aa r X i v : . [ nu c l - t h ] F e b Flow Effects on Jet quenching with Detailed Balance
Luan Cheng a,b and Enke Wang a,ba
Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China b Key Laboratory of Quark & Lepton Physics (Huazhong NormalUniversity), Ministry of Education, China
Abstract
A new model potential in the presence of collective flow describing the interaction of the hardjet with scattering centers is derived based on the static color-screened Yukawa potential. Theflow effect on jet quenching with detailed balance is investigated in pQCD. It turns out that thecollective flow changes the emission current and the LPM destructive interference comparing tothat in the static medium. Considering the collective flow with velocity v z along the jet direction,the energy loss is (1 − v z ) times that in the static medium to the first order of opacity. The flowdependence of the energy loss will affect the suppression of high p T hadron spectrum and anisotropyparameter v in high-energy heavy-ion collisions. . INTRODUCTION One of the most striking features of nucleus-nucleus collisions at the Relativistic Heavy IonCollider (RHIC) is the collective flow. In recent years this phenomenon has been a subject ofintensive theoretical and experimental studies[1–4]. It is believed that the medium produced innucleus-nucleus collsions at RHIC equilibrates efficiently and builds up a flow field.Gluon radiation induced by multiple scattering of an energetic parton propagating in a densemedium leads to induced parton energy loss or jet quenching. As discovered in high-energy heavy-ion collisions at RHIC, jet quenching is manifested in both the suppression of single inclusive hadronspectrum at high transverse momentum p T region[5] and the disappearance of the typical back-to-back jet structure in dihadron correlations[6]. Extentive theoretical investigation of jet quenchinghas been widely carried out in recent years[7–12]. Most of the jet quenching theory researchare studied in a static medium based on the static color-screened Yukawa potential proposedby Gyulassy and Wang[7]. However, the medium is not static, the collective flow need to beconsidered[13–15]. Later, the interaction between the jet and the target partons in the presenceof collective flow was modeled by a momentum shift q perpendicular to the jet direction inthe Gyulassy-Wang’s static potential[16], but this assumption lacks sufficient theoretical evidence.Recently, local transport coefficient ˆ q , which is related to the squared average transverse momentumtransfer from the medium to the hard parton per unit length, has been investigated in the presenceof transverse flow[17, 18]. However, when relating ˆ q with the energy density ε of the medium,ˆ q ≃ cε / , a problem appears that the determination of c is different from c = 2 to c > , · · · , q study with collective flow in Ref.[17]based on BDMPS energy loss calculation gives only a macroscopic result for parton energy loss.Many interesting properties in jet quenching theory such as the flow effects on the non-AbelianLaudau-Pomeranchuk-Migdal(LPM) interference effect and opacity can not be studied. On theother hand, only radiative energy loss can be considered in Ref.[17], the detailed balance effectwith gluon absorption cannot be included. It has been shown that the gluon absorption play animportant role for the intermediate jet energy region[19].In this letter, we report a first study of the parton energy loss with detailed balance in thepresence of collective flow in perturbative Quantum Chromodynamics (pQCD). We first determinethe model potential to describe the interaction between the energetic jet and the scattering targetpartons with collective flow of the quark-gluon medium using Lorentz boosts. Based on this newpotential, we then consider both the radiation and absorption induced by the self-quenching andmultiple scattering in the moving medium. We are led to the conclusion that to the zeroth orderopacity, the energy loss is dominated by the final-state thermal absorption, whose result is thesame as that in the static medium since the jet has no interaction with the medium. However, forthe case of rescattering with targets, the collective flow changes the emission current and the LPMdestructive interference. Overall, it reduces (enhances) the jet energy loss induced by rescatteringwith stimulated emission and thermal absorption depending on the direction of the flow in thepositive (negative) jet direction. II. THE POTENTIAL MODEL
To calculate the induced radiation energy loss of jet in a static medium, the interaction potentialis assumed in the Gyulassy-Wang’s static model [7] that the quark-gluon medium can be modeled IG. 1: View of kinematics in the quark-gluon medium with collective flow.by N well-separated color screened Yukawa potentials, V ai ( q i ) = 2 πδ ( q i ) 4 πα s q + µ e − i q · x i T a i ( j ) T a i ( i ) , (1)where µ is the Debye screening mass, T a i ( j ) and T a i ( i ) are the color matrices for the jet and targetparton at x i . In this potential, each scattering has no energy transfer ( q i = 0) but only a smallmomentum q transfer with the medium. If using the four-vector potential, the Gyulassy-Wang’sstatic potential can be denoted as A µ = ( V ai ( q i ) , A ( q i ) = 0).As is well known in Electrodynamics, the static charge produces a static Coulomb electric field,while a moving charge produces both electric and magnetic field. In analogy a moving target partonin the quark-gluon medium will produce color-electric and color-magnetic fields simutaneously dueto the collective flow. Therefore, the static potential model should be reconsidered.In the quark-gluon medium with collective flow, the rest frame fixed at target parton moveswith a velocity v relative to the observer’s system frame Σ ′ , as illustrated in Fig.1. We first take aLorentz transformation for four-momentum q , and then for four-vector potential A µ , we can thenwrite A µ = ( V ai ( flow ) ( q i ) , A ( flow ) ( q i )) in the observer’s system frame Σ ′ as ( V ai ( flow ) ( q i )=2 πδ ( q i − v · q ) e − i q · x i ˜ v ( q ) T a i ( j ) T a i ( i ) , A ( flow ) ( q i )=2 πδ ( q i − v · q ) v e − i q · x i ˜ v ( q ) T a i ( j ) T a i ( i ) , (2)where ˜ v ( q ) = 4 πα s / ( q − ( v · q ) + µ ). The new potential differs from Gyulassy-Wang’s staticpotential in that the collective flow of the quark-gluon medium produces a color-magnetic field andthe flow leading to non-zero energy transfer q i = v · q , which will affect jet energy loss as we willshow below.Elastic cross section for small transverse momentum transfer between jet and target partonscan be deduced as dσ el d q = C R C d A | ˜ v ( q ) | (2 π ) , (3)where C R and C are the Casimir of jet and target parton in fundamental representation in d R dimension, respectively. d A is the dimension of corresponding adjoint representation. Our resultagrees with the GLV elastic cross section in static potential when the flow velocity goes to zero [7]. II. FLOW EFFECT ON GLUON RADIATION
Consider a hard parton produced at ˜ z = ( z , x ⊥ ). The hard parton has initial energy E,interacts with the target parton at ˜ z = ( z , x ⊥ ) with flow velocity v by exchanging gluon withfour-momentum q , radiates a gluon with four-momentum k and polarization ǫ ( k ), and emergeswith final four-momentum p . In the light-cone components, k = [2 ω, k ⊥ ω , k ⊥ ] , (4) ǫ ( k ) = [0 , ǫ ⊥ · k ⊥ xE + , ǫ ⊥ ] , (5) p = [(1 − x ) E + + 2 v · q , p − , p + ⊥ ] , (6)where ω = xE , E + = 2 E ≫ µ .At zeroth order in opacity, the jet has no interaction with the target parton, we obtain the samefactorized radiation amplitude off a quark R (0) = 2 igT c k ⊥ · ǫ ⊥ k ⊥ , (7)as that in the static medium in Ref. [19], where T c is the color matrix. As shown in Ref.[19], thenet energy gain without rescattering can be expressed as∆ E (0) abs ≈− πα s C R T E (cid:20) ln 4 ETµ +2 − γ E + 6 ζ ′ (2) π (cid:21) , (8)where γ E ≈ . ζ ′ (2) ≈ − . T is the thermal finite temperature.However, at first order in opacity, consider the jet has the simplest case of elastic scattering.The radiation amplitude M (1) ∝ − i (2 p − q ) µ A µ ∝ πδ ( q i − v · q ) e − i q · x i T a i ( j ) T a i ( i )(2 E + v · q ) R (1) , (9)where R (1) = (1 − v z )˜ v ( q ), which is changed by the collective flow with a factor (1 − v z ).When the hard parton goes through the quark-gluon medium, it will suffer multiple scatteringwith the parton target inside the medium. Here we investigate the rescattering-induced radiation byconsidering the flow effect resulting from the moving parton target. We will work in the frameworkof opacity expansion developed by Gyulassy, L´evai and Vitev (GLV)[10] and Wiedemann[11]. Itwas shown by GLV that the higher order corrections contribute little to the radiative energy loss.So we will only consider the contributions to the first order in the opacity expansion. The opacityis defined as the mean number of collisions in the medium, ¯ n ≡ L/l = N σ el /A ⊥ . Here N , L , A ⊥ and l are the number, thickness, transverse area of the targets, and the average mean-free-path forthe jet, respectively.Based on our new potential in Eq.(2) by considering collective flow of the quark-gluon medium,assuming the flow velocity | v | ≪
1, we obtain the factorized radiation amplitude associated witha single rescattering, R ( S ) = 2 ig (cid:16) H T a T c + B e iω z − vz [ T c , T a ] − v z T a T c H (1 − e iω z − vz ) + C e i ( ω − ω z − vz [ T c , T a ] (cid:17) · ǫ ⊥ × (1 − v z ) , (10) here z = z − z , ω = k ⊥ ω , ω = ( k ⊥ − q ⊥ ) ω , (11) H = k ⊥ k ⊥ , C = k ⊥ − q ⊥ ( k ⊥ − q ⊥ ) , B = H − C . (12)Different from the static medium case, the single rescattering amplitude depends on the collectiveflow of the quark-gluon medium.The interference between the process of double scattering and no rescattering should also betaken into account to the first order in opacity [10]. Assuming no color correlation between differenttargets, the double rescattering corresponds to the “contact limit” of double Born scattering withthe same target [10]. Assuming the flow velocity | v | ≪
1, with our new potential in Eq.(2) theradiation amplitude can be expressed as R ( D ) = 2 igT c e iω z − vz (cid:16) − C R + C A H e − iω z − vz + C A B + 2 v z C R − C A H (1 − e − iω z − vz )+ C A C e − iω z − vz (cid:17) · ǫ ⊥ (1 − v z ) , (13)where C A is the Casimir of the target parton in adjoint representation in d A dimension. The doublerescattering amplitude also depends on the collective flow.To the first order in opacity, we then derive the induced radiation probability including boththe stimulated emission and thermal absorption as dP (1) dω = C πd A d R NA ⊥ Z dxx Z d k ⊥ (2 π ) Z d q ⊥ (2 π ) P ( ωE ) | R (1) | D T r h | R ( S ) | +2 Re (cid:16) R (0) † R ( D ) (cid:17)iEh (1+ N g ( xE )) δ ( ω − xE ) θ (1 − x )+ N g ( xE ) δ ( ω + xE ) i ≈ α s C C R C A d A π NA ⊥ Z dxx Z d k ⊥ k ⊥ Z d q ⊥ (2 π ) P ( ωE ) J eff ( k ⊥ , q ⊥ ) (cid:28) Re (1 − e iω z − vz ) (cid:29)h (1+ N g ( xE )) δ ( ω − xE ) θ (1 − x ) + N g ( xE ) δ ( ω + xE ) i , (14)where N g ( | k | ) = 1 / [exp( | k | /T ) −
1] is the thermal gluon distribution, v ( q ⊥ ) = 4 πα s / ( q ⊥ + µ ), α s = g / π is strong coupling constant. We have also included the splitting function P gq ( x ) ≡ P ( x ) /x = [1 + (1 − x ) ] /x for q → gq .The gluon formation factor 1 − exp( iω z / (1 − v z )) reflects the destructive interference ofthe non-Abelian LPM effect. The formation time of gluon radiation τ f ≡ (1 − v z ) /ω becomesshorter (longer), the LPM effect is reduced (enhanced) in the presence of collective flow in thepositive (negative) jet direction, respectively. The gluon formation factor must be averaged over thelongitudinal target profile, which is defined as h· · ·i = R dzρ ( z ) · · · . We take the target distributionas an exponential Gaussian form ρ ( z ) = exp( − z/L e ) /L e with L e = L/
2, the gluon formation factorcan be deduced as (cid:28) Re (1 − e iω z − vz ) (cid:29) = 2 L Z ∞ dz e − z /L (1 − e iω z − vz )= ( k ⊥ − q ⊥ ) L x E (1 − v z ) + ( k ⊥ − q ⊥ ) L . (15)Here the emission current J eff ( k ⊥ , q ⊥ ) in Eq.(14) is defined as J eff ( k ⊥ , q ⊥ ) = | R (1) | D T r h | R ( S ) | +2 Re (cid:16) R (0) † R ( D ) (cid:17)iE(cid:28) Re (1 − e iω z − vz ) (cid:29) (1 − v z ) k ⊥ · q ⊥ ( k ⊥ − q ⊥ ) . (16)It shows that the collective flow reduces the emission current with a factor (1 − v z ) when 0 < v z <<
1. The jet energy loss can be divided into two parts. The zero-temperature part corresponds tothe radiation induced by rescattering without detailed balance effect and can be expressed as∆ E (1) rad = Z dωω dP (1) dω (cid:12)(cid:12)(cid:12) T =0 = α s C R π Ll g E Z dx Z d k ⊥ k ⊥ Z d q ⊥ | ¯ v ( q ⊥ ) | P ( x ) J eff ( k ⊥ , q ⊥ ) (cid:28) Re (1 − e iω z − vz ) (cid:29) , (17)where l g = C R l/C A is the mean-free path of the gluon.The temperature-dependent part of energy loss induced by rescattering at the first order ofopacity comes from thermal absorption with partial cancellation by stimulated emission, in thepresence of flow it can be written as∆ E (1) abs = Z dω ω dP (1) dω − dP (1) dω (cid:12)(cid:12)(cid:12) T =0 ! = α s C R π Ll g E Z dx Z d k ⊥ k ⊥ Z d q ⊥ | ¯ v ( q ⊥ ) | N g ( xE ) J eff ( k ⊥ , q ⊥ )[ P ( − x ) (cid:28) Re (1 − e iω z − vz ) (cid:29) − P ( x ) (cid:28) Re (1 − e iω z − vz ) (cid:29) θ (1 − x )] , (18)where | ¯ v ( q ⊥ ) | is defined as the normalized distribution of momentum transfer from the scatteringcenters, | ¯ v ( q ⊥ ) | ≡ σ el d σ el d q ⊥ = 1 π µ eff ( q ⊥ + µ ) , (19)1 µ eff = 1 µ − q ⊥ max + µ , q ⊥ max ≈ Eµ . (20)To obtain a simple analytic result, we take the kinematic boundaries limit q ⊥ max → ∞ , theangular integral can be carried out by partial integration. In the limit of EL ≫ E ≫ µ , weobtain the approximate asymptotic behavior of the energy loss,∆ E (1) rad E =(1 − v z ) α s C R µ L λ g E (cid:20) ln 2 Eµ L − . (cid:21) + O ( | v | ) , (21)∆ E (1) abs E = − (1 − v z ) πα s C R LT λ g E " ln µ LT − γ E − ζ ′ (2) π + O ( | v | ) . (22)Our analytic result implies, although the formation time of gluon radiation becomes shorter, andLPM effects is reduced when v z >
0, the collective flow reduces the emission current more. To thefirst order in opacity, that the energy loss is changed by a factor (1 − v z ) for rescattering case withcollective flow compared to the static medium case. With collective flow velocity | v | = 0 . − . −
10 15 20 25 30 35 400.000.020.040.060.080.100.120.140.160.18
E/ E (1)abs, v z =0 E (0)abs, E (1)abs, v z =0.1 E (1)abs, v z =0.2 E (1)abs, v z =0.3 FIG. 2: The energy gain via gluon absorption with rescattering for v z = 0 , . , . , . E/µ .collective flow goes to zero. Since the parton energy loss is dominated by the first order opacitycontribution, our result also agrees with the result of ˆ q calculation in Ref.[17, 18].Shown in Fig. 2 are the energy gain via gluon absorption with rescattering for v z = 0 , . , . , . E/µ . For comparison, we take the same values for themedium thickness, the mean free path, and the Debye screen mass as in Refs.[10] and [19]. Theenergy gain without rescattering at very small
E/µ region is larger than that with rescattering if v z > .
2, but at smaller flow velocity or at higher jet energies, it becomes smaller than that withrescattering.
IV. CONCLUSION
In summary, we have derived a new potential for the interaction of a hard jet with the partontarget. It can be used to study the jet quenching phenomena in the presence of collective flow ofthe quark-gluon medium. With this new potential, we have investigated the effect of collective flowon jet energy loss with detailed balance. Collective flow along the jet direction leads to a reducedemission current square J eff , (1 − v z ) times that in static medium, and an increased LPM gluonformation factor, (1 + v z ) times that in static medium. The energy gain without rescattering is thesame as in the static medium, but the total energy loss to the first order of opacity is (1 − v z ) timesthat in the static medium. Compared to calculations for a static medium, our results will affectthe suppression of high p T hadron spectrum and anisotropy parameter v in high-energy heavy-ioncollisions. Our new potential can also be used for heavy quark energy loss calculation and willalter the dead cone effect of heavy quark jets. Our results shall have implications for comparisonsbetween theory and experiment in the future.We thank Fuqiang Wang and Xin-Nian Wang for helpful comments. This work was supportedby NSFC of China under Projects No. 10825523, No. 10635020 and No. 10875052, by MOEof China under Projects No. IRT0624, and by MOE and SAFEA of China under Project No. ITDU-B08033.[1] T. Cs¨og¨o, B. Lorstad, Phys. Rev. C , (1996) 1390.[2] S. A. Voloshin, W. E. Cleland, Phys. Rev. C , (1996) 896.[3] U.A. Wiedemann, Phys. Rev. C , (1997) 266.[4] S. Voloshin, R. Lednicky, S. Panitkin, Nu Xu, Phys. Rev. Lett. , (1997) 4766.[5] K. Adcox et al. , Phys. Rev. Lett. , (2001) 022301; C. Adler et al. , Phys. Rev. Lett. ,(2002) 202301.[6] C. Adler et al. , Phys. Rev. Lett. , (2003) 082302.[7] M. Gyulassy, X.-N. Wang, Nucl. Phys. B , (1994) 583; X.-N. Wang, M. Gyulassy,M. Plumer, Phys. Rev. D , (1995) 3436.[8] R. Baier, Y. L. Dokshitzer, S. Peigne, D. Schiff, Phys. Lett. B , (1995) 277; R. Baier,Y. L. Dokshitzer, A. H. Mueller, S. Peigne, D. Schiff, Nucl. Phys. B , (1997) 265.[9] B. G. Zakharov, JETP Lett. , (1996) 952; JETP Lett. , (1997) 615.[10] M. Gyulassy, P. Levai, I. Vitev, Phys. Rev. Lett. , (2000) 5535; Nucl. Phys. B , (2001)371.[11] U. A. Wiedemann, Nucl. Phys. B , (2000) 303.[12] X. Guo, X.-N. Wang, Phys. Rev. Lett. , (2000) 3591.[13] R. Baier, Y. L. Dokshitzer, A. H. Mueller, D. Schiff, Phys. Rev. C , (1998) 1706.[14] C. A. Salgado, U. A. Wiedemann, Phys. Rev. Lett. (2002) 092303.[15] E. Wang, X.-N. Wang, Phys. Rev. Lett. (2002) 162301.[16] N. Armesto, C. A. Salgado, U.A. Wiedemann, Phys. Rev. Lett. (2004) 242301.[17] R. Baier, A. H. Mueller, D. Schiff, Phys. Lett. B. (2007) 147.[18] H. Liu, K. Rajagopal, U. A. Wiedemann, JHEP (2007) 066.[19] E. Wang, X.-N. Wang, Phys. Rev. Lett. (2001) 142301.(2001) 142301.