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Equation of state of strongly interacting matter and intensity interferometry of thermal photons
Nuclear Physics A 862–863 (2011) 290–293 www.elsevier.com/locate/nuclphysa
Equation of state of strongly interacting matter and intensity interferometry of thermal photons Somnath Dea , Dinesh K. Srivastavaa , Rupa Chatterjeea,b a Variable b Department
Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India of Physics, P. O. Box 35 (YFL), FI-40014 University of Jyv¨askyl¨a, Finland
Abstract We find that an equation of state (EOS) for hot hadronic matter consisting of all mesons (baryons) having M < 1.5(2.0) GeV along with Hagedorn resonances in thermal and chemical equilibrium, matches rather smoothly with lattice EOS (p4 action, Nτ = 8) for T up to ≈ 200 MeV, when corrections are made for the finite volume of hadrons. Two equations of state, HHL and HHB are constructed where the above is matched to the lattice and bag model EoS respectively at a critical temperature T c = 165 MeV. We find that the particle and thermal photon spectra differ only marginally for the two equations of state at both RHIC and LHC energies. The intensity interferometry results, specially the outward correlations for thermal photons are found to be quite distinct for the two equations of state. Keywords: EOS, QGP, lattice, interferometry, particle spectra.
1. Introduction Recent results from the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory have provided significant evidences of the formation of Quark-Gluon Plasma (QGP) [1] or rather a strongly interacting QGP (sQGP) in collisions of heavy nuclei at relativistic energies. Important signatures of QGP formation include observations of jet quenching, elliptic flow, recombination of partons as a process of hadronization, and electromagnetic radiation. The system, produced in such collisions, behaves like a nearly ideal fluid [2] and relativistic hydrodynamics has been used quite successfully to explain its evolution. We consider that a quark-gluon plasma is formed in central collision of heavy nuclei at a very small initial formation time τ0 at the top RHIC and LHC energies. The plasma then expands, cools, and hadronizes either through a first order phase transition at T = T c , for a bag model equation of state or in a rapid cross-over as suggested by lattice QCD simulations. The hadronic matter part of the EOS consists of all mesons having M < 1.5 GeV and all baryons having M < 2 GeV along with or without Hagedorn resonances. The mass spectrum can be written as: ρ(m) = ρHG (m) + ρHS (m) , where ρHG =
gi δ(m − mi ) .
i
0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.06.008
(1) (2)
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S. De et al. / Nuclear Physics A 862–863 (2011) 290–293 20
0.4
15
15
0.3
lattice p4; Nτ=8 hadron gas hadron gas(vol. corr.)
5
2
Cs
10
ε/T
ε/T
4
4
20
10 lattice p4; Nτ=8 hadron+Hagedorn gas hadron+Hagedorn gas (vol. corr.)
0.2
HHB HHL 0.1
5 (c)
(a) 0 0
0.1
0.2
0.3 T (GeV)
0.4
0.5
(b) 0 0
0.1
0.2
0.3
0.4
0.5
0
0.0
0.5
T (GeV)
1.0 1/4
ε
1.5
2.0
2.5
3 1/4
(GeV/fm )
Figure 1: (a) ε/T4 for hadron gas, volume corrected hadron gas, along with lattice results. (b) Corresponding results for hadron + Hagedorn gas without and with correction for finite volume of the particles. (c) Square of speed of sound for the two EOS, HHB and HHL (for detail see the Ref. [4]).
In the above, the sum runs over all the discrete hadronic states and their corresponding degeneracies are given by gi . We take the density of Hagedorn states from Ref. [3]: ρHS (m) = A
exp(m/T H ) , (m2 + m20 )5/4
(3)
where, A = 0.5 GeV3/2 , m0 = 0.5 GeV, T H = 0.196 GeV, and m varies from M0 = 2 GeV to MMax = 12 GeV. It is to be noted that the inclusion of the Hagedorn resonances is responsible for the rapid chemical equilibration of hadrons produced in relativistic heavy ion collisions [3]. Only mesonic Hagedorn states are considered in the present study. Various themodynamic quantities like - pressure, energy density, entropy densiry are calculated using standard methods and compared to the same obtained from lattice calculations (for details see Ref. [4]). For the finite volume corrections, we have used the method advocated by Kapusta and Olive [5], which is believed to be thermodynamically consistent. It is found that the above quantitites agree best with recent lattice QCD results (p4 action; Nτ = 8) [6] by including Hagedorn resonances and the finite volume correction for the hadrons at zero baryonic chemical potential (Fig.1(a) and 1(b)). Next we construct two EOS, HHB and HHL, where the volume corrected hadron + Hagedorn gas is used for both in the temperature limit T < 165 MeV. The bag model EOS (HHB) admits a mixed phase at T = 165 MeV, whereas the lattice equation of state (HHL) shows a sharp cross-over for 180 < T < 190 MeV. This is clearly seen in the variation of speed of sound as a function of ε1/4 (Fig.1(c)) for the two EOS. We solve the hydrodynamic equations for central collisions of Au+Au (200 A GeV) and Pb+Pb (5.5 A TeV) for the two equations of state with identical initial conditions. The average initial energy densities ε0 are: RHIC : ε0 LHC : ε0
= 80.80 GeV/fm3 = 718.3 GeV/fm3
at τ0 = 0.2 fm/c, at τ0 = 0.1 fm/c.
The freeze-out is assumed to take place at T = 100 MeV and the particle spectra are obtained using the CooperFrye formula. The invariant photon yield is obtained by integrating the emission rates over the spacetime volume of the system. The rate of production of photons from the quark matter is taken from the complete leading order calculation of Arnold et al. [7]. The production of photons from the hadronic matter is estimated using the results from Turbide et al. [8]. In Fig. 2 we have shown the results for transverse momentum distribution of thermal pions and photons for central collision of gold nuclei at midrapidity, for the two EOS, at RHIC energy. The experimental data for 0–5% most central collisions for the pions (from Ref. [9]) are also shown for comparison. No attempts were made to adjust any parameters or normalizations. It is obvious that both the EOS describe the data qualitatively well but the HHL EOS is slightly preferred. We also note that the inverse slope of the spectra for HHL EOS is larger than the same for HHB EOS. The slope of thermal photon spectrum is quite close to the earlier calculations reported in Ref. [11]. Also the data for single photons for 0–20% most central collisions [10] are shown in Fig. 2 for a comparison. A complete description of photon data would involve addition of prompt contribution scaled by appropriate nuclear overlapping function. No significant difference is seen for the results at top LHC energy, as well (see Ref. [4]). Thus we conclude that the particle and photon spectra are seen to be only marginally dependent on the EOS.
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S. De et al. / Nuclear Physics A 862–863 (2011) 290–293
10
3
Thermal photon spectrum for Au+Au @RHIC
-2
dN/d kT dy (GeV )
1
HHL HHB 0-20% PHENIX
10
-1
2
10
QM+HM -3
10
-5
10 0
0.5
1
1.5
2
2.5
3
3.5
4
kT (GeV)
Figure 2: Thermal pion and photon pT spectra at RHIC for the HHB and HHL equations of state. The experimental data for hadrons (0–5% centrality bin) are taken from [9] and photon data for 0–20% centrality bin from [10]. The calculations are for the impact parameter b=0 fm (see the Ref. [4] for detail). 1.6
1.6
Au+Au@RHIC, γγ Intensity Correlation
Au+Au@RHIC, γγ Intensity Correlation 1.5
Thermal Photons, k1T=1.7 GeV
1.3
QM+HM
1.2
HHL HHB
1.3 1.2
1.1 1.0 0
Thermal Photons, k1T=1.7 GeV
1.4
HHL HHB C
C
1.4
1.5
QM+HM
1.1 0.04
0.08
0.12
0.16
1.0 0
qo (GeV)
0.04
0.08
0.12
0.16
ql (GeV)
Figure 3: The outward (left panel) and longitudinal correlation (right panel) functions for thermal photons produced in central collision of gold nuclei at RHIC (see the Ref. [4] for detail).
Next we calculate two photon intensity correlation which is found sensitive to the details of evolution dynamics of the system [12]. We calculate the outward, side-ward, and longitudinal correlation functions of thermal photons for a typical momentum KT ∼ 1.7 GeV. The correlation function between two photons with momenta k1 and k2 is defined as [12]: d4 x S (x, K)eix·q 2 1 C(q, K) = 1 + . (4) 2 d4 x S (x, k1 ) d4 x S (x, k2 ) In the above, S (x, K) is the spacetime emission function, and q = k1 − k2 , K = (k1 + k2 )/2 .
(5)
The space-time emission function S is approximated as photon production rate , EdN/d4 xd3 k, for the QGP and hadronic phases. In Fig. 3 we have plotted the correlation function for outward (qo ) and longitudinal (ql ) momentum differences (see e.g. [4]). We choose momenta of the interfering photons such that when qo 0, q s and ql are identically zero. Photons from QGP originate at early times and those from hadronic matter at a later time and interference occurs due to the spatiotemporal difference between the two sources. The outward momenta difference qo couples the spatial as well as temporal difference, ql depends on the temporal difference only. We find that the spacetime distribution of the source is very different for the two equations of state [4]. This is reflected in the outward correlation function which clearly distinguishes between the two equations of state. The longitudinal correlation also shows a small differnce at RHIC energy and the difference becomes notciable at LHC energy [4]. In summary, we have shown that inclusion of Hagedorn resonances and finite volume correction can give a rich description of haronic matter. Also the direct photon intensity interferometry can be a valuable probe for the equation of state of strongly interacting matter if verified experimentally.
S. De et al. / Nuclear Physics A 862–863 (2011) 290–293
Acknowledgements RC would like to thank the Finnish Academy Project (No. 130472) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
K. Adcox et al [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005). R. A. Lacey et al., Phys. Rev. Lett. 98, 092301 (2007). J. Noronha-Hostler, M. Beitel, C. Greiner, and I. Shovkovy, Phys. Rev. C 81, 054909 (2010). S. De, D. K. Srivastava and R. Chatterjee, J. Phys. G 37, 115004 (2010). J. I. Kapusta and K. A. Olive, Nucl. Phys. A 408, 478 (1983). A. Bazavov et al., Phys. Rev. D 80, 014504 (2009). P. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 12, 009 (2001). S. Turbide, R. Rapp, and C. Gale, Phys. Rev. C 69, 014903 (2004). S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 69, 034909 (2004). A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 104, 132301 (2010). E. Frodermann, R. Chatterjee and U. Heinz, J. Phys. G 34, 2249 (2007), and references there-in. D. K. Srivastava and R. Chatterjee, Phys. Rev. C 80, 054914 (2009) [Erratum-ibid. C 81, 029901 (2010)].
293
Equation of state of strongly interacting matter and intensity interferometry of thermal photons Somnath Dea , Dinesh K. Srivastavaa , Rupa Chatterjeea,b a Variable b Department
Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India of Physics, P. O. Box 35 (YFL), FI-40014 University of Jyv¨askyl¨a, Finland
Abstract We find that an equation of state (EOS) for hot hadronic matter consisting of all mesons (baryons) having M < 1.5(2.0) GeV along with Hagedorn resonances in thermal and chemical equilibrium, matches rather smoothly with lattice EOS (p4 action, Nτ = 8) for T up to ≈ 200 MeV, when corrections are made for the finite volume of hadrons. Two equations of state, HHL and HHB are constructed where the above is matched to the lattice and bag model EoS respectively at a critical temperature T c = 165 MeV. We find that the particle and thermal photon spectra differ only marginally for the two equations of state at both RHIC and LHC energies. The intensity interferometry results, specially the outward correlations for thermal photons are found to be quite distinct for the two equations of state. Keywords: EOS, QGP, lattice, interferometry, particle spectra.
1. Introduction Recent results from the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory have provided significant evidences of the formation of Quark-Gluon Plasma (QGP) [1] or rather a strongly interacting QGP (sQGP) in collisions of heavy nuclei at relativistic energies. Important signatures of QGP formation include observations of jet quenching, elliptic flow, recombination of partons as a process of hadronization, and electromagnetic radiation. The system, produced in such collisions, behaves like a nearly ideal fluid [2] and relativistic hydrodynamics has been used quite successfully to explain its evolution. We consider that a quark-gluon plasma is formed in central collision of heavy nuclei at a very small initial formation time τ0 at the top RHIC and LHC energies. The plasma then expands, cools, and hadronizes either through a first order phase transition at T = T c , for a bag model equation of state or in a rapid cross-over as suggested by lattice QCD simulations. The hadronic matter part of the EOS consists of all mesons having M < 1.5 GeV and all baryons having M < 2 GeV along with or without Hagedorn resonances. The mass spectrum can be written as: ρ(m) = ρHG (m) + ρHS (m) , where ρHG =
gi δ(m − mi ) .
i
0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.06.008
(1) (2)
291
S. De et al. / Nuclear Physics A 862–863 (2011) 290–293 20
0.4
15
15
0.3
lattice p4; Nτ=8 hadron gas hadron gas(vol. corr.)
5
2
Cs
10
ε/T
ε/T
4
4
20
10 lattice p4; Nτ=8 hadron+Hagedorn gas hadron+Hagedorn gas (vol. corr.)
0.2
HHB HHL 0.1
5 (c)
(a) 0 0
0.1
0.2
0.3 T (GeV)
0.4
0.5
(b) 0 0
0.1
0.2
0.3
0.4
0.5
0
0.0
0.5
T (GeV)
1.0 1/4
ε
1.5
2.0
2.5
3 1/4
(GeV/fm )
Figure 1: (a) ε/T4 for hadron gas, volume corrected hadron gas, along with lattice results. (b) Corresponding results for hadron + Hagedorn gas without and with correction for finite volume of the particles. (c) Square of speed of sound for the two EOS, HHB and HHL (for detail see the Ref. [4]).
In the above, the sum runs over all the discrete hadronic states and their corresponding degeneracies are given by gi . We take the density of Hagedorn states from Ref. [3]: ρHS (m) = A
exp(m/T H ) , (m2 + m20 )5/4
(3)
where, A = 0.5 GeV3/2 , m0 = 0.5 GeV, T H = 0.196 GeV, and m varies from M0 = 2 GeV to MMax = 12 GeV. It is to be noted that the inclusion of the Hagedorn resonances is responsible for the rapid chemical equilibration of hadrons produced in relativistic heavy ion collisions [3]. Only mesonic Hagedorn states are considered in the present study. Various themodynamic quantities like - pressure, energy density, entropy densiry are calculated using standard methods and compared to the same obtained from lattice calculations (for details see Ref. [4]). For the finite volume corrections, we have used the method advocated by Kapusta and Olive [5], which is believed to be thermodynamically consistent. It is found that the above quantitites agree best with recent lattice QCD results (p4 action; Nτ = 8) [6] by including Hagedorn resonances and the finite volume correction for the hadrons at zero baryonic chemical potential (Fig.1(a) and 1(b)). Next we construct two EOS, HHB and HHL, where the volume corrected hadron + Hagedorn gas is used for both in the temperature limit T < 165 MeV. The bag model EOS (HHB) admits a mixed phase at T = 165 MeV, whereas the lattice equation of state (HHL) shows a sharp cross-over for 180 < T < 190 MeV. This is clearly seen in the variation of speed of sound as a function of ε1/4 (Fig.1(c)) for the two EOS. We solve the hydrodynamic equations for central collisions of Au+Au (200 A GeV) and Pb+Pb (5.5 A TeV) for the two equations of state with identical initial conditions. The average initial energy densities ε0 are: RHIC : ε0 LHC : ε0
= 80.80 GeV/fm3 = 718.3 GeV/fm3
at τ0 = 0.2 fm/c, at τ0 = 0.1 fm/c.
The freeze-out is assumed to take place at T = 100 MeV and the particle spectra are obtained using the CooperFrye formula. The invariant photon yield is obtained by integrating the emission rates over the spacetime volume of the system. The rate of production of photons from the quark matter is taken from the complete leading order calculation of Arnold et al. [7]. The production of photons from the hadronic matter is estimated using the results from Turbide et al. [8]. In Fig. 2 we have shown the results for transverse momentum distribution of thermal pions and photons for central collision of gold nuclei at midrapidity, for the two EOS, at RHIC energy. The experimental data for 0–5% most central collisions for the pions (from Ref. [9]) are also shown for comparison. No attempts were made to adjust any parameters or normalizations. It is obvious that both the EOS describe the data qualitatively well but the HHL EOS is slightly preferred. We also note that the inverse slope of the spectra for HHL EOS is larger than the same for HHB EOS. The slope of thermal photon spectrum is quite close to the earlier calculations reported in Ref. [11]. Also the data for single photons for 0–20% most central collisions [10] are shown in Fig. 2 for a comparison. A complete description of photon data would involve addition of prompt contribution scaled by appropriate nuclear overlapping function. No significant difference is seen for the results at top LHC energy, as well (see Ref. [4]). Thus we conclude that the particle and photon spectra are seen to be only marginally dependent on the EOS.
292
S. De et al. / Nuclear Physics A 862–863 (2011) 290–293
10
3
Thermal photon spectrum for Au+Au @RHIC
-2
dN/d kT dy (GeV )
1
HHL HHB 0-20% PHENIX
10
-1
2
10
QM+HM -3
10
-5
10 0
0.5
1
1.5
2
2.5
3
3.5
4
kT (GeV)
Figure 2: Thermal pion and photon pT spectra at RHIC for the HHB and HHL equations of state. The experimental data for hadrons (0–5% centrality bin) are taken from [9] and photon data for 0–20% centrality bin from [10]. The calculations are for the impact parameter b=0 fm (see the Ref. [4] for detail). 1.6
1.6
Au+Au@RHIC, γγ Intensity Correlation
Au+Au@RHIC, γγ Intensity Correlation 1.5
Thermal Photons, k1T=1.7 GeV
1.3
QM+HM
1.2
HHL HHB
1.3 1.2
1.1 1.0 0
Thermal Photons, k1T=1.7 GeV
1.4
HHL HHB C
C
1.4
1.5
QM+HM
1.1 0.04
0.08
0.12
0.16
1.0 0
qo (GeV)
0.04
0.08
0.12
0.16
ql (GeV)
Figure 3: The outward (left panel) and longitudinal correlation (right panel) functions for thermal photons produced in central collision of gold nuclei at RHIC (see the Ref. [4] for detail).
Next we calculate two photon intensity correlation which is found sensitive to the details of evolution dynamics of the system [12]. We calculate the outward, side-ward, and longitudinal correlation functions of thermal photons for a typical momentum KT ∼ 1.7 GeV. The correlation function between two photons with momenta k1 and k2 is defined as [12]: d4 x S (x, K)eix·q 2 1 C(q, K) = 1 + . (4) 2 d4 x S (x, k1 ) d4 x S (x, k2 ) In the above, S (x, K) is the spacetime emission function, and q = k1 − k2 , K = (k1 + k2 )/2 .
(5)
The space-time emission function S is approximated as photon production rate , EdN/d4 xd3 k, for the QGP and hadronic phases. In Fig. 3 we have plotted the correlation function for outward (qo ) and longitudinal (ql ) momentum differences (see e.g. [4]). We choose momenta of the interfering photons such that when qo 0, q s and ql are identically zero. Photons from QGP originate at early times and those from hadronic matter at a later time and interference occurs due to the spatiotemporal difference between the two sources. The outward momenta difference qo couples the spatial as well as temporal difference, ql depends on the temporal difference only. We find that the spacetime distribution of the source is very different for the two equations of state [4]. This is reflected in the outward correlation function which clearly distinguishes between the two equations of state. The longitudinal correlation also shows a small differnce at RHIC energy and the difference becomes notciable at LHC energy [4]. In summary, we have shown that inclusion of Hagedorn resonances and finite volume correction can give a rich description of haronic matter. Also the direct photon intensity interferometry can be a valuable probe for the equation of state of strongly interacting matter if verified experimentally.
S. De et al. / Nuclear Physics A 862–863 (2011) 290–293
Acknowledgements RC would like to thank the Finnish Academy Project (No. 130472) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
K. Adcox et al [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005). R. A. Lacey et al., Phys. Rev. Lett. 98, 092301 (2007). J. Noronha-Hostler, M. Beitel, C. Greiner, and I. Shovkovy, Phys. Rev. C 81, 054909 (2010). S. De, D. K. Srivastava and R. Chatterjee, J. Phys. G 37, 115004 (2010). J. I. Kapusta and K. A. Olive, Nucl. Phys. A 408, 478 (1983). A. Bazavov et al., Phys. Rev. D 80, 014504 (2009). P. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 12, 009 (2001). S. Turbide, R. Rapp, and C. Gale, Phys. Rev. C 69, 014903 (2004). S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 69, 034909 (2004). A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 104, 132301 (2010). E. Frodermann, R. Chatterjee and U. Heinz, J. Phys. G 34, 2249 (2007), and references there-in. D. K. Srivastava and R. Chatterjee, Phys. Rev. C 80, 054914 (2009) [Erratum-ibid. C 81, 029901 (2010)].
293