# Properties of N, $\Delta$ Baryons with Screened Potential

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

###### Abstract

N and $\Delta$ baryons hold an important place towards understanding the quark
dynamics inside hadrons. The hypercentral Constituent Quark Model (hCQM) has
been employed in various studies ranging from light to heavy hadrons. In the
present article, screened potential has been used to study light baryon
resonances. The Regge trajectories have been plotted alongwith the details of
slopes and intercepts. The strong decay widths to pion have been calculated
for some channels using the present masses.

###### keywords:

Screened potential, light baryon, Pion decay width

##  1 Introduction

Hadron spectroscopy is an important tool to understand the quark dynamics
inside hadrons [1, 2, 3]. Most nuclear phenomena can be explained in terms of
the non-relativistic interactions between protons and neutrons, which make up
the nucleus’s elementary constituents. Quantum Chromodynamics (QCD), on the
other hand, is the theory behind nuclear forces and describes relativistic
quarks and gluons as the fundamental degrees of freedom [4]. Confinement
serves as QCD’s defining characteristic. It appears to prohibit the free
existence of isolated elementary quarks and gluons in nature. One of the
central issues in physics is how to understand how confinement arises.
Confinement results from the self-interactions of gluons, which act as the
strong force’s intermediaries between colored quarks and other gluons. Since
the light, u, and d quarks that make up the nuclei are many times lighter than
the proton, the majority of the visible mass in the universe is actually
produced by relativistic interactions between gluons.

$N^{*}$ corresponding to P(uud) and N(udd) with $J=\frac{1}{2}$ and isospin
$I=\frac{1}{2}$ has always been in discussion for decades [5]. N. Isgur has
nicely highlighted few facts as to why to study of $N^{*}$ have always been in
the priority as: The whole stable matter around us is made up of nucleons; it
being the simplest system to manifest the non-abelian nature of QCD. However,
inspite of being the lowest state, it has been known through understanding of
years, that hadrons are a really complex systems whose all the properties are
not known to us till date.

Over the years, many approaches have been implemented with the intention to
understand thoroughly the light baryon sector. Recently all light and strange
baryons have been studied through Bethe Ansatz method with U(7) by an
algebraic method [6]. The earlier algebraic method has been discussed by R.
Bijker et.al. to study baryon resonances in terms of string-like model [7]. A.
V. Anisovich et al. has reproduced the N and $\Delta$ spectrum using the
multichannel partial wave analysis of pion and photo-induced reactions [8].
The quark-diquark model using Gursey Radicati-inspired exchange interaction
has been studied [9, 10]. The semi-relativistic constituent quark model,
classification number describing baryon mass range [11], mass formula obtained
by Klempt [12], dynamical chirally improved quarks by BGR [13] are among
various models. Also, the relativistic study with quark-diquark model have
been studied for all sectors by Faustov et al. [14]. A recent study based on
HAL-QCD, decuplet baryons have been focused wherein interaction potentials
extracted from lattice QCD [15]. Regge phenomenology has also been employed in
the study of light, strange baryons using n and J plane linear curves [16].

The present study is based on a non-relativistic approach namely hypercentral
Constituent Quark Model (hCQM). Our earlier works deal with hCQM applied for
hadrons from light to heavy sector. The screened potential term is accompanied
with spin-dependent term in the present approach. The section 3 discusses mass
spectra of N and $\Delta$ baryons alongwith the experimental and theoretical
background known so far. Section 4 is dedicated to Regge trajectory for
(J,$M^{2}$) and (n,$M^{2}$). The strong decay channels with pion have been
studied in section 5.

##  2 Theoretical Framework

The choice of a hypercentral SU(6) invariant potential, sometimes referred to
as a potential whose value is exclusively governed by the hyperradius x, is
the foundation of the hypercentral Constituent Quark Model, abbreviated as
hCQM. In addition to making the process of finding a solution for the
Schrodinger equation more straightforward, the selection of a hypercentral
potential also has some intriguing ramifications in terms of the physical
world. As a result of the fact that the hyperradius x is dependent on each of
the three constituent coordinates at the same time, a hypercentral potential
is not just an interaction between two bodies but can also involve terms
involving three bodies [18, 17]. hCQM has been applied in various systems and
with a variety of potentials by our team for heavy hadrons [19, 20, 21, 22,
24, 23, 25, 26, 27] and exotics. The linear potential has been employed for
all octet, decuplet baryons in our earlier works [28, 29, 30, 31, 32].

Because of the non-abelian nature of QCD, which results in gluon-gluon
interactions, which, in turn, can produce three body forces, these terms have
the potential to play an important part in the description of hadrons. The
space component of the 3q wave function, on the other hand, can be expanded in
the hyperspherical harmonics basis, at which point the Schrodinger equation
transforms into a set of coupled differential equations. Additionally, it has
been shown that the low lying resonance states can be adequately described by
the hyperspherical approximation, which is the assumption that only the first
term in the expansion of the potential in terms of hyperspherical harmonics is
maintained. Keeping these considerations in mind, a hypercentral potential can
be interpreted in one of two ways: either as a standard two body potential or
as a three body potential, both of which are treated in the hypercentral
approximation. First of all, we introduce hyperspherical coordinates as
hyperradius x and hyperangle $\xi$ from Jacobi coordinates as [7],

$x=\sqrt{{\bf\rho^{2}}+{\bf\lambda^{2}}};\;\;\xi=arctan(\frac{\rho}{\lambda})$
(1)

The model itself suggets that the potential to be chosen should be
hypercentral i.e. depending only on hyperradius x. The hyperradius x depends
at the same time on all the three constituent coordinates, therefore an
hypercentral potential is not a pure two body interaction but can also contain
three body terms. These terms can play an important role in the description of
hadrons, since the non-abelian nature of QCD leads to gluon-gluon interaction
which can, in turn, produce three body forces. On the other hand the space
part of the 3q wave function can be expanded in the hyperspherical harmonics
basis and the Schrodinger equation becomes a set of coupled differential
equations. Thus, the spatial wave-function can be expressed as hyper-radial
part and hyperspherical harmonics as

$\psi_{space}=\psi(x)Y(\Omega_{\rho},\Omega_{\lambda},\xi)$ (2)
$L^{2}Y_{[\gamma]}l_{\rho}l_{\lambda}(\Omega_{\rho},\Omega_{\lambda},\xi)=-\gamma(\gamma+4)Y_{[\gamma]}l_{\rho}l_{\lambda}(\Omega_{\rho},\Omega_{\lambda},\xi)$
(3)

where, $\Omega_{\rho}$ and $\Omega_{\lambda}$ are the angle of hyperspherical
coordinates. $\vec{L}=\vec{L_{\rho}}+\vec{L_{\lambda}}$ is the total angular
momentum and $l_{\rho}$ and $l_{\lambda}$ are the angular momenta associated
with the Jacobi coordinates $\rho$ and $\lambda$ respectively.
$-\gamma(\gamma+4)$ gives the eigenvalues of $L^{2}$, where,
$\gamma=2n+\rho+\lambda$ is the grand angular momentum quantum number which
acquires positive integer value.

The hyper-radial equation whose solution is $\psi(x)$ is as follows,

$\left[\frac{d^{2}}{dx^{2}}+\frac{5}{x}\frac{d}{dx}-\frac{\gamma(\gamma+4)}{x^{2}}\right]\psi(x)=-2m[E-V_{3q}(x)]\psi(x)$
(4)

Another choice of hypercentral potential has brought us to screened potential
of the form as described below [33].

$V^{0}(x)=a\left(\frac{1-e^{-{\mu}x}}{\mu}\right)$ (5)

Such potential has been known to show good results using hCQM for heavy quark
system including mesons and baryons [34]. However, here we have attempted to
see such effect in light, strange systems as well. The screening parameter is
different in case of heavy and light systems. Also, the results have been
discussed for this study here. Based on a paper by R. Chaturvedi, the
screening parameter $\mu$ has been varied over a range and 0.3 has been
considered as the value obtain the spectra for all the systems considered
here.[35]
Also, this potential form cannot account for splittings of multiplet levels.
So, an additional hyperfine splitting term is to be incorporated. However, the
spin-dependent interaction $V_{SD}(x)$ consists of three types of
interactions, i.e. spin-orbit, spin-spin and tensor terms [36].

$\displaystyle V_{SD}(x)=V_{LS}(x)({\bf L\cdot
S})+V_{SS}(x)\left[S(S+1)-\frac{3}{2}\right]$ (6)
$\displaystyle+V_{T}(x)\left[S(S+1)-\frac{3({\bf S\cdot x})({\bf S\cdot
x})}{x^{2}}\right]$ (7)

Let, $V_{V}=\frac{\tau}{x}$ and $V_{S}=\alpha x$
The spin-orbit term,

$V_{LS}(x)=\frac{1}{2m_{\rho}m_{\lambda}x}\left(3\frac{dV_{V}}{dx}-\frac{dV_{S}}{dx}\right)$
(8)

The spin-spin term,

$V_{SS}(x)=\frac{1}{3m_{\rho}m_{\lambda}}\nabla^{2}V_{V}$ (9)

The tensor term,

$V_{T}(x)=\frac{1}{6m_{\rho}m_{\lambda}}\left(\frac{d^{2}V_{V}}{dx^{2}}-\frac{1}{x}\frac{dV_{V}}{dx}\right)$
(10)

L and S represent the angular momentum and total spin.

$\overrightarrow{s_{1}}\cdot\overrightarrow{s_{2}}=\frac{1}{2}\left[\vec{S}^{2}-s_{1}\left(s_{1}+1\right)-s_{2}\left(s_{2}+1\right)\right]$
(11) $\vec{L}\cdot\vec{S}=\frac{1}{2}[j(j+1)-l(l+1)-S(S+1)]$ (12)
$S_{12}=2\left[3\frac{(\vec{S}\cdot\vec{r})^{2}}{r^{2}}-\vec{S}^{2}\right]$
(13)

Here, $s_{1}$ and $s_{2}$ denote the spins of individual quarks inside the
baryon. L and S denote the total orbital angular momentum and spin quantum
numbers for a given state. All these have been carried out with Mathematica
notebook [37].

##  3 Results and discussions

The latest update of Particle Data Group (PDG) lists the most recent and
precise ground state mass to be $m_{p}=938.272MeV$ and $m_{N}=939.565MeV$. [5]
As for today, 28 total states are listed with PDG for $N^{*}$ which were a few
before a decade as shown in 1. ClAS experiment is focused to study nucleon
structure through KY production [38]. Unlike the ground state, excited state
masses for both isospins are categorised under a common $N^{*}$, as it can be
revealed by the relevant decay channel for the given mass[39]. The N(1440)
resonance in $J^{P}=\frac{1}{2}^{+}$, also known as the Roper resonance, is
one of the most interesting states among the nucleon resonances [40]. The
Roper resonance is lighter than the lowest negative-parity nucleon
excitations, i.e., N(1535) in $J^{P}=\frac{1}{2}^{-}$ [41] and N(1520) in
$J^{P}=\frac{3}{2}^{-}$, which cannot be easily explained if one assumes that
the Roper resonance is a radial excitation of nucleon as a three-quark system.
A promising physical interpretation is that the Roper resonance is the first
radial excitation of nucleon but consists of a dressed-quark core augmented by
a meson cloud. One of the most prominent examples of a baryon spectrum riddle
is presented here. However, the present study is not able to comment on the
nature of Roper resonance but the obtained result is well within the mass-
range.

Table 1:  Experimental Status of all known $N^{*}$ [5] **** | states | *** | states | ** | states | * states |
---|---|---|---|---|---|---|---
N(1440) | $\frac{1}{2}^{+}$ | N(1700) | $\frac{3}{2}^{-}$ | N(1860) | $\frac{5}{2}^{+}$ | N(2040) | $\frac{3}{2}^{+}$
N(1520) | $\frac{3}{2}^{-}$ | N(1875) | $\frac{3}{2}^{-}$ | N(1990) | $\frac{7}{2}^{+}$ |  |
N(1535) | $\frac{1}{2}^{-}$ | N(1880) | $\frac{1}{2}^{+}$ | N(2000) | $\frac{5}{2}^{+}$ |  |
N(1650) | $\frac{1}{2}^{-}$ | N(2060) | $\frac{5}{2}^{-}$ | N(2300) | $\frac{1}{2}^{+}$ |  |
N(1675) | $\frac{5}{2}^{-}$ | N(2100) | $\frac{1}{2}^{+}$ | N(2570) | $\frac{5}{2}^{-}$ |  |
N(1680) | $\frac{5}{2}^{+}$ | N(2120) | $\frac{3}{2}^{-}$ | N(2700) | $\frac{13}{2}^{+}$ |  |
N(1710) | $\frac{1}{2}^{+}$ | N(2600) | $\frac{11}{2}^{-}$ |  |  |  |
N(1720) | $\frac{3}{2}^{+}$ |  |  |  |  |  |
N(1895) | $\frac{1}{2}^{-}$ |  |  |  |  |  |
N(1900) | $\frac{3}{2}^{+}$ |  |  |  |  |  |
N(2190) | $\frac{7}{2}^{-}$ |  |  |  |  |  |
N(2220) | $\frac{9}{2}^{+}$ |  |  |  |  |  |
N(2250) | $\frac{9}{2}^{-}$ |  |  |  |  |  |
Table 2:  Resonance masses of $N^{*}$ using Screened potential (in MeV).

State | $J^{P}$ | $M_{scr}$ | $M_{exp}$
---|---|---|---
1S | $\frac{1}{2}^{+}$ | 939 | 938
2S | $\frac{1}{2}^{+}$ | 1420 | 1440
3S | $\frac{1}{2}^{+}$ | 1762 | 1710
4S | $\frac{1}{2}^{+}$ | 2090 | 2040*
5S | $\frac{1}{2}^{+}$ | 2422 |
$1^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1505 | 1535
$1^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1493 | 1520
$1^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1512 | 1650
$1^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1499 |
$1^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1482 | 1675
$2^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1882 | 1895
$2^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1868 | 1875
$2^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1890 |
$2^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1875 |
$2^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1856 |
$3^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2286 |
$3^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2270 |
$3^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2294 |
$3^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2278 |
$3^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2257 |
$4^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2709 |
$4^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2693 |
$4^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2717 |
$4^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2701 |
$4^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2679 |
$1^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 1816 | 1720
$1^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1792 | 1680
$1^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 1843 | 1880
$1^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 1825 | 1900
$1^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1801 | 1860
$1^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 1771 |

State | $J^{P}$ | $M_{scr}$ | $M_{exp}$
---|---|---|---
$2^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2215 |
$2^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2190 |
$2^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2243 |
$2^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2224 |
$2^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2199 |
$2^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2168 |
$3^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2635 |
$3^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2610 |
$3^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2663 |
$3^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2644 |
$3^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2619 |
$3^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2588 |
$1^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2143 | 2060
$1^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2107 | 2190
$1^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2183 | 2120
$1^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2154 |
$1^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2118 | 2190
$1^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2074 | 2250
$2^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2561 | 2570
$2^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2523 |
$2^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2602 |
$2^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2572 |
$2^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2534 |
$2^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2489 |
$1^{2}G_{7/2}$ | $\frac{7}{2}^{+}$ | 2487 |
$1^{2}G_{9/2}$ | $\frac{9}{2}^{+}$ | 2433 | 2220
$1^{4}G_{5/2}$ | $\frac{5}{2}^{+}$ | 2546 |
$1^{4}G_{7/2}$ | $\frac{7}{2}^{+}$ | 2501 |
$1^{4}G_{9/2}$ | $\frac{9}{2}^{+}$ | 2447 |
$1^{4}G_{11/2}$ | $\frac{11}{2}^{+}$ | 2383 |
$1^{4}H_{11/2}$ | $\frac{11}{2}^{-}$ | 2786 | 2600

The ground state parameters lead to the value of 938 MeV for P and 948 MeV for
N, whereas earlier both the masses were 939 MeV such that excited states
cannot be separated out. Table 2 describes the obtained mass using the above
phenomenological approach. The S-wave states are within a good range from
those of Particle Data Group (PDG). The next four star status N(1440) with
$J^{P}=\frac{1}{2}^{+}$ is the 2S state and present result 1420 is well within
the PDG range and other approaches too. Similarly the 3S and 4S states are
found to be with good agreement with experimental results.

In case of 1P states, the higher spin states have lower mass compared to their
respective spin states for a given angular momentum. This has been observed
has an intrinsic to hypercentral Constituent Quark Model (hCQM).
1P$\frac{1}{2}^{-}$ is 30 MeV less compared to 1535 MeV. All the spin states
for 1D are experimentally established. The variation of splitting from
$\frac{1}{2}$ to $\frac{7}{2}$ is around 80 MeV for the present masses. The
very first state in negative parity is N(1520) with $J^{P}=\frac{3}{2}^{-}$ is
reproduced as 1493. Also, this state is lower than its spin partner i.e.
$J^{P}=\frac{1}{2}^{-}$ N(1535) is also consistent with our results as the
model is predicting lower mass for higher spin state. Here, it is noteworthy
that N(1650) $J^{P}=\frac{1}{2}^{-}$ doesn’t appear in the present data.
N(1720)$\frac{3}{2}^{+}$ with four star label is 1816 in present results.
1D$\frac{1}{2}^{+}$ appears as a higher state than $\frac{3}{2}^{+}$ and
$\frac{5}{2}^{+}$. Also, states in 2D are found to vary within 100 MeV
difference compared to all.

All the negative parity states of 1F have been observed and assigned three and
four star status. In the present results, higher spin state with
$J^{P}=\frac{9}{2}^{-}$ is under-predicted compared to PDG. The two states
which doesn’t appear in earlier study have been calculated here. The N(2220)
1G $\frac{9}{2}^{+}$ obtained to be 2433 which is quite differing from PDG
range. Also, the N(2600) is assigned 1H $\frac{11}{2}^{-}$ is over-predicted
in the current results. Not all the models have resolved the hyperfine
splitting of masses and also that hierarchy is not maintained in all the
results. Recent studies have focused on N(1895) state and its decay to light
hyperons $\gamma p\rightarrow K^{+}\Lambda$ [42, 43]. Out of the four states
in the vicinity from 1875 to 1900, the negative parity states of our results
are in accordance PDG range. But for positive parity states lying in D states
cannot be precisely matched. This is true for other model comparisons as well.
It is noteworthy that for low-lying states the masses with screened potential
are in slightly increment which then falls with higher excited states.
However, as the experimental masses fall within a range the predictions of
screened and linear potential are not very far. The notable change comes into
the picture with higher order correction terms.

$\Delta$ baryon has played prime role towards the understanding of color
quantum number. Even till date, $\Delta$ is important candidate not only in
the field of high energy physics, but nuclear and astrophysical systems also
[44]. Over the course of many years, pion-nucleon decays and photoproduction
decays have allowed for the observation of $\Delta$s. In the field of
astrophysics, $\Delta$ isobars are investigated under the quark-meson coupling
model to determine whether if they could possibly be observed. Incorporating
the recent additions, 8 four star, 4 three star and many other experimental
status have been explored with the values ranging from $J=\frac{1}{2}$ to
$J=\frac{15}{2}$ and still many states are awaited of confirmation of
existence as listed by Particle Data Group (PDG).
Also, $\Delta$ being the lightest member with the presence of electric
quadrupole moment makes it interesting to dig deep into the shape and
structure of the baryon [45]. Also, the MicroBooNE collaboration has recently
reported $\Delta$(1232) radiative decay through neutrino induced neutral
current [46]. The pole positions for N and $\Delta$ resonances have been
investigated through photoproduction of K$\Sigma$ from coupled-channel study
[47].

Table 3: Resonance Masses of $\Delta$ baryon using Screened Potential (in MeV)

State | $J^{P}$ | $M_{scr}$ | $M_{exp}$
---|---|---|---
1S | $\frac{3}{2}^{+}$ | 1232 | 1232
2S | $\frac{3}{2}^{+}$ | 1602 | 1600
3S | $\frac{3}{2}^{+}$ | 1900 | 1910
4S | $\frac{3}{2}^{+}$ | 2200 |
5S | $\frac{3}{2}^{+}$ | 2499 |
$1^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1556 | 1620
$1^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1542 | 1700
$1^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1563 |
$1^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1549 |
$1^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1530 |
$2^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1911 | 1900
$2^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1895 |
$2^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 1919 |
$2^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 1903 | 1940
$2^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 1882 | 1930
$3^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2272 |
$3^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2257 |
$3^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2280 |
$3^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2265 |
$3^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2243 | 2350*
$4^{2}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2638 |
$4^{2}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2623 |
$4^{4}P_{1/2}$ | $\frac{1}{2}^{-}$ | 2646 |
$4^{4}P_{3/2}$ | $\frac{3}{2}^{-}$ | 2631 |
$4^{4}P_{5/2}$ | $\frac{5}{2}^{-}$ | 2610 |
$1^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 1841 |
$1^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1817 | 1905
$1^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 1867 |
$1^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 1850 | 1920
$1^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 1826 |
$1^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 1797 | 1950
$2^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2202 |
$2^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2178 |

State | $J^{P}$ | $M_{scr}$ | $M_{exp}$
---|---|---|---
$2^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2230 |
$2^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2211 |
$2^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2187 |
$2^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2156 |
$3^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2566 |
$3^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2543 |
$3^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2593 |
$3^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2575 |
$3^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2552 |
$3^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2523 |
$4^{2}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2934 |
$4^{2}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2912 |
$4^{4}D_{1/2}$ | $\frac{1}{2}^{+}$ | 2959 |
$4^{4}D_{3/2}$ | $\frac{3}{2}^{+}$ | 2942 |
$4^{4}D_{5/2}$ | $\frac{5}{2}^{+}$ | 2921 |
$4^{4}D_{7/2}$ | $\frac{7}{2}^{+}$ | 2893 |
$1^{2}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2131 |
$1^{2}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2095 | 2200
$1^{4}F_{3/2}$ | $\frac{3}{2}^{-}$ | 2170 |
$1^{4}F_{5/2}$ | $\frac{5}{2}^{-}$ | 2141 |
$1^{4}F_{7/2}$ | $\frac{7}{2}^{-}$ | 2106 |
$1^{4}F_{9/2}$ | $\frac{9}{2}^{-}$ | 2063 |
$1^{2}G_{7/2}$ | $\frac{7}{2}^{+}$ | 2420 |
$1^{2}G_{9/2}$ | $\frac{9}{2}^{+}$ | 2375 | 2300
$1^{4}G_{5/2}$ | $\frac{5}{2}^{+}$ | 2468 |
$1^{4}G_{7/2}$ | $\frac{7}{2}^{+}$ | 2431 | 2390
$1^{4}G_{9/2}$ | $\frac{9}{2}^{+}$ | 2387 |
$1^{4}G_{11/2}$ | $\frac{11}{2}^{+}$ | 2335 | 2420
$1^{2}H_{9/2}$ | $\frac{9}{2}^{-}$ | 2719 |
$1^{2}H_{11/2}$ | $\frac{11}{2}^{-}$ | 2657 |
$1^{4}H_{7/2}$ | $\frac{7}{2}^{-}$ | 2786 |
$1^{4}H_{9/2}$ | $\frac{9}{2}^{-}$ | 2732 |
$1^{4}H_{11/2}$ | $\frac{11}{2}^{-}$ | 2670 |
$1^{4}H_{13/2}$ | $\frac{13}{2}^{-}$ | 2600 | 2750

Similar to N, the mass spectra of $\Delta$ has been tabulated in table 3. The
S-wave mass predictions are very well matching with experimental data. The
1P(1556) state is 70 MeV lower compared to 1620 MeV. However, the
1P$\frac{3}{2}^{-}$ state is 150 MeV under-predicted. Few states in 2P show
good agreement. The 1D states are under-predicted compared to PDG. The 1G
$\frac{9}{2}^{+}$ is in accordance with PDG value of 2300 MeV by a difference
of 75 MeV. 1H $\frac{13}{2}^{-}$ (2750) is predicted to be 2600 MeV in the
present work.

As the screened potential has been observed in heavy quark systems, our
primary goal here to check if similar kind of effects apply to light quark
systems. Briefly concluding that screening effect in light systems is notable
at higher mass scale but not too suppressed as in the case of heavy systems.
Another important aspect is the higher order corrections do not resolve the
spin structure in case of screened potential.

##  4 Regge Trajectory

One of the helpful tools in spectroscopic research has been Regge
trajectories. Based on calculated data, figures depict a plot of total angular
momentum J and principle quantum number n versus the square of resonance mass
$M^{2}$. Many studies have found that the theoretical and experimental data
are consistent with the non-intersecting and linearly fitted lines [48]. A
correct spin-parity assignment for a state can perhaps be predicted using
these plots.

$\displaystyle J=aM^{2}+a_{0}$ (14a) $\displaystyle n=bM^{2}+b_{0}$ (14b)

Figure 1:  (n,$M^{2}$) Regge trajectory for $N$ baryon for screened potential.
Figure 2:  (J,$M^{2}$) Regge trajectory for N baryon for screened potential.
Figure 3:  (J,$M^{2}$) Regge trajectory for N baryon for screened potential.
Figure 4:  (n,$M^{2}$) Regge trajectory for $\Delta$ states for screened
potential Figure 5:  (J,$M^{2}$) Regge trajectory for $\Delta$ states for
screened potential Figure 6:  (J,$M^{2}$) Regge trajectory for $\Delta$ states
for screened potential

The Regge trajectories have been observed to be following linear nature as
expected. However, it is observed that all the lines are not equidistant. The
slopes and intercepts have been mentioned in the graphs itself.

##  5 Decay

In the case of nucleons, including $\Delta$, the prominent decay channel has
been observed to be either $N*$ or pion, depending on the charge of the
respective parent [49]. In addition to other constants, the transition
couplings of vector mesons have been obtained thanks to the work of Riska and
colleagues [50]. Lagrangian densities for the transition couplings involving
generalized Rarita-Schwinger vector spinors are used to define the coupling
constants of the vector-meson transition to nucleon resonances. The transition
coupling constants can be written in terms of the corresponding vector-meson
coupling constants to the nucleons by comparing the matrix elements of these
Lagrangians to the corresponding matrix elements in the quark model. The
latter are calculated using phenomenological boson exchange interaction models
and fitted to nucleon-nucleon scattering data, albeit with large uncertainty
margins. P- and D-shell and excited S-shell states can be related to the
ground state through these expressions involving SU(2) Clebsch-Gordan
coefficients and orbital matrix elements of quark wave functions. The latter
are dependent on the Hamiltonian model of the three-quark system. Here we use
a straightforward model for covariant harmonic oscillators where the confining
interaction is linear and the hyperfine interaction depends on the flavour and
spin of the particle. In the work that is being presented here, the constants
and decay widths that have been supplied by the Particle Data group have been
used to determine the decay width of certain resonance masses that have
already been established.

For $\Delta(1232)$, $\Delta(1600)$ decay to $N\pi$,

$\Gamma=\frac{1}{3}\frac{f^{2}}{4\pi}\frac{E^{{}^{\prime}}+m_{N}}{m_{\Delta}}\frac{k^{3}}{m_{\pi}^{2}}$
(15)

where, $E^{{}^{\prime}}$ is the energy of the final nucleon and k is pion
momentum.

$E^{{}^{\prime}}=\frac{m^{*2}-m_{\pi}^{2}+m_{N}^{2}}{2m^{*}}$ (16)
$k=\frac{\sqrt{[m^{*2}-(m_{N}+m_{\pi})^{2}][m^{*2}-(m_{N}-m_{\pi})^{2}]}}{2m^{*}}$
(17)

Here $m^{*}$ is resonance mass calculated using above model, $m_{N}$ is
nucleon mass 939 MeV and $m_{\pi}$ is pion mass 139 MeV. For $N(1535)$,
$N(1650)$ and $\Delta(1620)$ decaying to $N\pi$,

$\Gamma=\frac{f^{2}}{4\pi}\frac{E^{{}^{\prime}}+m_{N}}{m^{*}}\frac{k}{m_{\pi}^{2}}(m^{*}-m_{N})^{2}$
(18)

For $N(1520)$, $N(1700)$ and $\Delta(1700)$ decaying to $N\pi$,

$\Gamma=\frac{1}{3}\frac{f^{2}}{4\pi}\frac{E^{{}^{\prime}}-m_{N}}{m_{\Delta}}\frac{k^{3}}{m_{\pi}^{2}}$
(19)

Table 4: Pion decay widths $\Gamma$ for N and $\Delta$ channels Decay | Mass |  $\Gamma$(Present work) | [39] | [51] |
---|---|---|---|---|---
$\Delta$(1232)$\frac{3}{2}^{+}$ | 1232 | 119.855 |  | 118.7(0.6) |
$\Delta$(1600)$\frac{3}{2}^{+}$ | 1602 | 60.586 | 34(8) |  |
N(1440)$\frac{1}{2}^{+}$ | 1420 | 209.962 | 153 | 284(18) |
N(1535)$\frac{1}{2}^{-}$ | 1505 | 99.961 | 62(3) | 188.4(3.8) |
N(1650)$\frac{1}{2}^{-}$ | 1512 | 76.5586 | 86(6) | 115.4(2.8) |
$\Delta$(1620)$\frac{1}{2}^{-}$ | 1556 | 111.779 | 26(2) | 146.9(1.9) |
N(1520)$\frac{3}{2}^{-}$ | 1493 | 17.87 | 71(2) | 103.6(0.4) |
$\Delta$(1700)$\frac{3}{2}^{-}$ | 1542 | 17.85 | 3 | 375.5 |
N(1675)$\frac{5}{2}^{-}$ | 1482 | 16.29 | 53 | 146.5 |
N(1720)$\frac{3}{2}^{+}$ | 1816 | 27.80 | 41 | 210(22) |
N(1680)$\frac{5}{2}^{+}$ | 1792 | 178.76 | 84 |  |

The values for decay constant f varies with each decay channel. These values
are descriptively studied by Riska and Brown. Table 4 shows decay channels of
N and $\Delta$ baryon with respective decay widths obtained for our predicted
resonance masses. The comparison of decay widths with a recent experimental
findings as elaborated by Hunt et al. [39]. Our results are in good agreement
in few channels. Also, we have compared with another partial wave analysis
done by Ardnt et al. [51].

##  6 Conclusion

The present work summarizes the effect of screened type potential under
hypercentral Constituent Quark Model (hCQM) for N and $\Delta$ baryons. So
far, linear potential has been applied to light spectrum, whereas screened
potential provided reasonable results for heavy quark systems. The screening
parameter plays a role in determining the spin-split and mass at higher
angular momentum states. The obtained masses have been comparable with the
basis of experimental known values with different star status. The masses of
higher spin state for a given L value, decreases in hierarchy. The hyperfine
splitting is observed to be less with screened potential than those of linear
one.
The Regge trajectories show linear trend for all natural and unnatural parity
points. The strong decay widths for pion channel are also calculated in the
present study. The ongoing and upcoming experimental facilities namely HADES
[52], PANDA[53, 54, 55, 56, 57, 58], shall be providing us with new insights
in the understanding of light, strange baryons.

## Acknowledgment

Ms.Chandni Menapara acknowledges the support for pursuing this work under DST-
INSPIRE Fellowship Scheme.

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