L12.0pt

# Sea contribution to the charge radii and quadrupole moment of
$J^{P}=\frac{1}{2}^{+},\frac{3}{2}^{+}$ baryons

Preeti Bhall*, Meenakshi Batra, A. Upadhyay

(School of Physics and Material Science,
Thapar Institute of Engineering and Technology, Patiala, Punjab-147004
Dev Samaj College for Women, Sector 45, Chandigarh-160047
)

###### Abstract

An operator formalism is used on the wavefunction of baryons to compute their
charge radii and quadrupole moments. Total anti-symmetric wavefunction in
spin, color and flavor space is framed for $J^{P}=\frac{1}{2}^{+}$ nucleons
and $J^{P}=\frac{3}{2}^{+}$ hyperons. To understand the importance of sea,
statistical model is used in conjugation with the detailed balance principle.
Within the statistical approach, the importance of sea with quarks and gluons
are studied using the relevant probabilities that are associated with spin,
flavor, and color space. The present work also focuses on individual
contributions of valence and sea which contains terms of scalar, vector and
tensor sea. The obtained results are in agreement with available theories and
few experimental outcomes. Our computed results may provide important
information for upcoming experimental findings.

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##  1 Introduction

The main goal for particle physicists is to understand the structure of
hadrons. The study of electromagnetic properties of baryons over the last two
decades enhances our understanding of the internal structure of baryons and
gives new directions about the properties of matter. Various discoveries of
proton suggest that quarks-antiquarks carry 30$\%$ of the total proton spin.
Further, it has been predicted [1] that the orbital angular momentum of quarks
and gluons also make a significant contribution to the proton’s spin and other
static properties [2] of the baryons like magnetic moment, masses, etc. It is
also confirmed that there is a significant effect from the strange quarks [3,
4] which are assumed to be present in the sea. Usually, the ’sea’ is a
combination of virtual gluons and quark-antiquark pairs. It may contain light
quarks as well as heavy ones and plays a vital role in determining the
hadronic structure. Therefore, to go into deeper details of the structure of
baryons, it becomes mandatory to investigate their low energy properties. The
most common properties to study the structure of hadrons are magnetic moments,
masses, semi-leptonic decays, charge radii, quadrupole moment and spin
distribution etc. Although, Quantum Chromodynamics(QCD) is the fundamental
theory of strong interactions of hadrons but presently these properties can
not be predicted directly from QCD. The study of hadronic structure is
complicated due to the non-perturbative interaction between the fundamental
constituents (i.e. quarks and gluons). To get an interior image of the
baryons, the theorist depends upon phenomenological models and effective
theories. Over the past years [5, 6, 7], we have witnessed remarkable
discoveries in the area of phenomenology. The charge radii and quadrupole
moments are important observables of hadrons that contain information about
their quark-gluon structure. These are directly related to the spatial charge
and current distribution in baryons. Various scattering experiments [8] have
been made measurements for the charge radii of the proton, neutron and
$\Sigma^{-}$ giving values $r_{p}$ = 0.8409 $\pm$ 0.0004 fm, $r_{n}^{2}$ =
-0.115 $\pm$ 0.0017 fm2 and $\Sigma^{-}$ = 0.78 $\pm$ 0.10 fm listed in PDG
[7]. On reviewing the previous literature, many theoretical models like the
quark model [9, 10, 11], light cone QCD sum rules [12, 13], the Skyrme model
with bound-state approach [14], QCD parametrization method [15, 16], the 1/Nc
expansion method[17, 18], lattice QCD [19, 20] and chiral constituent quark
model ($\chi$QM) [21, 22, 23] analysed the various properties of baryons such
as magnetic moment, charge radii, octupole moments, quadrupole moments etc. By
using the QCD sum rule approach in ref. [12, 13], the author estimated the
magnetic dipole, electric quadrupole, and magnetic octupole moments of
$\Delta$-baryons. In ref. [17, 18], the authors predicted the charge radii and
quadrupole moment of decuplet baryons by 1/$N_{c}$ expansion method. In ref.
[19, 20], the charge radii of decuplet baryons have been studied in lattice
QCD. In ref. [21, 22, 23], the charge radii, quadrupole moment of
$(\frac{1}{2})^{+}$ octet, $(\frac{3}{2})^{+}$ decuplet baryons and spin
$(\frac{3}{2})^{+}\rightarrow(\frac{1}{2})^{+}$ transitions are calculated in
the framework of $\chi$CQM using general parameterization (GP) method. An
enormous amount of experimental information [7] is available now, which caters
to the need of detailed comparisons between theory and experimental data.
Also, the decuplet to octet transition moment plays an important role in
studying the internal structure of the spin. The quadrupole transition
moment$(\Delta^{+}\rightarrow N)$ measured by LEGS [24] and Mainz [25]
collaborations $(-0.018\pm 0.009\pm 0.034$ fm2 and $-0.0846\pm 0.0033$
fm${}^{2})$ respectively [26] brings to the conclusion that the nucleon and
$\Delta^{+}$ baryon are intrinsically deformed.
There are many theoretical approaches in the literature to study
electromagnetic properties. But the results from different theoretical models
are not consistent with each other. In order to have a clear understanding, we
study these electromagnetic properties (quadrupole moment and charge radii) in
the framework of the statistical model with the principle of detailed balance.
In this model, hadrons are assumed to consist of valence quarks which are
associated with a virtual dynamic sea where quark-antiquark pairs are multi-
connected through gluons. Singh & Upadhyay et al. [27] used the statistical
model to find the quark contribution to the spin of the nucleons. The
statistical model with the inclusion of ’sea quarks’ successfully explains the
low energy hadronic properties like masses [28], spin distribution [29],
magnetic moments [30] of octet and decuplet baryons, and importance of sea
contribution to nucleons [31]. As this approach has been proven to be quite
effective in the calculation of static properties of baryons. So, we extended
the statistical model along with the principle of detailed balance by
computing the quadrupole moments and charge radii of the ground state nucleons
and $\Delta$-hyperons.
This paper is organised in the following order: In sec. 2, a brief overview of
octet and decuplet baryons wavefunction with sea components is given. Sec. 3,
presents the explanation of quadrupole moment and charge radii operator. In
Sec. 4, the statistical model with the detailed balance principle is
discussed. The importance of statistical parameters to the static properties
of baryons is also explained. In Sec. 5, we discuss the calculated charge
radii and quadrupole moment, and compare the computed values to the recent
available experimental and theoretical data. The conclusion is given in the
sec. 6.

##  2  Preliminaries

The naive valence picture of the hadron structure is considered as a first-
order approximation to the real system. With the extension of the naive quark
model, baryons are assumed a composite system of valence quark (qqq) and a
’sea’, where sea contains gluons and virtual quark-antiquark pairs [32, 33].
Sea is characterized by its total quantum number consistent with the quantum
number of valence quarks. The valence quark wavefunction of the baryon [34] is
written as:

$\Psi=\Phi(|\phi>|\xi>|\chi>|\rho>)$

where $|\phi>$, $|\xi>$, $|\chi>$ and $|\rho>$ represent the flavor, spin,
color and space $q^{3}$ wave functions respectively. Here, the spatial part
$|\rho>$ is symmetric under the exchange of any two quarks. The other three
parts flavor-spin-color must be anti-symmetric in order to maintain the
overall anti-symmetrization of the baryons. The three valence quarks each
having spin-$\frac{1}{2}$ will lead to the possibility of
$J^{P}=\frac{1}{2}^{+},\frac{3}{2}^{+}$ represented as:

qqq:
$\frac{1}{2}\bigotimes\frac{1}{2}\bigotimes\frac{1}{2}=2(\frac{1}{2})\bigoplus\frac{3}{2}$

and being the ground state particles, these quarks (fermions) have positive
parity which is given by $(-1)^{l}$. We take into consideration a flavorless
sea having spin (0, 1, 2) and color (1, 8, $\bar{10}$). Let $H_{0,1,2}$ and
$G_{1,8,\bar{10}}$ represent the spin and color sea wave functions
respectively which satisfy the condition: $<H_{i}|H_{j}>=\delta_{ij}$,
$<G_{k}|G_{l}>=\delta_{kl}$. The ’sea’ is supposed to be comprised of a gluon
or a $q\bar{q}$ pair, or more complicated, with multi-gluon state (gg, ggg),
multi-$q\bar{q}$ pairs or gluon(s) plus ($q\bar{q}$) pairs. The spin (H) and
color (G) combination of sea having two gluons, $q\bar{q}$ or multi $q\bar{q}$
pairs can be present as:

Spin: gg: $1\bigotimes 1=0_{s}\bigoplus 1_{a}\bigoplus 2_{S}$
$q\bar{q}q\bar{q}:(\frac{1}{2}\bigotimes\frac{1}{2})\bigotimes(\frac{1}{2}\bigotimes\frac{1}{2})=(0_{a}\bigoplus
1_{s})\bigotimes(0_{a}\bigoplus 1_{s})$
= $2(0_{s})\bigoplus 1_{s}\bigoplus 2(1_{a})\bigoplus 2_{s}$

Color: gg: $8\bigotimes 8=1_{s}\bigoplus 8_{s}\bigoplus 8_{a}\bigoplus
10_{a}\bigoplus\bar{10_{a}}\bigoplus 27_{s}$
$q\bar{q}q\bar{q}:(3\bigotimes\bar{3})\bigotimes(3\bigotimes\bar{3})=(1_{a}\bigoplus
8_{s})\bigotimes(1_{a}\bigoplus 8_{s})$
= $2(1_{s})\bigoplus 2(8_{s})\bigoplus 2(8_{a})\bigoplus
10_{s}\bigoplus\bar{10_{s}}\bigoplus 27_{s}$

Similar treatment is extended for multiple gluon cases upto three. Subscripts
a and s denote the anti-symmetry and symmetry of the combined state. The total
flavor-spin-color wave function of baryon octet (decuplet) of three valence
quarks with the inclusion of sea components can be represented as:

$\displaystyle|\Phi_{1/2}^{(\uparrow)}>=$
$\displaystyle\frac{1}{N}[{\Phi_{1}}^{(\frac{1}{2}\uparrow)}H_{0}G_{1}+a_{8}({\Phi_{8}}^{(\frac{1}{2})}\otimes
H_{0})^{\uparrow}G_{8}+a_{10}{\Phi_{10}}^{(\frac{1}{2}\uparrow)}H_{0}G_{\bar{10}}+$
(1) $\displaystyle b_{1}({\Phi_{1}}^{(\frac{1}{2})}\otimes
H_{1})^{\uparrow}G_{1}+b_{8}({\Phi_{8}}^{(\frac{1}{2})}\otimes
H_{1})^{\uparrow}G_{8}+b_{10}({\Phi_{10}}^{(\frac{1}{2})}\otimes
H_{1})^{\uparrow}G_{\bar{10}}$
$\displaystyle+c_{8}({\Phi_{8}}^{(\frac{3}{2})}\otimes
H_{1})^{\uparrow}G_{8}+d_{8}({\Phi_{8}}^{(\frac{3}{2})}\otimes
H_{2})^{\uparrow}G_{8}$

where
$N^{2}=1+a_{8}^{2}+a_{10}^{2}+b_{1}^{2}+b_{8}^{2}+b_{10}^{2}+c_{8}^{2}+d_{8}^{2}$
For Decuplet baryons [30]:

$\displaystyle|\Phi_{3/2}^{(\uparrow)}>=$
$\displaystyle\frac{1}{N}[a_{0}{\Phi_{1}}^{(\frac{3}{2}\uparrow)}H_{0}G_{1}+b_{1}({\Phi_{1}}^{(\frac{3}{2})}\otimes
H_{1})^{\uparrow}G_{1}+b_{8}({\Phi_{8}}^{(\frac{1}{2})}\otimes
H_{1})^{\uparrow}G_{8}+$ (2) $\displaystyle
d_{1}({\Phi_{1}}^{(\frac{3}{2})}\otimes
H_{2})^{\uparrow}G_{1}+d_{8}({\Phi_{8}}^{(\frac{1}{2})}\otimes
H_{2})^{\uparrow}G_{8}$

where $N^{2}=a_{0}^{2}+b_{1}^{2}+b_{8}^{2}+d_{1}^{2}+d_{8}^{2}$
Here N is the normalisation constant. The symbol ‘$\Phi$’ contains the spin
and color of valence quarks. For e.g.- $\Phi_{1}^{(\frac{1}{2}\uparrow)}$
represent the spin-$\frac{1}{2}$ and color singlet state of valence quarks.
Similarly, $\Phi_{8}^{(\frac{1}{2}\uparrow)}$ specifies the spin-$\frac{1}{2}$
and color octet of valence quarks. So, the total wavefunction consists of
various combination of valence and sea part can be written as:
${\Phi_{1}}^{(\frac{1}{2}\uparrow)}H_{0}G_{1},({\Phi_{8}}^{(\frac{1}{2})}\otimes
H_{0})^{\uparrow}G_{8},{\Phi_{10}}^{(\frac{1}{2}\uparrow)}H_{0}G_{\bar{10}}$.
The terms $(\Phi_{1}^{(\frac{1}{2})}\otimes H_{1})^{\uparrow}$,
$(\Phi_{8}^{(\frac{1}{2})}\otimes H_{1})^{\uparrow}$,
$(\Phi_{1}^{(\frac{3}{2})}\otimes H_{1})^{\uparrow}$, etc. written with
suitable C.G. coefficients by considering the symmetry property of the
component wave function. The sea with spin 0, 1, and 2 represents scalar,
vector and tensor sea respectively. In octet wave function, the coefficients
$a_{8},a_{10}$ come from the scalar sea whereas the coefficients
$b_{1},b_{8},b_{10},c_{8}$ and $d_{8}$ are from the vector and tensor sea
respectively. Similarly, in the decuplet wavefunction, the coefficient $a_{0}$
comes with the contribution of scalar sea and the coefficients $b_{1},b_{8}$
and $d_{1},d_{8}$ come with the contribution of vector and tensor sea
respectively. These coefficients
($a_{0},a_{8},a_{10},b_{1},b_{8},b_{10},c_{8},d_{8}$) are associated with each
state in the wavefunction has their own importance. They contain information
about the various properties of hadrons like masses, magnetic moment, spin
distribution etc. The details of all the terms of above-mentioned wave
function can be found in refs. [34, 35].

##  3 Quadrupole Moment and Charge radii

Charge radii ($r_{B}^{2}$) and quadrupole moments ($Q_{B}$) are the lowest
order moments of the charge density $\rho$ in a low-momentum expansion. They
are used to characterize the total charge, spatial extension, and shape of the
baryons. Here, intrinsic quadrupole moment means the one obtained in a body-
fixed-coordinate system that rotates with the nucleon. The nucleonic vertex
function in terms of Electromagnetic Dirac and Pauli form factors
$F_{1}(Q^{2})$ and $F_{2}(Q^{2})$ is represented as:

$\Gamma^{\mu}=F_{1}(Q^{2})\gamma^{\mu}+\kappa
F_{2}(Q^{2}(i\frac{\sigma^{\mu\nu}q_{\nu}}{2m}))$ (3)

where $\kappa$ is the anomalous part of the magnetic moment, $\gamma^{\mu}$
are Dirac matrices and
$\sigma^{\mu\nu}=i(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu})/2$.
Further, the electric and magnetic Sach form factors $G_{E}(Q^{2})$ and
$G_{M}(Q^{2})$ [36] can be related as:

$G_{E}=F_{1}-\tau\kappa F_{2}$ (4) $G_{M}=F_{1}+\kappa F_{2}$ (5)

where $\kappa=(\frac{Q}{2m})^{2}$. The study of electromagnetic form factors
of hadrons has received a lot of attention over the past years. The Fourier
transform of the elastic form factors provide insights about the radial
variation of the charge $\rho(r)$ and current j(r) densities [37]. The
observation by the physicists suggests that the ratio between electric and
magnetic form factors decreases sharply for $1<Q^{2}<6GeV^{2}$ [38]. The
experimental evidence shows that there is a variation of charge distribution
from spherical symmetry. It was suggested earlier that the quadrupole moment
of nucleon should vanish on account of spin-$\frac{1}{2}$ nature which is the
subject of theoretical and experimental activity. In addition, quadrupole
moments of the decuplet baryons are still not clear. The transition amplitude
in the $\gamma+p\rightarrow\Delta^{+}$ gives us information about photon
absorption amplitudes, the magnetic dipole $G_{M1}$ and electric quadrupole
moment from $G_{E2}$ and $G_{C2}$ [39] respectively. Electric quadrupole
$(E_{2})$ matrix elements, including quadrupole moments provide a principal
measure of nuclear deformation, rotation, and related collective structure.
The recent data from the collaborations [41] like Jlab, Mainz predicts a non-
zero value of quadrupole moment. The ratio of electric quadrupole to the
magnetic dipole amplitude i.e. $\frac{E_{2}}{M_{1}}$= $-0.025\pm 0.005$.
Similarly, the measurements made by LEGS and Mainz collaboration [26, 42]
conclude that nucleon and $\Delta^{+}$ baryons have a deformity from spherical
symmetric distribution. The geometrical shape of baryons can be measured from
intrinsic quadrupole moments [43]:

$Q_{0}^{p}=\int d^{3}r\rho^{p}{(r)(3z^{2}-r^{2})}$ (6)

where $\rho^{p}(r)$ is not necessarily spherically symmetric charge density of
proton. If the charge density is concentrated along the z-direction (symmetry
axis of the particle), the term proportional to $3z^{2}$ dominates, $Q_{0}$ is
positive, and the particle is prolate. If the charge density is concentrated
in the equatorial plane perpendicular to z, the term proportional to $r^{2}$
prevails, $Q_{0}$ is negative, and the particle is oblate. The most general
form of the multipole expansion of the charge density $\rho$ in the spin-
flavor space can be expressed as:

$\rho=A\sigma_{i=1}^{3}e_{i}I-B$ (7)

In order to find the quadrupole moment of baryon, a general QCD unitary
operator and QCD eigen states $|B\rangle$ are defined explicitly in terms of
quark and gluons. The quadrupole moment operator in terms of spin-flavor space
can be expressed as [44]:

${\widehat{Q}_{B}=B\sum_{i\neq
j}^{3}e_{i}(3\sigma_{iz}\sigma_{jz}-\sigma_{i}.\sigma_{j})+C\sum_{i\neq j\neq
k}^{3}e_{i}(3\sigma_{jz}\sigma_{kz}-\sigma_{j}.\sigma_{k})}$ (8)

The z-component of the Pauli spin (isospin) matrix $\sigma_{i}$ is denoted by
$\sigma_{iz}$ and $e_{i}$ is the charge of the i-th quark where i = (u,d,s).
The expanded form of the quadrupole moment operator of octet and decuplet
baryons [22] can be written as:

$\widehat{Q}_{1/2}=3B\sum_{i\neq j}e_{i}\sigma_{iz}\sigma_{jz}+3C\sum_{i\neq
j\neq
k}e_{i}\sigma_{jz}\sigma_{kz}+(-3B+3C)\sum_{i}e_{i}\sigma_{iz}+3B\sum_{i}e_{i}$
(9) $\widehat{Q}_{3/2}=3B\sum_{i\neq
j}e_{i}\sigma_{iz}\sigma_{jz}+3C\sum_{i\neq j\neq
k}e_{i}\sigma_{jz}\sigma_{kz}+(-5B+5C)\sum_{i}e_{i}\sigma_{iz}+(3B-6C)\sum_{i}e_{i}$
(10)

In the operator formalism [18, 45], the charge radii operator can be expressed
in terms of the sum of one-, two-, and three-quark contributions-

$\widehat{r}_{B}^{2}=A\sum_{i}e_{i}.1+B\sum_{i\neq
j}e_{i}\sigma_{i}\sigma_{j}+C\sum_{i\neq j\neq k}e_{i}\sigma_{j}\sigma_{k}$
(11)

Further, for octet and decuplet baryons, the expression can be written as:

$\widehat{{r}}^{2}_{1/2}=(A-3B)\sum_{i}e_{i}+3(B-C)\sum_{i}e_{i}\sigma_{iz}$
(12)
$\widehat{{r}}^{2}_{3/2}=(A-3B+6C)\sum_{i}e_{i}+5(B-C)\sum_{i}e_{i}\sigma_{iz}$
(13)

Here $\widehat{{r}}^{2}_{1/2}$ and $\widehat{{r}}^{2}_{3/2}$represents the
charge radii operator for spin- $\frac{1}{2}$ and spin-$\frac{3}{2}$ particles
respectively. The unknown parameters A, B and C arise in the expansion of
quadrupole moment and charge radii operators including the contribution of
orbital and color space [16-18]. These constants (A, B and C) are determined
from the experimental data on charge radii and quadrupole moments.
For a better understanding of the internal structure of baryons, an approach
based on operator formalism is employed. In this formalism, a suitable
operator which is associated with the properties of baryons, are applied to
the terms of the wavefunction mentioned in equ. (1) $\&$ (2) represented as:

$\displaystyle\langle\Phi_{1/2,3/2}^{(\uparrow)}|{\widehat{O}}|\Phi_{1/2,3/2}^{(\uparrow)}\rangle=$
$\displaystyle\frac{1}{N^{2}}[{a_{0}}^{2}{\langle\Phi_{1}}^{(\frac{1}{2}\uparrow)}|\widehat{O}|{\Phi_{1}}^{(\frac{1}{2}\uparrow)}\rangle+{a_{8}}^{2}\langle{\Phi_{8}}^{(\frac{1}{2}\uparrow)}|\widehat{O}|{\Phi_{8}}^{(\frac{1}{2}\uparrow)}\rangle+$
(14)
$\displaystyle{a_{10}}^{2}\langle{\Phi_{10}}^{(\frac{1}{2}\uparrow)}|\widehat{O}|{\Phi_{10}}^{(\frac{1}{2}\uparrow)}\rangle+{b_{1}}^{2}\langle{\Phi_{1}}^{(\frac{1}{2}\uparrow)}|\widehat{O}|{\Phi_{1}}^{(\frac{1}{2}\uparrow)}\rangle+{b_{8}}^{2}\langle{\Phi_{8}}^{(\frac{1}{2}\uparrow)}|\widehat{O}|{\Phi_{8}}^{(\frac{1}{2}\uparrow)}\rangle$
$\displaystyle+{c_{8}}^{2}\langle{\Phi_{8}}^{(\frac{3}{2}\uparrow)}|\widehat{O}|{\Phi_{8}}^{(\frac{3}{2}\uparrow)}\rangle+{d_{8}}^{2}\langle{\Phi_{8}}^{(\frac{3}{2}\uparrow)}|\widehat{O}|{\Phi_{8}}^{(\frac{3}{2}\uparrow)}\rangle]$

where $\widehat{O}$ indicates the quadrupole moment and charge radii operator.
Based on this formalism, we are able to derive certain expressions for the
quadrupole moments of baryons listed in Table 1. Similarly, the obtained
expression for charge radii of octet and decuplet particles are displayed in
Table 2.

Baryon |  $\langle\Phi_{3/2}^{(\uparrow)}|Q_{B}^{2}|\Phi_{3/2}^{(\uparrow)}\rangle N^{2}$
---|---
p |  ${a_{0}}^{2}$(-0.333B) + ${a_{8}}^{2}$(0.822B) + ${a_{10}}^{2}$(B) + ${b_{1}}^{2}$(0.9464B - 2.22C) + ${b_{8}}^{2}$(0.176B - 0.296C) + ${b_{10}}^{2}$(0.0574B) + ${c_{8}}^{2}$(1.889B + 2.445C) + ${d_{8}}^{2}$(4.4B -0.06C)
n |  ${a_{0}}^{2}$(0.666B) + ${a_{8}}^{2}$(0.3779B) + ${b_{1}}^{2}$(-1.111B + 1.77C) + ${b_{8}}^{2}$(-0.341B + 0.126C) + ${c_{8}}^{2}$(-0.111B + 0.444C) + ${d_{8}}^{2}$(-0.466B + 0.7998C)
$\Delta^{++}$ |  ${a_{0}}^{2}$(2B +4C) + ${b_{1}}^{2}$(-3.523B + 8.121C) + ${b_{8}}^{2}$(-4.328B +7.856C) + ${d_{1}}^{2}$(-1.2B - 0.796C) + ${d_{8}}^{2}$(-1.894B -0.215C)
$\Delta^{+}$ |  ${a_{0}}^{2}$(B -2C) + ${b_{1}}^{2}$(-1.66B + 1.838C) + ${b_{8}}^{2}$(-2.041B + 1.816C) +${d_{1}}^{2}$(0.192B - 1.704C) + ${d_{8}}^{2}$(0.491B - 2.314C)
$\Delta^{0}$ |  ${b_{1}}^{2}$(0.622B - 0.355C) + ${b_{8}}^{2}$(0.933B - 0.533C) + ${d_{1}}^{2}$(-0.533B + 1.066C) + ${d_{8}}^{2}$(-0.533B + 1.066C)
$\Delta^{-}$ |  ${a_{0}}^{2}$(-B - 2C) + ${b_{1}}^{2}$(1.762B - 4.06C) + ${b_{8}}^{2}$(2.164B - 3.928C) + ${d_{1}}^{2}$(0.066B + 0.133C) + ${d_{8}}^{2}$(0.893B + 0.214C)
Table 1: Expression obtained after applying quadrupole moment operator to the baryon octet $J^{P}=\frac{1}{2}^{+}$ and decuplet $J^{P}=\frac{3}{2}^{+}$ particles Baryon |  $\langle\Phi_{3/2}^{(\uparrow)}|r_{B}^{2}|\Phi_{3/2}^{(\uparrow)}\rangle N^{2}$
---|---
p |  ${a_{0}}^{2}$(0.333A +0.666B -1.666C)+ ${a_{8}}^{2}$(0.333A +2.110B -3.110C)+ ${a_{10}}^{2}$(0.333A- B) +${b_{1}}^{2}$(0.647A -2.498B +0.555C) +${b_{8}}^{2}$(1.352A -4.613B +0.555C) +${b_{10}}^{2}$(0.646A - 1.940B) + ${c_{8}}^{2}$(0.549A +0.018B -1.666C) + ${d_{8}}^{2}$(0.066A- 1.266B +1.066C)
n |  ${a_{0}}^{2}$(-1.33B + 1.33C) + ${a_{8}}^{2}$(-0.178B +0.178C) + ${b_{1}}^{2}$(0.444B - 0.444C) + ${b_{8}}^{2}$(0.059B -0.059C) + ${c_{8}}^{2}$(-0.222B +0.222C) + ${d_{8}}^{2}$ (0.134B - 0.134C)
$\Delta^{++}$ |  ${a_{0}}^{2}$(2A + 4B + 2C) + ${b_{1}}^{2}$(0.910A + 5.269B - 2.539C) + ${b_{8}}^{2}$(0.420A + 6.739B - 5.478C) + ${d_{1}}^{2}$(0.4A +0.8B+ 0.533C) + ${d_{8}}^{2}$(-0.647A +2.942B -4.885C)
$\Delta^{+}$ |  ${a_{0}}^{2}$(A+2B+C) + ${b_{1}}^{2}$(0.168A+3.601B -3.098C) + ${b_{8}}^{2}$(0.210A + 4.036B - 3.406C c) + ${d_{1}}^{2}$(0.198A +0.406B -0.212C) + ${d_{8}}^{2}$(0.533A - 1.099B + 2.698C)
$\Delta^{0}$ |  ${b_{1}}^{2}$(-0.88B + 0.88C) + ${b_{8}}^{2}$ (-1.33B + 1.33C)
$\Delta^{-}$ |  ${a_{0}}^{2}$(-A -2B - C) +${b_{1}}^{2}$ (-0.168A - 3.163B + 2.66C) + ${b_{8}}^{2}$(-0.210A - 2.869B + 2.239C) + ${d_{1}}^{2}$(-0.21A - 0.4B -0.2C) + ${d_{8}}^{2}$(-0.533A+1.099B -2.699C)
Table 2: Expressions obtained after applying charge radii operator to the
baryon octet $J^{P}$= $\frac{1}{2}^{+}$ and decuplet $J^{P}$=
$\frac{3}{2}^{+}$ particles

Our derived expressions involved two input parameters as listed in Table 1 and
2. One corresponds to the parameters used in operator formalism (A, B and C).
The values of these parameters (A, B and C) are fitted by taking the available
experimental data on quadrupole moment and charge radii as input. The other
ones are statistical coefficients
($a_{0},a_{8},a_{10},b_{1},b_{8},b_{10},c_{8},d_{8}$) that plays a pivotal
role in determining the static properties of baryonic systems. As we know,
there is a dynamic sea present with the valence quarks in the structure of
baryons, where sea is a cluster consisting of gluons and quark-antiquarks
pairs. The contribution of sea in spin, flavor and color space are determined
in terms of these parameters
($a_{0},a_{8},a_{10},b_{1},b_{8},b_{10},c_{8},d_{8}$). In order to calculate
these parameters, we apply a statistical approach with the detailed balance
principle.

##  4 Statistical model

Hadrons are assumed to be an ensemble of valence quarks and sea containing
various quark-gluon Fock states. These quarks and gluons in the Fock states
are basically the ”intrinsic” partons that are non-perturbatively
multiconnected to the valence quarks. The complete set of quark-gluon Fock
states can be expressed in expanded form as:

$|Baryon\rangle=\sum_{i,j,k}C_{i,j,k}|(q^{3}),(i,j,k)\rangle$ (15)

Here $q^{3}$ represents the three valence quarks of the baryon, i is the
number of quark-antiquark $u\bar{u}$ pairs, j is the quark-antiquark
$d\bar{d}$ pairs and k is the number of gluons [27]. The probability of
finding the baryon in quark-gluon Fock state is $\rho_{i,j,k}=|C_{i,j,k}|^{2}$
and satisfy the condition of normalization: $\sum_{i,j,k}\rho_{i,j,k}=1$.
The statistical model assumes the decomposition of baryonic state in various
quark-gluon Fock states such as $|qqq\rangle|g\rangle,|qqq\rangle|gg\rangle$,
$|qqq\rangle|q\bar{q}\rangle$ with the possibility of gluon splitting into
$q\bar{q}$ pairs and the annihilation of quark-antiquark pairs into gluons.
The octet and decuplet wavefunction of baryons includes the flavor, color, and
spin space. To find the individual contribution of flavor, spin and color
space, we calculate their relative probabilities. For the relative probability
in flavor space, we depend upon the principle of detailed balance [46]. This
principle proposed by Zhang et al. [47], based on the assumption that every
two subensembles balance with each other in a way:

$\rho_{i,j,k}|\\{q^{3}\\},\\{i,j,k\\}\rangle\xLongleftrightarrow{balance}\rho_{i^{\prime},j^{\prime},k^{\prime}}|\\{q^{3}\\},\\{i^{\prime},j^{\prime},k^{\prime}\\}\rangle$

The basic assumption of balancing any two ensembles is that the probability of
finding hadrons in any Fock state should not change over time. The “go out”
probability balance the “come in” probability for any Fock state.

$\rho_{A}R_{A\rightarrow B}=\rho_{B}R_{B\rightarrow A}$

Here, A and B represent the complete set of all Fock states. $\rho_{A}$ is the
probability of finding the hadrons in Fock state A, $\rho_{B}$ is the
probability of finding the hadrons in Fock state B and $R_{A\rightarrow B}$,
$R_{B\rightarrow A}$ is the transition probability of Fock State from A to B
and B to A. Various transition subprocesses such as q $\Leftrightarrow$ qg, g
$\Leftrightarrow q\bar{q}$ and g $\Leftrightarrow$ gg are considered to
calculate the transition probability of different Fock states in flavor space
can be found in ref. [29, 47]. The transition probability talks about the
chances of gluon(s) $\Leftrightarrow$ $u\bar{u}$, $d\bar{d}$, $s\bar{s}$ or
$u\bar{u}\Leftrightarrow d\bar{d}$ states etc. This approach will change the
Fock state $\&$ hence change the probability. The transition probabilities
$R_{A\rightarrow B}$, $R_{B\rightarrow A}$ work on the same principle. This
principle has been successfully explain the flavor asymmetry of hadrons [27],
Parton distribution functions [48]. Moreover, the probabilities in spin and
color space are calculated from the multiplicities in the statistical model.
The multiplicities for all Fock states are computed in the form of
$\rho_{p,q}$ where the relative probability for valence part has ’Spin p’ and
sea carries ’Spin q’ such that the resultant spin should come out as 1/2 for
octet baryons and 3/2 for decuplet baryons. Similarly, the probabilities for
the color spaces can be calculated which yields the color singlet state.
Detailed calculations of these multiplicities are discussed in ref. [27].
Further, the product of probabilities in spin and color space can be written
in terms of a common multiplier “c” computed in the form of “nc”, where n is
the multiplicity factor for each Fock state [29]. The sum of the total
probabilities in spin and color space will give the coefficients
$a_{0},a_{8},a_{10},b_{1},b_{8},b_{10},c_{8},d_{8}$ to the total wavefunction.
These coefficients give us the contribution of sea quarks in computing the
different properties of baryons. The main part within the statistical model
which fascinates us that all the properties of baryons such as masses, spin
distribution, magnetic moment etc. are interrelated with these probabilities
(flavor, spin and color). Detailed information for the calculation of all the
parameters using the statistical model is given in Refs. [27, 35]. The
statistical model in combination with the principle of detailed balance is
reliable for describing the properties like flavor asymmetry, semi-leptonic
decays, masses, magnetic moments, spin distribution and axial coupling ratios
of baryon octet and decuplet particles [3, 28, 29, 30, 31].

##  5 Results and Discussion

Low-energy electromagnetic properties of baryons like charge radii and
quadrupole moments are very important to probe their internal structure. These
properties are computed by taking the effect of scalar (spin 0), tensor (spin
2) and vector (spin 1) sea. Individual contribution suggests the dominancy of
sea (scalar, vector $\&$ tensor) to the quadrupole moment and charge radii of
baryons. To examine the separate contribution of sea, we assume
$a_{0,8,10}\neq 0$ and $b_{1,8,10},c_{8},d_{8}=0$ for scalar sea; whereas for
vector sea $b_{1,8,10},c_{8}\neq 0$ and $a_{0,8,10},d_{8}=0$; for tensor sea
contribution, $d_{8}\neq 0$ and $b_{1,8,10},c_{8},a_{0,8,10}=0$ for octet
baryons. In similar manner, the individual contribution of sea (scalar, vector
and tensor) can be obtained for decuplet particles. It is worthy to mention
that the statistical coefficients ($a_{0,8,10},b_{1,8,10},c_{8},d_{1,8}$)
demonstrate the importance of scalar, vector and tensor sea for determining
the properties (charge radii and quadrupole moment) of baryons. These
parameters are used as an input in the present work which is taken from the
ref. [27, 35]. Different forms (Model C, P and D) of the statistical model
help to find the role of sea dynamics to several static properties. We
analyzed the properties (charge radii, quadrupole moment) by using Model D
which is stated by Singh $\&$ Upadhyay [27] that the contribution of states
with higher multiplicities is suppressed. The suppression is based on the
assumption that probability of a system in spin and color state is inversely
proportional to the multiplicity (both in spin and color spaces) of the state.
We would also like to mention that by taking the available experimental data
on charge radii and quadrupole moment as input, we fit the parameters A, B and
C by using $\chi^{2}$ minimization. The set of parameters (A, B and C)
obtained after fitting are presented in Table 3 and 4.

###  5.1 Charge radii

To study the internal structure of hadrons, we are concerned with the two
major crisis ’proton spin’ and ’proton charge radius’ which still remain
unresolved. The proton radius puzzle has motivated new theoretical and
experimental work. Recently PRad, an electron-proton scattering experiment at
Jefferson Lab [49], reported a proton charge radius of $0.831\pm
0.007_{stat.}\pm 0.012_{sys.}$ fm. Moreover, in the three most recent
H-spectroscopy measurements [50-52], two experiments found a small radius
($r_{p}=0.833\pm 0.010$ fm) [50,52] and another one supports a larger value of
charge radii ($r_{p}=0.877$ fm) of proton [51]. The electric charge radii are
related to the spatial distribution of charged constituents (quarks) and
provide information about the size of the hadrons. Using the statistical
method, we present the calculated numerical values of electric charge radii of
nucleons ($J^{P}=\frac{1}{2}^{+}$) and $\Delta$-hyperons
($J^{P}=\frac{3}{2}^{+}$) in Table 3. In order to get the significant
contribution of sea, the statistical parameters
($a_{0,8,10},b_{1,8,10},c_{8},d_{1,8}$) combine with valence part and the
value of charge radii of baryons are determined. For nucleons (p, n), the
dominancy of vector sea can be easily observed. Since the sea part is
dominated by the emission of virtual gluons, we can expect the major
contribution coming from the vector sea parameters ($b_{1},b_{8},c_{8}$) to
the charge radii of nucleons. For $J^{P}=\frac{3}{2}^{+}$ particles, we
observed that the scalar sea is an active contributor from the total sea. This
might be due to the larger multiplicities of valence quark spin states when
combined with the sea quarks spin (i.e. spin- 0,1 and 2). The probability of
having spin-$\frac{3}{2}$ is more when coupled with spin-0 (scalar sea) as
compared to spin-1 (vector sea). For both octet and decuplet baryons, the
tensor sea appears to be less dominating due to the quark spin-flip process
but its effect cannot be neglected. For $\Delta^{0}$, only vector sea
contributed while pure scalar and tensor sea impact is zero. Because the
contribution of valence quark across the scalar sea ($a_{0}$) and tensor sea
($d_{1},d_{8}$) term is zero.
The importance of sea lies in the suppression of higher multiplicities of the
quark-gluon Fock states. However in case, when the sea is excluded in the
statistical model, the charge radii deviates more than $50\%$ from the
computed value. The deviation is maximum for the proton and $\Delta^{++}$ is
evident in Table 3. On the contrary, if the total sea (scalar, vector, and
tensor) is taken, the charge radii of nucleons is pretty close to the
experimental value [7]. On comparison our predicted value of charge radii
shows good agreement in sign and magnitude with different phenomenological
models [18, 21-23, 53-55]. Various theoretical models [18, 21] show the same
sign for both particles $\Delta^{+},\Delta^{-}$. Although, the difference in
sign of charge radii of $\Delta^{+},\Delta^{-}$ matches with the Lattice QCD
predictions [56] within the error of 20$\%$. Also, the three parameters A, B
and C are essential for valence quark distribution in charge radii. In order
to understand the individual contribution of one-, two- and three-quark term,
we calculated the charge radii corresponding to one- and two-quark terms by
taking the value of C=0. After putting C=0, the results are affected, as
charge radii decreases in case of nucleons and for decuplet baryons, predicted
values are increased upto 40$\%$. It indicates that a small variation in ’C’
leads to a considerable change in charge radii values.

|  | Statistical Model with SU(3) symmetry |  |
---|---|---|---|---
Baryons |  Charge radii |  With |  |  |  |  |  |  |
A=0.754 |  |  |  |  |  |  |  |  |
B=0.062 |  |  |  |  |  |  |  |  |
C=-0.337 |  Scalar |  |  |  |  |  |  |  |
Sea |  Vector |  |  |  |  |  |  |  |
Sea |  Tensor |  |  |  |  |  |  |  |
Sea |  Without |  |  |  |  |  |  |  |
Sea |  With |  |  |  |  |  |  |  |
A=0.754 |  |  |  |  |  |  |  |  |
B=0.062 |  |  |  |  |  |  |  |  |
C=0 | Ref.[22] | Ref.[56] |  |  |  |  |  |  |
p | 0.756A -1.576B -0.693C | 0.707 | 0.352 | 0.402 | -0.020 | 4.126 | 0.472 | 0.766 | 0.685
n | -0.292B +0.292C | -0.116 | -0.02 | -0.084 | 0.002 | -0.436 | -0.018 | -0.116 | -0.158
$\Delta$++ | 1.909A + 3.862B+ 1.871C | 1.049 | 1.018 | 0.028 | 0.0 | 4.409 | 1.681 | 0.996 | -
$\Delta$+ | 0.951A + 1.940B + 0.894C | 0.537 | 0.509 | 0.028 | 0.0 | 3.165 | 0.839 | 0.983 | 0.410
$\Delta$0 | -0.0106B + 0.0106C | -0.004 | 0.0 | -0.004 | 0.0 | -0.882 | 0.0 | -0.025 | -
$\Delta$- | -0.965A - 1.951B - 0.943C | -0.531 | -0.518 | -0.0103 | 0.0 | -2.393 | -0.849 | 1.033 | -0.410
Table 3: Numerical values for the charge radii of nucleons and
spin-$(\frac{3}{2})^{+}$ decuplet baryons in the units of [fm2]

###  5.2 Quadrupole moment

Quadrupole moments are significant for providing evidence for the non-
sphericity of baryons. They are important for determining the shape of
baryons. Within the statistical approach, we present the computed values of
quadrupole moments in Table 4. The individual contributions of scalar, vector
and tensor sea are also presented in the Table 4. For nucleons (p, n), vector
sea is the only active contributor from the total sea. This means that the
quadrupole moment is mainly affected by vectorial sea parameters
($b_{1},b_{8},c_{8}$) having spin 1. For decuplet baryons
($\Delta^{++},\Delta^{+},\Delta^{-}$), the scalar sea contribution is greater
as compared to vector and tensor sea. Due to the quark spin-flip process, the
tensor sea has negligible contribution. The valence quark coupled with spin
0,1 and 2 (scalar, vector and tensor sea respectively) resulting into a large
no. of spin states. The estimated multiplicities from spin states suggests
that the higher multiplicity have a lower probability of survival. This might
indicate that the chances of having spin-$\frac{3}{2}^{+}$ is more with the
scalar sea (spin-0) as compared to vector and tensor sea. In case of
$\Delta^{0}$, neither vector nor scalar sea contributes, but a little bit of
contribution comes from the tensor sea.
In general, when the sea is completely excluded in the statistical model, the
value of quadrupole moment shows a large deviation from the computed value
which shows the active participation of sea quarks. A considerable change
occurs only in the magnitude of the quadrupole moment but not in the sign.
From this, we concluded that the impact of sea quarks do not directly affect
the shape of baryons but rather it influences the overall structure of
baryons. The sea contribution in quadrupole moment are given in Table 4 from
different models. As there is no experimental information available for the
quadrupole moments of baryons, we compare our findings with other theoretical
predictions [16, 55, 57, 58] and they are consistent with the ref. [21, 23].
In the statistical model, the value of quadrupole moment observed an oblate
shape for p, n, $\Delta^{++},\Delta^{0}$ baryons while a prolate shape for
$\Delta^{+},\Delta^{-}$ baryon. The quadrupole moment of
$\Delta^{+},\Delta^{-}$ show strange behavior with respect to the parameter
’C’ which includes the contribution of three-quark term. If we put C=0, a -ve
sign for $\Delta^{+}$ and a +ve sign for $\Delta^{-}$ observed which is
consistent with other theoretical models [16, 21, 53]. Although, a
considerable change in the value of quadrupole moment observed after putting
C=0. This shows the importance of three-quark term parameter ’C’ to the
quadrupole moment values. Similarly, by putting B=0, the contribution
corresponding to the three-quark term can be calculated. The results are
presented in the table below:

|  | Statistical model with SU(3) symmetry |  |
---|---|---|---|---
Baryons |  Quadrupole |  |  |  |  |  |  |  |
moment |  With |  |  |  |  |  |  |  |
B=-0.006 |  |  |  |  |  |  |  |  |
C=-0.094 |  Scalar |  |  |  |  |  |  |  |
Sea |  Vector |  |  |  |  |  |  |  |
Sea |  Tensor |  |  |  |  |  |  |  |
Sea |  Without |  |  |  |  |  |  |  |
Sea |  With |  |  |  |  |  |  |  |
B=-0.006 |  |  |  |  |  |  |  |  |
C=0 | Ref.[21,23] | Ref.[16] |  |  |  |  |  |  |
p | 0.707B + 0.292C | -0.032 | 0.0 | -0.030 | -0.001 | -0.043 | -0.004 | -0.032 | -
n | 0.4128B + 0.0241C | -0.038 | -0.001 | -0.034 | -0.002 | -0.326 | -0.055 | -0.019 | -
$\Delta$++ | 1.842B + 3.863C | -0.375 | -0.372 | -0.006 | 0.002 | -1.884 | -0.011 | -0.343 | -0.12
$\Delta$+ | 0.929B -1.939C | 0.177 | 0.171 | -0.002 | 0.007 | 0.064 | -0.005 | -0.171 | -0.06
$\Delta$0 | 0.018B + 0.046C | -0.004 | 0.0 | 0.00 | -0.004 | -0.120 | 0.0 | 0.0 | 0.0
$\Delta$- | -0.939B - 1.940C | 0.188 | 0.186 | 0.003 | 0.0 | 0.885 | 0.005 | 0.171 | 0.06
Table 4: Numerical values for the quadrupole moments of nucleons and
spin-$(\frac{3}{2})^{+}$ decuplet baryons in the units of [fm2]

##  6 Conclusion

In the present paper, a statistical approach with the principle of detailed
balance is used to calculate the electromagnetic properties i.e. charge radii
and quadrupole moments of the nucleons and $\Delta$-hyperons. Our main purpose
is to study the impact of sea quarks on the properties of baryons in the
framework of statistical model with detailed balance principle. It is
important to study the sea because we expect the sea to be a decisive factor
for determine the properties of baryons. A suitable anti-symmetric
wavefunction for baryons written in spin, color, flavor space is operated with
charge radii and quadrupole moment operator. Since the sea contains quark-
gluon Fock states, the statistical principle determine the probabilities for
these Fock states in terms of statistical parameters
($a_{0},a_{8},a_{10},b_{1},b_{8},b_{10},c_{8},d_{8}$). With the help of these
parameters, we deeply explore the individual sea contribution to the charge
radii and quadrupole moment of baryons. The parameter ’C’ considerably
contributes to the spin-spin interaction term of three-quarks. The quadrupole
moment and charge radii are associated with the spin contribution and spatial
charge distribution of the particles respectively. And the sea spin
contribution is reflected in the results of these properties. Based on our
analysis of the quadrupole moment, we conclude that the sea quarks have a
pronounced impact on the overall structure of baryons, without directly
inducing any changes in their shape. Also, our numerical analysis appreciated
the presence of sea (scalar, vector and tensor) and gives crucial information
about the geometrical structure of baryons. It is important to mention that
our calculations performed in a non-relativistic frame and hold good for the
hadronic energy scale of order 1 GeV2.

##  7 Acknowlegement

The authors gratefully acknowledge the financial support by the Department of
Science and Technology (SERB/F/9119/2020), New Delhi.

##  Appendix A Appendix 1

The expansion of quadrupole moment operator are given below:

$\widehat{Q}_{B}=B\sum_{i\neq
j}^{3}e_{i}(3\sigma_{iz}\sigma_{jz}-\mathbb{\sigma_{i}.\sigma_{j}})+C\sum_{i\neq
j\neq k}^{3}e_{i}(3\sigma_{jz}\sigma_{kz}-\mathbb{\sigma_{j}.\sigma_{k}})$
(16) $\widehat{Q}_{B}=B\left[\sum_{i\neq
j}^{3}3e_{i}\sigma_{iz}\sigma_{jz}-\sum_{i\neq
j}^{3}e_{i}(\mathbb{\sigma_{i}.\sigma_{j})}\right]+C\left[\sum_{i\neq j\neq
k}^{3}3e_{i}\sigma_{jz}\sigma_{kz}-\sum_{i\neq j\neq
k}^{3}e_{i}\mathbb{(\sigma_{j}.\sigma_{k})}\right]$ (17)

The spin operator terms included in above eq. can be written as [23]:

$\sum_{i\neq
j}e_{i}\mathbb{(\sigma_{i}.\sigma_{j})}=2J\sum_{i}e_{i}\sigma_{iz}-3\sum_{i}e_{i}\\\
$ (18) $\sum_{i\neq j\neq k}e_{i}\mathbb{(\sigma_{j}.\sigma_{k}})=\pm
3\sum_{i}e_{i}-\sum_{i\neq j}e_{i}\mathbb{(\sigma_{i}.\sigma_{j})}$ (19)

In equ. (19), + sign holds for J=$\frac{3}{2}$ states and the - sign holds for
J=$\frac{1}{2}$ states. After putting these operator values, equ. (17)
becomes:

$\widehat{Q}_{B}=B\sum_{i\neq j}^{3}3e_{i}\sigma_{iz}\sigma_{jz}-B\sum_{i\neq
j}^{3}\left(2J\sum_{i}e_{i}\sigma_{iz}+3\sum_{i}e_{i}\right)+C\sum_{i\neq
j\neq k}^{3}3e_{i}\sigma_{jz}\sigma_{kz}-C\sum_{i\neq j\neq k}^{3}\left(\pm
3\sum_{i}e_{i}-\sum_{i\neq j}e_{i}\mathbb{\sigma_{i}.\sigma_{j}}\right)$ (20)

Using the expectation value of operator $2J\sum_{i}e_{i}\sigma_{iz}$, the
operator in equ. (18) and (19) for octet and decuplet states written as:

For $\mathbb{J=\frac{1}{2}}$ For $\mathbb{J=\frac{3}{2}}$
$\sum_{i\neq
j}e_{i}\mathbb{(\sigma_{i}.\sigma_{j})}=3\sum_{i}e_{i}\sigma_{iz}-3\sum_{i}e_{i}$
$\sum_{i\neq
j}e_{i}\mathbb{(\sigma_{i}.\sigma_{j})}=5\sum_{i}e_{i}\sigma_{iz}-3\sum_{i}e_{i}$

$\sum_{i\neq j\neq
k}e_{i}\mathbb{(\sigma_{j}.\sigma_{k}})=-3\sum_{i}e_{i}\sigma_{iz}$
$\sum_{i\neq j\neq
k}e_{i}\mathbb{(\sigma_{j}.\sigma_{k}})=6\sum_{i}e_{i}-5\sum_{i}e_{i}\sigma_{iz}$

After substituting the above values in equ. (20), the expression for the
quadrupole moment operator of octet and baryons can be expressed as:

$\widehat{Q}_{1/2}=3B\sum_{i\neq j}e_{i}\sigma_{iz}\sigma_{jz}+3C\sum_{i\neq
j\neq
k}e_{i}\sigma_{jz}\sigma_{kz}+(-3B+3C)\sum_{i}e_{i}\sigma_{iz}+3B\sum_{i}e_{i}$
(21) $\widehat{Q}_{3/2}=3B\sum_{i\neq
j}e_{i}\sigma_{iz}\sigma_{jz}+3C\sum_{i\neq j\neq
k}e_{i}\sigma_{jz}\sigma_{kz}+(-5B+5C)\sum_{i}e_{i}\sigma_{iz}+(3B-6C)\sum_{i}e_{i}$
(22)

The expansion of charge radii operator given as:

$\widehat{r}_{B}^{2}=A\sum_{i}e_{i}.1+B\sum_{i\neq
j}e_{i}\sigma_{i}\sigma_{j}+C\sum_{i\neq j\neq k}e_{i}\sigma_{j}\sigma_{k}$
(23)

Substituting the operator value of equ. (18) and (19) in equ. (23), the
expression for charge radii of octet and decuplet baryons obtained:

$\widehat{{r}}^{2}_{1/2}=(A-3B)\sum_{i}e_{i}+3(B-C)\sum_{i}e_{i}\sigma_{iz}$
(24)
$\widehat{{r}}^{2}_{3/2}=(A-3B+6C)\sum_{i}e_{i}+5(B-C)\sum_{i}e_{i}\sigma_{iz}$
(25)

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