Политематический сетевой электронный научный журнал Кубанского государственного аграрного университета, 2012, №81

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1

УДК 531.9+539.12.01 
UDC 531.9+539.12.01 
 
 
МОДЕЛИРОВАНИЕ МАССЫ АДРОНОВ И 
ЭНЕРГИИ ВОЗБУЖДЕННЫХ СОСТОЯНИЙ 
АТОМНЫХ ЯДЕР В МОДЕЛИ ГЛЮОННОГО 
КОНДЕНСАТА  

SIMULATION OF HADRON MASSES AND 
ATOMIC NUCLEI EXITED STATES IN THE 
GLUON CONDENSATE MODEL  

 
 
Трунев Александр Петрович 
к.ф.-м.н., Ph.D. 
Alexander Trunev 
Cand.Phys.-Math.Sci., Ph.D. 
Директор, A&E Trounev IT Consulting, Торонто, 
Канада 
Director, A&E Trounev IT Consulting, Toronto, 
Canada  
 
В работе рассмотрена скалярная модель глюонного 
конденсата, в котором образуются глюболы. 
Показано, что масса известных адронов и энергия 
возбужденных состояний ядер описываются с 
приемлемой точностью интегралом от плотности 
конденсата по объему глюбола 
 

In this article we consider a scalar model of the gluon 
condensate, in which bubbles are formed - glue balls. 
It is shown that the mass of the known hadrons as well 
as nuclei exited states are described with the 
acceptable accuracy by the integral of the condensate 
density in terms of the glueball 

Ключевые слова: АДРОН, АТОМНОЕ ЯДРО, 
ВОЗБУЖДЕННЫЕ СОСТОЯНИЯ,  ГЛЮОННЫЙ 
КОНДЕНСАТ, ГЛЮБОЛ, МАССА, СКАЛЯРНОЕ 
ПОЛЕ   

Keywords: EXITED STATES, NUCLEI, GLUON 
CONDENSATE, GLUEBALL, HADRON, MASS, 
SCALAR FIELDS 

 

Introduction 

According to modern ideas hadrons consist of quarks interacting via 

vector gauge bosons - gluons. Quantum chromodynamics (QCD), which 

describes this kind of interaction is extremely complex theory, so the models of 

elementary particles that are based on QCD, are widely used to simplify and 

various numerical methods. Glueball is a hypothetical particle predicted by 

QCD [1]. It is assumed that only consists of glueball gluon condensate. 

According to the calculations made in the framework of lattice QCD [2], this 

type of a scalar particle has a mass of about 1730 MeV. 

In [3-4] and others have shown that the glueball is the result of the 

nonlinear interaction of two scalar fields, describing the state of the gluon 

condensate. In this paper we calculate the hadron masses and energy of the 

excited states of nuclei based on the model [3-4]. 

 

Simulation of hadron masses 

In this paper we used a scalar model of the gluon condensate, developed 

in [3-4]. This model, in the notation of [4] has the form 

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2

(
)
[
]

(
)
[
]
2
2
2
2

2
2
1
2

∞

∞

−
+
−
=
∂
∂

−
+
−
=
∂
∂

χ
χ
λ
φ
χ
χ

φ
φ
λ
χ
φ
φ

µ
µ

µ
µ
                                       (1) 

Here 
χ
φ,
describe the distribution of the scalar field condensate; 
2
1,λ
λ
 - 

model parameters; 
∞
∞ χ
φ ,
 - the eigenvalues of the problem. In the case of 

spherical symmetry the system (1) is reduced to 

    
(
)
[
]

(
)
[
]
2
2
2
2

2
2
1
2

2

2

∞

∞
−
+
=
′
+
′′

−
+
=
′
+
′′

χ
χ
λ
φ
χ
χ
χ

φ
φ
λ
χ
φ
φ
φ

ax
x

ax
x
                                   (2) 

Here we introduced the dimensionless variable
2
/
1
−
= ra
x
. The boundary 

conditions for the system (2) are: 

    
.0
)
0
(
,
)
0
(
,0
)
0
(
,1
)
0
(

0
=
′
=

=
′
=

χ
χ
χ

φ
φ

                                               (3) 

 The system (2) with boundary conditions (3) was solved using Wolfram 

Mathematica 8 [5] with the values of [4]:  

49273856
.1
;
6171579
.1
;1
;1.0
;1
2
1
=
=
=
=
=
∞
∞
χ
φ
λ
λ
a
.  

The results of calculations of functions 
χ
φ,
are shown in Figure 1. As can 

be seen from the data shown in Fig. 1 glueball is spherical formation with 

density dependent on coordinates. In theory [3-4], the density of condensate 

describes the effective Lagrangian 

Ai
A
i
Ai
A
i
eff
E
E
H
H
L
G
−
=
−
=
                                               (4) 

Here 

A
i
A
i
H
E ,
- chromoelectric and chromomagnetic field accordingly. 

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3

 
 Figure 1: Glueball parameters, calculated according to [4]. 

 

Expression of the condensate density as a function of the distribution of 

the scalar fields is given by [4] 

 
(
)
(
)
(
)
2
2
4
2
2
2
2
2
2
2
2
1
2
2

2
1

4
4
4
2
1
χ
φ
χ
λ
χ
χ
λ
φ
φ
λ
χ
φ
−
−
−
+
−
+
′
+
′
−
=
∞
∞
∞
G
   (5) 

In the particular case of the subgroup SU (2) the expression (5) reduces to 

 
(
)
2
2
2
1
2
)
2
(
4
2
1

∞
−
+
′
−
=
φ
φ
λ
φ
SU
G
                                       (6) 

Expressions (5) - (6) together with the solutions of the problem (2) - (3) 

were used to simulate the mass of hadrons - Fig. 2-3. Suppose that hadrons 

consist of a central core - glueball surrounded coat of quark-gluon fields. For 

each hadron glueball has a certain radius, and the mass of the glueball is 

determined by the integral of a linear combination of the functions (5) and (6). 

In addition, the glueball mass contributes surface tension caused by the finite 

size of the glueball. Thus, the mass is determined according to the glueball 

(
)
∫
+
+
=

0

0

2
)
2
(
2
/
3
/
4

x

SU
dx
x
x
k
bG
G
a
m
ρ
π
                                            (7) 

We have considered two models of density

2
2
χ
φ
ρ
+
=
 - Fig. 2, 
1
=
ρ
- 

Fig. 3. Both models have the same accuracy compared with the mass of hadrons, 

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4

which is apparently due to the behavior of functions 
χ
φ,
that remain constant 

over a wide range of variation of the radial coordinate. In addition, separately 

studied the functional mass of the SU (2) condensate 

 
(
)
∫
+
=

0

0

2
)
2
(
2
/
3
/
4

x

SU
dx
x
x
k
G
a
m
ρ
π
                                                 (8) 

Model (7) - (8) has been verified for the entire set of hadrons - Fig. 2-3. 

Assume that the mass of an individual hadron is proportional to the mass of it 

glueball therefore have 

  
Hm
mH =
                                                                      (9) 

By changing the parameters of the model, we can achieve matching 

dependencies (7) - (8) with tabular data hadron masses. To solve this problem, 

we used the built-in Wolfram Mathematica 8 [5] table of elementary particles 

with the parameters ParticleData ["Hadron", "Mass"]. The table is extracted data 

sheet, which adds a number of zero-particles - 175 for the model (7) and 100 for 

the model (8). These data allow us to combine the origin, in which the mass of a 

hadron and glueball mass is linear as it proposed in Eq. (9). Data for hadrons are 

normalized to the maximum element - mY = 11 019 MeV. Next is fitting the 

model parameters 
k
h
b
a
,
,
,
- for the model (7), and 
k
h
a
,
,
 for the model (8). 

The parameters 
49273856
.1
;
6171579
.1
;1
;1.0
2
1
=
=
=
=
∞
∞
χ
φ
λ
λ
are stored 

in all the glueball calculations   

This resulted in the following values of the model parameters (7): 

061
.0
;
3665
.0
;
792
.1
;
0003815
.0
:
1
;
0237
.0
,
3665
.0
;
792
.1
;
0003815
.0
:

,
4
/
/

2
2

=
=
=
=
=
=
=
=
=
+
=

=

k
h
b
a
k
h
b
a

hm
m
m
Y
H

ρ
χ
φ
ρ

π

         (10) 

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Figure 2: Comparison of the hadron masses with the glueball mass 

calculated from equations (7) - (8) with 

2
2
χ
φ
ρ
+
=
. Parameters of the model 

(7): a = 0.0003815; b = 1.792; k = 0.0237; h = 0.3665. Parameters of the model 

(8): a = 0.000536; k = 0.0164; h = 0.414; mY = 11019 MeV. 

 

Comparison of hadron masses with glueball mass calculated by the model 

(7) with data (10) is shown in Fig. 2-3. A satisfactory agreement between the 

calculated and experimental data begins with mass ρ - meson of 775.5 MeV and 

ends at the mass ψ  - meson of 4421 MeV. For hadrons smaller and larger mass 

the linear model (9) is not satisfied. 

  

Figure 3: Comparison the hadron masses with the mass of glueball 

calculated from equations (7) - (8) with 
1
=
ρ
. Parameters of the model (7): a = 

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0.0003815; b = 1.792; k = 0.061; h = 0.3665. Parameters of the model (8): a = 

0.000536; k = 0.042; h = 0.414; mY = 11019 MeV. 

For the model (8) we obtained the following parameters 

 

042
.0
;
414
.0
;
000536
.0
:
1
;
0164
.0
,
414
.0
;
000536
.0
:

,
4
/
/

2
2

=
=
=
=
=
=
=
+
=

=

k
h
a
k
h
a

hm
m
m
Y
H

ρ
χ
φ
ρ

π

(11) 

Note that the difference in the accuracy of the description of the 

experimental data between the models (7) and (8) is nominal, but the model (8) 

contains one less parameter. On the other hand, the difference in density models 

used to simulate the surface energy is also nominal and limited to a redefinition 

of the parameter k, while maintaining the values of other parameters of the 

model, as it follows from the expressions (10) - (11). 

 Consider the difference between the theoretical curve and the 

experimental data in the case of SU (2) condensate - Fig. 4. Here we normalize 

the mass of hadrons on the mass of proton. Data for the deviation from the 

theoretical curve given in absolute units, show that the contribution of the orbital 

motion of the quarks in the hadron mass is not more than 0.1 of the proton mass 

for light particles and a maximum of 0.15 of the proton mass for the heavy 

particles. Consequently, we can construct a perturbation theory, using as the 

main solution glueball and a perturbed motion – orbital motion of quarks. 

Thus, we have shown that the linear model (9), which relates the mass of 

hadrons with a mass of the central core - glueball is performed for a large part of 

the hadrons, whose mass is in the range from 775.5 MeV to 4421 MeV - about 

922 particles from a total of 973. This is evidence in favor of a model of the 

structure of elementary particles, in which, it is assumed that hadrons contain 

glueball central core and surrounding fields of quarks and gluons. 

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  Figure 4: Deviation of hadron masses from the theoretical curve in the 

case of SU (2) condensate. 

  

Modeling the energy of the excited states of nuclei 

According to modern concepts atomic nuclei consist of nucleons - protons 

and neutrons, which in turn are composed of quarks interacting via vector gauge 

bosons - gluons. To model the energy of the excited states of nuclei - Fig. 5-6, 

the model (7) - (9) and a built-in Wolfram Mathematica 8 [5] table of isotopes 

and associated parameters are used. For example, data on the left of Fig. 5 table 

of isotopes invoked as IsotopeData ["Ni58", "ExcitedStateEnergies"]. 

From the table of isotopes extracted data sheet, which adds a number of 

null states. These data allow us to combine the origin, in which the energy of the 

excited state and the glueball mass is linear as it proposed in Eq. (9). The data 

for the energy of the excited states are normalized to the maximum element. 

Next is fitting the model parameters 
k
h
b
a
,
,
,
- for the model (7), and 
k
h
a
,
,
for 

the model (8). 

 

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8

  

Figure 5: A comparison of the energy of the excited states of isotopes of 

nickel 
and 
glueball 
energy, 
calculated 
from 
equations 
(7) 
- 
(8) 

with

2
2
χ
φ
ρ
+
=
. Parameters of the model (7) for the isotope Ni58: k = 

0.01906; h = 0.2698; a = 0.003756; b = 1.94. Parameters of the model (8) for the 

isotope Ni59: a = 0.0068; k = 2.09; h = 0.3235. 

 

 
Figure 6: A comparison of the energy of the excited states of isotopes of 

copper and zinc with energy glueball calculated from equation (7) with. 

Parameters of the model (7) for the isotope Cu64: k = 0.0092; h = 0.44; a = 

0.01356; b = 2.085. Parameters of the model (7) for the isotope Zn65: a = 

0.02168; b = 1.962; k = 0.00984; h = 0.44. 

  

For each isotope chosen own settings, that indicating there is an individual 

scenario glueball in each case. For example, for the isotope Ni59 surface tension 

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9

parameter is not used in the form k / x, as for isotope Ni58, but in the form k/x2, 

which is apparently due to the influence of angular momentum, which is not 

considered in the model (1). 

Thus, we have shown that the linear model (9), which relates the mass of 

hadrons with a mass of the central core - glueball, also applies to the excited 

states of nuclei. In this case the glueball, apparently, should be considered as a 

bubble, formed in the quantum condensed due to nucleus excitation, just like 

pores are formed in the solid state and cavitation bubbles in a liquid under 

tension. 

To model the linear stage of the glueball excitation in atomic nuclei one 

can use, for example, the first equation (1), supplemented by terms that take into 

account the oscillations of the bubble. Such a model of the quantum harmonic 

oscillator is widely used in the modeling of nuclear shells [6-7]. In [8-9] for the 

nuclear shell model used a scalar wave equation in the five-dimensional space, 

which in a 4-dimensional space is reduced to the first equation (1). In this sense, 

the model [3-4] (and the model excited states of nuclei developed above) is an 

obvious non-linear generalization of linear shell model, consistent with the 

structure of hadrons. 

Finally, we note that the glueball is apparently the only one of the possible 

forms of the organization of hadronic matter, though, for example, Feynman 

[10] considered it as hadronic "bubbles" and even introduced a special symbol 

of hadron on Feynman diagrams in the form of bubble. Another possible form is 

a drop of a Fermi liquid, around which the nucleons in nuclei are concentrated, 

filling the shell [11]. It is possible that there is a third form, when a drop of 

quantum liquids is formed as a mixture of two Bose-Einstein condensates [12]. 

In all these cases in a quantum field there is a central body - bubble or drop, 

around which is organized the orbital motion of the quarks (in the case of 

hadrons) or nucleons in atomic nuclei.  

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10

 

The author expresses his gratitude to Professor VD Dzhunushaliev and Professor EV 

Lutsenko for useful discussions. 

              

References 
 

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http://arxiv.org/pdf/hep-ph/0312289v4.pdf  

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Политематический 
сетевой 
электронный 
научный 
журнал 
Кубанского 

государственного 
аграрного 
университета 
(Научный 
журнал 
КубГАУ) 

[Электронный ресурс]. – Краснодар: КубГАУ, 2012. – №02(76). С. 862 – 881. – 

Режим доступа: http://ej.kubagro.ru/2012/02/pdf/70.pdf 

9. A.P. Trunev. Nuclear shells and periodic trends// Политематический сетевой 

электронный 
научный 
журнал 
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государственного 
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университета (Научный журнал КубГАУ) [Электронный ресурс]. – Краснодар: 

КубГАУ, 
2012. 
– 
№05(79). 
С. 
414 
– 
439. 
– 
Режим 
доступа: 

http://ej.kubagro.ru/2012/05/pdf/29.pdf 

10. R.P. Feynman. Photon-Hadron Interactions. - Massachusetts, 1972.   

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condensates// arXiv:1204.2949v1 [cond-mat.quant-gas] 13 Apr 2012.