 2076 |  |  |  |  | 1926 | 1844 $\pm$ 43 |
| 2373 | 2386 | 2160 | 2241 | 2225 |  |  |  |  |  |  | 2758 $\pm$ 78 |
| 2929 | 2948 | 2210 | 2266 | 2285 |  |  |  |  |  |  |  |
| 2946 | 2966 | 2233 | 2387 | 2300 |  |  |  |  |  |  |  |
| 3577 | 3609 | 2261 | 2411 | 2320 |  |  |  |  |  |  |  |
| 3599 | 3632 |  | 2445 | 2380 |  |  |  |  |  |  |  |
$\frac{3}{2}^{-}$ | 1871 | 1873 | 1764 | 1780 | 1785 | 1828 | 1792 | 1861 | 1820 | 1759 | 1801 | 1906 $\pm$ 29 |
| 1879 | 1881 | 1798 | 1873 | 1880 | 1869 |  | 1971 |  | 1826 | 1918 | 1894 $\pm$ 38 |
| 2337 | 2347 | 1904 | 1924 | 1895 | 1932 |  |  |  |  | 1976 | 2497 $\pm$ 61 |
| 2349 | 2360 | 2245 | 2246 | 2240 |  |  |  |  |  |  | 2426 $\pm$ 73 |
| 2786 | 2813 | 2252 | 2284 | 2305 |  |  |  |  |  |  |  |
| 2894 | 2913 | 2350 | 2353 | 2330 |  |  |  |  |  |  |  |
| 2912 | 2931 | 2352 | 2384 | 2340 |  |  |  |  |  |  |  |
| 3426 | 3465 |  | 2416 | 2385 |  |  |  |  |  |  |  |
| 3532 | 3563 |  |  |  |  |  |  |  |  |  |  |
| 3554 | 3586 |  |  |  |  |  |  |  |  |  |  |
$\frac{5}{2}^{-}$ | 1859 | 1859 | 1853 | 1955 | 1900 | 1881 |  |  |  | 1883 | 1917 |  |
| 2318 | 2325 | 2333 | 2292 | 2345 |  |  |  |  |  |  |  |
| 2713 | 2736 | 2411 | 2409 | 2350 |  |  |  |  |  |  |  |
| 2733 | 2757 |  | 2425 | 2385 |  |  |  |  |  |  |  |
| 2865 | 2884 |  | 2438 |  |  |  |  |  |  |  |  |
| 3333 | 3368 |  |  |  |  |  |  |  |  |  |  |
| 3358 | 3394 |  |  |  |  |  |  |  |  |  |  |
| 3494 | 3524 |  |  |  |  |  |  |  |  |  |  |
$\frac{7}{2}^{-}$ | 2647 | 2666 | 2460 | 2320 | 2355 |  |  |  |  |  |  |  |
| 2667 | 2687 | 2474 |  | 2425 |  |  |  |  |  |  |  |
| 3249 | 3280 |  | 2464 |  |  |  |  |  |  |  |  |
| 3274 | 3306 |  | 2481 |  |  |  |  |  |  |  |  |
$\frac{9}{2}^{-}$ | 2588 | 2603 | 2502 | 2505 |  |  |  |  |  |  |  |  |
| 3173 | 3201 |  | 2570 |  |  |  |  |  |  |  |  |

An attempt has been made to summarize the calculated masses in present article
along with those by different models. The following tables 6, 13, 19, 7, 14
and 20 depict the range of masses for a given $J^{P}$ value irrespective of
assigned state in increasing order. It is evident that the low lying resonance
states are within considerable range for almost all the models and approaches
listed. However, the higher states have huge variations possibly because of
the fact that not a single model exactly predicts the spin-parity assignments
and in addition to it there are no experimental evidence for the states. Also,
the present calculations have included masses upto 4 GeV.

The Ref. [30] has employed relativistic quark-diquark model for the
calculation of strange baryon mass spectra. As the model considers both ground
and excited states of diquarks, the number of excited states are limited and
only confined to lower states. The another relativistic approach based on
three quark Bethe-Salpeter equation with instantaneous two and three body
forces is described by Ref. [31]. It has introduced instanton induced
hyperfine splitting of positive and negative parity states. Another approach
is well known relativised Capstick-Isgur model with higher order spin-
dependent potential terms in three quark system and has predicted masses for
nearly 2 GeV [32].

The approach by R.Bijker et al. [33] uses the algebraic model. It is based on
collective string-like qqq wherein the orbital excitations are treated as
rotations and vibrations of the strings. Low-lying states are established by
this model for octet and decuplet class but exact spin-parity are not assigned
in the case of $\Xi$. Ref. [34] has utilised relativistic constituent quark
model (RCQM) with Goldstone-boson exchange. Another relativistic quark-diquark
approach with Coulomb plus linear interaction along with an exchange term
which is inspired by Gürsey-Radicati has been employed in Ref. [35] for low-
lying resonance states of $\Xi$. Y. Oh [36] has studied the cascade and omega
baryons through Skyrme model which is based on the approximation of equal mass
splitting of hyperon resonances. The mass formula is developed with isospin
and spin in the soliton-kaon bound-state model. The $\Xi$ has also been
explored in large-$N_{c}$ limit [38, 39] as well as through QCD-SUM Rule
method [42]. M. Pervin [3] has obtained mass spectra using a non-relativistic
quark model approach. Y. Chen et al. has implemented a different approach with
non-relativistic quark model supplemented with hyperfine interaction of higher
order $O(\alpha_{s}^{2})$ [43]. Some low-lying states have been exploited
through dynamical chirally improved quarks by BGR Collaboration [44].

The ground state of $\Xi$ has been very well established with known spin-
parity at 1321 MeV and $J^{P}=\frac{1}{2}^{+}$ in the octet family. It is
evident from table 5 that ground state fits well for nearly all the models
owing the little variations to the assumptions of any given model. Another
state is 1532 MeV with $J^{P}=\frac{3}{2}^{+}$ holding a place in decuplet.
The mass for this state varies within 20 MeV among all the models discussed.
The only negative parity state by PDG is 1823 MeV at $J^{P}=\frac{3}{2}^{-}$
which is obtained as 1871 MeV and 1879 MeV for octet and decuplet $\Xi$.

$\Xi$(1690) is fairly known state in the PDG database however the spin-parity
assignments and exact mass predictions vary a lot. The BABAR Collaboration
concluded the state to be spin $\frac{1}{2}$ [7]. As shown in Table 6, various
models have predicted this state in a comparatively higher mass from PDG the
nearest being 1682 MeV. Also, due to intrinsic drawback of the present model,
it could not provide conclusive assignment of this state. The $\Xi$(2030) with
assigned angular momentum value to be $\frac{5}{2}$ is predicted here to be
2234 MeV with positive parity. Other models predictions vary within 200 MeV
range for the same spin-parity. $\Xi$(1620) appears in PDG with one star
status. However, any such state is not established in this work but one study
throws the light on the existence of $\Xi$(1620) and $\Xi$(1690) [47].

The PDG states $\Xi$(1950), $\Xi$(2250) and $\Xi$(2370) are three and two
starred however due to lack of spin-parity assignment, the comparison is not
reasonable. In addition, one study depicts the states of cascade around
$\Xi$(1950) within the range of 1900-2000 MeV into three different states
[48]. Few of the states of our results are comparatively near to the BGR work
[44] which don’t appear in other approaches.

Table 8: S-wave of $\Lambda$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
1S | $\frac{1}{2}^{+}$ | 1115 | 1115 | 1115
2S | $\frac{1}{2}^{+}$ | 1592 | 1589 | 1600
3S | $\frac{1}{2}^{+}$ | 1885 | 1892 | 1810
4S | $\frac{1}{2}^{+}$ | 2202 | 2220 |
5S | $\frac{1}{2}^{+}$ | 2540 | 2571 |

Table 9: P-wave of $\Lambda$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$ [2]
---|---|---|---|---
$1^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1546 | 1558 | 1670
$1^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1534 | 1544 | 1520
$1^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1553 | 1564 |
$1^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1540 | 1551 |
$1^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1524 | 1533 |
$2^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1834 | 1858 | 1800
$2^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1819 | 1841 | 1690
$2^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1841 | 1867 |
$2^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1827 | 1850 |
$2^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1807 | 1827 | 1830
$3^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2149 | 2186 |
$3^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2131 | 2166 |
$3^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2158 | 2196 |
$3^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2140 | 2176 |
$3^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2116 | 2149 |
$4^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2484 | 2536 |
$4^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2464 | 2513 |
$4^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2495 | 2548 |
$4^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2474 | 2525 |
$4^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2447 | 2494 |

Table 10: D-wave of $\Lambda$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 1769 | 1789 | 1890
$1^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1746 | 1767 | 1820
$1^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 1794 | 1814 |
$1^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 1777 | 1798 |
$1^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1755 | 1776 |
$1^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 1727 | 1748 |
$2^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2076 | 2113 | 2070
$2^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2051 | 2085 | 2110
$2^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2105 | 2144 |
$2^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2086 | 2123 |
$2^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2060 | 2096 |
$2^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2029 | 2061 | 2085
$3^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2408 | 2459 |
$3^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2378 | 2426 |
$3^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2441 | 2496 |
$3^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2419 | 2471 |
$3^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2389 | 2438 |
$3^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2352 | 2398 |

Table 11: F-wave of $\Lambda$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2005 | 2039 |
$1^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 1970 | 2002 | 2100
$1^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2043 | 2079 |
$1^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2015 | 2050 |
$1^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 1980 | 2013 |
$1^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 1939 | 1969 |
$2^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2329 | 2380 |
$2^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2291 | 2337 |
$2^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2371 | 2427 |
$2^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2341 | 2393 |
$2^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2303 | 2350 |
$2^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2257 | 2299 |
$3^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2676 | 2741 |
$3^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2632 | 2693 |
$3^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2723 | 2793 |
$3^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2689 | 2755 |
$3^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2645 | 2707 |
$3^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2593 | 2650 |

Table 12: G-wave of $\Lambda$ baryon (in MeV) State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}G_{7/2}$ | $\frac{7}{2}^{+}$ | 2253 | 2302 |
$1^{2}G_{9/2}$ | $\frac{9}{2}^{+}$ | 2204 | 2246 | 2350
$1^{4}G_{5/2}$ | $\frac{5}{2}^{+}$ | 2305 | 2363 |
$1^{4}G_{7/2}$ | $\frac{7}{2}^{+}$ | 2265 | 2316 |
$1^{4}G_{9/2}$ | $\frac{9}{2}^{+}$ | 2216 | 2260 |
$1^{4}G_{11/2}$ | $\frac{11}{2}^{+}$ | 2159 | 2195 |
Table 13: Comparison of masses with other predictions based on $J^{P}$ value for $\Lambda$ baryon (in MeV) $J^{P}$ | $Mass_{cal}1$ | $Mass_{cal}2$ | [30] | [31] | [32] | [33] | [34] | [35] | [40] | [41] | [43] | [44]
---|---|---|---|---|---|---|---|---|---|---|---|---
$\frac{1}{2}^{+}$ | 1115 | 1115 | 1115 | 1108 | 1115 | 1133 | 1136 | 1116 | 1116 | 1112 | 1113 | 1149 $\pm$ 18
| 1592 | 1589 | 1615 | 1677 | 1680 | 1577 | 1625 | 1518 | 1600 | 1695 | 1606 | 1807 $\pm$ 94
| 1794 | 1814 |  |  |  | 1799 | 1799 | 1666 | 1810 |  | 1764 | 2112 $\pm$ 54
| 1885 | 1892 | 1901 | 1747 | 1830 |  |  | 1955 |  |  | 1880 | 2137 $\pm$ 68
| 2105 | 2144 | 1972 | 1898 | 1910 |  |  | 1960 |  |  | 2013 |
| 2202 | 2220 | 1986 | 2077 | 2010 |  |  |  |  |  | 2173 |
| 2441 | 2496 | 2042 | 2099 | 2105 |  |  |  |  |  | 2198 |
| 2540 | 2571 | 2099 | 2132 | 2120 |  |  |  |  |  |  |
$\frac{3}{2}^{+}$ | 1769 | 1789 | 1854 | 1823 | 1900 | 1849 |  | 1896 | 1890 | 1903 | 1836 | 1991 $\pm$ 103
| 1777 | 1798 | 1976 | 1952 | 1960 |  |  |  | 2000 |  | 1958 | 2058 $\pm$ 139
| 2076 | 2113 | 2130 | 2045 | 1995 |  |  |  |  | 1993 |  | 2481 $\pm$ 111
| 2086 | 2123 | 2184 | 2087 | 2050 |  |  |  |  | 2061 |  |
| 2408 | 2459 | 2202 | 2133 | 2080 |  |  |  |  | 2121 |  |
| 2419 | 2471 |  |  |  |  |  |  |  |  | 2134 |
$\frac{5}{2}^{+}$ | 1746 | 1767 | 1825 | 1834 | 1890 | 1849 |  | 1896 | 1820 | 1846 | 1839 |
| 1755 | 1776 | 2098 | 1999 | 2035 | 2074 |  |  | 2110 | 2132 | 2008 |
| 2051 | 2085 | 2221 | 2078 | 2115 |  |  |  |  |  | 2103 |
| 2060 | 2096 | 2255 | 2127 | 2115 |  |  |  |  |  | 2129 |
| 2305 | 2363 | 2258 | 2150 | 2180 |  |  |  |  |  | 2155 |
| 2378 | 2426 |  |  |  |  |  |  |  |  |  |
| 2389 | 2471 |  |  |  |  |  |  |  |  |  |
$\frac{7}{2}^{+}$ | 1727 | 1748 | 2251 | 2130 | 2120 |  |  |  | 2020 |  | 2064 |
| 2029 | 2061 | 2471 | 2331 |  |  |  |  |  |  |  |
| 2253 | 2302 |  |  |  |  |  |  |  |  |  |
| 2265 | 2316 |  |  |  |  |  |  |  |  |  |
| 2352 | 2398 |  |  |  |  |  |  |  |  |  |
$\frac{9}{2}^{+}$ | 2204 | 2246 | 2360 | 2340 |  | 2357 |  |  | 2350 | 2360 |  |
| 2216 | 2260 |  |  |  |  |  |  |  |  |  |
$\frac{11}{2}^{+}$ | 2159 | 2195 |  |  |  |  |  |  | 2585 |  |  |
Table 14: Comparison of masses with other predictions based on $J^{P}$ value for $\Lambda$ baryon (in MeV) $J^{P}$ | $Mass_{cal}1$ | $Mass_{cal}2$ | [30] | [31] | [32] | [33] | [34] | [35] | [40] | [41] | [43] | [44] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\frac{1}{2}^{-}$ | 1546 | 1558 | 1406 | 1524 | 1550 | 1686 | 1556 | 1650 | 1670 | 1679 | 1559 | 1416 $\pm$ 81 |
| 1553 | 1564 | 1667 | 1630 | 1615 | 1799 | 1682 | 1732 |  | 1830 | 1656 | 1546 $\pm$ 110 |
| 1834 | 1858 | 1733 | 1816 | 1675 |  | 1778 | 1969 | 1800 |  | 1791 | 1713 $\pm$ 116 |
| 1841 | 1867 | 1927 | 2011 | 2015 |  |  | 1854 |  |  |  | 2075 $\pm$ 249 |
| 2149 | 2186 | 2197 | 2076 | 2095 |  |  | 1928 |  |  |  |  |
| 2158 | 2196 | 2218 | 2117 | 2160 |  |  |  |  |  |  |  |
| 2484 | 2536 |  |  |  |  |  |  |  |  |  |  |
| 2495 | 2548 |  |  |  |  |  |  |  |  |  |  |
$\frac{3}{2}^{-}$ | 1534 | 1544 | 1549 | 1508 | 1545 | 1686 | 1556 | 1650 | 1690 | 1683 | 1560 | 1751 $\pm$ 41 |
| 1540 | 1551 | 1693 | 1662 | 1645 |  | 1682 | 1785 | 1520 |  | 1702 | 2203 $\pm$ 106 |
| 1819 | 1841 | 1812 | 1775 | 1770 |  |  | 1854 | 2325 |  | 1859 | 2381 $\pm$ 87 |
| 1827 | 1850 | 2035 | 1987 | 2030 |  |  | 1928 |  |  |  |  |
| 2043 | 2079 | 2319 | 2090 | 2110 |  |  | 1969 |  |  |  |  |
| 2131 | 2166 | 2322 | 2147 | 2185 |  |  |  |  |  |  |  |
| 2140 | 2176 | 2392 | 2259 | 2230 |  |  |  |  |  |  |  |
| 2371 | 2427 | 2454 | 2290 | 2275 |  |  |  |  |  |  |  |
| 2464 | 2513 | 2468 | 2313 |  |  |  |  |  |  |  |  |
| 2474 | 2525 |  |  |  |  |  |  |  |  |  |  |
| 2723 | 2793 |  |  |  |  |  |  |  |  |  |  |
$\frac{5}{2}^{-}$ | 1524 | 1533 |  |  |  |  |  |  |  |  |  |  |
| 1807 | 1827 | 1861 | 1828 | 1775 | 1799 | 1778 | 1785 | 1830 | 1850 | 1803 |  |
| 2005 | 2039 | 2136 | 2080 | 2180 |  |  |  |  |  |  |  |
| 2015 | 2050 | 2350 | 2179 | 2250 |  |  |  |  |  |  |  |
| 2116 | 2149 |  |  |  |  |  |  |  |  |  |  |
| 2329 | 2380 |  |  |  |  |  |  |  |  |  |  |
| 2341 | 2393 |  |  |  |  |  |  |  |  |  |  |
| 2447 | 2494 |  |  |  |  |  |  |  |  |  |  |
| 2676 | 2741 |  |  |  |  |  |  |  |  |  |  |
| 2689 | 2755 |  |  |  |  |  |  |  |  |  |  |
$\frac{7}{2}^{-}$ | 1970 | 2002 | 2097 | 2090 | 2150 |  |  |  | 2100 | 2087 |  |  |
| 1980 | 2013 | 2583 | 2227 | 2230 |  |  |  |  |  |  |  |
| 2291 | 2337 |  |  |  |  |  |  |  |  |  |  |
| 2303 | 2350 |  |  |  |  |  |  |  |  |  |  |
| 2632 | 2693 |  |  |  |  |  |  |  |  |  |  |
| 2645 | 2707 |  |  |  |  |  |  |  |  |  |  |
$\frac{9}{2}^{-}$ | 1939 | 1969 | 2665 | 2370 |  |  |  |  |  |  |  |  |
| 2257 | 2299 |  |  |  |  |  |  |  |  |  |  |
| 2593 | 2650 |  |  |  |  |  |  |  |  |  |  |
Table 15: S-wave of $\Sigma$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
1S | $\frac{1}{2}^{+}$ | 1193 | 1193 | 1193
| $\frac{3}{2}^{+}$ | 1384 | 1384 | 1385
2S | $\frac{1}{2}^{+}$ | 1643 | 1643 | 1660
| $\frac{3}{2}^{+}$ | 1827 | 1827 |
3S | $\frac{1}{2}^{+}$ | 2083 | 2099 |
| $\frac{3}{2}^{+}$ | 2229 | 2236 |
4S | $\frac{1}{2}^{+}$ | 2560 | 2589 |
| $\frac{3}{2}^{+}$ | 2675 | 2693 |
5S | $\frac{1}{2}^{+}$ | 3067 | 3108 |
| $\frac{3}{2}^{+}$ | 3159 | 3189 |

Table 16: P-wave $\Sigma$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1720 | 1725 | 1620
$1^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1698 | 1702 | 1670
$1^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1731 | 1736 | 1750
$1^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1709 | 1713 |
$1^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1680 | 1683 | 1775
$2^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2128 | 2145 | 1900
$2^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2099 | 2114 | 1910
$2^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2142 | 2159 |
$2^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2114 | 2129 |
$2^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2076 | 2087 |
$3^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2580 | 2608 |
$3^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2545 | 2571 |
$3^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2598 | 2627 |
$3^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2563 | 2589 |
$3^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2516 | 2541 |
$4^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 3068 | 3111 |
$4^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 3027 | 3067 |
$4^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 3088 | 3133 |
$4^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 3047 | 3089 |
$4^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2994 | 3030 |
$5^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 3588 | 3641 |
$5^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 3541 | 3594 |
$5^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 3612 | 3664 |
$5^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 3565 | 3617 |
$5^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 3502 | 3555 |

Table 17: D-wave $\Sigma$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2040 | 2057 | 1940
$1^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1998 | 2013 | 1915
$1^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2086 | 2107 |
$1^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2055 | 2074 |
$1^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2014 | 2029 |
$1^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 1962 | 1974 | 2025
$2^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2481 | 2510 |
$2^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2432 | 2459 |
$2^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2536 | 2568 |
$2^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2499 | 2529 |
$2^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2451 | 2478 |
$2^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2390 | 2414 |
$3^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2962 | 3004 |
$3^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2905 | 2945 |
$3^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 3027 | 3072 |
$3^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2984 | 3027 |
$3^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2926 | 2967 |
$3^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2855 | 2892 |
$4^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 3476 | 3534 |
$4^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 3410 | 3464 |
$4^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 3549 | 3613 |
$4^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 3500 | 3560 |
$4^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 3435 | 3490 |
$4^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 3353 | 3403 |

Table 18: F-wave $\Sigma$ baryon (in MeV)

State | $J^{P}$ |  $Mass_{cal}$1 |  $Mass_{cal}$2 |  $Mass_{exp}$[2]
---|---|---|---|---
$1^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2386 | 2416 |
$1^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2318 | 2343 |
$1^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2461 | 2495 |
$1^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2406 | 2437 |
$1^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2338 | 2365 |
$1^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2257 | 2278 |
$2^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2858 | 2901 |
$2^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2781 | 2819 |
$2^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2943 | 2990 |
$2^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2881 | 2925 |
$2^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2804 | 2844 |
$2^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2712 | 2746 |
$3^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 3363 | 3417 |
$3^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 3278 | 3330 |
$3^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 3456 | 3512 |
$3^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 3388 | 3443 |
$3^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 3303 | 3356 |
$3^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 3202 | 3252 |

Table 19: Comparison of masses with other predictions based on $J^{P}$ value for $\Sigma$ baryon (in MeV) $J^{P}$ | $Mass_{cal}1$ | $Mass_{cal}2$ | [30] | [31] | [32] | [33] | [34] | [35] | [40] | [41] | [43] | [44] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\frac{1}{2}^{+}$ | 1193 | 1193 | 1187 | 1190 | 1190 | 1170 | 1180 | 1211 | 1193 | 1198 | 1192 | 1216 $\pm$ 15 |
| 1643 | 1643 | 1711 | 1760 | 1720 | 1604 | 1616 | 1546 | 1660 | 1656 | 1664 | 2069 $\pm$ 74 |
| 2083 | 2099 | 1922 | 1947 | 1915 |  | 1911 | 1668 | 1770 |  | 1924 | 2149 $\pm$ 66 |
| 2086 | 2107 | 1983 | 2009 | 1970 |  |  | 1801 | 1880 |  | 1986 | 2335 $\pm$ 63 |
| 2536 | 2568 | 2028 | 2052 | 2005 |  |  |  |  |  | 2022 |  |
| 2560 | 2589 | 2180 | 2098 | 2030 |  |  |  |  |  | 2069 |  |
| 3027 | 3072 | 2292 | 2138 | 2105 |  |  |  |  |  | 2172 |  |
| 3067 | 3108 | 2472 |  |  |  |  |  |  |  |  |  |
| 3549 | 3613 |  |  |  |  |  |  |  |  |  |  |
$\frac{3}{2}^{+}$ | 1384 | 1384 | 1381 | 1411 | 1370 | 1382 | 1389 | 1334 | 1385 | 1381 | 1383 | 1471 $\pm$ 23 |
| 1827 | 1827 | 1862 | 1896 | 1920 |  | 1865 | 1439 | 1560 |  | 1868 | 2194 $\pm$ 81 |
| 2040 | 2057 | 2025 | 1961 | 1970 |  |  | 1924 | 1690 |  | 1947 | 2250 $\pm$ 79 |
| 2055 | 2074 | 2076 | 2011 | 2010 |  |  |  | 1840 |  | 1993 | 2468 $\pm$ 67 |
| 2229 | 2236 | 2096 | 2044 | 2030 |  |  |  |  |  | 2039 |  |
| 2481 | 2510 | 2157 | 2062 | 2045 |  |  |  |  |  | 2075 |  |
| 2499 | 2529 | 2186 | 2103 | 2085 |  |  |  |  |  | 2098 |  |
| 2675 | 2693 |  | 2112 | 2115 |  |  |  |  |  | 2122 |  |
| 2962 | 3004 |  |  |  |  |  |  |  |  | 2168 |  |
| 2984 | 3027 |  |  |  |  |  |  |  |  |  |  |
| 3159 | 3189 |  |  |  |  |  |  |  |  |  |  |
$\frac{5}{2}^{+}$ | 1998 | 2013 | 1991 | 1956 | 1995 | 1872 |  | 2061 | 1915 | 1930 | 1949 |  |
| 2014 | 2029 | 2062 | 2027 | 2030 |  |  |  | 2070 |  | 2028 |  |
| 2432 | 2459 | 2221 | 2071 | 2095 |  |  |  |  |  | 2062 |  |
| 2451 | 2478 |  |  |  |  |  |  |  |  | 2107 |  |
| 2905 | 2945 |  |  |  |  |  |  |  |  | 2154 |  |
| 2926 | 2967 |  |  |  |  |  |  |  |  |  |  |
| 3410 | 3464 |  |  |  |  |  |  |  |  |  |  |
| 3435 | 3490 |  |  |  |  |  |  |  |  |  |  |
$\frac{7}{2}^{+}$ | 1962 | 1974 | 2033 | 2070 | 2060 | 2012 |  |  | 2030 | 2039 | 2002 |  |
| 2390 | 2414 | 2470 | 2161 | 2125 |  |  |  |  |  | 2106 |  |
| 2855 | 2892 |  |  |  |  |  |  |  |  |  |  |
| 3353 | 3403 |  |  |  |  |  |  |  |  |  |  |
Table 20: Comparison of masses with other predictions based on $J^{P}$ value for $\Sigma$ baryon (in MeV) $J^{P}$ | $Mass_{cal}1$ | $Mass_{cal}2$ | [30] | [31] | [32] | [33] | [34] | [35] | [40] | [41] | [43] | [44] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\frac{1}{2}^{-}$ | 1720 | 1725 | 1620 | 1628 | 1630 | 1711 | 1677 | 1753 | 1620 | 1754 | 1657 | 1603 $\pm$ 38 |
| 1731 | 1736 | 1693 | 1771 | 1675 |  | 1736 | 1868 | 1750 |  | 1746 | 1718 $\pm$ 58 |
| 2128 | 2145 | 1747 | 1798 | 1695 |  | 1759 | 1895 | 2000 |  | 1802 | 1730 $\pm$ 34 |
| 2142 | 2159 | 2115 | 2111 | 2110 |  |  |  |  |  |  | 2478 $\pm$ 104 |
| 2580 | 2608 | 2198 | 2136 | 2155 |  |  |  |  |  |  |  |
| 2598 | 2627 | 2202 | 2251 | 2165 |  |  |  |  |  |  |  |
| 3068 | 3111 | 2289 | 2264 | 2205 |  |  |  |  |  |  |  |
| 3088 | 3133 | 2381 | 2288 | 2260 |  |  |  |  |  |  |  |
| 3588 | 3641 |  |  |  |  |  |  |  |  |  |  |
| 3612 | 3664 |  |  |  |  |  |  |  |  |  |  |
$\frac{3}{2}^{-}$ | 1698 | 1702 | 1706 | 1669 | 1655 | 1711 | 1677 | 1753 | 1670 | 1697 | 1698 | 1861 $\pm$ 26 |
| 1709 | 1713 | 1731 | 1728 | 1750 | 1974 | 1736 | 1868 | 1940 | 1956 | 1790 | 1736 $\pm$ 40 |
| 2099 | 2114 | 1856 | 1781 | 1755 |  | 1759 | 1895 | 2250 |  | 1802 | 2394 $\pm$ 74 |
| 2114 | 2129 | 2175 | 2139 | 2120 |  |  |  |  |  |  | 2297 $\pm$ 122 |
| 2416 | 2495 | 2203 | 2171 | 2185 |  |  |  |  |  |  |  |
| 2545 | 2571 | 2300 | 2203 | 2200 |  |  |  |  |  |  |  |
| 2563 | 2589 |  |  |  |  |  |  |  |  |  |  |
| 2943 | 2990 |  |  |  |  |  |  |  |  |  |  |
| 3027 | 3067 |  |  |  |  |  |  |  |  |  |  |
| 3047 | 3089 |  |  |  |  |  |  |  |  |  |  |
| 3456 | 3512 |  |  |  |  |  |  |  |  |  |  |
| 3541 | 3594 |  |  |  |  |  |  |  |  |  |  |
| 3565 | 3617 |  |  |  |  |  |  |  |  |  |  |
$\frac{5}{2}^{-}$ | 1680 | 1683 | 1757 | 1770 | 1755 |  | 1736 | 1753 | 1775 | 1777 | 1743 |  |
| 2076 | 2087 | 2214 | 2174 | 2205 |  |  |  |  |  |  |  |
| 2386 | 2416 | 2347 | 2226 | 2250 |  |  |  |  |  |  |  |
| 2406 | 2437 |  |  |  |  |  |  |  |  |  |  |
| 2516 | 2541 |  |  |  |  |  |  |  |  |  |  |
| 2858 | 2901 |  |  |  |  |  |  |  |  |  |  |
| 2881 | 2925 |  |  |  |  |  |  |  |  |  |  |
| 2994 | 3030 |  |  |  |  |  |  |  |  |  |  |
| 3363 | 3417 |  |  |  |  |  |  |  |  |  |  |
| 3388 | 3443 |  |  |  |  |  |  |  |  |  |  |
| 3565 | 3617 |  |  |  |  |  |  |  |  |  |  |
$\frac{7}{2}^{-}$ | 2318 | 2343 | 2259 | 2236 | 2245 |  |  |  | 2100 |  |  |  |
| 2338 | 2365 | 2349 | 2285 |  |  |  |  |  |  |  |  |
| 2781 | 2819 |  |  |  |  |  |  |  |  |  |  |
| 2804 | 2844 |  |  |  |  |  |  |  |  |  |  |
| 3278 | 3330 |  |  |  |  |  |  |  |  |  |  |
| 3303 | 3356 |  |  |  |  |  |  |  |  |  |  |
$\frac{9}{2}^{-}$ | 2257 | 2278 | 2289 | 2325 |  |  |  |  |  |  |  |  |
| 2712 | 2746 |  |  |  |  |  |  |  |  |  |  |
| 3202 | 3252 |  |  |  |  |  |  |  |  |  |  |

For $\Lambda$ baryon, the ground state mass is 1115 MeV and for $\Sigma$ it is
1193 MeV and confinement parameters are determined accordingly. Here, the
constituent quark mass for u and d quarks is similar so, the charges of
$\Sigma$ are not distinguished. The four star status states are in good
agreement with the PDG masses as evident from the tables. As for excited
states of $\Lambda$ 2S(1600), the predicted masses are very near to almost all
the models’ mass. However, the first negative parity state $\Lambda$(1405)
$J^{P}=\frac{1}{2}^{-}$ is not established by the present model but the next
state with $J^{P}=\frac{3}{2}^{-}$ 1520 MeV is varies by 15 MeV from PDG. The
$J^{P}=\frac{5}{2}^{+}$ state of 1D(1769 MeV) too falls within the PDG mass
range of 1750-1850. The $J^{P}=\frac{9}{2}^{+}$ for 1G is somewhat under-
predicted owing to the limitations of the hCQM.

For the case of $\Sigma$ baryon, the early negative parity states are just one
star status. The later state with $J^{P}=\frac{1}{2}^{-}$ appearing as
$\Sigma$(1750) is predicted well here as 1720 MeV within the range 1700-1800.
The $J^{P}=\frac{5}{2}^{+}$ $\Sigma$(1915) is slightly higher predicted from
most of the models. The other higher ranged states results from hCQM are
comparable to those of BGR Collaboration results [44].

##  4 Regge Trajectories

Regge trajectories have been one of the useful tools in spectroscopic studies.
The plot of total angular momentum J and principle quantum number n against
the square of resonance mass $M^{2}$ are drawn based on calculated data. The
non-intersecting and linearly fitted lines have been in accordance with
theoretical and experimental data in many studies [49]. These plots might be
helpful in predicting the correct spin-parity assignment of a given state.

$\displaystyle J=aM^{2}+a_{0}$ (8a) $\displaystyle n=bM^{2}+b_{0}$ (8b)

Figure 1:  Regge trajectory $\Xi$ for $n\rightarrow M^{2}$ Figure 2:  Regge
trajectory $\Xi$ for $J^{P}\rightarrow M^{2}$ Figure 3:  Regge trajectory
$\Xi$ for $J^{P}\rightarrow M^{2}$

As it is evident from the graphs [1], [4] and [7], the Regge trajectory for n
against $M^{2}$ has been linearly fitted for the calculated resonance masses
which follows the expected trend. The trajectories for J against $M^{2}$ for
the natural parity also follows the linear nature which signifies that the
spin-parity assignment for the obtained states are in agreement. The overall
nature of the Regge trajectories observed for baryon studies agrees with the
calculated results for few of the states.

Figure 4:  Regge trajectory $\Sigma$ for $n\rightarrow M^{2}$ Figure 5:  Regge
trajectory $\Sigma$ for $J^{P}\rightarrow M^{2}$ Figure 6:  Regge trajectory
$\Sigma$ for $J^{P}\rightarrow M^{2}$ Figure 7:  Regge trajectory $\Lambda$
for $n\rightarrow M^{2}$ Figure 8:  Regge trajectory $\Lambda$ for
$J^{P}\rightarrow M^{2}$

##  5 Magnetic Moment

The study of electromagnetic properties of baryons is an active area for
theoretical as well as experimental work. This intrinsic property help reveal
the shape and other dynamics of transition in decay modes. In case of cascade
$\Xi$ and $\Sigma$ baryon, magnetic moment is to be determined for both the
spin configuration based on the respective spin-flavour wave-function. The
generalized form of magentic moment is

$\mu_{B}=\sum_{q}\left\langle\phi_{sf}|\mu_{qz}|\phi_{sf}\right\rangle$ (9)

where $\phi_{sf}$ is the spin-flavour wave function. The contribution from
individual quark appears as

$\mu_{qz}=\frac{e_{q}}{2m_{q}^{eff}}\sigma_{qz}$ (10)

$e_{q}$ being the quark charge, $\sigma_{qz}$ being the spin orientation and
$m_{q}^{eff}$ is the effective mass which may vary from model based quark mass
due to interactions. Here, is it noteworthy that magnetic moment shall have
contribution from many other effects within the baryon as sea quark, valence
quark, orbital etc. Various models have contributed to obtaining the octet and
decuplet baryons’ magnetic moment. The order of quarks in spin-flavour won’t
be affecting the magnetic moment calculation. The final spin-flavour wave
function along with calculated ground state magnetic moment in terms of
nuclear magneton ($\mu_{N}$) are mentioned in the following table [21].

Table 21: Magnetic moment for ground states Spin | Baryon | $\sigma_{qz}$ | Mass (MeV) |  $\mu$ ($\mu_{N}$)
---|---|---|---|---
$\frac{1}{2}$ |  $\Xi^{0}$(uss) | $\frac{1}{3}(4\mu_{s}-\mu_{u})$ | 1322 | -1.50
$\frac{1}{2}$ |  $\Xi^{-}$(dss) | $\frac{1}{3}(4\mu_{s}-\mu_{d})$ | 1322 | -0.46
$\frac{3}{2}$ |  $\Xi^{*0}$(uss) | $(2\mu_{s}+\mu_{u})$ | 1531 | 0.766
$\frac{3}{2}$ |  $\Xi^{*-}$(dss) | $(2\mu_{s}+\mu_{d})$ | 1531 | -1.962
$\frac{1}{2}$ |  $\Sigma^{+}$(uus) | $\frac{1}{3}(4\mu_{u}-\mu_{s})$ | 1193 | 2.79
$\frac{1}{2}$ |  $\Sigma^{0}$(uds) | $\frac{1}{3}(2\mu_{u}+2\mu_{d}-\mu_{s})$ | 1193 | 0.839
$\frac{1}{2}$ |  $\Sigma^{-}$(dds) | $\frac{1}{3}(4\mu_{d}-\mu_{s})$ | 1193 | -1.113
$\frac{3}{2}$ |  $\Sigma^{*+}$(uus) | $(2\mu_{u}+\mu_{s})$ | 1384 | 2.877
$\frac{3}{2}$ |  $\Sigma^{*0}$(uds) | $(\mu_{u}+\mu_{d}+\mu_{s})$ | 1384 | 0.353
$\frac{3}{2}$ |  $\Sigma^{*-}$(dds) | $(2\mu_{d}+\mu_{s})$ | 1384 | -2.171
$\frac{1}{2}$ |  $\Lambda^{0}$(uds) | $\mu_{s}$ | 1115 | -0.606
Table 22: Comparison of calculated magnetic moments with various models (All data in units of $\mu_{N}$) Baryon | $\mu_{cal}$ | Exp [2] | [50] | [50] | [51] | [52] | [53] | [53] | [54] | [55] | [56] | [57]
---|---|---|---|---|---|---|---|---|---|---|---|---
$\Xi^{0}$ | -1.50 | -1.25 | -1.41 | -1.39 |  | -1.3 |  |  | -1.25 | -1.37 |  |
$\Xi^{-}$ | -0.46 | -0.651 | -0.50 | -0.50 |  | -0.7 |  |  | -1.07 | -0.82 |  |
$\Xi^{*0}$ | 0.766 |  | 0.60 | 0.49 | 0.49 | 0.69 | 0.48 | 0.32 | 0.44 | 0.16 | 0.508 | 0.65
$\Xi^{*-}$ | -1.962 |  | -2.11 | -2.43 | -2.27 | -1.18 | -1.9 | -2.05 | -2.27 | -0.62 | -1.805 | -2.30
$\Sigma^{+}$ | 2.79 | 2.458 | 2.61 | 2.64 |  |  |  |  | 2.46 | 2.87 |  |
$\Sigma^{0}$ | 0.839 |  |  |  |  | 0.11 |  |  | 0.65 | 0.76 |  |
$\Sigma^{-}$ | -1.113 | -1.16 | -1.01 | -1.28 |  | -1.13 |  |  | -1.16 | -1.48 |  |
$\Sigma^{*+}$ | 2.877 |  | 3.02 | 3.07 | 2.85 |  | 2.56 | 2.54 | 2.63 | 1.27 | 3.028 |
$\Sigma^{*0}$ | 0.353 |  | 0.30 | 0.08 | 0.09 |  | 0.23 | 0.14 | 0.08 | 0.33 | 0.188 |
$\Sigma^{*-}$ | -2.171 |  | -2.41 | -2.92 | -2.66 |  | -2.10 | -2.19 | -2.43 | -1.88 | -2.015 |
$\Lambda^{0}$ | -0.606 | -0.613 | -0.59 | -0.60 |  | -0.11 |  |  | -0.51 | -0.70 |  |

Table [22] gives comparison of present magnetic moment with those of different
approaches. H. Dahiya et al. have presented octet and decuplet baryon magnetic
moments in chiral quark model with configuration mixing and generalizing the
Cheng-Li mechanism. Effective mass and screened charge scheme has been
employed in ref. [53] and both the results appear in the table. Light-cone sum
rules [54] and lattice QCD [55] have also been employed for octet and decuplet
magnetic moments. Recent study has focused on hyperonic medium at a finite
temperature using chiral mean field approach [56]. Our results are under-
predicted compared to experimental values of PDG by 0.5 and 0.4 respectively
for $\Xi$ octet baryons, however, there are large variations in case of
decuplet considering all the approaches. It is expected that experimental
values for decuplet shall be the deciding factor. In case of $\Lambda^{0}$,
the magnetic moment is nearly similar to those obatined by [2], [50] and [55].
For $\Sigma^{+}$ and $\Sigma^{-}$, results are within 0.5$\mu_{N}$ variation
for all the approaches.

Here, an effort has been made to determine the magnetic moment of low-lying
negative parity state of $\Xi$ which has been inspired by studies based on
N(1535). The hyperfine interactions between the constituent quarks induce
linear combinations of two states with the same angular momentum value. The
magnetic moment will now have contribution from spin as well as orbital
angular momentum as [58]

$\mu=\mu^{S}+\mu^{L}=\sum\frac{Q_{q}}{m_{q}}S_{q}+\sum\frac{Q_{q}}{2m_{q}}l_{q}$
(11)

The $J^{P}=\frac{1}{2}^{-}$, L=1 for $S=\frac{1}{2}$ and $S=\frac{3}{2}$,
calculated resonance masses are 1886 MeV and 1894 MeV. These states are not
experimentally established, however 1690 MeV is tentatively assigned by many
predictions to this state. The physical eigenvalues will be

$|\Xi(1886)\rangle=cos\theta|^{2}P_{1/2}\rangle-sin\theta|^{4}P_{1/2}\rangle$
(12)
$|\Xi(1894)\rangle=sin\theta|^{2}P_{1/2}\rangle+cos\theta|^{4}P_{1/2}\rangle$
(13)

From the approach of NCQM described in Ref. [59], we directly write the final
wave-function with mixing for $\Xi^{0}(uss)$ as

$(\frac{2}{9}\mu_{s}+\frac{1}{9}\mu_{u})cos^{2}\theta+(\mu_{s}+\frac{1}{3}\mu_{u})sin^{2}\theta-(\frac{8}{9}\mu_{s}-\frac{8}{9}\mu_{u})cos\theta
sin\theta$ (14)

The mixing angle is taken as $\theta=-31.7$. The magnetic moment for
$\Xi^{0}(1886)$ is obtained to be -0.695$\mu_{N}$ which is -0.99$\mu_{N}$ by
[59]. Similarly, for $\Xi^{-}(1886)$, magnetic moment is obtained as
-0.193$\mu_{N}$ as compared to -0.315$\mu_{N}$ [59]. The variation relies on
the fact that resonance mass is a model dependent and the effective mass used
in the calculated ultimately depends on resonance mass.

##  6 Transition Magnetic Moment and Radiative Decay Width

Transition magnetic moment as well as radiative decay width are also important
in the understanding of internal structure of baryon as well as magnetic and
electric transitions. Many approach have been utilized for the study of
radiative decay over years including some recent ones [60].Here,
$\Xi^{*0}\rightarrow\Xi^{0}\gamma$ as well as
$\Sigma^{*}\rightarrow\Sigma\gamma$ have been studied using the effective mass
obtained using hCQM approach. The generalized form for transition magnetic
moment is [61],

$\mu(B_{\frac{3}{2}^{+}}\rightarrow B_{\frac{1}{2}^{+}})=\langle
B_{\frac{1}{2}^{+}},S_{z}=\frac{1}{2}|\mu_{z}|B_{\frac{3}{2}^{+}},S_{z}=\frac{1}{2}\rangle$
(15)

The spin-flavour wave function obtained in a similar way as above mentioned
for decuplet and octet $\Xi$ is

$\frac{2\sqrt{2}}{3}(\mu_{u}^{eff}-\mu_{s}^{eff})$ (16)

The effective mass here is a geometric mean of those for spin $\frac{1}{2}$
and $\frac{3}{2}$. Our result comes out to be 2.378$\mu_{N}$ which is in good
agreement with various models implemented in [53, 61]. The radiative decay
width is obtained as [62],

$\Gamma_{R}=\frac{q^{3}}{m_{p}^{2}}\frac{2}{2J+1}\frac{e^{2}}{4\pi}|\mu_{\frac{3}{2}^{+}\rightarrow\frac{1}{2}^{+}}|^{2}$
(17)

where q is the photon energy, $m_{p}$ is the proton mass and J is the initial
angular momentum giving $\Gamma_{R}$ = 0.214 MeV. The total decay width
available from experiment is 9.1 MeV. Thus, the branching ratio
$\frac{\Gamma_{R}}{\Gamma_{total}}$ is 2.35% where the PDG data suggests
$<3.7\%$. Thus, it is in accordance with other results as well as from
experimental data. The radiative decay of $\Sigma$ baryons with transition
magnetic moments are summarized in the following table [23]. The obtained
results are consistent with experimental data from PDG as well as few other
theoretical approaches.

Table 23: $\Sigma^{*}$ Radiative Decays Decay | Wave-function | Transition moment(in $\mu_{N}$) |  $\Gamma_{R}$(in MeV)
---|---|---|---
$\Sigma^{*0}\rightarrow\Lambda^{0}\gamma$ | $\frac{\sqrt{2}}{\sqrt{3}}(\mu_{u}-\mu_{d})$ | 2.296 | 0.4256
$\Sigma^{*0}\rightarrow\Sigma^{0}\gamma$ | $\frac{\sqrt{2}}{3}(\mu_{u}+\mu_{d}-2\mu_{s})$ | 0.923 | 0.0246
$\Sigma^{*+}\rightarrow\Sigma^{+}\gamma$ | $\frac{2\sqrt{2}}{3}(\mu_{u}-\mu_{s})$ | 2.204 | 0.1404
$\Sigma^{*-}\rightarrow\Sigma^{-}\gamma$ | $\frac{2\sqrt{2}}{3}(\mu_{d}-\mu_{s})$ | -0.359 | 0.0037

##  7 Conclusion

The present work has been aimed at studying the strangeness -1 $\Lambda$,
$\Sigma$ and -2 $\Xi$ light baryon owing to its limited data. The non-
relativistic hypercentral Constituent Quark Model (hCQM) has been a tool for
obtaining large number of resonance masses with a linear term. The results
have been compared for with and without first order correction terms as well
wherein a few MeV difference for low-lying states and upto the order of 30 MeV
for higher excited states for $\Xi$. The octet and decuplet states have not
been exclusively distinguished due to lack of required data.

The mass-range have been compared to various theoretical approaches listed in
section 3. The state-wise comparison is not possible because no approach has
established spin-parity assignments. The overview of Tables 6 and 7 shows that
the low-lying resonance masses are in good agreement among each other
especially the four star states of PDG. For higher excited states, present
work over-estimates the results compared to other models. The $\Xi$(1820)
differs by 48 MeV from PDG mass. The $\Xi(2030)$ state with $J=\frac{5}{2}$
could find a place with either positive or negative parity a both have masses
in that range. $\Xi$(1620) and $\Xi$(1690) are not obtained in present
results. However, $\Xi$(1690) which is likely $\frac{1}{2}^{-}$ by BABAR
Collaboration, this study calculates it as 1886 MeV. Few differences in
results owes to the model dependent factors. As for $\Lambda$(1405), hCQM
could not establish the mass but predicts the other negative parity state with
good agreement. The four star states for $\Sigma$ as well as $\Lambda$ agree
to a good extent with many approaches too.

The Regge trajectories based on some resonance mass are plotted. The principle
quantum number n against the square of mass $M^{2}$ shows linear nature but
fitted lines are not exactly parallel. The angular momentum J versus the
square of mass $M^{2}$ plots also depicts the linearly of data points which
validates our spin-parity assignments to a particular state as well as might
be helpful in new experimental states.

The magnetic moment for spin $\frac{1}{2}$, $\Xi^{0}$ and $\Xi^{-}$ vary by
0.25$\mu_{N}$ and 0.19$\mu_{N}$ from PDG as well as other results. For spin
$\frac{3}{2}$, $\Xi^{*0}$ and $\Xi^{*-}$, our result vary from nearly
0.2$\mu_{N}$ to 0.4$\mu_{N}$ compared to all the approaches. The transition
magnetic moment is obtained for $\Xi^{0}$ 1886 MeV of our spectra which
differs by 0.30$\mu_{N}$ and $\Xi^{-}$ differs by 0.12$\mu_{N}$ from Ref [59].
However, difference of magnetic moment follows due to difference of resonance
masses used for the calculation. Similarly for $\Sigma^{+}$, $\Sigma^{-}$ and
$\Lambda^{0}$ results are very well in accordance with other approaches and
PDG.

The transition magnetic moment has been obtained as 2.378$\mu_{N}$ which
agrees with other models. Also, the radiative decay width branching fraction
comes out to be 2.35% in our case which is well within the PDG range of
$<3.7\%$. The transition magnetic moments and radiative decay widths for
$\Sigma$ falls similar to other results.

Thus, with some agreements and some discrepancies, this study with large
number of predicted resonances along with important properties, might be find
helpful for upcoming experimental facilities like PANDA which is expected to
intensively study the light strange baryons [11, 13].

## Acknowledgement

Ms. Chandni Menapara would like to acknowledge the support from the Department
of Science and Technology (DST) under INSPIRE-FELLOWSHIP scheme for pursuing
this work.

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