PoS(Confinement8)037 http://pos.sissa.it/ ∗ † [email protected] [email protected] [email protected] his work was supported by Grants-in-Aid for Scientific Research from "The Ministry of Education, Culture, Speaker. T Condensation of the Abelian monopoles andlow the temperature phase center of vortices Yang-Mills theory. leads We to stressfreedom confinement that are these of topological also color magnetic very in degrees important of inphase the transition deconfinement both regime: the at monopoles thetributing, and point in the of vortices particular, the are to deconfinement released thegluonic into equation medium. the of thermal state Thus, vacuum we and, con- argue definitely,other that to hand, a transport novel, it properties magnetic was of component demonstrated the playsbeginning that hot a with an high crucial effective temperatures, role. three-dimensional down On description tosystem the can the of be critical 3d temperature brought, by Higgs postulating fields.projection existence of of We the a propose 4d magnetic to vortices. identify Suchand the identification contributes fits 3d to well interpretation color-singlet the of 3d Higgs the properties field magnetic of component the with theory of the the 3d Yang-Mills plasma. † ∗ ang-Mills plasma Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. aboratoire de Mathematiques et Physique Theorique CNRS UMR 6083, Fédération Denis c

Sports, Science and Technology ofJapan, Japan" No. Nos. S-08035, by 17340080 the STINT and Institutionalof 20340055, grant Industry, IG2004-2 JSPS Science 025, Invitation and and Fellowship by the Technology forsupercomputer Federal No. at Research Program RCNP of in 40.052.1.1.1112. the at Russian Osaka Ministry The University, and numerical a simulations SR11000 machine were at performed Hiroshima using University. a SX-8 M. N. Chernodub L Abelian monopoles and center vortices in Y 8th Conference Quark Confinement and theSeptember Hadron 1-6 Spectrum 2008 Mainz, Germany Atsushi Nakamura Research Institute for Information Science andHigashi-Hiroshima, Education, 739-8527, Hiroshima Japan University, E-mail: V. I. Zakharov Institute of Theoretical and Experimental PhysicsMax-Planck ITEP, Institut 117259 für Moscow, Russia Physik, Föhringer RingE-mail: 6, 80805, München, Germany Poisson, Université de Tours, Parc de Grandmont,Institute F37200, of Tours, Theoretical France and Experimental PhysicsE-mail: ITEP, 117259 Moscow, Russia PoS(Confinement8)037 . N. Chernodub M ) gauge theories the c gauge theory. 2) ( SU , the transport properties of the plasma c T 2 part of a monopole-vortex chain in A : The liquid exists in the “strongly coupled” region in the deconfinement. c Figure 1: : The monopoles form a condensate in the confinement phase. c 2T T : As temperature increases the monopole liquid gradually evaporates into a gas. : The point of a gas–liquid crossover. . c c : The condensate melts into a monopole liquid at the phase transition. < c T 2T T 2T T < = ≈ & < c gauge theory the center vortex can be regarded as an Abelian vortex carrying the magnetic oth experimental observations at RHIC and numerical simulations of Yang-Mills theories 0 T T T T A general phase diagram of the monopole component is suggested to be as follows [2]: B Below we further discuss the picture [2, 3, 4, 5, 6, 7] according to which the unusual proper- The monopoles and the vortices are parts of a genuine non-Abelian object. For example, in There are two constituents of magnetic component [2]: a particlelike magnetic monopole and • • • • • (2) Abelian monopoles and center vortices in Yang-Mills plasma 1. Introduction indicate that the Yang–Mills[1]. plasma At possesses temperatures quite just unusual above the properties, critical for temperature a review see, e.g., correspond to an (ideal) fluid ratherof than the to gluon a plasma weakly lies interacting in a gas.explanation strong A of collective reason interaction this for of phenomena the the is gluons. liquid still nature A lacking self-consistent and theoretical the topic attracts great interest nowadays. The intermediate monopole liquid statethe was monopole also gas at discussed very in high Refs. temperatures [2, was predicted 4] in while Refs. the [12, formation 13]. of a stringlike magnetic vortex. Thesefields. constituents The appear magnetic as monopoles singular areconductor magnetic related defects mechanism to in [8]. the the color gluon In confinementpercolation via of the the the vortex vortices so picture [9]. called the dual quark super- confinement emerges asSU a result of a flux which is equalmagnetic to part a of half the ofis gluon the a energy total source density of magnetic around two fluxconservation vortex a of of fluxes the monopole a which vortex is must flux monopole. not be [10].which connected spherical: connect As The to a alternating distribution each other result, monopoles of there anti-monopole(s) monopole andmonopoles appears because the antimonopoles, a and of Figure closed vortices a 1. set form In of nets SU(N the(non-)supersymmetric [3]. non-Abelian vortex segments gauge Similar theories involving monopole-vortex various chains Higgs were fields [11]. found in numerous ties of the plasma appearAccording as to a this result picture of monopolelikeunusual a and thermodynamical (topological) vortexlike and magnetic topological transport component defects properties of arecomponent of the responsible of the gluon the for plasma. plasma plasma. the can A be brief found in review Ref. of [3]. the magnetic PoS(Confinement8)037 : p, µν 2.2) T (2.1) (2.3) ( (in units . T ) . 3 ) (T . N. Chernodub ε ... . c M T  + ) + 2 σρ g T/T 1 G p(T b cts σρ 2 + G 0 ) = as follows: µν (b anomaly from vortices from anomaly . For example, the pressure 2 δ θ s(T µ 1 4 A −g − = , νσ of the energy–momentum tensor 1 g ) µ G θ g (T lo µσ lo 0.2 0.0

θ reg -0.2 -0.4 d

dg G T 

) +

vort = 4 2Tr p(T 3 g (g) 3 = β 3) parts. In Figure 2 (right) we show the contribution c 2, ≡ µν ) = T 4, and the scaling properties of our results have not been 1, 3.0 (T (g) × = T/T ˜ ε 3 j β magnetic full electric , i, 305(8)MeV can be calculated from the trace , (in units of temperature) as a function of the temperature p = 2.5 c s 2 i j 3 T θ , , ) − E 1 4 ε ) 1 2.0 (T T (x ). The full anomaly (circles), and its electric (squares) and magnetic (triangles) θ c i ≡ µν 2 T 1 µ µ 1 T G T d hT 1.5 is the field strength tensor of the gluon fields , and entropy T a ε t Z ) = (g)Tr 4 ˜ µν a β T Yang–Mills theory is a conformal theory and therefore at the classical level the (T 3) and magnetic (G 1.0 D G θ 2, = = ) = 1.00 0.75 0.50 0.25 0.00 1, -0.25 ( bare θ µν = p(T i G , Thus, the vortex constituents of the monopole-vortex chains are relevant for the thermody- The This Section is based on results of Ref. [5]. Basic thermodynamical quantities of Yang-Mills 2 4i G contributions are shown. (right) The contribution of the magneticof vortices into the the magnetic trace vortices anomaly. intojust the above trace the anomaly phase of transition thetakes (i.e., gluon a in negative plasma. the value in monopole In agreement liquid the with region) deconfinement general the phase theoretical vortex-generated expectations anomaly [6]. namics. The monopole constituentsan are excess of also the important (magnetic forcontributes part to thermodynamics the of) [3, trace non-Abelian 5] anomaly action (2.3)Yang-Mills as density fields and, they (2.2). [14]. consequently, to carry the The pressure action and density, energy in density turn, of the where In Figure 2 (left) we show theperformed trace for anomaly one (2.3) lattice for geometry, SU(2) 18 gaugestudied theory. yet. The measurements The were trace ofto the energy-momentum tensor is subdivided into its electric (proportional Figure 2: the energy density of the critical temperature, Abelian monopoles and center vortices in Yang-Mills plasma 2. Thermodynamics of Yang-Mills theories and topological obje left) The trace anomaly energy–momentum tensor is traceless.energy–momentum tensor exhibits a However, trace because anomaly, of a dimensional transmutation the theory can be found from the expectation value of the trace PoS(Confinement8)037 (3.3) (3.2) (3.1) Ω(x) 0. i 6= . N. Chernodub – is the global hL Ω, serves as an i M r hL T 1 2 ≡ , 0. The success of the L 2 a ), 2 Π 2 Ω| Σ with hΣi 6= 3 , c . Namely, the surfaces become i c (|Tr i T + hL V 2 x) ) , 2 a + τ 2 ( (Π 0 2 Ω| A i c τ invariance of the Lagrangian and, second, D d + † 2 4 Z Σ |Ω 1/T 1 r 0 c T Z –symmetric lagrangian looks as [16] : 4 2 2 2 + − Z g T 2 a h Ω. To this end one introduces color triplet and color Π + 2 j b exp 2 i, P G + r 2 T Σ ≡ 1 1 2 b are organized in such a way that Σ. The Higgs-fields Lagrangian is then the standard kinetic terms can produce a non-trivial vacuum expectation value, ) = Ω(x) ) 1,2,3 a Ω) = 2 c , i and Π , Ω| a and do not percolate any longer from the 4d point of view. However, (A 1,2 (Σ, c Π b eff T V (|Tr L depends only on 3d variables and we are invited to consider 3d reduced, or V > T Ω(x) is the Euclidean time. The vacuum expectation value, τ n this Section we briefly compare the 4d, lattice-based picture of the magnetic component [2, We propose to identify the 3d color-singlet with the 3d projection of the magnetic vortices The vortices are 2d surfaces percolating at low temperatures through the vacuum. The 3d The Lagrangian (3.2) is nonrenormalizable in 3d. A renormalizable version of the effective Note that I It is quite common nowadays to assume that the dynamics of the Polyakov’s lines plays symmetry (for definiteness we consider SU(2) gauge group). 2 time-oriented at 3d effective theory (3.3)theory is [17] and impressive in both the clarity in of the terms underlying of symmetry-based argumentation its for(or, its numerical introduction. of match the to 4d theidentification magnetic allows original component at 4d least of relate the themagnetic plasma) newly component. made onto observations the to 3d the known space. properties of We the suggest 4d projection that of such the vortices an isare given by 1d intersections defects, of or the lines.were 2d These surfaces studied lines, with in or the detail trajectories, 3d [18]. areprojection time closed. depend slice It crucially The which was on properties shown the of that temperature these properties in 1d of the defects the vicinity of 2d and, consequently, of their 1d where the coefficients singlet 3d scalar fields plus the potential energy [17]: Lagrangian was suggested in Ref. [17]. Theelements idea inherent can be to represented as the follows.spontaneous construction There breaking are (3.2). of two the basic symmetry First, dueof to the local potential. fields, Both rather elements than can non-local be realized objects in terms where the potential Z effective theories. A particular form of such a order parameter, which is vanishing in theThe confinement symmetry phase – and which is is finite in violated the by deconfining a phase. nonvanishing vacuum expectation value Abelian monopoles and center vortices in Yang-Mills plasma 3. Effective 3d models of magnetic component where 5] with newly developedapproaches, 3d originally models developed of in theemphasize absolutely rather plasma. independent the ways. general picture We emerging The find rather presentation than a details. is close aimed relationa to crucial between role the in the two confinement-deconfinement phase transition [15, 16, 17] PoS(Confinement8)037 AIP (2000) (1999) PoS 101 73 (2000) B489 ects. Using (2006) 025011 . N. Chernodub D 61 M D74 (1999) 019 Phys. Lett. (1998) 30 Phys. Rev. Lett. 9903 Phys. Rev. 80 Phys. Rev. JHEP (2007) 1139 [hep-th/0703267]; (2008) 074021 [arXiv:0807.5012 Nucl. Phys. Proc. Suppl. 79 D78 (2003) 1 [hep-lat/0301023]. (1993) 176; M. N. Chernodub and 51 (2008) 125014 [arXiv:0801.1566 [hep-ph]]. (2003) 027 [hep-lat/0204003]. 30 (2001) 203 [hep-ph/0102022]; C. P. Korthals Phys. Rev. Lett. (2007) 082002 [hep-ph/0611228] and in 98 D77 0309 (2008) 241 [arXiv:0711.1266 [hep-lat]]; Phys. Rev. B608 (2005) 30 [nucl-th/0405013]; (2000) 033 [hep-lat/9907021]; V. I. Zakharov, Rev. Mod. Phys. 5 JHEP (2008) 045017, [arXiv:0707.1284 [hep-th]]. (2007) 276 [hep-ph/0612191]. B 799 0002 A 750 Phys.Rev. 168 D77 (2007) 054907, [hep-ph/0611131]; Nucl. Phys. JHEP Phys. Rev. Lett. Prog. Part. Nucl. Phys. C 75 (2005) 64 [hep-ph/0405066]. Nucl. Phys. (1978) 477; A. Vuorinen, L.G. Yaffe, Nucl. Phys. Proc. Suppl. Nucl. Phys. Phys. Rev. B72 A 750 Phys. Rev. we understand now a 3d field corresponding to the cluster of the mentioned “Confinement, duality, and nonperturbative aspects of QCD”, p. 387, ed. by P. M (2005) 182 [hep-ph/0501011]. Phys. Lett. φ Prog. Theor. Phys. Suppl. Nucl. Phys. (2007) [arXiv:0710.2547 [hep-lat]]. 756 174 “Minneapolis 2006, Continuous advances in QCD”, p. 266 [hep-ph/0607154]. 0, where by [hep-lat/9903023]; M. Engelhardt, K. Langfeld, H. Reinhardt, O. Tennert, 575 [hep-lat/9809158]; R. Bertle, M. Faber, J. Greensite, S. Olejnik, 054504 [hep-lat/9904004]; J. Gattnar, K. Langfeld, A. Schafke, H. Reinhardt, 251 [hep-lat/0005016]. D. Tong, arXiv:0809.5060 [hep-th]. Conf. Proc. Altes, in [hep-ph/0604100]. van Baal, Plenum Press, 1997 [hep-th/9710205]. [hep-lat/9706007]; M. N. Chernodub, A. D’Alessandro, M. D’Elia and V.I. Zakharov, in progress. LAT2007 PoS(Confinement8)127 [arXiv:0812.1867 [hep-lat]]. M. I. Polikarpov, in (2008) 162302, [arXiv:0804.0255 [hep-ph]]. [hep-lat]]; M.N. Chernodub, K. Ishiguro, A. Nakamura, T. Sekido, T. Suzuki, V.I. Zakharov, “Nagoya 2006, The origin of mass and strong coupling gauge theories”, p. 80 [hep-ph/0702245]. E. V. Shuryak, i 6= M [6] A. Gorsky and V.I. Zakharov, [7] A. D’Alessandro and M. D’Elia, [9] For a review see, e.g., J. Greensite, [8] For reviews see, e.g., T. Suzuki, [5] M.N. Chernodub, A. Nakamura and V.I. Zakharov, [3] M. N. Chernodub and V. I.[4] Zakharov, Phys.Atom.Nucl J. (2009), Liao [arXiv:0806.2874 and [hep-ph]]. E. Shuryak, [2] M. N. Chernodub and V. I. Zakharov, [1] M. Gyulassy and L. McLerran, φ [17] Ph. de Forcrand, A. Kurkela, A. Vuorinen, [18] M. N. Chernodub, M. I. Polikarpov, A. I. Veselov, M. A. Zubkov, [12] P. Giovannangeli, C. P. Korthals Altes, [11] For reviews see, e.g., M. Shifman and A. Yung, [16] R. D. Pisarski, [10] J. Ambjorn, J. Giedt and J. Greensite, [13] M. N. Chernodub, K. Ishiguro[14] and T. Suzuki, B. L. G. Bakker, M. N. Chernodub and M. I. Polikarpov, [15] A. M. Polyakov, Abelian monopoles and center vortices in Yang-Mills plasma as is emphasized in [18] the 3d percolation continues, now in terms of the 1d def 1d magnetic defects. References the percolation theory oneh can readily argue, that the lattice observations imply the inequality