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Screening mass andSU(3) gluon propagator at finite temperature
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 119 (2003) 493495
ELSEVIER
www.clscvier.com/locate/npc
Screening mass and SU(3) gluon propagator at finite temperature A. Nakamuraa, “Information bSchool “Faculty
I. Pushkinab. Media
of Biosphere of Education,
Center,
T. Saito* Hiroshima
Sciences, Hiroshima Yamagata
a and S. SakaiC University,
Higashi-Hiroshima
University,
University,
Yamagata
739-8521.
Higashi-Hiroshima 990-8560,
*Japan
739-8521,
Japan
Japan
We study screening masses of SU(3) gluon propagator at the temperature T, m 6T, using the stochastic gauge fixing method. We also check the gauge invariance for electric and magnetic masses. Finally we compare our non-perturbative numerical results with a hard-thermal-loop resummation calculation.
1. INTRODUCTION In a quark-gluon plasma(QGP) phase, gluons have quantum corrections from the medium and behave as the massive particle. The screening effect appears in the heavy quark potential and the study of gluon screening effects becomes essential to understand finite temperature &CD. The electric mass can be defined by perturbation thermal QCD: m,,o = gT (electric or soft scale). On the other hands, the definition of the magnetic mass (cut-off for an infrared problem) is difficult [ 11. 3D reduction argument provides[ l] m,, N g2T (magnetic or super-soft scale). In this study we report the lattice quenched QCD calculation for both screening masses. We mainly investigate the temperature and gauge dependences. Finally we compare our results with normal perturbation and hard-thermal-loop resummation [2].
2. MEASUREMENTS In order to obtain the screening mass we use the gluon correlation functions with finite momenta, Gwu(Pt, (nA,(P,,
J’z, py> z) = pz, J’u> z)-&-P,,
-J-‘z, -J’y,O)),
(‘I
APL(x) = (U,(x) where gauge potentials L$(x))/agi. The propagator with finite momenta *Talk presented by T. Saito at Lattice
enables us to construct transverse and longitudinal parts without ambiguity[3]; we define electric and magnetic gluon propagators[4] as G, (P,z)
r-u Gtt (&O;O.z)
+Gtt
(0, g,O:i),
G,(P,z)
NG~I:(O,‘~,O,Z)
+G,,($.O,O,z);
la)
which correspond to standard projected propagators in perturbative calculations[l]. They behave large disas G,(,)(z) 3; e-E,(,,~)2 at a sufficiently tance.
3. STOCHASTIC
QUANTIZATION
We employ stochastic gauge fixing Zwanziger[5] as the quantization method.
a
la
(3) Here 7- is the Langevin time, the second term of the r.h.s. is the gauge fixing term. and 71 represents Gaussian noise. a is the gauge parameter and , for example, LY = 0 corresponds to the Lorentz gauge. The lattice version was proposed in Ref.[6] as
Here, w means a gauge rotation driving force,
2002
0920-5632/03/$ - see front matter 0 2003 Elsevier Science B.V. All rights reserved doi:10.1016/S0920-5632(03)01593-7
matrix
and f a
A. Nukumura et al. /Nucleor
494
Physics B (Proc. Suppl.) I19 (2003) 493-495
The gauge rotation and Langevin step are executed alternately. There are two reasons for using the stochastic gauge fixing method here. Practical reasons: Contrary to the standard iterative method, there is no convergence problem of gauge fixing. Therefore we can precisely estimate CPU time. Moreover, since this algorithm may be flexible to change the gauge parameter CY,the check of gauge dependences becomes relatively easy. Conceptual reasons: we must deal with Gribov gauge copies when doing the gauge fixing in non-Abelian gauge Although this algorithm may not theories[7]. solve the Gribov problem completely, Zwanziger’s stochastic gauge fixing procedure maintains the gauge system inside the Gribov region[5] if we start from the trivial configuration {A,, = 0). 4. SIMULATION
Figure 1. The propagator behavior in confinement and deconfinement phase.
CONDITION
We use N,N,N,N, = 20 x 20 x 32 x 6 lattice with the standard Wilson gauge action. The temperature is changed by varying the gauge coupling with fixed Nt. We adopt the lattice cut-off appeared in Ref. [8]. The Runge-Kutta algorithm is applied to reduce the finite Langevin step AT dependence. We got small AT dependences and AT = 0 extrapolations can be reliably done[9]. The gauge parameter a is fixed to be one, except in the gauge dependence test. 5. RESULTS We observe a large fluctuation of gauge propagators, particularly at large distances. A typical number of simulations is 0.2 N 0.4 million steps. Fig.1 shows that the dynamics of a gluon propagator is completely different in confinement and deconfinement regions [lo]. To extract screening masses from the propagators we use the hyperbolic functions: G +)
Electric propagator
- cosh(E,(,)(p)(z-N,/2))
at z > l/7’.(6)
In Fig.2, we show the gauge parameter (Y dependence of electric and magnetic masses. The result strongly suggests that they are gauge independent and physical observables.
Fig.3 shows the behavior of electric and magnetic masses as a function of temperature. We study the temperature range , T/Tc = 1 - 6, which would be realized in high-energy heavy ion collision experiments such as RHIC or LHC. The magnetic part definitely has nonzero mass in this temperature region. We fit the data above T N 1.5T, by me/T = Ceg(T) and m,/T = C,g2(T). Here, we use two loop scale running couplings and we set ,LL= 21rT which is a Matsubara frequency, as the renormalization point and A = l.O3T, as the QCD mass scale. C, = 1.63(3), x2/NDF = 0.715 and C, = 0.482(31), x2/NDF = 0.979 are obtained. The scaling works well for the both screening masses. The hard-thermal-loop resummation technique gives a formula for the electric mass in the oneloop resummed perturbation theory[2],
>+ 1(7)
2m, 1 log - O(g2) 2rme,0 mm 2 ( [ In Fig.3, we depict this hard-thermal-loop resummation result together with the lowest order calculation. mz=m&
l+--
39 me
6. CONCLUSION We have measured gluon propagators and obtained the electric and magnetic masses by lat-
A. Nukamura et (11./Nucleur Physics B (Proc. Suppl.) 119 (2003) 493-495
T/T,=2.69, A~=0.0.5,
5
.I,!.?'#
1
1
r
495
0 Magnetic n Electric
4
s3 2
w
I 02
0
0
0
i
------
0.2
0.4
0.6 cx
0.8
1
tice QCD simulations in the quench approximation for SU(3) between T = T, and 6T,. Features of QGP in this temperature region will be extensively studied theoretically and experimentally in near future. We observed that the magnetic mass does not vanish there and it can be approximately fitted by Cmg2T. The electric mass is consistent with the hard-thermal-loop resummation calculation. We also confirmed that both electric and magnetic masses are gauge independent. 7. ACKNOWLEDGEMENT We would. like to thank A. Niegawa, S. Muroya and T. Inagaki for many helpful discussions. Numerical calculations were carried out on SX5 at RCNP, Osaka Univ., VPP5000 at SIPC, Tsukuba Univ. and HPC computer at INSAM, Hiroshima Univ.. This work is supported by Grants-inAid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology, Japan (No.11694085, No.11740159, and No. 12554008). REFERENCES 1. M. Le Bellac, Thermal Field Theory (Cambridge monographs on mathematical physics,
/
---
1
2
TkC
1.2
Figure 2. Gauge dependences of both screening masses are very slight in non-perturbative regions.
0'
Figure 3. Dotted line is a fit based on the scaling of mg N g2T. For the electric mass, the dashed and solid lines represent a leading order perturbation and the hard-thermal-loop resummation result, respectively.
2. 3. 4.
5. 6.
7.
8.
9. 10.
Cambridge University Press. Cambridge, 1996). A.K. Rebhan, Phys. Rev. D 48, R3967 (1993); Nucl. Phys. B430, 319 (1994). A. Nakamura and M. Plewnia, Phys. Lett. B 255, 274 (1991). U.M. Heller, F. Karsch and J. Rank, Phys. Lett. B 355, 511 (1995): Phys. Rev. D 57, 1438 (1998); A. Cucchieri, F. Karsch, P. Petreczky, Phys. Lett. B 497 80-84 (2001). D. Zwanziger, Nucl. Phys. B192, 259 (1981). Vistas in A. Nakamura and M. Mizutani, Astronomy (Pergamon Press), vol. 37, 305 (1993); M. Mizutani and A. Nakamura, Nucl. Phys. B(Proc. Suppl.) 34, 253 (1994). V.N. Gribov, Nucl. Phys. B139, 1 (1978); Gribov Theory of Quark Confinement, Ed. by J. Nyiri, (World Scientific, Singapore, 2001). QCDTARO collaboration, K. Akemi, et al., Phys. Rev. Lett. 71,3063 (1993): C.R. Allton, hep-lat/9610016 (1996). A. Nakamura: I. Pushkina, T. Saito and 8. Sakai, hep-lat/0208075. A. Nakamura, Prog. Theor. Phys. Suppl. No. 131, 585, 19 98.
ELSEVIER
www.clscvier.com/locate/npc
Screening mass and SU(3) gluon propagator at finite temperature A. Nakamuraa, “Information bSchool “Faculty
I. Pushkinab. Media
of Biosphere of Education,
Center,
T. Saito* Hiroshima
Sciences, Hiroshima Yamagata
a and S. SakaiC University,
Higashi-Hiroshima
University,
University,
Yamagata
739-8521.
Higashi-Hiroshima 990-8560,
*Japan
739-8521,
Japan
Japan
We study screening masses of SU(3) gluon propagator at the temperature T, m 6T, using the stochastic gauge fixing method. We also check the gauge invariance for electric and magnetic masses. Finally we compare our non-perturbative numerical results with a hard-thermal-loop resummation calculation.
1. INTRODUCTION In a quark-gluon plasma(QGP) phase, gluons have quantum corrections from the medium and behave as the massive particle. The screening effect appears in the heavy quark potential and the study of gluon screening effects becomes essential to understand finite temperature &CD. The electric mass can be defined by perturbation thermal QCD: m,,o = gT (electric or soft scale). On the other hands, the definition of the magnetic mass (cut-off for an infrared problem) is difficult [ 11. 3D reduction argument provides[ l] m,, N g2T (magnetic or super-soft scale). In this study we report the lattice quenched QCD calculation for both screening masses. We mainly investigate the temperature and gauge dependences. Finally we compare our results with normal perturbation and hard-thermal-loop resummation [2].
2. MEASUREMENTS In order to obtain the screening mass we use the gluon correlation functions with finite momenta, Gwu(Pt, (nA,(P,,
J’z, py> z) = pz, J’u> z)-&-P,,
-J-‘z, -J’y,O)),
(‘I
APL(x) = (U,(x) where gauge potentials L$(x))/agi. The propagator with finite momenta *Talk presented by T. Saito at Lattice
enables us to construct transverse and longitudinal parts without ambiguity[3]; we define electric and magnetic gluon propagators[4] as G, (P,z)
r-u Gtt (&O;O.z)
+Gtt
(0, g,O:i),
G,(P,z)
NG~I:(O,‘~,O,Z)
+G,,($.O,O,z);
la)
which correspond to standard projected propagators in perturbative calculations[l]. They behave large disas G,(,)(z) 3; e-E,(,,~)2 at a sufficiently tance.
3. STOCHASTIC
QUANTIZATION
We employ stochastic gauge fixing Zwanziger[5] as the quantization method.
a
la
(3) Here 7- is the Langevin time, the second term of the r.h.s. is the gauge fixing term. and 71 represents Gaussian noise. a is the gauge parameter and , for example, LY = 0 corresponds to the Lorentz gauge. The lattice version was proposed in Ref.[6] as
Here, w means a gauge rotation driving force,
2002
0920-5632/03/$ - see front matter 0 2003 Elsevier Science B.V. All rights reserved doi:10.1016/S0920-5632(03)01593-7
matrix
and f a
A. Nukumura et al. /Nucleor
494
Physics B (Proc. Suppl.) I19 (2003) 493-495
The gauge rotation and Langevin step are executed alternately. There are two reasons for using the stochastic gauge fixing method here. Practical reasons: Contrary to the standard iterative method, there is no convergence problem of gauge fixing. Therefore we can precisely estimate CPU time. Moreover, since this algorithm may be flexible to change the gauge parameter CY,the check of gauge dependences becomes relatively easy. Conceptual reasons: we must deal with Gribov gauge copies when doing the gauge fixing in non-Abelian gauge Although this algorithm may not theories[7]. solve the Gribov problem completely, Zwanziger’s stochastic gauge fixing procedure maintains the gauge system inside the Gribov region[5] if we start from the trivial configuration {A,, = 0). 4. SIMULATION
Figure 1. The propagator behavior in confinement and deconfinement phase.
CONDITION
We use N,N,N,N, = 20 x 20 x 32 x 6 lattice with the standard Wilson gauge action. The temperature is changed by varying the gauge coupling with fixed Nt. We adopt the lattice cut-off appeared in Ref. [8]. The Runge-Kutta algorithm is applied to reduce the finite Langevin step AT dependence. We got small AT dependences and AT = 0 extrapolations can be reliably done[9]. The gauge parameter a is fixed to be one, except in the gauge dependence test. 5. RESULTS We observe a large fluctuation of gauge propagators, particularly at large distances. A typical number of simulations is 0.2 N 0.4 million steps. Fig.1 shows that the dynamics of a gluon propagator is completely different in confinement and deconfinement regions [lo]. To extract screening masses from the propagators we use the hyperbolic functions: G +)
Electric propagator
- cosh(E,(,)(p)(z-N,/2))
at z > l/7’.(6)
In Fig.2, we show the gauge parameter (Y dependence of electric and magnetic masses. The result strongly suggests that they are gauge independent and physical observables.
Fig.3 shows the behavior of electric and magnetic masses as a function of temperature. We study the temperature range , T/Tc = 1 - 6, which would be realized in high-energy heavy ion collision experiments such as RHIC or LHC. The magnetic part definitely has nonzero mass in this temperature region. We fit the data above T N 1.5T, by me/T = Ceg(T) and m,/T = C,g2(T). Here, we use two loop scale running couplings and we set ,LL= 21rT which is a Matsubara frequency, as the renormalization point and A = l.O3T, as the QCD mass scale. C, = 1.63(3), x2/NDF = 0.715 and C, = 0.482(31), x2/NDF = 0.979 are obtained. The scaling works well for the both screening masses. The hard-thermal-loop resummation technique gives a formula for the electric mass in the oneloop resummed perturbation theory[2],
>+ 1(7)
2m, 1 log - O(g2) 2rme,0 mm 2 ( [ In Fig.3, we depict this hard-thermal-loop resummation result together with the lowest order calculation. mz=m&
l+--
39 me
6. CONCLUSION We have measured gluon propagators and obtained the electric and magnetic masses by lat-
A. Nukamura et (11./Nucleur Physics B (Proc. Suppl.) 119 (2003) 493-495
T/T,=2.69, A~=0.0.5,
5
.I,!.?'#
1
1
r
495
0 Magnetic n Electric
4
s3 2
w
I 02
0
0
0
i
------
0.2
0.4
0.6 cx
0.8
1
tice QCD simulations in the quench approximation for SU(3) between T = T, and 6T,. Features of QGP in this temperature region will be extensively studied theoretically and experimentally in near future. We observed that the magnetic mass does not vanish there and it can be approximately fitted by Cmg2T. The electric mass is consistent with the hard-thermal-loop resummation calculation. We also confirmed that both electric and magnetic masses are gauge independent. 7. ACKNOWLEDGEMENT We would. like to thank A. Niegawa, S. Muroya and T. Inagaki for many helpful discussions. Numerical calculations were carried out on SX5 at RCNP, Osaka Univ., VPP5000 at SIPC, Tsukuba Univ. and HPC computer at INSAM, Hiroshima Univ.. This work is supported by Grants-inAid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology, Japan (No.11694085, No.11740159, and No. 12554008). REFERENCES 1. M. Le Bellac, Thermal Field Theory (Cambridge monographs on mathematical physics,
/
---
1
2
TkC
1.2
Figure 2. Gauge dependences of both screening masses are very slight in non-perturbative regions.
0'
Figure 3. Dotted line is a fit based on the scaling of mg N g2T. For the electric mass, the dashed and solid lines represent a leading order perturbation and the hard-thermal-loop resummation result, respectively.
2. 3. 4.
5. 6.
7.
8.
9. 10.
Cambridge University Press. Cambridge, 1996). A.K. Rebhan, Phys. Rev. D 48, R3967 (1993); Nucl. Phys. B430, 319 (1994). A. Nakamura and M. Plewnia, Phys. Lett. B 255, 274 (1991). U.M. Heller, F. Karsch and J. Rank, Phys. Lett. B 355, 511 (1995): Phys. Rev. D 57, 1438 (1998); A. Cucchieri, F. Karsch, P. Petreczky, Phys. Lett. B 497 80-84 (2001). D. Zwanziger, Nucl. Phys. B192, 259 (1981). Vistas in A. Nakamura and M. Mizutani, Astronomy (Pergamon Press), vol. 37, 305 (1993); M. Mizutani and A. Nakamura, Nucl. Phys. B(Proc. Suppl.) 34, 253 (1994). V.N. Gribov, Nucl. Phys. B139, 1 (1978); Gribov Theory of Quark Confinement, Ed. by J. Nyiri, (World Scientific, Singapore, 2001). QCDTARO collaboration, K. Akemi, et al., Phys. Rev. Lett. 71,3063 (1993): C.R. Allton, hep-lat/9610016 (1996). A. Nakamura: I. Pushkina, T. Saito and 8. Sakai, hep-lat/0208075. A. Nakamura, Prog. Theor. Phys. Suppl. No. 131, 585, 19 98.