A Phenomenological Model of the Glasma and Photon Production
AA Phenomenological Model of the Glasma and PhotonProduction ∗ Larry McLerran (1 , ,
1. Physics Department, Brookhaven National Laboratory, Upton, NY11973, USA2. RIKEN BNL Research Center, Brookhaven National Laboratory, Up-ton, NY 11973, USA3. Physics Dept. Central China Normal University, Wuhan, China
I discuss a phenomenological model for the Glasma. I introduce overoccupied distributions for gluons, and compute their time evolution. Iuse this model to estimate the ratio of quarks to gluons and the entropyproduction as functions of time. I then discuss photon production at RHICand LHC, and how geometric scaling and the Glasma might explain genericfeatures of such production.
1. Introduction
There have been many talks at this meeting concerning the Color GlassCondensate[1]-[5] and the Glasma[6]-[13], so I will not present an extendedreview the subject in this talk. I will concentrate here on providing a sim-plified description of the evolution of the Glasma. The Glasma is a stronglyinteracting Quark Gluon Plasma. It is not thermalized. It is produced veryshortly after the collision of two nuclei, thought of as sheets of Color GlassCondensate, and evolves into the Thermalized Quark Gluon Plasma. TheGlasma is strongly interacting because the gluon distributiuons are over oc-cupied, and this overoccupation enhances the interaction strength due toBose coherence. There may or may not be a Bose condensate of gluonsin the Glasma, but this interesting feature will not be the subject of thistalk[14]-[22]. In fact, I will ignore the possibility of such condensation whenI analyze the Glasma, although the result I present may be generalized tothe case where condensation is present. ∗ Invited talk presented at the 54’th Cracow School of Theoretical Physics, Zakopane,Poland, June 2014 (1) a r X i v : . [ h e p - ph ] N ov zakopane printed on October 25, 2018
2. The Glasma
The Glasma is compoased of gluons that are highly coherent. The max-imal coherence occurs at some momentum scale Λ IR ( t ) where the gluondistributions have strength of order 1 /α s . At this infrared scale, the inter-actions of gluons are maximally strong since the 1 /α S in the gluon distribu-tions eats factors of α s due to gluon interactions. This is easily seen, since ifthe gluon field has strength 1 /g , the coupling strength scales out of classicalequations for the gluon fields.There is also an ultraviolet scale Λ UV ( t ) at which the gluon distributionrapidly goes to zero. In the early stages of the evolution of the Glasma, thegluons are characterized by only one scale, and the distribution functionsare maximally strong. We have the initial conditionsΛ IR ( t in ) ∼ Λ UV ( t in ) ∼ Q sat (1)where Q sat is the typical momentum scale associated with the field in theColor Glass Condensate, which determines the initial conditions for theGlasma.As time evolves, both the infrared and ultraviolet scales change. Ther-malization can oocur whenΛ IR ( t therm ) ∼ α s Λ UV ( t therm ) ∼ α s T init (2)where T init is the temperature when the system first thermalizes. One cansee this from thermal Bose distribution functions f BE = 1 e E/T − f ∼ T /E . At the UVscale of Eqn. 2, the distribution is of order 1, but at the IR, the distributionsare of order 1 /α S Usually in thermal field theory it is argued there is a magnetic massgenerated at the scale m mag ∼ α S T , which guarantees the distributions donot get too large at small momentum scales. It might however happen thatfor the highly occupied distributions typical of heavy ion collisions, that wemight genenate a chemical potential for the gluons. If there are too manygluons, this chemical potential would approach the gluon mass, and thegluon distribution would be infinite at zero momentum. If one integratesover the gluon distribution, the singularity is integrable, and the number ofgluons is a fixed number. If the number of gluons in our system exceeds thisnumber, then the remainder must go into a Bose condensate. Whether ornot such condensation occurs is a matter of much discussion and controversy, akopane printed on October 25, 2018 and depends upon dynamical details that are not yet understood[14]-[22].We will make the conservative assumption here that no such condensationoccurs, although our considerations may be generalized to the case withcondensation.In addition to the fascinating issue of Bose condensation, there are anumber of questions that should be asked about the Glasma that we do notyet have firm answers: • How long does it take to thermalize? • For a three dimensional Glasma expanding in 1 dimension, how dothe longitudinal and transverse pressures depend upon time[23]-[24]?This is the system of relevance for heavy ion collisions. How does sucha system approach isotropization? • Such systems have strong fluctuating electric and magnetic fields.Are there interesting non-perturbative phenomena generated in thisweakly coupled system[25]? For example, the strongly coupled fluc-tuating fields in the vacuum generate confinement. Might there berelated effects for the chaotic Glasma fields?
3. Evolution of the Glasma
There have been a number of attempts to simulate properties of theGlasma. Early simulations assumed that the Glasma was uniform in lon-gitudinal coordinate[9]-[12]. It was soon discovered that such uniformitywas destroyed by small fluctuations which led to developing a turbulentfluid[26]-[27]. The issue then became how to properly include the quantumfluctuations in the initial conditions which lead to the development of suchturbulence, and how fields with these initial conditions evolve in time. Atpresent there is consensus on how to set up such a computation[17]-[18], butnot broad consensus on the results of simulations of the evolution of thesefields[28]. Classical field methods have difficulties at largish times, and themethods of transport theory have difficulty incuding inelastic effects andproperly including condensation phenomena[29].I think that the results show the promise that although the Glasma maytake some time to thermalize, it may undergo hydrodynamic behaviour fromearly times. If so, this hydrodynamics will have a significant anisotropybetween longitudinal and transverse pressure[23]-[24]. This behaviour isnot seen just in Glasma simulations but also in computations employingAdSCFT methods with intrinsic strong coupling[30]-[32].In what follows, I will construct a simplifed model of the Glasma thatillustrates some simple features of the Glasma, and may be useful for phe- zakopane printed on October 25, 2018 nomenological applications[33]. I will assume that distributions are approx-imately isotropic, and again the considerations presented here might begeneralized to the anisotropic case.Let us begin with the definition of the gluon distribution function1 τ πR dNd p = f ( p ) (4)where R is the transverse size of the system, and τ is the proper time.For a non expanding system the proper time is just the time, but for alongitudinally expanding system τ = √ t − z . We take as initial conditions f ( p ) ∼ α S , p ≤ Q sat (5)and f ( p ) → , p ≥ Q sat (6)At some point the distribution function must go to zero and will have avalue of order 1, so we see that the UV scale is defined from f (Λ UV ) ∼ f >> dfdt ∼ α S f (8)Implicit in this relationship are integrations on the right hand side of theequation with weight associated with the scattering kernal. The factor of α S is the coupling strength. In scattering there are two particles in theinitial and two particles in the final state, so we would naively expect thatthe scattering term in the transport equations to be of order f , but thisleading term cancels in the forward and backward going processes leaving aterm of order f .Let us assume that the distribution function is classical for E << Λ UV ,then f ∼ α S Λ IR E (9)More generally we can write f ∼ α S Λ IR Λ UV f ( E/ Λ UV ) (10)Now plugging this into the transport equation and integrating over mo-mentum gives an equation ddt Λ IR Λ UV ∼ Λ IR Λ UV (11) akopane printed on October 25, 2018 Taking 1 /t ∼ IR Λ UV ddt Λ IR Λ UV (12)we can identify the scattering time as t scat ∼ Λ UV Λ IR (13)Note that the coupling constant has entirely disappeared from this equation.One can show that this form of the time dependence persists when one in-cludes higher order corrections associated with inelastic particle production[14].If there is a Bose condensate present then there is a term in the transportequation associated with scattering from a condensate. In this case, thedependence upon the infrared and ultraviolet scales for the scattering timeis different, but can also be explicitly obtained.The relationship between the dynamical scale and the scattering time, t ∼ t scat gives one equation determining the evolution of the scales. Theother equation is energy conservation. The energy density is (cid:15) ∼ α S Λ IR Λ UV (14)The solution to these equations in a fixed box or an expanding box givespower law dependences in time for the infrared and ultraviolet scale.
4. A Simple Model for the Glasma
It is useful to consider a simple model for the Glasma that is explicit andhas the properties described above. Let us take the the gluon distributionfunction to be an overoccupied Bose-Einstein distribution[33], f p ) = γ ( t ) e E/ Λ( t ) − UV The factor γ is the overoccupation factor for the Bose-Einstein distribution.For a thermally equilibrated distribution γ = 1. For the Glasma, we take γ = 1 α S Λ IR Λ UV (16)At some time in the evolution γ ( t ) = 1 (17) zakopane printed on October 25, 2018 At this time, the system is thermal, and the criterion of Eqn. 2 is satisfied.At this time, t th is determined from T = Λ UV ( t th ) (18)Beyond this time, γ ( t ) = 1, but the temperature may evolve.The entropy density of these overoccupied distributions is s = (cid:90) d p { (1 + f ) ln (1 + f ) − f ln ( f ) } ∼ Λ UV ln (cid:26) Λ IR α S Λ UV (cid:27) (19)On the other hand the number density of gluons is ρ ∼ α S Λ IR Λ UV (20)The entropy per particle becomes s/n ∼ α s Λ UV / Λ IR (21)This means that early on when the system is highly coherent, the entropyper particle is small. By the time of thermalization, the entropy per particlehas become of order 1.We can also estimate the quark to gluon number density. We take forthe quark distribution function f quark = 1 e E/ Λ( t ) + 1 (22)The quarks cannot be over occupied because they are fermions. We assumethe UV scale is the same for quarks and gluons. The total number of quarksis of order q ∼ Λ UV (23)This means that the ratio of quarks to gluons is q/g ∼ α S Λ UV / Λ IR (24)and like the entropy to gluon ratio, it begins small but at thermalization hasachieved a ratio of order one. This underabundance of quarks at early timeshas no relatiionship to the rate of quark production. It simply reflects theoverabundance of gluons, and that Fermi statistics forbid the overoccupationof fermions. akopane printed on October 25, 2018 (GeV/c) T p ) - G e V ( c dp γ N d E ev t N -7 -5 -3 -1 ± Au+Au 0-20% (PHENIX) n = 6.95 0.45 ±
200 GeV p+p (PHENIX) n = 5.24 1.88 ±
200 GeV d+Au (PHENIX) n = 5.30 0.27 ± Fig. 1. Measurements of invariant yields of direct photon production in nuclearcollisions below p T = 5 GeV/ c compared to power law parameterizations. Data aretaken from the PHENIX experiment at RHIC [35,36] and the ALICE experimentat the LHC [37]. The error bars represent the combined systematic and statisticaluncertainties of the measurements. Original figure is from Ref. [40].
5. Saturation, the Glasma, and Photons
If both the Glasma and the Thermalized Quark Gluon Plasma obey ap-proximate hydrodynamic behaviour, it will be difficult to disentangle whichis the source of bulk properties of matter produced in heavy ion collisions.As suggested by Shuryak many years ago[34], the internal dynamics of anevolving QGP might be best addressed by looking at penetrating probessuch as photons and dileptons. These particles can probe the internal dy-namics of the QGP and in principle resolve the difference between a Glasmaand a Thermalized QGP. It is not easy however, as most experimental ob-servables have siginificant contributions from other sources, such as thematter produced at late times as a hadron gas, and from the fragmentation zakopane printed on October 25, 2018 (GeV/c) T p ) - G e V ( c dp γ N d E ev t N -7 -5 -3 -1 ×
200 GeV Au+Au 0-20% 2345.4 ×
200 GeV p+p 238.3 ×
200 GeV d+Au 0.2 × Fig. 2. Geometrically scaled invariant yields of direct photon production below p T = 5 GeV/ c , the assumed common power law shape of p − . T has been fit tothe PHENIX AuAu data. The error bars represent the combined systematic andstatistical uncertainties of the measurements. Original figure is from Ref. [40] of produced jets into photons.Nevertheless, we can first try to see if saturation dynamics has anythingto do with photon production. We can first see whether or not the availablephoton data has geometric scaling[38]-[39]. This should be a generic featureof emission from the Color Glass Condensate and early time emission fromthe Glasma. In these cases, the only scale in the problem is the saturationmomentum We therefore expect that the distribution of photons will be ofthe form[40] 1 πR d Ndyd p T = F (cid:18) Q sat p T (cid:19) (25) akopane printed on October 25, 2018 The saturation momenum for nucleus-nucleus collisions is determined by Q sat = N / part (cid:18) Ep T (cid:19) δ (26)Here N part is the number of nucleon participants and δ ∼ . − .
28 is de-termined by both fits to deep inelastic scattering data and high energy pp interactions.The photon data are RHIC and LHC energies is shown in Fig. 1[40].Included are pp , dAu and AuAu data from RHIC [35]-[36] and
P bP b datafrom LHC[37]. Note that the range of variation of the photon rate is over 4orders of magnitude.When we rescale the data using geometric scaling, we obtain the re-markable results of Fig. 2. It is also true that data from RHIC for
AuAu collisions for varying multiplicity of produced particles also falls on thisscaling curve.The underlying mechanism behind this remarkable scaling behaviourmight be jet production and fragmentation into photons[41]-[43]. Such afragmentation process should be approximately scale invariant, and wouldpreserve the geometric scaling of the initial conditions in the Color GlassCondensate.We can also try to describe photon production using the Glasma. Schenkeand I used the known lowest order formula for photon production[44], withthe distribution functions replaced by the over-occupied distribution func-tions above[33]. The result is that one can obtain a good description ofthe spectrum of produced photons in the 1-4 GeV transverse momentumrange. To do this requires a factor fo 5-10 increase in the rates relative tothe computed rates. Similar results with related meachanisms are found inthe semi-QGP analysis of Ref. [45]. The Thermalized QGP computationswith realistic hydrodynamic simulation are off by a factor of 2-5, so this isa common problem for both computations.The remarkable result of the photon measurements at RHIC and LHCis the observation that photons flow almost like hadrons. This is difficultto achieve in Thermalized QGP computations of photon production. Thisis because the photons are produced early before much flow develops. Itmight be that such photons are produced late in the collision[46]-[47], butthen it would be difficult to explain the geometric scaling seen in the data.At very late times there are scales of order Λ
QCD which become important.The Glasma is producing significant entropy per gluon during its expansion,and therefore cools more slowly than does a Thermalized QGP. This allowsmore time for flow to develop. It is possible to get acceptable flow from theGlasma emission, at the expense as mentioned above, of reducing rates ofphoton emission which are already somewhat low. zakopane printed on October 25, 2018 Acknowledgements
I thank Michal Praszalowicz for organizing this wonderful meeting. Ialso thank the Theoretical Physics Institute at the University of Heidlebergwhere L. McLerran is a Hans Jensen Professor of physics, and where thistalk was written up. The research L. McLerran is supported under DOEContract No. DE-AC02-98CH10886.REFERENCES [1] L. D. McLerran and R. Venugopalan, Phys. Rev. D (1994) 3352 [hep-ph/9311205].[2] L. D. McLerran and R. Venugopalan, Phys. Rev. D (1994) 2233 [hep-ph/9309289].[3] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, Phys. Rev. D (1998) 014014 [hep-ph/9706377].[4] E. Iancu, A. Leonidov and L. D. McLerran, Nucl. Phys. A (2001) 583[hep-ph/0011241].[5] E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nucl. Phys. A (2002)489 [hep-ph/0109115].[6] A. Kovner, L. D. McLerran and H. Weigert, Phys. Rev. D (1995) 6231[hep-ph/9502289].[7] A. Kovner, L. D. McLerran and H. Weigert, Phys. Rev. D (1995) 3809[hep-ph/9505320].[8] A. Kovner, L. D. McLerran and H. Weigert, Phys. Rev. D (1995) 3809[hep-ph/9505320].[9] A. Krasnitz and R. Venugopalan, Phys. Rev. Lett. , 4309 (2000) [hep-ph/9909203].[10] A. Krasnitz and R. Venugopalan, Phys. Rev. Lett. , 1717 (2001) [hep-ph/0007108].[11] A. Krasnitz, Y. Nara and R. Venugopalan, Phys. Rev. Lett. , 192302 (2001)[hep-ph/0108092].[12] T. Lappi, Phys. Rev. C (2003) 054903 [hep-ph/0303076].[13] T. Lappi and L. McLerran, Nucl. Phys. A (2006) 200 [hep-ph/0602189].[14] J. P. Blaizot, F. Gelis, J. F. Liao, L. McLerran and R. Venugopalan, Nucl.Phys. A (2012) 68 [arXiv:1107.5296 [hep-ph]].[15] A. Kurkela and G. D. Moore, JHEP (2011) 044 [arXiv:1107.5050 [hep-ph]].[16] K. Dusling, T. Epelbaum, F. Gelis and R. Venugopalan, Nucl. Phys. A (2011) 69 [arXiv:1009.4363 [hep-ph]].[17] T. Epelbaum and F. Gelis, Phys. Rev. Lett. (2013) 232301[arXiv:1307.2214 [hep-ph]]. akopane printed on October 25, 2018 (2014) 122 [arXiv:1401.1666[hep-ph]].[19] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, Phys. Rev. D (2014) 074011 [arXiv:1303.5650 [hep-ph]].[20] J. P. Blaizot, J. Liao and L. McLerran, Nucl. Phys. A , 58 (2013)[arXiv:1305.2119 [hep-ph]].[21] X. G. Huang and J. Liao, arXiv:1303.7214 [nucl-th].[22] Z. Xu, K. Zhou, P. Zhuang and C. Greiner, arXiv:1410.5616 [hep-ph].[23] M. Martinez and M. Strickland, Nucl. Phys. A , 183 (2010)[arXiv:1007.0889 [nucl-th]].[24] M. Martinez and M. Strickland, Nucl. Phys. A , 68 (2011) [arXiv:1011.3056[nucl-th]].[25] T. Gasenzer, L. McLerran, J. M. Pawlowski and D. Sexty, Nucl. Phys. A(2014) [arXiv:1307.5301 [hep-ph]].[26] S. Mrowczynski, Phys. Lett. B , 118 (1993).[27] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. , 062302 (2006) [hep-ph/0510121].[28] J. Berges, B. Schenke, S. Schlichting and R. Venugopalan, arXiv:1409.1638[hep-ph].[29] T. Epelbaum, F. Gelis and B. Wu, Phys. Rev. D , 065029 (2014)[arXiv:1402.0115 [hep-ph]].[30] R. A. Janik and R. B. Peschanski, Phys. Rev. D , 045013 (2006) [hep-th/0512162].[31] R. A. Janik and R. B. Peschanski, Phys. Rev. D , 046007 (2006) [hep-th/0606149].[32] M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev. Lett. , 201602(2012) [arXiv:1103.3452 [hep-th]].[33] L. McLerran and B. Schenke, arXiv:1403.7462 [hep-ph].[34] E. V. Shuryak, Phys. Lett. B , 150 (1978) [Sov. J. Nucl. Phys. , 408(1978)] [Yad. Fiz. , 796 (1978)].[35] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. , 132301 (2010)[arXiv:0804.4168 [nucl-ex]].[36] A. Adare, S. S. Adler, S. Afanasiev, C. Aidala, N. N. Ajitanand, Y. Ak-iba, H. Al-Bataineh and A. Al-Jamel et al. , Phys. Rev. C , 054907 (2013)[arXiv:1208.1234 [nucl-ex]].[37] M. Wilde [ALICE Collaboration], Nucl. Phys. A , 573c (2013)[arXiv:1210.5958 [hep-ex]].[38] A. M. Stasto, K. J. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett. , 596(2001) [hep-ph/0007192].[39] L. McLerran and M. Praszalowicz, Acta Phys. Polon. B , 1917 (2010)[arXiv:1006.4293 [hep-ph]].2 zakopane printed on October 25, 2018 [40] C. Klein-Bsing and L. McLerran, Phys. Lett. B , 282 (2014)[arXiv:1403.1174 [nucl-th]].[41] H. Holopainen, S. Rasanen and K. J. Eskola, Phys. Rev. C , 064903 (2011)[arXiv:1104.5371 [hep-ph]].[42] R. Chatterjee, H. Holopainen, I. Helenius, T. Renk and K. J. Eskola, Phys.Rev. C , 034901 (2013) [arXiv:1305.6443 [hep-ph]].[43] G. Y. Qin, J. Ruppert, C. Gale, S. Jeon and G. D. Moore, Phys. Rev. C ,054909 (2009) [arXiv:0906.3280 [hep-ph]].[44] J. I. Kapusta, P. Lichard and D. Seibert, Phys. Rev. D , 2774 (1991)[Erratum-ibid. D , 4171 (1993)].[45] C. Gale, Y. Hidaka, S. Jeon, S. Lin, J.-F. Paquet, R. D. Pisarski, D. Satowand V. V. Skokov et al. , arXiv:1409.4778 [hep-ph].[46] H. van Hees, C. Gale and R. Rapp, Phys. Rev. C , 054906 (2011)[arXiv:1108.2131 [hep-ph]].[47] O. Linnyk, V. P. Konchakovski, W. Cassing and E. L. Bratkovskaya, Phys.Rev. C88