Virtual photon polarization in ultrarelativistic heavy-ion collisions
RRIKEN-QHP-251
Virtual photon polarization in ultrarelativistic heavy-ion collisions
Gordon Baym, a,b
Tetsuo Hatsuda, b,c , and Michael Strickland d a Department of Physics, University of Illinois, 1110 W. Green Street, Urbana, IL 61801-3080, United States b iTHES Research Group and iTHEMS Program, RIKEN, Wako, Saitama 351-0198, Japan c Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan d Department of Physics, Kent State University, Kent, OH 44242, United States
The polarization of direct photons produced in an ultrarelativistic heavy-ion collision reflects themomentum anisotropy of the quark-gluon plasma created in the collision. This paper presents ageneral framework, based on the photon spectral functions in the plasma, for analyzing the angulardistribution and thus the polarization of dileptons in terms of the plasma momentum anisotropies.The rates of dilepton production depend, in general, on four independent spectral functions, corre-sponding to two transverse polarizations, one longitudinal polarization, and – in plasmas in whichthe momentum anisotropy is not invariant under parity in the local rest frame of the matter – a newspectral function, ρ n , related to the anisotropy direction in the collision. The momentum anisotropyappears in the difference of the two transverse spectral functions, as well as in ρ n . As an illustration,we delineate the spectral functions for dilepton pairs produced in the lowest order Drell-Yan processof quark-antiquark annihilation to a virtual photon. PACS numbers: 25.75.Cj,12.38.Mh,11.10.Wx
I. INTRODUCTION
Direct photons, both real and virtual, are an impor-tant probe of the dynamics of ultrarelativistic heavy-ioncollisions. An average temperature of the quark-gluonplasma (QGP) formed in high-energy collisions has beenextracted from the transverse momentum spectrum of di-rect photons in the range q T ∼ l + l − ) have beenproposed to provide information on the early stages ofcollisions, before the onset of thermalization [4–6]. Whilerelativistic hydrodynamics provides a successful space-time description of the later stages of the collision dy-namics and associated hadronic and leptonic observables[7], important questions concerning the early dynamics,such as the degree of thermalization as well as isotropiza-tion of the QGP, have not been answered either exper-imentally or theoretically. Recently, Ref. [8] proposedusing the polarization of direct photons as a measure ofthe gluon anisotropy in collisions. While measuring di-rect photon polarization, involving external conversionto dilepton pairs, is very difficult experimentally, a morepromising approach to is measure polarization of virtualphotons, through the angular distribution of dileptonsproduced via internal conversion.The lowest order mechanism to produce dilepton pairsis the Drell-Yan process, Fig. 1, in which a quark andan antiquark annihilate to a virtual photon. The dilep-ton cross section dσ/d Ω can be parametrized as ∝ λ cos θ + µ sin 2 θ cos φ +( ν/
2) sin θ cos 2 φ , where θ and φ are the polar and azimuthal angles of one of the dileptonsin the dilepton rest frame measured in the Collins-Soperreference frame [9, 10]. High-energy p ¯ p and pp collisionsat the Tevatron and the LHC have confirmed the leading-order prediction λ (cid:39) µ = ν = 0, for q T < ∼ M l + l − (cid:39) M Z (see, e.g., [11] and references therein). The dilepton angular distribution inIn-In collisions has been measured by the NA60 experi-ment [12] at the CERN-SPS in the primary kinematicalrange 0 . < M l + l − < π + π − annihilationdominates Drell-Yan dileptons; the results are consistentwith λ = µ = ν = 0. FIG. 1: Lowest order Drell-Yan production of a lepton pair.
On the other hand, in ultrarelativistic heavy-ion colli-sions we expect the momentum-space anisotropy of theQGP to be detected most readily in dilepton productionin the range 1 GeV < q T < M l + l − < ∼ . a r X i v : . [ nu c l - t h ] F e b an illustration, we apply the formalism to virtual photonemission through the leading-order Drell-Yan process in aplasma with anisotropic distributions of the Romatschke-Strickland form [13]. II. DILEPTON PRODUCTION
Quite generally, the production rate, R , of a dileptonpair is proportional to the spectral function, ρ µν , of thein-medium photon polarization or self-energy operator,for momentum (cid:126)q and energy q ,Π µν ( (cid:126)q , z ) = e (cid:90) ∞−∞ dq π ρ µν ( (cid:126)q , q ) z − q , (1)times the squared matrix element L µν (L for leptons) fora virtual photon of 4-momentum q to produce a leptonof 4-momentum p and mass m and an antilepton of 4-momentum p (cid:48) , averaged over the spins of the leptons.Explicitly, dR l + l − d ¯ pd ¯ p (cid:48) = α π Q ρ µν ( q ) L µν ( p, p (cid:48) ) , (2)with the leptonic tensor, L µν ( q, s ) = 2 (cid:0) q µ q ν − g µν Q − s µ s ν (cid:1) , (3)where d ¯ p ≡ d p/ E p , d ¯ p (cid:48) ≡ d p (cid:48) / E p (cid:48) , q = p + p (cid:48) , s = p − p (cid:48) , Q ≡ q µ q µ >
0, and Q + s = 4 m with m the lepton mass.The spectral function ρ µν ( q ) is related to the cut, orimaginary part, of the photon polarization operator, il-lustrated in Fig 2. Its explicit form in the kinematicalregime q (cid:29) T (cid:29) (cid:112) Q has been previously evaluatedusing hard thermal loop effective theory for the isotropicquark-gluon plasma [15, 16]. The heart of the problem inthis paper is to determine the structure of ρ µν ( q ), to seehow the anisotropy of the gluon and quark distributionsis reflected in the final orientation of the dilepton pair. FIG. 2: Photon polarization tensor with hard thermal loopcorrections to the quark lines and vertices [15]. In the course of writing this paper we became aware of thework of Friman and collaborators [14] which does not includethe anisotropic terms ρ T − ρ T and ρ n , but otherwise arrives atresults in agreement with those given here; the approach of thesetwo treatments of the problems are complementary and will bediscussed in a future joint publication of the two groups. III. STRUCTURE OF ρ µν In a heavy-ion collision volume, the initial gluon andquark distributions are anisotropic in momentum spacewith a single preferred axis ˆ n , which we assume to bealong the beam direction [17] (we do not consider at thispoint possible multiple anisotropy axes). We define thefour vector n µ to have space component ˆ n in the localrest frame of the matter and time component, n = 0, n µ = (0 , ˆ n ) , (4)so that n = −
1. We also define, in the local rest frame,the two transverse polarization vectors ε µi = (0 , ˆ ε i ) , (5)where ˆ ε ≡ ( (cid:126)q × ˆ n ) × (cid:126)q/ | (cid:126)q × ˆ n | and ˆ ε ≡ (cid:126)q × ˆ n/ | (cid:126)q × ˆ n | . These polarization vectors are illustrated in the leftpanel of Fig. 3. In addition, we define the longitudinalpolarization vector ε µ L ≡ (cid:112) Q ( | (cid:126)q | , q ˆ q ) = ( | (cid:126) ˜ q | , ˜ q ˆ q ) . (6)where we write ˜ q µ = q µ / (cid:112) Q . Note that ε = ε = ε = −
1. The three polarization vectors are individuallyorthogonal to q µ : ( qε ) = 0 (where ( ab ) denotes the fourvector product of a and b ) and together with q µ form anorthogonal basis obeying; g µν = ˜ q µ ˜ q ν − ε µ ε ν − ε µ ε ν − ε µ L ε ν L . (7) q^ n^pspʼ q q zn^ e ~ q^ x^ n^ e ~ q^x( x )q^^ s ⏊ e e f s q^ ^^ q s FIG. 3: (Left) Virtual photon polarization vectors ˆ ε i andthe relative spatial momentum (cid:126)s between the lepton and an-tilepton. (Right) The relative lepton momentum in the planetransverse to the virtual photon momentum (cid:126)q . Thus the photon spectral function, ρ µν , is a sum ofterms of the form aε µ L ε ν L + bε µ ε ν + c ( ε µ ε ν L + ε µ L ε ν ) + dε µ ε ν . (8)There are no terms proportional to q µ q ν for Q (cid:54) = 0,since q is a zero eigenvector of ρ ; in addition ( nε ) = 0,so there are no ε ε L terms by symmetry. Since (cid:126)n can bewritten as the linear superposition, (cid:126)n = cos θ q ˆ q +sin θ q (cid:126)ε ,with θ q being the angle between ˆ n and ˆ q , and n = 0, weobtain n µ = cos θ q (˜ q ε µ L − | (cid:126) ˜ q | ˜ q µ ) + sin θ q ε µ . (9)We also introduce the four-vector N µ with the property( N q ) = 0 as, N µ ≡ ˜ q cos θ q ε µ L + sin θ q ε µ = n µ − (˜ qn )˜ q µ , (10)Thus the ε µ ε ν L + ε µ L ε ν term can be eliminated in favor of N µ N ν , ε µ ε ν , and ε µ L ε ν L . In addition, N = − (1 + ( n ˜ q ) ).The latter term, plus the explicit ε µ L ε ν L term in Eq. (8),can be eliminated using Eq. (7). The photon spectralfunction, again with the help of Eq. (7), assumes thegeneral form ρ µν = ε µ L ε ν L ρ L + ε µ ε ν ρ T1 + ε µ ε ν ρ T2 + N µ N ν ρ n (11)= − ( g µν − ˜ q µ ˜ q ν ) ρ L + ε µ ε ν ( ρ T1 − ρ L )+ ε µ ε ν ( ρ T2 − ρ L ) + N µ N ν ρ n . (12)The momentum-space anisotropy of the system leads tothe extra ρ n term, as well as a difference of ρ T1 and ρ T2 .The terms ρ n , ρ T2 can be extracted directly fromEqs. (11) and (10) as ε µ ρ µν ε L ν = − ˜ q cos θ q sin θ q ρ n , (13)and ε µ ρ µν ε ν = ρ T2 , (14)while ρ L is found from ε L µ ρ µν ε L ν = ρ L + (˜ q ) cos θ q ρ n . (15)Using Eq. (13), we find ρ T1 from the trace condition, ρ µµ = − ( ρ L + ρ T1 + ρ T2 + (1 + ( n ˜ q ) ) ρ n ) . (16)When the particle distribution functions are even un-der parity, so that (cid:126)n enters only as a special axis, nota special direction, the extra ρ n term must vanish. Tosee this we note that when the distribution functionsare parity invariant, both parity and the transformationˆ n → − ˆ n are independent symmetries, meaning that aparity transformation keeping (cid:126)n fixed is also a symme-try. But under such a transformation (cid:126)ε L transforms as avector, while (cid:126)ε transforms as a pseudovector; thus themixing of the two directions in ρ ij , the source of ρ n , can-not occur. For collisions of two identical nuclei, thereshould not be a special direction in the local rest frameof the matter. Below, when we write down the Drell-Yan rate in the medium in such a situation, we will seeexplicitly how this argument is realized. However, forasymmetric collisions, one expects a non-zero ρ n term inthe photon spectral function. In the following, we keepthe ρ n term in the general discussions. The various ρ depend separately on the local q , q ⊥ ,and (cid:126)q · ˆ n , where q ⊥ is the magnitude of the component of (cid:126)q orthogonal to ˆ n . Or expressed covariantly, they dependon Q , ( qu ), as well as on ( qn ), where u µ is the 4-velocityof the local rest frame. Note that ( nu ) ≡ (cid:126)q along ˆ n , the ρ n term vanishes and ρ T1 = ρ T2 , while for (cid:126)q ⊥ ˆ n the eigenvectors of ρ are the ε i , with the two eigenvalues, ρ T1 and ρ T2 . More gener-ally, in the local rest frame, the eigenvectors of ρ µν are q µ with eigenvalue 0, ε µ with eigenvalue ρ T2 , and two or-thogonal linear combinations of ε µ and ε µ L whose spatialcomponents lie in the ( (cid:126)q, ˆ n ) plane. In the isotropic limit,the eigenvector ε L has eigenvalue ρ L , and ε has eigen-value ρ T . In contrast, when the system is anisotropic, thetwo eigenfunctions describe propagation in a birefringentmedium with a mixing of the longitudinal (L) and trans-verse (T) polarizations. Furthermore, as (cid:126)q → ρ T i − ρ L must vanish as (cid:126)q , and thus, ρ T i = ρ L for i = 1 , IV. EMISSION RATE OF DILEPTONS ANDPHOTONS
To calculate the production rates of dilepton pairs wefirst note that quite generally, ( sq ) = 0, so that12 ρ µν L µν = − ( Q ρ µµ + s µ ρ µν s ν ) (17)= Q (cid:0) ρ T1 + ρ T2 + ρ n (cid:1) + 4 m ρ L − s ( ρ T1 − ρ L ) − s ( ρ T2 − ρ L )+(( qn ) − ( sn ) ) ρ n , (18)where we use the identity Q + s = 4 m , with m beingthe lepton mass, and we define s i ≡ ( sε i ) ( i = 1 ,
2) tobe the components of (cid:126)s transverse to (cid:126)q in the local restframe: (cid:126)s ⊥ = s (cid:126)ε + s (cid:126)ε .Equation (18) gives the dilepton production rate interms of the projections of s along the two transversepolarizations and n . The s and s terms contain theanisotropy produced by transverse virtual photons, whilefrom Eq. (9), we see that the s z term arises from the mix-ing of longitudinal and transverse ( (cid:126)ε ) virtual photons.As noted above, for symmetric collisions with parity in-variance in the local matter rest frame, ρ n should van-ish so then the final term Eq. (18) is absent. To bringout the anisotropic terms, we write s = | (cid:126)s ⊥ | cos φ s , and s = | (cid:126)s ⊥ | sin φ s ; the squared matrix elements (18) be-come 12 ρ µν L µν = 2 Q ¯ ρ T + (cid:0) s ⊥ + 4 m (cid:1) ρ L + (cid:0) Q + ( qn ) − ( sn ) (cid:1) ρ n −| (cid:126)s ⊥ | (cid:0) ¯ ρ T + δρ T cos 2 φ s (cid:1) , (19)where ¯ ρ T ≡ ( ρ T1 + ρ T2 ) / δρ T ≡ ( ρ T1 − ρ T2 ) /
2. Thecos 2 φ s as well as the ρ n terms are anisotropic. Formassless dileptons in the absence of ρ n , the right side ofEq. (19) becomes 2 Q ¯ ρ T + | (cid:126)s ⊥ | ( ρ L − ¯ ρ T − δρ T cos 2 φ s ).With θ s the angle between (cid:126)q and (cid:126)s , we see that this ex-pression is of the form ∝ λ s cos θ s + µ s sin 2 θ s cos φ s +( ν s /
2) sin θ s cos 2 φ s , with λ s = ¯ ρ T − ρ L ¯ ρ T (1 − s /(cid:126)s ) + ρ L ,ν s = − δρ T ¯ ρ T (1 − s /(cid:126)s ) + ρ L , (20)and µ s = 0. We note the similarity to the angular distri-bution fitted in the NA60 analysis [12], where the anglesare defined in the Collins-Soper frame; as noted above,NA60 finds when averaging over all lab directions of thevirtual photons, that λ , µ , and ν are consistent with zero.With the Jacobian from the variables p and p (cid:48) to Q , (cid:126)s ⊥ , rapidity y and (cid:126)q T , d ¯ p d ¯ p (cid:48) = 12 dQ dy d q T d s ⊥ (cid:112) Q ( Q − s ⊥ − m ) , (21)we finally obtain the dilepton emission rate dR l + l − dQ d s ⊥ dyd q T = α π Q ρ µν L µν / (cid:112) Q ( Q − s ⊥ − m ) , (22)where ρ µν L µν / sn ) and ( qn ) are not independent; theirdependence on the experimental variables, Q , s ⊥ , y , and q T is algebraic (but too complicated to quote here).It is instructive to connect the present formalism forvirtual photons to the calculation of the rate for realphotons, Q = 0 including possible polarization, as con-sidered by [18, 19] and [8]. To do so, we rewrite Eq. (12)as ρ µν = − ( g µν Q − q µ q ν ) ρ L Q + ε µ ε ν ( ρ T1 − ρ L )+ ε µ ε ν ( ρ T2 − ρ L ) + ( Q N µ )( Q N ν ) ρ n Q , (23)from which we see that as Q → ρ L vanishes as Q and ρ n vanishes as Q . Thus the rate to produce a realphoton with polarization ε µ is dR γ d ¯ q = α π ε ∗ µ ρ µν ε ν , = α π (cid:0) ¯ ρ T + ( | ( εε ) | − | ( εε ) | ) δρ T (cid:1) = α π (cid:0) ¯ ρ T + δρ T cos 2 φ ε (cid:1) , (24)where ( εε ) ≡ − cos φ ε , ( εε ) ≡ − sin φ ε , and d ¯ q = d q/ | (cid:126)q | . The anisotropy for real photons arises entirelyfrom the difference, δρ T , of ρ T1 and ρ T2 : the spectral func-tion ρ n does not enter. V. DRELL-YAN PROCESS IN THE MEDIUM
To give a specific illustration of the present formalismwe focus on the leading-order Drell-Yan production ofdilepton pairs where the squared matrix element for aquark and antiquark to produce a virtual photon is H µν ( q, t ) = 2( q µ q ν − g µν Q − t µ t ν ) , (25)with t = k − k (cid:48) the difference of the four momenta of thetwo incident quarks, k and k (cid:48) .In a heavy-ion collision, the anisotropy in the Drell-Yanprocess arises only from the distributions of the initialquarks and antiquarks. The imaginary part of the lowest-order photon polarization tensor in a heavy-ion collisionis 12 ρ µν ( q ) = ( q µ q ν − g µν Q ) (cid:104) (cid:105) − (cid:104) t µ t ν (cid:105) , (26)where (cid:104) X (cid:105) = N c (cid:88) f e π (cid:90) d ¯ kd ¯ k (cid:48) Xδ (4) ( q − k − k (cid:48) ) f (cid:126)k ¯ f (cid:126)k (cid:48) , (27)with the sum being over flavors, e f denoting the quarkcharge; +2 / u -quarks, − / d -quarks, etc., and N c = 3 is the number of colors. The generally anisotropicquark and antiquark distributions are denoted by f and¯ f , respectively. In this notation, the coefficients in the spectral distri-bution function are obtained by comparing Eqs. (11) and(26): ρ T1 + sin θ q ρ n = 2 Q (cid:104) (cid:105) − (cid:104) ( ε t ) (cid:105) ,ρ T2 = 2 Q (cid:104) (cid:105) − (cid:104) ( ε t ) (cid:105) ,ρ L + (¯ q ) cos θ q ρ n = 2 Q (cid:104) (cid:105) − (cid:104) ( ε L t ) (cid:105) , (28)with ρ n = − q sin 2 θ q (cid:104) ( ε t )( ε L t ) (cid:105) . (29)When the product of the distribution functions is invari-ant under parity as well as ˆ n → − ˆ n , ρ n must vanish,as argued after Eq. (16). Explicitly, the orthogonality( qt ) = 0 implies ( ε L t ) = − t / | (cid:126) ˜ q | , so that in Eq. (29), (cid:104) ε t )( ε L t ) (cid:105) ∼ (cid:126)ε ·(cid:104) t (cid:126)t (cid:105) . But (cid:104) t (cid:126)t (cid:105) can at most be propor-tional to a linear combination of (cid:126)q and ˆ n , and if ˆ n → − ˆ n is The photon polarization operator is not that for an equilibriumsystem at finite temperature, owing to the fact that the electro-magnetic sector in a heavy-ion collision never has adequate timeto come into thermal equilibrium. The inverse processes in whicha dilepton pair is absorbed, would lead to the distributions en-tering as f (cid:126)k ¯ f (cid:126)k (cid:48) + (1 − f (cid:126)k )(1 − ¯ f (cid:126)k (cid:48) ) = 1 − f (cid:126)k − ¯ f (cid:126)k (cid:48) in the thermalequilibrium photon spectral function in full thermal equilibrium. an invariance, the ˆ n term must vanish; then since (cid:126)q · (cid:126)ε = 0one has ρ n = 0. In this case, Eq. (28) implies δρ T = −(cid:104) ( (cid:126)ε · (cid:126)t ) − ( (cid:126)ε · (cid:126)t ) (cid:105) = − (cid:104) t z − (cid:126)t / (cid:105) sin θ q , (30)which has the structure of a second spherical harmonic,and will thus select out the second spherical harmonicanisotropy in the particle distribution functions.A simple approach to describing the anisotropy of thedistributions is to assume an angular dependent temper-ature, so that, e.g., the quark distribution of masslessquarks becomes (with the chemical potential suppressed) f (cid:126)k = 1 e β (ˆ k ) k + 1 . (31)The parametrization of the angular dependent inversetemperature given in Ref. [13] takes the form β (ˆ k ) = β (1 + ξ cos θ k ) / where θ k is the angle between thequark momentum (cid:126)k and the anisotropy ( z ) axis. Thefull calculation of the Drell-Yan dileptons in an ultra-relativistic heavy-ion collisions with such an anisotropictemperature has been given by Strickland et al. [20–23].To illustrate, we expand the quark distribution func-tions, assumed to be of the form Eq. (31), to lowest orderin the angular dependence of the temperature, writing f (cid:126)k (cid:39) f k − f k (1 − f k )( β (ˆ k ) − β ) k , (32)where f k is the distribution with β (ˆ k ) = β . We see thenthat for weak anisotropy, δρ T ∼ − (cid:104) t z − (cid:126)t / (cid:105) ∼ − β (33)where β = (cid:82) − d (cos θ ) β (ˆ k ) P (cos θ ) = β ξ/
15 is thesecond spherical harmonic component of the tempera-ture.The terms in the photon spectral function ρ µν be-yond the lowest order Drell-Yan contribution are foundfrom the imaginary part of the polarization diagram inFig. 2 with hard thermal loop corrections and distribu-tion anisotropies [24].In practice, in order to obtain the dilepton spectra,one should integrate the spectral function, as given byEq. (27), over the space-time volume of the collisionvlume with an underlying model for the dynamics of ξ ( x ) and β ( x ) such as viscous [25] or anisotropic hydro-dynamics [26]. Previous work has shown that the high-energy dilepton rate computed in this manner is sensitiveto the assumed initial momentum-space anisotropy of theplasma [23] and that the momentum-space anisotropy ofthe quark-gluon plasma induces suppression of forwarddilepton production [22]. VI. OUTLOOK
In this paper we have formulated the general struc-ture of the spectral functions that describe the rate ofvirtual photon (dilepton) production in a heavy ion col-lision that is locally anisotropic in momentum space. Aswe have demonstrated, momentum-space anisotropy in-duces new angular dependence in the transverse struc-ture functions and, in the case of a non-parity symmetricmomentum-space anisotropy, a new structure function ρ n appears. As an example, we delineated the formal-ism for the leading-order Drell-Yan process. The struc-ture derived is not limited simply to production of virtualphotons from a quark-gluon plasma, but encompasses allvirtual photon production processes in collisions.With this full framework in place for relating polariza-tion information in dilepton production to the underly-ing physical mechanisms, the next step will be to gener-alize prior calculations for real photon production in ananisotropic quark-gluon plasma [18, 19]; also [8]. A forth-coming publication will present a detailed calculation ofthe polarization of dileptons to order α s , the strong inter-action fine structure constant, including Compton scat-tering of a gluon on a quark or antiquark to a virtualphoton, and annihilation of a quark-antiquark pair intoa gluon and virtual photon [24]. In the computation ofthe spectral functions, we include space and time depen-dent anisotropic quark, antiquark, and gluon distribu-tions, hard thermal loops, and soft scale processes, withthe space-time evolution described by full 3+1 dimen-sional anisotropic hydrodynamics. Acknowledgments
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