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

Authors: Adamu Issifu; Francisco A. Brito
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

The glueballs lead to gluon and QCD monopole condensations as by-products of color confinement. A color dielectric functionG∣ϕ∣ coupled with a 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 the quark-gluon plasma (QGP) hadronization phase where the free quarks and gluons start clamping together to form hadrons. The QCD-like vacuum η2mη2FμνFμν, confining potential Vcr, string tension σ, penetration depth λ, superconducting and normal monopole densities (nsnn), and the effective masses (mη2 and mA2) will be investigated at finite temperature T. We also calculate the strong “running” coupling αs and subsequently the QCD β-function. The dual superconducting nature of the QCD vacuum will be investigated based on monopole condensation.


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## Body

## 1. Introduction

The Large Hadron Collider (LHC) is currently attracting the attention of particle physicists to investigate 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 substructures. Generally, jets are collimated bundles of hadrons constituted by quarks and gluons at short distances or high energies [1–3]. Jets played a significant role in the discovery of gluons (g) [4–7] and top quark (t) [8, 9]. These observations are 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 [10–13] 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 is reduced to about 1 GeV and below or at a separation of order 1 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 the formation of light pions. Hadron jet was first produced by annihilation of electron-positron to produce a two-jet event, e+e−⟶qq¯⟶2−jets. This two-jet structure [14] was first observed in 1975 at SPEAR (SLAC), and the spin 1/2 nature of the quarks was also established [15]. The jets are produced when qq¯ fly apart; more qq¯ pairs are produced which recombine with the existing pairs to form mesons and baryons as a by-product [16]. The first three-jet structure from the electron-positron annihilation process was observed by analysing 40 hadronic events at the center-of-mass energy 27.4 GeV from SPEAR [16]. These jets were subsequently observed clearly and separately [17, 18]. The discovery of the gluon through the three-jet process played a major role in the discovery of the Higgs particle [19–24] by ATLAS collaboration [25] and CMS collaboration [26] using the LHC at CERN. On the other hand, pion-pion annihilation serves as the principal source of dileptons (e−e+ and μ−μ+) produced from the hadron matter [27, 28]. Thus, the proper study of the main source of dilepton spectra observed experimentally proposes significant observables which help to understand the pion dynamics in a dense nuclear matter that exist at the beginning of the collision [29].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 gluong 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∣ϕ∣. 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 η after SSB, to give meaning to color confinement and its associated properties. While ϕ initiates the annihilation and production processes, η 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 [30, 31] that also suggested “infrared Abelian dominance.” Fast forward, it has also been shown that about 92% of the QCD string tension is Abelian [32], so an Abelian approximation will not be out of place. Further studies of Abelian dominance for color confinement in SU2 and SU3 lattice QCD can be found in Ref. [33–37]. An interested reader can also refer to Ref. [38] for more recent results on Abelian dominance in QCD theory. This approach has also been adopted in other confinement models—see [39–41] and references therein for more justifications.The annihilation ofϕ∗ϕ 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 U1 symmetry to an energy regime is suitable for describing color confinement with a glueball field η. Because the running coupling, αsQ2, of the strong interaction decreases with momentum increases (short distances), ϕ∗ϕ annihilation can be studied from perturbation theory (ϕ∗ϕ⟶ϕ∗ϕ) similar to annihilation and creation of pions (π−π+⟶π−π+γ) and Bhabha scattering (e−e+⟶e−e+) [42]. A low energy analyses of ϕ∗ϕ annihilation will be made to show color confinement, bound states of the gluons (glueball masses) [43–45], and the QCD vacuum [46, 47] 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 nonzero expectation value of the Higgs field. That notwithstanding, quark masses continue to be an input parameter in the standard model [48]. 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 nontrivial solution leads to the creation of quark masses resulting in “dynamical chiral symmetry breaking,” which is a consequence of confinement [49–54]. 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 spontaneous symmetry breaking [55]. Monopoles play important role in color confinement. Monopole condensation produces a 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 [56–58] and Polyakov loop [59] to establish its confinement properties. The involvement of monopoles in chiral symmetry has recently been investigated as well [60, 61]. The impact of monopoles in quark-gluon plasma (QGP) has also been studied in [62–64].The paper is organised as follows: In Section2, we introduce the Lagrangian density that will be the basics for this study. In Section 3, we present spontaneous symmetry breaking of the model presented in the previous sections; this section is divided into four subsections; in Section 3.1, we investigate confinement of glueballs; in Section 3.2, we study the effective masses; we dedicate Section 3.3 to gluon condensation, and in Section 4, we investigate the monopole condensation. We proceed to present the analysis in Section 5.1 and the final findings in Section 5.2. We have adopted the natural units c=ℏ=kB=1, except otherwise stated.


## 2. The Model

We start with a Lagrangian density developed by exploring gauge-invariant properties (see [39]):
(1)L=ημνDμϕDνϕ∗−14GϕFμνFμν−14F~μνF~μν−Vϕ=ημν∂μϕ+iqA~μϕ∂νϕ∗−iqA~νϕ∗−14GϕFμνFμν−14F~μνF~μν−Vϕ,where Dμ=∂μ+iqA~μ, F~μν=∂μA~ν−∂νA~μ, and Fμν=∂μAν−∂νAμ 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 ϕ∗ϕ and its transformation into glueballs η at low energies. The complex scalar fields ϕx and its conjugate ϕ∗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 F~μνF~μν are the suitable scenario to address a 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 GϕFμνFμν. The equations of motion are
(2)DμDμϕ+14∂Gϕ∂ϕ∗FμνFμν+∂Vϕ∂ϕ∗=0,(3)∂μF~μν=−iqϕ∗∂νϕ−ϕ∂νϕ∗+2q2ϕ∗ϕA~ν=jϕν,(4)∂μGϕFμν=0.We will adopt a complex scalar field potential of the form(5)Vϕ=ρ4α2ϕ2−a22,where
(6)ϕ=ϕ1+iϕ22.We can isolate the electromagnetic interactions among the fields [65] for study using Equation (3). This enables us to visualize how the fields annihilate and create identical particles [66, 67] through single-photon exchange, ϕp1ϕ∗p2⟶γ⟶ϕp1′ϕ∗p2′, in high energy regime [68].


## 3. Confinement, Effective Masses, Gluon Condensation, and Dual Superconductivity

To discuss the color confinement and its consequences, we need to follow spontaneous symmetry breaking of theU~1 symmetry in Lagrangian Equation (1) to give mass to the resulting glueball fields and their 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:
(7)ϕx⟶ϕ′x=eiqαxϕ,A~μx⟶A~′μx=A~μx−∂μαx.The vacuum expectation value of the potential Equation (5),
(8)ϕ0=±aα,breaks the U~1 symmetry spontaneously. Considering two real scalar fields ηr and ζr representing small fluctuations about the vacuum, a shift in the vacuum can be expressed as
(9)ϕ⟶ηr+ϕ0,where ϕ0≡ϕ0. Hence, the potential takes the form
(10)Vη=Vϕ∣ϕ∣0+V′ϕ∣ϕ∣0η+12V′′ϕϕ0η2=ρα2a2η2.Consequently, we can suitably parametrize the scalar field such that,(11)ϕ=eiζ/ϕ0ϕ0+η2≈ϕ0+ηr+iζr2.The Lagrangian in Equation (1) becomes
(12)L=12∂μη∂μη−2ρα2a2η2+12∂μζ∂μζ−2qϕ0A~μ∂μζ+q2ϕ02A~μA~μ−14GηFμνFμν−14F~μνF~μν+⋯,where η is associated with the glueball fields and ζ is also associated with the massless Goldstone bosons which will be swallowed by the massive gauge fields through the gauge transformation [39]:
(13)A~μ⟶A~′μ=A~μ−1qϕ0∂μζ.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 nonrelativistic particles. As a result, they can be represented by spin-0 scalar fields. In the nonrelativistic regime, they exist as Goldstone bosonsζ, signifying the breakdown of chiral symmetry [69]. Pions are generally produced through matter annihilations such as pp¯, NN¯, and ee+, which undergo transition from baryon structures to mesons. This is an interesting phenomenon in low energy hadron physics [70, 71]. Pions have a flavour structure and other quantum numbers that permit us to classify them as bound states of quark and an antiquark. However, the valence quarks which characterize the flavour structure are surrounded by gluons and quark-antiquark pairs. Physically, glue-rich components mix with pions (qq¯) forming an enriched spectrum of isospin-zero states and qq¯g hybrid states [72]. The glueball spectrum and their corresponding quantum numbers are known in lattice gauge theory predictions [43, 73]. Additionally, the existence of glueballs has not been decisive because of the fear of possible mixing with quark degrees of freedom [74]. 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 occurs when the separation distance between the valence gluons exceeds some thresholds (>1 fm) leading to hadronization.Consequently, the Lagrangian can be simplified as(14)L=12∂μη∂μη−Vη−14GηFμνFμν−14F~μνF~μν+q2ϕ022A~μA~μ,where Vη=ρα2a2η2. The equations of motion for this Lagrangian are
(15)∂μ∂μη+∂Gη∂ηFμνFμν+∂Vη∂η=0,(16)∂μF~μν=q2ϕ02A~ν=jϕ0ν,(17)∂μGηFμν=0.The Feynman propagator for the glueball fieldη in Equation (12) reads
(18)Dηp=ip2−mη2+iε,where mη2=2ρα2a2.


### 3.1. Confinement

Here, we are interested in calculating static confining potential, such that onlychromoelectric flux that results in confining electric field is present, while chromomagnetic flux influence is completely eliminated. Thus, in spherical coordinates, Equations (15) and (17) become, respectively,
(19)1r2ddrr2dηdr=12∂Gη∂ηE2−∂Vη∂η,(20)1r2∂∂rr2GηE=0⟶E=Λr2Gη,where Λ=q/4πε0 is the integration constant. We are also now setting Gη=ξ2Vη (similar process was adopted in Ref. [39–41]), where ξ2 is a dimensionful constant associated with the Regge slope 2πα′2 that absorbs the dimensionality of Vη, so Gη remains dimensionless and Equation (19) reads
(21)1r2ddrr2dηdr=−∂∂ηΛ221ξ2Vη1r4+V⟶η′′+2rη′+mη2η=0.In the last step, we have assumed that the particles are far away from the charge sourceq, and in such a limit, terms of the order O1/r4 can be disregarded. This equation has a solution
(22)ηr=asinmηrαmηr.Substituting this result into the electric field Equation (20),
(23)E=Λr2G=2Λα2a2ξ2sin2mηr,and using the well-known expression for determining the electrodynamic potential,
(24)Vr=∫Edr,to determine the confining potential Vcr, we get
(25)Vcr=−2Λα2cotmηra2mηξ2+c≃−2Λα2a2mηξ21mηr−mηr3−Or3+c≃2α2a2ξ2mη2−1r+mη2r3;so, setting
(26)a4ξ2ρ=1⟶ξ2=1a4ρleads to
(27)Vcr=−1r+mη2r3,with string tension
(28)σ=mη23=2ρα2a23.In the last step of Equation (25), we substituted Λ=q/4πε0=1 and c=0 for simplicity.


### 3.2. Effective Masses

Glueballs are simply bound states of gluons, and they are “white” or colorless in nature. They areflavour blind and are capable of decaying. Scalar glueball with quantum number JPC=0++ is observed by lattice QCD simulations as the lightest with a mass of about 1.7 GeV [43–45].The dispersion relation for the glueball and the gluon excitations including thermal fluctuations can be expressed as(29)Eη2=k2+mη∗2,EA2=k2+mA∗2.We can calculate the effective thermal fluctuating glueball mass by taking the second derivative of the Lagrangian density Equation (14) with respect to η:
(30)mη∗2=−∂2L∂η2=2ρα2a2+12ρα2a2ξ2FμνFμν.We redefine the glueball fieldη to include the thermal fluctuations as η=η¯+Δ, with restriction, Δ=0. The angle brackets represent thermal average, and η¯ is the mean field of the glueballs. This equation can be solved by determining explicitly the nature of FμνFμν and Δ2 using field quantum distributions. By the standard approach, we can express
(31)FμνFμν=−ν2π2∫0∞dkk4EAnBEA,Δ2=12π2∫0∞dkk2EηnBEη.Here, nBx=exβ−1−1 represents the Bose-Einstein distribution function and β=1/T, where T is the temperature. We can analytically solve Equation (31) at a high energy regime, where E≈kc, kBT≫mAc2 and mA is the gluon/screening mass. Therefore,
(32)FμνFμν=−ν2π2∫0∞dkk4EA1eβEA−1=−νT42π2∫0∞x3dxex−1=−4T4Tc4ξ2,where we have substituted x=k/T and used the standard integral
(33)∫x3dxex−1=π415.In the last step, we have substituted(34)Tc4ξ2=152π2ν,being Tc the critical temperature. Following the same analysis,
(35)Δ2=12π2∫0∞dkk2Eη1eβEη−1≃T22π2∫0∞xdxex−1≃T212,where we have assumed kBT≫mηc2, with mη being the glueball mass, and considered the standard integral,
(36)∫xdxex−1=π26.Thus, Equation (30) becomes
(37)mη∗2T=2ρα2a21−T4Tc4=mη201−T4Tc4.When we take a thermal average of Equation (15), we get ξ2mη2ηFμνFμν=−mη2η; therefore, η¯=0 and η¯=1 are exact solutions of the glueball fields [75]. We further demonstrate in Figures 1 and 2 that η¯ has solution η¯=0 at T=0 and increases steady as T increases to its maximum η¯≃1 at T=Tc, corresponding to no glueball fields and the melting of the glueballs.Figure 1
The gluon condensate with the mean glueball fieldη¯. The condensate has its maximum value at η¯=0 and vanishes at η¯≃1.Figure 2
The gluon condensate with varying temperatureT. At T=0, 2GηFμνFμνT/4∣εv∣=1 representing maximum condensate corresponding to η¯=0. Also, at T=Tc, 2GηFμνFμνT/4∣εv∣=0 representing minimum condensate corresponding to η¯≃1 [75].


### 3.3. Gluon Condensation

Classical gluodynamics is described by the Lagrangian density:(38)L′=−14FaμνFμνa,which is invariant under scale and conformal symmetries, x⟶λx, and also produces vanishing gluon condensate FaμνFμνa=0. Meanwhile, the symmetries are broken when quantum correction −εv is added to the Lagrangian, i.e.,
(39)L′=−14FaμνFμνa+εv,resulting in nonvanishing gluon condensate, FaμνFμνa>0. This correction is the consequence of the well-known scale anomaly observed in QCD energy-momentum (θμν) trace
(40)θμμ=βg2gFaμνFμνa,where βg is the “so-called” beta-function of the strong coupling g, with a leading term
(41)βg=−11g34π2.Accordingly, we have nonzero vacuum expectation, expressed as(42)θμμ=−4εv.The second term in Equation (14) acts similar to the quantum correction and explicitly breaks the scale and the conformal symmetries, so its energy-momentum tensor satisfies Equation (42) [40, 76–78]. We will compute the trace of the energy-momentum tensor of Equation (14) using the relation
(43)θμμ=4Vη+η□η.Substituting the equation of motion Equation (15) into Equation (43) yields
(44)θμμ=4Vη−η∂G∂ηFμνFμν−η∂V∂η=4V~−ηG′FμνFμν,where in the last step we have substituted V~=Vη−ηV′η/4 with G′ and V′ representing the derivative of G and V with respect to η, respectively. In order to make the expression easy to be compared with Equation (42), we rescale V~ as V~η⟶−εvV~, consequently,
(45)ηG′FμνFμν=4εv1−V~.We recover the classical expression forFμνFμν=0 when we set ∣εv∣⟶0. Following the potential defined below Equation (14), we can express
(46)V~=V−ηV′4=ρa2α2η22=mη2η24;thus, Equation (45) becomes
(47)2GηFμνFμν=4εv1−mη2η24.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 cannot 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 breakdown in the vacuum. Additionally, the gluon massmA can be derived from the vacuum as a function of the glueball mass as
(48)mA2=mη24.Generally, scalar glueball mass is related to the gluon mass as(49)m0++mA=6≅2.45;considering the leading order [79, 80] from Yang-Mills theory with an auxiliary field ϕ, here, the scalar glueball mass m0++ represents fluctuations around ϕ. The gluon mass is determined to be mA=600~700 MeV as obtained from lattice simulations [81–83]. Heavier gluon masses have also been observed in the range of ~1 GeV by phenomenological analysis [84, 85], lattice simulation [86], and analytical investigations [87]. The glueball mass mη is responsible for all the confinement properties and the chiral symmetry breaking in the vacuum. Observing that the string tension is σ~1 GeV/fm, we can identify mη2=3GeV2, corresponding to a gluon mass mA=0.87GeV, from the model framework.In terms of temperature, we use Equation (35), and the gluon condensate becomes
(50)2GηFμνFμν=4εv1−mη24ϕ02T2ϕ0212=4εv1−T2Tcη2,whereTcη=48mη21/2.Corresponding to a temperature fluctuating gluon mass(51)mA∗2=mη2T248ϕ02=g~2T2,where g~2=mη2/48ϕ02 is a dimensionless coupling constant. This expression looks like the Debye mass derived from the leading order of QCD coupling expansion [88–90]. It carries nonperturbative property of the theory. Comparing Equations (40) and (44), we identify
(52)βg2g=−ηG′η⟶β1r2=−2gηG′η.Additionally, we can determine the strong running couplingαs from the renormalization group theory [91]
(53)βQ2=Q2dαsdQ2;therefore,(54)βη≃−ηdGdη=−ξ2mη2η2r=2Gη=β1r2withg=1.Comparatively, the strong running coupling can be identified asαsη=Gη=αs1/r2, with Q≡1/r, as the space-like momentum associated with the four-vector momentum as Q2≡−q2. Thus, the color dielectric function is associated with the QCD β-function and the strong running coupling. We can write the Feynman propagator for the interaction by considering the left-hand side of Equation (45):
(55)FμνFμν=−2Aν□gμν−∂μ∂νAμ=−2Aν∂2gμν−1−1α∂μ∂νAμ=−2Aνkk2gμν−1−1αkμkνAμ−k⟶ηG′ηFμνFμν=−2AνkηG′ηk2gμν−1−1αkμkνAμ−k=Aνkβq2k2gμν−1−1αkμkνAμ−k, where we have added the gauge fixing term 2/α∂μAμ2 and made the Fourier transform
(56)Aμk=∫d4k2π4e−ikxAμx.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−2ηG′η=β1/r2≡βq2 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 [40], if we consider temperature fluctuations in the scalar field ϕ. In that case, we move away from the known form of field theory, i.e., perturbation about the vacuum, ϕ=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 ϕ⟶ϕ+ϕT into the potential in Equation (5), where ϕT plays a similar role as Δ introduced above. Thus,
(57)ϕT2=12π2∫0∞p2dpEeEβ−1≃T212;here, we have used the high-temperature approximation kBT≫mϕc2, E≈pc and the potential becomes
(58)Vϕ,T=ρ4α2ϕ2+α2ϕT2−a22=ρ4α2ϕ2−a21−T2Tcϕ22=ρ4α2ϕ2−a~22,where ϕT=0,
(59)Tcϕ2=12a2α2,a~2=a21−T2Tcϕ2.The glueball mass can be calculated from the fluctuations in Equation (58) around the vacuum, such that
(60)mϕ∗2T=∂2V∂ϕ2ϕ0=2ρα2a~2=2ρα2a21−T2Tcϕ2=mη201−T2Tcϕ2,where ϕ0=a~/α is the vacuum of the potential. We observe that mϕ∗20=mη∗20 at T=0, similar to Equation (37). The differences in the degree of the temperature in both equations arise because Equation (37) has temperature correction to the gauge field FμνFμν, while in Equation (37), we have temperature correction to the scalar field ϕT2. Following the same procedure as the one used in deriving Equation (47), we can express
(61)2GηFμνFμν=−4εmϕ∗2Tη24−1=−4εmϕ2041−T2Tcϕ2η2−1.Here, we have replacedmη∗2T with mϕ∗2T from Equation (47). The negative sign and η2 differences between this result and Ref. [40] might be due to the differences in the methods adopted for calculating the energy-momentum trace tensor θμμ. Also, if we consider the IR part of the potential in Equation (25) as studied in Ref. [40], the potential becomes
(62)Vcr=σr=mϕ23r⟶VcT,r=mϕ∗2T3r=mϕ2031−T2Tcϕ2r,with string tension
(63)σT=mϕ∗2T3,which is similar to the result in Ref. [40]. Therefore, the approach adopted in this paper accounts for the temperature corrections to both the glueball field Δ2 in Equation (35) and the gauge field FμνFμν, while in Ref. [40], only the thermal correction to the scalar field ϕT2 was considered.


## 3.1. Confinement

Here, we are interested in calculating static confining potential, such that onlychromoelectric flux that results in confining electric field is present, while chromomagnetic flux influence is completely eliminated. Thus, in spherical coordinates, Equations (15) and (17) become, respectively,
(19)1r2ddrr2dηdr=12∂Gη∂ηE2−∂Vη∂η,(20)1r2∂∂rr2GηE=0⟶E=Λr2Gη,where Λ=q/4πε0 is the integration constant. We are also now setting Gη=ξ2Vη (similar process was adopted in Ref. [39–41]), where ξ2 is a dimensionful constant associated with the Regge slope 2πα′2 that absorbs the dimensionality of Vη, so Gη remains dimensionless and Equation (19) reads
(21)1r2ddrr2dηdr=−∂∂ηΛ221ξ2Vη1r4+V⟶η′′+2rη′+mη2η=0.In the last step, we have assumed that the particles are far away from the charge sourceq, and in such a limit, terms of the order O1/r4 can be disregarded. This equation has a solution
(22)ηr=asinmηrαmηr.Substituting this result into the electric field Equation (20),
(23)E=Λr2G=2Λα2a2ξ2sin2mηr,and using the well-known expression for determining the electrodynamic potential,
(24)Vr=∫Edr,to determine the confining potential Vcr, we get
(25)Vcr=−2Λα2cotmηra2mηξ2+c≃−2Λα2a2mηξ21mηr−mηr3−Or3+c≃2α2a2ξ2mη2−1r+mη2r3;so, setting
(26)a4ξ2ρ=1⟶ξ2=1a4ρleads to
(27)Vcr=−1r+mη2r3,with string tension
(28)σ=mη23=2ρα2a23.In the last step of Equation (25), we substituted Λ=q/4πε0=1 and c=0 for simplicity.


## 3.2. Effective Masses

Glueballs are simply bound states of gluons, and they are “white” or colorless in nature. They areflavour blind and are capable of decaying. Scalar glueball with quantum number JPC=0++ is observed by lattice QCD simulations as the lightest with a mass of about 1.7 GeV [43–45].The dispersion relation for the glueball and the gluon excitations including thermal fluctuations can be expressed as(29)Eη2=k2+mη∗2,EA2=k2+mA∗2.We can calculate the effective thermal fluctuating glueball mass by taking the second derivative of the Lagrangian density Equation (14) with respect to η:
(30)mη∗2=−∂2L∂η2=2ρα2a2+12ρα2a2ξ2FμνFμν.We redefine the glueball fieldη to include the thermal fluctuations as η=η¯+Δ, with restriction, Δ=0. The angle brackets represent thermal average, and η¯ is the mean field of the glueballs. This equation can be solved by determining explicitly the nature of FμνFμν and Δ2 using field quantum distributions. By the standard approach, we can express
(31)FμνFμν=−ν2π2∫0∞dkk4EAnBEA,Δ2=12π2∫0∞dkk2EηnBEη.Here, nBx=exβ−1−1 represents the Bose-Einstein distribution function and β=1/T, where T is the temperature. We can analytically solve Equation (31) at a high energy regime, where E≈kc, kBT≫mAc2 and mA is the gluon/screening mass. Therefore,
(32)FμνFμν=−ν2π2∫0∞dkk4EA1eβEA−1=−νT42π2∫0∞x3dxex−1=−4T4Tc4ξ2,where we have substituted x=k/T and used the standard integral
(33)∫x3dxex−1=π415.In the last step, we have substituted(34)Tc4ξ2=152π2ν,being Tc the critical temperature. Following the same analysis,
(35)Δ2=12π2∫0∞dkk2Eη1eβEη−1≃T22π2∫0∞xdxex−1≃T212,where we have assumed kBT≫mηc2, with mη being the glueball mass, and considered the standard integral,
(36)∫xdxex−1=π26.Thus, Equation (30) becomes
(37)mη∗2T=2ρα2a21−T4Tc4=mη201−T4Tc4.When we take a thermal average of Equation (15), we get ξ2mη2ηFμνFμν=−mη2η; therefore, η¯=0 and η¯=1 are exact solutions of the glueball fields [75]. We further demonstrate in Figures 1 and 2 that η¯ has solution η¯=0 at T=0 and increases steady as T increases to its maximum η¯≃1 at T=Tc, corresponding to no glueball fields and the melting of the glueballs.Figure 1
The gluon condensate with the mean glueball fieldη¯. The condensate has its maximum value at η¯=0 and vanishes at η¯≃1.Figure 2
The gluon condensate with varying temperatureT. At T=0, 2GηFμνFμνT/4∣εv∣=1 representing maximum condensate corresponding to η¯=0. Also, at T=Tc, 2GηFμνFμνT/4∣εv∣=0 representing minimum condensate corresponding to η¯≃1 [75].


## 3.3. Gluon Condensation

Classical gluodynamics is described by the Lagrangian density:(38)L′=−14FaμνFμνa,which is invariant under scale and conformal symmetries, x⟶λx, and also produces vanishing gluon condensate FaμνFμνa=0. Meanwhile, the symmetries are broken when quantum correction −εv is added to the Lagrangian, i.e.,
(39)L′=−14FaμνFμνa+εv,resulting in nonvanishing gluon condensate, FaμνFμνa>0. This correction is the consequence of the well-known scale anomaly observed in QCD energy-momentum (θμν) trace
(40)θμμ=βg2gFaμνFμνa,where βg is the “so-called” beta-function of the strong coupling g, with a leading term
(41)βg=−11g34π2.Accordingly, we have nonzero vacuum expectation, expressed as(42)θμμ=−4εv.The second term in Equation (14) acts similar to the quantum correction and explicitly breaks the scale and the conformal symmetries, so its energy-momentum tensor satisfies Equation (42) [40, 76–78]. We will compute the trace of the energy-momentum tensor of Equation (14) using the relation
(43)θμμ=4Vη+η□η.Substituting the equation of motion Equation (15) into Equation (43) yields
(44)θμμ=4Vη−η∂G∂ηFμνFμν−η∂V∂η=4V~−ηG′FμνFμν,where in the last step we have substituted V~=Vη−ηV′η/4 with G′ and V′ representing the derivative of G and V with respect to η, respectively. In order to make the expression easy to be compared with Equation (42), we rescale V~ as V~η⟶−εvV~, consequently,
(45)ηG′FμνFμν=4εv1−V~.We recover the classical expression forFμνFμν=0 when we set ∣εv∣⟶0. Following the potential defined below Equation (14), we can express
(46)V~=V−ηV′4=ρa2α2η22=mη2η24;thus, Equation (45) becomes
(47)2GηFμνFμν=4εv1−mη2η24.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 cannot 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 breakdown in the vacuum. Additionally, the gluon massmA can be derived from the vacuum as a function of the glueball mass as
(48)mA2=mη24.Generally, scalar glueball mass is related to the gluon mass as(49)m0++mA=6≅2.45;considering the leading order [79, 80] from Yang-Mills theory with an auxiliary field ϕ, here, the scalar glueball mass m0++ represents fluctuations around ϕ. The gluon mass is determined to be mA=600~700 MeV as obtained from lattice simulations [81–83]. Heavier gluon masses have also been observed in the range of ~1 GeV by phenomenological analysis [84, 85], lattice simulation [86], and analytical investigations [87]. The glueball mass mη is responsible for all the confinement properties and the chiral symmetry breaking in the vacuum. Observing that the string tension is σ~1 GeV/fm, we can identify mη2=3GeV2, corresponding to a gluon mass mA=0.87GeV, from the model framework.In terms of temperature, we use Equation (35), and the gluon condensate becomes
(50)2GηFμνFμν=4εv1−mη24ϕ02T2ϕ0212=4εv1−T2Tcη2,whereTcη=48mη21/2.Corresponding to a temperature fluctuating gluon mass(51)mA∗2=mη2T248ϕ02=g~2T2,where g~2=mη2/48ϕ02 is a dimensionless coupling constant. This expression looks like the Debye mass derived from the leading order of QCD coupling expansion [88–90]. It carries nonperturbative property of the theory. Comparing Equations (40) and (44), we identify
(52)βg2g=−ηG′η⟶β1r2=−2gηG′η.Additionally, we can determine the strong running couplingαs from the renormalization group theory [91]
(53)βQ2=Q2dαsdQ2;therefore,(54)βη≃−ηdGdη=−ξ2mη2η2r=2Gη=β1r2withg=1.Comparatively, the strong running coupling can be identified asαsη=Gη=αs1/r2, with Q≡1/r, as the space-like momentum associated with the four-vector momentum as Q2≡−q2. Thus, the color dielectric function is associated with the QCD β-function and the strong running coupling. We can write the Feynman propagator for the interaction by considering the left-hand side of Equation (45):
(55)FμνFμν=−2Aν□gμν−∂μ∂νAμ=−2Aν∂2gμν−1−1α∂μ∂νAμ=−2Aνkk2gμν−1−1αkμkνAμ−k⟶ηG′ηFμνFμν=−2AνkηG′ηk2gμν−1−1αkμkνAμ−k=Aνkβq2k2gμν−1−1αkμkνAμ−k, where we have added the gauge fixing term 2/α∂μAμ2 and made the Fourier transform
(56)Aμk=∫d4k2π4e−ikxAμx.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−2ηG′η=β1/r2≡βq2 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 [40], if we consider temperature fluctuations in the scalar field ϕ. In that case, we move away from the known form of field theory, i.e., perturbation about the vacuum, ϕ=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 ϕ⟶ϕ+ϕT into the potential in Equation (5), where ϕT plays a similar role as Δ introduced above. Thus,
(57)ϕT2=12π2∫0∞p2dpEeEβ−1≃T212;here, we have used the high-temperature approximation kBT≫mϕc2, E≈pc and the potential becomes
(58)Vϕ,T=ρ4α2ϕ2+α2ϕT2−a22=ρ4α2ϕ2−a21−T2Tcϕ22=ρ4α2ϕ2−a~22,where ϕT=0,
(59)Tcϕ2=12a2α2,a~2=a21−T2Tcϕ2.The glueball mass can be calculated from the fluctuations in Equation (58) around the vacuum, such that
(60)mϕ∗2T=∂2V∂ϕ2ϕ0=2ρα2a~2=2ρα2a21−T2Tcϕ2=mη201−T2Tcϕ2,where ϕ0=a~/α is the vacuum of the potential. We observe that mϕ∗20=mη∗20 at T=0, similar to Equation (37). The differences in the degree of the temperature in both equations arise because Equation (37) has temperature correction to the gauge field FμνFμν, while in Equation (37), we have temperature correction to the scalar field ϕT2. Following the same procedure as the one used in deriving Equation (47), we can express
(61)2GηFμνFμν=−4εmϕ∗2Tη24−1=−4εmϕ2041−T2Tcϕ2η2−1.Here, we have replacedmη∗2T with mϕ∗2T from Equation (47). The negative sign and η2 differences between this result and Ref. [40] might be due to the differences in the methods adopted for calculating the energy-momentum trace tensor θμμ. Also, if we consider the IR part of the potential in Equation (25) as studied in Ref. [40], the potential becomes
(62)Vcr=σr=mϕ23r⟶VcT,r=mϕ∗2T3r=mϕ2031−T2Tcϕ2r,with string tension
(63)σT=mϕ∗2T3,which is similar to the result in Ref. [40]. Therefore, the approach adopted in this paper accounts for the temperature corrections to both the glueball field Δ2 in Equation (35) and the gauge field FμνFμν, while in Ref. [40], only the thermal correction to the scalar field ϕT2 was considered.


## 4. Dual Superconductivity

In this section, we will discuss brieflycolor superconductivity and proceed to elaborate on dual superconductivity in detail. The fields ϕ∗ϕ annihilate and create identical fields through a decay process, ϕp1ϕ∗p2⟶ϕp1′ϕ∗p2′. The dual gauge field F~μνF~μν which mediates the interaction of the scalar fields ϕ 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 [92]. In the context of this work, the 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, multiparticle states are formed in the high energy regime. In such a high particle density regime, the charged scalar fields form Bose-Einstein condensation [93, 94] similar to induced isospin imbalance systems leading to color superconductivity [95, 96]. With this type of condensation, there is a higher occupation number at the ground state than the excited states. So, temperature increase goes into reducing the occupation numbers [97]. There are several works on nonzero isospin chemical potential (μI) and baryon chemical potential (μB) in pion condensation [98–100], and they are related to the two flavour quarks that constitute the pions as μI=μu−μd/2 and μB=μu+μd/2, respectively. It has also been established [101] analytically at the quark level that the critical isospin chemical potential [102, 103] for pion superfluidity is precisely the pion mass μIc=mπ in vacuum. This behaviour and other related isospin chemical potentials in pion condensate have been investigated using the Nambu-Jona-Lasinio (NJL) model, ladder-QCD [98].In the model framework, the color superconducting property can be studied from the propagator,ϕϕ∗. The high density region where the particles are asymptotically free forms deconfined matter leading to Bose-Einstein condensation resulting in the color superconducting phase [96, 104, 105]. On the other hand, at the low energy region, the fields modify into glueballs with real mass, mη, 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 [106]. This phenomenon was first studied by Bardeen, Cooper, and Schrieffer (BCS) [107]. The BCS mechanism for pairing in general appears to be more robust in dense quark matter than superconducting metals due to the nature of interactions among quarks. The virtual fermions in the vacuum make it color diamagnetic preventing both the electric and magnetic fields from penetrating [2].We turn attention to thedual superconductivity in color confinement whose property we can derive from Equation (16). This subject was first proposed by Nambu, ‘t Hooft, and Mandelstam [56]. While the color superconducting phase takes place at the deconfined matter regime, the dual superconductivity is observed at the confining phase [108]. In the model framework, dual superconductivity is observed at the high glueball condensation region, η⟶0, i.e., a region of strong confinement. We will use the rest of this section to discuss dual superconductivity in detail within the model framework. In this phenomenon, the QCD vacuum is seen as a color magnetic monopole condensate resulting in 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 in the formation of a color flux tube between quark and antiquark used in describing string pictures of hadrons [38]. In this regard, magnetic monopole condensation is crucial to color confinement and dual superconductivity. Monopoles also play a role in chiral symmetry breaking [109] and strongly coupled QGP. They are involved in the decay of plasma formed immediately after relativistic heavy ion collision [62–64]. Even though monopoles have not been seen or proven experimentally, there are many theoretical bases for which one can believe their existence. The theoretical evidence for its existence is as strong as any undiscovered theoretical particle. Polyakov [110] and ‘t Hooft [111] 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. The 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 regionr⟶r∗, where ϕϕ∗~ϕ20. Hence, from Equation (16), we can express
(64)∂μF~μν=q2ϕ02A~ν=jmν.The equations of motion for the static fields are(65)∇·B~⟶=ρm,−∇×E~⟶=j⟶m;here, E~⟶ and B~⟶ are the dual versions of the static electric and magnetic fields, while j⟶m and ρm are the magnetic current and charge densities, respectively. The Lorentz force associated with these fields can be expressed as
(66)F⟶=qB~⟶−v⟶×E~⟶,where v⟶ is the speed of the particle within the fields and q is the monopole charge. The homogeneous equation ∇·E~⟶=0 represents the uniform electric field present at the confining phase. Combining the dual version of Ampere’s law on the right side of Equation (65) with the dual London equation responsible for persistent current generated by the monopoles
(67)∇×j⟶m=1λ2E~⟶,we obtain the expression
(68)∇2E~⟶=1λ2E~⟶.Noting thatE=∇×A, we can identify λ=q2ϕ02−1/2=q2mη2/2α4ρ−1/2 as the London penetration depth [112, 113]. Developing the Laplacian in Equation (68) in spherical coordinates yields
(69)d2E~rdr2+2rdE~rdr−E~rλ2=0;this equation has a solution given as
(70)E~r=c1e−r/λr,where c1 is a constant. Using dimensional analysis, we can fix the constant as c1=E~0; therefore,
(71)E~r=E~0e−r/λr,where E~=E~=E~r. This means that the electric field is exponentially screened from the interior of the vacuum with penetration depth λ, a phenomenon known as the color Meissner effect. Also, using London’s accelerating current relation,
(72)B~⟶=−λ2∂j⟶m∂t,and writing the continuity equation in the form of London equations,
(73)∂j⟶m∂t+∇ρm=0,together with the equation at the left side of Equation (65), we obtain
(74)∇2B~⟶=B~⟶λ2.This equation has a solution similar to Equation (69), i.e.,
(75)B~r=B~0e−r/λr.Thus, the magnetic field is equally screened exponentially from the interior of the vacuum by a penetration depthλ. In sum, both time-varying electric and magnetic fields are screened with the same depth λ.Now, comparingλ with the results in Refs. [112, 114], where
(76)λ2=mnse2,we rearrange the expression obtained for λ in the model framework,
(77)λ2=2α4ρq2mη2=mη22q2ρa4,to make it suitable for comparison. Juxtaposing these two equations leads to q=e, m=mη2/a and ns=2ρa3, where m,e,andns are mass, charge, and electron number density of a superconducting material. Hence, the electron charge e is equivalent to the monopole charge q [115, 116], and the electron number density and mass are related to the monopole density and mass, respectively. Combining Equation (67) with the electric field expression at the right side of Equation (65), we obtain a fluxoid quantization relation:
(78)∫E~⟶·dS⟶−λ2∮j⟶m·dl⟶=nΦe,where n∈ℤ and Φe is the quantum of the electric flux [112]. The penetration depth could also be expressed as a function of temperature
(79)λT≃2α4ρq2mη∗2T1/2≃λ01−T4Tc4−1/2.We retrieve the initial penetration depth below Equation (68), if we set T=0. Accordingly, the penetration depth increases with temperature until it becomes infinite at T=Tc, where deconfinement and restoration of chiral symmetry coincide [117, 118]. At this point, the field lines are expected to penetrate the vacuum causing it to lose its superconducting properties. This expression is precisely the same as the one obtained for metallic superconductors [114]. At T>Tc, the magnetic and electric field lines go through 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
(80)nsT=2ρa31−T4Tc43/2=n1−T4Tc43/2.The number density decreases sharply with temperatureT and vanishes at T=Tc. On the other hand, at T<Tc, similar to the two-fluid model where normal monopole fluid mixes with superconducting ones, the density of the monopoles can be calculated from the relation n=ns+nn, where nn is the normal monopole density [119]. Additionally, at T⟶0, ns⟶n, and in the limit T⟶Tc, ns⟶0. Consequently,
(81)nn=n1−1−T4Tc43/2.Thus, thecolor superconducting monopoles ns at T<Tc conduct with dissipationless flow; on the other hand, the normal monopoles nn conduct with finite resistance at the same temperature range. Consequently, ns decreases sharply with increasing T while nn increases steadily with increasing T.


## 5. Analysis and Conclusion

### 5.1. Analysis

At a high energy regime, there is high particle density which forms deconfined matter leading to hadronization [120, 121], ϕ∗p1ϕp2⟶γ⟶ϕ∗p1′ϕp2′ [68, 122–125]. In this regime, the process takes place at low temperatures 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, Equation (55) represents the gluon propagator which mediates the interaction of the glueballs in the IR regime of the model. Hadronization [126] comes into play in this regime when the momenta of the particles are reduced below a particular threshold set by the QCD scale, ΛQCD~250 MeV, or the separation distance between particle and antiparticle pairs is greater than ~1fm. 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~170MeV, or higher baryon densities above 5ρ0 where the free quarks and gluons start clamping together into hadrons. This is also observed at the initial stages of heavy ion collisions [121, 127, 128]. On the other hand, the string decay hadronization occurs due to nonperturbative effects. Under this picture, new hadrons are formed out of quark-antiquark pairs or through a gluon cascade [126].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(82)π=qq¯+qq¯qq¯+qq¯g+⋯,where the valence qq¯ and the nonvalence quark components are fully covered [129, 130]. Hence, there is much 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 ultrarelativistic heavy ion collision experiments in BNL Brookhaven and CERN Geneva were able to create matter under extreme conditions of temperature and densities necessary for phase transition [131]. Pion-pion annihilation is particularly necessary since pions are the second most abundant particles produced during the heavy ion collision from hot and dense hadron gas. Such annihilation also results 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 [10–12]. However, the leptons are capable of annihilating into quarks and gluons (partons) in a process popularly referred to as gluon bremsstralhaung [16, 17]. The free parameters of the model are ν, ξ, εv, ρ, α, a, and ϕ0, where ν is the degeneracy of the gluons and its acceptable value for massless gluons in SU3 representation is ν=16 while its value under SU2 representation is ν=6. It is known from QCD lattice simulation that σ~1GeV/fm; therefore, we can determine from Equation (28) that mη=1.73GeV [43–45, 132, 133] which is precisely the scalar glueball mass. For zero degeneracy, i.e., ν=0, the critical temperature Equation (34) goes to infinity and all the glueballs get melted leaving pure gluons [75]. We determined the gluon mass to be mA=870MeV in the model framework. This value lies within the values obtained from lattice simulation as referred to in Section 3.3. Also, εv 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 the sum rule for gluon condensate is 0.006±0.012GeV4 [134], and the Bag model is also quoted as 1454MeV4. From the model εv=FμνFμν=ξ−2=ρa4, recalling that ϕ0 is related to a and α as ϕ0=a/α, a has the dimension of energy and α is dimensionless. Again, when we combine Equations (28) and (37), we can express the string tension σ as a function of temperature T:
(83)σT=mη∗2T3=mη2031−T4Tc4.Here, we retrieve Equation (28) at T=0, σT=Tc=0, and at T>Tc, we move into the quark-gluon plasma (QGP) phase where the particles interact in a disorderly manner [46, 135, 136]. Thus, the potential in Equation (27) can be expressed in terms of temperature as
(84)Vcr,T=−1r+mη∗2T3r=−1r+mη2031−T4Tc4r,where we recover Equation (27) at T=0. Thermal deconfinement and the restoration of chiral symmetry occur at T=Tc [117, 118] due to the dissolution of the glueball mass, and we have the QGP phase at T>Tc [46, 135, 136] where the glueball mass becomes unstable. As established in SU2 group representation, it is known from the hadron spectrum that the SU2L×SU2R chiral symmetry is spontaneously broken due to the nonperturbative dynamics of the theory, leading to color confinement. Considering quark representation in SU2, 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.


### 5.2. Conclusion

The process of high energy annihilation ofϕ∗ϕ to the production of ϕ∗ϕ through the interaction ϕ∗ϕ⟶γ⟶ϕ∗ϕ during hadronization in the high particle density region is briefly addressed. Thus, we focus on the discussion to cover the low energy regime where U~1 symmetry is spontaneously broken through the Abelian Higgs mechanism to give mass to the resulting glueballs. The scalar field in this case plays the role of the Higgs field, which undergoes modification into glueballs with mass mη. This leads to color confinement of glueballs. We explored the behaviour of the scalar field potential V∣ϕ∣ and the glueball potential Vη with the decay constant fα, and the results are presented in Figures 3 and 4. 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 Figures 1–5, respectively. The glueball fields acquire their mass through SSB, and their mass remains the most relevant parameter throughout the confinement properties. The magnitude of the glueball mass mη is precisely the same as the observed lightest scalar glueball mass mη=1.73GeV. This mass is responsible for the string tension σ that keeps the particles in a confined state and appears in the monopole condensate as well. It also appears in the gluon condensate 2GηFμνFμν, 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. The second is at the low energy regime where the quark and antiquark pairs/valence gluons that form pions/glueballs get separated due to long separation distances, and the string tension that holds them together breaks leading to hadronization. We also investigated the effect of temperature on the effective scalar glueball mass mη∗2T, the gluon mass mA∗T, string tension σT, confining potential Vcr,T, gluon condensate 2GηFμνFμνT, the penetration depth λT, the superconducting monopole densities nsT, and the normal monopole density nnT, and their results are presented in Figures 6–11, respectively. We calculated the QCD β-function and the strong “running” coupling αs 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 was compared with lattice simulation results, 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, Equation (27), consistent with the Cornell potential model. The potential keeps growing linearly for T<Tc with thermal deconfinement phase at T≥Tc. However, most of the results on QCD lattice calculations point to a temperature correction of order −T2 to the string tension [137–139]; meanwhile, the model proposes correction in order of −T4 in Equation (83). This can be regularized at T≃Tc to obtain the correct order −T2. Indeed, we did not find such regularization necessary because there are some proposed phenomenological models that suggest −T4 correction [140] to the string tension as well. But we deem it necessary to provide some explanations based on the model. In the previous paper Ref. [40], we obtained a correction of order −T2 as elaborated in Equation (63). We observed that in Ref. [40], the temperature correction to the string tension came from the scalar field ϕ as elaborated in the latter part of Section 3.3. On the other hand, the temperature correction in the gauge field leads to a correction in the order of −T4 as discussed in Section 3.2 and elaborated in Section 5.1 in Equations (83) and (84).Figure 3
The scalar field potential for different decay constantsfα. The potential has degenerate vacua.Figure 4
The glueball potentialVη against the glueball field η for different values of fα. The potential has a nondegenerate and well-defined vacuum giving the glueballs a real mass mη.Figure 5
Confining Cornell-like potential. The perturbative and the nonperturbative nature of the potential is displayed. Below the Fermi scale, the gluons are asymptotically free, while above the Fermi scale, we only find confined color neutral particles.Figure 6
The screened electric (magnetic) fieldE~rB~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 λ=65nm; below it, we have a mix or an intermediate state with higher electric or magnetic field strength. Above the estimated λ, we have a superconducting state because the vacuum is pure diamagnetic with no penetrating electric or magnetic fields.Figure 7
Thermally fluctuating glueball mass with temperature. The glueball mass decreases sharply with increasingT until it vanishes at T=Tc, where the glueballs melt into gluons.Figure 8
Variation of the gluon massmA∗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 9
The string tensionσT/σ0 against temperature T/Tc. The string tension decreases sharply with T and breaks or vanishes at T=Tc representing hadronization.Figure 10
Temperature fluctuating Cornell-like potential. The gradient of the graph decreases with increasingT and flattens at T=Tc indicating deconfinement and hadronization phase.Figure 11
Penetration depthλ (a), superconducting ns, and normal nn monopole densities with temperature (b). The penetration depth remains constant at λ=λ0 representing pure diamagnetic vacuum before it starts rising steadily with T and becomes infinite at T=Tc, where the fields are expected to penetrate the vacuum. Thus, we have a superconducting state at low temperatures, and at relatively high temperatures, we have a mix or intermediate state. On the other hand, the superconducting monopole density ns decreases with increasing T while the normal monopole density nn increases with an increasing T.
(a)(b)


## 5.1. Analysis

At a high energy regime, there is high particle density which forms deconfined matter leading to hadronization [120, 121], ϕ∗p1ϕp2⟶γ⟶ϕ∗p1′ϕp2′ [68, 122–125]. In this regime, the process takes place at low temperatures 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, Equation (55) represents the gluon propagator which mediates the interaction of the glueballs in the IR regime of the model. Hadronization [126] comes into play in this regime when the momenta of the particles are reduced below a particular threshold set by the QCD scale, ΛQCD~250 MeV, or the separation distance between particle and antiparticle pairs is greater than ~1fm. 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~170MeV, or higher baryon densities above 5ρ0 where the free quarks and gluons start clamping together into hadrons. This is also observed at the initial stages of heavy ion collisions [121, 127, 128]. On the other hand, the string decay hadronization occurs due to nonperturbative effects. Under this picture, new hadrons are formed out of quark-antiquark pairs or through a gluon cascade [126].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(82)π=qq¯+qq¯qq¯+qq¯g+⋯,where the valence qq¯ and the nonvalence quark components are fully covered [129, 130]. Hence, there is much 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 ultrarelativistic heavy ion collision experiments in BNL Brookhaven and CERN Geneva were able to create matter under extreme conditions of temperature and densities necessary for phase transition [131]. Pion-pion annihilation is particularly necessary since pions are the second most abundant particles produced during the heavy ion collision from hot and dense hadron gas. Such annihilation also results 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 [10–12]. However, the leptons are capable of annihilating into quarks and gluons (partons) in a process popularly referred to as gluon bremsstralhaung [16, 17]. The free parameters of the model are ν, ξ, εv, ρ, α, a, and ϕ0, where ν is the degeneracy of the gluons and its acceptable value for massless gluons in SU3 representation is ν=16 while its value under SU2 representation is ν=6. It is known from QCD lattice simulation that σ~1GeV/fm; therefore, we can determine from Equation (28) that mη=1.73GeV [43–45, 132, 133] which is precisely the scalar glueball mass. For zero degeneracy, i.e., ν=0, the critical temperature Equation (34) goes to infinity and all the glueballs get melted leaving pure gluons [75]. We determined the gluon mass to be mA=870MeV in the model framework. This value lies within the values obtained from lattice simulation as referred to in Section 3.3. Also, εv 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 the sum rule for gluon condensate is 0.006±0.012GeV4 [134], and the Bag model is also quoted as 1454MeV4. From the model εv=FμνFμν=ξ−2=ρa4, recalling that ϕ0 is related to a and α as ϕ0=a/α, a has the dimension of energy and α is dimensionless. Again, when we combine Equations (28) and (37), we can express the string tension σ as a function of temperature T:
(83)σT=mη∗2T3=mη2031−T4Tc4.Here, we retrieve Equation (28) at T=0, σT=Tc=0, and at T>Tc, we move into the quark-gluon plasma (QGP) phase where the particles interact in a disorderly manner [46, 135, 136]. Thus, the potential in Equation (27) can be expressed in terms of temperature as
(84)Vcr,T=−1r+mη∗2T3r=−1r+mη2031−T4Tc4r,where we recover Equation (27) at T=0. Thermal deconfinement and the restoration of chiral symmetry occur at T=Tc [117, 118] due to the dissolution of the glueball mass, and we have the QGP phase at T>Tc [46, 135, 136] where the glueball mass becomes unstable. As established in SU2 group representation, it is known from the hadron spectrum that the SU2L×SU2R chiral symmetry is spontaneously broken due to the nonperturbative dynamics of the theory, leading to color confinement. Considering quark representation in SU2, 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.


## 5.2. Conclusion

The process of high energy annihilation ofϕ∗ϕ to the production of ϕ∗ϕ through the interaction ϕ∗ϕ⟶γ⟶ϕ∗ϕ during hadronization in the high particle density region is briefly addressed. Thus, we focus on the discussion to cover the low energy regime where U~1 symmetry is spontaneously broken through the Abelian Higgs mechanism to give mass to the resulting glueballs. The scalar field in this case plays the role of the Higgs field, which undergoes modification into glueballs with mass mη. This leads to color confinement of glueballs. We explored the behaviour of the scalar field potential V∣ϕ∣ and the glueball potential Vη with the decay constant fα, and the results are presented in Figures 3 and 4. 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 Figures 1–5, respectively. The glueball fields acquire their mass through SSB, and their mass remains the most relevant parameter throughout the confinement properties. The magnitude of the glueball mass mη is precisely the same as the observed lightest scalar glueball mass mη=1.73GeV. This mass is responsible for the string tension σ that keeps the particles in a confined state and appears in the monopole condensate as well. It also appears in the gluon condensate 2GηFμνFμν, 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. The second is at the low energy regime where the quark and antiquark pairs/valence gluons that form pions/glueballs get separated due to long separation distances, and the string tension that holds them together breaks leading to hadronization. We also investigated the effect of temperature on the effective scalar glueball mass mη∗2T, the gluon mass mA∗T, string tension σT, confining potential Vcr,T, gluon condensate 2GηFμνFμνT, the penetration depth λT, the superconducting monopole densities nsT, and the normal monopole density nnT, and their results are presented in Figures 6–11, respectively. We calculated the QCD β-function and the strong “running” coupling αs 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 was compared with lattice simulation results, 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, Equation (27), consistent with the Cornell potential model. The potential keeps growing linearly for T<Tc with thermal deconfinement phase at T≥Tc. However, most of the results on QCD lattice calculations point to a temperature correction of order −T2 to the string tension [137–139]; meanwhile, the model proposes correction in order of −T4 in Equation (83). This can be regularized at T≃Tc to obtain the correct order −T2. Indeed, we did not find such regularization necessary because there are some proposed phenomenological models that suggest −T4 correction [140] to the string tension as well. But we deem it necessary to provide some explanations based on the model. In the previous paper Ref. [40], we obtained a correction of order −T2 as elaborated in Equation (63). We observed that in Ref. [40], the temperature correction to the string tension came from the scalar field ϕ as elaborated in the latter part of Section 3.3. On the other hand, the temperature correction in the gauge field leads to a correction in the order of −T4 as discussed in Section 3.2 and elaborated in Section 5.1 in Equations (83) and (84).Figure 3
The scalar field potential for different decay constantsfα. The potential has degenerate vacua.Figure 4
The glueball potentialVη against the glueball field η for different values of fα. The potential has a nondegenerate and well-defined vacuum giving the glueballs a real mass mη.Figure 5
Confining Cornell-like potential. The perturbative and the nonperturbative nature of the potential is displayed. Below the Fermi scale, the gluons are asymptotically free, while above the Fermi scale, we only find confined color neutral particles.Figure 6
The screened electric (magnetic) fieldE~rB~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 λ=65nm; below it, we have a mix or an intermediate state with higher electric or magnetic field strength. Above the estimated λ, we have a superconducting state because the vacuum is pure diamagnetic with no penetrating electric or magnetic fields.Figure 7
Thermally fluctuating glueball mass with temperature. The glueball mass decreases sharply with increasingT until it vanishes at T=Tc, where the glueballs melt into gluons.Figure 8
Variation of the gluon massmA∗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 9
The string tensionσT/σ0 against temperature T/Tc. The string tension decreases sharply with T and breaks or vanishes at T=Tc representing hadronization.Figure 10
Temperature fluctuating Cornell-like potential. The gradient of the graph decreases with increasingT and flattens at T=Tc indicating deconfinement and hadronization phase.Figure 11
Penetration depthλ (a), superconducting ns, and normal nn monopole densities with temperature (b). The penetration depth remains constant at λ=λ0 representing pure diamagnetic vacuum before it starts rising steadily with T and becomes infinite at T=Tc, where the fields are expected to penetrate the vacuum. Thus, we have a superconducting state at low temperatures, and at relatively high temperatures, we have a mix or intermediate state. On the other hand, the superconducting monopole density ns decreases with increasing T while the normal monopole density nn increases with an increasing T.
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