The Lifetime of the Electric Flux Tubes near the QCD Phase Transition
TThe Lifetime of the Electric Flux Tubesnear the QCD Phase Transition
Cyrus Faroughy and Edward Shuryak
Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794 (Dated: October 22, 2018)Electric flux tubes are a well known attribute of the QCD vacuum in which they manifest con-finement of electric color charges. Recently, experimental results have appeared suggesting that notonly those objects persist at temperatures T ≈ T c near the QCD phase transitions, but their decayis suppressed and the resulting clusters in AuAu collisions are larger than in pp (i.e. in vacuum).This correlates well with recent theoretical scenarios that view the QCD matter in the T ≈ T c regionas a dual-magnetic plasma dominated by color-magnetic monopoles. In this view the flux tubes arestabilized by dual-magnetic currents and are described by dual-magnetohydrodynamics (DMHD).In this paper we calculate classically the dissipative effects in the flux tube. Such effects are associ-ated with rescattering and finite conductivity of the matter. We derive the DMHD solution in thepresence of dissipation and then estimate the lifetime of the electric flux tubes. The conclusion ofthis study is that a classical treatment leads to too short of a lifetime for the flux tubes. PACS numbers:
I. INTRODUCTIONA. Motivation
Non-perturbative phenomena in both the QCD vac-uum and in the finite temperature/density QCD mat-ter have been the subject of intense studies for a longtime. Various phenomenological approaches have beenproposed during the last few decades to tackle them. Onesimple example is the stochastic QCD vacuum model [1]which provided a picture of the gluonic ”condensate” andcorrelations as a constant fields, another is the instantonliquid model [2] that explained several phenomena relatedto the SU ( N F ) and the U A (1) symmetry breaking.Most researchers in the field hold to a generic pictureof confinement introduced by t’Hooft and Mandelstamin the 1980’s: the so-called “dual superconductivity”. Itsuggests that confinement is a dual Meissner effect, andthat a “coil” which prevents the color-electric flux tubefrom spreading out is produced by a magnetic supercur-rent. The condensate of Cooper pairs produced in usualsuperconductors is substituted by a Bose-Einstein con-densate of some objects possessing color magnetic charge.Presently, however, there is no concrete understandingof the field configurations that are accountable for colorconfinement. The effort for identifying the correspond-ing topological objects responsible for color confinementis still continuing to this date. Monopoles, dyons, ortheir composites, induced by fermions, have been shownto provide a mechanism for confinement and chiral sym-metry breaking in the N =2 SYM theory [3]. But it isstill unclear whether such objects can be concretely car-ried into the QCD-like theories without involving scalars.The seminal paper by Polyakov [12] set an early exampleof confinement in (2+1)D Yang-Mills theories by meansof gluomagnetic monopoles. Recently, Unsal [4] has pro-posed an interesting (3+1)D extension of the latter model of confinement for QCD-like theories, involving fermions,by invoking certain composite objects endowed with totalmagnetic and zero topological charges. However, here tooit remains to be seen whether such objects can be trulyresponsible for confinement, particularly in the frame-work of lattice gauge theories.In the last few years, this subject has been revital-ized by experimental studies at the Relativistic HeavyIon Collider (RHIC), where hot QCD matter is producedand studied. One conclusion stemming from the analysisof RHIC results is that the quark-gluon plasma (QGP) inthe 1-2 T c temperature interval is a strongly coupled fluid(sQGP). Liao and Shuryak [5] (a review on the subjectcan also be found in [6]) have related this finding with theso called “magnetic scenario”, arguing that while color-electric objects (quarks and gluons) interact stronger andstronger when approaching the critical temperature T c ,the Dirac condition implies that the color-magnetic ob-jects should become lighter and more weakly coupled.They conjectured that electric components dominate thematter at high- T while the magnetic components domi-nate it near T c . Furthermore, they gave arguments thatan equilibrium point in which both electric and mag-netic coupling constants are equal ( α m = α e = 1) ex-ists at T ≈ . T c . Lattice observations have confirmedthat monopoles form a relatively strongly coupled liquidwhere the magnetic coupling increases at high T (seediscussion in [7]). Lattice observations of monopoles alsostrongly support the idea that they form a Bose-Einsteincondensate at exactly T = T c [10].Recent two and three particle correlations in experi-ments at RHIC indicate that certain fluctuations occur-ring on top of the overall expanding quark-gluon plasma(QGP) have small or even zero expansion velocity in thenear- T c region, suggesting the existence of stabilized elec-tric flux tubes. In addition, the experimental resultsalso show that the resulting clusters in AuAu collisions a r X i v : . [ h e p - ph ] A p r are even larger than in pp collisions (i.e. in vacuum).The purpose of this paper is to use the classical dual-magnetohydrodynamic model to study their evolutionwhen the dissipative effects are included. We will alsoattempt to evaluate the stability of the flux tubes in thenear- T c region, where we consider the plasma to be dual-magnetized [5]. Specifically, uncondensed color-magneticmonopoles circulate and form solenoidal currents which,by dual-Faraday’s law, induce a color-electric field andcreate the flux tube. By studying the diffusion of thisfield in the medium we can calculate the halflife of theflux tube and see if the field is strong enough to accountfor its stability during this magnetized phase. The dis-sipative effects are included by allowing the conductivityof the matter to be finite. B. Dual-Magnetohydrodynamics
Magnetohydrodynamics (MHD) studies the dynamicsof electrically conducting fluids and is widely used inplasma physics. In other words, it studies the interactionbetween a magnetic field and a plasma, treating it as acontinuous medium. It is an approximation which keepsonly the magnetic field in Maxwell’s equations, while theelectric field is entirely screened. Furthermore, MHD isused by solar physicists to describe the overall effects ofelectric currents and magnetic fields in the Sun’s coronato describe the sun spots and flux tubes.As mentioned previously, one scenario has been pro-posed where the QCD matter near the T c region is dual-magnetized. In this region, the electric and magneticscreening masses are assumed to be M m > M e becausethe color-magnetic monopoles dominate the scene. (Thisis opposite to the case of high temperatures, in whichelectric particles – quarks and gluons – dominate, andtherefore M m ∼ g T (cid:28) M e ∼ gT , where g is the gaugecoupling, small at high T . The high-T value for M m kasbeen suggested by Polyakov [12]) It therefore is plausibleto use DMHD ignoring the magnetic fields.In the usual plasmas like in the Sun, the electric screen-ing mass is very high and thus electric fields are ignoredand magnetic fields are included. Under certain condi-tions the flux tubes can be formed as a result of solenoidalelectric currents. Analogously, in RHIC collisions, thereexist a ”dual corona” [8] in which the electric flux tubesare produced by the magnetic currents. In what followswe study the diffusion of the field for flux tubes in MHDand then translate the results in the context of DMHDto investigate the lifetime of the flux tubes in the near- T c region of the QCD matter. II. THE DIFFUSION EQUATION WITH FINITECONDUCTIVITY
The ideal MHD approximation is the limit of inf inite conductivity of the plasma σ → ∞ , analogous to the zero viscosity approximation in the case of ideal hydrodynam-ics. In this section, for self-consistency of the paper, weinclude the known derivation of the diffusion equationfor the magnetic field in standard MHD with finite con-ductivity in order to account for dissipative effects. Thefundamental equations of MHD are: − ∂ (cid:126)B∂t = (cid:126) ∇ × (cid:126)E (2.1) E = − (cid:126)v × (cid:126)B + (cid:126)jσ (2.2)and (cid:126)j = (cid:126) ∇ × (cid:126)B π (2.3)If the currents are orthogonal to the magnetic field (asthey are in an ideal solenoid), then (cid:126)v × (cid:126)B = 0 and theabove equations simplify into the Diffusion Equation: ∂ (cid:126)B∂t = η ∇ (cid:126)B (2.4)where we have used div (cid:126)B = 0 and η = 1 / πσ is themagnetic diffusivity. The magnetic diffusivity containsthe conductivity of the plasma which will be discussed inpart IV. The dual version of the diffusion equation is thesame as the latter with the magnetic field (cid:126)B replaced bythe dual-electric field ˜ E . In addition, instead of havingions and electrons with different masses ( m i >> m e ),there are two (uncondensed) color-magnetic monopolesand antimonopoles with equal mass and opposite charge g + and g − forming two currents of equal magnitude cir-culating along the cylinder in opposite directions. III. SOLVING THE DIFFUSION EQUATION
The flux tube is stabilized by magnetic currents circu-lating solenoidally on its surface. We place the solenoidin a cylindrical coordinate system with its vertical axisalong the z direction. At t=0 each current ± (cid:126)J flows cir-cularly in the ± (cid:126)ϕ direction at r = a o , the initial radius ofthe tube. The initial condition is taken to be a gaussianprofile of the field centered at the origin. At all times, thefield points only along the z axis. We assume azimuthalsymmetry so that there is no ϕ dependence and considerthe field to be constant along the z direction. Thereforethe field ˜ E ( r, t ) will only depend on the radial direction r and time t . The diffusion equation in cylindrical coor-dinates is: 1 r ∂∂r ( r ∂ ˜ E ( r, t ) ∂r ) = 1 η ∂ ˜ E ( r, t ) ∂t (3.1)where η = 14 πσ (3.2)is now the dual diffusivity and σ is the dual conductivityof the plasma. The initial profile of the field is assumedto be a Gaussian centered at the origin:˜ E ( r,
0) = ˜ E o e − r a o (3.3)Separation of variables ˜ E ( r, t ) = R ( r ) T ( t ) decouples thePDE (3 .
1) into two ordinary ODE’s, one with time de-pendence and one with spatial dependence: ∂T ( t ) ∂t + λ ηT ( t ) = 0 ∂∂r ( r ∂R ( r ) ∂r ) + λ rR ( r ) = 0The first equation gives us the time dependent part T ( t ) = e − λ ηt and the second equation is the Bessel equa-tion of zeroth order ( m = 0) and the solution is a super-position of Bessel functions of the first and second kind.We exclude the Bessel function of second kind since theyare not finite at r = 0 and obtain R ( t ) = AJ ( λr ). InSturm-Liouville conditions the general solution is:˜ E ( r, t ) = (cid:90) ∞ A ( λ ) J ( λr ) e − λ ηt dλ (3.4)To find the A ( λ )’s lets look at ˜ E ( r, t ) at t = 0:˜ E ( r,
0) = (cid:90) ∞ A ( λ ) J ( λr ) dλ Now multiply both sides by J ( βr ) r and integrate over r : (cid:90) ∞ ˜ E ( r, J ( βr ) rdr = (cid:90) ∞ A ( λ ) (cid:90) ∞ J ( βr ) J ( λr ) rdrdλ The closure relation for Bessel functions gives: (cid:90) ∞ J ( βr ) J ( λr ) rdr = 1 β δ ( β − λ )and we obtain: (cid:90) ∞ ˜ E ( r, J ( βr ) rdr = (cid:90) ∞ A ( λ ) 1 β δ ( β − λ ) dλ = 1 β A ( β ) A ( λ ) = λ (cid:90) ∞ ˜ E ( r, J ( λr ) rdr which for ˜ E ( r,
0) = ˜ E o e − r a o gives: A ( λ ) = λ ˜ E o (cid:90) ∞ e − r a o J ( λr ) rdr = 12 ˜ E λ a o e − λ a o Plugging this back in (3 .
4) we obtain an expression for˜ E ( r, t ):˜ E ( r, t ) = (cid:90) ∞
12 ˜ E λ a o e − λ a o J ( λr ) rdrJ ( λr ) e − λ ηt dλ and finally:˜ E ( r, t ) = ˜ E o a o a o + 4 ηt exp (cid:20) − r a o + 4 ηt (cid:21) (3.5)Comparing (3 .
3) with (3 . t >
0, the term a o ac-quires an effective time dependent form a o + 4 ηt thatshows an increase in the field’s squared radius. There-fore the field’s time dependent radius is given by a ( t ) =( a o + 4 ηt ) / . IV. ESTIMATES OF THE CONDUCTIVITYAND THE FATE OF THE FLUX TUBES
The standard electrodynamic plasma consists of elec-trons and ions having densities N e and N i respectively.Due to large difference in masses, m e (cid:28) m i the currentis assumed to be due to the motion of electrons only.The textbook expression for the conductivity of the usualplasmas, normal to the magnetic field [9] is: σ ⊥ = 3 √ πe N e √ mν e (4.1)for Z = 1 ions, and ν e = 4 πe L e N i m / T / (4.2)is the collision rate between electron and ions. L e is theCoulomb logarithm and it is equal to ln(1 /χ min ) where χ min is the magnitude of the smallest angles for whichthe scattering can still be regarded as Coulomb scatter-ing. The electron-electron collisions are ignored sincethey cannot change the total momentum of the electronsand thus do not modify the electron current. Our task isto translate these results into those appropriate for theQCD matter in the near- T c region.It consists of four main components. The first twoare positively and negatively charged monopoles. Theirdensities are denoted by N + and N − . As stressed before,the monopoles create counterdirected flows around theelectric flux tube. The medium produced has overall zeromagnetic charge, so that N + = N − = N m /
2. The lasttwo components are electric objects – quarks and gluons– with densities N q and N g respectively. As noted in theIntroduction, the electric components dominates at high T , in the Quark-Gluon Plasma, but not in the near- T c region, where they are suppressed. The total scatteringrate for the positive monopoles is ν + = ν + − + ν + g + ν + q (4.3)and, by symmetry, the total collision rate is then ν tot = ν + .It is important to note that the ++ scattering termis omitted for the same reason as the ee collisions areomitted in the electrodynamic plasmas: they would notchange the total current. The + − cross section is thetransport cross section due to magnetic Coulomb forces,so the + − collision rate is given by dual to the electron − ion rate above. It is obtained by the substitution of thecoupling e → g m / π (note the difference in 4 π resultingfrom two different ways the fields are defined in QED andQCD) and the density N e → N − .Scattering on electric objects is different, as discussedin details by Ratti and Shuryak [14]. It has similarRutherford-like scattering at small angles but the trans-port cross section is dominated by large (near-backward)scattering angles. In order to obtain expressions for thescattering rates and conductivity, we have to use certainempirical values of the parameters involved. All of themare, in principle, a function of the temperature T , butwe restrict our discussion to the vicinity of T ≈ T c .The values for the monopole density, magnetic Coulombcoupling and the mass that we use for the estimates areas follows:quantity value reference N m /T c ≈ g m / π ≈ / m/T c ≈ ν + g /T c ≈ ν + g ≈ ν + q [22]. Fromthese relations we obtain the +- dual Coulomb collisionrate: ν + − = g m L e N − πm / T / c = g m L e √ N − π T c = g m L e T c √ π And from equation (4 . ν tot = g m L e T c √ π + 2 T c + 2 T c = T c (cid:32) g m L e √ π π (cid:33) The conductivity (4 .
1) is then: σ = 3 √ πg m N − π √ mν tot = 3 √ πg m T c πν tot = 3 √ πg m T c g m L e √ π )The value we used for T c is the T c = 170 MeV fromregular QCD with quarks rather than the larger T c = 260MeV value from pure gauge. Using g m / π ≈ / χ min = 1 /
10 so that L e = ln(10) ≈ .
30, weget: ν + − = 16 πL e T c √ ≈ . M eV = 5 . f m − ν tot = T c ( 16 πL e √ ≈ . M eV = 9 . f m − σ = 3 √ πT c π ( L e √ π ) ≈ . M eV = 6 . × − f m − Finally, from equation (3 . η = 14 πσ ≈ . × − M eV − = 1 . f m Now we need to address the overall timing of the heavyion collisions at RHIC. According to (very successful)hydrodynamical simulations, the duration of the magne-tized phase of the collisions is τ M ≈ − f m/c , at anycentrality and any position of the fluid cell. According toour solution the mean square radius of the tube duringthis time is increasing by a o → a o + 4 ητ M (4.4)The energy per unit length (cid:15) diffused during the magneticphase is given by (cid:15) ( t ) = 2 π (cid:90) ∞ (cid:20) π ˜ E ( r, t ) + K ( r, t ) (cid:21) rdr Where K = (1 / ρv ∝ J ( r, t ) is the kinetic energydensity associated to the monopole current (cid:126)J ( r, t ). Usingequation (2.3) we see that the time component of K scaleslike the time component of the magnetic energy. As aresult, during the magnetic phase the total energy perunit length scales as : (cid:15) ( τ M ) (cid:15) (0) = a o a o + 4 η τ M We estimate a o ≈ . f m so that a o ≈ . f m andfrom the values above we get:4 ητ M ≈ . f m To gain more insight about the lifetime of the flux tube,let us calculate the half-life t / of the field at the ori-gin. We know that at the origin and at t = 0 we have˜ E (0 ,
0) = ˜ E o . So, by definition:˜ E (0 , t / ) = ˜ E o E o a o a o + 4 ηt / (4.5)and solving for t / : t / = 14 a o η = πσa o = 5 × − f m (4.6)which is many times shorter than the expected lifetimeof the observed flux tubes. V. SUMMARY AND DISCUSSION
In this paper we have used a classical dual-magnetohydrodynamic approach to calculate the fluxtube lifetime in the magnetic phase of the QGP nearthe QCD phase transition. More specifically, we havefound a solution for the flux tube including the dissi-pative “diffusive” term. We calculated the value forthe “dual magnetic diffusion constant” using a pictureof monopole-monopole and monopole-gluon rescatteringand found that this crude classical rescattering modelpredicts very strong dissipative effects that are way toostrong for the flux tubes to survive in the few fm/c time-frame of the magnetic phase. Yet, “ridge” correlations ofthe detected pions are found in experiments.One possible view on this, held e.g. by the BNLgroup [15, 16], is that the flux tubes do indeed decayvery quickly, as the estimates above suggest, and the ob-served “ridge” is nothing but the extra amount of en-tropy left behind. A problem with this interpretation(discussed by one of us in [8]) is that a spot of extrathermal entropy/energy would evolve hydrodynamicallyinto a cylinder of several fm radius, which is in direct con-tradiction with the rather narrow φ distribution width ofthe ridge.More likely, a classical approach to the flux tube dissi- pation is incorrect. First of all, unlike flux tubes usuallyconsidered by magnetohydrodynamics (e.g. in solar plas-mas) the QCD flux tubes under consideration are small insize and not larger than the quasiparticle Compton wavelength. This suggests that one should use a quantum-mechanical description, like the one in [11]. Quantumeffects in the monopole motion may provide two “super-currents” which propagate through each other without any dissipation. We know that this is the case in theconfining vacuum.Finally, let us also mention that there are additionalevidence for survival of the flux tubes in the magneticphase, at T ≈ T c , which do not come from RHIC experi-ments but from lattice numerical simulations of the QCDthermodynamics. Those indicate that baryonic states re-main under such conditions, and that their density ofstate is well described by Hagedorn-like “stringy” states,see [21]. Unfortunately we do not know the lifetime ofthese baryons as the lattice thermodynamics does notgive us such information. Acknowledgments
The work of ES is supported by the US DOE grantDE-FG-88ER40388. [1] Yu. Simonov, Phys. Usp.
313 (1996)[2] T. Schafer and E. Shuryak, Rev. Mod. Phys. , 484 (1994)[arXiv:hep-th/9408099].[4] M. Unsal, Int. J. Mod. Phys. A , 278 (2010).[5] J. Liao and E. Shuryak, Phys. Rev. C , 054907 (2007)[6] E. Shuryak, Prog. Part. Nucl. Phys. , 48 (2009)[7] Phys. Rev. Lett. , 162302 (2008)[8] E. Shuryak, Phys. Rev. C , 054908 (2009) [Erratum-ibid. C , 069902 (2009)] [arXiv:0903.3734 [nucl-th]].[9] E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics (volume 10 of Landau-Lifshitz Course of TheoreticalPhysics), Pergamon press, 1981.[10] A. D’Alessandro, M. D’Elia and E. Shuryak Phys. Rev.C , 064905 (2008)[11] J. Liao and E. Shuryak, Phys. Rev. C , 064905 (2008)[12] A. M. Polyakov, Phys. Lett. B 72, 477 (1978).[13] M. N. Chernodub and V. I. Zakharov, Phys. Rev. Lett. , 082002 (2007) [arXiv:hep-ph/0611228].[14] C. Ratti and E. Shuryak, Phys. Rev. D , 034004 (2009) [arXiv:0811.4174 [hep-ph]].[15] A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan,Nucl. Phys. A , 91 (2008) [arXiv:0804.3858 [hep-ph]].[16] S. Gavin, L. McLerran and G. Moschelli, arXiv:0806.4718[nucl-th].[17] M.Daugherity (for the STAR coll.), Anomalous centralityvariation..., QM08, J.Phys.G.Nucl/Part.Phys. 35 (2008)104090[18] G. I. Veres et al.et al.