The flux-tube phase transition and bound states at high temperatures
aa r X i v : . [ h e p - ph ] J a n The flux-tube phase transition and bound states at high temperatures
G.A. KozlovBogolyubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear Research,Joliot-Curie st. 6, Dubna, 141980 Moscow Region, Russia
We consider the phase transition in the dual Yang-Mills theory at finite temperature T . The phasetransition is associated with a change (breaking) of symmetry. The effective mass of the dual gaugefield is derived as a function of T -dependent gauge coupling constant. We investigate the analyticalcriterion constraining the existence of a quark-antiquark bound state at temperatures higher thanthe temperature of deconfinement. I. INTRODUCTION
Most expositions of dual model focus on its possible use as a framework for quark confinement in nature[1]. Rather than the confinement of color charges, we will here describe what one might call the phasetransition in the dual Yang-Mills (Y-M) theory at finite temperature T . It is believed that the energyrequired grows linearly with the distance between the color charge and anticharge due to the formationof a color electric flux tube. The idea is that a charge or anticharge is a source or sink, respectively, ofcolor electric flux, which is the analog of ordinary electric flux for the strong interactions. But unlikeordinary electric flux, the color electric flux is expelled from the vacuum and is trapped in a thin fluxtube connecting the color charge and anticharge. This is very similar to the way that a superconductorexpels magnetic flux and traps it in thin tubes called Abrikosov-Gorkov vortex lines.There is a general statement that the color confinement is supported by the idea that the vacuum ofquantum Y-M theory is realized by a condensate of monopole-antimonopole pairs [2]. In such a vacuumthe interacting field between two colored sources located in ~x and ~x is squeezed into a tube whoseenergy E tube ∼ | ~x − ~x | . This is a complete dual analogy to the magnetic monopole confinement in theType II superconductor. Since there is no monopoles as classical solutions with finite energy in a pureY-M theory, it has been suggested by ’t Hooft [3] to go into the Abelian projection where the gaugegroup SU(2) is broken by a suitable gauge condition to its (may be maximal) Abelian subgroup U(1). Itis proposed that the interplay between a quark and antiquark is analogous to the interaction between amonopole and antimonopole in a superconductor.It is known that the topology of Y-M SU ( N ) manifold and that of its Abelian subgroup [ U (1)] N − aredifferent, and since any such gauge is singular, one might introduce the string by performing the singulargauge transformation with an Abelian gauge field A µ [4] A µ ( x ) → A µ ( x ) + g π ∂ µ Ω( x ) , (1)where Ω( x ) is the angle subtended by the closed space-like curve described by the string at any point x = ( x , x ), and g = 2 π/e is responsible for the magnetic flux inside the string, e being the Y-Mcoupling constant. Here, a single string in the two-dimensional world sheet y x ( τ, σ ), is shown, for example.Obviously, the Abelian field-strength tensor F Aµν = ∂ µ A ν − ∂ ν A µ transforms as F Aµν ( x ) → F Aµν ( x ) + ˜ G µν ( x ) , where a new term (the Dirac strength string tensor)˜ G µν ( x ) = g π [ ∂ µ , ∂ ν ] Ω( x ) , is valid on the world sheet only [5]˜ G µν ( x ) = g ǫ µναβ Z Z dσ dτ ∂ ( y α , y β ) ∂ ( σ, τ ) δ [ x − y ( σ, τ )] . Actually, a gauge group element, which transforms a generic SU ( N ) connection onto the gauge fixingsurface in the space of connections, is not regular everywhere in space-time. The projected (or trans-formed) connections contain topological singularities (or defects). Such a singular transformation (1)may form the worldline(s) of magnetic monopoles. Hence, this singularity leads to the monopole current J monµ . This is a natural way of the transformation from the Y-M theory to a model dealing with Abelianfields. A dual string is nothing other but a formal idealization of a magnetic flux tube in the equilibriumagainst the pressure of surrounding superfluid (the scalar Higgs-like field) which it displaces [6,7].The lattice results, e.g., [8] give the promised picture that the monopole degrees of freedom can indeedform a condensate responsible for the confinement. By lattice simulations in quantum chromodynamics(QCD), it is observed that large monopole clustering covers the entire physical vacuum in the confinementphase, which is identified as a signal of monopole condensation being responsible for confinement. Theexpression for the static heavy quark potential, using an effective dual Ginzburg-Landau model [9], hasbeen presented in [10]. In the paper [11], an analytic approximation to the dual field propagator withoutsources and in the presence of quark sources, and an expression for the static quark-antiquark potentialwere established.The aim of this paper is to consider the phase transition in the four-dimensional model based on thedual description of a long-distance Y-M theory which shows some kind of confinement. We study themodel of Lagrangian where the fundamental variables are an octet of dual potentials coupled minimallyto three octets of monopole (Higgs-like) fields [12].In the scheme presented in this work, the flux distribution in the tubes formed between two heavycolor charges is understood via the following statement: the Abelian Higgs-like monopoles are excludedfrom the string region while the Abelian electric flux is squeezed into the string region.It is strongly believed to the possibility for a quark-antiquark pair to form a bound state at temperatureshigher that the critical one, T c , i.e., in the deconfinement state (see, e.g., [13] and the references therein).One of the aim of this article is to find the analytical criterium constraining the existence of bound statesat T > T c .In the model there are the dual gauge field ˆ C aµ ( x ) and the scalar field ˆ B ai ( x ) ( i = 1 , ..., N c ( N c − / a =1,...,8 is a color index) which are relevant modes for infrared behaviour. The local coupling of theˆ B i -field to the ˆ C µ -field provides the mass of the dual field and, hence, a dual Meissner effect. Althoughˆ C µ ( x ) is invariant under the local transformation of U (1) N c − ⊂ SU ( N c ), ˆ C µ = ~C µ · ~H is an SU ( N c )-gauge dependent object and does not appear in the real world alone ( N c is the number of colors and ~H stands for the Cartan superalgebra). The scope of commutation relations, two-point Wightman functionsand Green’s functions as well-defined distributions in the space S ( ℜ d ) of complex Schwartz test functionson ℜ d , the monopole- and dual gauge-field propagations, the asymptotic transverse behaviour of boththe dual gauge field and the color-electric field, the analytic expression for the static potential can befound in [12]. II. THE STRING-LIKE FLUX TUBE PHASE TRANSITION
Phase transitions in dual models are associated with a change in symmetry or more correctly thesetransitions are related with the breaking of symmetry. As a starting point we assume, for simplicity, thatthe model is characterized by the scalar order-parameter h ˆ B i ( x ) i = ˆ B for the scalar field ˆ B i ( x ) identifiedin the dual model as the Higgs-like field. The classical partition function looks like Z cl = Z D ˆ B i exp ( − Z β dτ Z d ~x L ( τ, ~x ) ) . (2)In (2) the sum is taken over fields periodic in Euclidean time τ with period β in thermal (heatbath)equilibrium in space-time at temperature T = β − . The dual description of the Y-M theory is simplyunderstood by switching on the dual gauge field ˆ C µ ( x ) (non-Abelian magnetic gauge potentials) and thethree scalar fields ˆ B i ( x ) (necessary to give the mass to the gauge field C aµ and carrying color magneticcharge) in the Lagrangian density (LD) L [14] L = T r (cid:20) −
14 ˆ F µν ˆ F µν + 12 (cid:16) D µ ˆ B i (cid:17) (cid:21) − W (cid:16) ˆ B i (cid:17) , (3)where ˆ F µν = ∂ µ ˆ C ν − ∂ ν ˆ C µ − ig [ ˆ C µ , ˆ C ν ] ,D µ ˆ B i = ∂ µ ˆ B i − ig [ ˆ C µ , ˆ B i ] . The Higgs-like fields develop their vacuum expectation values (v.e.v.) ˆ B i and the Higgs potential W ( ˆ B i )has a minimum at ˆ B i of the order O (100 MeV) defined by the string tension. In the confinement phasethe magnetic gauge symmetry is broken due to dual Higgs-like mechanism. All the particles becomemassive. The v.e.v. ˆ B i produce a color monopole generating current confining the electric color flux[12]: J monµ ( x ) = 23 ∂ ν G µν ( x ) , where G µν = ∂ µ C ν − ∂ ν C µ + ˜ G µν , (4)ˆ C µ = λ C µ ( λ a is the generator of SU(3)), ˜ G µν is the Dirac string tensor. The interaction of the dualgauge field with other fields (scalar nonobservable fields) is due to monopole current J monµ ( x ) in theHiggs-like condensate ( χ + B ) in terms of the dual gauge coupling g up to divergence of the local phaseof the Higgs-like field, ∂ µ f ( x ): g C µ ( x ) = J monµ ( x )4 g ( χ + B ) + ∂ µ f ( x ) . As a result, we obtained [12] that the dual gauge field is defined by the divergence of ˜ G µν shifted bythe divergence of the scalar Higgs-like field. For large enough ~x , the monopole field is going to its v.e.v.,while C µ ( ~x → ∞ ) → J monµ ( ~x → ∞ ) → m C µ with m being the mass of C µ field. The Higgs-likefields are associated with not individual particles but the subsidiary objects in the massive gauge theory.These fields cannot be experimentally observed as individual particles.It is believed that LD (3) can generate classical equations of motion carrying a unit of the z flux confinedin a narrow tube along the z -axis (corresponding to quark sources at z = ±∞ ). This is a dual analogyto the Abrikosov [15] magnetic vortex solution.The question is what happens with the flux tube in excited matter at nonzero temperature T . At T = 0the oscillations of the flux tube become visible up to the energy of excitation e β = s/β at certain entropydensity s . In Eq. (2) the sum is taken over fields ˆ B i periodic in Euclidean time τ with the period β , andthe thermal equilibrium in a flat space-time at temperature T is considered. The scalar Higgs-like fieldis only part of the dual picture, however it is the only field that is visible in the path integral (2). TheFourier expanding of ˆ B ( τ, ~x ) in the formˆ B ( τ, ~x ) = ˆ B ( ~x ) + X n =1 ˆ B n ( ~x ) exp(2 πinτ /β ) (5)can allow one to reflect the periodicity of ˆ B ( τ, ~x ) in imaginary time. Note, that the first term in (5) givesthe zero-temperature mode while the other ones count the ”heavy” high-temperature modes.We now move to a simple physical pattern: let us define the ”large” and ”small” systems. It is knownthat in classical mechanics, the stochastic processes in a dynamic ”small” system are under the weakaction of a ”large” system. ”Small” and ”large” systems are understood to mean that the number of thestates of freedom of the former is less than that of the later. The ”large” system is supposed to be inthe equilibrium state (thermostat with the temperature T ). We do not exclude the interplay betweentwo systems. The role of the ”small” system is played by the restricted region of confined charges, theflux tube. The stationary stochastic processes in the deconfined state are distorted by the random source˜ G µν ( x ) in the dual field tensor G µν ( x ) (4), and under the weak action of a ”large” system described bythe scalar field φ in the Lagrangian density term | ( ∂ µ − i g C µ ) φ | .As a result, in the dual Higgs model [12] the finite energy of the peace of the isolating string-like fluxtube of the length R keeps growing as RE ( R ) ≃ ~Q α π m R (12 . − µ R ) , (6)where ~Q α = e ~ρ α is the Abelian color electric charge, while ~ρ α is the weight vector of the SU(3) algebra;˜ µ is the infrared mass parameter.It is more and more attractive view of existence of (colorless) hadronic excitations even in the high-temperature phase contrary to the standard pattern of it. Certainly, it is commonly believed that thehigh - T phase is composed of free or weakly interacting quarks and gluons (deconfinement phase). It wasalready shown that in the deconfinement phase, the color-Coulomb string tension does not vanish evenfor temperatures which exceed the critical one (see the review by D. Zwanziger in [13] and the referencestherein).Let us introduce the canonical partition function Z c = X flux tube configurations X β exp[ − β E ( R )] D ( | ~x | , β ; M ) = X R X β N ( R ) exp[ − β E ( R )] D ( | ~x | , β ; M ) (7)for ensembles of systems with a single static flux tube, where N ( R ) is the number of configurationsof the flux tube of length R . Here, D ( | ~x | , β ; M ) defines the screening mass M ( β ) from the large dis-tance exponential fall-off of correlators of gauge-invariant time-reflection odd operators O [16] at highertemperatures h O ( τ, ~x ) O ( τ, i ∼ const | ~x | a D ( | ~x | , β ; M ) as | ~x | → ∞ , (8)where D ( | ~x | , β ; M ) = exp [ − M ( β ) | ~x | θ ( T − T c )] (9)and a is a constant depending on the choice for the operator O ( τ, ~x ). ¿From the physical point of view, | ~x | can be replaced by the characteristic scale L of the thermostat (the ”large” system) and θ in (9) is thestandard step function. Actually, D = 1 as T < T c . The examples of the choice for the operator O canbe found in [16, 17] where the main strategy is a non-perturbative determination of the screening massat high temperature limit, T > T c . In the deconfined state it is evident an existence of non-perturbativeeffects and even hadronic modes (as quasiparticles) having strong couplings. In this case the sum (7) doesnot divergent if T > T c . The number of configurations N ( R ) can be considered in the discrete space of ascalar Higgs-like condensate and one requires the flux tubes to lie along the links of a 3-dimensional cubiclattice of volume V with the lattice size l ∼ µ − = ( √ λ ˆ B ) − , where µ is the mass of the scalar Higgs-like field and λ is its coupling constant. In the rest of physics, for l << R the number of configurations N ( R ) is interpreted in terms of the entropy density s of the flux tube by a fundamental formula˜ N ( R ) = e ˜ s , (10)where ˜ N ( R ) = N ( R ) c l /V , ˜ s = s R/l , c is the positive constant of the order O (1) (see also [18]). Therelation (10) counts the flux tubes that do not intersect the volume boundary (bound state). They called”short” flux tubes. If the entropy density s , which was inferred from classical reasoning, is like everyother entropy density that we have met, then a flux tube has a very large number of configurations,roughly N ( R ) ∼ exp[ s ( R/l )]. Can one by some sort of calculation count the number of configurations offlux tube and reproduce the formula (10) for the entropy density? For this, we need a quantum theoryof confinement, so, at present at least, dual Y-M theory is the only candidate. Even in this theory, thequestion was out of reach for last three decades.Inserting (10) into Eq. (7) one gets Z c = Vl X R exp[ − β σ eff ( β ) R ] , (11)where σ eff ( β ) = ˜ σ eff ( β ) + σ D ( β ) . (12)In Eq. (12) ˜ σ eff ( β ) = ˜ σ − sl β (13)is the order parameter of the phase transition and˜ σ = σ (cid:18) −
14 ln ˜ µ m R (cid:19) , σ = 34 α ( Q ) m = 34 πg m (14)with α ( Q ) being the running coupling constant. Here, R in the logarithmic function in (6) has beenreplaced by the characteristic length R c ∼ /m R which determines the transverse dimension of the dualfield concentration, while ˜ µ is associated with the inverse ”coherent length” and the dual field mass m defines the ”penetration depth” in the Type II superconductor where m < µ . The second term in Eq.(12) σ D ( β ) = b th M ( β ) T, as T > T c . (15)is the important component of deconfined matter above the phase transition that gives the contributionto the physical properties of the strongly interacting matter. In formula (15), b th = L/R is the thermostatcriterion factor; M ( β ) in SU ( N ) for N f quark flavors is defined perturbatively [17,16] with the leadingorder screening mass M LO ( β ) as M LO ( β ) + N α T ln M LO ( β )4 πα T , M LO ( β ) = s πα (cid:18) N N f (cid:19) T, and within the non-perturbative regime as 4 πα c N T + higher order corrections , c N =3 = 2 . ± .
15 [16].Therefore, the result σ D ( β ) ∼ α T can be shown explicitly at T > T c with the non-perturbative regime.Formula (15) gives the evidence of magnetic component of the deconfinement phase state which is relatedto thermal abelian monopoles evaporating from the magnetic condensate which is present at low T .The spatial Wilson loop L W ilson has area law behavior below and above T c . For large enough L W ilson the spatial string tensor is determined by the effective action of the dual (magnetic) theory at
T > T c S eff ( L W ilson , T ) → L W ilson σ D ( β ) , as L W ilson → ∞ . Hence, the thermostat characteristic scale L is given by L = lim L Wilson →∞ (cid:18) S eff ( L W ilson , T ) L W ilson M ( β ) T (cid:19) R, M ( β ) ∼ O ( αT ) . At zero temperature we got σ ≃ . GeV [12] for the mass of the dual C µ -field m = 0 . GeV and α = e / (4 π )=0.37 obtained from fitting the heavy quark-antiquark pair spectrum [19]. The value σ above mentioned is close to a phenomenological one (e.g., coming from the Regge slope of the hadrons).Making the formal comparison of the result obtained in the analytic form, we recall the expression of theenergy per unit length of the vortex in the Type II superconductor [20,10] ǫ = φ m A π ln (cid:18) m φ m A (cid:19) , (16)where φ is the magnetic flux of the vortex, m A and m φ are penetration depth mass and the inversecoherent length, respectively. On the other hand, the string tension in Nambu’s paper (see the first ref.in [2]) is given by ǫ = g m m v π ln (cid:18) m s m v (cid:19) , (17)with m s and m v being the masses of scalar and vector fields and g m is a magnetic-type charge. It is clearthat for a sufficiently long string R >> m − the ∼ R -behaviour of the static potential is dominant; for ashort string R << m − the singular interaction provided by the second term in (6) becomes importantif the average size of the monopole is even smaller.The model presented here is characterized by a limiting temperature T c , and it is evident that T c = 34 1 s α ( Q ) m µ (cid:18) −
14 ln ˜ µ m R (cid:19) (18)for which ˜ σ eff ( T c ) = 0. The vacuum expectation value B is the threshold energy to excite the monopole(Higgs-like field) in the vacuum. It corresponds to the Bogolyubov particle in the ordinary superconduc-tor. In case if such excitations exist, the phase transition is expected to occur at T c ∼
200 MeV. Thevalue B ≃
276 MeV is regarded as the ultraviolet cutoff of the theory. At sufficiently high temperatureQCD definitely loses confinement and the flux tube definitely disappears. It is evident that at T → T c the flux tube becomes arbitrary long. As a result, the temperature-dependent mass m ( β ) of the dualgauge field ˆ C µ is derived as follows m ( β ) = 43 σ eff ( β ) α ( Q, β ) . (19)Obviously, m ( β ) → m as β → ∞ , and m ( β ) → /β → T c . The latter limit means that ∂ ν ˜ G µν ∼ ( m C µ + 4 m ∂ µ ¯ b ) → T → T c (here, ¯ b is the Higgs-like field). On the other hand, the divergence of ˜ G µν is just the currentcarried by a charge g moving along the path Γ: ∂ ν ˜ G µν ( x ) = − g Z Γ dz µ δ ( x − z ) . (21)Hence, ∂ ν ˜ G µν ( x ) → g →
0. Actually, formula (19) relates the confinement to the spontaneous break-ing of a magnetic symmetry induced by monopole condensation. The magnetic condensate disappearsat the deconfining phase transition. And the final remark concerning the zeroth value of m ( β ): recallthat m ( β ) ∼ g ( β ) δ (0), where δ (0) is the inverse cross section of the flux tube. This cross section isinfinitely large if m →
0. Actually, σ eff ( β ) is the effective measure of the phase transition when the fluxtube is produced. The fact that ˜ σ eff ( β c = T − c ) = 0 means the special phase where two color charges areseparated from each other by infinite distance. At T = T c the entropy and the total energy are related toeach other by s = E/T c . The level density of a system is e s , therefore s = E/T c implies an exponentiallyrising mass spectrum if one identifies E with the mass of a quark-antiquark bound state.It is assumed that the flux tube starts in a thermal exciting phase, a phase in which the flux tube isquasi-static and in thermal equilibrium at temperatures close to T c or even higher than T c . We assumethat the string coupling is sufficiently small and the local space-time geometry is close to the flat over thelength scale of the finite size box-block of volume v = r . In each block j the flux tube is homogeneousand isotropic with the energy E j . It was shown [21] that applying the string thermodynamics to a volume v = r in the string gas cosmology one can get the mean square mass fluctuation in a region of radius r evaluated at the temperature close to T c :. h ( δµ ) i = r R β − β c , β > β c . (22)At high temperatures T > T c , the mass m disappears and the main object is the screening mass M ( β ). Actually, in deconfined state the scale R of the real hadron at low temperature is replaced bythe thermostat (heat bath) scale L . The spectrum of physical ”quark-antiquark” bound states at T in SU (3) can be expanded as E ( T ) ∼ αL T "s πα (cid:18) N f (cid:19) + 3 ln s πα (cid:18) N f (cid:19) + 4 π c + ... , (23)where one sees the saving of the α = α ( Q, T )-dependence in both perturbative (two first terms in (23))and non-perturbative regimes. III. COUPLING CONSTANT
In gauge theories at T = 0 thermal fluctuations of the gluon act to screen the electric field componentof the gluon, through the development of temperature-dependent electric mass m el ∼ g T . Recent studiesshow remarkable facts that instantons are related to monopoles in the Abelian gauge although thesetopological objects belong to different homotopy group. It is known that both analytical and latticestudies can show a strong correlation between instantons and monopoles in the Abelian projected theoryof QCD. It can be postulated that at finite T the running coupling would be replaced by the staticscreened charge1 α ( Q, T ) = 1 α ( Q ) ((cid:20) − Π ( q = 0 , ~q → β ) ~Q (cid:21) + α ( Q )6 π (cid:18) N − N f (cid:19) ln ~Q M ) (24)for gauge group SU ( N ), where M is the renormalization energy scale and the inverse screening length isgiven by the gluon self-energy Π µν ( q ) at the lowest order of g in hot theory containing the quark fieldswith the mass m q (see, e.g., [22]) − Π ( q = 0 , ~q → β ) = g T (cid:20) N N F π T I F ( β, ¯ µ, m q ) (cid:21) = m el ( β ) , (25)where I F = Z ∞ dx x q x + m q [ n F ( x ) + ¯ n F ( x )] , (26) n F ( x ) = 1exp[( q x + m q − ¯ µ ) β ] + 1 , ¯ n F ( x ) = 1exp[( q x + m q + ¯ µ ) β ] + 1 , (27) N F is the number of quarks, the chemical potential ¯ µ is defined by the baryon density ρ in the formula ρ ∼ Z d x (2 π ) [ n F ( x ) − ¯ n F ( x )] . (28)The first term in (25) refers to pure SU ( N ) gauge theory. Hence, α − ( Q, T ) has the following expansionover ~Q /M and T / ~Q :1 α ( Q, T ) = 1 α ( Q ) + 16 π (cid:18) N − N f (cid:19) ln ~Q M + 4 π T ~Q (cid:20) N N F π T I F ( β, ¯ µ, m q ) (cid:21) , (29)where α ( Q, T ) → T → ∞ .Because quark confinement is considered here as the dual version of the confinement of magnetic pointcharges in Type-II superconductor (magnetic Abrikosov vortexes), the upper limit for T c is given by therequirement ( m/µ ) <
1, i.e., T c < α ( Q ) m (cid:18) −
14 ln ˜ µ m R (cid:19) . (30)Numerical estimation leads to T c <
222 MeV at B ≃
276 MeV, α = 0 .
37 and m = 0 .
85 GeV [12] for m R ∼ ˜ µ and s ∼ O (1). IV. COULOMB POTENTIAL IN DECONFINEMENT
It is known that the confinement of quarks is explained within the instantaneous part of the potential V defined by the ”time-time” component of gluon propagator D ( x = ( ~x, t )) (see, e.g., the paper by D.Zwanziger in [13])4 πα D ( x ) = V ( | ~x | ) δ ( t ) + non − instantaneous vacuum polarization term. (31)At the Gribov horizon [23] V ( R = | ~x | ) is caused by the long-range forces having confining properties V ( R → ∞ ) = ∞ . One of the aims of this article is also to understand the origin of the presence ofsome-range forces that confines ”quarks” in deconfined phase. To proceed for this one should restore thefollowing expression for Coulomb-like potential V c ( R, T ) at finite temperature in the form: V c ( R, T ) = − π Z d ~q α ( q , T ) q e − ~q ~R , (32)where ~q is the difference between momenta of a particle and an antiparticle confined by forces we areexploring here; α ( q , T ) is given by (29). At zero temperature, or even for low T , the integral in (32)diverges at the upper limit | ~q | → ∞ . At large T , this integral can be naturally regularized by introducingthe temperature-dependent soft regularization function Υ( q , T ) = 1 / (cid:2) q /M ( β ) (cid:3) (see also [24, 25])which has the properties: Υ( q , T ) → T → ∞ and Υ( q , T ) → Υ ( q ) as T →
0. At low tempera-tures, α ( q , T ) is rather slowly varying with q compared to sin ( q R ) / ( q R ) function in one-dimensionalrepresentation of the integral in (32) (the integrating over the angles is already done). On the otherhand, at high T > | ~q | the main contribution will be done by T -dependent term in α ( q , T ) expansion(29). Thus, from the mathematical point of view, the problem with divergence of the integral at theupper limit would be solved if α ( q , T ) is replaced by α ( T ). We get (¯ α = (4 / α ): V c ( R, T ) = − α ( T )3 π Z ∞ dq Υ( q , T ) sin q Rq R = ¯ α ( T ) R (cid:16) e − M ( β ) R − (cid:17) . (33)At short distances one gets that the Coulomb potential is consistent with a linear increase with R : V c ( R, T ) = σ c ( T ) R − α ( T ) M ( β ) , (34)where the Coulomb string tension σ c ( T ) = 0 . α ( T ) M ( β ) for strongly interacting particles in deconfine-ment is σ c ( T ) = 23 [4 π α ( T ) T ] " a N,N f + p α ( T ) b N ln a N,N f p α ( T ) ! + p α ( T ) c N + ... , (35)where a N,N f = s π (cid:18) N + N f (cid:19) , b N = N π . We found that V c ( R, T ) has the linear rising, σ c ( T ) > T > T c , where the physical (giving by theWilson loop) string tension ˜ σ eff ( T > T c ) = 0. The fact that the Coulomb string tension in deconfinementincreases with α T is consistent with magnetic mass having the behaviour as ∼ α T within the non-perturbative regime. V. FLUX TUBE SOLUTIONS
The temperature-dependent flux-tube solution for the dual gauge filed along the z-axis (within thecylindrical symmetry) has the following asymptotic transverse behaviour (for details see [12] at T =0)˜ C ( r, β ) ≃ n g ( β ) − r π m ( β ) r κ e − κm ( β ) r (cid:20) κ m ( β ) r (cid:21) , (36)where r is the radial coordinate (the distance from the center of the flux-tube), n is the integer numberassociated with the topological charge [26], κ = √ E inside the quark-antiquark bound state is given by the rotation of the dualgauge field ~E = ~ ∇ × ~C = 1 r d ˜ C ( r ) dr ~e z ≃ E z ( r ) · ~e z , (37)where ~e z is a unit vector along the z -axis, and the T -dependent E z ( r, β ) looks like [12] E z ( r, β ) = r π m ( β )2 κ r e − κm ( β ) r (cid:20) κ m ( β ) − r (cid:21) . (38)The lower bound on r = r can be estimated from the relation r > [2 κ m ( β )] − which leads to r > . T = 0. Obviously, r → ∞ as m ( β ) → T → T (deconfinement).In Fig. 1, we show the dependence of m as a function of the temperature T at different scale parameters M . No dependence found on quark current masses (we used m q = 7, 10 and 135 MeV). No essentialdependence found for different N f and N F . T , MeV ) , G e V b m ( =3 F =N f N = 7 MeV q mM = 0.5 GeVM = 1.0 GeVM = 2.0 GeV T , MeV ) , G e V b m ( =6 F =N f N = 7 MeV q mM = 0.5 GeVM = 1.0 GeVM = 2.0 GeV FIG. 1: The dual gauge boson mass m ( β ) shown as a function of T at different renormalization energy scale M and fixed value m q = 7 MeV with a) N f = N F = 3 b) N f = N F = 6 In Fig. 2 and Fig. 3, we show numerical solutions of the flux tube, namely, the profiles of the transversebehaviour of ˜ C ( r, β ) and the color electric field E z ( r, β ), respectively, as functions of radial variable r atdifferent temperatures. We found rather sharp increasing of ˜ C ( r, β ) at small values of r . No essentialdependence of r emerges in the region r > . E ( r, β ) disappears when the temperatureclose to T c . VI. SUMMARY
We were based on the dual gauge model of the long-distance Yang-Mills theory in terms of two-pointWightman functions. Among the physicists dealing with the models of interplay of a scalar (Higgs-like)field with a dual vector (gauge) boson field, where the vacuum state of the quantum Y-M theory isrealized by a condensate of the monopole-antimonopole pairs, there is a strong belief that the flux-tubesolution explains the scenarios of color confinement. Based on the flux-tube scheme approach of Abelian1 r , fm ) b ( r , C ~ T = 5 MeVT = 75 MeVT = 150 MeVT = 190 MeVn = 1M = 0.5 GeV
FIG. 2: The profiles of the dual gauge boson field ˜ C ( r, β ) shown as a function of the radial coordinate r at different T . r , fm ) , G e V b ( r , z E T = 5 MeVT = 75 MeVT = 150 MeVT = 190 MeVM = 0.5 GeV
FIG. 3: The profiles of the color-electric field E ( r, β ) as a function of the radial coordinate r at different T . dominance and monopole condensation, we obtained the analytic expressions for both the monopole anddual gauge boson field propagators [12]. These propagators lead to a consistent perturbative expansionof Green’s functions.The monopole condensation causes the strong and long-range interplay between heavy quark andantiquark, which gives the confining force, through the dual Higgs mechanism. The analytic expression2for the static potential at large distances grows linearly with the distance R apart from logarithmiccorrection.We observed that the flux tube can be produced abundantly when the phase transition emerges atthe temperature T = T c , obeying the condition ˜ σ eff ( T = T c ) = 0. We found that the phase transitiontemperature essentially depends on α ( Q ) and the mass of the dual gauge field m . The analytic criterionconstraining the existence of a quark-antiquark bound state at T > T c is obtained (see (12) and (15)).We find that the Coulomb string tension for strongly interacting particles in deconfinement increases with α T , and at short distances the Coulomb potential has the linear rising.It is observed [16] that in wide range of higher temperatures, T c < T < T c the non-perturbativescreening mass M ( β ) is rather constant for both the SU (2) and SU (3) cases, and this mass is definedby the leading order perturbative result M ( β ) ≃ M LO ( β ). This means an essential role of σ D ( β ) andthe existence of heavy quark-antiquark bound states at temperatures above the critical ones in theframework of the dual gauge theory. It is the only the question, whether this result can modify thestandard picture of finite-temperature gauge theory relevant to understanding of the quark-hadron phasetransition and existence of strong QCD effect in deconfinement state.I recall with pleasure stimulating discussions with N. Brambilla. [1] See the book G. Ripka, Dual Superconductor Models of Color Confinement, Springer, 2004 and the referencestherein.[2] Y. Nambu, Phys. Rev. D , 4262 (1974) ; S. Mandelstam, Phys. Rep. C , 245 (1976); Phys. Rev. D ,2391 (1979); A.M. Polyakov, Nucl. Phys. B , 429 (1977); G.’t Hooft, Nucl. Phys. B , 1 (1978).[3] G.’t Hooft, Nucl. Phys. B [FS3] , 455 (1981).[4] R.I. Nepomechie, M.A. Rubin and Y. Hozotani, Phys. Lett. B , 457 (1981).[5] P.A.M. Dirac, Phys. Rev. , 817 (1948).[6] H.B. Nielsen and P. Olesen, Nucl. Phys. B , 45 (1973).[7] See Y. Nambu’s paper in Ref. 2.[8] A. Di Giacomo et al., Phys. Rev. D , 034503; 034504 (2000).[9] T. Suzuki, Prog. Theor. Phys. , 929 (1988); , 752 (1989); S. Maedan and T. Suzuki, Prog. Theor. Phys. , 229 (1989).[10] H. Suganuma, S. Sasaki and H. Toki, Nucl. Phys. B , 207 (1995).[11] M. Baker et al., Phys. Rev. D , 034010 (1998).[12] G.A. Kozlov and M. Baldicchi, New J. Phys. , 16.1 (2002).[13] Yu.A. Simonov, Phys. Lett. B , 293 (2005); D. Zwanziger, ”Equation of state of gluon plasma from localaction”, hep-ph/0610021.[14] M. Baker et al., Phys. Rev. D , 2829 (1996).[15] A.A. Abrikosov, Sov. Phys. JETP , 1442 (1957).[16] K. Kajantie et al., Phys. Rev. Lett. , 3130 (1997).[17] P. Arnold and L.G. Yaffe, Phys. Rev. D , 7208 (1995).[18] E. Copeland, S. Holbraad, R. Rivers, ”Fluctuations and the Cosmic String Phase Transitions”, in Proc. ofthe 2nd Workshop on Thermal Field Theories and Their Applications, Tsukuba, Japan, July 23-27, 1990(Elsevier Science Publishers B.V., 1991, ed. by H. Ezawa, T. Arimitsu, Y. Hashimoto) p. 295 .[19] M. Baker, J.S. Ball and F. Zachariasen, Phys. Rev. D , 1968 (1995).[20] E.M. Lifshitz and L.P. Pitaevski, ”Statistical physics, P.2, Course of Theoretical Physics” v. 19 (Pergamon,Oxford, 1981).[21] A. Nayeri, R. H. Brandenberger and C. Vafa, Phys. Rev. Lett. , 021302 (2006).[22] S. Gao, B.Liu, W.Q. Chao, Phys. Lett. B , 23 (1996).[23] V.N. Gribov, Nucl. Phys. B , 1 (1978).[24] G.A. Kozlov et al., J. Phys. G: Nucl. Part. Phys. , 1201 (2004).[25] H.-C. Pauli, ”The hadronic potential at short distances”, hep-ph/0312198.[26] Y. Koma, H. Suganuma and H. Toki, Phys. Rev. D60