Topological objects contribute to thermodynamics of gluon plasma Katsuya Ishiguro, Toru Sekido,Tsuneo Suzuki (Kanazawa, Japan) A. Nakamura (Hiroshima University, Japan), V.I. Zakharov in collaboration with M.N. Chernodub (ITEP, Moscow) • Magnetic component in Yang-Mills theory at T = 0. - Models of color confinement. - Strings and monopoles at T = 0 • Magnetic component at T > 0: - Condensate-liquid-gas transition? - Existence of real(?) strings/monopoles at T > T c ? - Strings/monopoles and equation of state (lattice) ◮ K.Ishiguro, A.Nakamura, T.Sekido, T.Suzuki, V.I.Zakharov, M.N.Ch., Proceedings of Science (LATTICE 2007) 174 [arXiv:0710.2547] ◮ V.I.Zakharov, M.N.Ch., Phys.Rev.Lett. 1 98 (2007) 082002
Phase structure of pure gluons Nontrivial dynamics of QCD is determined by the gluon sector: [Talk by Frithjof Karsch ≈ 27 hours ago] Phase structure of SU ( N ) gluodynamics, N = 2 , 3 • T < T c : confinement of color • T > T c : deconfinement of color In the deconfinement phase: • T ∼ [ T c . . . 2 − 5 T c ] : plasma, strongly interacting gluons • higher T : predictions scale towards perturbation theory • T ≫ T c : perturbative electric gluons plus logarithmically decaying non-perturbative magnetic sector 2
T c < T < 2 T c Properties of gluon plasma are unexpected: • Similar to ideal(!) liquid review in, e.g., [Shuryak, hep-ph/0608177] • Shear viscosity of plasma is low , η/s ≈ 0 . 1 . . . 0 . 4 * interpretation of RHIC experiment [Teaney, 03] η * simulations of quenched QCD 3 16 8 �� � � 3 s 24 8 [= SU(3) lattice gauge theory] Perturbative [A.Nakamura, S.Sakai, 05] Theory [H.Meyer, ’07] KSS bound�� 1 1.5 2 2.5 3 T T c 3
T ≫ T c • Stefan-Boltzmann law seems to be reached at T → ∞ C SB = π 2 ε free = 3 P free = N d.f. C SB T 4 , 30 N d.f. = 2( N 2 c − 1) degrees of freedom numerical simulations [Karsch et al, NPB’96] [Bringoltz, Teper, PLB’05; Gliozzi, ’07] • Many features may be described by – perturbation theory – large- N c supersymmetric Yang–Mills theory a review can be found in [Klebanov, hep-ph/0509087] – quasiparticle models (work also around T ≈ T c ) [Rischke et al’ 90, Peshier et al’ 96; Levai, Heinz’ 98] 4
T < T c (confinement phase) Widely discussed mechanisms of color confinement: • Dual superconductor picture [’t Hooft, Mandelstam, Nambu, ’74-’76] * Based on existence of special gluonic configurations, called “magnetic monopoles” * Monopoles are classified with respect to the Cartan sub- group [ U (1)] N − 1 of the SU ( N ) gauge group • Center vortex mechanism [Del Debbio, Faber, Greensite, Olejnik, ’97] * a realization of spaghetti (Copenhagen) vacuum * Center strings are classified with respect to the center Z N of the SU ( N ) gauge group 5
Popular models of confinement of quarks • Condensation of magnetic monopoles Abelian Dominance [T.Suzuki, I. Yotsuyanagi ’00] Monopole Dominance [T.Suzuki, H.Shiba ’00] • Percolation of magnetic strings Center/Vortex Dominance [L. Del Debbio, M. Faber, J. Greensite, S. Olejnik ’97] • These approaches are related: – The percolation is related to the condensation (presence of the IR component in the density) – Monopoles are related to strings. numerical fact [Ambjorn, Giedt & Greensite ’00] required analytically [Zakharov ’05] compact gauge models [Feldmann, Ilgenfritz, Schiller & Ch. ’05] 6
T < T c (Dual Superconductor) * condensation of monopoles → dual Meissner effect * dual Meissner effect → chromoelectric string formation * chromoelectric string = (dual) analogue of Abrikosov string * quarks are sources of chromoelectric flux → confinement Abrikosov string chromoelectric string monopole condensate in superconductor in QCD vacuum (numerical results) [Di Giacomo & Paffuti’97] monopole condensate from [Polikarpov, Veselov & Ch. ’97] 7
T < T c (Dual Superconductor and QCD string) 4 2 0 -2 -4 -4 -2 0 2 4 electric field (theor.) monopole current (theor.) 2D-monopole curl (num.!) 0.008 0.6 0.03 E k 0.007 V-V 0 k θ curl E 0.5 V ab -V ab 0.025 0.006 0 0.4 0.005 0.02 0.3 0.004 V(R) 0.015 0.2 0.003 0.002 0.1 0.01 0.001 0 0.005 0 -0.1 0 0 1 2 3 4 5 6 7 -0.2 0 1 2 3 4 5 6 7 x 2 4 6 8 10 12 14 16 x R electric field, magn. current London (Ampere) equation quark-antiquark potential [Bali, Schlichter, Schilling ’98; Bali, Bornyakov, M¨ uller-Preussker, Schilling ’96] 8
Center/Monopole mechanisms are linked • non-oriented half-flux of magnetic field • monopoles are at points at which the flux alternates • vortices are chains of monopoles + [Ambjorn, Giedt, Greensite, ’00] + - + - - • a similar string–monopole structure + + + appears also in SUSY models - [Hanany, Tong, ’03; Auzzi et al ’03] - and in non-SUSY theories + [Feldmann, Ilgenfritz, Schiller & Ch. ’05] - - [Gorsky, Shifman, Yung, ’04 ... ’07] + • required analytically [Zakharov ’05] 9
Monopoles vs. vortices ◮ The vortices may organize the monopoles into dipole-like and chain- like structures, which are also present in compact Abelian models with doubly charged matter fields ◮ Examples of monopoles-vortex configurations: [results of numerical simulations are taken from Feldmann, Ilgenfritz, Schiller & M.Ch. ’05] ◮ Observation of monopoles in the vortex chains: monopole is a point defect, where the flux of the vortex alternates. 10
Confinement ( T < T c ) and plasma ( T > T c ) ◮ The monopoles are condensed at T < T c ... and not condensed at T > T c ◮ The magnetic strings are percolating at T < T c ... and not percolating at T > T c ◮ What happens with topological defects at finite temperature T > 0? ◮ SUGGESTION: Degrees of freedom condensed at T = 0 form a light component of the thermal plasma at T > 0. ◮ The magnetic monopoles and the magnetic vortices become real (thermal) particles at T > 0 [Zakharov, M.N.Ch., ’07] [Liao and Shuryak, ’07] 11
Use lattice simulations? ◮ Lattice simulations provide us with ensembles of magnetic defects. ◮ Which defect is real and which is virtual? t s=+1 s= 2 x - ◮ s : the wrapping number with respect to the compact T –direction. ◮ Properties of thermal particles are encoded in the wrapped trajec- tories, s � = 0, and the virtual particles are non-wrapped, s = 0. [Zakharov, M.N.Ch., ’07] 12
Example of a free scalar particle ◮ How to get thermal component of the density d 3 p � ρ th ( T ) = (2 π ) 3 f T ( p ) from the trajectories of the particles? ◮ The propagator of the scalar particle is: P x,y e − S cl [ P x,y ] � G ( x − y ) ∝ is the sum over all trajectories P x,y connecting points x and y . ◮ The propagator in momentum space, � d 3 x e − i ( p , x ) G ( x , t = s/T ) . G s ( p ) = where s is the wrapping number of trajectories in the T –direction. 13
Example of a free scalar particle ◮ Then the vacuum ( s = 0) part of propagator is divergent: G vac ≡ G 0 = 4 Λ 2 UV ω p ◮ ... while the ratio G wr ( p ) f T ( ω p ) = 1 G wr ≡ � G vac ( p ) , G s 2 s � =0 is finite as it gives the thermal distribution of the free particles 1 phys ) 1 / 2 ω p = ( p 2 + m 2 f T = , e ω p /T − 1 ◮ CONCLUSION: Wrapped trajectories in the Euclidean space correspond to real particles in Minkowski space. 14
Density of thermal particles • The average number of wrappings s in a time slice of volume V 3 d is directly related to the density of real particles ρ th ( T ) = n wr = �| s |� V 3 d Wrapped monopole density 4 [V.I.Zakharov, M.N.Ch., ’07] Density of thermal monopoles vs. T 3 ρ 1/3 /T c [Bornyakov, Mitrjushkin, M¨ uller-Preussker ’92] [T.Suzuki, S.Ejiri, ’95] 2 [T.Ejiri, ’96] β =2.30 β =2.51 1 First reliable lattice calculation: β =2.74 [A.D’Alessandro, M.D’Elia, ’07] 0 2 4 6 8 T/T c 15
Interpretation ◮ At T = T c : The condensed cluster → wrapped trajectories ◮ Wrapped trajectories correspond to real (thermal) particles: condensate + virtual condensate + thermal + virtual thermal + virtual ⇒ ⇒ ( T < 0) (0 < T < T c ) ( T > T c ) ◮ Analogy with superfluid Helium-4 [Zakharov & M.Ch’07] In He-4 at T = 1 K ≈ 0 . 5 T c only 7% of particles are in the condensate! The rest (93%) is thermal! 16
Liquid state of monopoles at finite T • Monopole correlations � ρ ± ( x ) ρ ± ( y ) � • Calculation in lattice Yang-Mills: [A.D’Alessandro & M.D’Elia, ’07] • Liquid state interpretation: [Liao, Shuryak ’08] + [talk by Shuryak ≈ 40 minutes ago] ◮ A gas parameter for the monopole gas in Yang-Mills theory: ◮ The static monopoles contribute to spatial string tension σ sp . ◮ If the monopoles form a gas, then σ sp( T ) R sp = λ D ( T ) ρ ( T ) = 8 [theory] ◮ We find: R sp ≈ 8 at T � 2 . 5 T c [Ishiguro, Suzuki, M.Ch ’03] 17
Thermodynamics • Free Energy ( T is temperature and V is spatial volume) F = − T log Z ( T, V ) • Pressure p = T ∂ log Z ( T, V ) = − F V = T V log Z ( T, V ) ∂ log V V • Energy density ε = T ∂ log Z ( T, V ) V ∂ log T • Entropy density s ( T ) = ε + p = ∂ p ( T ) T ∂T 18
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