Probing the linear polarization of photons in ultraperipheral heavy ion collisions

We propose to measure the linear polarization of the external electromagnetic fields of a relativistic heavy ion through azimuthal asymmetries in dilepton production in ultraperipheral collisions. The asymmetries estimated with the equivalent photon approximation are shown to be sizable.


II. AZIMUTHAL ASYMMETRIES IN DI-LEPTON PRODUCTION IN UPCS
Di-lepton production in UPCs is well described by two photons reaction at the lowest order QED, 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 the individual lepton transverse momenta p 1⊥ or p 2⊥ . Since there are two well separated scales in this process, the application of TMD factorization is justified. If the calculation is formulated in TMD factorization, the two leading power photon TMDs: the normal unpolarized photon TMD and the linearly polarized photon TMD contribute to the differential cross section. They are formally defined as the following, where two photon TMDs, f γ 1 and h ⊥γ 1 , are the unpolarized and linearly polarized photon distribution, respectively. This matrix element definition for photon TMDs bears much resemblance to those for the gluon ones [2]. However, one should note that there is no need to add gauge link for ensuring gauge invariance since photon does't carry charge. As such, the light cone singularity is absent for the photon TMD case.
One can easily recover the azimuthal dependent cross section for lepton pair production from the results for heavy quark pair production existed in the literatures [8,9]. It is of course also straightforward to compute the cross section at the lowest order QED, which reads, where φ is the angle between transverse momenta q ⊥ and P ⊥ = (p 1⊥ − p 2⊥ )/2. y 1 and y 2 are leptons rapidities, respectively. Q is the invariant mass of the lepton pair. The coefficients A, B and C contain convolutions of photon TMDs, and and wherek 1⊥ andq ⊥ are unit vectors defined ask 1⊥ = k 1⊥ /|k 1⊥ | andq ⊥ = q ⊥ /|q ⊥ | respectively. The incoming photons longitudinal momenta fraction are fixed by the external kinematics according to x 1 = P 2 ⊥ +m 2 s (e y1 + e y2 ) and with m being lepton mass.
When going beyond the lowest order QED, the Sudakov type logarithm terms αe 2π ln 2 Q 2 q 2 ⊥ will arise from the final state soft photon radiation in higher order calculation. In particular, at LHC energy, the logarithm terms are sizeable and need to be resummed to all orders to improve the convergence of the perturbation series. This can be achieved by applying the Collins-Soper-Sterman(CSS) [1] formalism. The CSS formalism is formulated in the impact parameter space in which the large logarithms are resummed into an exponentiation known as the Sudakov factor. By taking into account the Sudakov factor, the coefficients A and C after the Fourier transform can be rewritten as, where θ is the angle between q ⊥ and b ⊥ , and µ b = 2e −γE /|b ⊥ |. At LHC energy, one can neglect the contributions suppressed by the power of m 2 P 2 ⊥ in the hard part as shown in the above formulas. Note that cos 2φ asymmetry vanishes at LHC energy under this approximation because it is proportional to m 2 P 2 ⊥ . However, muon mass can not be neglected when computing both cos 2φ and cos 4φ asymmetries at RHIC energy. At one loop order, the Sudakov factor is given by [22], It has been shown that this Sudakov factor plays a crucial role in correctly reproducing the high q ⊥ tail observed by the ATLAS collaboration [22]. The distribution of photons coherently generated by the charge source inside relativistic nuclei is commonly computed with the Weizsäcker-Williams method. This quasi-classical method also can be used to determine the linearly polarized photon distribution following the similar derivation that relates the dipole amplitude to the various polarized gluon distributions [3,[28][29][30]. Supposing that a nuclei moves along P + direction, the dominant component of the gauge potential is A + and other components are suppressed by the Lorentz contraction factor γ. Based on this observation, after taking partial integration the photon field strength tensor is approximated as F µ +⊥ F ν +⊥ ∝ k µ ⊥ k ν ⊥ A + A + , which implies the relation, In the equivalent photon approximation, one then has [14,15], where Z is the nuclear charge number, and F is the nuclear charge form factor. M p is proton mass. The form factor is often parameterized using the Woods-Saxon distribution, where the radius R W S (Au: 6.38fm, pb: 6.62fm) and the skin depth d(Au.:0.535fm, Pb:0.546fm). ρ 0 is the normalization factor. Alternatively, one can use the form factor in momentum space from the STARlight MC generator [19], where R A = 1.1A 1/3 fm, and a = 0.7fm. This parametrization numerically is very close to the Woods-Saxon distribution, and will be used in our numerical evaluation. With all these ingredients, we are ready to perform numerical study of the azimuthal asymmetries in lepton pair production in UPCs.
The numerical results for the computed azimuthal asymmetries in the different kinematical regions are presented in Figs. [1][2][3][4]. Here the azimuthal asymmetries, i.e. the average value of cos(2φ) and cos(4φ) are defined as, cos(2φ) = dσ dP.S. cos(2φ)dP.S. dσ dP.S. dP.S. (13) cos(4φ) = dσ dP.S. cos(4φ) dP.S. dσ dP.S. dP.S. (14) As the cos(2φ) azimuthal asymmetry is suppressed by the power of m 2 /P 2 ⊥ , it is only sizable for di-muon production at RHIC energy. We plot the cos(2φ) asymmetry for muon pair production at mid-rapidity as the function of the total transverse momentum q ⊥ for three different invariant mass regions at the center mass energy √ s = 200GeV. Obviously, the asymmetry decreases with increasing invariant mass as its power behavior indicates. In the lowest invariant mass region M µµ ∈ [0.4, 0.76]GeV, the asymmetry reaches a maximal value of 7% percent around q ⊥ = 110MeV. For the same kinematical regions at RHIC, we also plot the cos 4φ asymmetry for electron pair and muon pair production. The asymmetry grows with increasing q ⊥ until it reaches a maximal value at total transverse momentum around 120MeV. The maximal value of the asymmetry is about 20% for electron pair production. The cos 4φ asymmetry for di-muon production is slightly smaller than that for electron pair production in the same kinematical region. One sees that the cos 4φ asymmetry drops rather fast at relatively large transverse momentum(> 120MeV).
The curve for the cos 4φ asymmetry for di-muon production at LHC is presented in Fig.4. The the q ⊥ dependence of the asymmetry is similar to these for RHIC energy. The maximal size of the asymmetry is about 9% for the invariant mass region GeV. We further found that the Sudakov suppression effect due to final state soft photon radiation reduce the asymmetry significantly at relatively large q ⊥ as compared to the lowest order calculation. This may serve as a very clean test of the resummation formalism for the QED case.

III. CONCLUSIONS
The unpolarized photon distribution used to compute physical observables in ultraperipheral heavy ion collisions is commonly determined using the classical external electromagnetic fields of a relativistic charged nuclei. Applying this quasi-classical method to the polarized case, one easily finds that the linearly polarized photon distribution is identical to the normal unpolarized photon distribution. The linearly polarized photon distribution can be cleanly probed through the cos 2φ and cos 4φ azimuthal asymmetries in lepton pair production in ultraperipheral heavy ion collisions, where φ is the angel between lepton pair total transverse momentum and individual lepton transverse momentum. We present numerical results for the azimuthal asymmetries in the kinematical regions where the experimental data for di-lepton production has been taken at RHIC and LHC. In these kinematical regions, the magnitude of the cos 4φ azimuthal asymmetry for both electron pair and muon pair production are rather large. And moreover, the cos 2φ azimuthal asymmetry in di-muon production at RHIC energy is sizable. These findings are very promising concerning a future extraction of h ⊥γ 1 in UPCs at RHIC and LHC. In our numerical estimation, we also took into account the Sudakov suppression effect which reduces the asymmetries significantly at relatively large lepton pair transverse momentum. The Sudakov suppression of the azimuthal asymmetry in this process would provide a clean way to test the resummation formalism in the QED case. Furthermore, one may expect that this mechanism also plays a role in generating azimuthal asymmetries in hadronic heavy-ion collisions. The study of such initial state effect thus would set a baseline for investigating the electromagnetic properties of the quark-gluon plasma created in hadronic heavy-ion collisions [21-24, 31, 32].