Virtual Photon Polarization and Dilepton Anisotropy in Relativistic Nucleus-Nucleus Collisions

Accepted Manuscript

Virtual photon polarization and dilepton anisotropy in relativistic nucleus-nucleus collisions

Enrico Speranza, Amaresh Jaiswal, Bengt Friman

PII: S0370-2693(18)30417-9 DOI: https://doi.org/10.1016/j.physletb.2018.05.053 Reference: PLB 33827

To appear in: Physics Letters B

Received date: 12 February 2018 Revised date: 15 May 2018 Accepted date: 20 May 2018

Please cite this article in press as: E. Speranza et al., Virtual photon polarization and dilepton anisotropy in relativistic nucleus-nucleus collisions, Phys. Lett. B (2018), https://doi.org/10.1016/j.physletb.2018.05.053

This is a PDF ﬁle of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its ﬁnal form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Virtual photon polarization and dilepton anisotropy in relativistic nucleus-nucleus collisions

Enrico Speranza Institute for Theoretical Physics, Goethe University, D-60438 Frankfurt am Main, Germany and GSI Helmholtzzentrum f¨urSchwerionenforschung, D-64291 Darmstadt, Germany

Amaresh Jaiswal School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India and Extreme Matter Institute EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany

Bengt Friman GSI Helmholtzzentrum fürSchwerionenforschung, D-64291 Darmstadt, Germany (Dated: May 21, 2018) The polarization of virtual photons produced in relativistic nucleus-nucleus collisions provides information on the conditions in the emitting medium. In a hydrodynamic framework, the resulting angular anisotropy of the dilepton final state depends on the flow as well as on the transverse momentum and invariant mass of the photon. We illustrate these effects in dilepton production from quark-antiquark annihilation in the QGP phase and π+π− annihilation in the hadronic phase for a static medium in global equilibrium and for a longitudinally expanding system.

I. INTRODUCTION of the virtual photon is reflected in anisotropies of the angular distribution of the lepton pair. Thus, different In relativistic heavy-ion collisions strongly interacting photon production mechanisms give rise to characteristic matter at very high temperatures and densities is cre- shapes for the dilepton angular distribution [11–14]. ated [1, 2]. At such extreme conditions, quarks and The angular distribution of the leptons originating gluons are deconfined and form a new phase of quan- from the decay of a virtual photon, expressed in the pho- tum chromodynamics (QCD), the quark-gluon plasma ton rest frame, is of the form [11, 12, 15, 16]: (QGP). Therefore relativistic nucleus-nucleus collisions dΓ N 2 provide a unique opportunity to study and characterize 4 = 1+λθ cos θ d qdΩ thermodynamic phases of QCD matter. 2 + λφ sin θ cos 2φ + λθφ sin 2θ cos φ Since, experimentally, the matter produced in nuclear ⊥ 2 ⊥ collisions can be probed only by observing and analyzing + λφ sin θ sin 2φ + λθφ sin 2θ sin φ , (1) the spectra of emitted particles, it is important to un- derstand their production mechanisms and interactions. dN where Γ ≡ d4x is the dilepton production rate per Hadronic observables interact strongly and do not probe unit volume, qμ the virtual photon momentum while the entire space-time volume of the collision because they θ and φ are the polar and azimuthal angles of, e.g., are emitted only from the surface and in the final state the negative lepton in the rest frame of the photon and of the medium. On the other hand, because their mean- dΩ = d cos θ dφ. The normalization N is independent ⊥ free paths in nuclear matter are much larger than nuclear of the lepton angles. The coefficients λθ, λφ, λθφ, λφ sizes, electromagnetic probes such as photons and dilep- ⊥ ⊥ ⊥ and λθφ are the anisotropy coefficients, λφ and λθφ be- tons are emitted throughout all stages of the collision ing non-zero only for processes that are not symmetric and escape from the system essentially without final- with respect to reflections in the production plane. state interactions. Therefore electromagnetic radiation The anisotropy coefficients depend on the choice of the carries direct information on the space-time evolution of quantization axis. In this work we employ two reference the matter created in such collisions [3–5]. frames (see Fig. 1). In the helicity frame, the quantiza- Recently, it was proposed that the polarization of real tion axis is along the photon momentum, while in the and virtual photons can be used to study the momentum Collins-Soper frame the quantization axis is the bisector anisotropy of the distributions of quarks and gluons [6–8]. of the angle formed by the beam and target momenta In a first measurement of the dilepton angular anisotropy, in the photon rest frame [16, 17]. We compute dilepton the NA60 Collaboration found that the anisotropy coef- emission from quark-antiquark annihilation in the QGP ficients in 158 AGeV In-In collisions are consistent with phase (the Drell-Yan process) and π+π− annihilation in zero [9], while the HADES Collaboration [10] finds a sub- the hadronic phase. stantial transverse polarization in Ar-KCl at 1.76 AGeV. The invariant mass spectrum and qT dependence of In general, multiply differential cross sections for dilep- low-mass dileptons (M<1 GeV) produced in heavy-ion ton emission provide information needed to disentangle collisions are consistent with an equilibrated, collectively the various production channels. The polarization state expanding source [18, 19]. Moreover, the lack of dilepton 2

y II. ANGULAR DISTRIBUTION

∗ Consider the annihilation process X1X2 → γ → + − zHX ,whereX1 and X2 denote a particle and its an- ∗ p tiparticle, while γ is the intermediate virtual photon, B which decays into a lepton-antilepton pair. The rate per pA δ unit volume for this process can be written as [21, 22] 4 3 3 zCS dΓ e d l+ d l− μν 4 = 4 3 3 W Lμν (2) d q q (2π) 2E+ (2π) 2E− (4) × δ (q − l+ − l−), 2 2 where E± = |l±| + m , m is the lepton mass, e its charge and Lμν the lepton tensor

FIG. 1. (Color online) Illustration of the two reference frames Lμν =2(−q2gμν + qμqν − ΔlμΔlν ). (3) employed in this paper. The production plane, indicated in μ μ gray, is spanned by the three momenta of the initial ions in Here l+ and l− are the antilepton and lepton momenta, the rest frame of the virtual photon, pA and pB ,andisor- μ μ μ while q ≡ l+ + l− is the virtual photon momentum, and thogonal to the y axis. The axes zHX and zCS define the he- μ μ μ Δl ≡ l+ − l−. licity and Collins-Soper frames, respectively. The two frames For dilepton emission from a medium in local thermo- are connected by a rotation through the angle δ about the y μν axis [16]. dynamic equilibrium, the tensor W , which describes the annihilation of X1 and X2 into a virtual photon, is given by an ensemble average anisotropy found in Ref. [9] has been interpreted as ev- W μν = wμν . (4) idence for a thermalized medium. However, as noted in Ref. [3], also a fully thermalized medium in general emits The tensor wμν , which describes the elementary process, polarized photons. In this letter, we present results on will be discussed below and in Sect. II B. The ensemble the anisotropy coefficients for dileptons emitted from a average, denoted by the brackets in (4), is of the form thermalized static medium and explore the effect of lon- 3 3 gitudinal (Bjorken) flow on the polarization observables. d p1 d p2 4 (4) − − A = 3 3 (2π) δ (q p1 p2) While in Ref. [4] the coefficient λθ has been discussed (2π) 2E1 (2π) 2E2 for some relevant reactions in vacuum as well as in a ther- 1 1 × A. (5) mal system, a systematic study of the dilepton anisotropy e(u·p1)/T ± 1 e(u·p2)/T ± 1 coefficients in heavy-ion collisions has so far not been per- μ μ formed. Here we present a general framework for study- Here p1 and p2are the momenta of X1 and X2, respec- 2 2 ing the angular anisotropy of dileptons emanating from tively, E1,2 = |p1,2| + m , T is the temperature and the decay of polarized virtual photons produced in a hot, expanding medium. In order to characterize the angular μ distribution of the dilepton final state, we introduce the u =(γ,γv), (6) anisotropy coefficients and demonstrate explicitly their dependence on the flow velocity and temperature profile is the four velocity of the medium. The plus and minus of the medium. We first consider the simplest case of signs in the distribution functions in (5) apply when the particles X1 and X2 are fermions or bosons, respectively. a static uniform medium and then incorporate the non- μν trivial dynamics of a longitudinally expanding system. The tensor w in Eq. (4) is given in terms of the QCD matter formed in high-energy heavy-ion collisions matrix elements of the electromagnetic current operator exhibits a striking collective behaviour and its space-time Jμ evolution can be described quite accurately with rela- MprodMprod∗ tivistic hydrodynamics (for a recent review, see [20]). We wμν = μ ν , (7) use one-dimensional longitudinal Bjorken flow to account pol for the expansion of the medium along the beam axis. We where find that in general the anisotropy coefficients in a thermalized medium are nonvanishing and depend on the flow prod Mμ = 0|Jμ|X1X2. (8) of the medium as well as on the transverse momentum and invariant mass of the virtual photon. The sum in (7) is over the spin states of X1 and X2. Throughout the text we use natural units, with c = The most general structure for W μν is composed of the μν = kB =1. metric tensor g and the two four vectors that we have 3

μ μ μν ¯ μ at our disposal, q and u . By requiring that W is f ef ψf γ ψf ,whereψ is the quark field and ef the symmetric and current conserving, one finds quark charges. Thus, the explicit expression for the wμν μ ν tensor of Eq. (7) is formally the same as the lepton tensor: μν μν q q W = W1 g − q2 μν 2 μν μ ν μ ν · · w =2Cq (−q g + q q − Δp Δp ), (14) μ − u q μ ν − u q ν + W2 u 2 q u 2 q , (9) q q μ μ where p1 and p2 are the quark and antiquark momenta μ μ μ where W1 and W2 are Lorentz invariant functions, similar and Δp = p1 − p2 . The sum over quark flavors and 2 to the structure functions in deep inelastic scattering, colors yields the factor Cq = Nc f ef ,whereNc is that depend on q2, u · q and the temperature T .The the number of colors. For two flavors and three colors μν Lorentz scalars in (9) are given by Cq =5/3. The scalar contractions α ≡ gμν W and μν β ≡ uμuν W are given by αa − β 3β − αa 1 2 W = ,W= 2 , (10) 2 2 2a 2a α =2Cq (−3q − Δp )1, (15) μν μν (u·q)2 · 2 − 2 − · 2 where α ≡ gμν W , β ≡ uμuν W and a =1− q2 . β =2Cq ((u q) q ) 1 (u Δp) , (16) where we use the notation introduced in Eq. (5). A. Dilepton production rate In the case of the pion annihilation process (i.e., π+π− → e+e−), the photon couples to the pion con- μ − μ + + μ − + vection current Jπ =(Φ∂ Φ − Φ ∂ Φ ), where Φ Starting from Eq. (2), we can write the angular dis- − tribution of the lepton pair in the photon rest frame as (Φ ) is the field of the positive (negative) pion. For the annihilation process, the current is proportional to the difference of the momenta of the two pions Δpμ.One 2 d Γ α 1 − 2 2 μν thus finds 4 = 4 4 1 4m /q Wμν L , (11) d qdΩ 32π q wμν =ΔpμΔpν , (17) where α = e2/4π. In the helicity frame the polar angle is measured relative to the three momentum of the photon which leads to in a reference frame, while the azimuthal angle is measured with respect to the plane formed by the beam axis α =Δp21,β= (u · Δp)2. (18) and the photon momentum, the production plane. 2 2 2 Using Eq. (3) and Eq. (9), the angular distribution We note that Δp =4m − q is a constant for a given becomes value of the photon invariant mass M = q2.Herem is 2 the mass of the incident particles. d Γ α 1 − 2 2 − 2 − 2 4 = 4 4 1 4m /q ( 3q Δl )W1 d qdΩ 16π q 2 2 2 +[−q +(u· q) − (u· Δl) ]W2 . (12) III. MEDIUM AND FLOW

The angular dependence is expressed in polar coordinates In this section, we compute the anisotropy coefficients in the helicity frame: for dilepton emission from a medium. We consider two cases: (i) a static medium with uniform temperature and μ |l| |l| |l| Δl =(0, 2 sin θ cos φ, 2 sin θ sin φ, 2 cos θ). (ii) a longitudinally expanding system with Bjorken flow. (13) We note that the only dependence on lepton angles in · 2 (12)isinthe(u Δl) term. The angular distribution is A. Static uniform medium parametrized in the general form (1). As mentioned in the introduction, the anisotropy co- In the photon rest frame, the collective velocity of the efficients depend on the choice of the quantization axis. fluid is in the direction opposite to the photon momen- The coefficients in different coordinate systems are re- tum in the fluid rest frame. Thus, in the helicity frame, lated by rotations [16]. For a fluid cell with a given ve- μ ⊥ ⊥ where the z-axis is along the photon momentum, the fluid locity u , the anisotropy coefficients λθ, λφ, λθφ, λ , λ φ θφ velocity is given by and the normalization N are obtained from Eq. (12). μ 1 u = γz(1, 0, 0,vz)= (Eγ , −q) (19) M B. Drell-Yan and pion annihilation 2 where γz ≡ 1/ 1 − vz = Eγ /M , vz = −|q|/Eγ , while 0 For the Drell-Yan process (i.e., quark-antiquark an- Eγ = q and q are the energy and momentum of the + − μ nihilation qq¯ → e e ), the current is given by Jq = virtual photon in the fluid rest frame, respectively. 4 √ In the photon rest frame, the phase-space integrals in τ ≡ t2 − z2. The fluid four-velocity is, in the center-of- Eq. (5) are, using the azimuthal symmetry, reduced to an momentum (c.m.) frame, given by integral over the angle ϑ between the momentum of one μ of the incident particles (q or π−) and the fluid velocity u = (cosh η, 0, 0, sinh η) , (23) −1 1 where η ≡ tanh (z/t) is the space-time rapidity. In 1 1 1 A = d(cos ϑ) χ A. the absence of viscosity, the temperature evolution of the (u·p1)/T ± (u·p2)/T ± 8π −1 e 1 e 1 system is given by the Bjorken scaling solution, (20) Here χ ≡ 1 − 4m2/M 2 and T ∝ τ −1/3. (24)

Eγ |q| Here a conformal equation-of-state is assumed, with = u · p1 = + χ cos ϑ, (21) ∝ 4 2 2 3P T ,where is the energy density and P the pres- sure of the system. Eγ |q| u · p2 = − χ cos ϑ. (22) The photon momentum can, in the c.m. frame, be writ- 2 2 ten as

For a static system, the multiplicity is given by an inte- μ gral of the rate d Γ over space-time. This yields an overall q =(MT cosh y, qT cos φγ ,qT sin φγ , factor Vte (volume × emission time), which cancels in the MT sinh y), (25) anisotropy coefficients. Owing to the azimuthal symmetry of the static uniform where y is the longitudinal rapidity of the photon, qT its ≡ 2 2 system in the helicity frame 1, the only non-vanishing transverse momentum and MT M + qT its transverse mass. In the rest frame of a cell, moving with fluid anisotropy coefficient is in this case λθ, corresponding to a tensor polarized virtual photon [23]. Moreover, in rapidity η, the same photon has momentum the limit q → 0, the distribution functions of the initial μ (q ) =(MT cosh(y − η),qT cos φγ ,qT sin φγ , state in Eq. (20) are spherically symmetric, leading to − zero polarization and vanishing anisotropy coefficients. MT sinh(y η)), (26) · · We note that in the Boltzmann limit, u p1,u p2 T , For a given fluid cell, the z-axis of the corresponding the angular dependence of the distribution functions in helicity frame is aligned with the momentum of the pho- (20) cancels, implying that W2 vanishes and that the vir- ton in the rest frame of the cell (26). Consequently, the tual photon is unpolarized [21]. Thus, the anisotropy of helicity frames for photons emitted from cells with dif- dileptons emitted from thermal systems in annihilation ferent fluid velocities do not coincide. However, in order processes is a consequence of quantum statistics. It fol- to define the polarization of the ensemble of photons, lows that the anisotropy coefficients approach zero for emitted from all fluid cells, one needs to choose a unique large photon momenta, or large invariant masses, where reference frame. A natural choice is the helicity frame the distribution functions of the initial particles are well of a cell, which is at rest in the c.m. frame. The z-axis approximated by the Boltzmann distribution. is then aligned with the spatial part of qμ,(25). In the Furthermore, for particles of non-zero mass m in the following this frame is denoted HX. It follows that the initial state, the anisotropy coefficients vanish also in the → helicity frame of a cell moving with rapidity η, HX ,is limit M 2m, since in this limit, the distribution func- rotated with respect to HX. The rotation angle equals tions in (20) are independent of the orientation of the μ μ 2 p p that between the spatial parts of q and (q ) ,(25)and momenta 1 and 2. (26). For a photon emitted transversely to the beam axis in the c.m. frame, i.e., with y = 0, this angle is given by ξ =arctan(−(MT /qT )sinhη). B. Longitudinal flow Another tenable choice for the reference frame is defined by choosing the z-axis perpendicular to the beam In the case of a purely-longitudinal boost-invariant ex- axis, irrespective of the direction of the photon. Clearly, pansion of a transversely homogeneous system first con- this frame coincides with HX for y =0. sidered by Bjorken [24], all scalar functions of space The Lorentz invariant structure functions W1 and W2 and time depend only on the longitudinal proper time are, for each cell, computed as in a static, uniform medium. Consequently, in the corresponding helicity frame (HX), the emitted photon is tensor polarized and the only non-zero anisotropy parameter is λθ.Thecontri- 1 We stress that, for a static system, the azimuthal symmetry is bution of the fluid cell to the angular distribution of the a general property, independent of the elementary emission pro- lepton pairs in HX is then obtained by evaluating (12) cess. μ 2 Since all averages defined by (20) vanish for χ → 0, this is the in that frame. Clearly, the fluid four-velocity u in HX case also for the cross section (1). However, the contributions to depends on the fluid rapidity η in the c.m. frame. The the dilepton anisotropy vanish with a higher power of χ, implying rotation from HX to HX yields non-zero contributions λ ∝ χ2 χ → that θ for 0. to λφ and λθφ [16]. 5

Finally, the integration of the dilepton rate over the + - space-time evolution of the system is in a symmetric, q q → e e M = 0.6 GeV y = 0 central collision performed using 0 λ Static

] θ λ dN -2 -2 θ Bjorken 2 2 (27) d qT dM dy λ Bjorken [10 τf ∞ 1 dN -4 = πR2 dτ τ dη , A 2 d4xd4q τi −∞ (a) -6 where τi and τf are initial and final proper time of the 2 + - + - 1/3 π π → specific phase we are considering and RA =1.2A is e e M = 0.6 GeV y = 0 the nuclear radius, A being the nucleon number of the 0 incident ions. The integrand in Eq. (27) depends on the ] -2 integration variables τ and η through T and u· q. -2 At this point, several comments are in order. i) The -4 λ Static [10 θ integration over fluid cells with different rapidities in λ -6 θ Bjorken Eq. (27), leads to non-vanishing λφ and λθφ in the HX λ 3 Bjorken frame . Thus, not only anisotropies in the momentum -8 (b) distribution of the emitting particles [8], but also collective flow is reflected in the polarization observables. ii) 0 1 2345 Even though the Bjorken solution is boost invariant, the qT [GeV] resulting anisotropy coefficients, defined in the reference frame HX, depend on the photon rapidity y.Thisisbe- FIG. 2. (Color online) Anisotropy coefficients as functions cause the quantization axis in the HX frameisaligned of the virtual photon transverse momentum at an invariant mass M =0.6 GeV for (a) the Drell-Yan process, and (b) with the photon momentum in the c.m. frame (25). Nev- pion annihilation. The red dashed lines refer to λθ in the case ertheless, for the photon emission rate per unit volume λ 4 of a static uniform medium, while the blue solid lines show θ dΓ/d q, obtained by integrating Eq. (1)overthelep- in the helicity frame for the longitudinal Bjorken expansion ton angles Ω, longitudinal boost invariance is recovered. and the green dot-dashed lines the corresponding λ˜. This implies that the combination N (1 + λθ/3), inte- grated over d4x, is independent of y. iii) Moreover, for 4 the Bjorken solution, the combination N (λθ +3λφ)is pion-annihilation processes . We compute these coeffi- invariant under longitudinal Lorentz boosts. cients for a static uniform medium and for a medium We stress that the above invariances follow from the as- with longitudinal Bjorken expansion. In the Bjorken sumptions of local thermodynamic equilibrium and lon- case, we evolve the system from the initial temperature, gitudinal boost invariance and are independent of the Ti = 500 MeV (at initial proper time τi =0.4 fm) to the production process. As noted above, the anisotropy co- final temperature Tf = 160 MeV for the Drell-Yan pro- efficients, λn (n = θ, φ etc.), are frame dependent, i.e., cess and from Ti = 160 MeV (τi =12.2fm)toTf = 120 they depend on the choice of quantization axis. How- MeV for the pion-annihilation process. In the case of a ever, it is possible to define frame-invariant combinations, static uniform medium, we use, in order to facilitate the like [16, 25] comparison between the scenarios, the relevant average temperature for each process, λθ +3λφ λ˜ ≡ . (28) 1 − λφ 3 Ti + Tf Tav = Ti Tf 2 2 , (29) 2 Ti + Ti Tf + Tf In the case of the Bjorken expansion, λ˜ is also invariant under Lorentz boosts along the beam axis. obtained using the Bjorken evolution, Eq. (24). In Fig. 2, we show the anisotropy coefficients λθ and λ˜ against the photon transverse momentum at an invariant IV. NUMERICAL RESULTS mass M =0.6 GeV. The coefficients are shown for the Drell-Yan and pion annihilation processes for two velocity In this section, we present numerical results for the profiles describing a static and uniform medium and a anisotropy coefficients λθ and λ˜ for the Drell-Yan and longitudinal Bjorken expansion. We observe that, in the static case, the anisotropy coefficient tends to zero for

3 For zero photon rapidity y, λθφ is an odd function of ﬂuid rapid- 4 ity and hence vanishes after integrating over η with the Bjorken In the pion-annihilation process we use mπ = 139.6 MeV, while solution. in the Drell-Yan process we set the quark masses to zero. 6

1 similar anisotropy patterns, although the photon polar- 0.6 GeV < qT < 2 GeV y = 0 izations in the corresponding elementary reactions are 0 + - q q → e e distinctly different. In the Drell-Yan process, the pho- ] -1 tons are purely transverse (λθ = 1), while in the pion- -2 annihilation process they are purely longitudinal (λθ = −1) in a frame where the z-axis is along the “beam” axis, [10 -2 λ θ Static λ Bjorken defined by the momenta of the incident particles in the -3 θ c.m. frame. Now, when the incident particles are drawn λ Bjorken (a) -4 from a Bose-Einstein distribution, momenta along the z axis of the helicity frame are preferred, while for a Fermi- Dirac distribution function, there is a slight preference 0.6 GeV < qT < 2 GeV y = 0 0 + - + - for momenta in the plane orthogonal to that axis. These π π → e e effects conspire to yield the resulting negative values for ] -2 -2 λθ in the helicity frame, shown in Figs. 2 and 3. λ θ Static In order to make a qualitative comparison with the [10 -4 λ θ Bjorken data of NA60 [9], we perform an integration over the invariant mass, the photon transverse momentum and -6 λ Bjorken (b) the rapidity in the intervals 0.4GeV

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