Exclusive production of rho^0 rho^0 pairs in gamma-gamma collisions at RHIC

We discuss exclusive electromagnetic production of two neutral $\rho$ mesons in coherent photon-photon processes in ultrarelativistic heavy-ion collisions. The cross section is calculated in the equivalent photon approximation (EPA). Both uncertainties of the flux factors and photon-photon cross sections are discussed in details. We show that inclusion of precise charge densities in nuclei is essential for realistic evaluations of the nuclear photon-photon cross sections. We find that the cross section, especially with realistic flux factors, is sensitive to low energy in the subsystem $\gamma \gamma \to \rho^0 \rho^0$. The experimental data for the $\gamma \gamma \to \rho^0 \rho^0$ cross section extracted from $e^+ e^-$ collisions are parametrized and used to estimate the nucleus-nucleus cross section. In addition, we include vector-dominance-model(VDM)--Regge contribution which becomes important at large photon-photon energy. Large nuclear cross sections are obtained. We discuss a possibility of focusing on the large-energy component. We find that both $\rho^0$ mesons are produced predominantly at midrapidities and could be measured by the STAR collaboration at RHIC.


I. INTRODUCTION
Exclusive production of elementary particles (lepton pairs, Higgs, etc.) or mesons (vector mesons, pair of pseudoscalar mesons, etc.) in ultrarelativistic heavy ion collisions is an interesting and quickly growing field [1,2,3] of theoretical investigation.On experimental side the situation is slightly different.So far only single-ρ 0 exclusive cross section AA → AAρ 0 was measured [4].Here the dominant mechanism is the photoproduction described by the photon-pomeron fusion.
FIG. 1: The reaction discussed in this paper.
In the present paper we consider exclusive production of ρ 0 ρ 0 pairs in γγ collisions.So far only integrated cross section for AA → AAρ 0 ρ 0 was roughly estimated in the literature [5], using a pQCD-inspired modelling of the Pomeron exchange between mesons for AA → AAJ/ΨJ/Ψ and AA → AAρ 0 J/Ψ and assuming Regge factorization.Uncertainties of almost three-orders of magnitude were found.As we discuss here, in the AA → AAρ 0 ρ 0 reaction one is sensitive rather to the close-to-threshold region for the γγ → ρ 0 ρ 0 subprocess where the specific pQCD approaches of [5] are not applicable at all.Furthermore the Reggefactorization for this hard and soft processes is likely broken.The ρ 0 ρ 0 pair will eventually decay into the final state of four charged pions.If the ρ 0 ρ 0 invariant mass is very large, the two π + π − -pairs will be separated by a large rapidity gap.It has been recently suggested that in this kinematics, charge asymmetries can be used to discover/study the QCD-Odderon exchange [6].
At the RHIC energies discussed here, we will be however dominated by γγ center-of-mass energies in the region of a few GeV.Here, experimental data for γγ → ρ 0 ρ 0 are known and were measured by several groups at e + e − colliders [7].Experimentally one observes a huge enhancement close to the threshold.The origin of this enhancement was never understood.Some speculations are discussed in Ref. [8].The huge cross section at W ≈ 1.5-2 GeV was interpreted as a tensor resonance decaying into ρ 0 ρ 0 channel [9].In Ref. [10] the Regge factorization was tested.In the following we shall see the consequences of the anomalous behaviour seen in the e + e − data for the two-photon production of the ρ 0 ρ 0 pair in the nucleus-nucleus collisions.

A. Equivalent photon approximation for nuclei
Let us consider the process AA → AAρ 0 ρ 0 depicted in Fig. 1.The cross section takes the familiar form of a convolution of equivalent photon fluxes and γγ-cross sections: Here we omitted helicity-dependent pieces, see below for more details.The effective photon flux is expressed through the electric field strengths of the ions and reads [11] (see also [12]): where the electromagnetic field strengths are given in terms of the charge form factor of the nucleus F em ( q 2 ) by: (bq)J 1 (bq) In practice F em ( q 2 ) is obtained as the Fourier transform of the nuclear charge densities extracted from electron-nucleus scattering data, for a useful compilation, see [13,14].By putting F em ≡ 1 one obtains the well-known result for a point-like charge (see e.g. the textbook [15]): Notice that our formula for the effective Weizsäcker -Williams flux of photons includes the impact-parameter dependent absorption factor S 2 abs (b).It represents the probability that no inelastic interaction between the nuclei occurs.At RHIC energies it could be for example estimated from Glauber theory using the Czyż-Maximon [16] approximation: where T A (b) is the optical thickness of the nucleus.We checked, that for our purposes it can be well approximated by the result for a black disc: The presence of the absorption factor also induces an azimuthal correlation between impact parameters b 1 and b 2 .Now, soft photons are polarized linearly in the transverse plane, and therefore a subtle helicity dependence arises.Strictly speaking, one should write where are the fluxes for photons with parallel ( ), and perpendicular (⊥) linear polarizations; σ and σ ⊥ are the corresponding γγ cross sections.The small effects associated with this helicity dependence have been discussed in [11] and will be neglected here.
A further comment on the practical uses of the equivalent photon approximation is in order.It is often pointed out that the Fermi-Weizsäcker-Williams idea of the equivalent photon flux represents indeed the archetypical parton-model concept.In this spirit, often flux factors of equivalent photons are calculated as for point-like particles with charge Ze, and the total cross section is then evaluated using a simple parton-model type formula: When calculating the two-dimensional integral it must be checked if is the flux of photons in an ultrarelativistic charge of energy γM A .For RHIC, at the cmsenergy √ s N N = 200 GeV we have γ ∼ 100.
Notice that such a factorized form is not borne out by the general formula (2.2) which accounts for the strong absorptive corrections.In particular, one cannot meaningfully improve the b-integrated Eq.(2.9) to effectively account for strong absorption, in particular this is not achieved by simply restricting the impact parameter integrals (2.10) of the individual photon fluxes.
When written in terms of photon energies ω i in the nucleus-nucleus center-of-mass, formula (2.1) takes the well-known form [11] σ AA → A(ρ 0 ρ 0 )A; We also use the γγ cms-energy W γγ , and the rapidity-type variable Y defined through The quantities used in the impact parameter calculation. (2.12) The σ (γγ → ρ 0 ρ 0 ) cross section can be calculated from models and/or taken from experimental data, if existing.Below we discuss separately the low-W γγ and high-W γγ case.
B. Low-energy γγ → ρ 0 ρ 0 cross section The cross section for this process was measured up to W γγ = 4 GeV [17].At low energy one observes a huge increase of the cross section.Several possible scenarios were discussed in this context (see e.g.Ref. [8]).For one real and one virtual photon induced processes the enhancement was interpreted as due to an isotensor meson decay [? ].We leave the difficult problem of the microscopic origin of the close-to-threshold bump for future studies and take here a pragmatic attitude of using directly experimental data.
In Fig. 3 we have collected the world data (see [7,17] and references therein).A huge rise of the cross section can be seen close to the threshold.Here we shall use rather directly experimental data in order to evaluate the cross section in nucleus-nucleus collisions.In Fig. 3 we show our fit to the world data.One can observe a small inconsistency of the data measured by different groups.This means that also our parametrization has about 20 % accuracy.
C. High-energy γγ → ρ 0 ρ 0 cross section The cross section above W γγ = 4 GeV was never measured in the past.It is well known that the cross section for γγ → hadrons can be well described in the VDM-Regge type model.Here we present a similar approach but for the specific, well defined, final state channel ρ 0 ρ 0 .
In the VDM-Regge approach the amplitude for the γγ → ρ 0 ρ 0 can be written as FIG. 3: The elementary cross section for the γγ → ρ 0 ρ 0 reaction.In the left panel we display the collection of the e + e − experimental data [7,17] and our fit.In the right panel we show our predictions based on the VDM-Regge model decribed in the text.For comparison we show by the dashed line in the left panel also the result when the form factor correcting for off-shell effect is ignored.
Above C γ→ρ 0 = f 2 ρ and we use f 2 ρ = α 2 em 2.54 .The later value is obtained in order to reproduce the decay width of the ρ 0 meson into dileptons.
For energies W γγ > 2-3 GeV the amplitude for the ρ 0 * ρ 0 * → ρ 0 ρ 0 proceses can be written in the Regge form1 : (2.14) Above we have introduced vertex form factors which, in general, are functions of exchanged pomeron/reggeon four-momentum and photon (ρ 0 meson) virtualities.We parametrize them in the factorized form: (2.15) The second term, which "corrects" for ρ 0 meson virtuality, is normalized to unity when ρ 0 meson is on mass shell.We expect the slope parameter of the order B ∼ 4 GeV −2 and the parameter responsible for off-shellness of ρ 0 mesons Λ ∼ 1 GeV.We take the powers of the pomeron (α IP ) and reggeon (α IR ) terms from the Donnachie-Landshoff fit to the total NN and πN cross sections [18].The parameters C IP and C IR are obtained assuming Regge factorization and assuming that σ(ρ 0 ρ 0 → ρ 0 ρ 0 ) = σ(π 0 π 0 → π 0 π 0 ) (see e.g.[19], they are: C IP = 8.56 mb, C IR = 13.39 mb).While this seems justified for the pomeron term, it is not so obvious if it is true for the reggeon terms.Consistent with our choice of normalization η IP (ŝ, t) and η IR (ŝ, t) are complex functions such that: η IP (ŝ, t = 0) ≈ i and η IR (ŝ, t = 0) ≈ i + 1.Standard signature functions [20] are normalized somewhat differently.The pomeron and reggeon trajectories are parametrized as α IP (t) = 1.088 + 0.25t and α IR (t) = 0.5 + 0.9t.The differential cross section can be obtained from the corresponding amplitude as: (2.16) The total cross section σ can be obtained by integrating (2.16) over t where t min and t max are ŝ-dependent kinematical limitations of t.
In Fig. 3 we present the corresponding t-integrated cross section together with existing experimental data taken from [7,17].Here the solid line includes the off-shell form factor while the dashed line does not.The vanishing of the VDM-Regge cross section at W γγ = 2m ρ is due to t min , t max limitations.It is obvious from Fig. 3 that the VDM-Regge model cannot explain the huge close-to-threshold enhancement.Up to now the origin of this enhancement remains unclear.The VDM-Regge model nicely describes the experimental data for W γγ ∼ > 2.5 GeV.

III. RESULTS
The main ingredient for the calculation of photon fluxes is the charge form factor of the nucleus.In Fig. 4 we show the charge form factor of 197 Au calculated from the realistic charge density as measured in electron scattering off nuclei [14].One can observe many oscillations characteristic for relatively sharp edge of the nucleus.This form factor is used to calculate flux of equivalent photons according to Eqs.(2.2) and (2.3).To illustrate the sensitivity of our calculation on details of the form factor, we also show a monopole form factor adjusted to the correct charge radius of the nucleus F ch ( q 2 ) = Λ 2 /(Λ 2 + q 2 ) , with Λ = 83 MeV.
In Fig. 5 we show the distribution of the cross section for the nucleus-nucleus scattering in photon-photon center-of-mass energy W γγ for both the low-energy component and highenergy VDM-Regge component.Below W γγ = 2 GeV the low-energy component dominates.The situation reverses above W γγ = 2 GeV.To study the high energy component one must impose an extra cut on W γγ , that is, the ρρ invariant mass M ρρ .However, at RHIC energies the high-energy tail of the photon spectrum is small, and the nuclear cross section drops quickly with increasing invariant mass of two-ρ mesons.By the solid lines we show the results for the realistic nuclear formfactor, while the dashed lines refer to the monopole form factor.The results obtained with the different form factors start to diverge at W γγ ∼ > 4 GeV.This is due to the increasing longitudinal momentum transfer involved for higher invariant mass of the produced system.The γ-factors at RHIC are not too large, so that the formfactor suppression is felt at such moderate invariant masses.The result for the point-like FIG.4: The modulus of the charge form factor F em (q) of the 197 Au nucleus for realistic charge distribution (solid).For comparison we show the monopole form factor often used in practical applications (dashed).
nucleus, which we show by the dotted line, overestimates the realistic cross section by more than an order of magnitude.In Fig. 6 we show the distribution in impact parameter b = |b 1 −b 2 | (see also Fig. 2).Again, we show distributions for the low-and high-energy components separately.Also shown are the distributions for the monopole form factor and for realistic charge density from [14].One can clearly see different results for different approaches to calculate flux factors of equivalent photons.The sharp cutoff at b = 2R A ∼ 14 fm is precisely the θ-function coming from absorptive corrections.
Finally in Fig. 7 we show distribution in rapidity-like variable Y .As far as small invariant masses of ρ 0 ρ 0 pairs dominate, we may refer to it as a rapidity of the ρ 0 ρ 0 pair.Compared to the point-like case, the distribution obtained with realistic charge density is concentrated at midrapidities, and configurations when both ρ 0 's are in very forward or both ρ 0 's are in very backward directions are strongly damped compared to the case with point-like nucleus charges.At larger rapidities, one can again see a substantial difference between results obtained with an approximate monopole form factor and with the exact one calculated from realistic charge density.

IV. CONCLUSIONS
We have calculated, for the first time, realistic cross sections for exclusive ρ 0 ρ 0 production in ultrarelativistic heavy-ion collisions at RHIC in the framework of equivalent photon approximation.We have discussed uncertainties related to the way how the nuclear photon flux is calculated.We have used realistic charge densities to calculate the nuclear charge form factors.The absorption effects have been included.
The low-energy part of the elementary γγ → ρ 0 ρ 0 process has been parametrized and the parameters have been fitted to the e + e − data while the high-energy part has been modeled in the vector-dominance Regge type model with parameters which are used to describe other hadronic processes.The model turned out to be consistent with the highest-energy data points (W ∼ 3-4 GeV) from e + e − collisions.
It was shown that a realistic calculation of both ingredients is necessary to make reliable estimates of the nucleus-nucleus exclusive production of the ρ 0 ρ 0 pairs.Large cross sections, of the order of fraction of milibarn, have been found.The bulk of the cross section is, however, concentrated in low photon-photon energies (low ρ 0 ρ 0 invariant masses).Making cuts on higher ρ 0 ρ 0 invariant masses one can easily select high-energy component.The ρ 0 mesons, decaying into π + π − , can be measured e.g. by the STAR detector at RHIC.A Monte Carlo study is necessary to analyse feasibility of a measurement of the process discussed here.
In the present analysis we have concentrated on processes with final nuclei in the ground state.It is very difficult, if not impossible, to measure such very forward/backward nuclei.The multiple Coulomb excitations associated with ρ 0 ρ 0 production may cause additional excitation of one or even both nuclei to the giant resonance region.The neutron emission from the giant resonances can be used then to tag the processes.The zero-degree calorimeters (ZDC) at RHIC can be used to measure such neutrons.We plan a detailed study of these processes in the future. )

FIG. 6 :
FIG.6:The Au + Au → Au + Au + ρ 0 ρ 0 cross section as a function of the impact parameter b for √ s N N = 200 GeV.

FIG. 7 :
FIG. 7: The Au + Au → Au + Au + ρ 0 ρ 0 cross section as a function of the rapidity of the ρ 0 ρ 0 pair Y for √ s N N = 200 GeV.In the left panel we show the decomposition into low-and high-energy contributions.The right panel shows the sensitivity to the nuclear form factor.