Impact parameter dependence of the azimuthal asymmetry in lepton pair production in heavy ion collisions

We investigate the impact parameter dependence of the $\cos 4\phi$ azimuthal asymmetry for electromagnetic lepton pair production in heavy ion collisions. The asymmetry induced by linearly polarized coherent photons exhibits strong impact parameter dependence.

parameter dependence of the cos 4φ azimuthal asymmetry. The paper is structured as follows. In the next section, we derive the b ⊥ dependent polarized cross section for dilepton production in heavy ion collisions. We present the numerical estimations for the asymmetries in the central, peripheral, and ultra-peripheral collisions at RHIC and LHC energy. The QED resummation effect is also included in our evaluations. The paper is summarized in Sec.III.

II. THE IMPACT PARAMETER DEPENDENCE OF cos 4φ AZIMUTHAL ASYMMETRIES
The dominant channel for dilepton production in the kinematical region where lepton pair transverse momentum is the order of the reverse of nucleus radius, is photon-photon fusion process, The leptons are produced nearly back-to-back in azimuthal with total transverse momentum q ⊥ ≡ p 1⊥ + p 2⊥ = k 1⊥ + k 2⊥ being much smaller than P ⊥ = (p 1⊥ − p 2⊥ )/2. When P ⊥ is sufficiently large, dilepton can be viewed as being produced locally. Since the location where dilepton is produced in the transverse plane is specified during our calculation, the incoming photons are no longer in the eigenstate of transverse momenta. Two incoming photon's momenta in the conjugate amplitude are denoted as x 1 P + k ′ 1⊥ and x 2P + k ′ 2⊥ respectively, with the constraint k ′ 1⊥ + k ′ 2⊥ ≡ q ⊥ . Following the method outlined in Refs. [42,43], the impact parameter dependent cross section computed at the lowest order QED reads, where φ is the angle between transverse momenta q ⊥ and P ⊥ . y 1 and y 2 are leptons rapidities, respectively. b ⊥ is the transverse distance between two colliding nuclei. Q is the invariant mass of the lepton pair. The B term is proportional to the lepton mass and thus very small. We do not present the detailed expression for B here. Instead, we focus on investigating the cos 4φ asymmetry in this work which depends only on the A and C terms. The coefficients A and C take form, and where The incoming photons longitudinal momenta fraction are fixed by the external kinematics according to with m being the lepton mass and s being the center mass energy. The nuclear charge form factor enters the , where M p is proton mass. Note that the lepton mass is ignored in the hard coefficients.
In comparison with our previous work where the calculation is formulated in TMD factorization, here we directly express the cross section as the convolutions of the form factor. Alternatively, one can factorize the impact parameter dependent cross section in terms of the photon Wigner distribution. However, this is beyond the scope of the present paper and will be addressed in a separate publication. Of course, if b ⊥ is integrated out in the above cross section formula, one can recover the cross section [27] derived using the equivalent photon approximation. This provides a nice consistency check. The cos 4φ azimuthal asymmetry is determined by the ratio between the terms A and C. In general, the size of the asymmetry can only be numerically calculated because of the complicated convolutions involved. However, quite remarkably, the analytical solution is available at b ⊥ = 0. After a few steps of algebraic manipulations, one finds that the convolutions in A and C terms turn out to be identical for b ⊥ = 0. The asymmetry is then simply proportional to, which indicates that the cos 4φ asymmetry for pure electromagnetic lepton pair production in the central collisions is independent of q ⊥ . This finding has been verified by the explicit numerical estimation as shown in Fig.1. However, one should keep in mind that hadronic background contribution in central collisions is significant. The size of the measured asymmetry could be far below the predicated value.
We now proceed to describe the ingredients needed in the numerical evaluations of the asymmetry. First, the form factor is taken from the STARlight MC generator [24], where a = 0.7 fm, and ρ 0 is a normalization factor. The nucleus radius is chosen to be R A = 1.1A 1/3 fm for Au and Pb targets, R A = 1.2A 1/3 fm for Ru target. This parametrization is very close to the Woods-Saxon distribution, and is used in our numerical evaluation. In order to take into account the effect of final state multiple soft photon radiation, a Sudakov factor has to be inserted in the differential cross section in r ⊥ space, where the Sudakov factor at one loop is given by [16], with µ r = 2e −γE /|r ⊥ |. The perturbative tail at relatively high q ⊥ is mainly generated by this Sudakov factor. The theoretical calculation is in good agreement with the ATLAS UPC high q ⊥ data [16]. In contrast, at intermediate lepton pair transverse momentum, roughly speaking 10 MeV < q ⊥ < 70 MeV, the q ⊥ shape is determined by the primordial photon distribution. At low q ⊥ (<10MeV), the smallness of the fine coupling constant is compensated by the large double logarithm ln 2 Q 2 , leading to a significant resummation effect, which, for example, has been clearly demonstrated in Fig.1.
The numerical results for the computed azimuthal asymmetries for the different collisions species and centralities are presented in Figs.2 and 3. Here the azimuthal asymmetries, i.e. the average value of cos 4φ are defined as, cos(4φ) = dσ dP.S. cos 4φ dP.S. dσ dP.S. dP.S.

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We compute the asymmetry for two deferent centrality classes as well as for the UPC and the tagged UPC cases. The corresponding impact parameter range for a given centrality class is determined using the Glauber model(see the review article [47] and references therein). For the UPC, the asymmetry is averaged over the impact parameter range [2R A , ∞]. However, STAR experiments at RHIC measure pair production cross section together with the double electromagnetic excitation in both ions. Neutrons emitted at forward angles by the fragmenting nuclei are measured, and used as a UPC trigger. Requiring lepton pair to be produced in coincidence with Coulomb breakup of the beam nuclei alters the impact parameter distribution compared with exclusive production. In order to incorporate the experimental conditions in the theoretical calculations, one can define a "tagged" UPC cross section, where the probability P (b ⊥ ) of emitting a neutron from the scattered nucleus is often parameterized as [48], As a matter of fact, the mean impact parameter is dramatically reduced in interactions with Coulomb dissociation. We plot the cos 4φ asymmetry for electron pair production at mid-rapidity as the function of the total transverse momentum q ⊥ at the center mass energy √ s = 200 GeV in Fig.2. The general trend is that the asymmetry increases when the impact parameter decreases. The overall q ⊥ and b ⊥ dependent behavior of the asymmetry for the different collision species(Au and Ru) are similar, except for that the curves are slightly more flat for the smaller nucleus. The asymmetry reaches a maximal value of 17%-22% percent around q ⊥ ≈ 30 MeV for the centrality classes [60%-80%], [80%-99.9%], and the tagged UPC. For the unrestricted UPC, the asymmetry is roughly twice smaller than that in the tagged UPC. The results obtained for di-muon production in Pb-Pb collisions at LHC energy shown in Fig.3 are rather close to these at RHIC energy.

III. CONCLUSIONS
We study the impact parameter dependence of the cos 4φ azimuthal asymmetry for purely electromagnetic lepton pair production in heavy ion collisions at low q ⊥ . This asymmetry arises from the correlation between the polarization vector of the electric field coherently generated by a fast moving heavy ion and the associated equivalent photon's transverse momentum. Such correlation reflects the nature of the boosted Coulomb potential. We found that the azimuthal asymmetry has a strong b ⊥ dependence. To be more specific, the asymmetry decreases with increasing impact parameter. Moreover, the q ⊥ dependent behavior of the azimuthal asymmetry is different in the different b ⊥ regions. We present numerical results for the b ⊥ and q ⊥ dependent asymmetry for the different collision species at various center mass energies. It would be interesting to test these theoretical predications at RHIC and LHC.
The study of such initial state effect in heavy ion collisions is not only important for facilitating the investigations of the electromagnetic properties of QGP, but also interesting in its own right. For instance, the polarization dependent observable can be used as a powerful tool to study QED processes in strong electromagnetic fields, particularly in view of the fact that no definitive conclusion on Coulomb correction has yet been drawn on experimental side. It has an advantage over the azimuthal angle averaged cross section in this regard because the absolute normalization which suffers various uncertainties are cancelled out when computing the asymmetry. Furthermore, comparing photon's linear polarization and gluon's linear polarization in the small x limit would be helpful for us to gain more insight into how the gluon polarization is affected by saturation effect.