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 f¨urSchwerionenforschung, 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 ther- malized medium are nonvanishing and depend on the flow prod Mμ = 0|Jμ|X1X2.