Research ArticleNonperturbative and Thermal Dynamics of Confined Fields inDual QCD

H. C. Chandola,1Deependra Singh Rawat ,1Dinesh Yadav,1H. C. Pandey,2 and H. Dehnen3

1Department of Physics (UGC-Centre of Advanced Study), Kumaun University, Nainital, India2Department of Physics, Birla Institute of Applied Sciences, Bhimtal, India3Fachbereich Physik, Universitat Konstanz, M677, 78457 Konstanz, Germany

Correspondence should be addressed to Deependra Singh Rawat; dsrawatphysics@gmail.com

Received 30 September 2019; Revised 30 May 2020; Accepted 11 June 2020; Published 9 July 2020

Academic Editor: Roelof Bijker

Copyright © 2020 H. C. Chandola et al. 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. Thepublication of this article was funded by SCOAP3.

In order to study the detailed dynamics and associated nonperturbative features of QCD, a dual version of the color gauge theorybased on a topologically viable homogeneous fibre bundle approach has been analysed taking into account its magnetic symmetrystructure. In the dynamically broken phase of magnetic symmetry, the associated flux tube structure on a S2 sphere in themagnetically condensed state of the dual QCD vacuum has been analyzed for the profiles of the color electric field using fluxquantization and stability conditions. The color electric field has its intimate association with the vector mode of themagnetically condensed QCD vacuum, and such field configurations have been analyzed to show that the color electric flux getslocalized towards the poles for a large sphere case while it gets uniformly distributed for the small sphere case in the infraredsector of QCD. The critical flux tube densities have been computed for various couplings and are shown to be in agreement withthat for lead-ion central collisions in the near infrared sector of QCD. The possible annihilation/unification of flux tubes undersome typical flux tube density and temperature conditions in the magnetic symmetry broken phase of QCD has also beenanalyzed and shown to play an important role in the process of QGP formation. The thermal variation of the profiles of thecolor electic field is further investigated which indicates the survival of flux tubes even in the thermal domain that leads thepossibility of the formation of some exotic states like QGP in the intermediate regime during the quark-hadron phase transition.

1. Introduction

It is widely believed that the quantum chromodynamics as anon-Abelian theory of gauge fields can be very well used forthe fundamental description of strong interactions [1, 2]between quarks and gluons especially in its high energysector. Apart from this, it is also known to exhibit manyimportant nonperturbative features, like confinement, chiralsymmetry breaking, and mass spectrum of the physical stateof hadrons as far as its low energy sector [3–6] is concerned.The color confinement in QCD is then naturally one of themost challenging issues, and it is directly linked with thephysical spectrum of the theory. In the low energy sector ofQCD, there have been many conjectures proposed for themechanism of color confinement. In this context, Nambu[7] and others [8, 9] have argued that the monopole conden-

sation leads the dual Meissner effect that could provide a wayfor the color confinement in QCD in a manner similar to theordinary superconductor where magnetic flux confinementoccurs due to the Meissner effect. To ensure the color con-finement through the dual Meissner effect [5, 10, 11], how-ever, one needs color monopoles as the most essentialdegrees of freedom. The appearence of color monopoles inQCD was proposed by ‘t Hooft [12] based on the Abeliangauge fixing that reduces SU(N) gauge theory to Uð1ÞNc−1

with color monopoles as a topological excitations of QCDvacuum. However, the mechanism of monopole condensa-tion in QCD vacuum remains far from clear. The chromo-electric flux between static quarks is then squeezed intostring or tube-like vortex structures in superconductingmedium, and such flux tube representation of hadrons leadsto a lineraly rising potential between pairs of static quarks at

HindawiAdvances in High Energy PhysicsVolume 2020, Article ID 4240512, 13 pageshttps://doi.org/10.1155/2020/4240512

zero temperature, characterizing the color confinementwhich, in turn, is strongly supported by the recent latticeQCD simulation studies [13–18]. Thus, the apparent domi-nance of color monopole condensation seems to provide aviable explanation of color confinement and related featuresof the infrared sector of QCD. However, inspite of the greatsuccess of lattice calculation, it is tempting to develop a for-mulation based on first principle of QCD, which may provideus with a clearer understanding of the physical picture of dualQCD vacuum and associated confinement mechanism andallows us to perform some analytical calculation also. Ananalytical effective theory of nonperturbative QCD is, there-fore, highly desirable which may incorporate the topologicalaspects, in addition to various glueballs and their interac-tions, which are the most practical degrees of freedom forthe study of the QCD phase transition. In this direction, atopologically effective magnetic symmetry based dual gaugeformulation [19–24] might prove a parallel to explain themysterious confining behaviour and associated features ofquarks inside the hadronic jails. Furthermore, in recent years,the collective behavior of color isocharges at a finite temper-ature and density has also attracted a great interest in thetheoretical as well as experimental investigations of QCDphase diagram [25]. In this way, the dual QCD formulationis expected to provide the relevant understanding of variousnonperturbative aspects of QCD, the QCD phase transitionand the possible formation of an intermediary state likequark-gluon plasma (QGP) [26–32] or any other exotic statesunder the extreme conditions of density and temperature[33–35] which might also be relevant to uncover the manyhidden aspects of the primordial structure of universe. Inthe present study, the mechanism of color confinement,associated confined field profiles, and the QCD phase transi-tion are studied by investigating the flux tube structure asso-ciated with dual QCD formulation based on topologicallyviable magnetic symmetry approach and analysing on ener-getic grounds. The typical dual QCD parameters of quark-hadron phase transition have been evaluated using energybalance conditions and analysed numerically. The study isfurther extended to investigate the profiles of color electricfield at different hadronic scales using energy minimizationand stabilty conditions. The critical flux tube densities havebeen computed and analyzed in view of the QGP formationin heavy-ion collision events. The field profiles have furtherbeen investigated for better understanding of the QCD phasetransition under varying thermal conditions.

2. Magnetic Symmetry Structure and ConfiningFeatures of Dual QCD

In order to analyze the typical nonperturbative features of thecolor gauge theory, let us briefly review the dual formulationof QCD [19–24, 36–40] in terms of the magnetic symmetryand the associated topological structure of dual QCD vac-uum. It is well-known that the non-Abelian theory of gaugefields can be viewed [19–24, 38–40] as the Einstein theoryof gravitation in a higher-dimensional space which unifiesthe four dimensional space-time with n-dimensional internalspace and allows the introduction of some additional internal

isometries. In this connection, the magnetic symmetry maybe introduced as a set of self-consistent Killing vector fieldsof the internal space which, while keeping the full gaugedegrees of freedom intact, restricts and reduces some of thedynamical degrees of freedom of the theory. For the case ofquark color symmetry, it, in turn, may be shown to establisha dual dynamics between the color isocharges and thetopological charges of the underlying gauge group. For thesimplest choice of the gauge group G ≡ SUð2Þ with its littlegroup H ≡Uð1Þ, the gauge covariant magnetic symmetrycondition may be expressed [19–24] in the following form,

Dμm̂ = 0⇒ ∂μ + gWμ ×� �

m̂ = 0, ð1Þ

where m̂ is a topological field that belongs to the adjoint rep-resentation of the gauge group G and Wμ is the associatedgauge potential of the underlying gauge group G. The condi-tion (1) thus implies that the magnetic symmetry imposes astrong constraint on the metric as well as connection andmay, therefore, be regarded as the symmetry of the potentials.The monopoles, therefore, emerge as the topological objectsassociated with the elements of the second homotopic groupΠ2ðG/HÞ. The typical gauge potential satisfying the condi-tion (1) is parametrised as

Wμ = Aμm̂ − g−1 m̂ × ∂μm̂� �

, ð2Þ

where m̂:Wμ = Aμ is the color electric potential unrestrictedby magnetic symmetry, while the second term is completelydetermined by magnetic symmetry and is topological inorigin. Thus, the virtue of the magnetic symmetry is that itcan be used to describe the topological structure of gaugesymmetry and the isolated singularities of m̂ may then beviewed as defining the homotopy of the mapping Π2ðS2Þon m̂ : S2R ⟶ S2 = SUð2Þ/Uð1Þ which describes the non-Abelian monopoles. IndeedWμ with vanishing Abelian com-ponent (Aμ = 0) and m̂ = r̂ describes precisely the Wu-Yangmonopole. The imposition of magnetic symmetry on thegauge group thus brings the topological structure into thedynamics explicitly. The associated field strength corre-sponding to the potential (2) is then given by

Gμν =Wν,μ −Wμ,ν + gWμ ×Wν ≡ Fμν + B dð Þμν

� �m̂, ð3Þ

where Fμν = Aν,μ − Aμ,ν and BðdÞμν = −g−1m̂:ð∂μm̂ × ∂νm̂Þ =

Bν,μ − Bμ,ν. The second part (Bμ), fixed completely by m̂, isthus identified as the magnetic potential associated with thetopological monopoles and the fields thus appear in acompletely dual symmetric way. Although, the group SU(2)is examined here as a simplest possible example, this wholeformulation can be established for an arbitrary gauge group(including SU(3) case). However, in all cases, the generalstructure of the formulation which includes the existence ofthe dual structure, gauge-independent separation of mag-netic degrees of freedom, the existence of different phases,

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and the magnetic confinement of the colored flux all remainidentical and intact.

However, in order to explain the dynamics of the result-ing dual QCD vacuum and its implications on confinementmechanism, let us start with the SU(2) chromodynamicLagrangian with a quark doublet source ψðxÞ, as given by

L = −14G

2μν + �ψ xð ÞiγμDμψ xð Þ −m0�ψ xð Þψ xð Þ: ð4Þ

In addition, in order to avoid the problems due to thepoint like structure and the singular behavior of the potentialassociated with monopoles, we use the dual magnetic poten-tial BðdÞ

μ coupled to a complex scalar field ϕðxÞ. Taking theseconsiderations into account, the modified form of the dualQCD Lagrangian (4) in quenched approximation may beexpressed as

L dð Þm = −

14B

2μν + ∂μ + i

4πgB dð Þμ

� �ϕ

��������2−V ϕ∗ϕð Þ, ð5Þ

with Vðϕ∗ϕÞ as a proper effective potential which is fixedby the requirements of the ultraviolet finiteness and infra-red instability of the Lagrangian as given by Coleman andWeinberg [41] by using the single loop expansion technique as

V ϕ∗ϕð Þ = 24π2

g4ϕ40 + ϕ∗ϕð Þ2 2 ln ϕ∗ϕ

ϕ20− 1

� �: ð6Þ

The effective potential is reliable in the far infrared regionwhere coupling becomes very intense (αs ⟶ 1) and confine-ment is strongly enforced. Since, in the present case, we arealso interested to investigate phase transition in dual QCDvacuum, the use of an effective potential reliable in relativelyweak coupling near-infrared regime is naturally desired and,therefore, the appropriate choice for inducing the dynamicalbreaking of magnetic symmetry is the quartic potential ofthe following form

Vpt ϕ∗ϕð Þ = 3λα−2s ϕ∗ϕ − ϕ20

� �2: ð7Þ

In order to analyze the nature of magnetically condensedvacuum and associated flux tube structure, let us investigatethe field equations led by the Lagrangian (5) in the followingform

∂μ + i4πg−1B dð Þμ

� �2ϕ + 6λα−2s ϕ∗ϕ − ϕ20

� �ϕ = 0, ð8Þ

∂νBμν + i4πg−1 ϕ∗∂μϕ − ϕ∂μϕ∗� �

− 8πα−1s B dð Þμ ϕϕ∗ = 0: ð9Þ

The unusual features of dual QCD vacuum responsiblefor its nonperturbative behaviour may become more trans-parent if we start with the Neilsen and Olesen [42] interpre-tation of vortex-like solutions. It leads to the possibility of

the existence of the monopole pairs inside the superconduct-ing vacuum in the form of thin flux tubes that may beresponsible for the confinement of any colored fluxes. Undercylindrical symmetry ðρ, φ, zÞ and the field ansatz given by

B dð Þφ xð Þ = B ρð Þ, B dð Þ

0 = B dð Þρ = B dð Þ

z = 0,ϕ xð Þ = exp inφð Þχ ρð Þ n = 0, ±1, ±2, −−ð Þ,

ð10Þ

the field equations (8) and (9) are transformed to the follow-ing form

1ρ

ddρ

ρdχdρ

� �−

"nρ+ 4πα−1s� �1/2

B ρð Þ� �2

− 6λα−2s χ2 − ϕ20� �#

χ ρð Þ = 0,ð11Þ

ddρ

ρ−1ddρ

ρB ρð Þð Þ� �

+ 16πα−1s� �1/2

� nρ−4πgB ρð Þ

� �χ2 ρð Þ = 0:

ð12Þ

Further, with these considerations, the form of the colorelectric field in the z-direction is given by

Em ρð Þ = −1ρ

ddρ

ρB ρð Þð Þ: ð13Þ

For a more convenient representation, we use the follow-ing dimensionless parameter

r = 2ffiffiffiffiffi3λ

pα−1s ϕ0ρ, F rð Þ = 4πα−1s

� �1/2ρB ρð Þ,H rð Þ = ϕ−10 χ ρð Þ,

ð14Þ

so that the field equations (11) and (12) (with λ = 1) arereduced to a more simpler form as

H″ + 1rH ′ − 1

r2n + Fð Þ2 + 1

2H H2 − 1� �

= 0, ð15Þ

F″ − 1rF ′ + α n − Fð ÞH2 = 0, ð16Þ

where α = 2παs/3λ and the prime stands for the derivativewith respect to r. Using the asymptotic boundary conditionsgiven by F ⟶ −n, H⟶ 1 as r⟶∞ along with equation(14), the asymptotic solution for the function F may beobtained in the following form

F ρð Þ = −n + Cρ1/2 exp −mBρð Þ, ð17Þ

where C = 2πBð3 ffiffiffi2

pλg−3ϕ0Þ

1/2and mB = ð8πα−1s Þ1/2ϕ0 is the

mass of the magnetic glueballs which appears as vectormode of the magnetically condensed QCD vacuum. Since,the function FðρÞ is associated with the color electric fieldgiven by equation (13) through gauge potential BðρÞ, it

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indicates the emergence of the dual Meissner effect leadingto the confinement of the color isocharges in the magneti-cally condensed dual QCD vacuum.

3. Critical Parameters of Phase Transition inDual QCD Vacuum

The formation of color electric-flux tubes and the confine-ment of color charges can be visualized more effectively onthe energetic grounds by evaluating the energy per unitlength of the flux tube configuration governed by the fieldequations (15) and (16) and the Lagrangian (5), which isobtained in the following form

k = 2πϕ20ð∞0rdr

6λg2

F ′� �2

r2+ n + Fð Þ2

r2H2

264

+ H ′� �2

+ H2 − 1� �2

4

375,

ð18Þ

which in view of equation (14) reduces to

k = 2πð∞0ρdρ

"n2g2

32π21ρ

dFdρ

� �2+ n2

ρ2F2 ρð Þχ2 ρð Þ

+ ∂χ∂ρ

� �2+ 48π2

g4χ2 ρð Þ − ϕ20� �2#

:

ð19Þ

It, in turn, plays an important role in the phase structureof QCD vacuum, if we take the multi-flux tube system on a S2

-sphere with periodically distributed flux tubes and intoducea new variables R on S2-sphere and express it as ρ = R sin θ.As a result, a number of flux tubes considered here inside ahadronic sphere of radius R pass through the two poles ofthe hadronic sphere. Under such prescription, the flux tubesolution governed by equations (11) and (12) correspondsto the case of large R limit ðR⟶∞Þ such that R≫ ρ andθ⟶ 0. With these considerations, the finite energy expres-sion given by the above equation (19) may be reexpressed as

k = εC + εD + ε0, εC = kCR2, εD = kDR

−2, ε0 = k0, ð20Þ

where the functions kC , kD, and k0 are given by

kC =6πα2s

ðπ0

χ2 θð Þ − ϕ20� �2

sinθdθ, ð21Þ

kD = n2αs4

ðπ0

1sin θ

∂F∂θ

� �2dθ, ð22Þ

k0 = 2πðπ0

n2F2 θð Þχ2 θð Þsin θ

+ sin θ∂χ∂θ

� �2" #

dθ: ð23Þ

The energy expression (20) provides a straightforwarddescription of the behavior of QCD vacuum at different

energy scales. At large hadronic distances, the first term (εC)in equation (20) dominates which increases at increasinghadronic distances and gets minimized when the monopolefield picks up its non-zero vacuum expectation value whichincidently acts as an order parameter to indicate the onsetof the dynamical breaking of magnetic symmetry. The asso-ciated magnetic condensation of QCD vacuum then forcesthe color electric field to localize in the form of the thin fluxtubes extending from θ = 0 to θ = π and the QCD vacuumis ultimately pushed to the confining phase. Further, theenergy expression (19) has its own implications for the eval-uation of critical parameters of phase transition and theirnumerical computation then leads to a deep significance forthe validity of field decomposition formulation of dualQCD. For the computation of such critical factors, we pro-ceed by evaluating the functions associated with the expres-sion (20) in terms of basic free parameters of the theory(viz., αs and mB) in the following way

kC =6πα2s

ðπ0

χ2 θð Þ − ϕ20� �2 sin θdθ⇒ kC =

3m4B

16π , ð24Þ

which shows the vector mode mass of the magnetically con-densed vacuum plays a crucial role in the confining phase ofthe QCD vacuum. On the other hand, the component ofenergy dominant over relatively short hadronic distances(εD), expressed in terms of the function kD given by equation(22), may also be evaluated in terms of free parameters of thetheory in the form as given below:

kD = πR4ðπ0E2m θð Þ sin θdθ, ð25Þ

where

Em θð Þ = ng

4πR2 sin θ

∂F∂θ

: ð26Þ

Using equations (25) and (26) alongwith the flux quanti-zation condition given byð

ρEm ρð Þdρ = ng4π ð27Þ

then leads to

kD = n2g2

8π = 12 n

2αs: ð28Þ

The contributions, in terms of the energy functions kCand kD, may be used to compute the critical parameters ofphase transition from magnetically dominated phase to elec-trically dominated one. For this purpose, let us evaluate theratio of εD and εC which is obtained as

εDεC

= η1R4

� �, η = kD

kC= 8πn2αs

3m4B

: ð29Þ

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For the confinement-deconfinement phase transition, wehave ðεD/εCÞ = 1 and R = Rc which leads to the critical radiusof phase transition in the following way

Rc =83πn

2αs

� �1/4m−1

B : ð30Þ

The corresponding critical density of the flux tubes (dc)inside the hadronic sphere is given by

dc =1

2πR2c

= 323 π3n2αs

� �−1/2m2

B: ð31Þ

These parameters are extremely important in exploringthe underlying mechanism and nature of QCD phase transi-tion. The equations (30) and (31) exhibit that the criticalradius and critical density of phase transition are clearlyexpressible in terms of free parameters of the QCD vacuum.In view of the running nature of QCD coupling constant, wecan estimate these critical factors associated with the QCDvacuum in its infrared sector using the numerical estimationsof glueball masses. For instance, for the optimal value of (αs)as αs ≡ 0:12 with the glueball masses mB = 2:102GeV andmϕ = 4:205GeV, equations (30) and (31) lead to,

Rc = 0:094 fm,dc = 18:003 fm−2:

ð32Þ

The deconfinement phase transition in the magneticallycondensed QCD vacuum therefore appears around theabovementioned critical values for a typical coupling ofαs = 0:12 in the near infrared sector of QCD. In this case,for Rc ≡ 0:094 fm, the flux tube density acquire its criticalvalue of 18.003 f m−2 and the first part of the energy expres-sion (20) dominates which demonstrates the confinementof color particles in the magnetically rich QCD vaccum.However, below Rc = 0:094 fm, the quarks and gluons appearas free states and the system stands near the boundary of theperturbative phase where the second part of the expression(20) becomes dominant leading to the deconfinement ofcolor isocharges. The flux tube density in this sector increasessharply and with sufficiently dense flux tube system, the fluxtube annihilation may takes place which then leads to thegeneration of dynamical quarks and gluons. The gluon self-interactions are then expected to play a major role in thethermalization of QCD system and create an intermediarystate of quark-gluon plasma (QGP). As a result of such fluxtube melting in the high momentum transfer sector ofQCD vacuum, the system is expected to evolve with an inter-mediary QGP phase. In addition, in the deep infrared sectorof QCD with higher couplings, (αs), e.g., 0.24, 0.48, and 0.96,a considerable enhancement in the critical radius (0.152 fm,0.226 fm, and 0.356 fm) is observed which leads to a formida-ble depletion in critical density (7.22, 3.10, and 1.24 fm-2,respectively) of flux tubes in QCD vacuum. As shown bycolor electric field profiles, the color electric field given byequation (26) obtained for the asymptotic solution (large R)

leads to uniform field when R decreases to approach itscritical value Rc where the flux tube density acquires itscritical value dc around which a number of flux tubes areexpected to get melted. These critical flux tube densitiesmay be compared to those produced in case of centralheavy-ion collisions for the creation of QGP. The flux tubedensities for such collisions [43] for the case of one flux tubeper nucleon-nucleon hard collision, is given by

dc =9A2/3

4πR20, ð33Þ

where A is the mass number of heavy ions and R0 is nucleusradius parameter (1.2 fm). For the case of Pb-Pb central col-lisions (A = 208), it leads to a value of 17.5 fm-2, which is invery good agreement with that given by equation (31) ofthe present dual QCD model and indicates the vital roleplayed by the flux tube number density and associated mono-pole condensate in QGP phase transition in QCD.

4. Confined Field Configurations in DualQCD Vacuum

In order to further discuss the phase structure of QCDvacuum in terms of the critical parameters, let us extendour study to the profiles of the color electric field in thefull infrared sector of dual QCD. Using the prescriptionfor the color electric field and potential as given in theprevious section, the total electric flux penetrating the area(S) surrounded by a closed loop around the upper sphereof radius R is given by

ϕ =ðSEm∙dS ≡

þB dð Þ∙dl = −2π

ð∞0ρEm ρð Þdρ

= −2πR2ðπ/20

Em θð Þ sin θdθ:

ð34Þ

Using the flux quantization condition along with thesubstitution of the variable, cos θ = p, it yields

ðπ/20

Em θð Þ sin θdθ =ð10Em pð Þdp = ng

4πR2 ≡N: ð35Þ

This along with equation (20) may be used to evaluatethe flux tube energy component εD in the following form

εD = πR2ð10

Em pð Þ −Nf g2dp +N2� �

: ð36Þ

The energy minimization condition then leads to

Em pð Þ =N = ng

4πR2 , ð37Þ

so that the color electric field is distributed uniformly inthe deconfinement region where εD dominates. The criticalvalue of such color electric field at the boundary of phase

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transition from the deconfined phase to the confined phasemay then be obtained by using R = Rc, in the following form

Ecm = ng

4πR2c

= 14π

ffiffiffi32

rm2

B, ð38Þ

which leads to its numerical value as Ecmð f m−2Þ = 23:16, 11.03,

5.81, 3.80, and 2.16 at strong couplings αs = 0:048, 0.12, 0.24,0.48, and 0.96, respectively, in the full infrared sector ofQCD. The variation of such critical color electric field at thephase transition boundary, as depicted in Figure 1, shows alarge reduction in the color electric flux spread out in deepinfrared sector on one hand and a considerable enhancementin its value on the other, in the transitional region where anumber of flux tubes are expected to lead a homogeneousQGP as a result of their annihilation.

Furthermore, the general form of color-electric field maybe evaluated by using equation (13) along with the asymp-totic solution (17) and is given as

Em ρð Þ = ngC8πρ3/2 1 − 2mBρð Þ exp −mBρð Þ: ð39Þ

For the case of multi-flux tube system on the S2-sphere,the flux tubes are periodically distributed over the sphere ofradius R and the associated color electric field passing verti-cally on the surface of sphere is obtained as,

Em θð Þ = ~Em exp −RmB sin θð Þ, ~Em

= nCα1/2s

4π1/2R3/2sin3/2θ1 − 2RmB sin θð Þ

ð40Þ

The profile of such color electric field as a function of thepolar angle θ for different values of radius (R) at different αsin the infrared sector of QCD has been presented by a 2-dgraphics given by Figure 2. It clearly shows that, in the infra-red sector of QCD, for a large sphere enclosing the flux tubes,the color electric flux gets localized or spreaded around thepoles (θ = 0 and π) while its gets uniformly distributed forthe small sphere case and acquires a constant value at thecritical radius Rc as given by equation (30). Similar resultscan be drawn from the 3-d graphics of Figure 3 for the colorelectric field as a function of the polar angle θ and radius R fordifferent values of coupling constant αs. In these graphicswith the increase in radius of sphere in the infrared sectorof QCD, it demonstrate the reduction and drifting of maximaof color electric flux towards the higher sphere radii and con-sequently a reduction in flux tube density in the far infraredsector of QCD. In the near-infrared sector, however, theincrease in flux tube density may lead to the annihilation ofthe neighbouring flux tubes and a large homogeneous QGPformation in the central region is expected.

5. Thermal Effects on Field Configurations andAssociated Critical Parameters

It has been argued that the multi-flux tube structure of dualQCD leads to a viable explanation for the low-energy confin-

ing features of QCD and may further be used for exploringthe phase structure of QCD under some unusual conditionslike those of high temperatures and high densities. Thebehavior of QCD at finite temperature is, in fact, expectedto play a vital role in understanding the dynamics of theQCD phase transition including QGP phase of nuclearmatter [22]. Hence, in view of these facts, starting from theLagrangian (5), let us use the partition functional approachalong with the meanfield treatment for the QCD monopolefield to evaluate the thermal contributions to the effectivepotential in the dual QCD. The partition functional, for thepresent dual QCD in the thermal equilibrium at a constanttemperature T , may be given by an Eucledian path integralover a slab of infinite spatial extent and β (≡T−1) temporalextent, as,

Z J½ � =ðD ϕ½ �D B dð Þ

μ

h iexp −S dð Þ

� �, ð41Þ

where SðdÞ is the dual QCD action and is given by

S dð Þ = −iðd4x L

mð Þd − J ϕj j2

� �, ð42Þ

whereLðmÞd is given by equation (5) and for the phase transi-

tion study in dual QCD vacuum, the effective potential reli-able in relatively weak coupling in near infrared regime isnaturally desired as given in the form of equation (7). Underthe thermal evolution of the QCD system, there are markedfluctuations in the monopole field and the effective potentialat finite temperatures then corresponds to the thermody-namical potential which leads to the vital informations forthe QCD phase transition. The thermal evolution of theQCD system as investigated [22] by taking into account theDolan and Jackiw approach [44] using high temperatureexpansion for the effective potential ultimately leads to,

V eff ϕ, Tð Þ = 3λα−2s ϕ2 − ϕ20� �2 − 7

90π2T4 + 4παs + λ

2α2s

� �T2ϕ2:

ð43Þ

0 1 2 3 4 5 60

5

10

15

20

25

30

Emc

mB

Figure 1: (color online) Variartion of critical color electric fieldEmðcriticalÞ with mB.

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Its minimization, in turn, leads to the thermally evolvingvector glueball masses as

m Tð ÞB = 8πα−1s

� �1/2ϕ0 1 − T

Tc

� �2" #1/2

, ð44Þ

where, the critical temperature of phase transition as obtainedby vanishing coefficient of terms quadratic in ϕ in effectivepotential (equation (43)) is given as (for λ = 1),

Tc = 2ϕ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

4παs + 1

s, ð45Þ

which for different values of the couplings (i.e., for 0.12,0.24, 0.48, and 0.96) leads to the critical temperatures as0.318GeV, 0.272GeV, 0.220GeV, and 0.172GeV, respec-tively. With such potential, the case λ = 1, in fact, correspondsvery nearly to that given by one-loop potential as it reproduces

the same ratio for the twomass scales (scalar and vectormodesof the magnetically condensed vacuum). In addition, for λ = 1,these mass scales become nearly identical to those for the one-loop potential in the near infrared sector (αs ≪ 1) of QCDwhere αs is given by the running coupling constant fixed bythe renormalization group equation. However, though the λis a free parameter, these mass scales start diverging for highervalues of λ. Further, the thermal response of dual QCD ana-lysed using the mean field approach shows that the abovemen-tioned confinement mechanism remains intact in the lowtemperature regime which is true for all gauge groups includ-ing SU(3). However, with the increase in temperature, there isa reduction in the monopole condensate along with the resto-ration of magnetic symmetry and the QCD system passesthrough a weakly bound phase of QGP before transiting to acompletely deconfined phase. The reduction in magnetic con-densate for the case of SU(3) is larger due to increase in thenumber of degrees of freedom in comparison to SU(2) case.Here, it is then imperative that the critical temperatures forthe case of SU(3) group must be considerably smaller thanthose obtained using SU(2) for all QCD phase transitions.With these considerations for the thermal evolution of the

0

0.2

0.4

0.6

0.8Em

(𝜃)

𝜋—2𝜋 0

0.2

0.4

0.6

0.8

Em

(𝜃)

𝜋—2𝜋

0

0.2

0.4

0.6

0.8

Em

(𝜃)

𝜋—2𝜋 0

0.2

0.4

0.6

0.8

Em

(𝜃)

𝜋—2𝜋

R = 0.085 fmR = 0.094 fmR = 0.3 fm

R = 0.12 fmR = 0.152 fmR =0.3 fm

R = 0.175 fmR = 0.226 fmR = 0.4 fm

R = 0.3 fmR = 0.356 fmR = 0.5 fm

𝛼s = 0.24𝛼

s = 0.12

𝛼s = 0.96𝛼

s = 0.48

𝜃 𝜃

𝜃 𝜃

Figure 2: (color online) Profiles of color electric field EmðθÞ as a function of θ for different R at αs = 0:12, 0.24, 0.48, and 0.96 respectively.

7Advances in High Energy Physics

QCD system, the color electric field as given by equation (40)evolves at finite temperature in the following form

Em θ, Tð Þ = ~Em Tð Þ exp −Rm Tð ÞB sin θ

� �ð46Þ

where

~Em Tð Þ = nCT 4παsð Þ1/28πR3/2sin3/2θ

1 − 2Rm Tð ÞB sin θ

h i, ð47Þ

with CT = 2Bð ffiffiffiffiffi18

pλπÞ1/2α−5/2s ðmðTÞ

B Þ1/2. The critical value ofsuch color electric field around phase transition boundarymay also be obtained using equation (38) in the following form

Ecm Tð Þ = Ec

m 1 − TTc

� �2" #

: ð48Þ

In addition, the associated critical radius of phase transi-tion and the critical flux tube density inside the hadronicsphere under high temperature in thermal QCD reduce tothe following form

Rc Tð Þ = Rc 1 − T2

T2c

� �−1/2,

dc Tð Þ = dc 1 − T2

T2c

� �,

ð49Þ

which show a large reduction in critical color electric field andthe flux tube density in the phase transition region aroundcritical temperature point and transition of the system todeconfined phase of large critical radii. The typical thermalprofiles of color electric field as given by equation (46) havebeen depicted in Figure 4 and 5 for different couplings ininfrared sector of QCD. A considerable reduction in the fieldat increasing temperatures has been demonstrated by both2-d and 3-d graphics. At temperatures, T ≥ Tc, the field tendsto get distributed uniformally around the centre of the spherewhile it gets localized around the two poles (θ = 0 and π) anddrops down with its minimum around θ = π/2. In any case,the system maintains perfect reflection symmetry aroundθ = π/2 plane. For the physically accessible intermediatecoupling infrared sector (αs = 0:48), the phase transition isexpected around 0.220GeV around which the flux tubedensity approaches to its minimum value and the magneticcondensate is evaporated into thermal monopoles.

0.2

0.0

0.1

0.2

0.3 0

0.2

0.4

0.6

Em

(𝜃)

𝛼s = 0.24𝛼

s = 0.12

𝛼s = 0.96𝛼

s = 0.48

R

𝜋—2

𝜋

𝜃

0.0

0.1

0.3 0

0.2

0.4

0.6

Em

(𝜃)

R

𝜋—2

𝜋

𝜃

0.00.1

0.30.2

0.40.5 0

0.2

0.4

0.6

Em

(𝜃)

R

𝜋—2

𝜋

𝜃

0.0

0.2

0.4

0.6 0

0.2

0.4

0.6Em

(𝜃)

R

𝜋—2

𝜋

𝜃

Figure 3: (color online) 3d profiles of color electric field EmðθÞ for αs = 0:12, 0.24, 0.48, and 0.96 as a function of R and θ.

8 Advances in High Energy Physics

6. Summary and Conclusions

In view of the fact that the topological degrees of freedomof non-Abelian gauge theories play a crucial role in under-standing the various nonperturbative features as well as theunderlying phase structure of QCD vacuum, the magneticsymmetry-based dual version of QCD has been utilizedwhich is shown to lead to a dual dynamics between color iso-

charges and topological charges imparting the dual super-conducting properties to QCD on dynamical breaking ofmagnetic symmetry and confining any color isocharge pres-ent. Although, the group SU(2) is examined in the presentpaper as the simplest possible example, the whole formula-tion can be established for an arbitrary gauge group (includ-ing SU(3) case). However, in all cases, the general structure ofthe formulation which includes the existence of dual

0.050.100.150.200.250.300.35

0.05

0.10

0.15

0.20

0.25

0.30

0.1

0.2

0.3

0.4

0.5

0.6

Em

(𝜃,T

)Em

(𝜃,T

)Em

(𝜃,T

)

0 𝜋—2𝜋

T = 0.2 GeVT = 0.3 GeVT = 0.310 GeV

0 𝜋—2𝜋

T = 0.2 GeVT = 0.3 GeVT = 0.305 GeV

0 𝜋—2𝜋

T = 0.1 GeVT = 0.235 GeVT = 0.250 GeV

𝛼s = 0.12, R = 0.3 fm

𝛼s = 0.12, R = 0.3 fm

𝛼s = 0.12, R = 0.009 fm

𝛼s = 0.12, R = 0.2 fm

0.0

0.300.25

0.200.15

0

0.20.40.6

Em

(𝜃,T

)

T

𝜋—2

𝜋

𝜃

𝜃

𝜃

𝜃

𝛼s = 0.12, R = 0.2 fm

0.0

0.300.25

0.200.15

0.10

0

0.20.40.6

Em

(𝜃,T

)

T

𝜋—2

𝜋

𝜃

𝛼s = 0.12, R = 0.009 fm

0.0

0.250.20

0.15

0

0.20.40.6

Em

(𝜃,T

)

T

𝜋—2

𝜋

𝜃

0.10

0.10

Figure 4: (color online) Thermal variation of color electric field Emðθ, TÞ at αs = 0:12:

9Advances in High Energy Physics

structure, gauge-independent separation of magnetic degreesof freedom, the existence of different phases, and the mag-netic confinement of colored flux all remain identical andintact. Further, for the group with rank > 1 (e.g., 2 forSU(3)) one can findmonopole solution by identifying the rel-evant homotopy group, e.g., Π2ðSUð3Þ/Uð1Þ ⊗U ′ð1ÞÞ forthe SU(3) group where U(1) and U ′ð1Þ are two Abelian sub-groups generated by λ3 and λ8, respectively. However, thetrue symmetry of the solutions remains λ3 like always as itautomatically generates λ8-like symmetry choosing its suit-

able symmetry product. The resulting monopole solutionsobtained by choosing a proper magnetic symmetry exhibit-ing abovementioned full homotopy class of mapping (Π2)may easily be used to establish the duality that exists inQCD which allows us to obtain the monopole condensationfor QCD vacuum leading to color confinement in the sameway as has been done with the present gauge group SU(2).A stable monopole condensation in SU(2) case has beenachieved after integrating out all the dynamical degrees offreedom of non-Abelian gauge potential and identifying the

0.1

0.2

0.3

0.4

0.5

0.6

,

,

,

0.1

0.2

0.3

0.4

0.5

0.6

Em

(𝜃,T

)Em

(𝜃,T

)

0.1

0.2

0.3

0.4

0.5

0.6

Em

(𝜃,T

)

0 𝜋—2𝜋

0 𝜋—2𝜋

0 𝜋—2𝜋

T = 0.10 GeVT = 0.15 GeVT = 0.20 GeV

𝛼s = 0.48, R = 0.35 fm

𝛼s = 0.48, R = 0.35 fm

𝛼s = 0.48, R = 0.4 fm

𝛼s = 0.48, R = 0.4 fm

𝛼s = 0.48, R = 0.5 fm

𝛼s = 0.48, R = 0.5 fm

0.0

0.20

0.15

0.10

0

0.20.4

0.6

Em

(𝜃,T

)

T

𝜋—2

𝜋

𝜃

0.0

0.20

0.15

0.10

0

0.20.4

0.6Em

(𝜃,T

)

T

𝜋—2

𝜋

𝜃

0.0

0.20

0.15

0.10

0

0.2

0.4

Em

(𝜃,T

)

T

𝜋—2

𝜋

𝜃

𝜃

𝜃

𝜃

Figure 5: (color online) Thermal variation of color electric field Emðθ, TÞ at αs = 0:48

10 Advances in High Energy Physics

non-Abelian topologyΠ2ðS2Þ in a purely gauge-independentway, and then, the same may easily be extended to realisticSU(3) case in an identical way. Hence, to address the basicissues of monopole dominance, topological structure, andimplications for color confinement, the SU(2) case has beenconsidered for the sake of simplicity keeping in mind thatthe essential features of SU(2) QCD remain the same inSU(3) QCD also. A unique periodically distributed flux tubestructure has been shown to emerge in magnetically con-densed QCD vacuum which has been analysed on a S2-sphere to compute the critical parameters of phase transitionusing energy-balance condition for the energy componentsgiven by equations (20)-(28). The higher value of critical fluxtube density around phase transition region in near infraredsector is shown to drive the system through QGP phase also.Further, the color electric flux quantization and the energyminimization conditions have been shown to lead to a uni-formally distributed color electric field given by equation(37) with its critical value as given by equation (38) in thedeconfinment region which immediately drops to more thanninety percent in the far-infrared sector of QCD. Using theasymptotic solutions for the dual potential, the analyticexpression of the color-electric field has been derived asequation (40) and the profiles of color electric field at differ-ent hadronic scales and different values of coupling constantare depicted as 2-d and 3-d graphics of Figures 2 and 3,respectively. It clearly demonstrates the localization of thecolor electric field around poles at large distance scales whichgoes homogeneous below the corresponding critical radiusand confined in the central region that leads to the possibilityof a homogeneous QGP formation due to increased flux tubedensity before transiting to completely deconfined phase inthe color screening region. The formation of this new kindof phase is expected as a result of the flux tube annihilationbecause of the increased flux tube density in the centralregion and, therefore, the flux tube density in dual QCD nat-urally plays a key role in creation of QGP in its infrared sec-tor. It is important to note that, in the present scenario, belowthe critical radius, the flux tube density does not completelyvanish in the color screening regime which, in turn, stronglysupports the formation of an intermediate phase before tran-siting to the completely deconfined region. In the near infra-red region physically relevant for the phase transition, thecritical flux-tube density has been shown to be in agreementwith that computed for Pb-Pb central heavy-ion collisionsand demonstrates the feasibility of multi-flux tube formula-tion of dual QCD in the study of phase structure of QCD.Further, the distribution of color electric field at or few sepa-ration below the critical radius still indicates the survival ofcolor flux tubes which points out the stability of QCD vac-uum as well as the strongly interacting [30, 31] behavior ofsuch intermediary phase in QCD.

Furthermore, since the thermal effects are extremelyimportant and play a dominant role in phase transition aswell as QGP formation process, the thermal evolution ofcolor electric field has also been investigated for differenthadronic scales which leads to the weakening of confiningforce with temperature and is expected to trigger the flux tubemelting before transiting to pure asymptotically free phase of

hadronic matter. Further, the thermal response of dual QCDanalysed using the mean field approach shows that theabovementioned confinement mechanism remains intact inthe low temperature regime which is true for all gauge groupsincluding SU(3). However, with the increase in temperature,there is a reduction in the monopole condensate along withthe restoration of magnetic symmetry and the QCD systempasses through a weakly bound phase of QGP before transit-ing to a completely deconfined phase. The reduction in mag-netic condensate for the case of SU(3) is larger due to theincrease in the number of degrees of freedom in comparisonto the SU(2) case. It is then imperative that the critical tem-peratures for the case of the SU(3) group must be consider-ably smaller than those obtained using SU(2) for all QCDphase transitions. Preliminary analysis for the SU(3) casesuggests a reduction in critical temperatures by a factor ofabout 28 percent and is thus expected to be around158MeV for the intermediate range of strong coupling(αs = 0:48) which is very near to that suggested by latticeQCD. Hence, the results with the present dual QCD analysisappear to be quite realistic when extended to the full SU(3)QCD case. The detailed analysis in this direction is in prog-ress. In the thermal environment, as the temperature isincreased and approached to its critical value, the amplitudeof the field in the interior of flux tube gets supressed but stillsupports the survival of flux tubes upto some extent beyondthe critical temperature in QCD vacuum which characterizesthe intermediatory and weakly interacting clusters of coloredparticles. On relatively higher temperature scales, it resem-bles with the evaporation of flux tubes and subsequently thetransition of system into completely deconfined state. Suchthermalization process with quarks and gluons and possibil-ity of QGP formation then becomes quite similar to that ofthe heavy-ion collisions where several interacting flux tubesare expected to overlap in the central region and lead to thesimilar kind of changes in the shape of the flux tube as a resultof their annihilation or unification and pushes the systeminto either the QGP phase or passes through the crossover.In addition, it is also important to extend the present thermaldynamics of QCD to the case of finite baryon density also fora detailed analysis of phase structure of QCD. This is, in fact,the part of a separate study and will be undertaken in ourforthcoming communication. However, it is worth mention-ing that such analysis may be performed using the tools of therelativistic statistical field theory where various thermody-namical quantities may be derived using the partitian func-tion approach. In this connection, various models play animportant role to discuss the QCD phase transition in viewof the fact that the non-perturbative confinement is basicallyaccounted by the bag constant. The bag constant when com-puted for the case of non-vanishing baryon chemical poten-tials shows an overall increase in its value and has importantimplications on QCD phase transition [45, 46]. The tempera-ture and chemical potential dependence of the bag constant, infact, show the thermal and baryon density influences of non-perturbative effects in QCD. Such increase in bag constantthen necessarily leads to the increase in the flux tube densityin the magnetically condensed QCD vacuum. For the case ofsmall chemical potentials (low baryon densities), the low

11Advances in High Energy Physics

density profiles of flux tube is expected to push the QCD vac-uum in type-II superconductor phase. On the other hand, forthe case of the large chemical potentials, it may lead to thefusion of the multi-flux tube structure leading to an intermedi-ate QGP phase transition before transiting to the completelydeconfined phase. A complementary behaviour of QCD sys-tem at high temperature or high baryon chemical potential istherefore expected in a natural way.

Data Availability

We have not used any available data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

One of the authors, D.S. Rawat, is thankful to the UniversityGrants Commission, New Delhi, India, for financial assis-tance in the form of UGC-BSR(RFSMS) fellowship. He is alsothankful to Prof. B.K. Patra for his invitation under TPSC vis-iting fellow programme of IIT Roorkee (India) for usefuldiscussions.

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