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Photon emission from out of equilibrium dissipative parton plasma Jitesh R. Bhatt∗ and V. Sreekanth† Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad, India - 380 009 (Dated: January 10, 2009) UsingthesecondorderIsrael-Stewart hydrodynamicswediscusstheeffectofviscosity onphoton productioninapartonplasmacreated inrelativistic heavyioncollisions. Wefindthatphotonpro- ductionratescanenhancebyseveralfactorsduetotheviscouseffectinachemicallynonequilibrated plasma. PACSnumbers: 9 Astronglycoupledquark-gluonplasma(sGGP)oramatterinaperfectfluidstateiswidelyexpectedtobeproduced 0 inrecentRelativisticHeavyIonCollider(RHIC)experiments. Recentmeasurementshasshownthatthematterflows 0 very rapidly at the time of its breakup into the freely streaming hadronic matter. Also the measurements of the 2 ellipticalflowparameterv showastrongcollectivityinthe flow[1,2,3]. Thiswouldimply thatthe QGPcanhavea 2 n verylowshearviscousstressandthe ratioofitsshearviscosityη tothe entropydensitysi.e. η/sshouldnotbemuch a largerthanthelowerbound1/4π [4]. ThisledtoaconjecturethattheQGPformedatRHICisthemostperfect-fluid J found in nature [5]. There has been a lot of attempts to determine the viscosity of sQGP [5, 6, 7, 8, 9, 10]. The 0 firstordertheoryofviscoushydrodynamicsisknowntogiveunphysicalresults. Forexample,whenthe Navier-Stokes 1 equations were applied to the one dimensional boost invariant expanding flow [11], one finds the expression for the temperature to be ] h p τ0 1/3 2η τ0 2/3 T(τ)=T 1+ 1 . - 0 τ 3sτ T − τ p (cid:16) (cid:17) (cid:20) 0 0 (cid:18) (cid:16) (cid:17) (cid:19)(cid:21) e h −3/2 This describes reheating of the flow and T has a maximum at time τ = τ 1 + sτ0T0 . This is an un- [ max 0 3 η 2 physical behaviour and the first order viscous dynamics is known to have such pr(cid:16)oblems [12,(cid:17)13]. The second order 1 hydrodynamics approach developed in the spirit of Israel and Stewarts [14] removes such artifacts. The second or- v der viscous hydrodynamics later developed and applied in the context of heavy-ions collisions by several authors 3 6 [15, 16, 17, 18, 19, 20, 21, 22]. 3 It would be interesting to study the role that viscosity can play on the plasma signals. Hard photons are one 1 such promising source that can provide information about the thermodynamical state of the plasma at time of their . production. The plasma created in the heavy-ion collisions is expected to be in a state of chemical nonequilibrium. 1 0 The photon emission from such a plasma has been studied within the framework of ideal hydrodynamics by earlier 9 workers [23, 24, 25, 26]. In this paper we study the photon production using causal hydrodynamics of Israel-Stewart 0 [14]. v: In the center of the fireball in a nuclear collision the viscous stress-energy tensor in the local comoving frame has i the form [16, 27, 28]: X r ε 0 0 0 a 0 P 0 0 Tµν = ⊥ (1)  0 0 P 0  ⊥ 0 0 0 P  z    with the transverse and longitudinal pressure 1 P = P + Φ ⊥ 2 P = P Φ (2) z − Here P denotes the (isotropic) pressure in thermal equilibrium, Φ denotes the non-equilibrium contributions to the pressure coming from shear stress. We ignore the bulk viscosity in the relativistic limit when the equation of state ∗ email: [email protected] † email: [email protected] 2 p = ǫ/3 is obeyed [29]. However, the bulk viscosity can be important near the critical temperature [30, 31]. The shear tensor in that frame takes the form πij =diag(Φ/2,Φ/2, Φ)consistent with the symmetries in the transverse − directions. To describeevolutionofthe energydensity andthe viscousstressφwe use secondorderdissipative hydrodynamics of Israel-Stewart[14, 33, 34, 35]: ∂ε 1 = (ε+P Φ), (3) ∂τ −τ − ∂Φ Φ 1 1 1 ∂ β 2 1 2 = Φ + T ( ) + , (4) ∂τ −τ − 2 τ β ∂τ T 3β τ π (cid:18) 2 (cid:19) 2 where β = 9/(4ε) and τ = 2β η denotes the relaxation time. Equations(3-4) are written in the local rest frame 2 π 2 using hydrodynamic velocity uµ = 1(t,0,0,z), where τ = √t2 z2 [11]. Equation of state is required to solve these τ − equations. We use ultra-relativistic equation of state : P = 1ǫ. 3 To describe the chemical non-equilibration while maintaining the kinetic equilibrium, one can use the parton distribution [36], 1 f(k,T) = λ (τ) (5) q,g q,g eu·k/T(τ) 1 ± where,uµ is thefour-velocityofthelocalcomovingreferenceframe. Thetemperature T isatime-dependent quantity andthedistributionismultipliedbytimeandanotherdependentquantitycalledfugacityλ (τ)todescribedeviations q,g from the chemical equilibrium. The fugacity parameter become unity when the chemical-equilibrium is reached and in general it has the range 0 λ 1. The scattering processes gg ggg and gg qq¯give the most dominant q,g ≤ ≤ ↔ ↔ mechanismfor the chemicalequilibration. The master equations describingevolutionthe partondensity are givenby 1 n 1 n2n˜2 ∂ (n uµ) = σ n2 1 g σ n2 1 q g , (6) µ g 2 3 g(cid:18) − n˜g(cid:19)− 2 2 g − n˜2qn2g! 1 n2n˜2 ∂ (n uµ) = σ n2 1 q g , (7) µ q 2 2 g − n˜2qn2g! wheren˜ (i=q,g)ispartondensitywithunitfugacity[37]andσ = σ(gg qq¯) andσ σ(gg ggg) arethermally i 2 3 h ↔ i h ↔ i averagedscattering cross sections. It should be noted here that when equation for n and n are added one gets the g q total number density n and the term with 1σ n2 1 n2qn˜2g will drop out. This is because due to the the scattering 2 2 g − n˜2qn2g process gg qq¯loss in the gluon density is equal(cid:16)to the ga(cid:17)in in quark density and vice verse. ↔ ǫ and n can be calculated using equation (5) as given below n=(λ a +λ b )T3, ǫ=(λ a +λ b )T4 (8) g 1 q 1 g 2 q 2 where a = 16ξ(3)/π2, a = 8π2/15 for the gluons and b = 9ξ(3)N /π2, b = 7π2N /20 for the quarks. Using 1 2 1 f 2 f equations (3-4,6-8)following evolution equations for T, λ and Φ can be obtained q,g T˙ 1 1λ˙ +b /a λ˙ Φ 1 g 2 2 q + = + , (9) T 3τ −4λ +b /a λ 4τ (a λ +b λ )T4 g 2 2 q 2 g 2 q Φ 8 Φ 1 T˙ λ˙ +b /a λ˙ Φ˙ + = [a λ +b λ ]T4 5 g 2 2 q , (10) 2 g 2 q τπ 27τ − 2 "τ − T − λg +b2/a2λq# λ˙ T˙ 1 λ2 g q +3 + = R (1 λ ) R 1 , (11) λg T τ 3 − g − 2 − λ2g! λ˙ T˙ 1 a λ λ q 1 g q +3 + = R (12) 2 λ T τ b λ − λ q 1 (cid:18) q g(cid:19) where, the rates R = 0.24N α2λ Tln(5.5/λ ) and R = 2.1α2T(2λ λ2)1/2 are defined as in Ref. [36, 37]. We 2 f s g g 3 s g − g would like to note that our gluon fugacity equation (11) differs from that given in Ref. [36, 37] by a fctor of two 3 in second term in right hand side. We believe this is a typographical error. In equation (9) the first term on left hand side is due to expansion of the plasma, while on the right hand side the first term describes effect of chemical nonequilibrium and second term is due to the presence of (causal) viscosity. The last term in parenthesis of equation (10) arises because of the chemical nonequilibrium process. It should be noted that equation (9) differ from that considered in Ref. [38]. In their treatment first order viscous hydrodynamics is used which does not require time evolution of Φ. However such treatment give unphysical results like reheating artifact [12] as mentioned before. Elastic(gg gg)aswellasnonelasticprocesseslikegg ggg cancontributetotheshearviscosity. Shearviscosity ↔ ↔ coefficient was recently calculated for the inelastic process in the presence of chemical nonequilibrium in Ref. [35]. It was shown that η/T3 n /T3 λ . From this one can write [34] g g ≃ ≃ 9 τ = λ T3. (13) π g 2ε It ought to be mentioned that this viscosity prescription was not considered considered in Ref.[38]. Kinetic theory without invoking nonequilibrium process gives τ =3/2πT. π Real photons are produced from the annihilation of a quark-antiquark pair into a photon and a gluon (qq¯ gγ) → andby absorptionofagluonbyaquarkemitting aphoton(qg qγ). Another sourceofphotonproductioncouldbe → the bremsstrahlungbutitseffectcanbe ignoredinthe lowestorderofaperturbationtheory. Inordertocompute the photonproductionratesoneneedstoknowtheunderlyingamplitude ofthebasicprocessinvolvingtheannihilation M or Compton scattering process and the parton distribution functions given by [39, 40] dN 1 d3p d3p d3p 1 2 3 = d4xd3p (2π)32E (2π)32E (2π)32E (2π)32E Z 1 2 3 n (E )n (E )[1 n (E )] (14) 1 1 2 2 3 3 × ± 2 (2π)4δ(P +P P P). 1 2 3 × h|M| i − − i X Here P and P are the 4-momenta of the incoming partons, P of the outgoing parton, and P of the pro- 1 2 3 duced photon. In equilibrium, the distribution functions n (E ) are given by the Bose-Einstein distribution, i i n (E )=1/[exp(E /T) 1],for gluonsandbythe Fermi-Diracdistribution, n (E )=1/[exp(E /T)+1],for quarks, B i i F i i − respectively. Thefactor 2 isthematrixelementofthebasicprocessaveragedovertheinitialstatesandsummed h|M| i overthe finalstates. The indicates the sumoverthe initial partonstates. The fugacity factorscanenter equation i (14) when equation (5) is considered P n (E)n (E)(1 n (e)) λ n λ n (1 λ n ). 1 2 3 1 1 2 2 3 3 ± 7→ ± This is can be rewritten as λ n λ n (1 λ n ) = λ λ λ n n (1 n ) (15) 1 1 2 2 3 3 1 2 3 1 2 3 ± ± + λ λ (1 λ )n n 1 2 3 1 2 − In carrying out the momentum integration it is useful to introduce a parameter k to distinguish between soft c and hard momenta of the quark [41]. For the hard part of the photon rate following [23] we take k2 = 2m2 = c q 0.22g2T2(λ +λ /2), where, m is the quark-thermal-mass which can be obtained from zero momentum limit of g q q quark self-energy in the high temperature limit. The first term on the right hand side of equation (15) can lead to the following photon rate [23] using the Boltzmann distribution functions instead of a quantum mechanical ones: 2E dn = 5ααsλ2qλgT2e−E/T ln 4ET 1.42 . (16) d3pd4x 9π2 k2 − (cid:18) (cid:19)1 (cid:20) (cid:18) c (cid:19) (cid:21) Here α and α are the electromagnetic and the strong interaction coupling constants. The second term in equation s (15) will give, under the Boltzmann approximation, the following contribution to the photon rate: dn 2E = (17) d3pd4x (cid:18) (cid:19)2 10αα sT2e−E/T λ λ (1 λ ) 1 2γ+2ln 4ET/k2 +λ λ (1 λ ) 2 2γ+2ln 4ET/k2 , 9π4 q g − q − c q q − g − − c (cid:8) (cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:0) (cid:1)(cid:3)(cid:9) 4 Thetotalphotonproductionrate2E dn canbeobtainedbyaddingequations(16-17),isrequiredtobeconvoluted d3pd4x with the space time evolution of the heavy-ion collision. We define [23] dn dn 2 = d4x 2E (18) d2p dy d3pd4x (cid:18) ⊥ (cid:19)y,p⊥ Z (cid:18) (cid:19) τ1 ynuc ′ dn = Q dτ τ dy 2E d3pd4x Zτ0 Z−ynuc (cid:18) (cid:19) wherethe times after the maximumoverlapofthe nucleiareτ andandτ andy is the rapidityofthe nuclei. Q 0 1 nuc isthetransversecross-sectionofthenucleiandp isthephotonmomentumindirectionperpendiculartothecollision ⊥ axis. For a Au nucleus Q 180fm2. The quantity 2E dn is Lorentz invariant and it is evaluated in the local ∼ d3pd4x rest frame in equation (18). The photon energy in(cid:16)this frame(cid:17), i.e in the frame comoving with the plasma, can be written as p cosh(y y′). ⊥ − 0.7 1.0 0.6 T RHIC T LHC 0.8 0.5 ΛΛLV,,gq0.4 ΛΛLV,,gq0.6 HTGe0.3 HTGe0.4 0.2 Λ Λg 0.2 g 0.1 Λq Λq 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 1 2 3 4 5 6 ΤHfm(cid:144)cL ΤHfm(cid:144)cL FIG. 1: Temperature, gluon fugacity and quark fugacity for RHIC and LHC. Solid lines indicate the case with the shear viscosity, while thedashed lines correspond to thecase without viscosity In Figure (1), we have shown T,λ ,λ as function of time. We have solved the equations (9-12) together with the g q initial conditions at τ from HIJING Monte Carlo model [42]. Which areλ0 = 0.09,λ0 = 0.02and T = 0.57GeV iso g q o for RHIC with τ = 0.31fm/c and λ0 = 0.14, λ0 = 0.03 and T = 0.83GeV for LHC with τ = 0.23fm/c. iso g q o iso Presence of the causal viscosity decreases the fall of temperature due to expansion and the chemical nonequilibrium. However if one considers the first order theory, there can be unphysical instability. Fugacity of gluons and quarks increasemoreslowlyduetothe presenceoftheviscositycomparedtothe caseswhennoviscouseffectswereincluded. This is because the chemical equilibration is reached here with falling of the temperature. The temperature can decrease due to the expansion and chemical nonequilibration. The lowering of T can help in attaining chemical equilibrium and which in turn will increase the rate at which the fugacities reach their equilibrium values. Inclusion of the viscosity will slowdown the falling rate of the temperature. Consequently the fugacities will take more time to reach their equilibrium values. LHC LHC 10-5 y=0..7 10-5 y=0..7 Q y=8 Q y=8 L(cid:144)pdyT10-6 y=9 L(cid:144)pdyT10-6 y=9 2H(cid:144)dnd 2H(cid:144)dnd 10-7 10-7 10-8 10-8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 pTHGeVL pTHGeVL FIG.2: (Leftpanel)PhotonratefordifferentrapiditiesinLHC(ynuc = 8.8). (Rightpanel)Samewiththeinclusionofviscosity. We plot photon spectra by using equation (13) for τ in solving equations (9-12). The figures (2-3) compare the π case without viscosity with the case of finite shear viscosity. Figure (2) shows the photon spectra emitted at fixed rapidities as a function of transverse momenta p . The ⊥ photon flux is normalized with the transverse size of the colliding nuclei(Q). For LHC we take: τ = 0.5fm/c, 0 5 10-6 10-6 RHIC RHIC y=0..4 y=0..4 10-7 y=5 10-7 y=5 y=6 y=6 2H(cid:144)L(cid:144)dndpdyQT10-8 y=7 2H(cid:144)L(cid:144)dndpdyQT10-8 y=7 10-9 10-9 10-101.0 1.5 2.0 2.5 3.0 10-101.0 1.5 2.0 2.5 3.0 pTHGeVL pTHGeVL FIG.3: (Leftpanel)Photon ratefordifferentrapiditiesinRHIC(ynuc=6.0). (Rightpanel)Samewiththeinclusionofviscosity. 10-6 LHC RHIC 2H(cid:144)L(cid:144)dndpdyQT1100--65 yyy===980..7 2H(cid:144)L(cid:144)dndpdyQT1100--87 yyyy====7650..4 10-7 10-9 10-8 1.0 1.5 2.0 2.5 3.0 10-10 1.0 1.5 2.0 2.5 3.0 pTHGeVL pTHGeVL FIG. 4: Photon rate for different rapidities in RHIC (ynuc=6.0) and LHC (ynuc=8.8) with kinetic viscosity. τ = 6.25fm/c and y = 8.8. We use equation (13) for τ in solving equations (9-12). The figure compares the 1 nuc π case without viscosity with the case of finite shear viscosity [35]. Figure (3) shows the comparison similar to that of figure (2) but with a set of initial conditions for RHIC: τ = 0 0.7fm/c,τ = 4fm/c and y = 6.0. For α = 0.3, shear viscosity to entropy density ratio η/s 0.29. Figures 1 nuc s ∼ (2-3) show that viscous effects enhance the photon flux by a factor (1.5-2). Finally, we compare the photon fluxes calculated using equation (13) with the fluxes calculated using the kinetic viscosity (τ =3/2πT). 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