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Hindawi Advances in High Energy Volume 2021, Article ID 5658568, 17 pages https://doi.org/10.1155/2021/5658568

Research Article An Effective Model for and Dual Superconductivity at Finite Temperature

Adamu Issifu 1 and Francisco A. Brito 1,2

1Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil 2Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil

Correspondence should be addressed to Francisco A. Brito; [email protected]

Received 21 May 2021; Accepted 17 July 2021; Published 10 August 2021

Academic Editor: Edward Sarkisyan Grinbaum

Copyright © 2021 Adamu Issifu and Francisco A. Brito. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The glueballs to and QCD monopole condensations as by-products of color confinement. A color dielectric function Gð∣ϕ ∣ Þ coupled with a Abelian gauge field is properly defined to mediate the at confining regime after spontaneous breaking (SSB) of the gauge symmetry. The are expected to form through the -gluon plasma (QGP) phase where the free and start clamping together to form . The QCD-like hη2 2 μν i fi ð Þ σ λ vacuum mη F Fμν , con ning potential Vc r , tension , penetration depth , superconducting and normal monopole ff 2 2 fi densities (ns nn), and the e ective (mη and mA) will be investigated at nite temperature T. We also calculate the strong “ ” α β running s and subsequently the QCD -function. The dual superconducting nature of the QCD vacuum will be investigated based on monopole condensation.

1. Introduction they decay quickly [10–13] making it easy to study the . Due to the large production of jets and pions at the LHC and The Large Collider (LHC) is currently attracting the Relativistic Heavy Ion Collider (RHIC), they are suitable can- attention of physicists to investigate the dynamics of didates for the search for new particles and the testing of the the (SM) at the TeV scale and to probe for a SM properties. Quarks and gluons behave as quasi-free parti- possible new physics. One of the major expectations at the cles at higher energies (short distances) due to asymptotically LHC is the observation of substructures. Generally, jets free nature of the QCD theory. When the energy among these are collimated bundles of hadrons constituted by quarks color particles is reduced to about 1 GeV and below or at a and gluons at short distances or high energies [1–3]. Jets separation of order 1 fm or higher, color confinement sets played a significant role in the discovery of gluons (g)[4–7] in, such that the quarks and the gluons coexist as hadrons. and (t) [8, 9]. These observations are featured At lower enough energies, hadronization is highly favoured prominently in classifying leading to the formation of pions. Hadron jet was first (QCD) as a theory for strong within the standard produced by of - to produce a model. The most abundant secondary particles produced in two-jet event, e+e− ⟶ qq ⟶ 2 − jets. This two-jet structure heavy ion collisions are pions which emerge from hot and [14] was first observed in 1975 at SPEAR (SLAC), and the dense hadron gas states. These pions can be studied from 1/2 nature of the quarks was also established [15]. The the projected quark-gluon plasma (QGP) and the deconfine- jets are produced when qq fly apart; more qq pairs are pro- ment phase through emitted and dileptons pro- duced which recombine with the existing pairs to form duced at this phase. Since the and the dilepton do and as a by-product [16]. The first three-jet not interact with the hadron through strong , structure from the electron-positron annihilation process 2 Advances in High Energy Physics was observed by analysing 40 hadronic events at the center- will shed light on monopole condensation in the QCD vac- of- energy 27.4 GeV from SPEAR [16]. These jets were uum and its role in screening the QCD vacuum from external subsequently observed clearly and separately [17, 18]. The electric and magnetic field penetrations similar to supercon- discovery of the gluon through the three-jet process played ductors. There is a long-standing belief that quark masses are a major role in the discovery of the Higgs particle [19–24] obtained through the spontaneous symmetry breaking mech- by ATLAS collaboration [25] and CMS collaboration [26] anism facilitated by the nonzero expectation value of the using the LHC at CERN. On the other hand, -pion anni- Higgs field. That notwithstanding, quark masses continue hilation serves as the principal source of dileptons (e−e+ and to be an input parameter in the standard model [48]. As a μ−μ+) produced from the hadron matter [27, 28]. Thus, the free parameter in the SM, the magnitude of the mass is deter- proper study of the main source of dilepton spectra observed mined phenomenologically and the result compared with experimentally proposes significant observables which help sum rules, lattice simulation results, other theoretical tech- to understand the pion dynamics in a dense nuclear matter niques, and experimental data. The Dyson-Schwinger inte- that exist at the beginning of the collision [29]. gral equation for the quark mass function has two known solutions, i.e., trivial and nontrivial solutions. The latter is We consider a complex field model coupled to obtained under low energy and nonperturbative conditions Abelian gauge fields in two different ways. At relatively low while the former to unphysical, massless quarks. Thus, energies, the particles undergo transformation into glueballs. the nontrivial solution leads to the creation of quark masses The emission of gluon g which mediates the strong interac- resulting in “dynamical ,” which is tions is a non-Abelian feature observed in the nonperturba- a consequence of confinement [49–54]. Monopoles can be tive regime of QCD theory. However, in the model said to be a product of grand unification theory (GUT). fi framework, the gluon emission is a result of the modi cation The mixing of the strong and the electroweak interactions of the Abelian gauge field by the color dielectric function. In due to the higher gauge symmetries in GUT breaks spontane- this approach, we approximate the non-Abelian gauge field ously at higher energies or extremely short distances. So, responsible for color confinement with an Abelian gauge physical features of monopoles such as size and mass are field coupled with a dimensionless color dielectric function studied through the energy of spontaneous symmetry break- Gð∣ϕ ∣ Þ. The color dielectric function is responsible for regu- ing [55]. Monopoles play important role in color confine- lating the long-distance dynamics of the photon , ment. Monopole condensation produces a dual so it does not decouple at higher wavelengths where glueballs superconductor which squeezes the uniform electric field at are expected to be in a confined state. The color dielectric the confinement phase into a thin flux tube picture. The flux function is defined in terms of the glueball field η after SSB, tube is formed between quark and an antiquark to keep them fi to give meaning to color confinement and its associated prop- con ned. This scenario has been extensively investigated in – erties. While ϕ initiates the annihilation and production pro- lattice [56 58] and Polyakov loop [59] to estab- fi cesses, η is the glueball field which brings about confinement lish its con nement properties. The involvement of mono- and gluon condensation. The Abelian approach to QCD the- poles in chiral symmetry has recently been investigated as ory was first proposed by ‘t Hooft to justify magnetic mono- well [60, 61]. The impact of monopoles in quark-gluon plasma (QGP) has also been studied in [62–64]. pole condensation in QCD vacuum leading to color The paper is organised as follows: In Section 2, we intro- superconductivity [30, 31] that also suggested “infrared Abe- duce the Lagrangian density that will be the basics for this lian dominance.” Fast forward, it has also been shown that study. In Section 3, we present spontaneous symmetry break- about 92% of the QCD string tension is Abelian [32], so an ing of the model presented in the previous sections; this sec- Abelian approximation will not be out of place. Further stud- tion is divided into four subsections; in Section 3.1, we ies of Abelian dominance for color confinement in SUð2Þ and investigate confinement of glueballs; in Section 3.2, we study SUð3Þ lattice QCD can be found in Ref. [33–37]. An inter- the effective masses; we dedicate Section 3.3 to gluon conden- ested reader can also refer to Ref. [38] for more recent results sation, and in Section 4, we investigate the monopole con- on Abelian dominance in QCD theory. This approach has densation. We proceed to present the analysis in Section 5.1 also been adopted in other confinement models—see [39– and the final findings in Section 5.2. We have adopted the 41] and references therein for more justifications. ∗ natural units c = ℏ = k =1, except otherwise stated. The annihilation of ϕ ϕ through “scalar QED” where the B particles are relatively free can be used to address hadroniza- 2. The Model tion due to the presence of high-density deconfined mater. The spontaneous symmetry breaking of the Uð1Þ symmetry fi We start with a Lagrangian density developed by exploring to an energy regime is suitable for describing color con ne- gauge-invariant properties (see [39]): ment with a glueball field η. Because the running coupling, α ð 2Þ s Q , of the decreases with momentum 1 1 μν ϕ∗ϕ L = ημν ϕ ϕ∗ − ðÞjjϕ μν − ~ ~ − ðÞjjϕ increases (short distances), annihilation can be studied Dμ Dν 4 G Fμν F 4 Fμν F V from perturbation theory (ϕ∗ϕ ⟶ ϕ∗ϕ) similar to annihila-  − + − + μν ~ ∗ ~ ∗ tion and creation of pions (π π ⟶ π π γ) and Bhabha = η ∂μϕ + iqAμϕ ∂νϕ − iqAνϕ − + ⟶ − + ϕ∗ scattering (e e e e ) [42]. A low energy analyses of 1 1 μν ϕ annihilation will be made to show color confinement, bound − ðÞjjϕ μν − ~ ~ − ðÞjjϕ , 4 G Fμν F 4 Fμν F V states of the gluons (glueball masses) [43–45], and the QCD vacuum [46, 47] responsible for the gluon condensate.We ð1Þ Advances in High Energy Physics 3

~ ~ ~ ~ where Dμ = ∂μ + iqAμ, Fμν = ∂μAν − ∂νAμ, and Fμν = ∂μAν (5), − ∂νAμ are the covariant derivative, strength of the dual a fi fi hijjϕ =± , ð8Þ gauge eld, and the gauge eld, respectively. This model 0 α can be used to study the annihilation and creation of identical ∗ particles ϕ ϕ and its transformation into glueballs η at low breaks the U~ ð1Þ symmetry spontaneously. Considering two energies. The complex scalar fields ϕðxÞ and its conjugate fi ηð Þ ζð Þ fl ϕ∗ð Þ real scalar elds r and r representing small uctuations x describe particle and its with the same about the vacuum, a shift in the vacuum can be expressed as mass but different charges, respectively. The electromagnetic fi interactions among the scalar elds with an exchange of sin- ϕ ⟶ ηðÞr + ϕ , ð9Þ gle photon produced from the dual gauge field dynamics 0 ~ ~μν Fμν F are the suitable scenario to address a high energy where ϕ0 ≡ hjϕji0. Hence, the potential takes the form limit. However, we shall focus on the strong interactions, where the scalar fields modify to glueball fields at relatively 1 η = ϕ + ′ ϕ η + ′′ ϕ η2 = ρα2 2η2: fi VðÞ VðÞjhi∣ϕ∣ V ðÞhi∣ϕ∣ V ðÞ a low energies, which will lead to color con nement and its 0 0 2 hijjϕ associated properties mediated by the modified gauge field 0 μν dynamics GðjϕjÞFμν F . The equations of motion are ð10Þ

1 ∂GðÞjjϕ ∂VðÞjjϕ Consequently, we can suitably parametrize the scalar μϕ + μν + =0, ð2Þ fi DμD 4 ∂ϕ∗ Fμν F ∂ϕ∗ eld such that, ðÞϕ + η ðÞϕ + ηðÞ+ ζðÞ iζ/ϕ 0 0 r i r ~μν ∗ ν ν ∗ 2 ∗ ~ν ν ð3Þ ϕ = e 0 pffiffiffi ≈ pffiffiffi : ð11Þ ∂μ F = −iq½Šϕ ∂ ϕ − ϕ∂ ϕ +2q ϕ ϕA = jϕ, 2 2

μν ∂콊GðÞjjϕ F =0: ð4Þ The Lagrangian in Equation (1) becomes hi fi 1 ÂÃ1 μ We will adopt a complex scalar eld potential of the form L = ∂ η∂μη − 2ρα2 2η2 + ∂ ζ∂μζ − 2 ϕ ~ ∂μζ + 2jjϕ 2 ~ ~ 2 μ a 2 μ q 0Aμ q 0 AμA Âà 1 1 μν ρ 2 2 2 2 − η μν − ~ ~ +⋯, ðÞϕ = α jjϕ − , ð5Þ GðÞFμν F Fμν F V 4 a 4 4 ð12Þ where where η is associated with the glueball fields and ζ is also ϕ + iϕ associated with the massless Goldstone which will ϕ = 1pffiffiffi 2 : ð6Þ fi 2 be swallowed by the massive gauge elds through the gauge transformation [39]: We can isolate the electromagnetic interactions among 1 ~ ~′ ~ fi Aμ ⟶ A μ = Aμ − ∂μζ: ð13Þ the elds [65] for study using Equation (3). This enables us qϕ to visualize how the fields annihilate and create identical par- 0 ϕð Þϕ∗ð Þ ticles [66, 67] through single-photon exchange, p1 p2 Pions are strongly interacting elementary particles with ⟶ γ ⟶ ϕð ′Þϕ∗ð ′Þ p1 p2 , in high energy regime [68]. an spin. Therefore, they are bosons which are not governed by Pauli’s exclusion principle and can exist as rela- 3. Confinement, Effective Masses, Gluon tivistic or nonrelativistic particles. As a result, they can be Condensation, and Dual Superconductivity represented by spin-0 scalar fields. In the nonrelativistic regime, they exist as Goldstone bosons ζ, signifying the fi To discuss the color con nement and its consequences, we breakdown of chiral symmetry [69]. Pions are generally pro- ~ need to follow spontaneous symmetry breaking of the Uð1Þ duced through matter such as pp, NN , and ee+, symmetry in Lagrangian Equation (1) to give mass to the which undergo transition from structures to mesons. resulting glueball fields and their associated products such This is an interesting phenomenon in low energy hadron as the monopole condensation and gluon condensation physics [70, 71]. Pions have a flavour structure and other which form the basics for color confinement. We will proceed quantum numbers that permit us to classify them as bound with the symmetry breaking from the transformations: states of quark and an antiquark. However, the valence quarks which characterize the flavour structure are sur- α ϕðÞx ⟶ ϕ′ðÞx = eiq ðÞx ϕ, rounded by gluons and quark-antiquark pairs. Physically, ð7Þ glue-rich components mix with pions (qq) forming an ~ ðÞ⟶ ~′ ðÞ= ~ ðÞ− ∂ αðÞ: Aμ x A μ x Aμ x μ x enriched spectrum of isospin-zero states and qqg hybrid states [72]. The glueball spectrum and their corresponding The of the potential Equation quantum numbers are known in 4 Advances in High Energy Physics [43, 73]. Additionally, the existence of glueballs away from the source q, and in such a limit, terms of has not been decisive because of the fear of possible mixing the order Oð1/r4Þ can be disregarded. This equation has a with quark degrees of freedom [74]. In the model framework, solution the glueballs coexist with the Goldstone bosons which are fi ÀÁ subsequently swallowed by the massive gauge elds leaving a sin mηr ηðÞ= : ð22Þ out the glueballs (gg) for analysis. The glueball decay occurs r α when the separation distance between the valence gluons mηr exceeds some thresholds (>1 fm) leading to hadronization. Consequently, the Lagrangian can be simplified as Substituting this result into the electric field Equation (20), 1 1 1 2 ϕ 2 μ μν ~ ~μν q jj0 ~ ~μ L = ∂μη∂ η − VðÞη − GðÞη Fμν F − Fμν F + AμA , 2 4 4 2 Λ 2Λα2 = = ÀÁ, ð23Þ ð14Þ E 2 2 2 2 r G a ξ sin mηr where VðηÞ = ρα2a2η2. The equations of motion for this Lagrangian are and using the well-known expression for determining the electrodynamic potential, ∂ ðÞη ∂ ðÞη ∂ ∂μη + G μν + V =0, ð15Þ ð μ ∂η Fμν F ∂η VrðÞ= Edr, ð24Þ μν ν ∂ ~ = 2jjϕ 2 ~ = ν , ð16Þ μ F q 0 A jϕ0 to determine the confining potential V ðrÞ, we get μν c ∂콊GðÞjjη F =0: ð17Þ ÀÁ "# 2Λα2 cot 2 mηr 2Λα 1 mηr ÀÁ3 The Feynman propagator for the glueball field η in Equa- V ðÞr = − + c ≃− − − Or + c c 2 2 2 2 3 a mηξ a mηξ mηr tion (12) reads "# 2α2 1 m2r ≃ − + η ; i 2ξ2 2 3 DηðÞp = , ð18Þ a mη r p2 − m2 + iε η ð25Þ

2 2 2 where mη =2ρα a . so, setting 3.1. Confinement. Here, we are interested in calculating static confining potential, such that only chromoelectric flux that 4 2 2 1 a ξ ρ =1⟶ ξ = ð26Þ results in confining electric field is present, while chromo- a4ρ magnetic flux influence is completely eliminated. Thus, in spherical coordinates, Equations (15) and (17) become, leads to respectively,  2 1 η 1 ∂ ðÞη ∂ ðÞη 1 mηr d 2 d G 2 V ðÞ= − + , ð27Þ r = E − , ð19Þ Vc r r2 dr dr 2 ∂η ∂η r 3

1 ∂ ÂÃ2 Λ with string tension r GðÞη E =0⟶ E = , ð20Þ r2 ∂r r2GðÞη m2 2ρα2 2 σ = η = a : ð28Þ where Λ = q/4πε0 is the integration constant. We are also 2 3 3 now setting GðηÞ = ξ VðηÞ (similar process was adopted in – ξ2 Ref. [39 41]), where is a dimensionful constant associated In the last step of Equation (25), we substituted Λ = q/4 ′ 2 with the Regge slope ð2πα Þ that absorbs the dimensionality πε0 =1and c =0for simplicity. of VðηÞ,soGðηÞ remains dimensionless and Equation (19) ff reads 3.2. E ective Masses. Glueballs are simply bound states of gluons, and they are “white” or colorless in nature. They "#  2 fl 1 d dη ∂ Λ 1 1 2 are avour blind and are capable of decaying. Scalar glueball 2 = − + ⟶ η′′ + η′ + 2η =0: PC ++ 2 r 2 4 V mη with quantum number J =0 is observed by lattice QCD r dr dr ∂η 2 ξ VðÞη r r simulations as the lightest with a mass of about 1.7 GeV ð21Þ [43–45]. The dispersion relation for the glueball and the gluon In the last step, we have assumed that the particles are far excitations including thermal fluctuations can be expressed Advances in High Energy Physics 5 as ball mass, and considered the standard integral,

2 2 ∗2 ð 2 E = k + m , xdx π η η ð29Þ = : ð36Þ ex − 1 6 2 = 2 + ∗2: EA k mA Thus, Equation (30) becomes We can calculate the effective thermal fluctuating glueball   mass by taking the second derivative of the Lagrangian den- 4 4 η ∗2 =2ρα2 2 1 − T = 2 0 1 − T : ð37Þ sity Equation (14) with respect to : mη ðÞT a 4 mηðÞ 4 T T *+ c c ∂2L 1 ∗2 = − =2ρα2 2 + ρα2 2ξ2 μν : ð30Þ mη 2 a a F Fμν When we take a thermal average of Equation (15), we get ∂η 2 2 2 μν 2 ξ mηhηF Fμνi = −mηhηi; therefore, η =0and η =1are exact solutions of the glueball fields [75]. We further demonstrate We redefine the glueball field η to include the thermal in Figures 1 and 2 that η has solution η =0 at T =0 and fluctuations as η = η + Δ, with restriction, hΔi =0. The angle increases steady as T increases to its maximum η ≃ 1 at T = η fi fi brackets represent thermal average, and is the mean eld Tc, corresponding to no glueball elds and the melting of of the glueballs. This equation can be solved by determining the glueballs. μν 2 explicitly the nature of hF Fμνi and hΔ i using field quan- tum distributions. By the standard approach, we can express 3.3. Gluon Condensation. Classical gluodynamics is described by the Lagrangian density: ð ∞ 4 μν ν k 1 F Fμν = − dk n ðÞE , ′ aμν a 2π2 B A L = − F Fμν, ð38Þ 0 EA 4 ð ð31Þ 1 ∞ 2 ÀÁ Δ2 = k : 2 dk nB Eη which is invariant under scale and conformal symmetries, x 2π 0 Eη ⟶ λx, and also produces vanishing gluon condensate h aμν a i =0 −1 F Fμν . Meanwhile, the symmetries are broken when Here, n ðxÞ = ðexβ − 1Þ represents the Bose-Einstein distri- −jε j B quantum correction v is added to the Lagrangian, i.e., bution function and β =1/T, where T is the temperature. We can analytically solve Equation (31) at a high energy regime, 1 2 ′ aμν a ≈ ≫ L = − F Fμν + jjε , ð39Þ where E kc, kBT mAc and mA is the gluon/screening 4 v mass. Therefore, aμν a ð ð resulting in nonvanishing gluon condensate, hF Fμνi >0. ν ∞ 4 1 ν 4 ∞ 3 4 4 μν = − k = − T x dx = − T , This correction is the consequence of the well-known scale F Fμν 2 dk β 2 x 4 2 μν 2π 0 E EA − 1 2π 0 e − 1 ξ θ A e Tc observed in QCD energy-momentum ( ) trace ð32Þ βðÞ μ g aμν a θμ = F Fμν, ð40Þ where we have substituted x = k/T and used the standard 2g integral βð Þ “ ” ð where g is the so-called beta-function of the strong cou- 3 π4 pling g, with a leading term x dx = : ð33Þ ex − 1 15 11 3 β = − g : ð41Þ ðÞg 2 In the last step, we have substituted ðÞ4π

4 2 15 Accordingly, we have nonzero vacuum expectation, T ξ = , ð34Þ c 2π2ν expressed as DE being T the critical temperature. Following the same analy- μ c θμ = −4jjε : ð42Þ sis, v ð ð 1 ∞ 2 1 2 ∞ 2 The second term in Equation (14) acts similar to the Δ2 = k ≃ T xdx ≃ T , ð35Þ 2 dk β 2 quantum correction and explicitly breaks the scale and the 2π Eη − 1 2π x − 1 12 0 Eη e 0 e conformal symmetries, so its energy-momentum tensor sat- isfies Equation (42) [40, 76–78]. We will compute the trace ≫ 2 where we have assumed kBT mηc , with mη being the glue- of the energy-momentum tensor of Equation (14) using the 6 Advances in High Energy Physics

μν 2 h i =0 〈2 GF 〉 We recover the classical expression for F Fμν 4 | | when we set ∣ε ∣ ⟶0. Following the potential defined below �V v Equation (14), we can express 1.0 2 2 2 2 2 ηV ′ ρa α η mηη 0.8 ~ = − = = ; ð46Þ V V 4 2 4

0.6 thus, Equation (45) becomes *+ 0.4 m2η2 2 ðÞη μν =4jjε 1 − η : ð47Þ G F Fμν v 4 0.2

� The QCD vacuum is seen as a very dense 0.2 0.4 0.6 0.8 1.0 comprising gauge fields and quarks that interact in a haphaz- ard manner. These characteristics are hard to be seen exper- Figure 1: The gluon condensate with the mean glueball field η. The fi η =0 η ≃ 1 imentally because quark and gluon elds cannot be observed condensate has its maximum value at and vanishes at . directly; only the color neutral hadrons are observable. The appearance of the mass term in the condensate is also signif- icant because it leads to chiral symmetry breakdown in the 2 2 〈 GF 〉T vacuum. Additionally, the gluon mass mA can be derived 4 | � | V from the vacuum as a function of the glueball mass as 1.0 2 mη 0.8 2 = : ð48Þ mA 4 0.6 Generally, scalar glueball mass is related to the gluon 0.4 mass as mðÞ0++ pffiffiffi 0.2 = 6 ≅ 2:45 ; ð49Þ mA T 0.2 0.4 0.6 0.8 1.0 Tc� considering the leading order [79, 80] from Yang-Mills the- ory with an auxiliary field ϕ, here, the scalar glueball mass Figure 2: The gluon condensate with varying temperature T.AtT ð0++Þ fl ϕ μν m represents uctuations around . The gluon mass is =0, h2GðηÞF Fμνi /4 ∣ ε ∣ =1 representing maximum T v determined to be m = 600 ~ 700 MeV as obtained from lat- η =0 = A condensate corresponding to . Also, at T Tc, – μν tice simulations [81 83]. Heavier gluon masses have also h2GðηÞF Fμνi /4 ∣ ε ∣ =0 representing minimum condensate T v been observed in the range of ~1 GeV by phenomenological corresponding to η ≃ 1 [75]. analysis [84, 85], lattice simulation [86], and analytical inves- tigations [87]. The glueball mass mη is responsible for all the relation confinement properties and the chiral symmetry breaking in the vacuum. Observing that the string tension is σ ~1 μ 2 2 θμ =4VðÞη + η□η: ð43Þ GeV/fm, we can identify mη = 3 GeV , corresponding to a =0:87 GeV gluon mass mA , from the model framework. Substituting the equation of motion Equation (15) into In terms of temperature, we use Equation (35), and the Equation (43) yields gluon condensate becomes "# "# 2 ϕ2 2 2 μ ∂G μν ∂V μν μν mη 0T T ~ ′ 2GðÞη F Fμν =4jjε 1 − =4jjε 1 − , where T η θμ =4VðÞη − η F Fμν − η =4V − ηG F Fμν, ð44Þ v 4 ϕ212 v 2 c ∂η ∂η 0 Tcη !1/2 48 = : ~ = ðηÞ − η ′ð 2 where in the last step we have substituted V V V mη ηÞ/4 ′ ′ with G and V representing the derivative of G and V ð50Þ with respect to η, respectively. In order to make the expres- ~ sion easy to be compared with Equation (42), we rescale V Corresponding to a temperature fluctuating gluon mass ~ðηÞ ⟶ −jε j~ as V v V, consequently, 2 2 DE ∗2 mηT 2 2 ′ μν ~ m = = g~ T , ð51Þ ηG F Fμν =4jjε 1 − V : ð45Þ A 2 v 48ϕ0 Advances in High Energy Physics 7

2 2 2 where g~ = mη/48ϕ0 is a dimensionless . about the vacuum and the thermal distributions in the scalar fi ϕ ⟶ ϕ + ϕ This expression looks like the Debye mass derived from the eld. So, we can substitute T into the potential in ϕ Δ leading order of QCD coupling expansion [88–90]. It carries Equation (5), where T plays a similar role as introduced nonperturbative property of the theory. Comparing Equa- above. Thus, tions (40) and (44), we identify ð 1 ∞ 2 2  ϕ2 = ÀÁp dp ≃ T ; ð57Þ βðÞg 1 T 2π2 Eβ − 1 12 = −ηG′ðÞη ⟶ β = −2gηG′ðÞη : ð52Þ 0 Ee 2g r2

here, we have used the high-temperature approximation kB Additionally, we can determine the strong running cou- ≫ 2 ≈ α T mϕc , E pc and the potential becomes pling s from the theory [91] "# ! 2 2 ÀÁ dα ρ ÂÃ 2 ρ β 2 = 2 s ; ð53Þ ðÞϕ, = α2jjϕ 2 + α2 ϕ2 − 2 = α2jjϕ 2 − 2 1 − T Q Q 2 V T 4 T a 4 a 2 dQ Tcϕ ρ ÂÃ = α2jjϕ 2 − ~2 2, therefore, 4 a  ð58Þ dGðÞ 2 2 2 1 βηðÞ≃−η = −ξ mηη ðÞr =2GðÞη = β with g =1: dη r2 hϕ i =0 where T , ð54Þ 12 2 2 = a , Comparatively, the strong running coupling can be iden- Tcϕ α2 fi α ðηÞ = ðηÞ = α ð1/ 2Þ ≡ 1/ ! ð59Þ ti ed as s G s r , with Q r, as the space- 2 like momentum associated with the four-vector momentum ~2 = 2 1 − T : 2 2 a a 2 as Q ≡−q . Thus, the color dielectric function is associated Tcϕ with the QCD β-function and the strong running coupling. We can write the Feynman propagator for the interaction The glueball mass can be calculated from the fluctuations by considering the left-hand side of Equation (45): in Equation (58) around the vacuum, such that  hi 1 "# "# μν = −2 ν □ − ∂ ∂ μ = −2 ν ∂2 − 1 − ∂ ∂ μ 2 2 2 F Fμν A gμν μ ν A A gμν μ ν A ∗2 ∂ V 2 2 2 2 T 2 T α m ðÞT = =2ρα ~a =2ρα a 1 − = m ðÞ0 1 − ,  ϕ ∂ϕ2 2 η 2 1 ϕ Tcϕ Tcϕ = −2 ðÞ 2 μν − 1 − μ ν ðÞ− ⟶ η ′ðÞη μν 0 Aν k k g α k k Aμ k G F Fμν  ð60Þ 1 = −2 ðÞη ′ðÞη 2 μν − 1 − μ ν ðÞ− Aν k G k g α k k Aμ k  where ϕ = ~a/α is the vacuum of the potential. We observe ÀÁ 1 0 = ðÞβ 2 2 μν − 1 − μ ν ðÞ− , ∗2ð0Þ = ∗2ð0Þ =0 Aν k q k g α k k Aμ k that mϕ mη at T , similar to Equation (37). The differences in the degree of the temperature in both equations ð55Þ arise because Equation (37) has temperature correction to the μν gauge field hF Fμνi, while in Equation (37), we have tem- μ 2 where we have added the gauge fixing term ð2/αÞð∂μA Þ fi hϕ2 i perature correction to the scalar eld T . Following the and made the Fourier transform same procedure as the one used in deriving Equation (47), we can express ð 4 = d k −ikx : ð56Þ *+ AμðÞk 4 e AμðÞx ðÞ2π m∗2ðÞT η2 2 ðÞη μν = −4jjε ϕ − 1 G F Fμν 4 In the last step, the term in the square brackets is pre- *+"# ð61Þ m2 ðÞ0 2 cisely the photon propagator normally associated with Abe- = −4jjε ϕ 1 − T η2 − 1 : lian gauge fields, but the multiplicative factor 4 2 Tcϕ −2ηG′ðηÞ = βð1/r2Þ ≡ βðq2Þ is defined such that the photon propagator does not decouple at higher wavelengths. This ∗2ð Þ ∗2ð Þ fl Here, we have replaced mη T with mϕ T from Equa- in uences the photons to behave like gluons. 2 Also, one can retrieve all the major results obtained in tion (47). The negative sign and η differences between this [40], if we consider temperature fluctuations in the scalar result and Ref. [40] might be due to the differences in the field ϕ. In that case, we move away from the known form of methods adopted for calculating the energy-momentum θμ field theory, i.e., perturbation about the vacuum, hϕi =0,to trace tensor μ. Also, if we consider the IR part of the poten- account for the density of particles and their interactions tial in Equation (25) as studied in Ref. [40], the potential 8 Advances in High Energy Physics becomes quark combinations do not form color singlet, Cooper pair "#condensate will break the local color symmetry giving rise to m2 m∗2ðÞT m2 ðÞ0 2 [106]. This phenomenon was first ðÞ= σ = ϕ ⟶ ðÞ, = ϕ = ϕ 1 − T , Vc r r 3 r Vc T r 3 r 3 2 r studied by Bardeen, Cooper, and Schrieffer (BCS) [107]. The T ϕ c BCS mechanism for pairing in general appears to be more ð62Þ robust in dense quark matter than superconducting metals due to the nature of interactions among quarks. The virtual with string tension in the vacuum make it color diamagnetic preventing both the electric and magnetic fields from penetrating [2]. ∗2 mϕ ðÞT We turn attention to the dual superconductivity in color σðÞ= , ð63Þ T 3 confinement whose property we can derive from Equation (16). This subject was first proposed by Nambu, ‘t Hooft, which is similar to the result in Ref. [40]. Therefore, the and Mandelstam [56]. While the color superconducting phase approach adopted in this paper accounts for the temperature takes place at the deconfined matter regime, the dual super- corrections to both the glueball field hΔ2i in Equation (35) conductivity is observed at the confining phase [108]. In the μν and the gauge field hF Fμνi, while in Ref. [40], only the ther- model framework, dual superconductivity is observed at the η ⟶ 0 mal correction to the scalar field hϕ2 i was considered. high glueball condensation region, , i.e., a region of T strong confinement. We will use the rest of this section to dis- 4. Dual Superconductivity cuss dual superconductivity in detail within the model frame- work. In this phenomenon, the QCD vacuum is seen as a In this section, we will discuss briefly color superconductivity color condensate resulting in one- and proceed to elaborate on dual superconductivity in detail. dimensional squeezing of the uniform electric flux that form ∗ The fields ϕ ϕ annihilate and create identical fields through a on the surface of the quark and antiquark pairs by dual ∗ ′ ∗ ′ Meissner effect. This results in the formation of a color flux decay process, ϕðp1Þϕ ðp2Þ ⟶ ϕðp1Þϕ ðp2Þ. The dual gauge ~μν ~ tube between quark and antiquark used in describing string field F Fμν which mediates the interaction of the scalar pictures of hadrons [38]. In this regard, magnetic monopole fi ϕ elds during the annihilation process is also responsible condensation is crucial to color confinement and dual super- for the monopole condensation. In the model framework, conductivity. Monopoles also play a role in chiral symmetry fi the scalar eld can be seen as point-like, asymptotically free breaking [109] and strongly coupled QGP. They are involved and degenerate at the high energy regime [92]. In the context in the decay of plasma formed immediately after relativistic of this work, the high energy regime is in reference to the heavy ion collision [62–64]. Even though monopoles have fi phase at which the scalar elds are annihilating while low not been seen or proven experimentally, there are many the- energy regime is after the symmetry breaking or the glueball oretical bases for which one can believe their existence. The ff regime. In e ect, multiparticle states are formed in the high theoretical evidence for its existence is as strong as any undis- energy regime. In such a high particle density regime, the covered theoretical particle. Polyakov [110] and ‘t Hooft fi charged scalar elds form Bose-Einstein condensation [93, [111] discovered that monopoles are the consequences of 94] similar to induced isospin imbalance systems leading to the general idea of unification of fundamental interactions color superconductivity [95, 96]. With this type of condensa- at short distances (high energies). Dirac on the other hand tion, there is a higher occupation number at the ground state demonstrated the presence of monopoles in QED. Mono- than the excited states. So, temperature increase goes into poles in general play an insightful role in understanding color reducing the occupation numbers [97]. There are several confinement. They also provide some useful explanations to μ works on nonzero isospin chemical potential ( I) and baryon the features of and supersymmetric quan- μ – chemical potential ( B) in pion condensation [98 100], and tum theory where the use of duality is a common phenome- fl they are related to the two avour quarks that constitute non. The discovery of these monopoles some day will be μ = ðμ − μ Þ/2 μ = ðμ + μ Þ/2 the pions as I u d and B u d ,respec- interesting and possibly revolutionary since almost all exist- tively. It has also been established [101] analytically at the ing technologies are based on and magnetism. quark level that the critical isospin chemical potential [102, Monopole condensation can now be exploited from the c 103] for pion superfluidity is precisely the pion mass μ = mπ ∗ I condensation of particles in the region r ⟶ r∗, where hϕϕ in vacuum. This behaviour and other related isospin chemical 2 i ~ hjϕj i . Hence, from Equation (16), we can express potentials in pion condensate have been investigated using the 0 Nambu-Jona-Lasinio (NJL) model, ladder-QCD [98]. μν ν ∂ ~ = 2jjϕ 2 ~ = ν : ð64Þ In the model framework, the color superconducting μ F q 0 A jm ∗ property can be studied from the propagator, hϕϕ i. The high density region where the particles are asymptotically The equations of motion for the static fields are free forms deconfined matter leading to Bose-Einstein con- densation resulting in the color superconducting phase [96, ! ∇· B~ = ρ , 104, 105]. On the other hand, at the low energy region, the m ð65Þ fields modify into glueballs with real mass, mη, leading to chi- ! ! −∇ × ~ = ; ral symmetry breaking and color confinement. Since quark- E j m Advances in High Energy Physics 9

V ~~ Er Br ~~ E0 B0 0.25 1.0 f = 0.29 0.20 � 0.8 0.15 0.6 0.10 f = 0.5 � 0.4 0.05 f = 1 � 0.2 | � | –2 –1 1 2 r Figure 3: The scalar field potential for different decay constants f α. 0.2 0.4 0.6 0.8 1.0 The potential has degenerate vacua. Figure fi ~ ð~ Þ 6: The screened electric (magnetic) eld Er Br against r. This graph holds for both electric and magnetic fields; thus, both fi V elds are screened from the interior of the vacuum. It was sketched for λ =65nm; below it, we have a mix or an intermediate 0.5 f = 0.29 � state with higher electric or magnetic field strength. Above the estimated λ, we have a superconducting state because the vacuum 0.4 is pure diamagnetic with no penetrating electric or magnetic fields.

0.3 ated with these fields can be expressed as f = 0.5 �  0.2 ! ! ! ! F = q B~ − v × E~ , ð66Þ f = 1 0.1 � ! where v is the speed of the particle within the fields and q is ! � ~ –2 –1 1 2 the monopole charge. The homogeneous equation ∇· E =0 represents the uniform electric field present at the confining Figure 4: The glueball potential VðηÞ against the glueball field η for phase. Combining the dual version of Ampere’s law on the ff di erent values of f α. The potential has a nondegenerate and well- right side of Equation (65) with the dual London equation fi de ned vacuum giving the glueballs a real mass mη. responsible for persistent current generated by the mono- poles

Vc ! 1 ! 10 ∇ × j = E~ , ð67Þ m λ2

5 we obtain the expression ! 1 ! ∇2 ~ = ~ : ð68Þ E 2 E r λ 2 4 6 8 10 = ∇× λ = ð 2jϕ j2Þ−1/2 –5 Noting that E A, we can identify q 0 2 2 4 −1/2 = ðq mη/2α ρÞ as the London penetration depth [112, 113]. Developing the Laplacian in Equation (68) in spherical coordinates yields Figure 5: Confining Cornell-like potential. The perturbative and the nonperturbative nature of the potential is displayed. Below the 2~ 2 ~ ~ d Er + dEr − Er =0; ð69Þ Fermi scale, the gluons are asymptotically free, while above the 2 2 Fermi scale, we only find confined color neutral particles. dr r dr λ

! ! this equation has a solution given as here, E~ and B~ are the dual versions of the static electric and ! −r/λ magnetic fields, while j and ρ are the magnetic current c1e m m E~ = , ð70Þ and charge densities, respectively. The Lorentz force associ- r r 10 Advances in High Energy Physics

2 ( ⁎) Vc m � 2 10 m� T = 0 1.0 5 T = 0.7T 0.8 c

0.6 r 2 4 6 8 10 = 0.4 T Tc –5 0.2

T Figure fl 0.2 0.4 0.6 0.8 1.0 Tc 10: Temperature uctuating Cornell-like potential. The gradient of the graph decreases with increasing T and flattens at T Figure fl = fi 7: Thermally uctuating glueball mass with temperature. Tc indicating decon nement and hadronization phase. The glueball mass decreases sharply with increasing T until it = vanishes at T Tc, where the glueballs melt into gluons. where c1 is a constant. Using dimensional analysis, we can fix ~ the constant as c1 = E0; therefore, ( )⁎ mA g ~ −r/λ E0e 1.0 E~ = , ð71Þ r r 0.8 ~ = j~j = ~ fi where E E Er. This means that the electric eld is expo- 0.6 nentially screened from the interior of the vacuum with pen- etration depth λ, a phenomenon known as the color Meissner 0.4 effect. Also, using London’s accelerating current relation,

0.2 ! ! ∂ ~ = −λ2 j m , ð72Þ B ∂ T t 0.2 0.4 0.6 0.8 1.0

Figure ∗ ð Þ/ and writing the continuity equation in the form of London 8: Variation of the gluon mass mA T g with T. The gluon mass rather increases with temperature, because this phenomenon equations, occurs at a temperature where the bound states of gluons or ! glueballs melt into gluons. ∂ j m +∇ρ =0, ð73Þ ∂t m � �0 1.0 together with the equation at the left side of Equation (65), we obtain 0.8 ! ! ~ 2 B 0.6 ∇ B~ = : ð74Þ λ2 0.4 This equation has a solution similar to Equation (69), i.e., 0.2

~ −r/λ T ~ = B0e : ð75Þ T Br 0.2 0.4 0.6 0.8 1.0 c r

Figure 9: The string tension σðTÞ/σ0 against temperature T/T . c Thus, the magnetic field is equally screened exponentially The string tension decreases sharply with T and breaks or λ vanishes at T = T representing hadronization. from the interior of the vacuum by a penetration depth .In c sum, both -varying electric and magnetic fields are screened with the same depth λ. Advances in High Energy Physics 11

�

�0

1.8

1.6

1.4

1.2

T 0.2 0.4 0.6 0.8 1.0 Tc (a)

ns nn n0 n0 1.0

0.8 ns

0.6

0.4

0.2 nn

T 0.2 0.4 0.6 0.8 1.0 Tc

(b) Figure λ 11: Penetration depth (a), superconducting ns, and normal nn monopole densities with temperature (b). The penetration depth λ = λ fi = remains constant at 0 representing pure diamagnetic vacuum before it starts rising steadily with T and becomes in nite 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.

Now, comparing λ with the results in Refs. [112, 114], equivalent to the monopole charge q [115, 116], and the elec- where tron and mass are related to the monopole density and mass, respectively. Combining Equation (67) λ2 = m , ð76Þ with the electric field expression at the right side of Equation 2 fl nse (65), we obtain a uxoid relation: ð þ λ ! ! ! ! we rearrange the expression obtained for in the model ~ · − λ2 · = Φ , ð78Þ framework, E dS j m dl n e

2α4ρ m2 λ2 = = η , ð77Þ where n ∈ ℤ and Φ is the quantum of the electric flux [112]. 2 2 2 2ρ 4 e q mη q a The penetration depth could also be expressed as a function of temperature to make it suitable for comparison. Juxtaposing these two 2 3 !1/2  equations leads to q = e, m = m /a and n =2ρa , where m, e 4 4 −1/2 η s 2α ρ T , and λðÞ≃ ≃ λðÞ0 1 − : ð79Þ ns are mass, charge, and electron number density of a T 2 ∗2 4 q mη ðÞT T superconducting material. Hence, the electron charge e is c 12 Advances in High Energy Physics

We retrieve the initial penetration depth below Equation Hagedorn temperature, T ~ 170 MeV, or higher baryon den- (68), if we set T =0. Accordingly, the penetration depth sities above 5ρ0 where the free quarks and gluons start increases with temperature until it becomes infinite at T = clamping together into hadrons. This is also observed at the fi Tc, where decon nement and restoration of chiral symmetry initial stages of heavy ion collisions [121, 127, 128]. On the coincide [117, 118]. At this point, the field lines are expected other hand, the string decay hadronization occurs due to to penetrate the vacuum causing it to lose its superconduct- nonperturbative effects. Under this picture, new hadrons ing properties. This expression is precisely the same as the are formed out of quark-antiquark pairs or through a gluon > one obtained for metallic superconductors [114]. At T Tc, cascade [126]. the magnetic and electric field lines go through the vacuum. Theoretically, Light-Front (LF) is one of the suitable the- This phenomenon is referred to as the Meissner effect. The ories for describing relativistic interactions due to its natural number density of the superconducting monopoles can also association with the light cone. Hence, under this theory, one be expressed in terms of temperature as can perform Fock expansion

4 3/2 4 3/2 ðÞ=2ρ 3 1 − T = 1 − T : ð80Þ jiπ = jiqq + jiqqqq + jiqqg +⋯, ð82Þ ns T a 4 n 4 Tc Tc where the valence jqqi and the nonvalence quark compo- The number density decreases sharply with temperature nents are fully covered [129, 130]. Hence, there is much T and vanishes at T = T . On the other hand, at T < T , sim- c c information on the partonic structure of hadrons stored in ilar to the two-fluid model where normal monopole fluid the pions produced during hadronization that can be further mixes with superconducting ones, the density of the mono- = + exploited. These processes give an insight into the transition poles can be calculated from the relation n ns nn, where to the hypothetical QGP phase of matter where deconfine- nn is the normal monopole density [119]. Additionally, at T ⟶ 0 ⟶ ⟶ ⟶ 0 ment and chiral symmetry restoration are expected. These , ns n, and in the limit T Tc, ns . Conse- studies gained more attention when ultrarelativistic heavy quently, ion collision experiments in BNL Brookhaven and CERN () Geneva were able to create matter under extreme conditions 4 3/2 T of temperature and densities necessary for phase transition n = n 1 − 1 − : ð81Þ n 4 [131]. Pion-pion annihilation is particularly necessary since Tc pions are the second most abundant particles produced dur- ing the heavy ion collision from hot and dense hadron gas. < Thus, the color superconducting monopoles ns at T Tc Such annihilation also results in the production of photon fl conduct with dissipationless ow; on the other hand, the nor- and dilepton which form the basics for probing quark- fi mal monopoles nn conduct with nite resistance at the same gluon plasma phase and further hadronizations. These parti- temperature range. Consequently, ns decreases sharply with cles are known to leave the dense matter phase quickly with- increasing T while nn increases steadily with increasing T. out undergoing strong interactions [10–12]. However, the are capable of annihilating into quarks and gluons 5. Analysis and Conclusion (partons) in a process popularly referred to as gluon brems- stralhaung [16, 17]. The free parameters of the model are ν, 5.1. Analysis. At a high energy regime, there is high particle ξ jε j ρ α ϕ ν density which forms deconfined matter leading to hadroniza- , v , , , a, and 0, where is the degeneracy of the gluons ∗ ∗ and its acceptable value for massless gluons in SUð3Þ repre- tion [120, 121], ϕ ðp Þϕðp Þ ⟶ γ ⟶ ϕ ðp′Þϕðp′Þ [68, 122– 1 2 1 2 sentation is ν =16while its value under SUð2Þ representation 125]. In this regime, the process takes place at low tempera- is ν =6. It is known from QCD lattice simulation that σ ~1 tures but in high momentum and particle density. This leads GeV/fm; therefore, we can determine from Equation (28) to an behaviour among the particles that mη =1:73 GeV [43–45, 132, 133] which is precisely the because color confinement depends on how far or close the scalar glueball mass. For zero degeneracy, i.e., ν =0, the crit- particles are together. Thus, higher density means the parti- fi cles are closer together leading to deconfinement. Addition- ical temperature Equation (34) goes to in nity and all the glueballs get melted leaving pure gluons [75]. We determined ally, Equation (55) represents the gluon propagator which = 870 MeV mediates the interaction of the glueballs in the IR regime of the gluon mass to be mA in the model framework. This value lies within the values obtained from lattice simula- the model. Hadronization [126] comes into play in this jε j regime when the momenta of the particles are reduced below tion as referred to in Section 3.3. Also, v is the magnitude of the QCD vacuum at the ground state. It has different values a particular threshold set by the QCD scale, ΛQCD ~ 250 depending on the model under consideration. Its value esti- MeV, or the separation distance between particle and anti- ~1fm mated from the sum rule for gluon condensate is 0:006 ± particle pairs is greater than . For the sake of clarity, 0:012 GeV4 there are two forms of hadronization mentioned here, one [134], and the Bag model is also quoted as ð145Þ4 MeV4 jε j = h μν i = ξ−2 = ρ 4 arising from high energy QGP formation and the other from . From the model v F Fμν a , QCD string decay into hadrons in the low energy regime. The recalling that ϕ0 is related to a and α as ϕ0 = a/α, a has the QGP hadronization is believed to have been formed immedi- dimension of energy and α is dimensionless. Again, when ately after the when the QGP starts cooling down to we combine Equations (28) and (37), we can express the Advances in High Energy Physics 13 string tension σ as a function of temperature T: in the low energy regime. In this paper, two forms of hadro- nization were discussed; the first one takes place at the high  m∗2ðÞT m2ðÞ0 4 energy regime where the individual quarks and gluons σðÞ= η = η 1 − T : ð83Þ T 3 3 4 recombine to form hadrons/pions. The second is at the low Tc energy regime where the quark and antiquark pairs/valence gluons that form pions/glueballs get separated due to long =0 σð = Þ =0 Here, we retrieve Equation (28) at T , T Tc , separation distances, and the string tension that holds them > and at T Tc, we move into the quark-gluon plasma together breaks leading to hadronization. We also investi- (QGP) phase where the particles interact in a disorderly gated the effect of temperature on the effective scalar glueball manner [46, 135, 136]. Thus, the potential in Equation (27) ∗2ð Þ ∗ ð Þ σð Þ mass mη T , the gluon mass mA T , string tension T , can be expressed in terms of temperature as confining potential V ðr, TÞ, gluon condensate μν c  h2GðηÞF Fμνi , the penetration depth λðTÞ, the supercon- ∗2 2 0 4 T 1 mη ðÞT 1 mηðÞ T ð Þ V ðÞr, T = − + r = − + 1 − r, ð84Þ ducting monopole densities ns T , and the normal monopole c 3 3 4 ð Þ – r r Tc density nn T , and their results are presented in Figures 6 11, respectively. We calculated the QCD β-function and the “ ” α where we recover Equation (27) at T =0. Thermal deconfine- strong running coupling s through = theory to enhance the discussions on gluon . ment and the restoration of chiral symmetry occur at T Tc [117, 118] due to the dissolution of the glueball mass, and we Additionally, the glueball mass and the gluon masses were > calculated, and the outcome was compared with lattice simu- have the QGP phase at T Tc [46, 135, 136] where the glue- ball mass becomes unstable. As established in SUð2Þ group lation results, analytical study, or phenomenological analyses representation, it is known from the hadron spectrum that to ascertain their reasonability. Finally, the model produces SUð2Þ ×SUð2Þ the correct behaviour of confining potential at T =0, where the L R chiral symmetry is spontaneously bro- ken due to the nonperturbative dynamics of the theory, lead- the potential has linear growth with r, Equation (27), consis- ing to color confinement. Considering quark representation tent with the Cornell potential model. The potential keeps < fi in SUð2Þ, the color charges may be screened by the gauge growing linearly for T Tc with thermal decon nement ≥ degrees of freedom. Under this representation, we can possi- phase at T Tc. However, most of the results on QCD lattice 2 bly have quark-gluon color singlet thermionic states, quark- calculations point to a temperature correction of order −T antiquark color singlets, or gluon-gluon color singlets. Gen- to the string tension [137–139]; meanwhile, the model pro- 4 erally, in this paper, we investigate glueballs which will fall poses correction in order of −T in Equation (83). This can ≃ − 2 under the gluon-gluon color singlet states. be regularized at T Tc to obtain the correct order T . Indeed, we did not find such regularization necessary because 5.2. Conclusion. The process of high energy annihilation of ∗ ∗ ∗ there are some proposed phenomenological models that sug- ϕ ϕ ϕ ϕ ϕ ϕ 4 to the production of through the interaction gest −T correction [140] to the string tension as well. But we ⟶ γ ⟶ ϕ∗ϕ during hadronization in the high particle fl deem it necessary to provide some explanations based on the density region is brie y addressed. Thus, we focus on the dis- model. In the previous paper Ref. [40], we obtained a correc- ~ ð1Þ 2 cussion to cover the low energy regime where U symme- tion of order −T as elaborated in Equation (63). We try is spontaneously broken through the Abelian Higgs observed that in Ref. [40], the temperature correction to the mechanism to give mass to the resulting glueballs. The scalar string tension came from the scalar field ϕ as elaborated in field in this case plays the role of the Higgs field, which fi the latter part of Section 3.3. On the other hand, the temper- undergoes modi cation into glueballs with mass mη. This ature correction in the gauge field leads to a correction in the fi 4 leads to color con nement of glueballs. We explored the order of −T as discussed in Section 3.2 and elaborated in fi ð∣ϕ ∣ Þ behaviour of the scalar eld potential V and the glue- Section 5.1 in Equations (83) and (84). ball potential VðηÞ with the decay constant f α, and the results are presented in Figures 3 and 4. The model explains other confining (IR) properties such as gluon condensation, glue- Data Availability ball mass, gluon mass, string tension, and dual superconduc- fi tivity through monopole condensation. The results for color The data used to support the ndings of this study have been confinement, screened electric (magnetic) field due to the made already available in the literature. monopoles, and the gluon condensate are presented in Figures 1–5, respectively. The glueball fields acquire their Conflicts of Interest mass through SSB, and their mass remains the most relevant parameter throughout the confinement properties. The mag- The authors declare that there are no conflicts of interest nitude of the glueball mass mη is precisely the same as the regarding the publication of this article. observed lightest scalar glueball mass mη =1:73 GeV. This mass is responsible for the string tension σ that keeps the Acknowledgments particles in a confined state and appears in the monopole condensate as well. It also appears in the gluon condensate We would like to thank CNPq, CAPES, and CNPq/PRO- μν h2GðηÞF Fμνi, capable of hadronizing to form light pions NEX/FAPESQ-PB (Grant no. 165/2018), for partial financial 14 Advances in High Energy Physics support. FAB also acknowledges support from CNPq (Grant [18] S. L. Wu and H. Zobernig, “A three-jet candidate (run 447, no. 312104/2018-9). event 13177),” TASSO Note No. 84, 1979. [19] F. Englert and R. Brout, “Broken symmetry and the mass of gauge vector mesons,” Letters, vol. 13, no. 9, References pp. 321–323, 1964. [20] P. W. Higgs, “Broken symmetries, massless particles and [1] S. Marzani, G. Soyez, and M. Spannowsky, Lecture Notes in fi ” Physics, 2019. gauge elds, Physical Review, vol. 12, no. 2, pp. 132-133, 1964. [2] A. Ali and G. 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