# Phenomenology of Strong Interactions — Towards an Effective Theory for Low
Energy QCD

Adamu Issifu<EMAIL_ADDRESS>Departamento de Física, CFM—Universidade
Federal de Santa Catarina,
Caixa Postal 476, 88040-900 Florianópolis, SC, Brazil  Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa,
Paraíba, Brazil  Francisco A. Brito<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João
Pessoa, Paraíba, Brazil  Departamento de Física, Universidade Federal de
Campina Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil

###### Abstract

In this paper, we develop models applicable to phenomenological particle
physics by using the string analogy of particles. These theories can be used
to investigate the phenomenology of confinement, deconfinement, chiral
condensate, QGP phase transitions, and even the evolution of the early
universe. Other confining properties such as scalar glueball mass, gluon mass,
glueball-meson mixing states, QCD vacuum, and color superconductivity can also
be investigated in these model frameworks. We use one of the models to
describe the phenomenon of color confinement among glueballs at the end of the
paper. The models are built based on the Dirac-Born-Infeld (DBI) action
modified for open strings with their endpoints on a D$p$-brane or brane-anti-
brane at a tachyonic vacuum.

##  I Introduction

String theory was conceived in the late 1960s to provide an explanation for
the behavior of nuclear matter such as protons and neutrons Schwarz ; Schwarz1
. Even though the theory was not successful in explaining the characteristics
of quarks and gluons at its inception, it promised an interesting intervention
in other areas of physics Green ; Polchinski2 ; Johnson ; Zwiebach1 ; Becker .
It showed capability in giving an insight into cosmology, astrophysics, and
unification of the fundamental forces of nature which has been on the table of
physicists for some time now. Consequently, the theory of Quantum
Chromodynamics (QCD) was developed in the early 1970s to give a comprehensive
explanation of nuclear matter Gross ; Politzer . The QCD theory is now
accepted as the standard theory for strong interactions. Interestingly, recent
development shows that string theory and QCD describe the same physics.

The formulation of Quantum Electrodynamics (QED) and QCD are almost the same.
They are both formulated from field theory based on gauge theory which forms
the foundation of the highly acceptable Standard Model (SM). The fundamental
particles of these theories are studied based on the gauge boson they
interchange: Photons for electrodynamic force, $W^{\pm}$ and $Z^{0}$ bosons
for electroweak force and gluon for strong nuclear force Quigg ; Collins1 ;
Shaw . Subsequently, gravitational force does not fall under this category, so
they are investigated under Einstein’s general theory of relativity.
Physicists have bunged their hope on string theory to unify all the
fundamental forces, this will fall under ’physics beyond the SM’ Aharony ;
Maldacena ; Grana ; Polchinski3 ; Haro . Upon the similarities, quarks and
gluons are color particles classified under three conventional colors (red,
blue, and green) whilst photons are color neural bosons that mediate
electrically charged leptons. Also, gluons self-interact due to their color
charges but photons do not Quigg ; Collins1 .

Additionally, QCD falls short in explaining color-neutral particles such as
bound states of gluons (glueballs) and quarks (hadrons and mesons), so string
theory can be resorted for further description. The string-like description of
hadrons arising from the quark model Amsler  is an important phenomenon in
applying string theory. Under this picture, when a quark and an antiquark are
pulled apart, they behave as if they are connected by a rubber band (gluon)
which becomes increasingly difficult to separate when the separation distance
between them keeps increasing. This analogy gradually fails when the particles
are brought closer and closer together. In this regime string theory fails and
QCD theory becomes viable. Now, looking at the particles in terms of
fundamental strings, string theory describes hadrons quite well and provides
the background for the unification of the fundamental forces. In string
theory, the strings are treated as particles, where different particles are
associated with different string oscillations. The masses of the particles are
also associated with the energy of the oscillating string. The intrinsic spin
of the particle is associated with the two perpendicular oscillations of the
string with its endpoints fixed on D-branes, similar to the direction of
electric and magnetic fields in a photon. Hence, photons and gluons can be
identified in terms of the open strings as spin-1 particles. The clockwise and
anti-clockwise movement of the closed strings makes it possible to classify it
among spin-2 bosons such as graviton. Since, QCD involves color fields, to
study it under string theory we assume that the endpoints of the open strings
on a D-brane serve as the source and sink of the color charge Bigazzi .

It has been conjectured that open bosonic strings studied at a tachyonic
vacuum behave as if they are closed strings with no D-branes. However, soliton
solutions in this region point to the presence of lower dimensional branes
Sent2 ; Bardakci . This projection is also corroborated in superstring
theories Sent1 ; Sen1  and evident in the first Recknagel ; Sen2 ; Harvey  and
second Kostelecky ; Sen ; Zwiebach2 ; Kostelecky1 ; Berkovits ; Harvey1 ;
Gopakumar ; Gerasimov  quantizations in string theories Sent4 . At the
tachyonic vacuum, the negative energy density $V(T_{0})$ of the tachyons in
the vacuum exactly cancels Witten1  the energy density of the D-branes
$\varepsilon_{p}$ i.e. $V(T_{0})+\varepsilon_{p}=0$. In non-BPS D-branes
$\varepsilon_{p}=\tau_{p}$ where $\tau_{p}$ is the D-brane tension Lindstrom ;
Gustafsson ; Sent3 . It should be noted that at $|T|=T_{0}$ the total energy
density of the tachyons and the brane tension vanishes identically, signifying
the absence of open strings at that point. In this view, at a tachyonic
vacuum, there will be no physical open string excitations because there are no
D-branes. On the contrary, the ’usual’ field theory gives an alternative
explanation, because shifting the vacuum and expanding around its ’true’
minimum can change a negative square mass of the tachyons to a positive one
even though it does not completely remove all the states. Since it will not
cost any energy to adjust the fundamental strings on the worldvolme of the
D-branes at a tachyonic vacuum, it will be difficult to notice their presence.
Hence, fluctuations around the vacuum represent the presence of lower
dimensional D-branes Sent2 ; Bardakci . These phenomena have been investigated
in references Sen ; Zwiebach2 ; Kostelecky  using open string field theory
Witten2 .

In this paper, we modify the Dirac-Born-Infeld (DBI) action, so that the
associated open strings with their endpoints on the remaining lower-
dimensional branes at the tachyonic vacuum can be analyzed. The objective is
to develop models that can mimic QCD theory both in the UV and IR regions with
UV safety. The models can be applied in developing potential models such as
the linear confining and Cornell potential models. In the analysis, we
consider that the string worldsheet falls inside the D-brane worldvolume
making it easy for the endpoints of the strings to be connected by the flux
line on the worldvolume to form a close string suitable for modeling color
confinement in a flux-tube like picture. The dynamics of the strings
tangential to the D-brane worldvolume is represented by the gauge field
$F_{\mu\nu}$ and the component transverse to the worldvolume is represented by
a massless scalar field $X^{a}$. The net flux involved is determined by the
source and sink of the flux carried by the endpoints of the string on the
lower dimensional remnants of the original D-brane in the vacuum. Also, the
condition for minimum energy warrants that the flux does not spread because
the source and sink of the flux emanate from a pointlike object on the D-brane
worldvolume.

The paper is organized as follows: In Sec. II we review Tachyons and D-branes,
divided into two subsections, under Sec. II.1 we review Tachyons, and in Sec.
II.2 we review D-branes. We present the Modification of the Dirac-Born-Infeld
Action which serves as the bases for the study in Sec. III. Also, in
subsection III.1 we review the Dimensional Reduced $\text{U}(1)$ Yang-Mills
theory and its intended consequences in relation to the Standard Model (SM) of
Particle Physics. Section IV and subsequent sections contain our original
contributions to the subject. We present the Bosonic String at Tachyonic
Vacuum in Sec. IV, where we present the details of dimensional reduction from
$10D$ Dirac-Born-Infeld action to $4D$ conducive for describing SM particles.
We present Gauge Theories Modified with $G(\phi)$, in Sec. V, this section was
divided into two. We studied Fermions Coupled to Fundamental Strings at
Tachyonic Vacuum in Sec. V.1 and Non-Abelian Gauge Theory in Sec. V.2. Section
VI contains the Phenomenon of Gluon Confinement, divided into four
subsections. We present The Model in Sec. VI.1, Confining Potentials in Sec.
VI.2, Gluon Condensation in Sec. VI.3 and strong Running Coupling and QCD
$\beta$-function in Sec. VI.4. we present our findings and Conclusions in Sec.
VII.

##  II Review of Tachyons and D$p$-branes

###  II.1 Tachyons

Generally, tachyons are classified as particles that travel faster than light
or weakly interacting superluminal particles. Relativistically, single
particle energy is expressed as $E^{2}=p^{2}c^{2}+m^{2}c^{4}$, where $p$ is
the spatial momentum and $m$ is the mass of the particle. For a particle to be
faster than light, the relativistic velocity $\beta=pc/E>1$. Hence, for
tachyons with real $p$, $m$ must necessarily be imaginary Bilaniuk . However,
this analogy does not make a strong and convincing case for the tachyons.
Rather, Quantum Field Theory (QFT) provides some satisfactory explanation to
the dynamics of tachyons. QFT suggests that particles that travel faster than
light does not exist in nature. So tachyons are simply unstable particles that
decay. Based on this understanding, we consider a scalar field, say $\phi$,
with the usual kinetic term and a potential $V(\phi)$ whose extremum is at the
origin. If one carries out perturbation quantization about $\phi=0$ up to the
quadratic term and ignores the higher order terms in the action, we obtain a
particle-like state at $V^{\prime\prime}(0)$. These results have two
interesting interpretations; if $V^{\prime\prime}(0)>0$ we have a particle of
mass $m^{2}_{\phi}$ but for $V^{\prime\prime}(0)<0$ we have a tachyonic state
with a negative $m^{2}_{\phi}$. In this case, tachyons can be given a physical
meaning. So far, we know that the tachyons have negative $m^{2}_{\phi}$ and
potential whose maximum is at the origin, thus a small displacement of $\phi$
at the origin will cause it to grow exponentially with time towards the true
minimum. By the above description, tachyons can be represented by a potential
such as

$V(\phi)=-\dfrac{1}{2}m_{\phi}^{2}\phi^{2}+c_{3}\phi^{3}+c_{4}\phi^{4}\cdots,$
(1)

where $c_{3}\,\text{and}\,c_{4}$ are constants. Hence, tachyonic fields are
associated with the ’usual’ Higgs fields where the Higgs particles acquire
negative square mass at the true minimum of their potential. Accordingly,
tachyons in QFT are related to instability that breaks down the perturbation
theory in normal field theory. The usual quantization where the cubic and
higher order terms are considered small corrections to the quadratic term is
no longer tenable. Since $V(\phi)$ has its maximum at $\phi=0$, it renders it
classically unstable at that point. So one cannot guarantee that the
fluctuation at that point is small. This behavior comes with an inbuilt
solution i.e. one can expand the potential about the true minima $\phi_{0}$ up
to the quadratic term and proceed with the perturbation quantization about
that point. Thus, the cubic and higher-order terms in the expansion can be
discarded. This process will lead to the creation of a particle with a
positive mass $m^{2}_{\phi}$ in the spectrum. This process removes the
tachyonic modes in the spectrum. Additionally, in D$p$-brane systems, the
theory must be invariant under $Z_{2}$ symmetry, i.e. $\phi\rightarrow-\phi$
and in the presence of brane-anti-brane, the theory must necessarily be
invariant under phase symmetry $\phi\rightarrow e^{i\alpha}\phi$. Indeed,
there are some benefits in working in tachyonic modes otherwise we can define
a new field, $\eta=\phi-\phi_{0}$, and express the potential in terms of the
new field,

$V(\eta)=V(\eta+\phi_{0}).$ (2)

Working with this potential from the onset will remove the tachyonic mode from
the spectrum because $V^{\prime\prime}(\eta=0)$ will be positive. However,
there are some benefits to working with tachyonic fields, for instance, they
possess high symmetry. This symmetry might not be explicit in $V(\eta)$ as it
is in $V(\phi)$. The high symmetry in tachyonic fields leads to the phenomenon
of spontaneous symmetry breaking, where the potential has more than one minima
i.e. $V(\phi)=V(-\phi)$ corresponding to $\pm\phi_{0}$. This phenomenon is
well known in elementary particle physics Sent3 .

###  II.2 D$p$-branes

Recent advances in string theory have provided some explanations regarding
nonperturbative features of QCD theory, thanks to the discovery of
D$p$-branes. The study of D$p$-branes gives insight into physically relevant
systems such as black holes, supersymmetric gauge theories, and the connection
between Yang-Mills theories and quantum gravity. Upon the numerous progress, a
consistent nonperturbative background-independent construction of the theory
is yet to be developed. This poses a challenge in directly addressing
cosmological problems. There are five known ways for which supersymmetric
closed strings can be quantized, i.e. IIA, IIB, I, heterotic $\text{SO}(32)$
and heterotic $E_{8}\times E_{8}$ superstring theories. Individually, they
give a perturbative description of quantum gravity. However, they are
connected through duality symmetries Hull ; Witten . Indeed, some of these
dualities are nonperturbative in nature because the string coupling $g_{s}$ in
one theory may have an inverse relation $1/g_{s}$ with the other. The
superstring theories together with M-theory (a theory that unifies superstring
theories) consist of an extended object of higher dimensions, called
D$p$-branes. Each of these theories contains branes but in different
complements. Particularly, IIA/IIB superstring theories consist of even/odd
D$p$-brane dimensional configurations. Using the appropriate duality
transformations, one brane can be mapped onto the other even with the strings
connecting them. Thus, none of the branes are seen to be more fundamental to
others. This process is commonly referred to as ’brane democracy’.

Before moving into the specific kind of strings set out for this study, we
will shed light on the features of open and closed strings. Closed strings are
topologically the same as a circle $S^{1}$, i.e. $[0,2\pi]$. They give rise to
a massless set of spacetime fields, identified as graviton $g_{\mu\nu}$,
dilaton $\varphi$, antisymmetric two-form $B_{\mu\nu}$ and an infinite set of
massive fields upon quantization. Supersymmetric closed strings are also
related to massless fields identified with graviton supermultiplets such as
Ramond-Ramond $p$-form fields $A^{(p)}_{\mu,\cdots,\mu_{p}}$ and gravitino
$\psi_{\mu\alpha}$. Consequently, the quantum theory of closed strings is
naturally related to the theory of gravity in spacetime. On the other hand,
open strings are topologically similar to the interval $[0,\pi]$. They give
rise to massless gauge field $A_{\mu}$ in spacetime upon quantization.
Supersymmetric open strings are also associated with massless gaugino
$\psi_{\alpha}$. Accordingly, open strings have their ends on Dirichlet
$p$-branes (D$p$-brane) and the gauge field lives on the worldvolume of the
D$p$-brane. Upon these differences, the physics of closed and open strings are
related at the quantum level. Closed strings were first observed through the
one-loop process of an open string. Under this process, close strings appeared
as poles in nonplanar one-loop open string diagrams Gross1 ; Lovelace . The
open strings have their endpoints on the D$p$-branes whilst close strings have
no endpoints. The degree of freedom of the open strings is associated with
standing wave modes on the fields while the close strings correspond to left-
moving and right-moving waves. The boundary conditions for open strings on the
bosonic field $X^{M}$ are Neumann (freely moving endpoints) and Dirichlet
boundary conditions (fixed endpoints). The close string on the other hand
corresponds to periodic and anti-periodic boundary conditions.

Some D$p$-branes have unstable configurations both in the supersymmetric or
the nonsupersymmetric string theories. The instability is attributed to
tachyonic modes with negative square mass $m^{2}_{\phi}\,<\,0$ in the open
string spectrum, we are interested in investigating the effects of the
tachyonic modes Sen ; Zwiebach . Some D$p$-brane configurations with open
strings containing tachyons are:

Brane-antibrane; it is a type IIA or IIB string theory with parallel
D$p$-branes separated by $d\,<\,l_{s}$ ($l_{s}$ is the string length scale).
They also carry tachyons in their open string spectrum. The difference in
their orientation leads to opposite Ramond-Ramond (R-R) charges Hughes . So
the brane and anti-brane pair can annihilate leaving a neutral vacuum because
the net R-R charge will be zero.

Wrong-dimension branes; the D$p$-brane with wrong dimension for type IIA/IIB
with odd/even spatial dimension for $p$ instead of even/odd dimensions carry
no charges under classical IIA/IIB supergravity fields. They have tachyons in
their open string spectrum. Such branes can annihilate to form a vacuum
without violating charge conservation.

Bosonic D$p$-branes; just as the wrong-dimension branes of type IIA/IIB string
theory, the D$p$-brane of any particular dimension in the bosonic string
theory have no conserve charge and has tachyons in their open string spectrum.
Also, they can annihilate to form a neutral vacuum without violating charge
conservation.

Again, even though the non-BPS D$p$-branes of type IIA/IIB string theory are
unstable owing to the presence of tachyonic modes on their worldvolume Sen1 .
We can obtain stable non-BPS branes by taking orientifolds/orbifolds of the
theory which projects out the tachyonic modes Sen1 ; Bergman ; Sen2 .

##  III Modification of Dirac-Born-Infeld Action

We begin the study with the Born-Infeld action (BI) Born  sometimes referred
to as Dirac-Born-Infeld action (DBI) Gibbons ; Dirac . The focus will be on
D$p$-branes Polchinski ; Taylora  which are nonperturbative states on which
open strings live. They are equally coupled with closed strings, Ramond-Ramond
states and other massive fields. The nonperturbative nature of the action
makes it possible to describe low energy degrees of freedom of the D$p$-branes
Leigh  making it possible for application in low energy QCD. The distinction
between the DBI action, other $p$-branes and supermembrane theories Hughes ;
Bergshoeff ; Townsend  is the presence of gauge field in the worldvolume of
DBI. The gauge field is associated with virtual open string states.
Consequently, we obtain confinement of the fundamental strings with their
endpoints fixed on the branes. Generally, the action can be expressed as

$S=-T_{p}\int d^{p+1}\xi
e^{-\varphi}\sqrt{-\text{det}\left(G_{\mu\nu}+B_{\mu\nu}+(2\pi\alpha^{\prime})F_{\mu\nu}\right)}+S_{CS}+\text{fermions},$
(3)

where $G_{\mu\nu}$, $B_{\mu\nu}$ and $\varphi$ are the induced metric tensor,
antisymmetric tensor, and dilaton field to the D$p$-brane worldvolume
respectively. Also, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
is the worldvolume electromagnetic field strength of $A_{\mu}$ and $S_{CS}$ is
a set of Chern-Simon terms while

$\tau_{p}=\dfrac{T_{p}}{g_{s}}=\dfrac{1}{g_{s}\sqrt{\alpha^{\prime}}(2\pi\sqrt{\alpha^{\prime}})^{p}},$
(4)

is the brane tension and $g_{s}=e^{\langle\varphi\rangle}$ is the string
coupling, with associated string tension

$T_{\text{string}}=\dfrac{1}{2\pi\alpha^{\prime}}.$ (5)

In type IIA/IIB string theory with $p$ even/odd are associated with quantum
open strings containing massless $A_{\mu},\,\mu=0,1,\cdots,p$ and
$X^{M},\,M=p+1,\cdots,9$ fields. These fields are the consequence of the gauge
field living on the hypersurface and $X^{\mu}(\xi)$ transverse excitations.
The geometry of D$p$-brane is not flat, so we generally define the embedding
$X^{\mu}(\xi)$, where $\xi^{\alpha}$ represent $p+1$ coordinates on the
D$p$-brane worldvolume $\sum_{(p+1)}$, and $X^{\mu}$ is the ten functions
mapping from $\sum_{(p+1)}$ onto the spacetime manifold $\mathbb{R}^{9,1}$.
Introducing the scalar field into the action, it becomes invariant under
diffeomorphism and Abelian gauge transformations. So, a way of fixing the
freedom of the former is to adopt a ’static gauge’ such that

$\displaystyle X^{\mu}\equiv\xi^{\mu}\qquad\text{for}\qquad 0\,\leq\,\mu\leq
p.$ (6)

The remaining fields are

$X^{\mu}\equiv X^{M}\qquad\text{for}\qquad(p+1)\,\leq\,M\,\leq d-p,$ (7)

which are the transverse coordinates to the worldvolume Tseytlin ; Taylor ;
Taylor1  where $d$ is a spatial dimension. Thus, one can choose $d=9$ for
superstring theory and $d=25$ for bosonic string theory. Under this gauge, the
originally $d+1$-dimensional global poincaré symmetry spontaneously breaks
down to a product of $p+1$ dimensional poincaré group with $d-p$ dimensional
rotational symmetry group i.e.
$\text{SO}(1,d)\rightarrow\text{SO}(1,p)\times\text{SO}(d-p)$. For a
D$p$-brane possessing $p<d$ extends over $p$-dimensional subspace of
$d$-dimensional space. The focus will be on a D$p$-branes with $p$-dimensional
hyperplanes in $d$-dimensional space. Again, according to the static gauge
fixed above, there are two possible consistent truncations;

  * •

$X^{M}=0$; corresponds to pure BI theory Born  in $\mathbb{E}^{p,1}$ and

  * •

$F_{\mu\nu}=0$; also corresponds to Dirac’s theory Dirac  of minimal timelike
submanifolds of $\mathbb{E}^{d,1}$.

We can introduce the transverse scalar fluctuations by defining the induced
metric as

$G_{\mu\nu}\approx\eta_{\mu\nu}+\eta_{MN}\partial_{\mu}X^{M}\partial_{\nu}X^{N},$
(8)

that will approximate the D$p$-brane worldvolume to near flat. In this study,
we are interested in understanding the dynamics of bosonic open strings with
tachyonic modes in their spectrum, so we will proceed systematically toward
that objective. The worldvolume theory of non-BPS D$p$-brane in IIA/IIB string
theory corresponds to a massless $\text{U}(1)$ vector field, transverse
oscillating scalar field $X^{M}$ and a tachyonic field $T$ Sent . Accordingly,
the leading order of the action results in dimensional reduction
$10$-dimensional $\text{U}(1)$ Yang-Mills theory. Besides, higher-order
corrections, $\alpha^{\prime}=l_{s}^{2}$, to the order of the string scale are
also possible. Since we are proceeding with the assumption that the massless
fields are slowly varying compared to the string length $l_{s}$, i.e. we
discard the higher order derivatives and write the DBI action Leigh ; Garousi
; Callan  in a simple form without the explicit presence of the tachyons.
Therefore, Eq.(3) takes the form

$S=-T_{p}\int d^{p+1}\xi
e^{-\varphi}\sqrt{-\text{det}\left(\eta_{\mu\nu}+\eta_{MN}\partial_{\mu}X^{M}\partial_{\nu}X^{N}+B_{\mu\nu}+(2\pi\alpha^{\prime})F_{\mu\nu}\right)}.$
(9)

Now we will express the extended form of the DBI by including the dynamics of
the tachyons as studied in Garousi . Accordingly,

$S=-T_{p}\int d^{p+1}\xi
e^{-\varphi}V(T)\sqrt{-\text{det}\left(\eta_{\mu\nu}+\partial_{\mu}T\partial_{\nu}T+\eta_{MN}\partial_{\mu}X^{M}\partial_{\nu}X^{N}+B_{\mu\nu}+(2\pi\alpha^{\prime})F_{\mu\nu}\right)},$
(10)

where $V(T)$ is the tachyon potential and $\partial_{\mu}T\partial_{\nu}T$ is,
as usual, the kinetic energy of the tachyons Garousi1 . Under this
conjuncture, the action vanishes at the minimum of the tachyon potential
$V(T_{0})=0$ Sent2 ; Sent1 ; Recknagel . We will now continue the discussion
by approximating that the D$p$-branes are nearly flat, with constant dilaton,
and vanishing antisymmetric two-form term $B_{\mu\nu}$ to remove the close
string quanta in the system.

###  III.1 Dimensional Reduction $\text{U}(1)$ Yang-Mills Theory

Studying D$p$-branes under Yang-Mills theory enables us to understand the
physics of D$p$-branes without necessarily applying any complex string theory
artifacts. Detailed analyses show there is enough evidence that super Yang-
Mills theory carries a lot of information concerning string theory than one
may possibly imagine Banks . Besides, recent developments in high-energy
physics investigations have shown that string theory gives insight into low-
energy field theories in the nonperturbative region Taylor . This has been
conjectured to be equivalent to QCD where confinement can be realized in color
fluxtube picture. From Eq.(9), in the limit of vanishing $B_{\mu\nu}$ and
further assuming that the D$p$-branes are almost flat, close to the
hypersurface i.e. $X^{M}=0$, $M>p$. Again, we suppose that the fields are
slowly varying such that $\partial_{\mu}X^{M}\partial^{\mu}X_{M}$ and
$2\pi\alpha^{\prime}F_{\mu\nu}$ are in the same order. Therefore, the action
can be expanded as

$S=-\tau_{p}V_{p}-\dfrac{1}{4g^{2}_{YM}}\int
d^{p+1}\xi\left(F_{\mu\nu}F^{\mu\nu}+\dfrac{2}{(2\pi\alpha^{\prime})^{2}}\partial_{\mu}X^{M}\partial^{\mu}X_{M}\right)+{{\cal
O}((\partial_{\mu}X^{M})^{4},F^{4})}.$ (11)

Here, $V_{p}$ is the $p$-brane worldvolume and $g_{YM}$ is the Yang-Mills
coupling given by,

$g_{YM}^{2}=\dfrac{1}{(2\pi\alpha^{\prime})^{2}\tau_{p}}=\dfrac{g}{\sqrt{\alpha^{\prime}}}\left(2\pi\sqrt{\alpha^{\prime}}\right)^{p-2}.$
(12)

The second term in Eq.(11) corresponds to $\text{U}(1)$ gauge theory in $p+1$
dimension coupled with $9-p$ scalar fields. Introducing fermions fields, as
mentioned below Eq.(3) into the action, we recover the supersymmetric
$\text{U}(1)$ Yang-Mills theory in $10\,d$ $\mathcal{N}=1$ Super Yang-Mills
action,

$S=\dfrac{1}{g_{YM}^{2}}\int
d^{10}\xi\left(-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}+\dfrac{i}{2}\bar{\psi}\Gamma^{\mu}\partial_{\mu}\psi\right).$
(13)

In the case of N parallel D$p$-branes, the p-dimensional branes must be
distinctively labeled from $1\,\text{to}\,\text{N}$. Subsequently, the
massless scalar fields living on the individual D$p$-brane worldvolume are
related to $\text{U}(1)^{N}$ gauge group. The fields arising from the strings
stretching from one brane to the other are labeled as $A^{\mu}_{i,j}$, where
$i,j$ specifies the individual branes that carry the endpoints of the strings.
The strings are oriented such that they consist of $N(N-1)$ fields,
corresponding to $A_{\alpha}=(A_{\mu},\,X^{M})$ individual fields. The mass of
the strings is proportional to the separation distances between the branes. So
the strings become massless Witten1 ; Taylora  when the D$p$-branes get very
close to each other. Hence, the open strings can transform under the adjoint
representation U(N). Thus, the corresponding fields can be decomposed
similarly to adjoint supersymmetric gauge field
$\text{U(N)}=\text{SU}(N)\times\text{U}(1)$ in $p+1$-dimensions. With the
stacks of D$p$-branes crossing each other at some angles, one can also break
the U(N) symmetry into SM particle physics i.e.
$\text{SU}(3)\times\text{SU}(2)\times\text{U}(1)$ Lust . In sum, the DBI
action can be generalized into a non-Abelian gauge group by considering stacks
of D$p$-branes instead of a D$p$-brane.

A major motivation for string dual to QCD is derived from the ’t Hooft large
$N_{c}$-limit Hoofta2 ; Witten4 . Though, $\text{SU}(N_{c})$ and
$\text{U}(N_{c})$ have different representations, in the limit of
$N_{c}\rightarrow\infty$, the difference can be overlooked. Thus, the number
of gluons can be approximated as $~{}N_{c}^{2}$. This is more than the quark
degree of freedom $N_{f}N_{c}$, therefore, we expect the dynamics of the
gluons to dominate in this regime Mateos .

##  IV Bosonic Strings at Tachyonic Vacuum

Generally, there are two known boundary conditions associated with open
strings. The Dirichlet (fixed) boundary condition is where the coordinates are
normal to the brane, and under Neumann (free) boundary condition is where the
coordinates are parallel to the brane Dai ; Leigh ; Polchinski1 . It has been
established Callan1  that a small disturbance normal to the string and the
brane are likely to reflect back with a phase shift of $\pi$, corresponding to
a Dirichlet boundary condition. Some study in this regard has been carried out
in Rey ; Lee  using Nambu-Goto action for strings with their endpoints on a
supergravity background of D$3$-branes. Since the strings attached to the
$3$-branes manifest themselves as electric charges, the Neumann boundary
condition where the endpoints of the strings are freely oscillating will lead
to the production of electromagnetic dipole radiation at the asymptotic outer
area of the brane.

In this section, we will consider Eq.(10) which includes the dynamics of
tachyons on open strings. Accordingly, we will keep all the arguments made for
the $10$-dimensional DBI action, but we will reduce the spacetime dimension
from $10$ to $4$. One of the major differences between superstring theory and
particle theory is that the former lives on $10$-dimensional spacetime and the
latter on $4$-dimensional spacetime. Nonetheless, this discrepancy can be
dealt with using compactification scheme Gell-Mann ; Guendelman ; Randall ,
where the spacetime is divided into an external non-compact spacetime
$\mathcal{M}_{10-d}$ and an internal compact spacetime $\mathcal{M}_{d}$. We
can combine these two phenomena into a single expression as

$\mathcal{M}_{10}=\mathcal{M}_{10-d}\times\mathcal{M}_{d}.$ (14)

We adopt a physically realistic phenomenon where $d=6$. Additionally, we can
set a compact scale $M_{c}=1/R$, where $R$ is the radius of the internally
compact spacetime, smaller than the string mass $M_{s}=1/l_{s}$ Grana ;
Shiraishi . As a result, the energy $E$ required for this work must lie in the
range $E\ll M_{c}\ll M_{s}$. So, the worldvolume coordinate becomes
$\xi=(x^{\mu},x^{6})$, where $\mu=0,1,2,3$. Consequently, we will reduce the
dimension to $1+3$-dimension with $\eta_{\mu\nu}=\text{diag}(+,-,-,-)$ metric
signature. We will also decouple the $6$ available transverse fluctuating
scalar fields and the antisymmetric tensor i.e., $X^{M}=B_{\mu\nu}=0$. Keeping
the dilaton field constant, we get

$\displaystyle S$ $\displaystyle=-\tau_{p}\int
d^{4}xV(T)\sqrt{-\text{det}\left(\eta_{\mu\nu}+\partial_{\mu}T\partial_{\nu}T+(2\pi\alpha^{\prime})F_{\mu\nu}\right)}$
$\displaystyle=-\tau_{p}\int
d^{4}xV(T)\sqrt{1-\eta^{\mu\nu}\partial_{\mu}T\partial_{\nu}T+\dfrac{1}{2}(2\pi\alpha^{\prime})^{2}F_{\mu\nu}F^{\mu\nu}+\cdots}$
$\displaystyle=-\tau_{p}\int
d^{4}xV(T)\left[1-\dfrac{1}{2}\partial_{\mu}T\partial^{\mu}T+\dfrac{1}{4}(2\pi\alpha^{\prime})^{2}F_{\mu\nu}F^{\mu\nu}+\cdots\right].$
(15)

In the above expression we calculate the determinant up to the second-order
derivative discarding higher-order corrections. In view of field theory at
tachyonic vacuum, the D-branes do not vanish completely rather, there are
lower dimensional D-branes present. So, the endpoints of the strings sitting
on the low dimensional D-branes behave like point-like particles which serve
as the source and sink of the flux carrying the color particles. For instance,
expanding around the ’true’ minimum of the potential gets rid of the tachyons
leading to a particle with a positive square mass.

We consider a field configuration such that $T(r)=f(\phi(r))$, in this case,
the potential can be expressed as

$V(T(\phi))=\left(\dfrac{\partial\phi}{\partial T}\right)^{2},$ (16)

so

$\displaystyle\dfrac{1}{2}V(T)\partial_{\mu}T\partial^{\mu}T=\dfrac{1}{2}\left(\dfrac{\partial\phi}{\partial
T}\dfrac{\partial T}{\partial
x}\right)^{2}=\dfrac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi.$ (17)

As a result, the Lagrangian of the system becomes,

$\tau^{-1}_{p}\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)-\frac{1}{4}G(\phi)F_{\mu\nu}F^{\mu\nu},$
(18)

here, we have introduced a dimensionless quantity
$G(\phi)=(2\pi\alpha^{\prime})^{2}V(\phi)$ which will be referred to as a
color dielectric function, subsequently. If we set $(2\pi\alpha^{\prime})=1$
the string tension becomes $T_{string}=1$ and $G(\phi)=V(\phi)$ Brito ; Adamu
; Issifu ; Issifu1 ; Issifu2 . It is important to mention at this point that
the potential of the field, $\phi$, follows the same discussion as contained
in Sec. II.1. It has its minimum at $V(\phi=\langle\phi\rangle_{0})=0$, where
$\langle\phi\rangle_{0}$ is the ’true’ vacuum of the potential. To apply this
theory to asymptotically free systems, the potential must satisfy additional
conditions,

$V(\phi=\langle\phi\rangle_{0})=0\qquad{,}\qquad\dfrac{\partial
V}{\partial\phi}|_{\phi=\langle\phi\rangle_{0}}=0\qquad{\text{and}}\qquad\dfrac{\partial
V}{\partial\phi}|_{\phi=0}=0,$ (19)

necessary for stabilizing its vacuum Rosina ; Adamu ; Kharzeev . These
restrictions make it possible to apply Eq.(18) in investigating asymptotically
free particles such as gluons in a QCD-like fashion. Here, the modified
Abelian gauge mimics the role of the non-Abelian gauge in the ’usual’ QCD
theory. We treat the endpoints of the open string on the lower dimensional
branes as the source of quark and an anti-quark (if fermions are present) or
valence gluons and the string connecting them as gluons that mediate the
interactions. Also, the potential $V(\phi)$ plays a similar role as a quantum
correction in gluodynamics theory that breaks the conformal symmetry to bring
about gluon condensation Kharzeev ; Gaete ; Issifu1 . This model has been used
to study the color confinement of glueballs at a finite temperature in
Ref.Issifu .

Considering brane-anti-brane systems, we note that the model must be invariant
under the global rotation $\phi\rightarrow e^{i\alpha}\phi$. This warrants an
introduction of a complex scalar field $\phi$ with potential $V(|\phi|)$ Sen3
. So, we can redefine the scalar field in a form,

$\phi=\dfrac{\phi_{1}+i\phi_{2}}{\sqrt{2}},$ (20)

to incorporate the D$p$-brane and anti-D$p$-brane dynamics. Supposing that the
original gauge field $F_{\mu\nu}$ is on the worldvolume of the D$p$-brane
represented by a complex scalar field $\phi$, we will have a dual gauge field
$\tilde{F}_{\mu\nu}$ also on the wordvolume of the anti-D$p$-brane represented
by the conjugate field $\phi^{*}$. Therefore, the string here has its
endpoints on remnants of D$p$-brane and the anti-D$p$-brane parallel to each
other. Imposing gauge invariance on the scalar sector, we can define an
Abelian covariant derivative

$\partial_{\mu}\rightarrow D_{\mu}\equiv\partial_{\mu}-iq\tilde{A}_{\mu},$
(21)

where $\tilde{A}_{\mu}$ is a gauge field which is dual to $A_{\mu}$.
Accordingly, the Lagrangian in Eq.(18), can be extended as

$\tau_{p}^{-1}\mathcal{L}=D_{\mu}\phi
D^{\mu}\phi^{*}-V(|\phi|)-\frac{1}{4}G(|\phi|)F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu},$
(22)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$\tilde{F}_{\mu\nu}=\partial_{\mu}\tilde{A}_{\nu}-\partial_{\nu}\tilde{A}_{\mu}$
are two independent Abelian field strengths. Aside from the original
$\text{U}(1)$ gauge invariance of the Lagrangian, it is also invariant under
the $\tilde{\text{U}}(1)$ gauge transformation,

$\phi(x)\rightarrow\phi^{\prime}(x)=e^{-iq\alpha(x)}\phi\qquad\text{and}\qquad\tilde{A}_{\mu}(x)\rightarrow\tilde{A}^{\prime}_{\mu}(x)=\tilde{A}_{\mu}(x)-\partial_{\mu}\alpha(x).$
(23)

This Lagrangian can undergo a spontaneous symmetry breaking (SSB) process
(when we choose an appropriate potential) similar to the usual Abelian Higgs
mechanism Quigg ; Collins1 . In this case, $\phi$ plays a role similar to the
’usual’ Higgs field in the standard model of particle physics. This process
will also lead to the observation of Goldston boson (most likely $\pi^{0}$ due
to the $\tilde{\text{U}}(1)$) corresponding to the number of unbroken
symmetries signaling confinement as well Issifu ; Nielsen . This model has
been exploited in detail to study glueballs and color superconductivity in
Issifu2 . It is important to mention that (22) is valid when we consider
D$p$-brane and an anti-D$p$-brane system in the framework of DBI action Erkal
; Senn . It is suitable for describing massless particles such as glueballs or
gluon confinement.

##  V Gauge Theories Modified with $G(\phi)$

###  V.1 Fermions Coupled To Fundamental Strings at Tachyonic Vacuum

In this section, we introduce Dirac’s Lagrangian for free particles adopted by
Maxwell in the unification of electric and magnetic field interactions. We
introduce the dynamics of the fermions which were dropped in the previous
discussions as stated below Sec. III. However, it will be introduced here
through Dirac’s equation modified by $G(\phi)$ while taking into consideration
all gauge invariance properties. We start with the well-known Dirac’s
Lagrangian for free particles

$\mathcal{L}_{0}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi.$
(24)

Even though this Lagrangian is well known in QED, it is also used to describe
free nucleons in terms of their composite fermions; protons, and neutrons in
strong interaction. It is invariant under global phase rotation,

$\psi\rightarrow\psi^{\prime}=e^{i\alpha}\psi.$ (25)

Noticing that the fields in the Lagrangian are color neutral in nature, to
apply them in studying color particles such as the ones considered here, we
need to modify the fields. Hence, we will modify the bispinors with the
$G(\phi)$ in order to give them some color features. Thus, we perform the
transformations

$\psi\rightarrow\psi^{\prime}\equiv
G^{1/2}\psi\qquad\text{and}\qquad\bar{\psi}\rightarrow\bar{\psi}^{\prime}\equiv\bar{\psi}G^{1/2},$
(26)

where $G$ is a function therefore, the gradient in Dirac’s equation will
transform as

$\partial_{\mu}\psi\rightarrow\partial_{\mu}\psi^{\prime}=\left[(\partial_{\mu}G^{1/2})+G^{1/2}\partial_{\mu}\right]\psi,$
(27)

and Lagrangian (24) becomes

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\psi}^{\prime}\left[i\gamma^{\mu}\partial_{\mu}-m\right]\psi^{\prime}$
$\displaystyle=\bar{\psi}\left[i\gamma^{\mu}G\partial_{\mu}+i\gamma^{\mu}G^{1/2}(\partial_{\mu}G^{1/2})-mG\right]\psi.$
(28)

The local gauge invariance is violated by $G(\phi)$ and the gradient function
$(\partial_{\mu}G^{1/2})$. To ensure local gauge invariance is satisfied, we
modify all variables involving derivatives, and color neutral fields such as
the Dirac $\gamma$-matrix with $G(\phi)$ and also introduce the
electromagnetic field $A_{\mu}(x)$ in other to make the equation gauge
invariant. Consequently, we will adopt the transformations,

$\gamma^{\mu}\rightarrow\gamma^{\prime\mu}=G^{-1}\gamma^{\mu}\qquad{\text{and}}\qquad\partial_{\mu}\rightarrow
D_{\mu}\equiv\partial_{\mu}-iqA_{\mu},$ (29)

where $D_{\mu}$ is the gauge-covariant derivative and $q$ is the electric
charge. This type of derivative corresponds to momentum transformation,
$p_{\mu}\rightarrow p_{\mu}-qA_{\mu}$. By this transformation, the gauge field
also gets modified and enters the Lagrangian as $GA_{\mu}$. It also introduces
a coupling between the electromagnetic field and matter in a form
$D_{\mu}\psi$. So Eq.(V.1) becomes

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+i\gamma^{\mu}\partial_{\mu}(\ln
G^{1/2})+qA_{\mu}\gamma^{\mu}-mG\right]\psi$
$\displaystyle=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+q\gamma^{\mu}A_{\mu}-mG\right]\psi,$
(30)

consequently, the electromagnetic field transforms as

$A_{\mu}\rightarrow A^{\prime}_{\mu}\equiv
A_{\mu}+\dfrac{i}{q}\partial_{\mu}(\ln G^{1/2}).$ (31)

Equation (V.1) looks similar to Dirac’s Lagrangian with interaction term that
couples the gauge field $A_{\mu}(x)$ to the conserve current
$j^{\mu}=q\bar{\psi}\gamma^{\mu}\psi$ with a modified mass term
$M(r)=mG(\phi)$. Also, $\bar{\psi}D_{\mu}\psi$ becomes invariant under local
phase rotation, granting interaction between $\bar{\psi}$, $\psi$ and
$A_{\mu}$ with momentum $p_{\mu}\rightarrow i\partial_{\mu}$. In this way, the
Lagrangian has been modified, but we ensure that all conservation laws are
duly respected.

Now, to derive a complete Lagrangian that can mimic QCD theory, we include the
kinetic energy term of the gauge field which has been derived in Eq.(18),
thus,

$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}+q\gamma^{\mu}A_{\mu}-mG(\phi)\right)\psi+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)-\frac{1}{4}G(\phi)F_{\mu\nu}F^{\mu\nu}.$
(32)

This expression looks similar to the usual QED Lagrangian with $\phi$ as
intermediate field/particle and a mass term modified by color dielectric
function, $G(\phi)$. As a result, this expression approximates the non-Abelian
QCD theory with an Abelian one. This was motivated by the projection that the
confining regime of QCD is mostly Abelian dominated Hoofta1 ; Ezawa ; Shiba ;
Suzuki ; Sakumichi . The spinor fields $\psi$ and $\bar{\psi}$ represent the
quarks and antiquarks and the modified gauge field $A_{\mu}$ also describes
gluons. All the long-distance behavior of the gluons is absorbed in $G(\phi)$.
The observed mass of the system $M(r)=mG(\phi)$ is expected to have a fixed
value at the beginning of the interaction $r\rightarrow r_{0}$ and at the end
of the interaction $r\rightarrow r_{*}$, so it can be measured precisely. That
is, the dielectric function should be such that $G(\phi(r\rightarrow
r_{0}))=G(\phi(r\rightarrow r_{*}))=\text{constant}$, where $1/r_{0}$ is the
energy at the beginning of the interaction and $1/r_{*}$ is the energy at the
end of the interaction. Thus, $M(r)=mG(\phi)$ is the constituent quark mass
function of the system. We have presented a detailed study of this theory in
Refs.Issifu1 ; Adamu

Since the renormalization theory remains the systematic approach for which UV
divergences can be resolved Neubert , we will compare the result with the
renormalized QED Lagrangian to properly identify the nature of the dielectric
function in the context of renormalization factor $Z(\mu)$, where $\mu$ is a
scale that comes from the dimensional regularization scheme Bollini ; Hoofta .
The objective is to ensure that the result obtained in Eq.(32) does not pose
any UV divergences. When the real QED Lagrangian is written in terms of the
renormalized factors, it takes the form,

$\mathcal{L}_{\text{QED}}=Z_{\psi}\bar{\psi}(i\gamma_{\mu}\partial^{\mu}-m_{0})\psi-\dfrac{Z_{A}}{4}F^{\mu\nu}F_{\mu\nu}-Z_{1}e\bar{\psi}\gamma_{\mu}A^{\mu}\psi,$
(33)

where

$\psi_{0}=Z_{\psi}^{1/2}\psi\quad\text{,}\quad
A_{0}^{\mu}=Z_{A}^{1/2}A^{\mu}\quad\text{,}\quad
m_{0}=\dfrac{Z^{\prime}_{\psi}}{Z_{\psi}}m\quad\text{and}\quad
e=\dfrac{Z_{\psi}}{Z_{1}}Z_{A}^{1/2}e_{0},$ (34)

the two equations bear some resemblance, so we can compare them. With $Z_{3}$
the gluon propagator renormalization factor, $Z_{1}$ the quark-quark-gluon
vertex renormalization, and $Z_{2}$ the quark self-energy renormalization
factor. Additionally, the covariant derivative can be expressed in terms of
the renormalized factors as
$D^{ren}_{\mu}\,=\,\partial_{\mu}-ie(Z_{1}/Z_{\psi})A_{\mu}$, gauge invariance
requires that $Z_{1}\,=\,Z_{\psi}$. We have substituted the ’conventional’
representation of the renormalization factors $Z_{2}$ and $Z_{3}$ with
$Z_{\psi}$ and $Z_{A}$ respectively, and $Z_{1}\,=\,Z_{\psi}Z_{A}^{1/2}$ to
make the distinction more obvious relative to fermions and the gauge field.
The renormalized fields are without subscript ’0’ Itzykson ; Pascual ; Peskin
; Collins ; Weinberg ; Weinberg1 ; Schwartz . Comparing the results in
Eqs.(32) and (33), we identify $Z_{A}\,=\,Z_{\psi}\,=\,G$ and the gauge
invariance warrant that, $Z_{A}\,=\,1$. Thus, in addition to the color
properties carried by the color dielectric function, it also absorbs the UV
divergences. Consequently, the dielectric function follows the restrictions,

$G\left(\phi(r)\right)\rightarrow\begin{cases}1&\text{for}\,\;r\,\rightarrow\,0,\;\text{deconfinement/Coulombian
regime}\\\ 0&\text{for}\,\;r\,\rightarrow\,r_{*},\;\text{confinement
regime}\end{cases}.$ (35)

From the gauge conditions adopted above, a gluon mass term,

$\mathcal{L}_{\gamma}=\dfrac{1}{2}M^{2}(r)A_{\mu}A^{\mu},$ (36)

will not be invariant under the local gauge transformation in Eq.(31) because

$A_{\mu}A^{\mu}\rightarrow
A^{\prime}_{\mu}A^{\prime\mu}\equiv\left(A_{\mu}+\dfrac{i}{q}\partial_{\mu}(\ln
G^{1/2})\right)\left(A^{\mu}+\dfrac{i}{q}\partial^{\mu}(\ln
G^{1/2})\right)\neq A_{\mu}A^{\mu}.$ (37)

Consequently, local gauge invariance accounts for the existence of massless
photons Quigg .

###  V.2 Non-Abelian Gauge Theory

In this section, we will focus on constructing a non-Abelian gauge theory
traditionally used for describing the strong nuclear force. Additionally, we
buy into the projection that proton and neutron have the same mass and are
charge independent due to the strong nuclear force. Therefore, there is an
agreement that isospin is conserved in strong interactions. As in the case of
QED theory, we will base the discussions on $\text{SU}(2)$-isospin gauge
theory introduced by Yang and Mills Yang , elaborated by Shaw Shaw . Here, we
will consider Eq.(24) as the Lagrangian for free nucleons with composite
fermions, protons (p), and neutrons (n)

$\psi\equiv\begin{pmatrix}p\\\ n\end{pmatrix}.$ (38)

The Lagrangian is invariant under global spin rotation

$\psi\rightarrow\psi^{\prime}=e^{i\vec{\tau}\cdot\vec{\alpha}/2}\psi,$ (39)

also, isospin current, $j^{\mu}=\bar{\psi}\gamma^{\mu}({\tau}/{2})\psi$, is
conserved therefore, proton and neutron can be treated symmetrically in the
absence of electromagnetic interactions. In that case, the distinction between
proton and neutron is arbitrary and conventional under this representation.

To maintain the differences between the Abelian theory treated in Sec. V.1 and
the non-Abelian theory intended for this section, we will replace $A_{\mu}$
from the Abelian covariant derivative, $D_{\mu}$, expressed in Eq.(29) with
its non-Abelian counterpart $B_{\mu}$ i.e.,

$D_{\mu}\equiv\partial_{\mu}-igB_{\mu}.$ (40)

We have also replaced $q$ with $g$, the strong coupling constant to make the
analysis distinct from Sec. V.1 and befitting for describing strong
interactions. Despite the similar global gauge invariance satisfied by both
theories, they also exhibit some major differences; the algebra of the non-
Abelian group is more complex and the associated gauge bosons self-interact
due to the structure of the non-Abelian group. Accordingly, the field can be
expressed as,

$B_{\mu}=\dfrac{1}{2}\vec{\tau}\cdot\vec{b}_{\mu}=\dfrac{1}{2}\tau^{a}b^{a}_{\mu}=\dfrac{1}{2}\left({\begin{array}[]{cc}b_{\mu}^{3}&b^{1}_{\mu}-ib^{2}_{\mu}\\\
b^{1}_{\mu}+ib^{2}_{\mu}&-b^{3}_{\mu}\\\
\end{array}}\right)=\dfrac{1}{2}\left({\begin{array}[]{cc}b_{\mu}^{3}&\sqrt{2}\,b^{+}_{\mu}\\\
\sqrt{2}\,b^{-}_{\mu}&-b^{3}_{\mu}\\\ \end{array}}\right),$ (41)

where the three non-Abelian gauge fields are
$\vec{b}_{\mu}=(b_{\mu}^{1},b_{\mu}^{2},b_{\mu}^{3})$. The generators
$\tau^{a}$ $(a=1,\,2$ and $3$) are associated with Pauli’s matrices,
$b_{\mu}^{\pm}=(b_{\mu}^{1}\mp ib_{\mu}^{2})/\sqrt{2}$ are the charged gauge
bosons and the isospin step-up and step-down operators $1/2(\tau_{1}\pm
i\tau_{2})$ exchange $p\leftrightarrow n$ following the absorption of
$b_{\mu}^{\pm}$ boson. The gradient in Eq.(24) will then transform as,

$\displaystyle D_{\mu}\psi\rightarrow D\psi^{\prime}$
$\displaystyle=G^{1/2}\partial_{\mu}\psi+(\partial_{\mu}G^{1/2})\psi-
igB_{\mu}(G^{1/2}\psi)$
$\displaystyle=G^{1/2}\left(\partial_{\mu}-igB^{\prime}_{\mu}\right)\psi,$
(42)

thus,

$\displaystyle-igG^{1/2}B^{\prime}_{\mu}$
$\displaystyle=-igB_{\mu}G^{1/2}+(\partial_{\mu}G^{1/2})\rightarrow$
$\displaystyle B^{\prime}_{\mu}$
$\displaystyle=G^{1/2}B_{\mu}G^{-1/2}+\dfrac{i}{g}(\partial_{\mu}G^{1/2})G^{-1/2}$
$\displaystyle=G^{1/2}\left(B_{\mu}+\dfrac{i}{g}G^{-1/2}(\partial_{\mu}G^{1/2})\right)G^{-1/2}.$
(43)

Again, following similar procedure as adopted for Eq.(V.1) using the necessary
transformations, we obtain

$\displaystyle\mathcal{L}^{\prime}=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+gB_{\mu}\gamma^{\mu}-mG\right]\psi,$
(44)

with gauge invariant transformation similar to Eq.(31),

$B_{\mu}\rightarrow B^{\prime}_{\mu}\equiv
B_{\mu}+\dfrac{i}{g}(\partial_{\mu}\ln G^{1/2}).$ (45)

By comparing Eq.(V.2) and Eq.(45) we can deduce,

$B_{\mu}\rightarrow B^{\prime}_{\mu}=G^{1/2}B_{\mu}G^{-1/2}.$ (46)

The $\text{SU}(2)$-isospin generators can be expressed in terms of commutator
relation,

$\left[\tau^{j},\tau^{k}\right]=2i\varepsilon_{jkl}\tau^{l}.$ (47)

Conventionally, $\tau_{i}$ do not commute with others in different spatial
directions, so only one component can be measured at a time and this is taken
to be the third component $T_{3}=1/2\tau_{3}$, generally in $z$-direction.

We will now develop the field strength tensor that will form the kinetic term
of the gauge field. Starting with the electromagnetism gauge, we can
construct,

$F_{\mu\nu}=\dfrac{1}{2}{\bf
F}_{\mu\nu}\cdot{\bf\tau}=\dfrac{1}{2}F^{a}_{\mu\nu}\tau^{a}\qquad{\text{where}}\qquad
tr\left(\tau^{a}\tau^{b}\right)=2\delta^{ab}.$ (48)

From these transformations, we can develop a gauge invariant field,

$\mathcal{L}^{\prime}_{gauge}=-\dfrac{1}{4}{\bf F}_{\mu\nu}\cdot{\bf
F}^{\mu\nu}=-\dfrac{1}{2}tr\left(F_{\mu\nu}F^{\mu\nu}\right),$ (49)

where the field strength tensor transforms under the dielectric function as

$F_{\mu\nu}\rightarrow F^{\prime}_{\mu\nu}\equiv G^{1/2}F_{\mu\nu}G^{-1/2}.$
(50)

We know from the QED relation constructed in Sec. V.1 that the field strength
can be expressed as

$\displaystyle F^{\prime}_{\mu\nu}$
$\displaystyle=\partial_{\nu}B^{\prime}_{\mu}-\partial_{\mu}B^{\prime}_{\nu}=\partial_{\nu}\left[G^{1/2}B_{\mu}G^{-1/2}+\dfrac{i}{g}(\partial_{\mu}G^{1/2})G^{-1/2}\right]-\partial_{\mu}\left[G^{1/2}B_{\nu}G^{-1/2}+\dfrac{i}{g}(\partial_{\nu}G^{1/2})G^{-1/2}\right]$
$\displaystyle=\left\\{(\partial_{\nu}G^{1/2})B_{\mu}G^{-1/2}+G^{1/2}(\partial_{\nu}B_{\mu})G^{-1/2}+G^{1/2}B_{\mu}(\partial_{\nu}G^{-1/2})+\dfrac{i}{g}\left[\partial_{\nu}(\partial_{\mu}G^{1/2})G^{-1/2}+(\partial_{\mu}G^{1/2})(\partial_{\nu}G^{-1/2})\right]\right\\}$
$\displaystyle-\left\\{(\partial_{\mu}G^{1/2})B_{\nu}G^{-1/2}+G^{1/2}(\partial_{\mu}B_{\nu})G^{-1/2}+G^{1/2}B_{\nu}(\partial_{\mu}G^{-1/2})+\dfrac{i}{g}\left[\partial_{\mu}(\partial_{\nu}G^{1/2})G^{-1/2}+(\partial_{\nu}G^{1/2})(\partial_{\mu}G^{-1/2})\right]\right\\}.$
(51)

We adopted the expression for $B^{\prime}_{\mu}$ defined in Eq.(V.2),
regrouping the terms in the above equation yields,

$\displaystyle F^{\prime}_{\mu\nu}$
$\displaystyle=G^{1/2}\left[\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right]G^{-1/2}+\left[(\partial_{\nu}G^{1/2})B_{\mu}-(\partial_{\mu}G^{1/2})B_{\nu}\right]G^{-1/2}+G^{1/2}\left[B_{\mu}(\partial_{\nu}G^{-1/2})-B_{\nu}(\partial_{\mu}G^{-1/2})\right]$
$\displaystyle+\dfrac{i}{g}\left[(\partial_{\mu}G^{1/2})(\partial_{\nu}G^{-1/2})-(\partial_{\nu}G^{1/2})(\partial_{\mu}G^{-1/2})\right]$
$\displaystyle\neq
G^{1/2}\left[\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right]G^{-1/2},$ (52)

higher derivative terms were discarded. We can cast this result in a slightly
symmetric form by using the identity $G^{1/2}G^{-1/2}=\mathbb{I}$, so

$\displaystyle\partial_{\mu}(G^{1/2}G^{-1/2})$
$\displaystyle=(\partial_{\mu}G^{1/2})G^{-1/2}+G^{1/2}(\partial_{\mu}G^{-1/2})=0\rightarrow$
$\displaystyle G^{1/2}(\partial_{\mu}G^{-1/2})$
$\displaystyle=-(\partial_{\mu}G^{1/2})G^{-1/2}.$ (53)

Using this identity appropriately leads to,

$\displaystyle F^{\prime}_{\mu\nu}$
$\displaystyle=G^{1/2}\left(\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right)G^{-1/2}+\left((\partial_{\nu}G^{1/2})B_{\mu}-B_{\mu}(\partial_{\nu}G^{1/2})\right)G^{-1/2}+\left(B_{\nu}(\partial_{\mu}G^{1/2})-(\partial_{\mu}G^{1/2})B_{\nu}\right)G^{-1/2}$
$\displaystyle+\dfrac{i}{g}\left((\partial_{\nu}G^{1/2})G^{-1/2}(\partial_{\mu}G^{1/2})-(\partial_{\mu}G^{1/2})G^{-1/2}(\partial_{\nu}G^{1/2})\right)G^{-1/2}$
$\displaystyle=G^{1/2}\left(\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right)G^{-1/2}+G^{1/2}\left\\{\left[G^{-1/2}(\partial_{\nu}G^{1/2}),B_{\mu}\right]-\left[G^{-1/2}(\partial_{\mu}G^{1/2}),B_{\nu}\right]\right\\}G^{-1/2}$
$\displaystyle+\dfrac{i}{g}G^{1/2}\left[G^{-1/2}(\partial_{\nu}G^{1/2}),G^{-1/2}(\partial_{\mu}G^{1/2})\right]G^{-1/2}.$
(54)

This equation shows the additional terms that come from the non-vanishing
commutators due to the non-Abelian group structure. By this expression, we
deduce that a term can be added to
$\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}$ to modify $F_{\mu\nu}$ to
achieve the desired transformation property we require. With this inspiration,
the observed electromagnetic field strength tensor can be modified to read,

$F_{\mu\nu}=\dfrac{1}{iq}\left[D_{\nu},D_{\mu}\right].$ (55)

Substituting the definition for $D_{\mu}$ in Eq.(29) into the above
expression, we obtain

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+iq\left[A_{\nu},A_{\mu}\right].$
(56)

The commutator vanishes for Abelian theories. Applying this transformation to
Eq.(50) yields,

$F^{\prime}_{\mu\nu}=\partial_{\nu}B^{\prime}_{\mu}-\partial_{\mu}B^{\prime}_{\nu}+ig\left[B^{\prime}_{\nu},B^{\prime}_{\mu}\right].$
(57)

Expanding the commutator using the transformation Eq.(V.2) to enable us to
compare the outcome to the nonvanishing commutator relations in Eq.(V.2) leads
to,

$\displaystyle ig\left[B^{\prime}_{\nu},B^{\prime}_{\mu}\right]$
$\displaystyle=ig\left[\left(G^{1/2}B_{\nu}G^{-1/2}+\dfrac{i}{g}(\partial_{\nu}G^{1/2})G^{-1/2}\right),\left(G^{1/2}B_{\mu}G^{-1/2}+\dfrac{i}{g}(\partial_{\mu}G^{1/2})G^{-1/2}\right)\right]$
$\displaystyle=igG^{1/2}\left[B_{\nu},B_{\mu}\right]G^{-1/2}-G^{1/2}\left\\{\left[G^{-1/2}(\partial_{\nu}G^{1/2}),B_{\mu}\right]-\left[G^{-1/2}(\partial_{\mu}G^{1/2}),B_{\nu}\right]\right\\}G^{-1/2}$
$\displaystyle-\dfrac{i}{g}G^{1/2}\left[G^{-1/2}(\partial_{\nu}G^{1/2}),G^{-1/2}(\partial_{\mu}G^{/2})\right]G^{-1/2}.$
(58)

The commutator relations that come after the first term in the last step are
the exact terms required to cancel the extra terms in Eq.(V.2). Therefore, the
field strength tensor expressed in Eq.(57) has the required structure under
local gauge transformation. Combining Eqs.(44) and (49) gives rise to modified
Yang-Mills Lagrangian Quigg ; Yang ,

$\mathcal{L}_{\text{YM}}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}+g\gamma^{\mu}B_{\mu}-mG\right)\psi-\dfrac{1}{2}tr\left(F_{\mu\nu}F^{\mu\nu}\right),$
(59)

where $M(r)=mG(\phi)$ is the modified nucleon mass. This Lagrangian is also
invariant under local gauge transformations and does not permit the existence
of mass term $M^{2}(r)B_{\mu}B^{\mu}$. It should also be noted that
introducing the non-Abelian gauge leads to the automatic cancellation of the
color dielectric function. Hence, we can infer that the color dielectric
function attached to the Abelian gauge induces strong interaction properties.
Using the expression in Eq.(41) and the commutator relation in Eq.(47), we can
rewrite the transformed version of Eq.(57) as

$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}+ig\left[B_{\mu},B_{\nu}\right]\rightarrow$
$\displaystyle F^{l}_{\mu\nu}$
$\displaystyle=\dfrac{1}{2}\left(\partial_{\mu}(\tau^{l}b^{l}_{\nu})-\partial_{\nu}(\tau^{l}b^{l}_{\mu})\right)+\dfrac{ig}{4}\left(-2i\varepsilon_{jkl}(b^{j}_{\nu}b^{k}_{\mu}\tau^{l})\right)$
$\displaystyle=\partial_{\mu}b^{l}_{\nu}-\partial_{\nu}b^{l}_{\mu}+g\varepsilon_{jkl}b^{j}_{\nu}b^{k}_{\mu},$
(60)

in the last step we have dropped the three isospin generators $\tau^{l}$ of
the gauge field because they are linearly independent. Generally, non-Abelian
gauge groups that do not fall under $\text{SU}(2)$, the Levi-Civitá symbol
$\varepsilon_{jkl}$ is replaced with the antisymmetric structure constant
$f_{jkl}$.

The introduction of the $\text{SU}(2)$-isospin symmetry by Heisenberg
Heisenberg  preceded the development of the quark model. That notwithstanding,
the only known fundamental components of the nucleon describing strong
interactions are up-quark ($u$) and down-quark ($d$). While the proton is
composed of two $u$-quarks and a $d$-quark, the neutron is also composed of
two $d$-quarks and an $u$-quark. Indeed, these particles remain the only known
constituents of proton and neutron with almost the same mass and coupling
force. Hence, Eq.(59) can be used to study the behavior of quarks and gluons
inside hadrons. In that case, the up/down quarks are treated as having the
same mass and $0$-charge, so they are seen as similar particles with different
isospin states, $I_{3u/d}=\pm 1/2$, same as the nucleon. Interestingly, the
spin addition of the quark constituents of proton and neutron agree with
$I_{3}=1/2$ and $I_{3}=-1/2$ respectively. Similar to the isospin
representation of the nucleon field $\psi$ in Eq.(38) the quark field can be
represented as

$\psi_{q}\equiv\begin{pmatrix}u\\\ d\end{pmatrix}.$ (61)

Nevertheless, the mathematical structure of this theory is the same as QCD,
the theory that describes the characteristics of quarks and gluons inside the
hadrons.

Again, the strong interaction is well known for its invariance under quark
color permutations, so we can express the quark wave function for color
triplet representation as

$\psi_{q}\equiv\begin{pmatrix}R\\\ G\\\ B\end{pmatrix},$ (62)

where $\text{R}\equiv\text{red}$, $\text{G}\equiv\text{green}$ and
$\text{B}\equiv\text{blue}$. This is invariant under $\text{SU}(3)$
transformation of the form

$\psi\rightarrow\psi^{\prime}\equiv\exp\left(i\dfrac{1}{2}\lambda_{j}\alpha_{j}\right)\psi,$
(63)

where $\alpha_{j},\,j=1,\,2,\,3,\cdots,8$ are the eight phase angles and
$\lambda_{j}$ is a $3\times 3$ matrix representing eight independent traceless
Hermitian generators of the group. Ignoring the differences in quark masses,
it can also be applied in studying u, d and s quark systems,

$\psi_{q}\equiv\begin{pmatrix}u\\\ d\\\ s\end{pmatrix}.$ (64)

The generators are fundamentally equivalent to Pauli’s matrices for the
$\text{SU}(2)$ representation and they satisfy the Lie algebra,

$\left[\lambda_{i},\lambda_{j}\right]=2if_{ijk}\lambda_{k},$ (65)

similar to Eq.(47), $f_{ijk}$ is the structure constant. Consequently, to
generalize the relation in Eq.(V.2) for QCD under the $\text{SU}(3)$ color
symmetry, we substitute the antisymmetric tensor $\varepsilon_{ijk}$ for the
antisymmetric structure constant $f_{ijk}$ Collins1 .

Finally, the models exhibit the expected asymptotic free Gross ; Politzer
behavior at high energy regions while at low energies color confinement and
hadronization sets in. Here, $M(r)$ represents constituent quark mass function
Atkinson  while $m$ is the bare quark mass. As analyzed in Sec. V.1, the
constituent mass of the quarks in the asymptotically free region can be
determined as $M(r\rightarrow r_{0})=M_{0}$, while the constituent mass at the
nonperturbative (low energy) region where color confinement is expected, will
be $M(r\rightarrow r_{*})=M_{*}$. Hence, it is possible to have the same
constituent quark mass in both the UV ($r\rightarrow r_{0}$) and the IR
($r\rightarrow r_{*}$) regimes, if indeed the bare quark mass $m$ in both
regimes are the same Adamu ; Eichten , because the $G(\phi)$ behaves as,
$G(\phi(r\rightarrow r_{0}))=G(\phi(r\rightarrow r_{*}))=\text{constant}$.
However, there is evidence that $m$ is small in the IR regime and large in the
UV regime Adamu ; Issifu1 . Accordingly, if we require color confinement in
both regions, $M_{0}\,>\,M_{*}$ because higher bare quark mass $m$ is required
to obtain confinement in the UV region than it is required in the IR region.

##  VI Phenomenology of Glueball Confinement

In this section we will use one of the models built in Sec. IV specifically,
Eq.(18) to build a model that describes the features of confining glueballs.
Apart from the insight into the behavior of the glueballs in both the IR and
the UV regions, it will serve as a test to the models developed earlier. The
model will be based on electric field confinement, commonly referred to as the
chromoelectric flux confinement. In that light, we will define the indices of
the gauge field such that the chromomagnetic flux is eliminated from the
system (i.e. $j^{\nu}=(\rho,\,\vec{0})$) and only the static sector of the
scalar field, i.e. $\mu=j$, is available for analysis. This ensures that the
color particles that generate the gluons are static. This section will enable
us to see how the Abelian gauge can be used to approximate a non-Abelian
theory. Additionally, the color dielectric function $G(\phi)$ absorbs the
long-distance dynamics of the gluons such that the photon propagator emanating
from the Abelian gauge field does not decouple at longer wavelengths. We will
further demonstrate that the $G(\phi)$ is directly related to the QCD
$\beta$-function and the strong running coupling constantly.

###  VI.1 The Model

The equations of motion for Eq.(18) are,

$\partial_{\mu}\partial^{\mu}\phi+\dfrac{1}{4}\dfrac{\partial
G(\phi)}{\partial\phi}F_{\mu\nu}F^{\mu\nu}+\dfrac{\partial
V}{\partial\phi}=0,$ (66)

and

$\partial_{\mu}[G(\phi)F^{\mu\nu}]=0.$ (67)

Expressing the above equations in spherical coordinates,

$\dfrac{1}{r^{2}}\dfrac{d}{dr}\left(r^{2}\dfrac{d\phi}{dr}\right)=-\dfrac{1}{2}\dfrac{\partial
G(\phi)}{\partial\phi}E^{2}+\dfrac{\partial V(\phi)}{\partial\phi},$ (68)

where, $F^{\mu\nu}F_{\mu\nu}=F^{0j}F_{0j}+F^{i0}F_{i0}=-2E^{2}$ (only electric
field components are considered) and $V=G$ as discussed in Sec. IV.
Accordingly,

$\displaystyle\dfrac{1}{r^{2}}\dfrac{d}{dr}\left(r^{2}\dfrac{d\phi}{dr}\right)$
$\displaystyle=\dfrac{\partial}{\partial\phi}\left[\dfrac{\lambda^{2}}{2}\dfrac{1}{V(\phi)}\dfrac{1}{r^{4}}+V\right]\rightarrow$
$\displaystyle\nabla^{2}\phi$
$\displaystyle=\dfrac{\partial}{\partial\phi}\left[\dfrac{\lambda^{2}}{2}\dfrac{1}{V(\phi)}\dfrac{1}{r^{4}}+V\right],$
(69)

where

$E=\dfrac{\lambda}{r^{2}G(\phi)},$ (70)

and

$\lambda=\dfrac{q}{4\pi},$ (71)

$\lambda$ is an integration constant, we also substituted $\varepsilon_{0}=1$
to achieve the desired objective. Now we choose a potential that satisfies all
the conditions expressed in Sec. IV thus,

$V(\phi)=\dfrac{\rho}{4}[(\alpha\phi)^{2}-a^{2}]^{2},$ (72)

where $\rho$ and $\alpha=1/f_{\alpha}$ are dimensionless constants,
$f_{\alpha}$ is the tachyon decay constant. This potential contains tachyonic
mode at $V^{\prime\prime}(0)$, so the fields cannot be quantized around this
point — see Sec. II.1 for detailed discussions. However, we can remove the
tachyonic modes by shifting the vacuum $\phi\rightarrow\phi_{0}+\eta$ and
quantizing around the true minimum, $\phi_{0}=\pm a/\alpha$. Consequently,

$m_{\phi}^{2}=\dfrac{\partial^{2}V}{\partial\phi^{2}}\bigg{|}_{\phi=\phi_{0}}=2\rho\alpha^{2}a^{2},$
(73)

and the potential can be expanded for the small perturbation $\eta$, which
will be referred to as a glueball field with a real square mass,
$m_{\phi}^{2}$, hence,

$\displaystyle V(\eta)=G(\eta)$
$\displaystyle=V(\phi)\bigg{|}_{\phi_{0}}+\dfrac{\partial
V}{\partial\phi}\bigg{|}_{\phi_{0}}\eta+\dfrac{1}{2}\dfrac{\partial^{2}V}{\partial\phi^{2}}\bigg{|}_{\phi_{0}}\eta^{2}$
$\displaystyle=\dfrac{1}{2}m_{\phi}^{2}\eta^{2}.$ (74)

Considering that the particles are sufficiently separated such that color
confinement can be observed, we ignore the $1/r^{4}$ term in Eq.(VI.1) and
simplify it as

$\displaystyle\nabla^{2}(\eta)$ $\displaystyle=\dfrac{\partial
V}{\partial\phi}\bigg{|}_{\phi_{0}}+\dfrac{\partial^{2}V}{\partial\phi^{2}}\bigg{|}_{\phi_{0}}\eta\rightarrow$
$\displaystyle\nabla^{2}\eta$
$\displaystyle=\dfrac{\partial^{2}V}{\partial\phi^{2}}\bigg{|}_{\phi_{0}}\eta,$
(75)

leading to

$\eta^{\prime\prime}(r)+\dfrac{2}{r}\eta^{\prime}(r)-m^{2}_{\phi}\eta=0.$ (76)

This equation has several solutions but we choose two of such solutions
suitable for the analysis,

$\eta(r)=\dfrac{a\cosh(m_{\phi}r)}{\alpha
m_{\phi}r}\qquad\text{and}\qquad\eta(r)=\dfrac{a\sin(m_{\phi}r)}{\alpha
m_{\phi}r}.$ (77)

Each solution corresponds to the characteristics of the particle in a
particular regime i.e., IR and UV regimes respectively. The solution at the IR
regime will give rise to a linear confining potential and the UV solution will
lead to a Cornell-like potential.

###  VI.2 Confining Potentials

In this section we will present the confining potentials derived from the
model by considering the electrodynamic potential,

$V=\mp\int Edr.$ (78)

Substituting the equation at the left side of the solution Eq.(77) and Eq.(70)
into Eq.(78) leads to

$\displaystyle V_{c}(r)$
$\displaystyle=\dfrac{2\lambda\alpha^{2}\tanh(m_{\phi}r)}{a^{2}m_{\phi}}+c$
$\displaystyle=m_{\phi}\lambda\tanh(m_{\phi}r)\qquad\text{for}\qquad
a^{4}\rho=1$ (79)

where $c$ is an integration constant that is set to zero in the last step.
Considering that $m_{\phi}r\ll 1$, $c=0$ and $\lambda=1$ corresponding to the
positive part of the potential $V$, we can deduce the QCD string tension
$\sigma$ to be,

$\sigma_{L}={m_{\phi}^{2}}.$ (80)

Here, we can infer that confinement is occasioned by the magnitude of the
glueball mass, in the limit of vanishing glueball mass there will be no
confinement. Furthermore, this potential leads to linear growth in $r$, and at
some critical distance, $r_{*}=1/\sqrt{\sigma_{L}}$ Bali  the potential begins
to flatten up leading to hadronization. It is known from the flux tube models
for confining color particles that, $r\gg r_{*}$ Bali .

Figure 1: Flux tube-like potential

Increase in distance $r$ increases the strength of confinement until $r\geq
r_{*}$ where the curve is expected to start flattening, signaling
hadronization.

On the other hand, taking the solution at the right side of Eq.(77) and
following the same process as followed above, we get

$\displaystyle V_{s}(r)$
$\displaystyle=-\dfrac{2\lambda\alpha^{2}\cot(m_{\phi}r)}{a^{2}m_{\phi}}+c$
$\displaystyle\simeq-\dfrac{2\lambda\alpha^{2}}{a^{2}m_{\phi}}\left[\dfrac{1}{m_{\phi}r}-\dfrac{m_{\phi}r}{3}-{\cal
O}(r^{3})\right]+c$
$\displaystyle\simeq\dfrac{2\lambda\rho\alpha^{2}a^{2}}{\rho
a^{4}m^{2}_{\phi}}\left[-\dfrac{1}{r}+\dfrac{m_{\phi}^{2}r}{3}\right],$ (81)

in the last step, we set the integration constant $c=0$, also choosing the
positive part of the potential corresponding to $\lambda=1$ and $\rho
a^{4}=1$, we arrive at the Cornell-like potential for confining heavy quarks
i.e.,

$V_{s}(r)=-\dfrac{1}{r}+\dfrac{m^{2}_{\phi}r}{3},$ (82)

corresponding to a string tension

$\sigma_{s}=\dfrac{m_{\phi}^{2}}{3}.$ (83)

It is important to recognize that the critical distance,
$r_{*s}=1/\sqrt{\sigma_{s}}$, in this regime marks the transition from the
asymptotic freedom region to the confining region.

Figure 2: A graph of Cornell-like potential

An increase in the distance also leads to an increase in the strength of
confinement.

We know from Sec. III that the string tension
$T_{\text{string}}\sim\sigma_{L}\sim\sigma_{s}\sim 1\,\text{GeV}/\text{fm}$,
hence the glueball mass in the IR regime becomes $m_{L}\approx 1\,\text{GeV}$
corresponding to glueball mass of isoscalar resonance $f_{0}(980)$ Tanabashi .
The commonly known ratio of $m(0^{++})/\sqrt{\sigma_{L}}$ in QCD theory in
$\text{SU}(\infty)$ limit Albanese ; Bacilieri ; Teper , at this regime, can
be determined as $m_{L}/\sqrt{\sigma_{L}}\approx 1$. Likewise in the UV
regime, we have a glueball of mass $m_{s}\approx 1.73\,\text{GeV}$
corresponding to the lightest scalar glueball mass of resonance $f_{0}(1710)$.
The result obtained here is precisely the same as the results obtained from
QCD lattice calculations Tanabashi ; Morningstar ; Loan ; Chen ; Lee1 ; Bali1
also, $m_{s}/\sqrt{\sigma_{s}}\approx 1.73$. The critical distances become
$r_{*s}=r_{*}=1\,\text{fm}$ for both the IR and the UV regimes. While critical
distance in the IR regime $r_{*}$ refers to the transition from confinement to
hadronization regions, critical distance in the UV regime $r_{*s}$ refers to
the transition from the asymptotically free region to the confining region.

###  VI.3 Gluon Condensation

Classical theory for gluodynamics is invariant under the scale transformation
$x\rightarrow\lambda x$, this leads to a scale current $s_{\mu}$ which is
related to the energy momentum tensor trace $\theta^{\mu}_{\mu}(x)$ as

$\partial^{\mu}s_{\mu}=\theta^{\mu}_{\mu}.$ (84)

In the absence of quantum corrections, $\theta^{\mu}_{\mu}=0$, the theory
remains conformally invariant. This will lead to a vanishing gluon
condensation $\langle F_{\mu\nu}^{a}F^{a\mu\nu}\rangle=0$. On the other hand,
when quantum correction, $-|\varepsilon_{v}|$, is introduced, the conformal
symmetry is broken leading to non-vanishing gluon condensate $\langle
F_{\mu\nu}^{a}F^{a\mu\nu}\rangle\neq 0$ and energy-momentum trace anomaly
comes to play

$\theta^{\mu}_{\mu}=\dfrac{\beta(g)}{2g}F^{a}_{\mu\nu}F^{a\mu\nu},$ (85)

with vacuum expectation,

$\langle\theta^{\mu}_{\mu}\rangle=-4|\varepsilon_{v}|.$ (86)

The leading term of the QCD $\beta$-function is known to be,

$\beta=-\dfrac{11g^{3}}{(4\pi)^{2}}.$ (87)

Now, calculating the energy-momentum tensor trace of Eq.(18) for the glueball
field $\eta$ using the relation,

$\theta^{\mu}_{\mu}=4V(\eta)+\eta\square\eta,$ (88)

we get,

$\displaystyle\theta^{\mu}_{\mu}$ $\displaystyle=4V(\eta)-\eta\dfrac{\partial
G}{\partial\eta}F^{\mu\nu}F_{\mu\nu}-\eta\dfrac{\partial V}{\partial\eta}$
$\displaystyle=4\tilde{V}-\eta G^{\prime}(\eta)F^{\mu\nu}F_{\mu\nu},$ (89)

where $G^{\prime}$ and $V^{\prime}$ represent first derivative with respect to
$\eta$ and $\tilde{V}(\eta)=V(\eta)-\eta V^{\prime}(\eta)/4$. Also, rescaling
$\tilde{V}(\eta)$ with the energy density $-|\varepsilon_{v}|$ i.e.
$\tilde{V}(\eta)\rightarrow-|\varepsilon_{v}|\tilde{V}(\eta)$ together with
the vacuum expectation value in Eq.(86) we get,

$\left\langle\eta
G^{\prime}(\eta)F_{\mu\nu}F^{\mu\nu}\right\rangle=4|\varepsilon_{v}|\langle
1-\tilde{V}\rangle,$ (90)

with this equation, we recover the classical result in the limit
$|\varepsilon_{v}|\rightarrow 0$ i.e. $\langle F_{\mu\nu}F^{\mu\nu}\rangle=0$.
Using the potential expressed in Eq.(VI.1) we can determine,

$\displaystyle\tilde{V}$ $\displaystyle=V-\dfrac{\eta V^{\prime}}{4}$
$\displaystyle=\dfrac{m_{\phi}^{2}\eta^{2}}{4},$ (91)

as a result, Eq.(90) can be expressed as

$\left\langle
2G(\eta)F_{\mu\nu}F^{\mu\nu}\right\rangle=4|\varepsilon_{v}|\left\langle
1-\dfrac{m^{2}_{\phi}\eta^{2}}{4}\right\rangle,$ (92)

we can identify the gluon mass Issifu2 ,

$m_{A}^{2}=\dfrac{m_{\phi}^{2}}{4}.$ (93)

Furthermore, taking the expectation value of Eq.(66) in terms of the glueball
field $\eta$, we can express

$\displaystyle\left\langle\dfrac{m^{2}_{\phi}\eta}{4}F^{\mu\nu}F_{\mu\nu}\right\rangle-\left\langle
m^{2}_{\phi}\eta\right\rangle=0,$ (94)

consequently, the mean glueball field $\bar{\eta}$ has two possible solution
i.e. $\bar{\eta}=0$ and $\bar{\eta}=1$. Higher glueball condensate corresponds
to $\bar{\eta}=0$ whilst lower glueball condensate corresponds to
$\bar{\eta}=1$ Carter .

Figure 3: Gluon condensation

An increase in the mean glueball field $\bar{\eta}$ decreases the gluon
condensate until it vanishes at maximum $\bar{\eta}$.

###  VI.4 Strong Running Coupling $\alpha_{s}$ and QCD $\beta$-Function

Comparing Eqs.(85) and (VI.3), we can relate

$\displaystyle\dfrac{\beta(g)}{2g}=-\eta G^{\prime}(\eta)\rightarrow$
$\displaystyle\beta(1/r^{2})=-2g\eta G^{\prime}(\eta).$ (95)

We can also extract the strong running coupling $\alpha_{s}$ using the
renormalization group theory Deur ,

$\displaystyle\beta(Q^{2})=Q^{2}\dfrac{d\alpha_{s}}{dQ^{2}},$ (96)

comparatively;

$\beta(\eta)\simeq-\eta\dfrac{d(G)}{d\eta}=-m_{\phi}^{2}\eta^{2}(r)=\beta(1/r^{2})\qquad\text{we
set}\qquad g=1.$ (97)

Therefore, the strong coupling can be identified as
$\alpha_{s}(\eta)=G(\eta)=\alpha_{s}(1/r^{2})$. QCD $\beta$-function is
naturally a negative quantity showing the asymptomatic freedom nature of the
strong coupling. It also reveals the anti-screening behavior of the theory at
higher energies. The strong running coupling, on the other hand, gives an
insight into the growing precision of hadron scattering experiments at high
energy limits. And at low energy limits, within the scale of hadron mass, it
enhances understanding of hadron structure, color confinement, and
hadronization. Now, substituting the solution of the glueball field $\eta$ and
expanding it for $r\rightarrow 0$ we obtain,

$\displaystyle\alpha_{s}(1/r^{2})=\left[1-\dfrac{m_{\phi}^{2}r^{2}}{3}\right].$
(98)

Also, we can associate the spacelike momentum $Q$ with $Q\equiv 1/r$, then

$\alpha_{s}(Q^{2})=\left[1-\dfrac{m_{\phi}^{2}}{3Q^{2}}\right].$ (99)

In terms of the four-vector momentum i.e. $Q^{2}\equiv-q^{2}$, the strong
coupling becomes

$\alpha_{s}(q^{2})=\left[1+\dfrac{m_{\phi}^{2}}{3q^{2}}\right],$ (100)

and the $\beta$-function becomes,

$\beta(q^{2})=-2\left[1+\dfrac{m_{\phi}^{2}}{3q^{2}}\right].$ (101)

We observe that in the limit $q^{2}\rightarrow 0$ the strong coupling shows a
singularity, generally referred to as the Landau singularity. It marks the
failure of perturbative QCD. The singularity is attributed to the self-
interacting gluons with hadron degrees of freedom in the IR regime leading to
color confinement Olive . In that case, the gluons dynamically acquire mass at
$q^{2}\rightarrow 0$ i.e. $q^{2}\cong m_{A}^{2}$ Badalian ; Yu  which
increases the coupling infinitely. Hence, the singularity can be removed by
fixing a freezing point Badalian ; Badalian1  to the strong running coupling
at $q^{2}\cong m_{A}^{2}$ i.e.,

$\alpha_{s}(q^{2})=\left[1+\dfrac{m_{\phi}^{2}}{3(q^{2}+m^{2}_{A})}\right],$
(102)

and

$\beta(q^{2})=-2\left[1+\dfrac{m_{\phi}^{2}}{3(q^{2}+m^{2}_{A})}\right].$
(103)

Thus, the ’so called’ gluon mass is more pronounced at $q^{2}\rightarrow 0$
and its effect gradually fades off in the limit where $q^{2}\rightarrow\infty$
Cornwall1 . A recent analysis of this subject is contained in Deur .

(a) Left Panel (b) Right Panel

Figure 4: Strong Running Couple $\alpha_{s}$ (left) and $\beta$-Function
(right) against $q$, with a Landau Ghost Pole

The graphs show an unphysical behavior at $q\rightarrow 0$, this is due to the
presence of dynamically generated gluon mass. The self-interacting gluons and
the strong force that exist between them are capable of creating bound states
with hadron degrees of freedom. These graphs depict the behavior of
$\alpha_{s}$ and $\beta$ observed from pQCD.

(a) Left Panel (b) Right Panel

Figure 5: Strong Running Coupling $\alpha_{s}$ (left) and $\beta$-Function
(right) against $q$ with a freezing point at $q\rightarrow 0$

Here, the Landau singularity has been fixed by introducing the gluon mass
$m_{A}$. The presence of the gluon mass is more pronounced at $q\rightarrow 0$
and gradually vanishes in the limit $q\rightarrow\infty$. Consequently,
$\alpha(0)\simeq 2.3$ and $\beta(0)\simeq-4.7$ from the graphs.

##  VII Conclusion

We modified the DBI action to develop models that are capable of mimicking the
phenomenon of QCD theory using an Abelian gauge field. The models were based
on the behavior of opened string with its endpoints on the D$p$-brane. In
studying color particles, the endpoints of the string serve as the source and
sink of the color charges. Additionally, the models are efficient in
investigating glueballs when the tachyons condense and transform into
glueballs with real square masses that keep them confined. Without fermions,
the models are suitable for studying the bound states of gluons and the
dynamics of glueballs. To study the dynamics of quarks, we showed how the
model can be coupled with standard model fermions systematically. Here, the
particles involved are glueball-fermion-mix in a confined state. Moreover, we
demonstrated that the color dielectric function coupled with the Abelian gauge
capable of causing color confinement vanishes automatically when we introduced
the non-Abelian gauge field in Sec. V.2. Consequently, the presence of
$G(\phi)$ coupled with the Abelian gauge field was to induce non-Abelian
characteristics.

We also developed one of the models to demonstrate its ability to explain some
basic characteristics of strong interactions. We derived the linear and
Cornell-like potentials that are used to describe color particles in
phenomenological QCD. The linear potential is motivated by the string model of
hadrons whilst the Cornell potential is motivated by Lattice QCD calculations.
The Cornell potential is particularly important in QCD because it shows both
the asymptotic freedom and color-confining behavior exhibited by the model. We
also calculated the strong running $\alpha_{s}$ coupling and the QCD
$\beta$-function and compared their behavior with the traditional QCD theory.
However, in the model framework, we are able to fix the non-physical Landau
ghost pole that occurs at the low energy region of the model by assuming the
existence of gluon mass $m_{A}$ at low momentum region, $q^{2}\sim m^{2}_{A}$.
The model leads to the determination of gluon condensate and how the glueball
fields contribute to the condensate.

Furthermore, the models can be discretized using path integral formalism and
investigated under lattice field theory with the availability of the required
computational artifacts. As observed in the model developed in Sec. VI, the
linear and the Cornell-like potentials can be used to study hadron and
quarkonia spectra. Other hadron properties can also be studied from these
models when the appropriate spin contributions to the potential are added.
Extending the models to study the characteristics of particles at a finite
temperature will pave way for understanding, chiral symmetry breaking and
restoration, confinement/deconfinement, and quark-gluon-plasma phase
transitions. Finally, the models can be applied in investigating physical
systems such as pions.

###### Acknowledgements.

This work was supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) project No.: 168546/2021-3, Brazil. F.A.B. would like to
thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant No. 165/2018), for partial
financial support. F.A.B. also acknowledges support from CNPq (Grant No.
312104/2018-9), Brazil.

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