Thermo-field dynamics and quark–hadron phase transition in QCD

Nuclear Physics A 829 (2009) 151–169 www.elsevier.com/locate/nuclphysa

Thermo-field dynamics and quark–hadron phase transition in QCD H.C. Chandola ∗ , D. Yadav Department of Physics, Kumaun University, Nainital-263002, India Received 2 October 2008; received in revised form 12 August 2009; accepted 13 August 2009 Available online 19 August 2009

Abstract Based on the topological structure of gauge theory, an effective dual version of QCD has been reviewed and analyzed for the phase structure and color confining properties of QCD by invoking the dynamical magnetic symmetry breaking. The multi-flux-tube configuration of condensed QCD vacuum has been explored and associated glueball masses and inter-quark potential have been derived. Thermal response of QCD vacuum has been analyzed using path-integral formalism alongwith the mean-field approach and associated thermodynamical potential is used to derive thermal form of glueball masses, monopole condensate, inter-quark potential and monopole density which then lead to an estimate of the critical temperature of QCD phase transition. During its thermal evolution, a smooth transition of hadronic system via a weakly bound QGP phase to the fully deconfined phase is established and the thermal evolution profiles of various parameters are shown to indicate a second-order deconfinement phase transition and the restoration of magnetic symmetry. Monopole density calculations have been shown to lead to gradual evaporation of magnetic condensate into thermal monopoles during QCD phase transition. © 2009 Elsevier B.V. All rights reserved. PACS: 11.30.Jw; 12.38.Nh; 12.38.Lg; 12.38.Aw Keywords: Dual QCD; Monopole; Confinement; Finite temperature QCD; QGP

1. Introduction Since the discovery of asymptotic freedom in QCD, it is believed that the complete dynamics of hadron physics may be developed by using QCD as a fundamental gauge field theory [1,2]. * Corresponding author. Tel.: +91 05942 236928; fax: +91 05942 237327.

E-mail address: [email protected] (H.C. Chandola). 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.08.006

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However, since a long time, it has been realized that there should be a qualitative change in the properties of hadron matter as the temperature or density is increased and, therefore, QCD must have a more detailed phase structure under such extreme conditions [3–5]. At low temperatures and low densities, the colored particles like quarks and gluons are confined inside hadrons due to the highly non-perturbative nature of QCD vacuum [6–15]. The QCD vacuum, in fact, is nontrivial at zero-temperature. As the temperature increases above a certain critical point, the color degrees of freedom in hadrons are defrozen and the QCD vacuum is expected to undergo a phase transition to the deconfined phase or a quark–gluon plasma (QGP) phase [16–18] where quarks and gluons appear weakly interacting. Recently [5,16–26], there has been a considerable interest in the properties of hadronic matter under high temperature conditions from several point of view. One of them is the attempt of a number of physicists to create QGP in high energy heavy-ion collisions at BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC). After two heavy ions collide and pass through each other, the huge energy deposition at central region is expected to lead to QGP, which may provide the opportunity to study the nuclear matter under extreme conditions of large density and very high temperatures. Furthermore, in view of the quite unusual properties of QGP in the RHIC temperature domain [16,18,23], it becomes necessary to attend the peculiar issues of strongly coupled QGP in quasi-conformal regime and the associated physics near deconfinement transition in QCD. This is further supplemented by the important question of the existence of an electric–magnetic equilibrium at high temperature [18,24,25] in the magnetically condensed vacuum picture of QCD. On the other hand, in the big-bang scenario of the evolution of the universe, the QCD phase transition is said to have occurred as a real physical event which strongly influenced the nucleosynthesis afterwards. In the big-bang cosmology of early universe, at the time when the temperature dropped to a few hundred MeV and for a few microsecond after the birth of the Universe, matter might have been in a state of quark–gluon plasma [4,5,19]. In addition, since certain intractable aspects of quark–gluon strong interactions have driven us to the computer, the QCD phase transitions have also been investigated using lattice formulations. In the resulting lattice QCD, the Monte Carlo simulation studies have also indicated the QCD phase transition within a range of critical temperatures for both the pure gauge and full QCD cases [26–30]. The situation is more subtle in the realistic QCD with dynamical quarks because of the large sensitivity of the transition to the quark masses [31–33]. There are serious problems as to the continuum limit and the finite-size effects inherent in numerical lattice study. In fact, we do not have yet an effective theory to study the finite temperature QCD based on the systematic treatment of the phase transitions except for some lattice QCD simulations, where the nature of the transition has not yet been determined due to the inaccessibility for all the necessary physical quantities at the moment. In view of these facts, it is therefore very much desired to perform an analytical study to understand some new features of QCD phase transition, such as, the dynamics of the confinement–deconfinement phase transitions under the extreme conditions of temperature and density in the infrared effective flux tube model of dual QCD. The present paper deals mainly with the analysis of the thermal response of QCD vacuum and the study of the QCD phase transition. Section 2 reviews a dual version of the color gauge theory developed in terms of the magnetic symmetry which is introduced as an additional internal isometry. The dynamical breaking of magnetic symmetry then leads to the dual superconducting nature to QCD vacuum which confines all the color isocharges by developing a unique flux tube structure in QCD vacuum. The associated glueball masses and the inter-quark potential have been computed and used as the initial parameters for the thermal QCD version. Section 3 deals with the study of thermal response of QCD vacuum where the thermodynamical potential has been

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evaluated using the path-integral formalism. Using the mean-field treatment of finite temperature dual superconductor, the thermal contributions to the effective potential have been derived and used to evaluate the thermal form of the magnetic condensate, glueball masses, monopole density, inter-quark potential and the confining force in the full infrared sector of QCD. The resulting large reduction in their values is used to estimate the critical temperature of phase transition. The graphical representation of thermal effective potential around the critical temperature in the infrared sector of QCD has been shown to lead to the restoration of magnetic symmetry in the domain of high temperatures. A continuous fallout of the linear confining force and the interquark potential during their thermal evolution has been demonstrated and, as a result of the gradual weakening of the confining force, a smooth transition of hadronic system via a weakly bound QGP phase to the fully deconfined phase is predicted. The evaluation of the monopole density in the condensed QCD vacuum has also been shown to lead to the gradual evaporation of the magnetic condensate into thermal monopoles around the point of QCD phase transition. 2. Dual QCD with magnetic symmetry and its non-perturbative features For the analysis of the non-perturbative and phase structure of QCD, let us start with a brief review of dual QCD formulation [10,34–37] using magnetic symmetry [35,36] in a (4 + n)dimensional metric manifold gAB (A, B = 0, 1, 2, . . . , 3 + n) with the unified space P . With the gauge symmetry G as an n-dimensional isometry and P /G ≡ M as base manifold, P may be identified as a principal fiber bundle P (M, G) over space–time with a canonical projection, Π : P → M. Since a connection on P (M, G) admits a left isometry, the magnetic symmetry may be introduced [35,36] as an additional internal isometry H (with Killing vector fields, ma ; a = 1, . . . , k = dim H ) which is the Cartan’s subgroup of G and commutes with it. The Killing condition, Lma gAB = 0, then leads to a gauge covariant magnetic symmetry condition as given by, Dμ m ˆ = 0,

i.e. (∂μ + gWμ ×)m ˆ = 0,

(1)

where m ˆ is a scalar multiplet which belongs to the adjoint representation of the gauge group G. The magnetic symmetry thus restricts the dynamical degrees of freedom of the system while keeping the full gauge degrees of freedom intact and Eq. (1) then yields an exact solution for the gauge potential which for the simplest choice of G ≡ SU(2) and its little group H ≡ U (1), is given by, Wμ = Aμ m ˆ − g −1 (m ˆ × ∂μ m), ˆ

(2)

where, Aμ ≡ m ˆ · Wμ is the Abelian component unrestricted by the magnetic symmetry. On the other hand, the second part which is determined completely by the magnetic symmetry, is of topological origin since the multiplet m ˆ may be viewed to define the homotopy of the mapping ˆ : SR2 → S 2 = SU(2)/U (1), which identifies the point-like monopoles of the nonΠ2 (S 2 ) as, m Abelian symmetry. The topological structure of the theory may then be brought into dynamics explicitly by performing the separation of gauge fields in the magnetic gauge obtained by rotating m ˆ to a prefixed direction in isospace (say ξ3 ) using a gauge transformation (U ) such that, U

m ˆ −→ξ3 = (0, 0, 1)T which, in turn, leads to the redefinition of the gauge potential (2) and the corresponding gauge fields as, U Wμ −→(Aμ + Bμ )ξˆ3 ,

 U    (d) (d) ˆ m ˆ −→ Fμν + Bμν Gμν = Fμν + Bμν ξ3 ,

(3)

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where, Fμν = Aν,μ − Aμ,ν and Bμν = Bν,μ − Bμ,ν = −g −1 m ˆ · (∂μ m ˆ × ∂ν m). ˆ The second part Bμ , fixed completely by m, ˆ is thus identified as the magnetic potential associated with the topological monopoles and the fields thus appear in a completely dual symmetric way. The Lagrangian obtained by using the above-mentioned gauge fields then yields completely dual symmetric field equations which have an intimate connection with the color electric flux confinement in QCD. Furthermore, in order to avoid the problems due to point-like structure and the singular behavior of the potential associated with monopoles, we use the regular dual mag(d) netic potential (Bμ ) for the topological part and introduce a complex scalar field φ(x) for the monopole. With these considerations, the built-in dual structure of the Lagrangian may be shown to impart it the flux confining features which becomes more clear if we express it in the quenched approximation as, (d)

  2 1 2 = − Bμν +  ∂μ + i4πg −1 Bμ(d) φ  − V (φ). (4) 4 With V (φ) as a proper effective potential, it induces the dynamical breaking of the magnetic symmetry which leads to the monopole condensation in QCD vacuum and imparts it the (dual) superconducting nature in a way identical to that with GL Lagrangian of superconductivity. Since, in the present case, we are interested in the phase transition study of dual QCD vacuum, the use of an effective potential reliable in relatively weak coupling near-infrared regime is naturally desired and, therefore, we choose the following familiar quadratic potential for inducing the magnetic condensation of QCD vacuum,  2 (5) Vpt (φ) = 3λαs−2 φ ∗ φ − φ02 . (m)

£d

With this potential, the Lagrangian (4) then leads to the field equations as given below,       −1 (d)μ μ ∂μ + i 4παs−1 Bμ(d) φ + 6λαs−2 φ ∗ φ − φ02 φ = 0, ∂ − i 4παs B   ↔  Bμν, ν + i 4παs−1 φ ∗ ∂ μ φ − 8παs−1 Bμ(d) φφ ∗ = 0.

(6) (7)

Under cylindrical symmetry (ρ, ϕ, z) and the field ansatz given by, Bϕ(d) = −B(ρ),

(d)

Bρ(d) = Bz(d) = 0 = B0

and φ = χ(ρ) exp(inϕ),

(8)

alongwith a more convenient representation of the dimensionless parameters given by, √ r = 2 3λαs−1 φ0 ρ,

1  F (r) = 4παs−1 2 ρB(ρ),

H (r) = φ0−1 χ(ρ),

(9)

Eqs. (6) and (7) transform to the following form,  1 1 1  (10) H  + H  + 2 (n + F )2 + H H 2 − 1 = 0, r 2 r 1 (11) F  − F  − α(n + F )H 2 = 0, r where, α = 2παs /3λ and the primes stand for the derivative with respect to r. Using the asympr→∞ r→∞ totic boundary conditions given by, F −→ − n, H −→ 1, alongwith Eq. (9), the asymptotic solution for the function F may then be obtained in the following form, √ (12) F (ρ) = −n + C ρ exp(−mB ρ),

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√ 1 1 where, C = 2πB(3 2λg −3 φ 0 ) 2 and mB = (8παs−1 ) 2 φ0 is the mass of the magnetic glueballs which appear as vector mode of the magnetically condensed vacuum. Since, the function F (ρ) d [ρB(ρ)]) through gauge potenis associated with the color electric field (Em (ρ) = −ρ −1 dρ tial B(ρ), it indicates the emergence of dual Meissner effect and the confinement of the color isocharges in magnetically condensed dual QCD vacuum. Further, the vector and scalar glueball mass ratio, as fixed by the effective potential (5) and given by,

mB 2παs = , (13) mφ 3λ √ then leads to, mφ = 2αs−1 3λφ0 , as the scalar glueball mass which represents the threshold for magnetic condensation of QCD vacuum. In addition, the energy per unit length of the associated flux tube configuration, governed by the field equations (10) and (11) and the Lagrangian (4), may also be obtained in the following form, ∞ k

= γ φ02 ;

γ = 2π

6λ (F  )2 (n + F )2 2 (H 2 − 1)2  2 . r dr 2 2 + H + (H ) + 4 g r r2

(14)

0

In view of the relationship of k with Regge slope parameter, k = (2πα  )−1 with α  = 0.90 GeV−2 , and using the numerical computation of equation for γ in the same way as done with one-loop potential [35,38], we obtain the vector and scalar glueball masses for some typical values of strong coupling in full infrared sector of QCD at zero-temperature and are given as, ⎫ αs = 0.22; γ = 7.891, φ0 = 0.149 GeV, mB = 1.51 GeV, mφ = 2.22 GeV, ⎪ ⎬ αs = 0.47; γ = 6.28, φ0 = 0.167 GeV, mB = 1.21 GeV, mφ = 1.22 GeV, ⎪ ⎭ αs = 0.96; γ = 5.40, φ0 = 0.181 GeV, mB = 0.929 GeV, mφ = 0.655 GeV. (15) Furthermore, following the approach of Ball, Caticha, Baker et al. [10,12], the inter-quark potential in dual QCD may also be derived using the dual QCD Lagrangian (4) with the phenomenological potential (5) and the dual gauge field solution given by Eq. (12), and is given as, ∞ V ( ) = 0



 (m)  αs exp(−2mB ) αs m2B + ρ dρ −£d = C 2 − 16 2

≡ Vy ( ) + Vc ( ),

(16)

which has a combination of Yukawa and linear confining terms. The values of the cut-off parameter (C) in the present case, for some typical values of strong coupling in infrared sector of QCD, may then be obtained by using the well established value of string tension of 1 GeV/fm, in the following form, C(αs = 0.22) 0.835,

C(αs = 0.47) 0.751,

C(αs = 0.96) 0.689.

(17)

The linear part of the inter-quark potential is, in turn, responsible for the quark confinement and explains the non-perturbative confining features of QCD in its infrared region due to the onset of

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the confining forces in dual QCD vacuum. The magnitude of such inter-quark confinement force may also be derived using Eq. (16) in the following form,    ∂Vc ( )  1  = αs C 2 m2 . (18) Fc =  B ∂  2 Such linear confining strong force appropriately contributes to the string tension for the resulting flux tube configuration in the magnetically condensed phase of the QCD vacuum. The present dual QCD model, therefore, strictly supports a flux-tube configuration and indicates a linear confinement of the color isocharges. At moderate couplings, such system has been shown [36,37] to favor type-II superconducting phase and consequently, a number of color flux tubes are expected to be produced during the passage of heavy ions in heavy ion collisions which may split thermally resulting in quark-pairs creation. The thermalized system may then switch over to the QGP phase where the non-perturbative properties of QCD vacuum are expected to get largely modified. As such, the estimates presented here may further be utilized (as initial zerotemperature parameters) for analyzing the behavior of QCD at finite temperatures and its phase structure. 3. Thermal aspects of dual QCD and dynamics of QCD phase transition As has been argued in the previous section, the multi-flux tube picture of dual QCD leads to a viable physical explanation for the low-energy confining features of QCD and may be used for exploring the phase structure of QCD under unusual conditions like those of high temperatures and high densities. The behavior of QCD at finite temperature is also expected to be very much useful in understanding the dynamics of QGP-phase of the matter [16–19,21,22]. For this purpose, starting from our dual QCD Lagrangian (4), let us use the partition function approach alongwith the mean-field treatment for the QCD monopole field to evaluate the thermal contributions to the effective potential in dual QCD. The partition functional, for the present dual QCD in thermal equilibrium at a constant temperature T , may be given by an Euclidian path-integral over a slab of infinite spatial extent and β(≡ T −1 ) temporal extent, as,      Z[J ] = D[φ]D Bμ(d) exp −S (d) , (19) where, S (d) is the dual QCD action and is given by,    (m) S (d) = −i d 4 x £d − J |φ|2 ,

(20)

(m)

where £d is given by Eq. (4) and a quadratic source term, which retains the symmetry of the classical potential, is introduced [39]. As the temperature is raised, there are marked fluctuations in the monopole field and the effective potential at finite temperatures then corresponds to the thermodynamical potential which leads to the vital informations about the QCD phase transition. Hence, for investigating the evolution of the system with temperature, we follow the Dolan and Jackiw approach [40] to compute the effective potential at finite temperatures. For this purpose, let us use the mean-field treatment and separate the fluctuation part of the QCD-monopole field ˜ φ(x) from its mean value φ(x) as, ˜ exp (iξ ), φ −→ (φ + φ) so that the integrand in action (20) may then be expressed as,

(21)

H.C. Chandola, D. Yadav / Nuclear Physics A 829 (2009) 151–169 (m)

£d

 2  2 1 2 1 − J |φ|2 = −3λαs−2 φ 2 − φ02 − J φ 2 − Bμν + m2B Bμ(d) 4 2   ˜ 2 − (mφ φ) ˜ 2 + (∂μ φ)   2     + 4παs−1 Bμ(d) 2φ φ˜ + φ˜ 2 − 3λαs−2 φ˜ 4 + 4φ φ˜ 3     ˜ − 12λαs−2 φ 2 − φ02 φ + 2J φ φ,

157

(22)

˜ which vanishes for the value of J given where, the last term on right-hand side is linear in φ, by, J = −6λαs−2 (φ 2 − φ02 ). It leads to the masses of the vector and scalar modes of glueballs at finite temperature in magnetically condensed QCD vacuum in the following form, m2B = 8παs−1 φ 2

and m2φ = 12λαs−2 φ 2 .

(23)

(d)

Integrating Eq. (19) over Bμ and φ upto one-loop level, the partition function may be expressed as, 

    4 −2 2 2 2 2 Z[J ] = exp i d x −3λαs φ − φ0 − J φ   −1 −1   −1 −1/2 Det iΔ (φ, k) × Det iDμν (B, k) ,

(24)

where the vector and scalar field propagators in magnetically condensed QCD vacuum are given by,   kμ kν i i , Δ(φ, k) = − g . (25) − Dμν (B, k) = μν 2 2 2 2 k − mB + i mB k − m2φ + i The functional Legendre transformation [41], then leads to the effective action in the following form,  Seff = −i ln Z[J ] + d 4 x J φ 2   2   −1   + i ln Det iDμν = d 4 x −3λαs−2 φ 2 − φ02 (B, k)   i ln Det iΔ−1 (φ, k) . (26) 2 The effective potential, which corresponds to the thermodynamical potential at finite temperature, may then be obtained as [41], +

Veff (φ) = − 

Seff d 4x  2 2 φ − φ02 + 3



  d 4k ln m2B − k 2 − i 4 i(2π)   2  1 d 4k ln mφ − k 2 − i . + 4 2 i(2π)

= 3λαs−2

(27)

The partition functional Z[J ], for analyzing a finite temperature system, is described in the imaginary time formalism (with x0 = −iτ ) and the four-vector k has its temporal component given as, ωn = 2πn −iβ (β being the upper bound of τ -integral). The k0 -integration, then becomes

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  d4k   d3k the infinite sum over the Matsubara frequency [42], k = − iβ1 n i(2π) , with 3 = i(2π)4 n = 0, ±1, ±2, . . . . It then leads to the effective potential at finite temperature as,

  ∞    3  2πn 2 d 3k −2 2 2 2 2 2 Veff (φ, β) = 3λαs φ − φ0 + ln + k + mB β n=−∞ i(2π)3 β

  ∞  1  2πn 2 d 3k 2 2 (28) + ln + k + m φ , 2β n=−∞ i(2π)3 β where, β = T −1 (T being the temperature). In view of the reliability of one-loop calculation upto β −2 -terms, we retain the terms only upto β −2 in the high temperature expansion of (28) and the effective potential at finite temperature may then be expressed as,  2 (29) Veff (φ, T ) ≡ 3λαs−2 φ 2 − φ02 + 6V T (mB , T ) + V T (mφ , T ), where, T V (mB , T ) = 2 π

∞

T

    dk k 2 ln 1 − exp − k 2 + m2B T ,

0

and T V (mφ , T ) = 2 π

∞

T

    dk k 2 ln 1 − exp − k 2 + m2φ T .

0

When the particle mass is much smaller than the temperature around the deconfinement phase transition, the effective potential may be well approximated by the high T -expansion. Since the particle masses are almost zero for |φ| 0 (which is very much true near the deconfinement phase transition point) as shown by Eq. (23), the high T -expansion for the effective potential may be performed around |φ| = 0. The validity of such expansion, of course, is mostly limited to the near Tc region as it becomes less reliable in the fully confined low temperature sector of QCD. As such, the temperature-dependent part of the effective potential for bosons may then be calculated in the following way, T4 V (mB , T ) = 2 π

∞

T

    , dy y 2 ln 1 − exp − y 2 + a 2

(30)

0

≡ with, powers of a as, y2

k 2 /T 2

and

a2

≡ m2B /T 2 . In its high-T expansion, V T (mB , T ) may be expanded in

   ∂V  V T (mB , T ) = V a 2 (φ) a 2 =0 + 2  a2 + · · · ∂a a 2 =0 ∞   T4 ≈ 2 dy y 2 ln 1 − exp(−y) π 0

T4 + 2π 2

∞ dy y 2  0

  1   a2 + · · · .  y 2 + a 2 (exp( y 2 + a 2 ) − 1) a 2 =0 1

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The shifting and truncation procedure [40] then gives a quadratic part with respect to each field as,   π 2 T 4 T 4 mB 2 V T (mB , T ) = − + . (31) 45 12 T Similarly, the contribution V T (mφ , T ) may also be calculated in an identical way which leads to,   π 2 T 4 T 4 mφ 2 V T (mφ , T ) = − + . (32) 45 12 T Using Eqs. (31), (32) and (23), the effective potential (28) at finite temperature in dual QCD reduces to the following form, Veff (φ, T ) = V0 (φ) + VT (φ, T ),

(33)

where,  2 V0 (φ) = 3λαs−2 φ 2 − φ02 and VT (φ, T ) = −

  7 2 4 4παs + λ T 2φ2. π T + 90 2αs2

The effective potential at finite temperature (33) may then be used for the evaluation of temperature dependent VEV of the monopole field, glueball masses, monopole density and confining potential, which play an important role in identifying the critical temperature and other parameters of QCD-phase transition. 4. Critical parameters of QCD phase transition and phase structure of Hot QCD In order to investigate the phase structure of the QCD vacuum at finite temperature and the associated dynamics, let us study the thermal evolution of the QCD vacuum and the associated phase transition parameters. Such studies form an important part not only of the present day heavy-ion collisions at extremely high energies in most recent colliders [43–46] but also of the evolutionary scenario of the universe where the deconfinement and QGP phase transitions have been expected to be realized as a real physical event [4,5,19,47]. For the analysis of the vital phase transition parameters of QCD, let us, therefore, first evaluate the vacuum expectation value of φ-field at finite temperature which may be obtained by applying the minimization condition on effective potential (33) as, 

   ∂Veff (φ, T ) 4παs + λ −2 2 2 T2 , (34) = 0 = φ 12λαs φ − φ0 + (T ) ∂φ αs2 φ= φ 0

which gives, ) φ (T 0 = 0,  (T )

φ 0

=

T  Tc ,   4παs + λ T 2 2 , φ0 − λ 12

(35) T < Tc .

(36)

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(T )

Fig. 1. The vacuum expectation value φ 0 strong couplings (αs ).

of monopole condensate as a function of temperature T for the different

The minimization condition (34) thus reveals the disappearance of the QCD monopole condensate at sufficiently high temperatures/energies which, in turn, indicates the restoration of the magnetic symmetry and the deconfinement of the quarks in such high temperature region. Such asymptotically free nature thus appears as a result of the reduction in self interactions among QCD monopoles in high temperature region. The situation prevails upto a critical value of the temperature (Tc ) below which the QCD monopole condensate starts picking up a non-zero-VEV which signals the onset of the confinement phase transition. In addition, a large reduction in monopole condensate is also indicated by Eq. (36) when the temperature falls below a critical (T ) value (Tc ). The variation of φ 0 (for λ = 1) with temperature for three different coupling val(T ) ues in infrared sector is shown in Fig. 1. At vanishing temperature, φ 0 reduces to its VEV value φ0 which shows a slight increase with the increase of the strong coupling. However, at (T ) sufficiently high temperatures, the φ 0 falls off more rapidly for higher couplings then the (T ) same at relatively lower couplings. In any case, the φ 0 value in high temperature region, ultimately vanishes at some typical characteristic value of temperature, called the critical temperature (Tc ) of phase transition, which is obtained as 0.268 GeV, 0.220 GeV and 0.173 GeV for (T ) the strong coupling values of 0.22, 0.47 and 0.96 respectively. The disappearance of φ 0 at Tc may then be identified with the melting of the condensate which ultimately leads to its evaporation into thermal monopoles. This is in agreement with the results of Liao and Shuryak [48] and is consistent with the idea of normal monopole liquid beyond Tc which also expels any electric flux [18] and leads to the persistence of the flux tubes in plasma phase as demonstrated in [49]. With increasing temperature, one thus has a picture of an evolving magnetic medium where the monopole ensemble changes from a dense monopole condensate to an equally dense normal component of thermally excited monopoles around Tc after which they start yielding to the regular electric quasi-particles. Similarly, the variation of the finite temperature effective potential given by Eq. (33) as a function of monopole condensate, around the critical temperature values for the case of above-mentioned three couplings, is depicted in Fig. 2 which demonstrates that

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Fig. 2. The behavior of the finite temperature effective potential Veff (φ, T ) as a function of monopole condensate φ around critical temperature are plotted for the couplings, (a) αs = 0.22, (b) αs = 0.47, (c) αs = 0.96.

at higher temperatures (T > Tc ), the magnetic symmetry tends to get restored and the system enters into the deconfinement phase. However, below Tc , the magnetic symmetry is dynamically broken thereby pushing the system to the confined phase where the local minima of effective potential corresponds to the physical stable vacuum state. Furthermore, for the temperatures below critical temperatures, Eq. (36) leads to the temperature dependent expression of scalar and vector glueball masses in the following form,

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Fig. 3. The variation of glueball mass mTB with the temperature T for the different strong couplings.

 1  √ 4παs + λ T 2 2 −1 2 = 2 3λαs φ0 − , λ 12  1   4παs + λ T 2 2 (T ) −1 2 mB = 8παs φ0 − . λ 12

(T ) mφ

(37) (38)

The thermal vector glueball masses alone as well as with the thermal monopole condensates are depicted in Fig. 3 and Fig. 4 respectively for different couplings in infrared sector of QCD. These plots demonstrate a continuous decrease in glueball masses with temperature which is, of course, much faster in deep infrared region of QCD where the couplings are sufficiently large. Similar continuous reduction in monopole condensate is also demonstrated by Fig. 4. With these variations, the vanishing of the glueball masses and monopole condensate leads to the critical temperature which corresponds to the confinement–deconfinement QCD phase transition and takes the values identical to those given earlier (Fig. 1). Such continuous reduction in glueball masses and monopole condensate and their resulting disappearance near critical temperatures, is indicative of a second-order phase transition in dual QCD. The expression for the critical temperature of QCD confinement–deconfinement phase transition may be obtained by vanishing the coefficient of the terms quadratic in φ in the effective potential given by Eq. (33), which for the case of λ = 1 leads to,  3 . (39) Tc = 2φ0 4παs + 1 Substituting this value in Eq. (36), clearly demonstrates that the QCD monopole condensate vanishes completely at T = Tc which ultimately forces the glueballs to disappear from the QCD vacuum. For different values 0.22, 0.47 and 0.96 of strong couplings in infrared sector, it leads to an estimate for the specific values of the critical temperatures as 0.268 GeV, 0.221 GeV and 0.174 GeV respectively which are in fairly good agreement with those obtained by the graphical plots (Fig. 3 and Fig. 4) for glueball masses and QCD monopole condensate. Furthermore, the

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(T )

163

Fig. 4. Plots of the VEV φ 0 of condensed monopole and the glueball mass mTB as a function of temperature T for the couplings (a) αs = 0.22, (b) αs = 0.47, (c) αs = 0.96, for the identification of critical temperature of phase transition.

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Fig. 5. The 3-d profile of inter-quark potential V ( , T ) as a function of temperature T and inter-quark separation for the couplings (a) αs = 0.22, (b) αs = 0.47, (c) αs = 0.96.

inter-quark potential governing the dynamics of quarks in QCD, may also be computed in the finite temperature case using Eq. (16), where the temperature dependence is reflected through the modified vector glueball masses given by Eq. (38), and has the following form,

(T ) αs exp(−2mB ) 1  T 2 (40) V ( , T ) = C 2 − + αs mB . 16 2 This expression leads to the dependence of inter-quark potential on temperature and the interquark distance and a 3-d profile of such potential has been created using Mathematica as given in terms of Fig. 5. The upper sheet in Fig. 5 corresponds to the linear confining part of the potential which shows a small curvature in the high temperature region because of phase transition around the critical temperature. The increasing curvature of the upper surface (V –T plane) with

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Fig. 6. The linear confining force Fc (T ) as the function of temperature T for the different strong couplings (αs ).

increased couplings in infrared sector shows a decreased critical temperature of phase transition in agreement with the values given by expression (39). Furthermore, the linear confining force, derived and computed using the second part of the inter-quark potential (40), as plotted in terms of Fig. 6, also reflects the temperature dependency of the confining force and indicates its disappearance near the critical temperatures where it pushes the QCD vacuum into a completely deconfined phase. In addition, the expression (40) may also be used to evaluate the relative value of the confining force at non-zero-temperatures using the vector glueball masses (given by Eq. (38)) and is obtained as,  2 (T ) (mB )2 FcT T . (41) = = 1− 0 0 2 Tc Fc (mB ) The model, therefore, leads to a continuous fallout of the confining force and the string tension and predicts a second-order phase transition. However, to account for the first-order QCD phase transition with a discontinuity at the critical temperature, the above expression may be modified to,  2 T FcT = a 1 − b , (42) 0 Tc Fc where the coefficients a and b are allowed to deviate from unity and their best fit values to the data [50] may be used as, a = 12.1 for b = 0.99, leading to the ratio, FcT =Tc /Fc0 = 0.121. It then leads to the value of physical mass gap at the transition point as given by, =Tc mTphys

Tc

=

FcT =Tc , Tc2

(43)

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M Fig. 7. The relative monopole density nM T /n0 as a function of temperature T .

which takes its values as 0.34, 0.5 and 0.81 for αs = 0.22, 0.47 and 0.96 respectively. This is quite =Tc compatible with the earlier results of the dedicated analysis leading to, mTphys /Tc = 0.4–0.8 as summarized in [51]. Furthermore, in view of the solution for the function given by Eq. (12) associated with the color electric field, the mass of the vector mode of the magnetically condensed QCD vacuum (mB ) may be used as the screening mass in dual QCD which leads to the condensed monopole density (at T = 0), as conjectured by Polyakov [52], and is given by, 3 nM 0 = CM mB ,

(44)

CM being a constant. Using the expression (38) for mB at finite temperature, it leads to the condensed monopole density at finite temperature in the following form,  2 3 2  (T ) 3 T M M = n0 1 − , (45) nT = CM mB Tc and, hence, the ratio,  2 3 2 nM T T = 1− . (46) M Tc n0 In view of the arguments related to the similar ratio for the confining force given by Eq. (41), the expression (46) may further be modified to,  2 3 2 nM T T = a 1 − b , (47) Tc nM 0

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with the identical values for the coefficients a and b. The relative (condensed) monopole density may then be graphically expressed for its variation with the temperature as given by Fig. 7 which demonstrates that, at the transition point, the condensed monopole density drops to about 1.2% and so the 98.8% monopole condensate turns into the thermal monopoles. This pattern is in agreement with the picture of the electric dominated phase of the Yang–Mills theories at the high temperatures in which the magnetic component appears to be strongly interacting [18,53]. 5. Summary and discussion In view of the fact that the topological features of non-Abelian gauge theories might play an important role in understanding the phase structure of QCD, a dual version of QCD, using its magnetic symmetry structure, has been reviewed and analyzed for its thermal extension. The dual dynamics between color isocharges and topological charges and the dynamical breaking of magnetic symmetry leads to a unique color confining flux-tube configuration in magnetically condensed state of QCD vacuum, the energy of which leads to an estimate for the glueball masses and the inter-quark potential with a definite linear confining contribution in infrared sector of QCD. The finite temperature effects of QCD vacuum have been studied by deriving the thermal form of effective potential using the mean-field treatment of finite temperature dual superconductor. The quadratic source terms introduced in the thermal version of the Lagrangian (22), in fact, maintain the reality of the scalar mass mode of the condensed QCD vacuum even in the negative curvature region of the classical potential and lead to the effective action (26) for the whole range of order parameter. High temperature expansion for the effective potential is performed around the boundary of confinement–deconfinement phase transition where the scalar and vector particle masses in the magnetically condensed QCD vacuum become vanishingly small and which leads to the expression (33) for the thermodynamical potential in dual QCD convenient for evaluating the critical temperature of phase transition, T-dependent masses and other QCD phase transition parameters. At high temperatures, a large reduction in QCD monopole condensate and its more rapid decay in far infrared region in comparison to that in near infrared region indicates the temperature dependent tendency of the self-coupling parameter λ which imply a sizable reduction in monopole interactions at high temperatures as indicated by many authors [19,53]. In addition, the variation of thermodynamical potential with monopole condensate around Tc values in full infrared sector of QCD, also leads to the restoration of magnetic symmetry (with a global minimum) in the domain of high temperatures where a deconfinement phase transition is predicted in a natural way. Furthermore, the thermal evolution of glueball masses and magnetic condensate shows a continuous decay pattern and a rapid evaporation of glueballs in deep infrared sector of QCD where the magnetically dominated dual superconducting QCD vacuum (in T < Tc region) then smoothly transforms to an electrically dominated QCD vacuum in postconfinement (T > Tc ) region. However, the flux-tube or string related physics is by no-means fully terminated at the critical point and, therefore, QGP near Tc is expected to contain not only the electrically charged particles but the magnetically charged one as well which is in agreement with the recent lattice studies [18,53–57]. The magnetic condensate thus appears to turn into the thermal monopoles as evident from the monopole density calculations also and which agrees with the results of various other authors [29,50,53] also. The thermal evolution of the inter-quark potential and the linear confining force also leads to similar results and indicates a second-order deconfinement phase transition in pure gauge SU(2) QCD which is in agreement with the qualitative observations of Monte Carlo [58] and others [59–62]. Below the critical temperature, magnetic symmetry is broken dynamically and QCD vacuum enters into a strongly coupled non-perturbative (confining)

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