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Hadronic spectral functions above the QCD phase transition
Nuclear
ELStiVlER
Physics
A7 15 (2003)
863c-866~ www.elsevier.com/locatelnpe
Hadronic X
Spectral
Asakawa”,
Functions
T. Hatsudab
above the QCD Phase Transition
*
and Y. Nakahara”
“Department
of Physics,
Kyoto
University,
‘Department
of Physics,
University
“Dppartment
of Physics,
Nagoya
Kyoto
of Tokyo, University,
606-8502,
Japan
Tokyo
113-0033,
Japan
Nagoya,
464-8602,
Japan
We extract the spectral functions in the scalar, pseudo-scalar, vector, and axial-vector channels above the deconfinement phase transition temperature (Z’,) using the maximum entropy method (MEM). We use anisotropic lattices, 323 x 32, 40, 54, 72, 80, and 96 (corresponding to T = 2.3T, + 0.8T,), with the renormalized anisotropy < = 4.0 to have c,uough temporal data points to carry out the MEM analysis. Our result suggests that the ypcc,trxl functions continue to possess non-trivial structures even above T, and in addition that there is a qualitative change in the state of the deconfined matter between 1.5T, and 2T,. 1.
INTRODUCTION
The spectral functions (SPFs) of hadronic operators play an important role in QCD. The modification of hadrons, which has been suggested, for example, by the dilepton production enhancement in high energy nuclear collisions observed at CERN SPS, can be formulated in terms of SPFs. Although there have been numerous theoretical attempts to underst,and the modification of SPFs at finite temperature and/or density [l], the exact nat,ure of the hadronic modes in matter is not understood well. Recently, the maximum entropy method (MEM) has been used to extract SPFs from latt,ice QCD data for the first time by the present authors [2,3]. The Euclidean 2-point functions in the temporal direction and the associated SPFs are related by the Laplacr transform. On the lattice, only a finite number of data points with statistical noise are availahlr in the t,rmporal direction. Therefore, direct Laplace inversion from lattice dat,n to SPF is an ill-posed problem. MEM is a method to evade such difficulty on the basis of thcl Bayes’ theorem in the theory of statistical inference [3].
2. HOW
MANY
DATA
POINTS
ARE NECESSARY
FOR MEM
?
As we have discussed in detail in (31, the result of MEM depends strongly on the numbtl of tc’mporal data points. The more data points are used, the closer the MEM result is t.o tllc, true SPF. Moreover, there is the minimum number of data points N to perform the reliable MEM analysis. ‘Presented
by M. Asakawa
0375-9474/03/$ - see Front matter doi:lO.l016/S0375-9474(02)01526-9
Q 2003 Elsevier
Science
B.V
All rights
reserved.
M. Asukuwa
864~
et al. /Nuclear
Physics
A715
(2003)
863ce866~
In order to find this minimum N at T = 0 on the lattice, we have carried out the following analysis [4]. First we calculated hadronic correlators with the quenched approximation on an isotropic 403 x 30 lattice at p = 6.47 [4,5]. The number of gauge’ configurations is 160. Then, MEM analysis has been done for the vector channel by using A data points (at 7 = 2,... ,N/2+1 and 31-N/2;.. ,30) out of the total 30 data points. The result is shown in Fig. 1. The figure clearly shows that, SPF changes considerably as ~1’ and t,hat at least about 30 data points are necessary for the convergence of the result. This minimum N would depend on p, and on whether one employs anisotropic lat,ticcs, ~III~)~(I\YY~actions, and so on. Nevertheless, we use this number as a practical guide in the tollo~villg.
Figure
1. MEM
results
3. ANISOTROPIC
in the vector channel
LATTICE
with
10 different
N’s,
AT T # 0
At finite temperature (T), because of the relation T = l/N,a, (IV? and (I, being thcx tc~mporal lntt,icc size and spacing, respectively), less and less data points are avail;lbl<, a5 7’ iucrrascs if nT is fixed. Requiring that N, > 30 holds even at t,hc highest T, e.g. Z.TiT,.. \IC’ iu~‘ iiirvit,ablv lc,d to use an anisotropic lat,tice.
M. Asukuwu
et al. /Nuclear
Physics
A715
(2003)
863c-866~
865~
M:e have used the bare anisotropy (0 = 3.5 and /J = 7.0 with the naive plaqucttc a&on. For the quark part, the standard Wilson action with the quenched approximation is used. The corresponding renormalized anisotropy is [ = 4.0 [6] and the lattice spacing is 0, = 1/4.u, = 9.75 x 10e3 fm. Simulations are done on 32” x N, lattices with IV, = 32. 10. 54. 72. 80, and 96. This corresponds to T Y 2.3T,, l.YT,, 1.4T,, l.O4T,., 0.93T,, ad 0.7HT,., respectively. More than 100 configurations are generated for each N,. The details of’ the lattice parameters used in the calculation will be given in [4]. Ll;c have studied 2-point correlation functions in the scalar (S), pseudo-scalar (PS). vector (V), and axial-vector (AV) channels. The default models, motivat,ed by pertllrbativc QCD, in each channel are m = 0.60, 1.15, 0.40, and 0.35, respectively. Tht latt,ic,r data points used in our MEM analysis are T = 5.“. ,21 and N, - 19;‘. , N, - 3 for -2; = 54. 72. 80. and 96. For N, = 32 and I(L)I v--w 40. T = 5, .‘. 1N, - 3 are used. When t,he Oi4 ---~ . WL~K~ and sink are closer than [a, in the “’ temporal direction. artifact could arise due O1 to unphysical modes at w > r/
4. RESULTS
OF MEM
111 Fig. 3, we show the results of the MEM analysis for SPF p(w) in the S, PS, V. ~ti AV channels on 32” x 54 lattice (T N 1.4T,). In this calculation, we have used the hopping parameter corresponding to m?r/rr+, LX 0.7 at T = 0. If the quark-gluon plasma is such an object as described as a free gas of massive quarks. anti-quarks and gluons, p(w) would show a flat behavior with a smooth rise from zero above the qq threshold as a function of w. To the contrary, Fig. 3 shows that SPFs possess nontrivial structures even above T, in all four channels. There are a sharp peak at about J - 2 GclV and two bumps above the peak. Also SPFs in all channels almost degenrrwt,r. Lvllic 11shows a strong signature of the restoration of chiral symmetry. The peak and bump strllc.1 urt’s. and the suppression of SPFs below w = 2 GeV are statist,ically significant ;lc,c.ordiug to OLW error analysis. We have used iv, twice as large as that employctl iu [7].
866~
M. A.sulcuwa
et 01. /Nuclear
Physics
A715
(2003)
H63c-X66c
As shown ill Scc.2. t,his is necessary for reliable MEM analysis. 111 Fig. 4% w sllow the results at N, = 40, i.e., T z 1.9T,. There is au apparent peak Molultf d = 0. A l)ossible explanat,ion of the peak is the effect of Landau da.mping [8]. Iion-r\-rr. this p~k is, at the moment, not statistically significant yet. The peaks A~~OIIII~ 2 r 4.5 Cc\’ i~r(~ significantly broader than those around 2 GrV at, 7‘ 2 1.9T,..
P(Wl
1
s-
;/I
5
10
Figure 3. Spectral (T zz l.4Tc)
15
20
25
30
0
5
10
15
20
o[GeV]
Functions
for N, = 54
Figure 4. Spectral (T = 1.9T,)
Functions
25
30 o [GeV]
for !V, = 40
\\‘r have carried out an error analysis for the average of the SPFs in small regions wrolmd tht peaks and found that the broadening is indeed statistically significant. SPFs iu the case of N, = 32 (T = 2.3T,) have similar structures as those at T Y 1.9T, except tllc shift of the peak positions. This suggests a possibility of a qualitative change ill the state of the deconfined phase between 1.5T, and 2T,. Acknowledgement This work was supported in part by the Grants-in-Aid by Minist,ry of Education (No. 12640263, No. 12640296, and No. 14540255). Lattice calculations llavt~ been carried out with the CP-PACS computer under the “Large-scale Numerical Siul~~lation Program” of Center for Computational Physics, University of Tsukuba. R.EFERENCES 1.
For example, S. H. Lee and T. Hatsuda, Phys. Rev. C46 (1992) R34; nl. Asakawn ct al., Phys. Rev. C46 (1992) R1159; R. Rapp and J. Wambach, Adv. Nncl. Pllys. 25 (2000) 1. 2. Y. Nakahara. 1~1.Asakawa, and T. Hatsuda, Phys. Rev. D 60 (1999) 091503. 3. .\I. Asakawa, T. Hatsuda, and Y. Nakahara, Prog. Part. Nucl. Phys. 46 (2001) 459. -1. hl. Asakawa, T. Hatsuda, and Y. Nakahara, in preparation. 5. S. Aoki et al., Phys. Rev. Lett. 84 (2000) 238. 6. .J. Engels, F. Karsch, and T. Scheideler, Nucl. Phys. B564 (2000) 303. 7. F. Karsch et al., Phys. Lett. B530 (2002) 147. 8. T. Hatsuda, Y. Koike, and S. H. Lee, Nucl. Phys. B394 (1993) 221.
ELStiVlER
Physics
A7 15 (2003)
863c-866~ www.elsevier.com/locatelnpe
Hadronic X
Spectral
Asakawa”,
Functions
T. Hatsudab
above the QCD Phase Transition
*
and Y. Nakahara”
“Department
of Physics,
Kyoto
University,
‘Department
of Physics,
University
“Dppartment
of Physics,
Nagoya
Kyoto
of Tokyo, University,
606-8502,
Japan
Tokyo
113-0033,
Japan
Nagoya,
464-8602,
Japan
We extract the spectral functions in the scalar, pseudo-scalar, vector, and axial-vector channels above the deconfinement phase transition temperature (Z’,) using the maximum entropy method (MEM). We use anisotropic lattices, 323 x 32, 40, 54, 72, 80, and 96 (corresponding to T = 2.3T, + 0.8T,), with the renormalized anisotropy < = 4.0 to have c,uough temporal data points to carry out the MEM analysis. Our result suggests that the ypcc,trxl functions continue to possess non-trivial structures even above T, and in addition that there is a qualitative change in the state of the deconfined matter between 1.5T, and 2T,. 1.
INTRODUCTION
The spectral functions (SPFs) of hadronic operators play an important role in QCD. The modification of hadrons, which has been suggested, for example, by the dilepton production enhancement in high energy nuclear collisions observed at CERN SPS, can be formulated in terms of SPFs. Although there have been numerous theoretical attempts to underst,and the modification of SPFs at finite temperature and/or density [l], the exact nat,ure of the hadronic modes in matter is not understood well. Recently, the maximum entropy method (MEM) has been used to extract SPFs from latt,ice QCD data for the first time by the present authors [2,3]. The Euclidean 2-point functions in the temporal direction and the associated SPFs are related by the Laplacr transform. On the lattice, only a finite number of data points with statistical noise are availahlr in the t,rmporal direction. Therefore, direct Laplace inversion from lattice dat,n to SPF is an ill-posed problem. MEM is a method to evade such difficulty on the basis of thcl Bayes’ theorem in the theory of statistical inference [3].
2. HOW
MANY
DATA
POINTS
ARE NECESSARY
FOR MEM
?
As we have discussed in detail in (31, the result of MEM depends strongly on the numbtl of tc’mporal data points. The more data points are used, the closer the MEM result is t.o tllc, true SPF. Moreover, there is the minimum number of data points N to perform the reliable MEM analysis. ‘Presented
by M. Asakawa
0375-9474/03/$ - see Front matter doi:lO.l016/S0375-9474(02)01526-9
Q 2003 Elsevier
Science
B.V
All rights
reserved.
M. Asukuwa
864~
et al. /Nuclear
Physics
A715
(2003)
863ce866~
In order to find this minimum N at T = 0 on the lattice, we have carried out the following analysis [4]. First we calculated hadronic correlators with the quenched approximation on an isotropic 403 x 30 lattice at p = 6.47 [4,5]. The number of gauge’ configurations is 160. Then, MEM analysis has been done for the vector channel by using A data points (at 7 = 2,... ,N/2+1 and 31-N/2;.. ,30) out of the total 30 data points. The result is shown in Fig. 1. The figure clearly shows that, SPF changes considerably as ~1’ and t,hat at least about 30 data points are necessary for the convergence of the result. This minimum N would depend on p, and on whether one employs anisotropic lat,ticcs, ~III~)~(I\YY~actions, and so on. Nevertheless, we use this number as a practical guide in the tollo~villg.
Figure
1. MEM
results
3. ANISOTROPIC
in the vector channel
LATTICE
with
10 different
N’s,
AT T # 0
At finite temperature (T), because of the relation T = l/N,a, (IV? and (I, being thcx tc~mporal lntt,icc size and spacing, respectively), less and less data points are avail;lbl<, a5 7’ iucrrascs if nT is fixed. Requiring that N, > 30 holds even at t,hc highest T, e.g. Z.TiT,.. \IC’ iu~‘ iiirvit,ablv lc,d to use an anisotropic lat,tice.
M. Asukuwu
et al. /Nuclear
Physics
A715
(2003)
863c-866~
865~
M:e have used the bare anisotropy (0 = 3.5 and /J = 7.0 with the naive plaqucttc a&on. For the quark part, the standard Wilson action with the quenched approximation is used. The corresponding renormalized anisotropy is [ = 4.0 [6] and the lattice spacing is 0, = 1/4.u, = 9.75 x 10e3 fm. Simulations are done on 32” x N, lattices with IV, = 32. 10. 54. 72. 80, and 96. This corresponds to T Y 2.3T,, l.YT,, 1.4T,, l.O4T,., 0.93T,, ad 0.7HT,., respectively. More than 100 configurations are generated for each N,. The details of’ the lattice parameters used in the calculation will be given in [4]. Ll;c have studied 2-point correlation functions in the scalar (S), pseudo-scalar (PS). vector (V), and axial-vector (AV) channels. The default models, motivat,ed by pertllrbativc QCD, in each channel are m = 0.60, 1.15, 0.40, and 0.35, respectively. Tht latt,ic,r data points used in our MEM analysis are T = 5.“. ,21 and N, - 19;‘. , N, - 3 for -2; = 54. 72. 80. and 96. For N, = 32 and I(L)I v--w 40. T = 5, .‘. 1N, - 3 are used. When t,he Oi4 ---~ . WL~K~ and sink are closer than [a, in the “’ temporal direction. artifact could arise due O1 to unphysical modes at w > r/
4. RESULTS
OF MEM
111 Fig. 3, we show the results of the MEM analysis for SPF p(w) in the S, PS, V. ~ti AV channels on 32” x 54 lattice (T N 1.4T,). In this calculation, we have used the hopping parameter corresponding to m?r/rr+, LX 0.7 at T = 0. If the quark-gluon plasma is such an object as described as a free gas of massive quarks. anti-quarks and gluons, p(w) would show a flat behavior with a smooth rise from zero above the qq threshold as a function of w. To the contrary, Fig. 3 shows that SPFs possess nontrivial structures even above T, in all four channels. There are a sharp peak at about J - 2 GclV and two bumps above the peak. Also SPFs in all channels almost degenrrwt,r. Lvllic 11shows a strong signature of the restoration of chiral symmetry. The peak and bump strllc.1 urt’s. and the suppression of SPFs below w = 2 GeV are statist,ically significant ;lc,c.ordiug to OLW error analysis. We have used iv, twice as large as that employctl iu [7].
866~
M. A.sulcuwa
et 01. /Nuclear
Physics
A715
(2003)
H63c-X66c
As shown ill Scc.2. t,his is necessary for reliable MEM analysis. 111 Fig. 4% w sllow the results at N, = 40, i.e., T z 1.9T,. There is au apparent peak Molultf d = 0. A l)ossible explanat,ion of the peak is the effect of Landau da.mping [8]. Iion-r\-rr. this p~k is, at the moment, not statistically significant yet. The peaks A~~OIIII~ 2 r 4.5 Cc\’ i~r(~ significantly broader than those around 2 GrV at, 7‘ 2 1.9T,..
P(Wl
1
s-
;/I
5
10
Figure 3. Spectral (T zz l.4Tc)
15
20
25
30
0
5
10
15
20
o[GeV]
Functions
for N, = 54
Figure 4. Spectral (T = 1.9T,)
Functions
25
30 o [GeV]
for !V, = 40
\\‘r have carried out an error analysis for the average of the SPFs in small regions wrolmd tht peaks and found that the broadening is indeed statistically significant. SPFs iu the case of N, = 32 (T = 2.3T,) have similar structures as those at T Y 1.9T, except tllc shift of the peak positions. This suggests a possibility of a qualitative change ill the state of the deconfined phase between 1.5T, and 2T,. Acknowledgement This work was supported in part by the Grants-in-Aid by Minist,ry of Education (No. 12640263, No. 12640296, and No. 14540255). Lattice calculations llavt~ been carried out with the CP-PACS computer under the “Large-scale Numerical Siul~~lation Program” of Center for Computational Physics, University of Tsukuba. R.EFERENCES 1.
For example, S. H. Lee and T. Hatsuda, Phys. Rev. C46 (1992) R34; nl. Asakawn ct al., Phys. Rev. C46 (1992) R1159; R. Rapp and J. Wambach, Adv. Nncl. Pllys. 25 (2000) 1. 2. Y. Nakahara. 1~1.Asakawa, and T. Hatsuda, Phys. Rev. D 60 (1999) 091503. 3. .\I. Asakawa, T. Hatsuda, and Y. Nakahara, Prog. Part. Nucl. Phys. 46 (2001) 459. -1. hl. Asakawa, T. Hatsuda, and Y. Nakahara, in preparation. 5. S. Aoki et al., Phys. Rev. Lett. 84 (2000) 238. 6. .J. Engels, F. Karsch, and T. Scheideler, Nucl. Phys. B564 (2000) 303. 7. F. Karsch et al., Phys. Lett. B530 (2002) 147. 8. T. Hatsuda, Y. Koike, and S. H. Lee, Nucl. Phys. B394 (1993) 221.