aa r X i v : . [ nu c l - t h ] M a r Thermal photon v with slow quark chemical equilibration Akihiko Monnai ∗ RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: June 28, 2018)Elliptic flow of direct photons in high-energy heavy ion collisions has been a topic of great interestas it is experimentally found larger than most hydrodynamic expectations. I discuss the implicationof possible late formation of quark component in the hot QCD medium on the photon elliptic flowas quarks are the source of thermal photons in the deconfined phase. Hydrodynamic equations arenumerically solved with the evolution equations for quark and gluon number densities. The numer-ical results suggest that thermal photon v is visibly enhanced by the slow chemical equilibration ofquarks and gluons, reducing the aforementioned problem. PACS numbers: 25.75.Cj, 25.75.-q, 25.75.Nq, 25.75.Ld
I. INTRODUCTION
QCD matter under extreme temperatures has beenstudied extensively in high-energy nucleus-nucleus colli-sions. One of the most remarkable achievements in heavyion physics is the realization of a deconfined phase calledquark-gluon plasma (QGP) at the Relativistic HeavyIon Collider (RHIC) in Brookhaven National Laboratory[1] and the Large Hadron Collider (LHC) in EuropeanOrganization for Nuclear Research [2]. The observedQGP is characterized with large azimuthal momentumanisotropy in hadronic particle spectra compared withthe spacial anisotropy of the system originating from thecollision geometry, which is believed to be the indicationof the existence of a strongly-coupled medium near ther-mal equilibrium [3]. Recent relativistic hydrodynamicanalyses provide quantitative description of hadronic el-liptic flow v h by including the effects of viscosity andgeometrical fluctuation [4]. On the other hand, directphoton elliptic flow v γ turned out to be a few times largerthan most of the hydrodynamic predictions in the exper-iments [5, 6]. It should be noted that thermal emissionof photons from anisotropic medium – thermal photons – rather than photon production at the time of colli-sion – prompt photons – would be the origin of directphoton v because experimental data indicate that thehot medium is chromodynamically opaque but electro-magnetically transparent. There are no conclusive un-derstanding of this discrepancy, though it has been ap-proached from various prominent perspectives [7–19].A heavy ion system at high energies goes through sev-eral stages after the collision. The color glass condensatepicture [20] indicates that the colliding nuclei is describedas saturated gluons. It is then speculated to quicklyisotropize, thermalize, and chemically equilibrate in a rel-atively short period of time ∼ O (1) fm/ c after the col-lision. Since the elliptic flow, or momentum anisotropy,of the bulk medium is not fully-developed in the earlystages of hydrodynamic evolution, v γ is na¨ıvely expected ∗ [email protected] to be smaller than v h , which is calculated using the flowprofile at the last stage of the evolution.In this paper, I investigate possible thermal photon v enhancement by late quark chemical equilibration be-cause thermal photons in the QGP phase are emittedby quarks. Most modern estimation for their emissionrate is calculated for a completely equilibrated medium,though there is an indication that chemical equilibrationof quarks and gluons could be slower than thermaliza-tion, a requirement for the onset of hydrodynamic flow,in several theoretical models [21–23]. The photon emis-sion from such systems is discussed extensively in termsof transverse momentum spectra [24, 25]. It is naturallyexpected that v γ can become large if the thermal pho-tons are created mainly in the middle-late stages wherethere already is a sizable momentum anisotropy in themedium. Thus late production of quark component in aheavy ion system can contribute to larger v γ .The paper is organized as follows. Sec. II describesthe formulation of relativistic hydrodynamics with theevolution of the quark and the gluon number densities.Thermal photon emission mechanism is embedded in thehydrodynamic model. In Sec. III, I present numericalresults for the thermal photon elliptic flow for chemi-cally equilibrated and non-equilibrated systems. Sec. IVis devoted for discussion and conclusions. The natu-ral unit c = ~ = k B = 1 and the Minkowski metric g µν = diag(+ , − , − , − ) is used throughout this paper. II. HYDRODYNAMIC MODEL FORQUARK-GLUON MIXTURE SYSTEM
In this section, I would like to develop a hydrodynamicmodel for chemically non-equilibrated systems with ther-mal photon emission. The time evolution equations forthe quark and the gluon number densities are coupled tothe energy-momentum conservation. The parton numberdensities are simply assumed to vanish at quark-hadroncrossover. Thermal photon emission from the QGP andthe hadron phases are considered for the estimation ofphoton elliptic flow. The purpose of the paper is to in-vestigate qualitative implications of late chemical equi-libration on the aforementioned quantity and precisionanalyses is beyond the scope of current discussion.The background bulk medium for the thermal photonemission is the flow field provided by a hydrodynamicmodel. The energy-momentum conservation for multi-component hydrodynamics of quark-gluon mixture sys-tem in inviscid case would be given as ∂ µ T µνq + ∂ µ T µνg = 0 , (1)where T µνq = ( e q + P q ) u µq u νq − P q g µν , (2) T µνg = ( e g + P g ) u µg u νg − P g g µν . (3)Here I make an ansatz that thermal equilibration is com-plete at the beginning of hydrodynamic evolution andassume the flows are common, i.e. , u µ ≡ u µg = u µq . Thenone can define the energy density e = e q + e g and thehydrostatic pressure P = P q + P g . Here the subscripts q and g denote the quantities for quarks and gluons, re-spectively. The net baryon number is neglected assum-ing the numbers of quarks and antiquarks are equal inhigh-energy heavy ion systems. The matter-antimatterdegrees of freedom are included in T µνq .The particle numbers are not conserved as the systemis not chemically equilibrated. Since the parton numberchanging processes are (a) g ⇋ g + g , (b) g ⇋ q + ¯ q ,and (c) q (¯ q ) ⇋ q (¯ q ) + g , the evolution equations for thequark and the gluon number currents for the 1-to-2/2-to-1 processes would be phenomenologically given as ∂ µ N µq = 2 r b n g − r b n eq g ( n eq q ) n q , (4) ∂ µ N µg = ( r a − r b ) n g − r a n eq g n g + r b n eq g ( n eq q ) n q , + r c n q − r c n eq g n q n g , (5)where r a , r b , and r c are the reaction rates for gluon split-ting, quark pair production, and gluon emission from aquark, respectively. n q is the quark number density and n g is the gluon number density. The subscript “eq” de-notes the quantity in chemical equilibrium. The num-ber currents can be decomposed as N µq = n q u µ and N µg = n g u µ in the inviscid case. The reaction ratesare dependent on the temperature T in thermal equilib-rium. It should be noted that they are valid only abovecrossover temperature T > T c before quarks and gluonsare confined in hadrons. Below the temperature the sys-tem is assumed to be hadronic, i.e. , n q = n g = 0 andchemically equilibrated. One may in principle considera chemical equilibrating hadronic system and write sep-arate equations for each component but it is beyond thescope of this paper as our interest is the modification ofthermal photon v by the late quark production.The reaction rates are parametrized as r i = c i T ( i = a, b, c ). Here c i is the dimensionless parameter that deter-mines relaxation time scale. It is implied that c b > c a , c c leads to slower chemical equilibration when the initialmedium is gluon rich, as would be the case for high-energy heavy ion systems. Let us consider an extremecase where c a = c c = 0 for non-expanding systems, i.e. , u µ = (1 , , , n q ( t ) = 2 n g (0)[1 − exp ( − r b t )] , (6) n g ( t ) = n g (0) exp ( − r b t ) , (7)for n q ≪ n q (0) = 0, which implies that τ chem ≡ /r b could be defined as a typical time scale for chemicalequilibration of the system. Note that the actual equili-bration time can be longer due to the decrease of r b bythe cooling effect of expanding systems and the presenceof gluon number increasing processes. More detailed dis-cussion which includes the quark recombination effect insemi-dense regions can be found in Appendix.The parton gas picture yields n eq q = 3 ζ (3) d q T / π and n eq g = ζ (3) d g T /π where ζ (3) ∼ . d q = 24 and d g = 16 when N f = 2.The color glass picture with the running coupling α s =0 . Q s = 2 GeV suggests n g ( τ ) ∼ Q s / π where Q s is chosen so that n g ( τ ) = n eq g + n eq q / τ is the initial time.Thermal photon emission rates from the QGP and thehadron phases are calculated based on Refs. [27] and [28],respectively. Since the emission rates are functions ofthe quark distribution, it is simply factored by n q /n eq q for chemically non-equilibrated systems. The rate in theQCD crossover region is interpolated with a hyperbolicfunction as E dR γ d p = 12 (cid:18) − tanh T − T c ∆ T (cid:19) E dR γ hadron d p + 12 (cid:18) T − T c ∆ T (cid:19) E dR γ QGP d p , (8)where the parametrization is set to T c = 0 .
17 GeV and∆ T = 0 .
017 GeV. The photon emission below p = 0 . dN γ dφ p p T dp T = Z dx dR γ dφ p p T dp T , (9)where φ p is the azimuthal angle in momentum space and p T is the transverse momentum. The energy density isgiven by p µ u µ . Then the differential photon v is definedas the second harmonics in Fourier expansion: v γ ( p T ) = R π dφ p cos(2 φ p ) dN γ dφ p p T dp T R π dφ p dN γ dφ p p T dp T . (10)In this work, the hydrodynamic and the parton evolu-tion equations are solved with a newly-developed (2+1)-dimensional hydrodynamic model, which assumes boost-invariance in the longitudinal direction. A smooth ini-tial condition is considered for the present hydrodynamicanalyses. The energy density profile is the one fromRef. [29] based on the wounded nucleon density with b = 7 fm for demonstrative purposes. The normaliza-tion is chosen so that e = 30 GeV/fm when b = 0 fm.The initial time is set to τ = 0 . c . The equationof state (EoS) in a chemically non-equilibrated system isnon-trivial because most of the lattice QCD simulationsare for equilibrated systems. Here it is simply approxi-mated by interpolating the entropy density of the piongas s hadron = 3 × π T /
90 and that of the parton gas s QGP = 37 × π T /
90 at N f = 2 with the same hyper-bolic function as the one for the photon emission rate.This preserves consistency with the parton density pic-ture introduced in the equilibrium values n eq q and n eq g forthe deconfined phase. III. RESULTS
The differential photon elliptic flow v γ ( p T ) in chemi-cally non-equilibrated QCD medium is shown in compar-ison with the one in chemical equilibrium for the trans-verse momentum range 0 . < p T < c a = c c = 1 . c b = 0 . T f = 0 .
15 GeVis taken into account.One can see that the elliptic flow is enhanced for thelate production of the thermal photon emission sources.Considering that most of the hydrodynamic analysesso far under-predicted photon momentum anisotropy,the result would provide an important viewpoint onhow to interpret the phenomenon in terms of pre- andpost-equilibrium physics, i.e. , quark pair production insplitting-based picture, e.g. , Ref. [23] and relativistic hy-drodynamics. The mean transverse momentum of ther-mal photons is also enhanced. One has to be carefulbecause the results are sensitive to the EoS, the durationtime of hydrodynamic phase, and the initial conditions.It should be noted that validity of background hydrody-namic flow analyses should not be na¨ıvely assumed formid-high p T regions above ∼ p T regions.Magnitude of chemical equilibration of the system atthe center of the hot medium x = y = 0 fm is shown inFig. 2 before the system reaches T = 0 .
17 GeV where theparton picture would no longer be valid. The longitudinalexpansion of the system leads to quick reduction in thenumber density during the time evolution. The systemeventually approaches the one in equilibrium. The effec-tive relaxation time for chemical equilibration is ∼ c with the current choice of reaction rate parameters,which is roughly in agreement with the estimation withEq. (6) because τ chem = 1 /c b T ∼ c for c b = 0 . T ∼ . n eq q below the (GeV) T p γ v γ v =1.5) a,c =0.5, c b (c γ v FIG. 1. The thermal photon elliptic flow v γ as a function oftransverse momentum for a chemically-equilibrated medium(solid line) and a non-equilibrated medium with c a = c c = 1 . c b = 0 . (fm/c) τ ) n ( G e V eqq n q n FIG. 2. The quark number densities in chemical equilibrium(solid line) and in dynamical evolution with c a = c c = 1 . c b = 0 . T c = 0 .
17 GeV. The dotted and the dash-dotted lines denotethat the system is near the crossover region
T < T c + ∆ T where ∆ T = 0 .
017 GeV. crossover region because the EoS is no longer from theone for the parton gas model, representing the breakdownof the quark-gluon picture.Fig. 3 shows the case where only the quark produc-tion process is present, i.e. , c a = c c = 0 and c b = 0 . v γ . The resultagain shows visible enhancement and is also not so dif-ferent from the previous calculation with gluon emissionprocesses. This can be interpreted that the chemical re-laxation time scale is mostly determined by the slowestprocess. By choosing larger quark production rate r b , i.e. , setting the shorter chemical relaxation time, one re- (GeV) T p γ v γ v =0) a,c =0.5, c b (c γ v FIG. 3. The thermal photon elliptic flow v γ as a function oftransverse momentum for a chemically-equilibrated medium(solid line) and a non-equilibrated medium with c a = c c = 0and c b = 0 . covers the result in equilibrium. IV. DISCUSSION AND CONCLUSIONS
I have developed a hydrodynamic model with quarkand gluon number density evolution and estimated ther-mal photon emission from the expanding medium withboost invariance at mid-rapidity. The numerical resultsshow that elliptic flow for thermal photons can be visi-bly enhanced by the slower quark chemical equilibration.This can be one of the mechanisms for explaining thelarge photon v problem in high-energy heavy ion colli-sions [5, 6]. There is possible overestimation in the cur-rent analyses because the quark number should be finiteat the beginning of the hydrodynamic stage while it issimply assumed to be vanishing in the present estima-tion. Also more microscopic treatments would be nec-essary for determining the parameters for the chemicalequilibration processes without ambiguities.It should be noted that late quark production leadsto the reduction in photon p T spectra. Since the spec-tra is often employed to experimentally constrain thetemperature of the medium [30, 31], this would lead tohigher medium temperature for chemically-equilibratingsystems for consistency. Introduction of more realisticoff-equilibrium equation of state and fine-tuning of ini-tial conditions, including thermalization times, would beessential for more quantitative discussion.Further future prospects include estimation of the ef-fects of shear and bulk viscosities, especially since it isknown that in addition to the flow modification duringthe hydrodynamic evolution, the distortion of distribu- tion can have non-trivial effects on particle spectra andelliptic flow of hadrons at freezeout [32–34]. Triangularand higher-order flow of thermal photons v γn are inter-esting observables because they would play an importantrole in distinguishing the origin of the large elliptic flowfor photons, i.e. , one should be able to distinguish if it isbased on internal medium properties of QCD systems asdiscussed in the present paper or on external properties ofheavy ion geometry such as strong magnetic fields. Onecan expect that the former scenario would lead to large v γ while the latter would not. Also if one considers separateflows for quarks and gluons, then interesting interplay ofthe intrinsic medium properties and the heavy ion ge-ometry should be observed as only the former would bedirectly affected by the strong magnetic field. ACKNOWLEDGMENTS
The work is inspired by fruitful discussion withB. M¨uller. The author would like to thank for valuablecomments by Y. Akiba and L. McLerran on the paper.The work of A.M. is supported by RIKEN Special Post-doctoral Researcher program.
Appendix: ANALYTIC EXPRESSION OF QUARKAND GLUON NUMBER DENSITIES
The equations (4) and (5) can be analytically solvedwithout neglecting the quark recombination terms fornon-expanding geometry when c a = c b = 0. The quarknumber and gluon number densities at a given time t are n q = ( n eq q ) n eq g (cid:20) I tanh (cid:18) I r b t + I (cid:19) − (cid:21) , (A.1) n g = n eq g + n eq q − ( n eq q ) n eq g (cid:20) I tanh (cid:18) I r b t + I (cid:19) − (cid:21) , (A.2)where I = 1 + 4 n eq g n eq q , (A.3) I = tanh − I , (A.4)when n q ( t = 0) = 0. One has I ∼ . N f = 2, which implies that the typical timescale for equilibration in the presence of the recombina-tion process would be slightly shorter the inverse reactionrate 1 /r b in non-expanding medium. This would be be-cause equilibration does not require complete conversionof gluons into quarks as is the case for systems with norecombination. The actual chemical equilibration wouldbecome longer than this estimation because of the gluonnumber increasing processes for non-vanishing c a and c c . [1] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys.A , 184 (2005); J. Adams et al. [STAR Collabora-tion], Nucl. Phys. A , 102 (2005); B. B. Back et al. [PHOBOS Collaboration], Nucl. Phys. A , 28 (2005);I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys. A , 1 (2005).[2] K. Aamodt et al. [The ALICE Collaboration], Phys. Rev.Lett. , 252302 (2010).[3] P. F. Kolb, P. Huovinen, U. W. Heinz and H. Heiselberg,Phys. Lett. B , 232 (2001); P. Huovinen, P. F. Kolb,U. W. Heinz, P. V. Ruuskanen and S. A. Voloshin,Phys. Lett. B , 58 (2001); P. F. Kolb, U. W. Heinz,P. Huovinen, K. J. Eskola and K. Tuominen, Nucl.Phys. A , 197 (2001); D. Teaney, J. Lauret, andE. V. Shuryak, Phys. Rev. Lett. , 4783 (2001);arXiv:nucl-th/0110037; T. Hirano, Phys. Rev. C ,011901 (2002); T. Hirano and K. Tsuda, Phys. Rev. C , 054905 (2002).[4] B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. ,042301 (2011); Phys. Lett. B , 59 (2011); Phys. Rev.C , 024901 (2012).[5] A. Adare et al. [PHENIX Collaboration], Phys. Rev.Lett. , 122302 (2012).[6] D. Lohner [ALICE Collaboration], J. Phys. Conf. Ser. , 012028 (2013).[7] R. Chatterjee, E. S. Frodermann, U. W. Heinz andD. K. Srivastava, Phys. Rev. Lett. , 202302 (2006).[8] R. Chatterjee and D. K. Srivastava, Phys. Rev. C ,021901 (2009).[9] R. Chatterjee, H. Holopainen, T. Renk and K. J. Eskola,Phys. Rev. C , 054908 (2011).[10] H. Holopainen, S. Rasanen and K. J. Eskola, Phys. Rev.C , 064903 (2011).[11] R. Chatterjee, H. Holopainen, I. Helenius, T. Renk andK. J. Eskola, Phys. Rev. C , 034901 (2013).[12] R. Chatterjee, D. K. Srivastava and T. Renk,arXiv:1401.7464 [hep-ph].[13] F. -M. Liu, T. Hirano, K. Werner and Y. Zhu, Phys. Rev.C , 034905 (2009).[14] H. van Hees, C. Gale and R. Rapp, Phys. Rev. C ,054906 (2011). [15] M. Dion, J. -F. Paquet, B. Schenke, C. Young, S. Jeonand C. Gale, Phys. Rev. C , 064901 (2011).[16] C. Shen, U. W. Heinz, J. -F. Paquet, I. Kozlov andC. Gale, arXiv:1308.2111 [nucl-th].[17] A. Bzdak and V. Skokov, Phys. Rev. Lett. , 192301(2013).[18] B. M¨uller, S. -Y. Wu and D. -L. Yang, Phys. Rev. D ,026013 (2014).[19] O. Linnyk, V. P. Konchakovski, W. Cassing andE. L. Bratkovskaya, Phys. Rev. C , 034904(2013); O. Linnyk, W. Cassing and E. Bratkovskaya,arXiv:1311.0279 [nucl-th].[20] L. D. McLerran and R. Venugopalan, Phys. Rev. D ,2233 (1994); D , 3352 (1994).[21] K. Geiger, Phys. Rev. D , 4965 (1992); 4986 (1992).[22] T. S. Biro, E. van Doorn, B. M¨uller, M. H. Thoma andX. N. Wang, Phys. Rev. C , 1275 (1993).[23] A. Monnai and B. M¨uller, in preparation.[24] B. Kampfer and O. P. Pavlenko, Z. Phys. C , 491(1994).[25] M. Strickland, Phys. Lett. B , 245 (1994).[26] J. D. Bjorken, Phys. Rev. D , 140 (1983).[27] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP ,009 (2001).[28] S. Turbide, R. Rapp and C. Gale, Phys. Rev. C ,014903 (2004).[29] P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Rev. C , 054909 (2000).[30] A. Adare et al. [PHENIX Collaboration], Phys. Rev.Lett. , 132301 (2010).[31] M. Wilde [ALICE Collaboration], Nucl. Phys. A , 573c (2013).[32] D. Teaney, Phys. Rev. C , 034913 (2003).[33] A. Monnai and T. Hirano, Phys. Rev. C , 054906(2009).[34] G. S. Denicol, T. Kodama, T. Koide and P. .Mota, Phys.Rev. C80