PoS(Lattice 2011)326 http://pos.sissa.it/ mportant in the deconfinement ortices. The equation of state of the ices after the phase transition. attice 2011 Sciences Unit, -cho, Kochi, 780-8520, Japan e Commons Attribution-NonCommercial-ShareAlike Licence. ∗ [email protected] Speaker. We show the evidence that magneticphase degrees by of a freedom lattice numerical are simulation sogluon in plasma i magnetic has center a large v contribution obtained from the vort ∗ Copyright owned by the author(s) under the terms of the Creativ c

Takuya Saito Research and Education Faculty, Natural Sciences Cluster, The center magnetic vortex and itsphysical influence quantities on in the plasma Integrated Information Center, Kochi University, Akebono E-mail: July 10-16, 2011 Squaw Valley, Lake Tahoe, California The XXIX International Symposium on Lattice Theory - L PoS(Lattice 2011)326 (2.1) riables, ter vortex singularity Takuya Saito and the change y, the magnetic nishes and then r that the system role for a spatial hase give also a oreover, a spatial unsolved problem enter vortex on the a weakly-coupling e importance of the equired an infrared such strong interac- ence of the magnetic m the assumption that the many phenomeno- lue energy-momentum deconfing (because this t of view of the topolog- e hot vortices is responsible cosity, which is observed to be [6, 7]). The center vortex mech- , 2 c  T ) x , the magnetic and/or spatial compo- ( c µ T U  r T x ∑ t ) of the transition from the hadron to the - 2 c 1 T VN ( = are freely moving, so that the conventional perturbation c R T quenched-lattice calculation. Removing center vortices from ) 2 ( SU Maximize We here use direct maximal center projection (MCP) [12], so as to identify c We here study the equation of state (EOS) of the gluon plasma in terms of the cen The heavy-ion collision experiment (RHIC and LHC) produces the new matte In many cases of the non-Abelian above One of the interesting ideas to describe such a peculiar sQGP is to focus on a degrees of freedom by using an 2. Formulations 2.1 Center vortex projection the original lattice enables usof to the investigate correlators the of the effect transport ofnearly coefficient, them zero, particular meaning on for that shear the QGP vis gluon is EOS, almost perfect fluid. anism also holds well overtensor hot is Yang-Mills largely theory[8]; affected [9]. as Those afor non-vanishing result, contribution the from magnetic the th singularity hot-g of thethermal gluon monopole (connected [10]. by In vortices) the has similar been way, discussed th [11]. nents are linked to some (infrared-singular)ical confining defect objects. on the In lattice, the the poin (magnetic) confinement, magnetic while monopole the and/or temporal vortex (electric) degrees playis of an freedom confirmed important is by the behavior of the time-like Polyakov line near Vortices in the gluon plasma 1. Introduction lattice. This method is implemented by realizing the following condition to updated link va coming from the magneticgluon degrees of in freedom thethe [1]. zero-momentum perturbation limit completely In diverges fails. if theregulator. Therefore, its thermal the The effective QCD lattice non-zero mass theor results magnetic va Wilson-loop show mass that and is the r a magnetic color-Coulomb massnon-zero instantaneous is string potentials not tension, in zero which the [2, ismass deconfining 3]. [4, in p 5]. M proportional to the temperature depend temperature exceeds the critical temperature theory is valid in the deconfinement phase. However, it is found, through logical analyses and lattice computations, thatplasma the discovered but QGP strongly-interacting matter is QGP not (sQGP)what mechanism matters. in quantum Therefore, chromodynamics it (QCD)tions. in is the still hot an phase causes the deconfining and above gluon plasma (QGP) phases. At first, a study of the QGP physics starts fro PoS(Lattice 2011)326 (2.6) (2.4) (2.3) (2.5) (2.2) gauge quette ) laquette 2 SB). The ( χ carries non- Takuya Saito rapidly goes SU µ S Z is plus, while if ∆ ensation are lost lattice configura- ) x pectation values of ( eaking ( µ 30), surements calculated . links. Z finement phase transi- 2 = quently prepare new con- β vortex pierces in this square . 2 Z , ) t ) , , ∈ 0 . S x )  for the vortex removal and normal lattice ( ∆ x S ) ′ 0 ( x β β ( µ U t − d µ 0 U N T 0 β = µ U = ∏ t β  t Z 3 S Z Tr ( = 1 2 as 4 )= t 0 x sgnTr β β N β (

′ µ 4 )= = p x can be defined as U T )= S ( x ) ∆ P ( 2 ( µ Z SU values of a function of S to the center projected link as [13] ∆ are the expectation values on the anti-symmetric and symmetric lattices ) x > ( µ 0 and Z S ) < a t N ( and / 1 > = T S T Moreover, in order to investigate the critical phenomena one can use the ex The equation of state of the gluon plasma on the lattice can be calculated via the pla We here carry out the Monte Carlo simulation with standard plaquette action of In addition, one can remove the center vortex contribution from the original In Fig.1 we plot the < [14, 15]. where This enables us to obtain a new non-confing theory consisting of these new and configurations. For the normal ones, after the phase transition (above perturbative information upon thelattice color simulations confinement show and that chiralafter non-zero the string br center removal tension [12, as 13]. well as the2.2 chiral Measurements cond the Polyakov line which is defined by Vortices in the gluon plasma so that a center field function on The plaquette value forming from four links is plus if all the corresponding This is usually considered astion. an order parameter on the confinement/decon 3. Numerical results the plaquette includes a minus linkis an a odd non-trivial times then negative it energyregion becomes of also gluon minus. fields; This therefore, minus p the center expectation value with the lattice coupling theory. We apply the MCP to allfigurations gauge after configurations the updated, removal and of subse centerby vortices, so the as normal to lattice compare ones. with the mea tion by multiplying PoS(Lattice 2011)326 lattice en well ) 2.9 lynomial N ( Takuya Saito s. This lattice 2.8 SU 2.30-2.90 0) the remnant plotting on the tric modes are ory. However, . ylakov line as a e dependence or 3 'R12' using 1:2:3 gluon plasma by β lowly as the beta 2.7 − quation of state. , the electric gluons es vortex degrees of 0 c . ents still are confing. T 2 ns of contributions from the eaking. We calculated 2.6 nfining) modes are also fter the center removal) ar = rom magnetic vortices. ger contributions than the c T 2.5 / T 2.4 2.3 2.2 6 4 2 0 -2 -4 calculated from the normal and center vortex 4 T / 4 p 2.9 6 which is approximately . becomes a minus; this contributes negative to EOS. There- 2 S 2.8 = 2.30-2.90 ∆ becomes larger than the normal. This means that in addition to β 'N12' using 1:2:3 S 3. The dashed-lines are numerically determined by using a po . In the confinement phase the Poylakov line becomes non-zero 2.7 . ∆ 2 β on the removal (of center vortices) and normal configuration ≈ 2.6 regions, c β β β 2.5 2.4 goes higher it asymptotically approaches zero axis. This behavior has be lattice gauge simulations. The center vortex is a of ) β 2.3 2 as a function of ( 4 and the critical S ∆ × SU 2.2 0 1 3 0.8 0.6 0.4 0.2 -0.2 -0.4 We have investigated the effect of the center magnetic vortices in the quark- Besides, in the higher In Fig.2 we show a graph of the pressures In Fig.3 we plot the numerical data of the expectation values of the electric-Po . β using the are also affected strongly due to the magnetic degrees freedom resulting f 4. Summary theory and is responsible forthe the equation color of confinement state and of chiral QGP symmetry after br the phase transition and found very large of after removing center vorticescompletely seems free to objects for vanishes. such temperature regions; This on may the other means side, that ne the elec the both data betweenincreases. the At high removal beta and (about over normal lattices approach each other s if removing vortices from the confining lattice; namely, it is a non-confining the size is 16 Fig. 1. One seesnormal that one. the This equation means of thatare state the important of magnetic modes center-vortex-free even degrees has in of the lar freedom deconfining (released phase. a known by a lot offreedom the from previous the original calculations. lattice, On the other hand, if one remov the electric degrees of freedom afterreleased; the note phase that transition, even the in magneticThus the (co the center-removing deconfining lattice phase, configuration the yields large spatial contribution gluon to compon the e function of the lattice coupling Figure 1: Vortices in the gluon plasma up and as the removal configurations. This has been done by integrating the fitting functio fore one needs to do an extensive lattice simulation to investigate such volume siz higher temperature limit. PoS(Lattice 2011)326 r vortices xpectation Takuya Saito high temperature as to obtain a final luon propagators in P. ingular objects even in upon the normal and vortex removed β 2.7 2.8 2.9 x4 3 Normal Removal L Normal; L=12 Removal; L=12 Normal; L=16 Removal; L=16 upon the normal and vortex removed configu- 2.5 2.6 β 2.7 2.8 2.9 3 β β 5 ) versus 4 T 2.5 2.6 / p ~ 2.3 c β 1.8 1.9 2 2.1 2.2 2.3 2.4 2.2 2.3 2.4 1 1 0 2 0

2.5 1.5 0.5 0.8 0.6

0.4 0.2

-0.2 p/T

4 ; it is found that from the confining phase to the deconfining phase, cente β The expectation values of the electric Polyakov line vs. The comparison of the pressures ( Our calculations presented here are preliminary numerical results, and so (magnetic modes) play an important role for understanding the physics of QG center vortices. This means that the magnetic degrees of freedom are still s configurations. the deconfining phase. This phenomenonthe emerges thermal as a medium. magnetic The mass similar ofvalues behavior versus the is g also given by plotting the Poylakov line e rations. Figure 3: Figure 2: Vortices in the gluon plasma conclusion one needs to doregions. more extensive simulations on the larger lattice and PoS(Lattice 2011)326 Takuya Saito , 094504 (2006). 73 er, Phys. Rev. D . Zakharov, Phys. Rev. D83,114501,2011. 506; A. Nakamura, I. Pushkina, T. Saito, 3-188. 3-743. b, Atsushi Nakamura, V.I. Zakharov, aya, N. Ukita, T. Umeda (WHOT-QCD 82 02,2007 ik, Phys.Rev.D58,094501,1998 8; J. Liao, E. Shuryak, J. Phys. , 189 (2006). 6 115 S. Sakai, Phys.Lett. B549 (2002) 133-138 Collaboration), Phys.Rev.D81:091501,2010. Phys.Rev.D78,074021,2008. G35,104058,2008 [3] Y. Maezawa, S. Aoki, S. Ejiri, T. Hatsuda, N. Ishii, K. Kan [1] A.D. Linde, Phys. Lett. B96, 289. [2] A. Nakamura, T. Saito, S. Sakai, Phys.Rev. D69 (2004) 014 [4] A. Nakamura and T. Saito, Prog. Theor. Phys. [5] Y. Nakagawa, A. Nakamura, T. Saito, H. Toki and D. Zwanzig [6] A. Nakamura and T. Saito, Prog.Theor.Phys. 112 (2004) 18 [7] A. Nakamura and T. Saito, Prog.Theor.Phys. 111[8] (2004) 73 M.N. Chernodub, V.I. Zakharov, Phys. Rev. Lett.[9] 98,0820 in equation of state of Yang-Mills plasma" M.N. Chernodu Vortices in the gluon plasma References [10] M.N. Chernodub, Y. Nakagawa, A. Nakamura, T. Saito, V. I [11] J. Liao and E. Shuryak, Phys.Rev.Lett.101,162302,200 [12] L. Del Debbio, M. Faber, J. Giedt, J. Greensite, S. Olejn [13] Ph. de Forcrand, M. D’Elia, Phys.[14] Rev. Lett. 82 G. (1999) Boyd, 45 et al., Nucl. Phys.[15] B469 (1996) M. 410-444. Okamoto, et al., Phys. Rev. D V60 (1999) 094510.