# An Effective Model for Glueballs and Dual Superconductivity at Finite
Temperature

Adamu Issifu<EMAIL_ADDRESS>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

The glueballs lead to gluon and QCD monopole condensations as by-products of
color confinement. A color dielectric function $G(|\phi|)$ coupled with
Abelian gauge field is properly defined to mediate the glueball interactions
at confining regime after spontaneous symmetry breaking (SSB) of the gauge
symmetry. The particles are expected to form through quark gluon plasma (QGP)
hadronization phase where the free quarks and gluons start clamping together
to form hadrons. The QCD-like vacuum
$\langle\eta^{2}m_{\eta}^{2}F^{\mu\nu}F_{\mu\nu}\rangle$, confining potential
$V_{c}(r)$, string tension $\sigma$, penetration depth $\lambda$,
superconducting and normal monopole densities ($n_{s}\,n_{n}$) and the
effective masses ($m_{\eta}^{2}$ and $m_{A}^{2}$) will be investigated at
finite temperature $T$. We also calculate the strong ‘running’ coupling
$\alpha_{s}$ and subsequently the QCD $\beta$-function. Dual superconducting
nature of the QCD vacuum will be investigated based on monopole condensation.

##  I Introduction

The Large Hadron Collider (LHC) is currently attracting the attention of
particle physicists to investigating the dynamics of the Standard Model (SM)
at the TeV scale and to probe for a possible new physics. One of the major
expectations at the LHC is the observation of jet substructure. Generally,
jets are collimated bundles of hadrons constituted by quarks and gluons at
short distances or high energies Marzani ; Ali ; Kogler . Jets played a
significant role in the discovery of gluons (g) JADE ; PLUTO ; MARK ; TASSO
and top quark (t) DO ; CDF . These observations featured prominently in
classifying Quantum Chromodynamics (QCD) as a theory for strong interaction
within the standard model. The most abundant secondary particles produced in
heavy ion collisions are pions which emerge from hot and dense hadron gas
states. These pions can be studied from the projected quark-gluon-plasma (QGP)
and the deconfinement phase through emitted photons and dileptons produced at
this phase. Since the photon and the dilepton do not interact with the hadron
matter through strong force, they decay quickly Rapp ; Li ; Schulze ; Volkov
making it easy to study the pions. Due to the large production of jets and
pions at the LHC and Relativistic Heavy-Ion Collider (RHIC) they are suitable
candidates for the search for new particles and the testing of the SM
properties.

Quarks and gluons behave as quasi free particles at higher energies (short
distances) due to asymptotically free nature of the QCD theory. When the
energy among these color particles are reduced to about $1\,\text{GeV}$ and
below or at a separation of order $1\,\text{fm}$ or higher, color confinement
sets in, such that, the quarks and the gluons coexist as hadrons. At lower
enough energies hadronization is highly favoured leading to formation of light
pions. Hadron jet was first produced by annihilation of electron-positron to
produce two jet event, $e^{+}e^{-}\rightarrow q\bar{q}\rightarrow
2-\text{jets}$. This two jet structure Hanson  was first observed in 1975 at
SPEAR (SLAC) and the spin $1/2$ nature of the quarks also established
Schwitters . The jets are produced when the $q\bar{q}$ fly apart, more
$q\bar{q}$ pairs are produced which recombine with the existing pairs to form
mesons and baryons as a by-product Wu . The first three jet structure from
electron-positron annihilation process was observed by analysing $40$ hadronic
events at the center-of-mass energy $27.4\,\text{GeV}$ from SPEAR Wu . These
jets were subsequently observed clearly and separately Wiik ; Zobernig . The
discovery of the gluon through the three jet process played a major role in
the discovery of Higgs particle Englert  by ATLAS collaboration ATLAS  and CMS
collaboration CMS  using the LHC at CERN. On the other hand, pion-pion
annihilation serves as the principal source of dileptons ($e^{-}e^{+}$ and
$\mu^{-}\mu^{+}$) produced from the hadron matter Koch ; Haglin . Thus, the
proper study of the main source of dilepton spectra observed experimentally
proposes significant observables which help to understand the pion dynamics in
dense nuclear matter that exist at the beginning of the collision Anchishkin .

We consider a complex scalar field model coupled to Abelian gauge fields in
two different ways. At relatively low energies the particles undergo
transformation into glueballs. The emission of gluon $g$ which mediates the
strong interactions is a non-Abelian feature observed in the nonperturbative
regime of QCD theory. However, in the model framework, the gluon emission is a
result of the modification of the Abelian gauge field by the color dielectric
function. In this approach, we approximate the non-Abelian gauge field
responsible for color confinement with an Abelian gauge field coupled with a
dimensionless color dielectric function $G(|\phi|)$. The color dielectric
function is responsible for regulating the long distance dynamics of the
photon propagator, so it does not decouple at higher wavelengths where
glueballs are expected to be in a confined state. The color dielectric
function is defined in terms of the glueball field $\eta$ after SSB, to give
meaning to color confinement and its associated properties. While $\phi$
initiates the annihilation and production processes, $\eta$ is the glueball
field which brings about confinement and gluon condensation. The Abelian
approach to QCD theory was first proposed by ’t Hooft to justify magnetic
monopole condensation in QCD vacuum leading to color superconductivity Hooft1
; Ezawa  that also suggested ‘infrared Abelian dominance’. Fast forward, it
has also been shown that about 92% of the QCD string tension is Abelian Shiba
, so an Abelian approximation will not be out of place. Further studies of
Abelian dominance for color confinement in $\text{SU}(2)$ and $\text{SU}(3)$
lattice QCD can be found in Ref.Suzuki . An interested reader can also refer
to Ref.Sakumichi  for more recent results on Abelian dominance in QCD theory.
This approach has also been adopted in other confinement models —see A-B ;
Adamu ; Issifu  and references therein for more justifications.

The annihilation of $\phi^{*}\phi$ through ‘scalar QED’ where the particles
are relatively free can be used to address hadronization due to the presence
of high density deconfined mater. The spontaneous symmetry breaking of the
$\text{U}(1)$ symmetry to an energy regime is suitable for describing color
confinement with a glueball field $\eta$. Because the running coupling,
$\alpha_{s}(Q^{2})$, of the strong interaction decreases with momentum
increase (short distances), $\phi^{*}\phi$ annihilation can be studied from
perturbation theory ($\phi^{*}\phi\,\rightarrow\,\phi^{*}\phi$) similar to an
annihilation and creation of pions
($\pi^{-}\pi^{+}\,\rightarrow\,\pi^{-}\pi^{+}\gamma$) and Bhabha scattering
($e^{-}e^{+}\rightarrow e^{-}e^{+}$ ) Louis . A low energy analyses of
$\phi^{*}\phi$ annihilation will be made to show color confinement, bound
states of the gluons (glueball masses) Morningstar ; Loan ; Chen  and the QCD
vacuum Pasechnik ; Schaefer  responsible for the gluon condensate. We will
shed light on monopole condensation in the QCD vacuum and its role in
screening the QCD vacuum from external electric and magnetic field
penetrations similar to superconductors.

There is a long standing belief that quark masses are obtained through the
spontaneous symmetry breaking mechanism facilitated by the non zero
expectation value of the Higgs field. That notwithstanding, quark masses
continue to be an input parameter in the standard model Melnikov . As a free
parameter in the SM, the magnitude of the mass is determined
phenomenologically and the result compared with sum rules, lattice simulation
results, other theoretical techniques and experimental data. The Dyson-
Schwinger integral equation for the quark mass function has two known
solutions i.e., trivial and nontrivial solutions. The latter is obtained under
low energy and nonperturbative conditions while the former leads to
unphysical, massless quarks. Thus, the non trivial solution leads to the
creation of quark masses resulting into ‘dynamical chiral symmetry breaking’,
that is a consequence of confinement Fomin ; Higashijima ; Acharya ; Bando .

Monopoles can be said to be a product of grand unification theory (GUT). The
mixing of the strong and the electroweak interactions due to the higher gauge
symmetries in GUT breaks spontaneously at higher energies or extremely short
distances. So physical features of monopoles such as size and mass are studied
through the energy of the spontaneous symmetry breaking John . Monopoles play
important role in color confinement. Monopole condensation produces dual
superconductor which squeezes the uniform electric field at the confinement
phase into a thin flux tube picture. The flux tube is formed between quark and
an antiquark to keep them confined. This scenario has been extensively
investigated in lattice gauge theory Nambu  and Polyakov loop Iwazaki  to
establish its confinement properties. The involvement of monopoles in chiral
symmetry has recently been investigated as well Giacomo . The impact of
monopoles in quark gluon plasma (QGP) has also been studied in Liao .

The paper is organised as follows: In Sec. II we introduce the Lagrangian
density that will be the basics for this study. In Sec. III we present
spontaneous symmetry breaking of the model presented in the previous sections;
this section is divided into four subsections, in Sec. III.1 we investigate
confinement of glueballs, in Sec. III.2 we study the effective masses, we
dedicate Sec. III.3 to gluon condensation and in Sec. IV we investigate the
monopole condensation. We proceed to present the analysis in Sec. V.1 and the
final findings in Sec. V.2. We have adopted the natural units
$c=\hbar=k_{B}=1$, except otherwise stated.

##  II The Model

We start with a Lagrangian density developed by exploring gauge invariant
properties — see Adamu —,

$\displaystyle\mathcal{L}$ $\displaystyle=\eta^{\mu\nu}D_{\mu}\phi
D_{\nu}\phi^{*}-\dfrac{1}{4}G(|\phi|)F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}-V(|\phi|)$
$\displaystyle=\eta^{\mu\nu}(\partial_{\mu}\phi+iq\tilde{A}_{\mu}\phi)(\partial_{\nu}\phi^{*}-iq\tilde{A}_{\nu}\phi^{*})-\dfrac{1}{4}G(|\phi|)F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}-V(|\phi|),$
(1)

where $D_{\mu}=\partial_{\mu}+iq\tilde{A}_{\mu}$,
$\tilde{F}_{\mu\nu}=\partial_{\mu}\tilde{A}_{\nu}-\partial_{\nu}\tilde{A}_{\mu}$
and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ are the covariant
derivative, strength of the dual gauge field and the gauge field respectively.
This model can be used to study the annihilation and creation of identical
particles $\phi^{*}\phi$ and its transformation into glueballs $\eta$ at low
energies. The complex scalar fields $\phi(x)$ and its conjugate $\phi^{*}(x)$
describe particle and its antiparticle with the same mass but different
charges respectively. The electromagnetic interactions among the scalar fields
with an exchange of single photon produced from the dual gauge field dynamics
$\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}$ is the suitable scenario to address
high energy limit. However, we shall focus on the strong interactions, where
the scalar fields modify to glueball fields at relatively low energies, which
will lead to color confinement and its associated properties mediated by the
modified gauge field dynamics $G(|\phi|)F_{\mu\nu}F^{\mu\nu}$. The equations
of motion are

$\displaystyle D_{\mu}D^{\mu}\phi+\dfrac{1}{4}\dfrac{\partial
G(|\phi|)}{\partial\phi^{*}}F_{\mu\nu}F^{\mu\nu}+\dfrac{\partial
V(|\phi|)}{\partial\phi^{*}}=0,$ (2)

$\partial_{\mu}\tilde{F}^{\mu\nu}=-iq[\phi^{*}{\partial^{\nu}}\phi-\phi{\partial^{\nu}}\phi^{*}]+2q^{2}\phi^{*}\phi\tilde{A}^{\nu}=j^{\nu}_{\phi},$
(3)

and

$\displaystyle\partial_{\mu}[G(|\phi|)F^{\mu\nu}]=0.$ (4)

We will adopt a complex scalar field potential of the form,

$V(\phi)=\dfrac{\rho}{4}[\alpha^{2}|\phi|^{2}-a^{2}]^{2},$ (5)

where

$\phi=\dfrac{\phi_{1}+i\phi_{2}}{\sqrt{2}}.$ (6)

We can isolate the electromagnetic interactions among the fields Dadi  for
study using Eq.(3). This enables us to visualize how the fields annihilate and
create identical particles Mandl ; Peskin  through single photon exchange,
$\phi(p_{1})\phi^{*}(p_{2})\rightarrow\gamma\rightarrow\phi(p^{\prime}_{1})\phi^{*}(p^{\prime}_{2})$,
in high energy regime Langacker .

##  III Confinement, Effective Masses, Gluon Condenstaion and Color
Superconductivity

To discuss the color confinement and its consequences we need to follow
spontaneous symmetry breaking of the $\tilde{\text{U}}(1)$ symmetry in
Lagrangian Eq.(II) to give mass to the resulting gueball fields and its
associated products such as the monopole condensation and gluon condensation
which form the basics for color confinement. We will proceed with the symmetry
breaking from the transformations,

$\displaystyle\phi(x)\rightarrow\phi^{\prime}(x)=e^{iq\alpha(x)}\phi$
$\displaystyle\tilde{A}(x)\rightarrow\tilde{A}^{\prime}(x)=\tilde{A}(x)-\partial_{\mu}\alpha(x).$
(7)

The vacuum expectation value of the potential Eq.(5)

$\langle|\phi|\rangle_{0}=\pm\dfrac{a}{\alpha},$ (8)

breaks the $\tilde{\text{U}}(1)$ symmetry spontaneously. Considering two real
scalar fields $\eta(r)$ and $\zeta(r)$ representing small fluctuations about
the vacuum, a shift in the vacuum can be expressed as

$\phi\rightarrow\eta(r)+\phi_{0},$ (9)

where $\phi_{0}\equiv\langle|\phi|\rangle_{0}$. Hence, the potential takes the
form

$\displaystyle V(\eta)$
$\displaystyle=V(\phi)|_{\langle|\phi|\rangle_{0}}+V^{\prime}(\phi)|_{\langle|\phi|\rangle_{0}}\eta+\dfrac{1}{2}V^{\prime\prime}(\phi)|_{\langle|\phi|\rangle_{0}}\eta^{2}$
$\displaystyle={\rho\alpha^{2}a^{2}}\eta^{2}.$ (10)

Consequently, we can suitably parametrize the scalar field such that,

$\displaystyle\phi$
$\displaystyle=e^{i\zeta/\phi_{0}}\dfrac{(\phi_{0}+\eta)}{\sqrt{2}}$
$\displaystyle\approx\dfrac{(\phi_{0}+\eta(r)+i\zeta(r))}{\sqrt{2}}.$ (11)

The Lagrangian in Eq.(II) becomes,

$\mathcal{L}=\dfrac{1}{2}[\partial_{\mu}\eta\partial^{\mu}\eta-2\rho\alpha^{2}a^{2}\eta^{2}]+\dfrac{1}{2}[\partial_{\mu}\zeta\partial^{\mu}\zeta-2q\phi_{0}\tilde{A}_{\mu}\partial^{\mu}\zeta+{q^{2}|\phi_{0}|^{2}}\tilde{A}_{\mu}\tilde{A}^{\mu}]-\dfrac{1}{4}G(\eta)F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}+\cdots,$
(12)

where $\eta$ is associated with the gueball fields and $\zeta$ is also
associated with the massless Goldstone bosons which will be swallowed by the
massive gauge fields through the gauge transformation Adamu ,

$\tilde{A}_{\mu}\rightarrow\tilde{A}^{\prime}=\tilde{A}_{\mu}-\dfrac{1}{q\phi_{0}}\partial_{\mu}\zeta.$
(13)

Pions are strongly interacting elementary particles with an integer spin.
Therefore, they are bosons which are not governed by Pauli’s exclusion
principle and can exist as relativistic or non-relativistic particles. As a
result, they can be represented by spin-0 scalar fields. In the non-
relativistic regime, they exist as Goldstone bosons $\zeta$, signifying the
breakdown of chiral symmetry Low . Pions are generally produced through matter
annihilations such as $p\bar{p}$, $N\bar{N}$, $ee^{+}$ and so on, which
undergo transition from baryon structure to mesons. This is an interesting
phenomenon in low energy hadron physics Dondi ; Klempt . Pions have flavour
structure and other quantum numbers that permits us to classify them as bound
states of quark and an antiquark. However, the valence quarks which
characterizes the flavour structure are surrounded by gluons and quark-
antiquark pairs. Physically, glue-rich components mix with pions ($q\bar{q}$)
forming an enriched spectrum of isospin-zero states and $q\bar{q}g$ hybrid
states Ochs . The glueball spectrum and their corresponding quantum numbers
are known in lattice gauge theory predictions Morningstar ; Close .
Additionally, the existence of glueballs have not been decisive because of the
fear of possible mixing with quark degrees of freedom Brodsky1 . In the model
framework, the glueballs coexist with the Goldstone bosons which are
subsequently swallowed by the massive gauge fields leaving out the glueballs
($gg$) for analysis. The glueball decay occur when the separation distance
between the valence gluons exceeds some thresholds ($>1\,\text{fm}$) leading
to hadronization.

Consequently, the Lagrangian can be simplified as,

$\mathcal{L}=\dfrac{1}{2}\partial_{\mu}\eta\partial^{\mu}\eta-V(\eta)-\dfrac{1}{4}G(\eta)F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}+\dfrac{q^{2}|\phi_{0}|^{2}}{2}\tilde{A}_{\mu}\tilde{A}^{\mu},$
(14)

where $V(\eta)=\rho\alpha^{2}a^{2}\eta^{2}$. The equations of motion for this
Lagrangian are

$\partial_{\mu}\partial^{\mu}\eta+\dfrac{\partial
G(\eta)}{\partial\eta}F_{\mu\nu}F^{\mu\nu}+\dfrac{\partial
V(\eta)}{\partial\eta}=0,$ (15)
$\partial_{\mu}\tilde{F}^{\mu\nu}=q^{2}|\phi_{0}|^{2}\tilde{A}^{\nu}=j_{\phi_{0}}^{\nu},$
(16)

and

$\partial_{\mu}[G(|\eta|)F^{\mu\nu}]=0.$ (17)

The Feynman propagator for the glueball field $\eta$ in Eq.(12) reads,

$D_{\eta}(p)=\dfrac{i}{p^{2}-m^{2}_{\eta}+i\varepsilon},$ (18)

where $m^{2}_{\eta}=2\rho\alpha^{2}a^{2}$.

###  III.1 Confinement

Here, we are interested in calculating static confining potential, such that
only chromoelectric flux that results in confining electric field is present,
whiles chromomagnetic flux influence is completely eliminated. Thus, in
spherical coordinates Eq.(15) and Eq.(17) become, respectively,

$\dfrac{1}{r^{2}}\dfrac{d}{dr}\left(r^{2}\dfrac{d\eta}{dr}\right)=\dfrac{1}{2}\dfrac{\partial
G(\eta)}{\partial\eta}E^{2}-\dfrac{\partial V(\eta)}{\partial\eta},$ (19)

and

$\displaystyle\dfrac{1}{r^{2}}\dfrac{\partial}{\partial
r}[r^{2}G(\eta)E]=0\rightarrow$ $\displaystyle E$
$\displaystyle=\dfrac{\Lambda}{r^{2}G(\eta)},$ (20)

where $\Lambda=q/4\pi\varepsilon_{0}$ is the integration constant. We are also
now setting $G(\eta)=\xi^{2}V(\eta)$ (similar process was adopted in Ref.Adamu
; A-B ; Issifu ), where $\xi^{2}$ is a dimensionful constant associated with
the Regge slope $(2\pi\alpha^{\prime})^{2}$ that absorbs the dimensionality of
$V(\eta)$, so $G(\eta)$ remains dimensionless and Eq. (19) reads

$\displaystyle\dfrac{1}{r^{2}}\dfrac{d}{dr}\left(r^{2}\dfrac{d\eta}{dr}\right)=-\dfrac{\partial}{\partial\eta}\left[\dfrac{\Lambda^{2}}{2}\dfrac{1}{\xi^{2}V(\eta)}\dfrac{1}{r^{4}}+V\right]\rightarrow$
$\displaystyle\eta^{\prime\prime}+\dfrac{2}{r}\eta^{\prime}+m^{2}_{\eta}\eta=0.$
(21)

In the last step, we have assumed that the particles are far away from the
charge source $q$, and in such a limit terms of the order ${\cal O}(1/r^{4})$
can be disregarded. This equation has a solution

$\eta(r)=\dfrac{a\sin(m_{\eta}r)}{\alpha m_{\eta}r}.$ (22)

Substituting this result into the electric field Eq.(III.1),

$\displaystyle E$ $\displaystyle=\dfrac{\Lambda}{r^{2}G}$
$\displaystyle=\dfrac{2\Lambda\alpha^{2}}{a^{2}\xi^{2}\sin^{2}(m_{\eta}r)},$
(23)

and using the well known expression for determining the electrodynamic
potential,

$V(r)=\int{Edr},$ (24)

to determine the confining potential $V_{c}(r)$, we get

$\displaystyle V_{c}(r)$
$\displaystyle=-\dfrac{2\Lambda\alpha^{2}\cot(m_{\eta}r)}{a^{2}m_{\eta}\xi^{2}}+c$
$\displaystyle\simeq-\dfrac{2\Lambda\alpha^{2}}{a^{2}m_{\eta}\xi^{2}}\left[\dfrac{1}{m_{\eta}r}-\dfrac{m_{\eta}r}{3}-{\cal
O}(r^{3})\right]+c$
$\displaystyle\simeq\dfrac{2\alpha^{2}}{a^{2}\xi^{2}m^{2}_{\eta}}\left[-\dfrac{1}{r}+\dfrac{m^{2}_{\eta}r}{3}\right],$
(25)

so, setting

$a^{4}\xi^{2}\rho=1\rightarrow\xi^{2}=\dfrac{1}{a^{4}\rho}$ (26)

leads to,

$V_{c}(r)=-\dfrac{1}{r}+\dfrac{m_{\eta}^{2}r}{3},$ (27)

with string tension

$\sigma=\dfrac{m^{2}_{\eta}}{3}=\dfrac{2\rho\alpha^{2}a^{2}}{3}.$ (28)

In the last step of Eq.(III.1) we substituted
$\Lambda=q/4\pi\varepsilon_{0}=1$ and $c=0$ for simplicity.

###  III.2 Effective Masses

Glueballs are simply bound states of gluons and they are ‘white’ or colourless
in nature. They are flavour blind and are capable of decaying. Scalar glueball
with quantum number $J^{PC}=0^{++}$ is observed by lattice QCD simulations as
the lightest with mass of about $1.7\,\text{GeV}$ Morningstar ; Loan ; Chen .

The dispersion relation for the glueball and the gluon excitations including
thermal fluctuations can be expressed as

$E_{\eta}^{2}=k^{2}+m^{*2}_{\eta}\qquad{\text{and}}\qquad
E^{2}_{A}=k^{2}+m_{A}^{*2}.$ (29)

We can calculate the effective thermal fluctuating glueball mass by taking the
second derivative of the Lagrangian density Eq.(14) with respect to $\eta$,

$\displaystyle
m^{*2}_{\eta}=-\left\langle\dfrac{\partial^{2}\mathcal{L}}{\partial\eta^{2}}\right\rangle=2\rho\alpha^{2}a^{2}+\dfrac{1}{2}\rho\alpha^{2}a^{2}\xi^{2}\langle
F^{\mu\nu}F_{\mu\nu}\rangle.$ (30)

We redefine the glueball field $\eta$ to include the thermal fluctuations as
$\eta=\bar{\eta}+\Delta$, with restriction, $\langle\Delta\rangle=0$. The
angle brackets represent thermal average and $\bar{\eta}$ is the mean field of
the glueballs. This equation can be solved by determining explicitly the
nature of $\langle F^{\mu\nu}F_{\mu\nu}\rangle$ and $\langle\Delta^{2}\rangle$
using field quanta distributions. By the standard approach, we can express,

$\langle
F^{\mu\nu}F_{\mu\nu}\rangle=-\dfrac{\nu}{2\pi^{2}}\int_{0}^{\infty}{dk\dfrac{k^{4}}{E_{A}}n_{B}(E_{A})}\quad{\text{and}}\quad\langle\Delta^{2}\rangle=\dfrac{1}{2\pi^{2}}\int_{0}^{\infty}{dk\dfrac{k^{2}}{E_{\eta}}n_{B}(E_{\eta})},$
(31)

here, $n_{B}(x)=(e^{x\beta}-1)^{-1}$ represents the Bose-Einstein distribution
function and $\beta=1/T$, where $T$ is the temperature. We can analytically
solve Eq.(31) at high energy regime, where $E\approx kc$, $k_{B}T\gg
m_{A}c^{2}$ and $m_{A}$ is the gluon/screening mass. Therefore

$\displaystyle\langle F^{\mu\nu}F_{\mu\nu}\rangle$
$\displaystyle=-\dfrac{\nu}{2\pi^{2}}\int_{0}^{\infty}{dk\dfrac{k^{4}}{E_{A}}\dfrac{1}{e^{\beta
E_{A}}-1}}$ $\displaystyle=-\dfrac{\nu
T^{4}}{2\pi^{2}}\int_{0}^{\infty}{\dfrac{x^{3}dx}{e^{x}-1}}$
$\displaystyle=-\dfrac{4T^{4}}{T^{4}_{c}\xi^{2}},$ (32)

where we have substituted $x=k/T$ and used the standard integral

$\int{\dfrac{x^{3}dx}{e^{x}-1}}=\dfrac{\pi^{4}}{15}.$ (33)

In the last step we have substituted

${T^{4}_{c}\xi^{2}}=\dfrac{15}{2\pi^{2}\nu},$ (34)

being $T_{c}$ the critical temperature. Following the same analysis,

$\displaystyle\langle\Delta^{2}\rangle$
$\displaystyle=\dfrac{1}{2\pi^{2}}\int_{0}^{\infty}{dk\dfrac{k^{2}}{E_{\eta}}\dfrac{1}{e^{\beta
E_{\eta}}-1}}$
$\displaystyle\simeq\dfrac{T^{2}}{2\pi^{2}}\int_{0}^{\infty}{\dfrac{xdx}{e^{x}-1}}$
$\displaystyle\simeq\dfrac{T^{2}}{12},$ (35)

where we have assumed $k_{B}T\gg m_{\eta}c^{2}$, with $m_{\eta}$ being the
glueball mass, and considered the standard integral,

$\int{\dfrac{xdx}{e^{x}-1}}=\dfrac{\pi^{2}}{6}.$ (36)

Thus, Eq.(30) becomes

$\displaystyle m_{\eta}^{*2}(T)$
$\displaystyle=2\rho\alpha^{2}a^{2}\left[1-\dfrac{T^{4}}{T^{4}_{c}}\right]$
$\displaystyle=m_{\eta}^{2}(0)\left[1-\dfrac{T^{4}}{T^{4}_{c}}\right].$ (37)

When we take a thermal average of Eq.(15), we get
$\xi^{2}m_{\eta}^{2}\langle\eta
F^{\mu\nu}F_{\mu\nu}\rangle=-m_{\eta}^{2}\langle\eta\rangle$, therefore,
$\bar{\eta}=0$ and $\bar{\eta}=1$ are exact solutions of the glueball fields
Carter . We further demonstrate in Figs. 5 and 10 that $\bar{\eta}$ has
solution $\bar{\eta}=0$ at $T=0$ and increases steady as $T$ increases to its
maximum $\bar{\eta}\simeq 1$ at $T=T_{c}$. Corresponding to no glueball fields
and the melting of the glueballs.

###  III.3 Gluon Condensation

Classical gluodynamics is described by the Lagrangian density,

$\mathcal{L}^{\prime}=-\dfrac{1}{4}F^{a\mu\nu}F^{a}_{\mu\nu},$ (38)

which is invariant under scale and conformal symmetries, $x\rightarrow\lambda
x$, and also produces vanishing gluon condensate $\langle
F^{a\mu\nu}F^{a}_{\mu\nu}\rangle=0$. Meanwhile, the symmetries are broken when
quantum correction $-|\varepsilon_{v}|$ is added to the Lagrangian i.e.

$\mathcal{L}^{\prime}=-\dfrac{1}{4}F^{a\mu\nu}F^{a}_{\mu\nu}+|\varepsilon_{v}|,$
(39)

resulting in non-vanishing gluon condensate, $\langle
F^{a\mu\nu}F^{a}_{\mu\nu}\rangle>0$. This correction is the consequence of the
well known scale anomaly observed in QCD energy-momentum ($\theta^{\mu\nu}$)
trace

$\theta_{\mu}^{\mu}=\dfrac{\beta(g)}{2g}F^{a\mu\nu}F^{a}_{\mu\nu},$ (40)

where $\beta(g)$ is the ‘so called’ beta-function of the strong coupling $g$,
with a leading term

$\beta(g)=-\dfrac{11g^{3}}{(4\pi)^{2}}.$ (41)

Accordingly, we have non-zero vacuum expectation, expressed as

$\langle\theta^{\mu}_{\mu}\rangle=-4|\varepsilon_{v}|.$ (42)

The second term in Eq.(14) acts similar to the quantum correction and
explicitly breaks the scale and the conformal symmetries, so its energy-
momentum tensor satisfies Eq.(42) A-B ; Shffman ; Kharzeev ; Migdal . We will
compute the trace of the energy-momentum tensor of Eq.(14) using the relation,

$\theta_{\mu}^{\mu}=4V(\eta)+\eta\square\eta.$ (43)

Substituting the equation of motion Eq.(15) into Eq.(43), yields

$\displaystyle\theta_{\mu}^{\mu}$ $\displaystyle=4V(\eta)-\eta\dfrac{\partial
G}{\partial\eta}F^{\mu\nu}F_{\mu\nu}-\eta\dfrac{\partial V}{\partial\eta}$
$\displaystyle=4\tilde{V}-\eta G^{\prime}F^{\mu\nu}F_{\mu\nu},$ (44)

where in the last step we have substituted $\tilde{V}=V(\eta)-\eta
V^{\prime}(\eta)/4$ with, $G^{\prime}$ and $V^{\prime}$ representing the
derivative of $G$ and $V$ with respect to $\eta$ respectively. In order to
make the expression easy to be compared with Eq.(42) we rescale $\tilde{V}$ as
$\tilde{V}(\eta)\rightarrow-|\varepsilon_{v}|\tilde{V}$, consequently,

$\langle\eta G^{\prime}F^{\mu\nu}F_{\mu\nu}\rangle=4|\varepsilon_{v}|\langle
1-\tilde{V}\rangle.$ (45)

We recover the classical expression for $\langle
F^{\mu\nu}F_{\mu\nu}\rangle=0$ when we set $|\varepsilon_{v}|\rightarrow 0$.
Following the potential defined bellow Eq.(14), we can express

$\displaystyle\tilde{V}$ $\displaystyle=V-\dfrac{\eta V^{\prime}}{4}$
$\displaystyle=\dfrac{\rho a^{2}\alpha^{2}\eta^{2}}{2}$
$\displaystyle=\dfrac{m^{2}_{\eta}\eta^{2}}{4},$ (46)

thus, Eq.(45) becomes

$\langle 2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle=4|\varepsilon_{v}|\left\langle
1-\dfrac{m^{2}_{\eta}\eta^{2}}{4}\right\rangle.$ (47)

The QCD vacuum is seen as a very dense state of matter comprising gauge fields
and quarks that interact in a haphazard manner. These characteristics are hard
to be seen experimentally because quark and gluon fields can not be observed
directly, only the color neutral hadrons are observable. The appearance of the
mass term in the condensate is also significant because it leads to chiral
symmetry break down in the vacuum. Additionally, the gluon mass $m_{A}$ can be
derived from the vacuum as a function of the glueball mass as

$m_{A}^{2}=\dfrac{m_{\eta}^{2}}{4}.$ (48)

Generally, scalar glueball mass is related to the gluon mass as

$\dfrac{m(0^{++})}{m_{A}}=\sqrt{6}\cong 2.45,$ (49)

considering the leading order Cornwall  from Yang-Mills theory with an
auxiliary field $\phi$, here, the scalar glueball mass $m(0^{++})$ represents
fluctuations around $\phi$. The gluon mass is determined to be $m_{A}=600\sim
700\,\text{MeV}$ as obtained from lattice simulations Leinweber . Heavier
gluon masses have also been observed in the range of $\sim 1\,\text{GeV}$ by
phenomenological analysis Field , lattice simulation Leinweber1  and
analytical investigations Kogan . The glueball mass $m_{\eta}$ is responsible
for all the confinement properties and the chiral symmetry breaking in the
vacuum. Observing that the string tension is $\sigma\sim 1\,\text{GeV/fm}$, we
can identify $m^{2}_{\eta}=3\,\text{GeV}^{2}$, corresponding to a gluon mass
$m_{A}=0.87\,\text{GeV}$, from the model framework.

In terms of temperature, we use Eq.(III.2) and the gluon condensate becomes,

$\displaystyle\langle 2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle$
$\displaystyle=4|\varepsilon_{v}|\left[1-\dfrac{m^{2}_{\eta}}{4}\left(\dfrac{\phi^{2}_{0}T^{2}}{\phi^{2}_{0}12}\right)\right]$
$\displaystyle=4|\varepsilon_{v}|\left[1-\dfrac{T^{2}}{T^{2}_{c\eta}}\right]\qquad\text{where}\qquad
T_{c\eta}=\left(\dfrac{48}{m_{\eta}^{2}}\right)^{1/2}.$ (50)

Corresponding to a temperature fluctuating gluon mass

$\displaystyle m^{*2}_{A}$
$\displaystyle=\dfrac{m^{2}_{\eta}T^{2}}{48\phi^{2}_{0}}$
$\displaystyle=\tilde{g}^{2}T^{2},$ (51)

where $\tilde{g}^{2}=m^{2}_{\eta}/48\phi^{2}_{0}$ is a dimensionless coupling
constant. This expression looks like the Debye mass derived from the leading
order of QCD coupling expansion Kajantie ; Nadkarni ; Manousakis . It carries
nonperturbative property of the theory. Comparing Eqs.(40) and (III.3), we
identify

$\displaystyle\dfrac{\beta(g)}{2g}=-\eta G^{\prime}(\eta)\rightarrow$
$\displaystyle\beta(1/r^{2})=-2g\eta G^{\prime}(\eta).$ (52)

Additionally, we can determine the strong running coupling $\alpha_{s}$ from
the renormalization group theory Deur

$\displaystyle\beta(Q^{2})=Q^{2}\dfrac{d\alpha_{s}}{dQ^{2}},$ (53)

therefore,

$\beta(\eta)\simeq-\eta\dfrac{d(G)}{d\eta}=-\xi^{2}m_{\eta}^{2}\eta^{2}(r)=2G(\eta)=\beta(1/r^{2})\qquad\text{with}\qquad
g=1.$ (54)

Comparatively, the strong running coupling can be identified as
$\alpha_{s}(\eta)=G(\eta)=\alpha_{s}(1/r^{2})$, with $Q\equiv 1/r$, as the
space-like momentum associated with the four vector momentum as
$Q^{2}\equiv-q^{2}$. Thus, the color dielectric function is associated with
the QCD $\beta$-function and the strong running coupling.

We can write the Feynman propagator for the interaction by considering the
left hand side of Eq.(45),

$\displaystyle F^{\mu\nu}F_{\mu\nu}$
$\displaystyle=-2A^{\nu}[\square-\partial_{\mu}\partial_{\nu}]A^{\mu}$
$\displaystyle=-2A^{\nu}\left[\partial^{2}g^{\mu\nu}-\left(1-\dfrac{1}{\alpha}\right)\partial_{\mu}\partial_{\nu}\right]A^{\mu}$
$\displaystyle=-2A_{\nu}(k)\left[k^{2}g^{\mu\nu}-\left(1-\dfrac{1}{\alpha}\right)k^{\mu}k^{\nu}\right]A_{\mu}(-k)\rightarrow$
$\displaystyle\eta G^{\prime}(\eta)F^{\mu\nu}F_{\mu\nu}$
$\displaystyle=-2A_{\nu}(k)\eta
G^{\prime}(\eta)\left[k^{2}g^{\mu\nu}-\left(1-\dfrac{1}{\alpha}\right)k^{\mu}k^{\nu}\right]A_{\mu}(-k)$
$\displaystyle=A_{\nu}(k)\beta(q^{2})\left[k^{2}g^{\mu\nu}-\left(1-\dfrac{1}{\alpha}\right)k^{\mu}k^{\nu}\right]A_{\mu}(-k),$
(55)

where we have added the gauge fixing term
$(2/\alpha)(\partial_{\mu}A^{\mu})^{2}$, and made the Fourier transform

$A_{\mu}(k)=\int{\dfrac{d^{4}k}{(2\pi)^{4}}e^{-ikx}A_{\mu}(x)}.$ (56)

In the last step, the term in the square brackets is precisely the photon
propagator normally associated with Abelian gauge fields but the
multiplicative factor $-2\eta
G^{\prime}(\eta)=\beta(1/r^{2})\equiv\beta(q^{2})$ is defined such that the
photon propagator does not decouple at higher wavelengths. This influences the
photons to behave like gluons.

Also, one can retrieve all the major results obtained in A-B , if we consider
temperature fluctuations in the scalar field $\phi$. In that case, we move
away from the known form of field theory i.e. perturbation about the vacuum,
$\langle\phi\rangle=0$, to account for the density of particles and their
interactions about the vacuum and the thermal distributions in the scalar
field. So, we can substitute $\phi\rightarrow\phi+\phi_{T}$ into the potential
in Eq.(5), where $\phi_{T}$ plays a similar role as $\Delta$ introduced above.
Thus,

$\langle\phi^{2}_{T}\rangle=\dfrac{1}{2\pi^{2}}\int_{0}^{\infty}{\dfrac{p^{2}dp}{E(e^{E\beta}-1)}}\simeq\dfrac{T^{2}}{12},$
(57)

here, we have used the high temperature approximation $k_{B}T\gg
m_{\phi}c^{2}$, $E\approx pc$ and the potential becomes,

$\displaystyle V(\phi,T)$
$\displaystyle=\dfrac{\rho}{4}[\alpha^{2}|\phi|^{2}+\alpha^{2}\langle\phi^{2}_{T}\rangle-a^{2}]^{2}$
$\displaystyle=\dfrac{\rho}{4}\left[\alpha^{2}|\phi|^{2}-a^{2}\left(1-\dfrac{T^{2}}{T^{2}_{c\phi}}\right)\right]^{2}$
$\displaystyle=\dfrac{\rho}{4}[\alpha^{2}|\phi|^{2}-\tilde{a}^{2}]^{2},$ (58)

where $\langle\phi_{T}\rangle=0$,

$T^{2}_{c\phi}=\dfrac{12a^{2}}{\alpha^{2}}\qquad{\text{and}}\qquad\tilde{a}^{2}=a^{2}\left(1-\dfrac{T^{2}}{T^{2}_{c\phi}}\right).$
(59)

The glueball mass can be calculated from the fluctuations in Eq.(III.3) around
the vacuum, such that,

$\displaystyle m_{\phi}^{*2}(T)$
$\displaystyle=\dfrac{\partial^{2}V}{\partial\phi^{2}}|_{\phi_{0}}=2\rho\alpha^{2}\tilde{a}^{2}$
$\displaystyle=2\rho\alpha^{2}a^{2}\left[1-\dfrac{T^{2}}{T^{2}_{c\phi}}\right]$
$\displaystyle=m^{2}_{\eta}(0)\left[1-\dfrac{T^{2}}{T^{2}_{c\phi}}\right],$
(60)

where $\phi_{0}=\tilde{a}/\alpha$ is the vacuum of the potential. We observe
that $m_{\phi}^{*2}(0)=m^{*2}_{\eta}(0)$ at $T=0$, similar to Eq.(III.2). The
differences in the degree of the temperature in both equations arises because
Eq.(III.2) has temperature correction to the gauge field $\langle
F^{\mu\nu}F_{\mu\nu}\rangle$, while in Eq.(III.3) we have temperature
correction to the scalar field $\langle\phi^{2}_{T}\rangle$. Following the
same procedure as the one used in deriving Eq.(47) we can express

$\displaystyle\langle 2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle$
$\displaystyle=-4|\varepsilon|\left\langle\dfrac{m^{*2}_{\phi}(T)\eta^{2}}{4}-1\right\rangle$
$\displaystyle=-4|\varepsilon|\left\langle\dfrac{m_{\phi}^{2}(0)}{4}\left[1-\dfrac{T^{2}}{T^{2}_{c\phi}}\right]\eta^{2}-1\right\rangle.$
(61)

Here, we have replaced $m_{\eta}^{*2}(T)$ with $m_{\phi}^{*2}(T)$ from
Eq.(47). The negative sign and $\eta^{2}$ differences between this result and
Ref.A-B  might be due to the differences in the methods adopted for
calculating the energy-momentum trace tensor $\theta_{\mu}^{\mu}$. Also, if we
consider the IR part of the potential in Eq.(III.1) as studied in Ref.A-B ,
the potential becomes

$\displaystyle V_{c}(r)$ $\displaystyle=\sigma
r=\dfrac{m_{\phi}^{2}}{3}r\rightarrow$ $\displaystyle V_{c}(T,r)$
$\displaystyle=\dfrac{m_{\phi}^{*2}(T)}{3}r=\dfrac{m_{\phi}^{2}(0)}{3}\left[1-\dfrac{T^{2}}{T^{2}_{c\phi}}\right]r,$
(62)

with string tension

$\sigma(T)=\dfrac{m_{\phi}^{*2}(T)}{3},$ (63)

which is similar to the result in Ref.A-B . Therefore, the approach adopted in
this paper accounts for the temperature corrections to both the glueball field
$\langle\Delta^{2}\rangle$ in Eq.(III.2) and the gauge field $\langle
F^{\mu\nu}F_{\mu\nu}\rangle$, while in Ref.A-B  only the thermal correction to
the scalar field $\langle\phi_{T}^{2}\rangle$ was considered.

##  IV Dual Superconductivity

In this section we will discuss briefly color superconductivity and proceed to
elaborate on dual superconductivity in detail. The fields $\phi^{*}\phi$
annihilate and create identical fields through a decay process,
$\phi(p_{1})\phi^{*}(p_{2})\rightarrow\phi(p^{\prime}_{1})\phi^{*}(p^{\prime}_{2})$.
The dual gauge field $\tilde{F}^{\mu\nu}\tilde{F}_{\mu\nu}$ which mediates the
interaction of the scalar fields $\phi$ during the annihilation process is
also responsible for the monopole condensation. In the model framework, the
scalar field can be seen as point-like, asymptotically free and degenerate at
the high energy regime Lan . In the context of this work, high energy regime
is in reference to the phase at which the scalar fields are annihilating while
low energy regime is after the symmetry breaking or the glueball regime. In
effect, multi-particle states are formed in the high energy regime. In such a
high particle density regime, the charged scalar fields form Bose-Einstein
condensation Liu1 ; Adhikari  similar to induced isospin imbalance systems
leading to color superconductivity He ; Alford1 . With this type of
condensation, there is higher occupation number at the ground state than the
excited states. So temperature increase goes into reducing the occupation
numbers Brito . There are several works on nonzero isospin chemical potential
($\mu_{I}$) and baryon chemical potential ($\mu_{B}$) in pion condensation
Barducci2 ; Nishida ; Hands  and they are related to the two flavor quarks
that constitute the pions as $\mu_{I}=(\mu_{u}-\mu_{d})/2$ and
$\mu_{B}=(\mu_{u}+\mu_{d})/2$ respectively. It has also been established He1
analytically at the quark level that the critical isospin chemical potential
Toublan ; Barducci  for pion superfluidity is precisely the pion mass
$\mu_{I}^{c}=m_{\pi}$ in vacuum. This behaviour and other related isospin
chemical potential in pion condensate have been investigated using Nambu-Jona-
Lasinio (NJL) model, ladder-QCD Barducci2 .

In the model framework, the color superconducting property can be studied from
the propagator, $\langle\phi\phi^{*}\rangle$. High density region where the
particles are asymptotically free, forming deconfined matter leading to Bose-
Einstein condensation resulting into color superconducting phase Sadzikowski ;
Alford1 ; Rapp1 . On the other hand, at low energy region, the fields modify
into glueballs with real mass, $m_{\eta}$, leading to chiral symmetry breaking
and color confinement. Since quark-quark combinations do not form color
singlet, Cooper pair condensate will break the local color symmetry giving
rise to color superconductivity Alford . This phenomenon was first studied by
Bardeen, Cooper, and Schrieffer (BCS) Cooper . The BCS mechanism for pairing
in general, appears to be more robust in dense quark mater than
superconducting metals due to the nature of interactions among quarks. The
virtual fermions in the vacuum makes it color diamagnetic preventing both the
electric and magnetic fields from penetrating Ali .

We turn attention to the dual superconductivity in color confinement whose
property we can derive from Eq.(16). This subject was first proposed by Nambu,
’t Hooft and Mandelstam Nambu . While the color superconducting phase takes
place at the deconfined matter regime, the dual superconductivity is observed
at the confining phase Carmona . In the model framework, dual
superconductivity is observed at high glueball condensation region,
$\eta\rightarrow 0$, i.e. a region of strong confinement. We will use the rest
of this section discussing dual superconductivity in detail within the model
framework. In this phenomena the QCD vacuum is seen as a color magnetic
monopole condensate resulting into one dimensional squeezing of the uniform
electric flux that form on the surface of the quark and antiquark pairs by
dual Meissner effect. This results into the formation color flux tube between
quark and an antiquark used in describing string picture of hadrons Sakumichi
. In this regard, magnetic monopole condensation is crucial to color
confinement and dual superconductivity. Monopoles also play a role in chiral
symmetry breaking Ramamurti  and strongly coupled QGP. They are involved in
the decay of plasma formed immediately after relativistic heavy ion collision
Liao .

Even though monopoles have not been seen or proven experimentally, there are
many theoretical bases for which one can belief their existence. The
theoretical evidence for its existence is as strong as any undiscovered
theoretical particle. Polyakov Polyakov  and ’t Hooft Hooft  discovered that
monopoles are the consequences of the general idea of unification of
fundamental interactions at short distances (high energies). Dirac on the
other hand demonstrated the presence of monopoles in QED. Monopoles in general
play an insightful role in understanding color confinement. They also provide
some useful explanations to the features of superstring theory and
supersymmetric quantum theory where the use of duality is a common phenomenon.
Discovery of these monopoles some day, will be interesting and possibly
revolutionary since almost all existing technologies are based on electricity
and magnetism.

Monopole condensation can now be exploited from the condensation of particles
in the region $r\rightarrow r_{*}$, where
$\langle\phi\phi^{*}\rangle\sim\langle|\phi|^{2}\rangle_{0}$. Hence, from
Eq.(16) we can express,

$\partial_{\mu}\tilde{F}^{\mu\nu}=q^{2}|\phi_{0}|^{2}\tilde{A}^{\nu}=j^{\nu}_{m}.$
(64)

The equations of motion for the static fields are,

$\nabla\cdot\vec{\tilde{B}}=\rho_{m}\qquad{\text{and}}\qquad-\nabla\times\vec{\tilde{E}}=\vec{j}_{m},$
(65)

here, $\vec{\tilde{E}}$ and $\vec{\tilde{B}}$ are the dual versions of the
static electric and magnetic fields, while $\vec{j}_{m}$ and $\rho_{m}$ being
the magnetic current and charge densities respectively. The Lorentz force
associated with these fields can be expressed as

$\vec{F}=q(\vec{\tilde{B}}-\vec{v}\times\vec{\tilde{E}}),$ (66)

where $\vec{v}$ is the speed of the particle within the fields and $q$ is the
monopole charge. The homogeneous equation $\nabla\cdot\vec{\tilde{E}}=0$,
represents the uniform electric field present at the confining phase.
Combining the dual version of the Ampere’s law on the right side of Eq.(65)
together with the dual London equation responsible for persistent current
generated by the monopoles

$\nabla\times\vec{j}_{m}=\dfrac{1}{{\lambda}^{2}}\vec{\tilde{E}},$ (67)

we obtain the expression

$\nabla^{2}\vec{\tilde{E}}=\dfrac{1}{{\lambda}^{2}}\vec{\tilde{E}}.$ (68)

Noting that $\tilde{{\bf E}}\,=\,\nabla\times{\bf A}$, we can identify,
${\lambda}=(q^{2}|\phi_{0}|^{2})^{-1/2}=(q^{2}m^{2}_{\eta}/2\alpha^{4}\rho)^{-1/2}$
as the London penetration depth Singh ; Bazeia . Developing the Laplacian in
Eq.(68) in spherical coordinates yields,

$\dfrac{d^{2}{\tilde{E}}_{r}}{dr^{2}}+\dfrac{2}{r}\dfrac{d\tilde{E}_{r}}{dr}-\dfrac{\tilde{E}_{r}}{{\lambda}^{2}}=0,$
(69)

this equation has a solution given as

$\tilde{E}_{r}=\dfrac{c_{1}e^{-r/{\lambda}}}{r},$ (70)

where $c_{1}$ is a constant. Using dimensional analysis, we can fix the
constant as $c_{1}=\tilde{E}_{0}$, therefore,

$\displaystyle{\tilde{E}}_{r}$
$\displaystyle=\dfrac{\tilde{E}_{0}e^{-r/{\lambda}}}{r},$ (71)

where ${\tilde{E}}=|{\bf\tilde{E}}|=\tilde{E}_{r}$. This means the electric
field is exponentially screened from the interior of the vacuum with
penetration depth ${\lambda}$, a phenomenon known as the color Meissner
effect. Also, using London’s accelerating current relation,

$\vec{\tilde{B}}=-{\lambda}^{2}\dfrac{\partial\vec{j}_{m}}{\partial t},$ (72)

and writing the continuity equation in the form of London equations,

$\dfrac{\partial\vec{j}_{m}}{\partial t}+\nabla\rho_{m}=0,$ (73)

together with the equation at the left side of Eq.(65), we obtain,

$\nabla^{2}\vec{\tilde{B}}=\dfrac{\vec{\tilde{B}}}{{\lambda}^{2}}.$ (74)

This equation has a solution similar to Eq.(69) i.e.,

$\tilde{B}_{r}=\dfrac{\tilde{B}_{0}e^{-r/\lambda}}{r}.$ (75)

Thus, the magnetic field is equally screened exponentially from the interior
of the vacuum by a penetration depth $\lambda$. In sum, both time varying
electric and magnetic fields are screened with the same depth $\lambda$.

Now, comparing ${\lambda}$ with the results in Refs.Singh ; Tinkham , where

${\lambda}^{2}=\dfrac{m}{n_{s}e^{2}},$ (76)

we rearrange the expression obtained for $\lambda$ in the model framework,

$\lambda^{2}=\dfrac{2\alpha^{4}\rho}{q^{2}m_{\eta}^{2}}=\dfrac{m^{2}_{\eta}}{2q^{2}\rho
a^{4}},$ (77)

to make it suitable for comparison. Juxtaposing these two equations leads to,
$q\,=\,e$, $m\,=\,m_{\eta}^{2}/a$ and $n_{s}\,=\,2\rho a^{3}$, where
$m,\,e,\,\text{and}\,n_{s}$ are mass, charge and electron number density of a
superconducting material. Hence, the electron charge $e$ is equivalent to the
monopole charge $q$ Dirac ; Schwinger  and the electron number density and
mass are related to the monopole density and mass respectively. Combining
Eq.(67) with the electric field expression at the right side of Eq.(65) we
obtain a fluxoid quantization relation

$\int{\vec{\tilde{E}}\cdot d\vec{S}}-{\lambda}^{2}\oint{\vec{j}_{m}\cdot
d\vec{l}}=n\Phi_{e},$ (78)

where $n\in\mathbb{Z}$ and $\Phi_{e}$ is the quantum of the electric flux
Singh . The penetration depth could also be expressed as a function of
temperature,

$\displaystyle\lambda(T)$
$\displaystyle\simeq\left(\dfrac{2\alpha^{4}\rho}{q^{2}m_{\eta}^{*2}(T)}\right)^{1/2}$
$\displaystyle\simeq\lambda(0)\left[1-\dfrac{T^{4}}{T_{c}^{4}}\right]^{-1/2}.$
(79)

We retrieve the initial penetration depth below Eq.(68), if we set $T=0$.
Accordingly, the penetration depth increases with temperature until it becomes
infinite at $T=T_{c}$, where deconfinement and restoration of chiral symmetry
coincides Gupta ; Gottlieb . At this point the field lines are expected to
penetrate the vacuum causing it to loose its superconducting properties. This
expression is precisely the same as the one obtained for metallic
superconductors Tinkham . At $T\,>\,T_{c}$ the magnetic and electric field
lines goes trough the vacuum. This phenomenon is referred to as the Meissner
effect.

The number density of the superconducting monopoles can also be expressed in
terms of temperature as

$\displaystyle n_{s}(T)$ $\displaystyle=2\rho
a^{3}\left[1-\dfrac{T^{4}}{T_{c}^{4}}\right]^{3/2}$
$\displaystyle=n\left[1-\dfrac{T^{4}}{T_{c}^{4}}\right]^{3/2}.$ (80)

The number density decreases sharply with temperature $T$ and vanishes at
$T=T_{c}$. On the other hand, at $T<T_{c}$, similar to the two fluid model
where normal monopole fluid mix with superconducting ones, the density of the
monopoles can be calculated from the relation $n=n_{s}+n_{n}$, where $n_{n}$
is the normal monopole density Poole . Additionally, at $T\rightarrow 0$,
$n_{s}\rightarrow n$, and in the limit $T\rightarrow T_{c}$, $n_{s}\rightarrow
0$. Consequently,

$n_{n}=n\left\\{1-\left[1-\dfrac{T^{4}}{T_{c}^{4}}\right]^{3/2}\right\\}.$
(81)

Thus, the color superconducting monopoles $n_{s}$ at $T\,<\,T_{c}$, conduct
with dissipationless flow, on the other hand, the normal monopoles $n_{n}$
conduct with finite resistance at the same temperature range. Consequently,
$n_{s}$ decreases sharply with increasing $T$ while $n_{n}$ increases steadily
with increasing $T$.

##  V Analysis and Conclusion

###  V.1 Analysis

At high energy regime, there is high particle density which form deconfined
matter leading to hadronization Letessier ; Senger ,
$\phi^{*}(p_{1})\phi(p_{2})\rightarrow\gamma\rightarrow\phi^{*}(p^{\prime}_{1})\phi(p^{\prime}_{2})$
Langacker ; Linnyk ; Liu ; Bratkovskaya ; Srinivasan . In this regime, the
process takes place at low temperature but in high momentum and particle
density. This leads to an asymptotic freedom behaviour among the particles
because color confinement depends on how far or close the particles are
together. Thus, higher density means the particles are closer together leading
to deconfinement. Additionally, Eq. (III.3) represents the gluon propagator
which mediates the interaction of the glueballs in the IR regime of the model.
Hadronization Webber  comes into play in this regime when the momenta of the
particles are reduced below a particular threshold set by the QCD scale,
$\Lambda_{QCD}\sim 250\,\text{MeV}$, or the separation distance between
particle and antiparticle pairs is grater than $\sim 1\,\text{fm}$. For the
sake of clarity, there are two forms of hadronization mentioned here, one
arising from high energy QGP formation and the other from QCD string decay
into hadrons in the low energy regime. The QGP hadronization is believed to
have been formed immediately after the Big Bang when the QGP starts cooling
down to Hagedorn temperature, $T\sim 170\,\text{MeV}$, or higher baryon
densities above $5\rho_{0}$ where the free quarks and gluons start clamping
together into hadrons. This is also observed at the initial stages of heavy-
ion-collisions Senger ; Braun1 ; Wagner . On the other hand, the string decay
hadronization occur due to nonperturbative effects. Under this picture, new
hadrons are formed out of quark-antiquark pairs or through gluon cascade
Webber .

Theoretically, Light-Front (LF) is one of the suitable theories for describing
relativistic interactions due to its natural association with the light-cone.
Hence, under this theory one can perform Fock expansion

$|\pi\rangle=|q\bar{q}\rangle+|q\bar{q}\,q\bar{q}\rangle+|q\bar{q}\,g\rangle+\cdots,$
(82)

where the valence $|q\bar{q}\rangle$ and the non valence quark components are
fully covered Brodsky ; Salme . Hence, there is many information on the
partonic structure of hadrons stored in the pions produced during
hadronization that can be further exploited. These processes give an insight
into the transition to the hypothetical QGP phase of matter where
deconfinement and chiral symmetry restoration are expected. These studies
gained more attention when ultra-relativistic heavy ion collision experiment
in BNL Brookhaven and CERN Geneva were able to create matter under extreme
conditions of temperature and densities necessary for phase transition Braun .
Pion-pion annihilation is particularly necessary since pions are the secondary
most abundant particles produced during the heavy ion collision from hot and
dense hadron gas. Such annihilation also result in the production of photon
and dilepton which form the basics for probing quark-gluon plasma phase and
further hadronizations. These particles are known to leave the dense matter
phase quickly without undergoing strong interactions Rapp ; Li ; Schulze .
However, the leptons are capable of annihilating into quarks and gluons
(partons) in a process popularly referred to as gluon bremsstralhaung Wiik ;
Wu .

The free parameters of the model are $\nu$, $\xi$, $|\varepsilon_{v}|$,
$\rho$, $\alpha$, $a$ and $\phi_{0}$, where $\nu$ is the degeneracy of the
gluons and its acceptable value for massless gluons in $\text{SU}(3)$
representation is $\nu=16$ while its value under $\text{SU}(2)$ representation
is $\nu=6$. It is known from QCD lattice simulation that $\sigma\sim
1\,\text{GeV/fm}$, therefore, we can determine from Eq.(28) that
$m_{\eta}=1.73\,\text{GeV}$ Morningstar ; Loan ; Chen ; Lee ; Bali  which is
precisely the scalar glueball mass. For zero degeneracy i.e. $\nu=0$, the
critical temperature Eq.(34) goes to infinity and all the glueballs get melted
leaving pure gluons Carter . We determined the gluon mass to be
$m_{A}=870\,\text{MeV}$ in the model framework. This value lies within the
values obtained from lattice simulation as referred to in Sec. III.3. Also
$|\varepsilon_{v}|$ is the magnitude of the QCD vacuum at the ground state. It
has different values depending on the model under consideration. Its value
estimated from sum rule for gluon condensate is $0.006\pm
0.012\,\text{GeV}^{4}$ Ioffe  and the Bag model is also quoted as
$(145)^{4}\,\text{MeV}^{4}$. From the model $|\varepsilon_{v}|=\langle
F^{\mu\nu}F_{\mu\nu}\rangle=\xi^{-2}=\rho a^{4}$, recalling that $\phi_{0}$ is
related to $a$ and $\alpha$ as $\phi_{0}=a/\alpha$, $a$ has the dimension of
energy and $\alpha$ is dimensionless.

Again, when we combine Eqs.(28) and (III.2) we can express the string tension
$\sigma$ as a function of temperature $T$

$\displaystyle\sigma(T)=\dfrac{m^{*2}_{\eta}(T)}{3}=\dfrac{m_{\eta}^{2}(0)}{3}\left[1-\dfrac{T^{4}}{T^{4}_{c}}\right].$
(83)

Here, we retrieve Eq.(28) at $T=0$, $\sigma(T=T_{c})=0$, and at $T\,>\,T_{c}$
we move into quark-gluon-plasma (QGP) phase where the particles interact in a
disorderly manner Yagi ; Pasechnik . Thus, the potential in Eq.(27) can be
expressed in terms of temperature as

$\displaystyle V_{c}(r,T)$
$\displaystyle=-\dfrac{1}{r}+\dfrac{m_{\eta}^{*2}(T)}{3}r$
$\displaystyle=-\dfrac{1}{r}+\dfrac{m_{\eta}^{2}(0)}{3}\left[1-\dfrac{T^{4}}{T^{4}_{c}}\right]r,$
(84)

where we recover Eq.(27) at $T=0$. Thermal deconfinement and the restoration
of chiral symmetry occur at $T=T_{c}$ Gupta ; Gottlieb  due to the dissolution
of the glueball mass, and we have QGP phase at $T\,>\,T_{c}$ Pasechnik ; Yagi
where the glueball mass becomes unstable. As established in $\text{SU}(2)$
group representation, it is known from hadron spectrum that the
$\text{SU}(2)_{L}\times\text{SU}(2)_{R}$ chiral symmetry is spontaneously
broken due to the nonperturbative dynamics of the theory, leading to color
confinement. Considering quarks representation in $\text{SU}(2)$, the color
charges may be screened by the gauge degrees of freedom. Under this
representation, we can possibly have quark-gluon color singlet thermionic
states, quark-antiquark color singlets or gluon-gluon color singlets.
Generally, in this paper, we investigate glueballs which will fall under the
gluon-gluon color singlet states.

Figure 1: The scalar field potential for different decay constants
$f_{\alpha}$

The potential has degenerate vacua.

Figure 2: The glueball potential $V(\eta)$ against the glueball field $\eta$
for different values of $f_{\alpha}$

The potential has a non degenerate and well defined vacuum giving the
glueballs a real mass $m_{\eta}$.

Figure 3: Confining Cornell-like potential

The perturbative and the nonperturbative nature of the potential is displayed.
Bellow the Fermi scale the gluons are asymptotically free while above the
Fermi scale we only find confined color neutral particles.

Figure 4: The screened electric(magnetic) field $\tilde{E}_{r}(\tilde{B}_{r})$
against $r$

This graph holds for both electric and magnetic fields, thus, both fields are
screened from the interior of the vacuum. It was sketched for
$\lambda=65\text{nm}$ below it, we have a mix or an intermediate states with
higher electric or magnetic field strength. Above the estimated $\lambda$ we
have a superconducting state because the vacuum is pure diamagnetic with no
penetrating electric or magnetic fields.

Figure 5: The gluon condensate with the mean glueball field $\bar{\eta}$

The condensate has its maximum value at $\bar{\eta}=0$ and vanishes at
$\bar{\eta}\simeq 1$.

Figure 6: Thermally fluctuating glueball mass with temperature

The glueball mass decreases sharply with increasing $T$ until it vanishes at
$T=T_{c}$, where the glueballs melt into gluons.

Figure 7: Variation of the gluon mass $m_{A}^{*}(T)/g$ with $T$

The gluon mass rather increases with temperature, because this phenomenon
occurs at a temperature where the bound states of gluons or glueballs melt
into gluons.

Figure 8: The string tension $\sigma(T)/\sigma_{0}$ against temperature
$T/T_{c}$

The string tension decreases sharply with $T$ and breaks or vanishes at
$T=T_{c}$ representing hadronization.

Figure 9: Temperature fluctuating Cornell-like potential.

The gradient of the graph decreases with increasing $T$ and flattens at
$T=T_{c}$ indicating deconfinement and hadronization phase.

Figure 10: The gluon condensate with varying temperature $T$

At $T=0$, $\langle
2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle_{T}/4|\varepsilon_{v}|=1$ representing
maximum condensate corresponding to $\bar{\eta}=0$. Also, at $T=T_{c}$,
$\langle 2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle_{T}/4|\varepsilon_{v}|=0$
representing minimum condensate corresponding to $\bar{\eta}\simeq 1$ Carter .

(a) Left Panel (b) Right Panel

Figure 11: Penetration depth $\lambda$ (left), superconducting $n_{s}$ and
normal $n_{n}$ monopole densities with temperature (right)

The penetration depth remains constant at $\lambda=\lambda_{0}$ representing
pure diamagnetic vacuum before it starts rising steadily with $T$ and becomes
infinite at $T=T_{c}$, where the fields are expected to penetrate the vacuum.
Thus, we have superconducting state at low temperatures and at relatively high
temperatures we have a mix or an intermediate states. On the other hand, the
superconducting monopole density $n_{s}$ decreases with increasing $T$ while
the normal monopole density $n_{n}$ increases with an increasing $T$.

###  V.2 Conclusion

The process of high energy annihilation of $\phi^{*}\phi$ to the production of
$\phi^{*}\phi$ through the interaction,
$\phi^{*}\phi\rightarrow\gamma\rightarrow{\phi}^{*}{\phi}$ during
hadronization in high particle density region is briefly addressed. Thus, we
focus on the discussion to cover the low energy regime where
$\tilde{\text{U}}(1)$ symmetry is spontaneously broken through Abelian Higgs
mechanism to give mass to the resulting glueballs. The scalar field in this
case plays the role of Higgs field, which undergoes modification into
glueballs with mass $m_{\eta}$. This leads to color confinement of glueballs.
We explored the behaviour of the scalar field potential $V(|\phi|)$ and the
glueball potential $V(\eta)$ with the decay constant $f_{\alpha}$ and the
results presented in Figs. 1 and 2. The model explains other confining (IR)
properties such as gluon condensation, glueball mass, gluon mass, string
tension and dual superconductivity through monopole condensation. The results
for color confinement, screened electric (magnetic) field due to the monopoles
and the gluon condensate are presented in Figs. 3, 4 and 5 respectively. The
glueball fields acquire their mass through SSB and their mass remain the most
relevant parameter throughout the confinement properties. The magnitude of the
glueball mass $m_{\eta}$ is precisely the same as the observed lightest scalar
glueball mass $m_{\eta}=1.73\,\text{GeV}$. This mass is responsible for the
string tension $\sigma$ that keeps the particles in a confined state and
appears in the monopole condensate as well. It also appears in the gluon
condensate $\langle 2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle$, capable of
hadronizing to form light pions in the low energy regime. In this paper two
forms of hadronization were discussed, the first one takes place at the high
energy regime where the individual quarks and gluons recombine to form
hadrons/pions. Secondly, at the low energy regime where the quark and an
antiquark pairs/valence gluons that form pions/glueballs get separated due to
long separation distances and the string tension that hold them together
breaks leading to hadronization. We also investigated the effect of
temperature on the effective scalar glueball mass $m_{\eta}^{*2}(T)$, the
gluon mass $m_{A}^{*}(T)$, string tension $\sigma(T)$, confining potential
$V_{c}(r,T)$, gluon condensate $\langle
2G(\eta)F^{\mu\nu}F_{\mu\nu}\rangle_{T}$, the penetration depth $\lambda(T)$,
the superconducting monopole densities $n_{s}(T)$, the normal monopole density
$n_{n}(T)$ and their results presented in Figs. 6, 7, 8, 9 10 and 11
respectively. We calculated the QCD $\beta$-function and the strong ‘running’
coupling $\alpha_{s}$ through renormalization group theory to enhance the
discussions on gluon mass generation. Additionally, the glueball mass and the
gluon masses were calculated and the outcome compared with lattice simulation
result, analytical study or phenomenological analyses to ascertain their
reasonability.

Finally, the model produces the correct behaviour of confining potential at
$T=0$, where the potential has linear growth with $r$, Eq.(27), consistent
with Cornell potential model. The potential keeps growing linearly for
$T\,<\,T_{c}$ with thermal deconfinement phase at $T\,\geqslant\,T_{c}$.
However, most of the results on QCD lattice calculations point to a
temperature correction of order $-T^{2}$ to the string tension Kaczmarek ;
Pisarski ; Forcrand  meanwhile, the model proposes correction in order of
$-T^{4}$ in Eq.(83). This can be regularized at $T\simeq T_{c}$ to obtain the
correct order $-T^{2}$. Indeed, we did not find such regularization necessary
because there are some proposed phenomenological models that suggest $-T^{4}$
correction Boschi-Filho  to the string tension as well. But we deem it
necessary to provide some explanations based on the model. In the previous
paper Ref.A-B  we obtained a correction of order $-T^{2}$ as elaborated in
Eq.(63). We observed that in Ref.A-B  the temperature correction to the string
tension came from the scalar field $\phi$ as elaborated in the latter part of
Sec. III.3. On the other hand, the temperature correction in the gauge field
leads to a correction in the order of $-T^{4}$ as discussed in Sec. III.2 and
elaborated in Sec. V.1 in Eqs.(83) and (V.1).

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant no.
165/2018), for partial financial support. FAB also acknowledges support from
CNPq (Grant no. 312104/2018-9).

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# Prediction of Influenza A Virus Infections in Humans using an Artificial
Neural Network Learning Approach

Charalambos Chrysostomou1∗, Harris Partaourides2 and Huseyin Seker3
1Computation-based Science and Technology Research Center, The Cyprus
Institute, 20 Konstantinou Kavafi Street, 2121, Aglantzia, Nicosia,
Cyprus2Department of Electrical Engineering, Computer Engineering and
Informatics, Cyprus University of Technology, 30 Archbishop Kyprianou Str.,
3036 Limassol, Cyprus3Department of Computer and Information Sciences, Faculty
of Engineering and Environment, The University of Northumbria at Newcastle,
NE1 8ST, Newcastle-upon-Tyne, The United<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Author

###### Abstract

The Influenza type A virus can be considered as one of the most severe viruses
that can infect multiple species with often fatal consequences to the hosts.
The Haemagglutinin (HA) gene of the virus has the potential to be a target for
antiviral drug development realised through accurate identification of its
sub-types and possible the targeted hosts. In this paper, to accurately
predict if an Influenza type A virus has the capability to infect human hosts,
by using only the HA gene, is therefore developed and tested. The predictive
model follows three main steps; (i) decoding the protein sequences into
numerical signals using EIIP amino acid scale, (ii) analysing these sequences
by using Discrete Fourier Transform (DFT) and extracting DFT-based features,
(iii) using a predictive model, based on Artificial Neural Networks and using
the features generated by DFT.

In this analysis, from the Influenza Research Database, 30724, 18236 and 8157
HA protein sequences were collected for Human, Avian and Swine respectively.
Given this set of the proteins, the proposed method yielded 97.36% ($\pm$
0.04%), 97.26% ($\pm$ 0.26%), 0.978 ($\pm$ 0.004), 0.963 ($\pm$ 0.005) and
0.945 ($\pm$ 0.005) for the training accuracy validation accuracy, precision,
recall and Mathews Correlation Coefficient (MCC) respectively, based on a
10-fold cross-validation. The classification model generated by using one of
the largest dataset, if not the largest, yields promising results that could
lead to early detection of such species and help develop precautionary
measurements for possible human infections.

###### Index Terms:

Artificial Neural Network, Amino Acid Indices, Discrete Fourier Transform
(DFT), Hemagglutinin (HA) Protein

##  I INTRODUCTION

The Influenza type A virus can be considered one of the most severe virus that
can infect both mammals and birds. The genome of the Influenza virus is
composed of eight segments that can encode more than 11 proteins [1]. One of
the most important proteins is the Haemagglutinin (HA), which is an essential
glycoprotein and a principal surface antigen which is responsible for
attaching the virions to hosts, deciding the pathogenicity and virulence [1].
Until now, 18 distinct Influenza A HA subtypes have been identified [2, 3].

The Influenza type A virus continually evolves due to the high mutation rate
and the constant changes to its genome. This constant adaptation usually makes
any new strain of virus more pathogenic than the previous. Furthermore, these
mutations also provide the virus with the ability to cross the species barrier
and may also affect the binding pattern of a virus, with catastrophic
consequences to the concerned species [4].

In the literature, previous efforts and analysis have been performed to
analyse and characterise the phylogenetic diversity, and discover mechanisms
that define the severity and distribution of influenza type A virus [5, 6, 7].
Additionally, as the authors concluded, classification and characterisation of
all the sequences with the proposed methods, was difficult [5, 6, 7], thus a
more advanced method is needed. Computational studies exist that tries to
characterise and analyse the Influenza type A with promising results [8]. In
the proposed method a computational, Artificial Neural Network (ANN) learning
based approach is created to predict if a particular virus has the capability
to infect humans, by only analysing the HA protein sequence.

The paper is organised as follows: Section II presents the methods and
materials developed and used, while Section III presents the results obtained.
Finally, concluding remarks are outlined in Section IV.

##  II METHODS AND MATERIAL

###  II-A Influenza A Hemagglutinin Proteins Data Set

For the proposed analysis 57117 HA Influenza type A protein sequences are
collected from the Influenza Research Database [9], for three species, Human,
Avian and Swine. More specifically, as Table I shows 30724, 18236 and 8157 HA
protein sequences were collected for Human, Avian and Swine respectively.
Furthermore, Table I shows the specific number of sequences for each class of
HA 1-18. Finally, figure 1 illustrates the percentage of HA proteins per class
and per species. For the analysis, classification of the HA protein sequences,
based on the ability of the virus to infect human hosts, the data were
separated into two groups. The first group contained all the sequences from HA
1-18 for the viruses that have the ability to infect the Humans hosts, and the
second group with the sequences that have the potential to infect the Avian
and Swine hosts. For the first and second groups, the total number of 30724
and 26393 HA protein sequences were used respectively.

TABLE I: Number of HA Protein Sequences HA Subtype | Human | Avian | Swine
---|---|---|---
H1 | 16145 | 650 | 5714
H2 | 96 | 462 | 2
H3 | 14055 | 1621 | 2138
H4 | 0 | 1478 | 5
H5 | 269 | 4242 | 32
H6 | 0 | 1529 | 2
H7 | 104 | 1809 | 1
H8 | 0 | 121 | 0
H9 | 13 | 3281 | 20
H10 | 4 | 834 | 1
H11 | 0 | 547 | 0
H12 | 0 | 178 | 0
H13 | 0 | 200 | 0
H14 | 0 | 17 | 0
H15 | 0 | 13 | 0
H16 | 0 | 150 | 0
H17 | 0 | 0 | 0
H18 | 0 | 0 | 0
Mixed | 38 | 1104 | 242
Total | 30724 | 18236 | 8157
Figure 1: HA Protein Sequences

###  II-B Data conversion and Normalisation

In this paper digital signal processing techniques are used to extract
information that can be directly used to characterise the HA proteins. In the
literature, various methods used signal processing in bioinformatics for
analysing and characterising protein sequences [10, 11, 12, 13, 14] such as
Complex resonant recognition model in analysing influenza a virus subtype
protein sequences [10], CISAPS: Complex informational spectrum for the
analysis of protein sequences [13] and Structural classification of protein
sequences based on signal processing and support vector machines [14].
Furthermore, previous studies [15] where signal processing was used to analyse
influenza A HA proteins aimed to identify new therapeutic targets for drug
development by better understanding the interaction of the influenza virus and
its receptors.

For the proposed analysis, signal processing methods are used, and more
specifically Discrete Fourier Transform (DFT), as shown in equations 1-3. The
analysis was performed directly to absolute spectrum. Before applying DFT to
the HA protein sequences, Electron-ion interaction potential (EIIP) [16, 17]
amino acid index, was used to convert alphanumerical sequences. The complete
list of the EIIP amino acid index can be found in Table II.

Discrete Fourier Transform (DFT)

$X(n)=\sum^{N-1}_{m=0}{x(m)e^{-j(2\pi/N)nm}}\>\>\>\>\>n=0,1,...,N-1$ (1)

where $X(n)$ are the DFT coefficients, N is the total number of points in the
series and $x(m)$ is the $m$th member of the numerical series. As the DFT
coefficients contain two mirror parts, only the $(N/2)$ points of the series
will be used.

The output of DFT is a complex sequence and can be formulated as

$X(n)=(R(n)+jI(n)),\>\>\>\>n=0,1,...,(N-1)/2$ (2)

where $R(n)$ and $I(n)$ are the Real and Imaginary parts of the sequence,
respectively. The absolute spectrum ($S_{(n)}$) can be formulated as

$S_{(n)}=X(n)X^{*}(n)=\left|X(n)\right|^{2},\>\>\>\>n=0,1,...,(N-1)/2$ (3)

where $X(n)$ are the DFT coefficients of the series $x(n)$, $X*(n)$ are the
complex conjugates.

The coefficients from the absolute spectrum will be used as a feature set to
represent the characteristics of different classes of proteins’ secondary
structure. The HA influenza A virus proteins sequences have different lengths,
and zero-padding was used to extend all the protein sequences to $N=1024$
before applying DFT. After DFT is applied the output of the absolute spectrum
includes 513 features. These features are used as input to the ANN model.

TABLE II: EIIP Values Amino acid | EIIP Values
---|---
Leucine | 0.0000
Isoleucine | 0.0000
Asparagine | 0.0036
Glycine | 0.0050
Glutamic acid | 0.0057
Valine | 0.0058
Proline | 0.0198
Histidine | 0.0242
Lysine | 0.0371
Alanine | 0.0373
Tyrosine | 0.0516
Tryptophan | 0.0548
Glutamine | 0.0761
Methionine | 0.0823
Serine | 0.0829
Cysteine | 0.0829
Threonine | 0.0941
Phenylalanine | 0.0946
Arginine | 0.0959
Aspartic acid | 0.1263

###  II-C Artificial Neural Network - Experimental Evaluation

Artificial Neural Networks (ANN) [18] are a computational method, based on an
extensive collection of artificial neurons, which mirrors the process a living
brain solves problems. Each neuron connects to multiple other neurons, which
can enforce or repress the impact on the activation event of the connected
neurons. The ANNs are considered as self-trained, rather than explicitly
programmed, and employed in research fields where the discovery of features
and classification is challenging in traditional classification systems.

For the proposed work, the ANN receives an input of 513 features derived from
the influenza type A virus pre-processing and returns the probability of the
virus infecting humans. For the proposed work, the binary classification
consist of 57117 samples of which 30724 can infect humans.

The network setup consists of a single hidden layer of 128 units, Glorot-style
uniform for initialization and rectified linear units for the activation
function. In order to train the ANN, the Adam optimizer [19] was used with
mini batch size of 128 for 200 epochs. We use 10 fold cross-validation and
show the network performance based on accuracy. The model was implemented by
utilising the Tensorflow [20] and Keras [21] libraries. A visual
representation of the model can be seen in Figure 2.

Figure 2: Artificial Neural Network Model

The high performance of both training and testing shows that for this type of
problem more advanced Neural Network models (such as Deep Neural Networks) and
regularization techniques are not needed.

##  III RESULTS AND DISCUSSION

In this paper, a classification model is presented, based on Artificial Neural
Networks, for the analysis and classification Influenza type A based upon the
ability to infect a human host solely by using the HA protein sequence. To
ensure that the proposed classification model is accurate and the results can
be generalised, 10-fold cross-validation was used. The total accuracy of the
predictive model with average training accuracy, testing accuracy, precision,
recall and MCC of 97.36% ($\pm$ 0.04%), 97.26% ($\pm$ 0.26%), 0.978 ($\pm$
0.004), 0.963 ($\pm$ 0.005) and 0.945 ($\pm$ 0.005) for the training accuracy
validation accuracy, precision, recall and Mathews Correlation Coefficient
(MCC) respectively. As the results show, the proposed model can distinguish HA
protein sequences with extremely high accuracy whenever the virus under
investigation will have the capability to infect human hosts. Detailed results
can be found in Table III.

TABLE III: Accuracy Results for the Prediction of Influenza A Virus Infections Fold | Training Accuracy | Validation Accuracy | Validation Precision | Validation Recall | Validation MCC
---|---|---|---|---|---
1 | 0.974 | 0.970 | 0.976 | 0.959 | 0.940
2 | 0.973 | 0.971 | 0.982 | 0.955 | 0.941
3 | 0.973 | 0.970 | 0.973 | 0.962 | 0.940
4 | 0.974 | 0.971 | 0.974 | 0.964 | 0.942
5 | 0.974 | 0.970 | 0.982 | 0.954 | 0.941
6 | 0.973 | 0.975 | 0.981 | 0.966 | 0.950
7 | 0.974 | 0.972 | 0.973 | 0.967 | 0.944
8 | 0.973 | 0.978 | 0.982 | 0.971 | 0.957
9 | 0.974 | 0.975 | 0.980 | 0.966 | 0.950
10 | 0.974 | 0.972 | 0.976 | 0.963 | 0.944
Average | 97.36% ($\pm$ 0.04%) | 97.26% ($\pm$ 0.26%) | 0.978 ($\pm$ 0.004) | 0.963 ($\pm$ 0.005) | 0.945 ($\pm$ 0.005)

##  IV CONCLUSIONS

The paper presents a highly successful predictive model to identify and
differentiate Influenza type A virus, which can and cannot infect Humans,
based on the HA gene, which is considered as a highly potential antiviral drug
candidate. The classification model was created by utilising one of the
largest possible datasets, if not the largest, in order to better generalise
the model. The classification model obtained over the 10-fold cross validation
yielded the average training accuracy, testing accuracy, precision, recall and
MCC of 97.36% ($\pm$ 0.04%), 97.26% ($\pm$ 0.26%), 0.978 ($\pm$ 0.004), 0.963
($\pm$ 0.005) and 0.945 ($\pm$ 0.005) respectively. Also, by applying signal
processing technique, namely Discrete Fourier Transform, on the protein
sequences, it was found that useful spectral characteristic features can be
distinguished that are capable of representing the protein groups and thus
further enhanced using the Artificial Neural Network based classifier.

As reported in the literature, there are over 600 amino acid indices, where
each represents a unique physicochemical feature of the protein [13], in
contrast to the one amino acid index used throughout this study. Future
studies are required to identify any potential amino acid indices that are
capable of representing, characterising and classify the Influenza type A HA
protein sequences. A computational tool that is capable of classifying and
distinguishing potentially dangerous viruses that have the capability to
infect Human hosts will be critical and required to monitor future outbreaks.
Future studies will investigate additional machine learning approaches, to
explore the efficiency of the methodology, using signal processing methods to
encode protein sequences.

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