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Gluon Propagators and Center Vortices in Gluon Plasma

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Gluon Propagators and Center Vortices in Gluon Plasma PHYSICAL REVIEW D 83, 114501 (2011) Gluon propagators and center vortices in gluon plasma M. N. Chernodub,1,2,* Y. Nakagawa,3 A. Nakamura,4 T. Saito,5 and V.I. Zakharov6,7 1CNRS, Laboratoire de Mathe´matiques et Physique The´orique, Universite´ Franc¸ois-Rabelais, Fe´de´ration Denis Poisson—CNRS, Parc de Grandmont, Universite´ de Tours, 37200 France 2Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium 3Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan 4Research Institute for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8521, Japan 5Integrated Information Center, Kochi University, Kochi, 780–8520, Japan 6Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia 7Max-Planck Institut fu¨r Physik, Fo¨hringer Ring 6, 80805, Mu¨nchen, Germany (Received 30 March 2011; published 1 June 2011) We study electric and magnetic components of the gluon propagators in quark-gluon plasma in terms of center vortices by using a quenched simulation of SUð2Þ lattice theory. In the Landau gauge, the magnetic components of the propagators are strongly affected in the infrared region by removal of the center vortices, while the electric components are almost unchanged by this procedure. In the Coulomb gauge, the time-time correlators, including an instantaneous interaction, also have an essential contribution from the center vortices. As a result, one finds that magnetic degrees of freedom in the infrared region couple strongly to the center vortices in the deconfinement phase. DOI: 10.1103/PhysRevD.83.114501 PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Gc, 12.39.Mk I. INTRODUCTION in the temporal direction—give a nonconfining screened potential of the Debye type, with a finite electric mass The Relativistic Heavy-Ion Collider (RHIC) at gðTÞT [17–20]. Brookhaven National Laboratory produces a new state of In addition, from the viewpoint of the Gribov-Zwanziger matter which may exceed the critical temperature Tc of the (GZ) confinement scenario [21–23], a color-Coulomb phase transition from the hadron phase to the quark-gluon instantaneous interaction between a quark and an antiquark plasma (QGP) phase. Many phenomenological studies and provides—even in the nonconfined QGP phase—a confin- lattice computations suggest that the QGP is a strongly ing potential which rises linearly as the function of the interacting plasma [1], for which we cannot apply the early quark-antiquark separation [24–27]. As a consequence, the arguments based on the perturbative approach with a small thermal string tensions obtained from the spatial-Wilson coupling constant. Furthermore, the recent Pb-Pb heavy- and the color-Coulomb potentials are nonzero. They de- ion collision experiment at the LHC has created QGP pend on the temperature and obey a magnetic scaling matter at even higher temperatures: This shows us an law [g2ðTÞT]. Extending this line of considerations, obvious jet-quenching event [2] and a larger elliptic flow Zwanziger has approximately reconstructed the equation [3] compared to the RHIC’s Au-Au collision. Therefore, it of state of QGP using the Gribov-type dispersion relation is indispensable for us to explore the mechanism which for the massive gluons [28]. drives such strong interactions using a nonperturbative These interesting aspects of the non-Abelian gauge first-principles approach in lattice simulations. theory may be related to center (magnetic) vortices—i.e., One of the most important ideas to describe a strongly to the topological defects associated with the nontrivial ½ ð Þ Zð Þ Zð Þ interacting QGP (sQGP) is to focus on an infrared singu- homotopy group 1 SU N = N N —which are larity arising from magnetic degrees of freedom [4,5]. The responsible for certain nonperturbative phenomena of magnetic component of the gluon propagator is fully inac- QCD. One can identify the center vortices on the lattice cessible by the perturbative calculation, but its infrared using a numerical technique [29] and also remove these divergence may cause an emergence of a nonperturbative vortices from the original gauge fields [30]. It turns out that magnetic mass that plays a cutoff role and can cure thermal the removal of the center vortices destroys the color con- QCD in the infrared region. The lattice simulations [6–8] finement property and restores the chiral symmetry. prove that the magnetic gluons have a nonvanishing mass Moreover, the lattice center-vortex density exhibits a scal- at finite temperature. Furthermore, it is well known that a ing consistent with the asymptotic freedom [31]. spatial Euclidean Wilson loop (which is not extended to the In terms of the vortex degrees of freedom, the QCD temporal dimension) bears a confining potential above Tc deconfinement phase transition can be considered as a [9–17], while the correlators of a Polyakov line—wrapped depercolation transition of the vortex lines in the direction of the Euclidean time [32]. As a result, we can naturally *On leave from Institute of Theoretical and Experimental understand the survival of the spatial confinement above Tc Physics, Moscow, Russia. because the center vortices remain intact in the spatial 1550-7998=2011=83(11)=114501(6) 114501-1 Ó 2011 American Physical Society CHERNODUB et al. PHYSICAL REVIEW D 83, 114501 (2011) space. Moreover, a typical center-vortex configuration is This propagator corresponds to the one in Eq. (2) 0 ¼ 00 located at the Gribov horizon in the gauge space. Thus, the at t t . Note that there is no q0 dependence in Eq. (2) removal of the center vortices results in the dilution of the and that the q0 ¼ 0 term is removed from the sum. ð Þ¼ ð ðqÞÞ lowest eigenvalues of the Faddeev-Popov operator. These An equal-timepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi propagator reads D q 1= 2! , eigenvalues—according to the GZ confinement scenario— where ! ¼ q2 þ m2 is the dispersion relation. cause confinement of color [25,33]. In the finite-temperature system, the electric and mag- Recently, three of us have argued that the center-vortex netic gluons have different effects due to breaking of the mechanism is also important in the hot phase of the Yang- Euclidean Lorentz invariance. One can define the spatially Mills theory because the center vortices carry information transverse (PT) and spatially longitudinal (PL) projection about the magnetic degrees of freedom [34,35]. The center operators as follows: vortices are related to Abelian magnetic monopoles, and qiqj the latter are expected to explain some of the interesting 00 ¼ 0i ¼ i0 ij ¼ ij À PT PT PT ;PT 2 ; (5) properties of the quark-gluon plasma as well [36]. qi In this paper we study a connection between the center vortices and the infrared properties of the gluon propaga- q q tors at finite temperature. To this end we study the behavior P ¼ À À P ; (6) L q2 T of the electric and magnetic components of the gluon propagators by removing the vortices from the original with the properties gauge configurations and comparing the result with the ð Þ2 ¼ ð Þ2 ¼ ¼ original one. We use the quenched SUð2Þ lattice simula- PT PT; PL PL;PTPL 0: (7) tions in the Landau and Coulomb gauges. In Sec. II we Both spatially transverse and spatially longitudinal projec- define gluon propagators on the lattice. In Sec. III a nu- tors correspond to the transverse states in momentum merical technique used to make a center projection is space: summarized. Our numerical results are presented in ¼ ¼ Sec. IV, while the last section is devoted to the summary qPT qPL 0: (8) of this work. Using these relations, the gluon propagators at finite tem- perature in a Landau-type gauge can be separated into two II. GLUON PROPAGATORS independent parts: In this study, we work in the SUð2Þ lattice gauge theory. 1 1 The gauge potential A is expressed via the SUð2Þ matrix D ¼ P þ P : (9) G þ q2 T F þ q2 L link variable UðxÞ as follows: X The electric component of the gluon propagator is 1 a ¼ Aðx;tÞ¼ Tr Uðx;tÞ; (1) given by the spatially longitudinal projection De D00, 2 ððq Þ¼ Þ¼ 2 a and the electric mass is given by F ;q0 0 me ðgðTÞTÞ2. The spatially transverse projection gives us the where a are the Pauli matrices. The correlation functions magnetic propagator Dm ¼ Dii. The magnetic mass is of the gauge fields (1) in momentum space are ððq Þ¼ Þ¼ 2 ð 2ð Þ Þ2 expected to be G ;q0 0 mm g T T , where 1 X gðTÞ is a running QCD coupling defined at the scale of D ðq;tÞ¼ hA ðx;t0ÞA ðy;t00ÞieiqðxÀyÞ; (2) temperature T. 3V x;y ð¼ Þ where V NxNyNz is the three-dimensional volume and III. MAXIMAL CENTER PROJECTION t ¼ t0 À t00 is the Euclidean time difference. In Landau-gauge fixing we study the static correlators of We employ a direct maximal center projection (MCP) ¼ gluon fields with q0 0: [29] in order to identify the center vortices on the lattice. The corresponding gauge is defined by the condition 1 X Dðq;q0 ¼ 0Þ¼ Dðq;tÞ; (3) X N 1 2 t t maximize R ¼ Tr½UðxÞ : (10) VNt x where Nt is the lattice size in the Euclidean temporal direction. In the Coulomb gauge, it is more appropriate The center gauge field, to investigate an equal-time gluon propagator in the Z ðxÞ¼sgn Tr½U ðxÞ 2 Z ; (11) following form: 2 X allows us to identify the center vortices. If the center eq 1 iqðxÀyÞ DðqÞ¼ hA ðx;tÞA ðy;tÞie : (4) plaquette is not equal to a trivial element (unity) then a 3VN t x;y;t center vortex goes through this plaquette.
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