Gluon propagators in the quark-gluon plasma
Content of my talk Part1: Lattice calculations of electric and magnetic gluons in the QGP phase by the SU(3) lattice simulations. Part2: Lattice calculations of infrared behavior of gluons in
center vortex mechanism above Tc in the SU(2) lattice simulations. T. Saito (Kochi Univ.)
XQCD2009 Soul 世宗大学 1 Part1: Lattice calculations of electric and magnetic gluons in the QGP phase by the SU(3) lattice simulations
1.Strongly interacting QGP
2.Color-screening effect in the QGP
3.Lattice calculations of color-screening masses
XQCD2009 Soul 世宗大学 2 Quark gluon plasma (QGP)
TT c TT c Hadron ( confinement ) phase QGP ( deconfinement ) phase New matter made by the Heavy ion collision experiments, RICH, 2004 Points: I. QGP is not a weekly interacting gas; non-perturbative method is still important. II. Through many studies (hydrodynamical model, jet-quenching, transport coeffcient, etc.), a picture of strongly interacting quark-gluon plasma (sQGP) has been established. III. sQGP may provide a great hint for understanding of non-abelian Yang-Mills theory. XQCD2009 Soul 世宗大学 3 Gluons at finite temperature (1)
Study of QGP in Lattice QCD : Critical temperature, nature of phase transition, equation of state, etc. have been studied extensively.
Screened gluons Gluons would have finite masses in the QGP phase Heat-bath system breaks Lorentz invariance and favors some rest frame.
Electric gluon Color-Debye screening and Yukawa-type potential among quarks. (imporant for model calculation, J/psi suppression, etc.) Perturbatively, the electric mass is calculable gauge-invariantly
me,0 ~ g(T)T T 0 HTL calculation (Bratten and Pisarsky) T 0 2 231g mmee2 2 mee m,0 1 log O ( g ) 22 mmem,0 eM e,0R m : magnetic mass V (R) ~ m R
XQCD2009 Soul 世宗大学 4 Gluons at finite temperature (2)
Magnetic gluon Magnetic screening for QCD ( do not exist in QED) Perturbatively not calculable ( depend on a gauge parameter) Thermal perturbation theory spoils if the magnetic mass vanishes; therefore we hope it has a non-zero value. Non-zero magnetic masses would have non-perturbative origin. 3D reduction arguments 2 mm ~ g (T )T
Energy hierarchy Simple perturbative picture fails. g(T) O(1) Emergence of energy hierarchy of QGP: Temperature scale 1/ me Electric scale 1/ m Magnetic scale m 1/T T, gT, g 2T,
XQCD2009 Soul 世宗大学 5 Color-screening potentials between two quarks
Polyakov correlators
V (T ) TrLL e T ~ TrL 2
Stochastic quantization with the Lorentz gauge. We have calculated singlet, octet, symmetric and antisymmetric potentials. All the color channels 33 18 are screened at finite T. 33 63 A. Nakamura and T. Saito, PTP. 111, (2004), 733; PTP. 112, (2004), 183
XQCD2009 Soul 世宗大学 6 Electric gluon propagators
Stochastic quantization with the Lorentz gauge.
* D (x,t) A (x,t)A (x,t)
DE (k y , z) ~ Dtt (k y , z)
DE,M (k, z) ~ exp(E(k)z) for large z
XQCD2009 Soul 世宗大学 7 Magnetic gluon propagators
Stochastic quantization with the Lorentz gauge.
DM (k, z) ~ Dxx(k y , z) Dyy (kx , z)
DE,M (k, z) ~ exp(E(k)z) for large z
XQCD2009 Soul 世宗大学 8 Screening masses vs. temperature
T/Tc=1.0~6.0, whose regions might be reached by RHIC or LHC. At most regions, electric mass is larger than magnetic one, and magnetic mass has non-zero value. Scalings work well (black dots line). These change monotonically with the temperature. m m e C g(T) m C g 2 (T) T e T m Dashed line (green) means LOP and Solid (orange) HTL.
A. Nakamura, S. Sakai, and I. Pushkina PLB549,133(2002); PRD69,014506(2004) XQCD2009 Soul 世宗大学 9 Short summary
In the QGP phase, we obtained the color-screened quark potentials (electric part) by using a Polyakov line correlator in each color channel. That gives a base of the previous studies of the QGP. (J/psi suppression, etc.)
The gluon propagators above Tc also were calculated and they give finite screening masses in electric and magnetic parts. Those values changes with the temperature monotonically. It seems that we have no drastic change for gluon dynamics of QGP on the temperature regions T/Tc =1-6 (RHIC and LHC)
Question: From these observations, it is hard to go to the understanding of strongly interacting QGP. Therefore, on the next, we would like to discuss infrared behavior of magnetic gluon propagators more, which may be relevant to the non-perturbative properties surviving above Tc such as the spatial Wilson loop with no-zero string tension, an instantaneous force in the Coulomb gauge, etc.
XQCD2009 Soul 世宗大学 10 Part2: Lattice calculations of infrared behavior of gluons
in center vortex mechanism above Tc in the SU(2) lattice simulations. 1.Spatial Wilson loop in the QGP phase
2.Center vortex mechanism
3.Gluon propagators after removal of center vortices.
For this study, in collaboration with M.N. Chernodub, A. Nakamura, V.I. Zakharov
XQCD2009 Soul 世宗大学 11 Magnetic degree of freedom in the QGP phase Question: How to understand properties of strongly interacting Quark Gluon Plasma. Some lattice calculations (spatial Wilson loop, instantaneous potential, etc. ) show a confining behavior above Tc. Are magnetic degrees of freedom important in the QGP phase?
As one of the approaches, for magnetic plasma made of monopoles (center vortices), there are some references: I. Physics of strongly coupled quark-gluon plasma; Liao and Shuryak, PPNP62(2009)48;PRC75(2007)054907. II. Magnetic component of Yang-Mills Plasma;Chernodub and Zakharov, PRL98(2007)082002. III. Center vortices, the functional Schrodinger equation, and CSB. J.M.Cornwall, arXiv:0812.0359 [hep-ph] IV. Manifestations of magnetic vortices in equation of state of Yang-Mills plasma, M.N.Chernodub, A. Nakamura, V.I. Zakharov, Phys.Rev.D78:074021,2008 V. EOS in the magnetic monopole and center voretex; Chernodub, Ishiguro, Nakamura, Sekido, Suzuki, Zakharov, PoS(LATTICE 2007)174.
However, we have no evidence that such topological objects are directly related to the magnetic degree of freedom of QCD ( or magnetic gluons ).
XQCD2009 Soul 世宗大学 12 Spatial Wilson loop in the QGP
Spatial Wilson loop (gauge invariant) gives a linearly rising potential in the QGP phase. G.S. Bali, et. al, PRL71,3059(1993)
2 W(R,S) ~ exp( s RS) s (T) cg (T)T
T Tc
Fitted by magnetic scaling
XQCD2009 Soul 世宗大学 13 Example of the Coulomb gauge QCD
Instantaneous potential In the Coulomb gauge QCD, one can obtain a linearly rising potential in the QGP phase. The non-vanishing string tension is fitted by magnetic scaling.
Properties of color-Coulomb string tension, Y. Nakagawa, A. Nakamura, TS, H. Toki, D. Zwanziger, PRD73,094504(2006)
XQCD2009 Soul 世宗大学 14 Center vortex We focus on center vortex mechanism, which provides a good numerical tool to understand various non-perturbative characteristics of QCD such as color confinement and chiral symmetry breaking.
Center vortex is topological object and can be defined via center group Z(N). ( originally, by t’Hooft, Mack, Cornwall in 1970’s)
Recently, Debbio, et. al (PRDv58,094501), center vortex degree can be identified using the center projection.
Some important points: I. Center-projected lattice configurations are confining, II. After center removal, no confining linear potential exists. Center vortices play an important role for understanding of confinement.
Here, we investigate gluon propagators for electric and magnetic parts, through the lens of center vortex picture, in order to study physics of the sQGP.
XQCD2009 Soul 世宗大学 15 Maximal center projection
Numerical technique
Direct Maximal Center Projection (MCP) by Debbio, et. al, PRDv58,094501 Greensite, Olejnik, Zwanziger, PRD69, 074506 (2004)
We apply the MCP to all configurations of the SU(2) gauge field
1 2 All the Us I Maximize R Tr U (x,t) VT x,t
Z (x) sgn Tr U (x)
Removing center vortex from the projected configurations
U (x) U (x) Z (x)U (x)
Numerical procedures
Link Update MCP Gauge fixing Measure gluon propagators
XQCD2009 Soul 世宗大学 16 Landau-gauge propagators
T /Tc ~1.40
The projected-lattice configurations reproduce the original gluon propagators completely. After the center vortices were removed, The magnetic propagator display a suppression at low momenta. The electric propagator at low momenta changes slightly.
XQCD2009 Soul 世宗大学 17 Results at high temperature
T /Tc ~ 6.0
At high temperature (may be relevant to LHC experiment), we get the same result. ( Note that the lattice size is very small compared with the simulations
of T/Tc~1.40.) XQCD2009 Soul 世宗大学 18 Summary of this talk
We studied gluon propagators and its screening masses.
I. Those (electric and magnetic) values are finite and they depend on the temperature monotonically, so that there is no drastic change at least up to the LHC temperature regions.
We investigated infrared properties of gluons at finite temperature, through the lens of center vortex picture, which are responsible for a non- perturbative physics.
I. After removal of center vortices, the magnetic gluon propagators change drastically at low momenta and may be relevant to strongly interacting QGP, while the electric gluon does not change significantly.
II. At high temperature (T/Tc=6.0) we obtained the same results as at low temperature. XQCD2009 Soul 世宗大学 19 Buck-up slides
XQCD2009 Soul 世宗大学 20 Example for center vortex removal
Numerical study of Coulomb gauge QCD via center vortex (Greensite, Olejnik, Zwanziger, PRD69, 074506 (2004) )
Gribov-Zwanziger scenario in the Coulomb gauge QCD: Instantaneous interaction (link-link correlator on the lattice ) produces a confining potential even in the QGP phase.
T /Tc ~1.40
Large difference, Polyakov line correlators because this is magnetic.
Small difference,
because this is V(R,0) log(U(R,0)U (0,0)) electric.
XQCD2009 Soul 世宗大学 21 Algorithm 1.) Stochastic Gauge fixing a dA S 1 ab b a a D (A) A d A
Virtual Langevin time ab D (A) Covariant derivative Gauge parameter a Gaussian Random Number
Stochastic Gauge fixing ： XQCD2009 Soul 世宗大学 D.Zwanziger,Nucl.Phys.B192(1981)22 Algorithm ２.) Lattice version
a a U (x, ) (x, )exp(if t )U (x, )(x ˆ, )
a S a Driving force f a A a a Gauge rotation exp(i t / )
Lattice generalization of stochastic gauge fixing：A.Nakamura and M.Mizutani, Vistas in Astronomy(Pergamon Press,1993), vol.37p.305., M.Mizutani and A.Nakamura, Nucl.Phys.B(Proc.Suppl.)34(1994),253.
XQCD2009 Soul 世宗大学 23 ３.) Gribov Copy A (x) 0 Standard gauge fixing can not fix the configuration completely, and then there may exist equal configuration after doing so. → Gribov gauge copies A (x) For example, when catching the copies the propagator is distorted. How to fix a gauge on lattice simulation ? 1. A style of Wilson and Mandula. 2. Stochastic Gauge fixing. This method has a remarkable feature, i.e., gauge configurations are induced to Gribov region. Not complete solution, but more effective method Gribov region ( A 0, F.P. 0) XQCD2009 Soul 世宗大学 24 4.) Speed of this algorithm Advantage: ～ Monte Carlo Quantization～ Gauge fixing procedure of usual Path Integral A Monte Carlo Steps Monte Carlo requires iterative method, and we cannot predict a convergence time. Moreover, for
larger lattice size the performance quickly grows Gauge Transformation worse. On the other hands, Stochastic gauge fixing procedure becomes relatively light work. →gain statistics by less computer resource It is easy to change a gauge parameter. →very convenient to check a gauge invariance ～ Stochastic Quantization ～ Comparing another method, it is not so necessary to take care of Gribov ambiguity.
Disadvantage: Langevin steps Langevin step Δτ dependence
XQCD2009 Soul 世宗大学 25 How to fit the data
1. Fitting by the perturbative scalings m m Scalings : electric : e C g(T) magnetic : m C g 2 (T) T e T m
2 1b1 log(2log( / )) Running Coupling: g ( ) 1 2 2bb00 log( / ) 2 log( / )
2T Tc
2 2 Results : Ce 1.63(3), 0.715 Cm 0.482(31), 0.979
2. Comparison with LOP and HTL calculation.
LOP me,0 g (T)T 3g m 2m 1 2 2 e e 2 HTL (Rebhan,1993) me me,0 1 log O(g ) 2 me,0 mm 2
XQCD2009 Soul 世宗大学 26 At higher temperature and on the large lattice
T/Tc=1.0~16.0 32x32x48x6 lattice Magnetic mass has strong finite size effect.
From this lattice, 2 Ce 1.69(4), 0.66 2 Cm 0.549(16), 1.27 about 10 % larger From the previous small lattice, 2 Ce 1.63(3), 0.715 2 Cm 0.482(31), 0.979
m m e C g(T) m C g 2 (T) T e T m
XQCD2009 Soul 世宗大学 27 Gluon propagators at finite temperature
Self-energy and propagators
1 1 k k GP FP D P P 0 : Landau gauge T L G k 2 T F k 2 L k 2 k 2 Projection operators
k ik j k k P00 P0i Pi0 , Pij ij , P P , T T T T k 2 L k 2 T
2 2 (PT ) PT , (PL ) PL , PT PL 0
Electric and magnetic gluon propagators in the momentum space
1 1 D (k,k 0) D00 , D (k,k 0) Dii E 0 F k 2 M 0 G k 2 2 F(0,0) mE ~ g(T)T G(0,0) mM ~ g (T)T
XQCD2009 Soul 世宗大学 28 Gluon propagators on the lattices
Gauge potentials and correlators
a a * A (x,t) Tr U (x,t) D (x,t) A (x,t)A (x,t)
Unequal-time propagators ( t t ' t '' )
1 a ' a '' iq (x y) D (q,t) 2 A (x,t )A (y,t )e V (Nc 1) x
Sum of t with q0=0,
1 D (q,q0 0) D (q,t) Nt t
XQCD2009 Soul 世宗大学 29 Gluon propagators on the lattices
Gauge potentials and correlators
a a * A (x,t) Tr U (x,t) D (x,t) A (x,t)A (x,t)
Equal-time propagators
1 a a iq (x y) D (q) 2 A (x,t)A (y,t)e V (Nc 1) x,a Unequal-time propagators ( t t ' t '' )
1 a ' a '' iq (x y) D (q,t) 2 A (x,t )A (y,t )e V (Nc 1) x
Sum of t with q0=0, 1 D (q,q0 0) D (q,t) Nt t
XQCD2009 Soul 世宗大学 30 Numerical parameters
SU(2) lattice calculation with quenched Wilson-gauge action Landau (Coulomb) gauge on the lattice in the path-integral formula satisfies the following condition: 1 A (x,t) 0 Maximize R Re TrU (x,t) VT x,t 2 a ˆ eps Tr U (x) U (x ) 10
Wilson-Mandula Method (PLB185,127(1987)) Parameters:
β Temp. Size Eps of GF Confs. 12x12x12x4 10-8 30 20x20x20x4 10-8 30 β=2.40 T/T ~1.40 c 32x32x32x4 10-8 30 48x48x48x4 10-8 10 -8 β=2.88 T/Tc~6.00 32x32x32x4 10 10 XQCD2009 Soul 世宗大学 31 Coulomb-gauge propagators
Preliminary
We have the same results as in the case of the Landau-gauge. At the configuration with removed vortices, the magnetic gluon is affected crucially, while the electric gluon is not. In the later case, the removal affects the overall normalization.
XQCD2009 Soul 世宗大学 32 Landau-gauge propagators volume dependence
Preliminary equal-time propagators
The volume dependence seems to be not significant. If the magnetic gluons are confining, should their low-momentum propagator vanish
XQCD2009 Soul 世宗大学 33