Topological objects contribute to thermodynamics of 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 > Tc? - Strings/monopoles and equation of state (lattice) I K.Ishiguro, A.Nakamura, T.Sekido, T.Suzuki, V.I.Zakharov, M.N.Ch., Proceedings of Science (LATTICE 2007) 174 [arXiv:0710.2547] I V.I.Zakharov, M.N.Ch., Phys.Rev.Lett. 98 (2007) 082002 1 Phase structure of pure

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 < Tc: confinement of color • T > Tc: deconfinement of color

In the deconfinement phase: • T ∼ [Tc ... 2 − 5Tc]: plasma, strongly interacting gluons • higher T : predictions scale towards perturbation theory • T À Tc : perturbative electric gluons plus logarithmically decaying non-perturbative magnetic sector

2 Tc < T < 2Tc

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]

3 η 16 8 * simulations of quenched QCD 3 24 8 s [= SU(3) lattice ] Perturbative Theory [A.Nakamura, S.Sakai, 05]

KSS bound [H.Meyer, ’07]

1 1.5 2 2.5 3 T Tc

3 T À Tc

• Stefan-Boltzmann law seems to be reached at T → ∞ π2 ε = 3P = N C T 4 ,C = free free d.f. SB SB 30 2 Nd.f. = 2(Nc − 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-Nc supersymmetric Yang–Mills theory a review can be found in [Klebanov, hep-ph/0509087] – quasiparticle models (work also around T ≈ Tc) [Rischke et al’ 90, Peshier et al’ 96; Levai, Heinz’ 98]

4 T < Tc (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 ZN of the SU(N) gauge group

5 Popular models of confinement of

• 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 < Tc (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 < Tc (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 kθ 0.007 V-V0 curl E 0.5 ab ab 0.025 0.006 V -V0 0.4 0.005 0.02 0.3 0.004 0.015 0.003 0.2 V(R)

0.01 0.002 0.1 0.001 0 0.005 0 -0.1 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 -0.2 x x 2 4 6 8 10 12 14 16 R electric field, magn. current London (Ampere) equation -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

I 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

I Examples of monopoles-vortex configurations:

[results of numerical simulations are taken from Feldmann, Ilgenfritz, Schiller & M.Ch. ’05]

I Observation of monopoles in the vortex chains: monopole is a point defect, where the flux of the vortex alternates.

10 Confinement (T < Tc) and plasma (T > Tc)

I The monopoles are condensed at T < Tc ... and not condensed at T > Tc

I The magnetic strings are percolating at T < Tc ... and not percolating at T > Tc

I What happens with topological defects at finite temperature T > 0?

I SUGGESTION: Degrees of freedom condensed at T = 0 form a light component of the thermal plasma at T > 0.

I 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?

I Lattice simulations provide us with ensembles of magnetic defects.

I Which defect is real and which is virtual?

t

x s=+1 s=- 2

I s: the wrapping number with respect to the compact T –direction.

I Properties of thermal particles are encoded in the wrapped trajec- tories, s 6= 0, and the virtual particles are non-wrapped, s = 0. [Zakharov, M.N.Ch., ’07] 12 Example of a free scalar particle

I How to get thermal component of the density Z d3p ρth(T ) = f (p) (2π)3 T from the trajectories of the particles?

I The of the scalar particle is: X G(x − y) ∝ e−Scl[Px,y] Px,y is the sum over all trajectories Px,y connecting points x and y.

I The propagator in momentum space, Z 3 −i(p,x) Gs(p) = d x e G(x, t = s/T ) . where s is the wrapping number of trajectories in the T –direction.

13 Example of a free scalar particle

I Then the vacuum (s = 0) part of propagator is divergent: 2 vac 4 ΛUV G ≡ G0 = ωp

I ... while the ratio 1 Gwr(p) X f (ω ) = , Gwr ≡ G T p vac s 2G (p) s6=0 is finite as it gives the thermal distribution of the free particles

1 2 2 1/2 fT = , ωp = (p + m ) eωp/T − 1 phys

I 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 V3d is directly related to the density of real particles

th h|s|i ρ (T ) = nwr = V3d Wrapped monopole density 4 [V.I.Zakharov, M.N.Ch., ’07]

Density of thermal monopoles vs. T 3 c

/T [Bornyakov, Mitrjushkin, M¨uller-Preussker’92] 1/3 ρ [T.Suzuki, S.Ejiri, ’95] 2 [T.Ejiri, ’96] β=2.30 β=2.51 1 β=2.74 First reliable lattice calculation: [A.D’Alessandro, M.D’Elia, ’07] 0 2 4 6 8 T/Tc

15 Interpretation

I At T = Tc: The condensed cluster → wrapped trajectories I Wrapped trajectories correspond to real (thermal) particles: condensate + virtual condensate + thermal + virtual thermal + virtual ⇒ ⇒ (T < 0) (0 < T < Tc) (T > Tc) I Analogy with superfluid Helium-4 [Zakharov & M.Ch’07]

In He-4 at T = 1K ≈ 0.5Tc only 7% of particles are in the condensate! The rest (93%) is thermal! 16 Liquid state of monopoles at finite T

• Monopole correlations hρ±(x)ρ±(y)i • Calculation in lattice Yang-Mills: [A.D’Alessandro & M.D’Elia, ’07] • Liquid state interpretation: [Liao, Shuryak ’08] + [talk by Shuryak ≈40 minutes ago]

I A gas parameter for the monopole gas in Yang-Mills theory: I The static monopoles contribute

to spatial string tension σsp. I If the monopoles form a gas, then σsp(T ) Rsp = = 8 [theory] λD(T )ρ(T )

I We find: Rsp ≈ 8 at T & 2.5Tc [Ishiguro, Suzuki, M.Ch ’03]

17 Thermodynamics

• Free Energy (T is temperature and V is spatial volume)

F = −T log Z(T,V )

• Pressure T ∂ log Z(T, V ) F T p = = − = log Z(T, V ) V ∂ log V V V • Energy density T ∂ log Z(T, V ) ε = V ∂ log T

• Entropy density ε + p ∂ p(T ) s(T ) = = T ∂T

18 Thermodynamics: Trace

• Trace anomaly of the energy–momentum tensor Tµν ∂ p(T ) θ(T ) = hT µi ≡ ε − 3p = T 5 µ ∂T T 4 • Pressure via trace anomaly

ZT 4 d T1 θ(T1) p(T ) = T 4 T1 T1 • Energy density via trace anomaly

ZT 4 d T1 θ(T1) ε(T ) = 3 T 4 + θ(T ) T1 T1 • Trace anomaly is a key quantity

19 Trace Anomaly for pure gluons

• Partition Function Z X 1 Z(T,V ) = DU exp{−β SP [U]} ,SP [U] = (1− Tr UP ) P 2 • Trace Anomaly ∂ log Z(T,V ) θ(T ) = T 5 ∂T T 3V 3 • Asymmetric Ns Nt lattice: 3 T = 1/(Nta) ,V = (Nsa)

• Trace anomaly on the lattice à ! θ(T ) ∂β(a) = 6 N4 · (hS i − hS i ) T 4 t ∂ log a P T P 0

20 Trace Anomaly from monopoles

diag off I Fix Maximal Abelian gauge Dµ Aµ = 0 diag I Define particular singular gluon objects (monopoles) kµ = ∂νF˜µν I Determine the monopole action by inverse Monte Carlo algorithm [Shiba, Suzuki ’95] I Partition function and the Trace of energy–momentum tensor:

Z = ZmonZrest , θ = θmon + θrest 1 3 5 I Monopole partition function:   6 X  X Xn  Zmon = exp − f (β)Smon(k) 2 4  i i  monopoles,k x i=1 7

I Contribution of monopoles into the trace anomaly: Ã ! µ ¶ X h i mon 4 ∂β ∂fi(β) mon mon 21 θ = Nt a hSi iT − hSi i0 ∂a i ∂β Are monopoles thermodynamically important? I Trace anomaly of energy momentum tensor pure SU(3) glue contribution from monopoles 10

SUH3L gauge theory 8 æ æ Maximal Abelian gauge ææ 163´4 lattice 6 æ Θ 4 4 æ T æ æ 2 æ æ æ æ æ 0 æ æ

-2 1 2 3 4 5 TTc from [G.Boyd, J.Engels, F.Karsch, E.Laermann, Yes, there is a contribution C.Legeland, M.L¨utgemeier,B.Petersson, ’96]

22 Trace Anomaly from vortices

(Ω) 2 I Fix Maximal Center gauge minΩ (Tr Uxµ ) I Define singular string-like gluon objects (vortices) Q with the worldsheet current σP = l∈∂P Zl I Separate all plaquettes into two sets:

i) σP = −1 (belong to the vortices)

ii) σP = +1 (outside the vortices) I Action splits trivially: X X X SP = SP + SP P P ∈vort P 6=vort I Trace anomaly splits as well:

θ = θvort + θoutside 23 Are vortices thermodynamically important? I Trace anomaly of energy momentum tensor pure SU(2) glue contribution from vortices

æ æ outside vortices 4 æ æ æ H95...98L% of volume æ æ 2 æ SUH2L glue, Θ ìììì ì ì ì total ìæ ì 4 ììà ìæ T 0 ìæà à à

-2 à à à à à vortices_occupy à à H2...5L% of volume -4 1.0 1.5 2.0 2.5 3.0 TTc from [J.Engels, J.Fingberg, K.Redlich, Yes, there is a contribution H.Satz, and M.Weber, ’88] Negative sign: [Gorsky, Zakharov ’07] 24 Discussion/Conclusion

I String-like and monopole-like magnetic gluonic configurations must be present as thermal excitations in the YM plasma.

I Evolution of the magnetic component of the YM vacuum:

I Strong contribution of magnetic component to the trace anomaly, and, consequently, to the equation of state of the Yang-Mills plasma. 25 26 SU(2) chains SU(3) nets 27 28 Phase structure of QCD

early universe Various phases: ALICE quark-gluon plasma • quark-gluon phase RHIC <ψψ> ∼ 0 Tc ~ 170 MeV crossover • hadron phase T SPS quark matter • superconducting phases

<ψψ> > 0 Low-µ structure confirmed crossover hadronic fluid superfluid/superconducting in lattice simulations phases ? n = 0 n > 0 2SC <ψψ> > 0 BB CFL [Fodor & Katz (2002),...] vacuum nuclear matter neutron star cores µ ∼ Reviews: arXiv:0711.0661, o 922 MeV µ arXiv:0711.0656, arXiv:0711.0336

May be observable in heavy-ion collision experiments!

29 Order of the µ = 0 transition

N f = 2 Pure ∞ Gauge Pure gauge: 2nd order O(4) ? 1st • N = 0 or m = ∞ 2nd order f u,d,s Z(2) • weak 1st order phase transition tric m s N f = 3 • order parameter: Polyakov loop ? phys. crossover N f = 1 Two light quarks: point m ? s • Nf = 2, mu,d = 0, ms = ∞ 2nd order Z(2) • 2nd order phase transition 1st • order parameter: chiral condensate 0 0 ∞ m u , m d

In pure gauge case Tc ≈ 265(1) MeV [from lattice simulations]

30 Center vortex mechanism

T < Tc Vortices are percolating = Vortices are condensed x x = Center disorder

t T > Tc No percolation t y y confined phase deconfined phase • Interaction with quarks: Aharonov–Bohm effect “magnetic (center) flux links with particle (quark) x trajectories” Percolation transition [Engelhardt, Langfeld, Reinhardt, Tennert ’99]

Aharonov-Bohm mechanism [Polikarpov, Veselov, Zubkov, Ch. ’98]

31 Percolation vs. condensation

General structure of ensembles: many small (UV) clusters plus one big (IR) cluster

ρ = ρUV + ρIR

Percolation: probability to find two points x and y separated by the distance R and connected by any trajectory C

P P x,y C δC(x)δC(y)δ(|x − y| − R) P (R) = P P ) x,y C δ(|x − y| − R)

Condensation: P (R) ' P∞ + P0 exp{−µR} with P∞ > 0 32 Dimensional Reduction at T À Tc

I Non-perturbative magnetodynamics: 3D YM with the coupling 2 2 g3d(T ) = g4d(T ) · T ∼ T/ log T

2 I All dimensional quantities are expressed in terms of g3d(T ) only. I The monopole density is à !3 6 T ρ(T ) = Cρ g3d(T ) ∝ T À Tc log T/ΛQCD found also in [Giovannangeli, Korthals Altes, ’05] I Reproduced by T -dependent chemical potential

µ ∼ 3T log log T/Λ

33