Quark and Spring Potential 1
QUARK POTENTIAL AND CLASSICAL SPRING POTENTIAL
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Quark and Spring Potential 2
Quark Potential and Classical Spring Potential
Introduction
Quarks constitute significant spin-1/2 particles during strong interactions [1]. From
relativity, Dirac spinors   ( x) consisting of four components (  =1… 4), that are functions of
space-time coordinates xu  (t , x, y, z ) can be used to describe quarks. They obey the free Dirac
equation when they do not interact with other associated fields or particles [2].
(i  m) ( x)  0
(1.1)
Where m denotes the “free” mass. The equation of standard plane waves is also used to
describe these quarks as shown
 k , ( x)  u(k ,  )ei ( Et  xk )
(1.2)
Where, k u  ( E, k ) and  represents polarisation as well as the four-momentum. u (k ,  )
represents the component of the momentum space wave function. Equation 1.1 can also be
derived based on the Lagrangian density function as shown:
( x)   ( x)(i  m) ( x)
(1.3)
Due to quark confinement, it is not possible to observe free quarks in a laboratory
experiment or isolated states [3]. It is a significant property in studying the low energy dynamics
of robust physical interactions. In high energy experiments, six flavours of quarks such as up (u ) ,
bottom (b) , charm (c) , down (d) , top (t ) , and strange (s) were found to form three families under
the influence of weak interactions [4]. The first family comprises of u and d , the second is
composed of c and s while t and b develops the third generation [5]. They have similar
quantum numbers; however, their physical significance is undefined. For instance, the magnitude
Quark and Spring Potential 3
of electric charge present in u , c and t quarks are equivalent to two-thirds of the electric charge
1
on a proton while the electric charges of d , s and b are equal to  [6].
3
The energy that binds quarks together to form hadrons is known as quantum
chromodynamic (QCD) binding energy. It is associated with the energy fields generated by the
strong forces regulated by gluons. The hadron’s mass is mainly composed of energy produced by
both motion and interactions in a mass-energy equivalence [7]. Quarks possess potential energy
resulting from interactions with conservative fields resulting in forces such as nuclear, gravity,
and electromagnetism. Strong forces are dominant and largely influence the property of these
quarks, such as clustering to form groups; for instance, electrons are controlled by
electromagnetic forces around these clusters of quarks [3, 4].
The force of gravity is the weakest since it requires more considerable distances and
massive objects as galaxies to generate sufficient potential to influence the behaviour of quarks
[8]. However, gravitational force particles can carry a charge, which affects the particles
interacting with it. For instance, an electron can be changed into a neutrino; an up quark changed
to a down quark and vice versa [5]. In classical spring potential, Hooke’s law describes the
relationship between force and displacement [9] as shown:
F  kx
(1.4)
At equilibrium, the unit vector iˆ points to the direction in which the object would move
when the spring is stretched. The force acting on the spring’s mass is then given by
F s  Fxs iˆ  kxiˆ
(1.5)
The magnitude of displacement and scalar product [10] are then obtained as shown:
dr  dxiˆ
(1.6)
Quark and Spring Potential 4
F  dr  kxiˆ  dxi  kxdx
(1.7)
The work done is then calculated as shown [11]:
Ws  
x x f
x  xi
F  dr  
1 xx f 1
1
 (kx)dx   k ( x 2f  xi2 )

2 x xi 2
2
(1.8)
Thus, the change in the spring’s potential energy when moving from the initial position to
a final position can be determined as:
1
U s  U s ( x f )  U s ( xi )  W s  k ( x 2f  xi2 )
2
(1.9)
Hence, any degree of stretch or deformation of the spring from equilibrium would change
its potential energy by:
1
U s  U s ( x f )  U s (0)  kx 2
2
(1.10)
The above equation shows that a zero-reference potential exists in the system, where the
potential energy is always zero [12]. Using a three-coordinate system, with the equilibrium point
taken as xi  0 . The magnitude of deformation ( x  0) or stretch ( x  0) from the equilibrium
position, the difference in potential energy can be determined as shown;
x
1
U s ( x)  U s ( xi )   Fxs dx  k ( x 2  xi2 )
xi
2
(1.11)
From the fundamental theorem of Calculus [13],
x'  x
U ( x)  U ( xi )   '
x  xi
dU '
dx
dx'
(1.12)
Comparing equations 1.11 and 1.12, the magnitude of the force is the negative derivative of the
spring’s potential energy.
Fxs  
dU s ( x)
dx
(1.13)
Quark and Spring Potential 5
The potential energy function can then be formed using the zero-reference potential, [14]
at equilibrium U s (0)  0 to give rise to:
1
U s ( x)  kx 2
2
(1.14)
From equation 1.14, the force law is obtained:
dU s ( x)
d 1
F 
  ( kx 2 )  kx
dx
dx 2
s
x
(1.15)
Quark Potential Energy Function
The proton is considered the most stable hadron [15], assuming the gluon has a uniform,
symmetrical field of radius, R within the proton where the gluons are usually combined
particles, quark confinement, as well as asymptotic freedom, can be analysed as shown [16];
V f (r )  
g 2 M (r )m f (r )
4
Rr
(1.16)
In which valence quarks q f with the flavour index, r the distance from the centre of the
proton to the quark q f ,0  r  R . The limits of the above equation show that the field acting
around the gluon is limited and confined within the proton.
Quark-quark strong interactions resemble a harmonic oscillator in which a pull exerciser
is grabbed and expanded which leads to greater forces in pulling the handles apart. A constant
point is attained in which no further expansion occurs, and the distance of expansion remains
constant. These properties are analogous to spring since the quarks cannot be eliminated from the
nucleon due to the strong forces that are similar to potential energy in spring [17]. In equation
1.18, k can be used as a spring constant in determining the potential energy of a spring. The
Coulomb potential is attributed to strong interactions in QCD with asymptotic degrees of
freedom at smaller and colour confinement at greater distances. The value of r is analogous to
Quark and Spring Potential 6
spring displacement x . It is indirectly proportional to the magnitude of Coulomb potential [18,
19].
Quark-quark interactions are similar to atoms in a spring, the nucleus of an atom
corresponds to the heavy quark while the role of electrons is performed by the light degrees of
freedom [20]. The main difference between hadronic and spring (atomic) systems lies in how the
light degrees of freedom are configured. In hadronic systems, it is impossible to calculate these
light degrees of freedom due to the non-perturbative capability of the strong interactions in QCD
[21]. However, it is possible to predict the range and application of these light degrees of
freedom using light hadrons [22].
Potential coefficient of quark-antiquark and radius of charge
Mesons are a combination of quarks and antiquarks controlled by QCD [23]. Most of
these are light quarks such as u, d and s which are inherently relativistic since the magnitude of
binding energies is higher compared to their masses [24]. An analysis of quark and an antiquark
at close ranges shows a coulomb potential while at greater distances, quark confinement is taken
into account [25]. Strong interactions are mainly controlled by QCD and the SU (3) gauge
theory. The former creates millions of hadrons that constitute various features such as oscillation
potential and “colour” confinement. The Schrodinger equation is used to analyse mesons with
heavy quarks, whereas mesons of light quarks are analysed using the Dirac equation [26]. Static
properties of central potentials such as oscillation, coulomb, and confinement are used to
examine light mesons. The wave function of a meson depends on two coordinates r1 and r2
representing the quark and antiquark, respectively. However, their internal motions are analysed
using relative coordinates [27].
r  r2  r1
(1.17)
Quark and Spring Potential 7
The central potentials are analysed with the coulombic potential associated with colour
charge as shown [28, 29];
V (r ) 
k s
c

r
r
(1.18)
At more considerable distances, the linear term in the above equation defines the quark
confinement [30],
V (r )  br
(1.19)
The oscillator potential, [27] can also be defined as
V  r2
(1.20)
Hence, the total sum of potential acting on the meson is
v(r )   r 2  br 
c
r
(1.21)
1
2
Taking the meson charge radius to be p with limits  r  p , the radius can be
0
2
cm
expressed as

(r 2 )q   r 2  (r )  (r )d 3r
(1.22)
0
The analytical solution of quark confinement shows that the complete interaction
involving the three potentials is sufficient to predict the position of the meson. Central potentials
are useful in determining steady states which contribute realistic quarks to prove that static
properties depend on the wave function [29, 31].
The above quark-antiquark pair represents a singlet colour state [22, 32]. Colour
confinement is essential in QCD at low energies as any strong interactions at zero temperatures,
and low densities consist of a coloured singlet at distances greater than ( 1/ QCD ) [33, 34, 35].
Hence, a consequence of quark confinement [36] attempts to separate the antiquark from the
Quark and Spring Potential 8
quark by pulling them apart creates stronger interactions between them as the distance also
increases. It is a similar scenario to a spring; once the elastic limit is exceeded, the spring breaks
into two [37].
In comparison to the quark-antiquark pair, in the event they break into two, new quarkantiquark pair is produced. The energy used in pulling the pair apart is transformed into the
development of a new pair; hence, it is not possible to have quarks as freely existing particles.
The quark-antiquark potential, coupled with QCD simulations, can be used to determine the
potential energy formed when the quarks are pulled apart [38, 39]. The quark potential is directly
proportional to the distance of separation of fermions, which is analogous to the spring potential
and displacement.
Quark and spring potentials both possess kinetic energy. For instance, the potential
energy of the spring can be transformed into kinetic energy once it is released after being
stretched based on the principle of energy conservation. In quarks, the potential energy can be
transferred by gluons that have spin, colour, and eight degrees of freedom. Differences between
the potentials include deformation in springs once the elastic limit has been exceeded, whereas,
in quarks, deformation can be attributed to strong magnetic fields, extreme densities, and high
temperatures.
Quark and Spring Potential 9
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