# Study of Baryon Spectroscopy Using a New Potential Form

Authors: L. I. Abou-Salem
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

In the present work, the nonrelativistic quark model is applied to study baryon systems, where the constituent quarks are bound by a suitable hyper central potential. We proposed a new phenomenological form of the interaction potential, digamma-type potential. Using the Jacobi coordinates, the three-body wave equation is solved numerically to calculate the resonance states of theN,Δ,Λ, and Σ baryon systems. The present model contains only two adjustable parameters in addition to the quark masses. Our theoretical calculations are compared to the available experimental data and Cornell potential results. The description of the spectrum shows that the ground states of the considered light and strange baryon spectra are in general well reproduced.


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## Body

## 1. Introduction

Hadron spectroscopy is very important to study its structures and the nature of the interaction forces between its constituents. In the previous works [1–5], the heavy meson spectroscopy is studied by using different potential models. Many authors have been studying the baryon spectroscopy. An analytical solution was provided in case of harmonic and anharmonic potentials for a system consists of three identical particles [6, 7]. Some authors have used the Cornell potential (Coulomb-type plus linear term) to study the resonance states of N and Δ baryons [8–14]. The harmonic oscillator potential has been used to study the interaction between three identical particles [15]. In the present work, the digamma-function is used as the interaction potential between constituent quarks of baryon systems. The three-body wave equation is solved, numerically, by using the Jacobi method to calculate the resonance state masses of N, Δ, Λ, and Σ baryons. These baryons consist of identical or nonidentical quarks. Our theoretical results will be investigated in Section 3.


## 2. The Used Model

The nonrelativistic Schroedinger equation for a system consists of three particles is given as(1)[-ℏ22m1∇r12-ℏ22m2∇r22-ℏ22m3∇r32+V12(r12)+V23(r23)+V31(r31)ℏ22m3]Ψ(r1,r2,r3)=EΨ(r1,r2,r3),
where m1, m2, and m3 are the three-quark masses and rij is the relative separation vector.In the quark model, a baryon is a three-body bound state made of quarks. The mathematical description of a three-body system is more complicated than that of a two-body system. Several methods have been used by different authors to solve three-body problems [11, 15]. In order to describe the baryon as a bound state of three constituent quarks, we define the configuration of three particles by the center of mass, R, and two Jacobi coordinates ρ and λ as
(2)R→=m1r1→+m2r2→+m3r3→M,ρ→=c1(r1→-r2→),λ→=c2(m1r1→+m2r2→M′-r3→),
where M=m1+m2+m3; M′=m1+m2;  c1 and c2 are constants.Using natural units(ℏ=C=1) and letting the coefficients of ∇ρ2 and ∇λ2 be equal to A, therefore, (1) can be separated into the following two equations:
(3)-12M∇R2ΨC.M(R)=E1ΨC.M(R),(4)[-A(∇ρ2+∇λ2)+V(x)]Ψ(ρ→,λ→)=E2Ψ(ρ→,λ→),
where
(5)E=E1+E2,A=c122μ12=c222μ12,3,
where
(6)μ12=m1m2m1+m2,μ12,3=m3(m1+m2)m1+m2+m3.
The wave function Ψ(ρ→,λ→) can be expressed as [8, 16, 17]
(7)Ψ(ρ→,λ→)=∑γΨγ(x)Yγ(Ω),
where Ψγ(x) and Yγ(Ω) are the hyper radial part of the wave function and the hyper spherical harmonic function, respectively. The symbol γ is called the grand angular quantum number and is given by
(8)γ=2υ+lρ+lλ,
where lρ and lλ are the angular momenta associated with the ρ and λ variables and υ is a nonnegative integer number.The hyperradiusx is given as
(9)x=ρ2+λ2=13(r122+r232+r312).
The potential V(x) can be assumed to depend on the hyperradius x and e space wave function is factored similar to the central potential [8].Equation (4) can be rewritten as
(10)d2Ψγ(x)dx2+D-1xdΨγ(x)dx-L(Ω)2x2Ψγ(x)+1A(E-V(x))Ψγ(x)=0,
where D represents the dimension of the x⃑ and L(Ω)2 is the angular momentum operator whose eigenfunctions are [6, 9, 11]
(11)L(Ω)2Yγ=-γ(γ+D-2)Yγ.
In the present work, the interaction potential is assumed as digamma-type as follows:
(12)V(x)=CΓ′(x)Γ(x)(13)=C∫0∞(e-tt-e-xt1-e-t)dt,
where C is an adjustable parameter and Γ(x) is the gamma-function.Now, applying the following transformation(14)Ψγ(x)=uγ(x)x(D-1)/2,
(10) can be rewritten as
(15)[d2dx2+1A(((D-1)(D-3)4x2+γ(γ+D-2)x2)E-V(x)-Ahhhhhhhhhh×((D-1)(D-3)4x2+γ(γ+D-2)x2))]×uγ(x)=0
With the new boundary conditions,
(16)uγ(0)=uγ(∞)=0.
Using
(17)λ=EA,φ(x)=V(x)+(D-1)(D-3)4x2+γ(γ+D-2)x2.Equation (12) can be rewritten as
(18)d2uγ(x)dx2+(λ-φ(x))uγ(x)=0.To transform the range ofx from (∞,0) to (0,1), I introduce the dimensionless variables, g, g=1/(1+(x/x0)), and assume that
(19)ρ(g)=gu(g),
where xo=1 GeV−1.Equation (18) can be rewritten as
(20)d2ρ(g)dg2+[x04g4](λ-φ(g))ρ(g)=0.
The new boundary conditions are given as
(21)ρ(0)=ρ(1)=0.
To transform (20) into a matrix form, one can divide the range of (g) from (0,1) into (n+2) points with equal interval, h, labeled by subscript (J).The new form of the boundary conditions is given as(22)ρn+1=ρ0=0.
Using the finite difference approximation [19], the second derivative term in (20) is given as
(23)d2ρ(g)dg2=112h2[-ρJ-2+16ρJ-1-30ρJ+16ρJ+1-ρJ+2]+O(h4),
where the term O(h4) represents the expected error.Using the symmetry properties of the radial wave function, one can assume that [15]
(24)ρ-1=(-1)γρ1+O(h2);ρn+2=(-1)γ+1ρn+O(h3).
Using (21), (22), and (23) in (18) and taking x0=1GeV-1, one gets
(25)(ρJ-2-16ρJ-1+30ρJ-16ρJ+1+ρJ+2)+12h2(Jh)4(φ(Jh)-λ)ρJ=0.
This equation is rewritten in the following matrix form:
(26)(A-λI)ρ=0,
where A is n×n symmetric matrix and I is n×n unitary diagonal matrix. Using the Jacobi method [20, 21], (26) can be solved numerically and the eigenvalues, E, are determined. The nonrelativistic resonance mass, M, for each state is given by
(27)M=m1+m2+m3+E.


## 3. Results and Discussions

In the present work, the spin-spin interaction is not considered and we deal with the spin averaged stats. The predicted values of theN, Δ, Λ, and Σ resonance masses are calculated through solving (26) numerically using the Jacobi method. The present potential form is proposed as the analytical form of the digamma-function; see (12) and (13). The χ2-test is used to determine the parameter values
(28)χ2=1N∑i=1n((Mitheo-Miexp⁡)ei)2,
where N is the total number of states, M is the resonance mass, and ei is the experimental error in the ith State. The parameter values which are used in the present calculations are given in Table 1. The uncertainties in these parameters may be due to relativistic effects and ignoring the spin interaction terms.Table 1
The values of the parameters which are taken in our calculations.
Parameters
Values
m
u
0.035 ± 0.005 GeV
m
d
0.035 ± 0.005 GeV
m
s
0.092 ± 0.008 GeV
A
0.40 ± 0.04 GeV−1
C
0.85 ± 0.08 GeVThe behavior of the digamma-type and Cornell potentials versus the hyperradius,x, is shown in Figure 1. From this figure one notices that the suggested potential behaves nearly like Cornell potential at small x (one gluon exchange part) and the Cornell potential is more confined than the digamma-type potential at large x.Figure 1
The digamma-type and Cornell potentials versus the hyperradiusx.Table2 contains the present calculations of N resonance states using both digamma-function potential and a Coulomb like hyper central potential plus the linear term (Cornell Model [10]) in comparison to the experimental values [18]. From this table, one notices that the observed seven experimental states are calculated using the suggested potential form while the results of Cornell potential are 4 states only and the present calculations give more satisfied agreement with the experimental data.Table 2
The present calculations ofN baryon states in comparison to the Cornell  potential results [10] and the experimental data [18] in (MeV).
Baryon state
M
exp
⁡  ref. [18]
γ
Ref. [10]
Present calculations
N (938) P11
938.272013 ± 0.000023
0
938
938.25
N (1440) P11
1445 ± 25
1463
1448.43
N (1710) P11
1710 ± 30
1752
1702.24
N (1535) S11
1535 ± 10
1
1524
1544.67
N (1905) S11
1905
—
1902.34
N (2090) S11
2150 ± 50
—
2130.87Table3 contains the present calculations of Δ baryon states and Cornell potential results [10] in comparison to the available experimental data [18]. From this figure one notices that the observed 5 states are calculated using the present potential form while the calculated states in [10] are three states only. In case of N and Δ baryons, one can notice that the calculated resonance masses using the digamma-type potential agree more with the experiment than the Cornell potential model results [10].Table 3
The present calculations ofΔ baryon states in comparison with the Cornell  potential results [10] and the experimental data [18] in (MeV).
Baryon  state
M
exp
⁡ ref. [18]
γ
Ref. [10]
Present calculations
Δ (1232) P33
1232 ± 1
0
1232
1233.24
Δ (1600) P33
1625 ± 75
1727
1594.57
Δ (1620) S31
1630 ± 30
1
1573
1612.95
Δ (1900) S31
1900
—
1896.32
Δ (2150) S31
2150
—
2160.73The present calculations ofΛ and Σ baryon states compared with the experimental values [21] are given in Tables 4 and 5. One notices that the suggested potential model can reproduce the experimental results of the ground states of Λ and Σ baryon systems.Table 4
The present calculations ofΛ baryon states in comparison to the experimental data [18]  in (MeV).
Baryon state
M
exp
⁡ ref. [18]
γ
Present calculations
Λ (1116) P01
1115.683 ± 0.006
0
1116.85
Λ (1600) P01
1630 ± 70
1689.46
Λ (1810) P01
1800 ± 50
1832.86
Λ (1670) S01
1670 ± 10
1
1665.21
Λ (1800) S01
1785 ± 65
1851.75
Λ (1890) P03
1880 ± 30
2
1915.79Table 5
The present calculations ofΣ baryon states in comparison to the experimental data [18] in (MeV).
Baryon state
M
exp
⁡ ref. [18]
γ
Present calculations
Σ (1193) P11
1192.642 ± 0.024
0
1193.15
Σ (1660) P11
1660 ± 30
1629.47
Σ (1880) P11
1880
1868.13
Σ (1620) S11
1620
1
1617.39
Σ (1750) S11
1765 ± 35
1809.35


## 4. Conclusions

A nonrelativistic quark model is used to study the spectra of the considered baryon systems (N, Δ, Λ, and Σ baryons). Using the hyper central approach, we have simplified the three-body problem and solved the Schrodinger equation numerically to obtain the ground-state energy eigenvalues and eigenfunctions. In the present work, we proposed a new phenomenological form of the interaction potential, digamma-type potential, between the quark constituents to study baryon systems. The behavior of this analytical function may be the same as the quark-antiquark interaction inside quarkonium systems. The description of the spectrum shows that the light and strange baryon ground states are in general well reproduced. The suggested potential model may be used to describe the energies of the higher excited states and negative-parity resonances through introducing the spin interaction terms as perturbed terms. Finally, this approach is useful because it allows for predictions of baryon states without extended lattice calculations and provides useful information about the short-distanceone gluon exchangeand the long-distanceconfinement interactions that can be useful in understanding the nature of the quark-quark force generated by QCD.



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