Formation Time of QGP from Thermal Photon Elliptic Flow
aa r X i v : . [ nu c l - t h ] S e p Formation Time of QGP from Thermal Photon Elliptic Flow
Rupa Chatterjee and Dinesh K. Srivastava
Variable Energy Cyclotron Centre, 1 / AF Bidhan Nagar, Kolkata 700 064, India
Abstract
We show that the transverse momentum dependent elliptic flow v ( p T ) of thermal photons isquite sensitive to the initial formation time ( τ ) of Quark Gluon Plasma (QGP) for semi-centralcollision of gold nuclei at RHIC [1]. A smaller value of the formation time or a larger initialtemperature leads to a significant increase in the thermal photon radiation from QGP phase,which has a smaller v . The elliptic flow of thermal photon is dominated by the contributionfrom the quark matter at intermediate and high p T range and as a result sum v decreases withsmaller τ for p T ≥ . τ .
1. Introduction
Heavy ion collisions at relativistic energies lead to formation of Quark-Gluon Plasma, adeconfined novel state of quarks and gluons in local thermal equilibrium. Interesting resultsof jet quenching due to parton energy loss in the medium [2] and elliptic flow of identifiedparticles [3] at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab haveprovided significant evidence of the formation of QGP. However, one of the most importantissues in this study, the initial formation time of the plasma, or the onset of collectivity andthermalization in the system, beyond which the powerful method of hydrodynamics can be usedto describe its evolution is still not known precisely.In a very simple treatment it is assumed that, the partons produced in the collisions have anaverage energy h E i , and thus their formation time τ ∼ / h E i . Looking at the various treatmentsand perhaps complementary models available in the literature, a direct measurement of the for-mation time would be very desirable. We show that the elliptic flow of thermal photons usingideal hydrodynamics can be very useful to estimate the value of τ accurately [1]. τ sensitivity of the elliptic flow parameter at RHIC It is quite well accepted that, photons are one of the most e ffi cient probes to explore theproperties of the hot and dense system produced in the collision of heavy nuclei at relativisticenergies. Being electromagnetic in nature, they do not su ff er any final state interactions and carryundistorted information about the circumstances of their production directly to the detector. Also,photons are emitted throughout the life time of the evolving system, whereas hadrons are emittedonly from the surface of freeze-out.In our earlier interesting work on elliptic flow of thermal photons at RHIC, we reported thatthe v ( p T ) of thermal photons using ideal hydrodynamics exhibits a completely di ff erent nature Preprint submitted to Nuclear Physics A November 7, 2018 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 p T (GeV/c) -4 -3 -2 -1 d N / d p T d y ( c G e V - ) Au+Au@RHIC, b=6 fm ρ p T (GeV/c) v ( p T ) Au+Au@200 A GeV; b=6 fm ρ Figure 1: p T spectra [left panel] and elliptic flow [right panel] of ρ mesons considering di ff erent initial formation time τ of the plasma for 200A GeV Au + Au collisions at RHIC and at b = compared to the elliptic flow of hadrons at intermediate and large p T [4]. Thermal photon v from quark matter is very small at large p T or early times, it gradually rises with smaller valuesof p T and then falls again as p T → v fromhadronic matter photons rises monotonically with p T , which is similar to the elliptic flow ofhadrons predicted by hydrodynamics. The sum v tracks the v ( p T ) from quark matter at large p T (inspite of very large v values from hadronic matter) as the quark matter radiation dominatesthe spectra beyond p T value of about 1 GeV / c . Thus, the thermal photon v at large p T reflectsthe momentum anisotropies of the partons produced at early times, soon after the collisions.We follow the same treatment and initial conditions as used in Ref. [4] to calculate the p T spectra and elliptic flow of thermal photons at di ff erent τ for semi-central collision of Au nucleiat √ s NN =
200 GeV at RHIC. Cooper-Fry formulation is used to calculate the p T spectra fordi ff erent hadrons considering the same initial conditions as for thermal photons. We assume thatthe system reaches a state of maximum entropy at the point of thermalization at a time τ andthen follows an isentropic expansion. The initial entropy density s is obtained by combining‘hard’ and ‘soft’ contributions which are the contributions from binary collisions ( n b ) woundednucleons ( n w ) respectively, and it follows the relation [5], s ( τ , x , y , b ) = κ [ α n w ( x , y , b ) + ( 1 − α ) n b ( x , y , b ) ] . (1) κ and α ( = n b and n w at di ff erent points in the transverse plane for a particular impact parameter b.The value of κ is obtained considering the initial entropy density at x = y = b = − when τ is 0.6 fm / c . [6]. We use boost invariant and impact parameter dependent azimuthallyanisotropic hydrodynamics to model our system, where the plasma experiences a first order phasetransition at a temperature of about 164 MeV and the freeze-out energy density is considered at0.075 GeV / fm . Thermal photon emission from the quark matter and the hadronic matter areobtained by integrating the rates of emission over the space time volume d x ( = dx dy τ d τ d η ).Photons from quark matter are calculated considering a complete leading order photon produc-tion rate from Arnold et al. [7] and we use the latest results by Turbide et al. [8] for thermalradiation from the hot hadronic gas. The elliptic flow parameter v , which directly reflects the2 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 p T (GeV/c) v ( p T ) e f = 0.045 GeV/fm e f = 0.135 GeV/fm πρ b = 6 fm τ = p T (GeV/c) -11 -9 -7 -5 -3 -1 d N / d p T d y ( c G e V - ) Thermal Photons; HM
Au+Au@200 AGeV b = 6 fm
Figure 2: [Left panel] Elliptic flow of primary hadrons with changing freeze-out density. [Right panel] Thermal photonsfrom hadronic phase (only) at di ff erent τ . rescattering among the particles produced in the collisions, is obtained using the relation, dN ( b ) d p T dy = dN ( b )2 π p T d p T dy (cid:2) + v ( p T , b ) cos(2 φ ) + ... (cid:3) . (2) at di ff erent τ The p T spectra and v ( p T ) of several hadrons (nearly complete list available in the particledata book) are calculated at di ff erent τ (ranging from 0.2 to 1.0 fm / c , in steps of 0.2 fm / c ) by keeping the total entropy of the system fixed. Particle spectra and elliptic flow resultsfor ρ mesons at di ff erent τ for semi-central collision of Au nuclei are shown in Fig. 1. Theemission of hadrons is mainly governed by the conditions of the freeze-out hyper-surface or,in particular, the temperature at freeze-out ( T f ∼
120 MeV). We see that the heavier particlesrespond strongly to radial flow with changing τ , whereas the elliptic flow results for all thehadrons remain almost una ff ected with changing initial formation time of the plasma. Earlystart of flow drives the freeze-out to happen sooner and hence the overall flow does not changesignificantly with changing τ for them. We find the same insensitivity to τ for all the hadronsin the particle data book. The e ff ect of changing freeze-out conditions on the hadron spectra andelliptic flow are checked and resultant elliptic flow for π and ρ mesons are shown in left panelof Fig. 2. We find that the flow parameter for hadrons essentially acquire its final value at somelarger temperature and does not change significantly with changing freeze-out conditions. ff erent τ We see that, the thermal photon spectra from hadronic phase at di ff erent τ are almost inde-pendent of the value of τ for p T < . / c . Only at very large p T ( ∼ / c ) values, thespectra at a smaller τ is flatter than that at a larger τ [see right panel of Fig. 2]. We know thatthe total entropy of the system is related to the particle number density and initial parameters bythe relation, S ( η ) ∝ dN / dy ∝ T τ , where, T is the initial temperature of the plasma at time τ . With smaller values of τ , the radiation from QGP phase increases significantly as the initialtemperature of the system increases. As a result, the photon spectra at τ = . / c is muchflatter than the same at τ = . / c and the two results di ff er by a few order of magnitudes at3 .0 1.0 2.0 3.0 4.0 5.0 6.0 p T (GeV/c) -9 -7 -5 -3 -1 d N / d p T d y ( c G e V - ) Thermal & Prompt Photons
Au+Au@200 AGeV
PHENIX; 10 - 20% most central
NLO pQCD p T (GeV/c) v ( p T ) Au+Au@200 AGeV
Thermal Photons b = 6 fm QM QM+HM
Figure 3: [Left panel] Thermal photon spectra at di ff erent formation time at RHIC. [Right panel] Di ff erential elliptic flowof thermal photon for di ff erent initial time τ . the intermediate and high p T range. Our results from hydrodynamics along with prompt photonresults using NLO pQCD [9] and experimental data from PHENIX [10] are shown in left panelof Fig. 3. Although the thermal photon yield from quark matter increases with smaller τ , the de-velopment of elliptic flow is not substantial at very early times and hence the v ( p T ) from quarkmatter decreases with smaller τ . As the v from hadronic matter is not a ff ected significantlywith changing τ , the sum v decreases with τ beyond a p T value of about 1.5 GeV / c as shownin right panel of Fig. 3.In conclusion, we show that the thermal photon v is quite sensitive to the formation time ofQGP and its value can be estimated precisely with the help of experimental determination of theflow parameter v , whereas the v ( p T ) for hadrons depends only marginally on the value of τ . Acknowledgments
RC would like to thank the QM09 organizing committee for allowing her to present the talkvia telephone. Special thanks to David Silvermyr and Vince Cianciolo for making the arrange-ment possible.
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