The $\cos 2\phi$ azimuthal asymmetry in $\rho^{0}$ meson production in ultraperipheral heavy ion collisions

Ultraperipheral collisions (UPCs) of heavy ions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) offer a great opportunity to explore nuclear structure with beams of quasi-real photons before the Electron Ion Collider (EIC) era. In UPCs the strong hadronic interaction is suppressed, and the photon-nucleus ($\gamma$A) interactions involving photons emitted from one of the colliding nuclei are expected to be dominant. Due to the large flux of quasi-real photons, $\gamma$A interactions are enhanced by a factor $Z^{2}$ as compared to those in proton-nucleus (pA) or electron-nucleus (eA) collisions where $Z$ is the nuclear charge number. Among many exciting directions of UPC studies, see for example Hagiwara:2017fye ; Hatta:2019ocp , diffractive vector mesons photoproduction on nuclei provide access to the three dimensional gluon tomography of nucleus as well as stringent tests of the color glass condensate (CGC) description of saturation physics. Because of this, such processes have been extensively studied from both theoretical Donnachie:1987pu ; Ryskin:1992ui ; Brodsky:1994kf ; Nemchik:1996cw ; Klein:1999qj ; Munier:2001nr ; Kopeliovich:2001xj ; Kowalski:2003hm ; Kowalski:2006hc ; Rebyakova:2011vf ; Guzey:2013qza ; Lappi:2010dd ; Guzey:2016piu ; Xie:2016ino ; Cai:2020exu and experimental Breitweg:1998nh ; Chekanov:2002xi ; Khachatryan:2016qhq ; Adamczyk:2017vfu ; STAR:2019yox ; Sirunyan:2019nog ; Acharya:2020sbc sides during the past few decades.

Recently, significant $\cos 2\phi$ and $\cos 4\phi$ asymmetries for $\rho^{0}$ meson production in UPCs have been observed by STAR collaboration Daniel , where $\phi$ is the angle between the produced $\rho^{0}$ meson’s transverse momentum and its decay product pion’s transverse momentum. As the angular distribution of final state decayed pions contains the information of the polarization of $\rho^{0}$, the observed angular correlation between $\rho^{0}$ and pion can be converted into the correlation between the transverse spin vector and the transverse momentum for $\rho^{0}$, thus the $\phi$ asymmetry can serve as the meson’s spin analyzer. The investigations of such polarization dependent observable in vector meson production certainly open a new window to study the small-$x$ structure of heavy nuclei as well as the associated nontrivial QCD dynamics.

Motivated by the recent measurement by STAR collaboration at RHIC Daniel , we carry out a detailed analysis of the $\cos 2\phi$ azimuthal asymmetry for diffractive meson production in UPCs. The underlying physics of $\cos 4\phi$ asymmetry is rather different and will be addressed in a future work. Our calculation is formulated in a conventional method: the quasi-real photon is treated as the color dipole of a quark-antiquark pair which recombines to form a vector meson after scattering off the CGC state inside a nucleus. To account for the $\cos 2\phi$ asymmetry within the dipole model, the key insight is that the incident photon is highly linearly polarized along the direction of its transverse momentum. The correlation between the initial state photon’s polarization and transverse momentum will be transferred to that for the final state vector meson. Notice that the Eikonal approximation employed in the dipole approach plays a crucial role in preserving spin information after the quark-antiquark pair experiences multiple gluon re-scattering.

As a matter of fact, the gauge bosons (photons/gluons) being highly linearly polarized in the small $x$ limit have been recognized as a common feature of the gauge theories in a series of publications Metz:2011wb ; Li:2019yzy ; Li:2019sin . It was shown in Refs. Metz:2011wb ; Dominguez:2011br ; Boer:2010zf ; Akcakaya:2012si ; Dumitru:2015gaa ; Kotko:2015ura ; Boer:2017xpy ; Marquet:2017xwy ; Gutierrez-Reyes:2019rug ; Scarpa:2019fol ; Kishore:2018ugo that the linear polarization of photons/gluons can be probed through the azimuthal asymmetries in two particles correlations. For instance, the QED calculations Li:2019yzy ; Li:2019sin predict a sizable $\cos 4\phi$ azimuthal asymmetry for pure electromagnetic dilepton production in heavy ion collisions. Such $\cos 4\phi$ modulation has been clearly seen in a recent STAR measurement Adam:2019mby . In particular, the computed impact parameter dependent asymmetry is in excellent agreement with the experimental data for the UPC case, while the QED calculation in peripheral collisions slightly overestimates the asymmetry in the centrality region $60-80\%$. With it being experimentally confirmed, the linearly polarized quasi-real photon beam in heavy ion collisions can be used as a powerful tool to explore the novel QCD phenomenology as well. The current work represents the first effort towards this direction.

The paper is organized as follows. In Sec. 2, we derive the joint impact parameter and transverse momentum dependent cross section in UPCs including both the coherent and incoherent vector meson photoproduction contributions. In Sec. 3, we present numerical estimations of polarization averaged and $\cos 2\phi$ azimuthal asymmetries for diffractive $\rho^{0}$ production at RHIC and LHC energies. Reasonable good agreement with the STAR measurements are reached. Finally, the paper is summarized in Sec. 4.

We proceed to perform the numerical estimations of the $\cos 2\phi$ asymmetry using Eq. (14) in this section. First of all, let us collect all ingredients that are necessary for numerical calculations. We start with introducing the parametrization for the dipole scattering amplitude whose formal operator definition is given by,

The dipole amplitude is usually obtained by solving the BK equation with the initial condition being fitted to the experimental data or derived from the MV model. However, the numerical implementation of the impact parameter dependent BK equation is a highly non-trivial task. For simplicity, we instead use a phenomenological parametrization for the $b_{\perp}$ dependence of the dipole amplitude Kowalski:2003hm ; Kowalski:2006hc ,

where, as mentioned before, ${\cal N}(r_{\perp})$ is the dipole-nucleon scattering amplitude. The nuclear thickness function $T_{A}(b_{\perp})$ is determined with the Woods-Saxon distribution in our numerical calculation. Note that $4\pi B_{p}{\cal N}(r_{\perp})=\sigma_{\text{dip}}^{p}(r_{\perp})$ is the total dipole-proton cross section for a dipole of size $r_{\perp}$. In literatures there are many parameterizations available for the dipole-proton cross section. Here we adopt a modified IPsat model in which the impact parameter dependence of the dipole-nucleon scattering amplitude has been factorized out Lappi:2010dd

where $G$ is proportional to the DGLAP evolved gluon distribution in the Bartels, Golec-Biernat and Kowalski (BGBK) parametrization Bartels:2002cj . In our numerical estimations, we adopt a simpler parametrization for the gluon distribution known as the Golec-Biernat and Wüsthoff (GBW) model Golec-Biernat:1998js ; Golec-Biernat:1999qd ,

where $Q_{s}(x_{g})=(x_{0}/x_{g})^{\lambda_{GBW}/2}$ GeV is the saturation scale. We use the parameters $x_{0}=3\times 10^{-4}$ and $\lambda_{GBW}=0.29$ Kowalski:2006hc which were determined by fitting to HERA data.

For the scalar part of the vector meson wave function, we use “Gaus-LC” wave function also taken from Refs. Kowalski:2003hm ; Kowalski:2006hc

where $\beta=4.47$, $R_{\perp}^{2}=21.9~{}\text{GeV}^{-2}$ for $\rho$ meson. An alternative parametrization, the “boosted Gaussian” wave function is also widely used in the study of exclusive production of vector meson. The existing HERA data is reasonably well described by estimations of vector meson photoproduction employing either wave function model.

The photon distribution $f(x,k_{\perp})$ at low transverse momentum is commonly computed with the equivalent photon approximation, also often referred to as the Weizs$\ddot{a}$cker-Williams method, in which the photon flux is calculated by treating the fields of charged relativistic heavy ions as external, i.e., classical electromagnetic field. This method has been widely used to compute UPC observables, see for example Refs. Klein:2018fmp ; Zha:2018tlq . The photon distribution derived in the equivalent photon approximation is given by Bertulani:1987tz ; Vidovic:1992ik

where $F$ is the nuclear charge form factor, $M_{p}$ is the proton mass. Similarly, one has ${\cal F}(Y,k_{\perp})=\frac{Z\sqrt{\alpha_{e}}}{\pi}|k_{\perp}|\frac{F(k_{\perp}^{2}+x^{2}M_{p}^{2})}{(k_{\perp}^{2}+x^{2}M_{p}^{2})}$. The nuclear charge form factor is commonly determined with the Woods-Saxon distribution,

where $R_{WS}$ (Au: 6.38 fm, pb: 6.62 fm) is the nuclear radius and $d$ (Au: 0.535 fm, Pb: 0.546 fm) is the skin depth, $C^{0}$ is the normalization factor. Alternatively, one can use the form factor in momentum space from the STARlight MC generator Klein:2016yzr ,

where $R_{A}=1.1A^{1/3}$ fm, and $a=0.7$ fm. This parametrization numerically is very close to the Woods-Saxon distribution, and will be used in our numerical evaluation. Due to the neutron skin effect and the surrounding pion cloud, the effective nuclear strong interaction radius is larger than its EM radius. To fit RHIC data Adamczyk:2017vfu , we compute the thickness function $T_{A}(b_{\perp})$ with the radius $R_{A}=6.9$ fm and the depth $d=0.64$ fm for a gold target. For a lead target, we simply re-scale these numbers by multiplying a factor $A_{\text{lead}}^{1/3}/A_{\text{gold}}^{1/3}$. We determine $e_{q}$ by noticing that the $\rho^{0}$ meson wave function reads $\frac{1}{\sqrt{2}}\left(|u\bar{u}\rangle-|d\bar{d}\rangle\right)$. This would imply a replacement of $e_{q}$ by $e_{q}\rightarrow\frac{1}{\sqrt{2}}(e_{u}-e_{d})$. The effective charge $e_{q}$ for $\rho^{0}$ then is $1/\sqrt{2}$.

For the unrestricted UPC case, the asymmetry is averaged over the impact parameter range $[2R_{A},\infty]$. However, RHIC-STAR measures $\rho^{0}$ photoproduction cross section together with the double electromagnetic excitation in both ions. Neutrons emitted at forward angles from the scattered nuclei are detected by zero-degree calorimeters (ZDCs), and used as a UPC trigger. Requiring that UPCs are accompanied by forward neutron emission alters the impact parameter distribution compared with unrestricted UPC events. In order to incorporate the experimental conditions in the theoretical calculations, one can define a “tagged” UPC cross section

Where the probability $P(\tilde{b}_{\perp})$ of emitting a neutron from the scattered nucleus is often parameterized as Baur:1998ay

which is denoted as the “1n” event, while for emitting any number of neutrons (“Xn” event), the probability is given by

with $P_{1n}(\tilde{b}_{\perp})=5.45\times 10^{-5}\frac{Z^{3}(A-Z)}{A^{2/3}\tilde{b}_{\perp}^{2}}\ \text{fm}^{2}$. As a matter of fact, the mean impact parameter is dramatically reduced in interactions with Coulomb dissociation.

With all these ingredients, we are ready to perform numerical study of the azimuthal asymmetries for $\rho^{0}$ meson production in heavy ion collisions. To test the theoretical calculation, we first compute the azimuthal averaged cross section for coherent photoproduction of $\rho^{0}$ and compare with experimental data from the STAR collaboration Adamczyk:2017vfu . In particular, we calculate the differential cross section $d\sigma/dt$ with the Mandelstam variable $t\approx-q_{\perp}^{2}$, and the rapidity is integrated out in the region $|Y|\leq 1$ to match the STAR measurement. Notice that the incoherent component has been subtracted out in STAR measurement. Therefore, we exclude the first term in Eq. (14) and integrate over the azimuthal angle $\phi$, namely only the first term in Eq. (15) needs to be considered. As shown in Fig. 3, our theoretical result represented by blue curve describes the experimental data perfectly in identifying the minima and peaks, as well as the overall shapes.

The numerical results for the azimuthal asymmetries for $\rho^{0}$ at RHIC and LHC energies are presented in Fig. 4, where the azimuthal asymmetry, i.e., the average value of $\cos 2\phi$ is defined as,

We use exactly the same setups as that in the unpolarized case but including both the coherent and incoherent components. Since we are considering the average value of $\cos 2\phi$, only the second term in Eq. (15) contributes. We can see clearly the diffractive pattern with two minima visible in $q_{\perp}$ distribution, such characteristic feature is also identified in the STAR preliminary measurement. The $q_{\perp}$ distribution for the average value of $\cos 2\phi$ can be easily understood as the asymmetry is almost entirely generated in the coherent scattering, while both the coherent and incoherent production contribute to the azimuthal averaged cross section. At the relative large transverse momentum($q_{\perp}>100{\text{M}eV}$), most of $\rho^{0}$ meson’s transverse momentum originates from the nucleus. Based on this observation, one, after few steps of algebraic manipulations, can show that the asymmetry is proportional to the slope of $\Delta_{\perp}$ distribution which gets very large near the first minima of the diffractive pattern. This can be clearly seen from our numerical result, where the second peak of the asymmetry located at the first minima of the unpolarized cross section. Nevertheless, as the first attempt, our result shown in the left plot in Fig. 4(solid line) describes the STAR preliminary data Daniel reasonably well in terms of finding the correct depths of the dips. However, slightly larger $q_{\perp}$ for the locations of the dips are found from our theoretical calculation comparing to those in STAR preliminary data, which suggests an increase in effective nuclear size in our calculation when considering polarized case. In order to investigate the impact parameter dependence, we also show in Fig. 4 the comparison between UPC and peripheral collisions at RHIC energy $\sqrt{s}=200$ GeV in Au-Au collisions and at LHC energy $\sqrt{s}=5020$ GeV in Pb-Pb collisions, we take $70-90\%$ centralities as an example in peripheral collisions ²²2For peripheral collisions with relative large impact parameter, the coherent photon-nucleus interaction still dominates over hadronic reactions in vector meson production STAR:2019yox . . With the increase of impact parameter, we see slightly shift of the location for the dips. We also predict measurable difference between UPC and peripheral collisions at both RHIC and LHC.

In summary, we have studied the $\cos 2\phi$ azimuthal angular correlation in vector meson production in ultraperipheral heavy ion collisions, where $\phi$ is defined as the angle between vector meson’s transverse spin vector and its transverse momentum. The asymmetry essentially results from the linear polarization of incident coherent photons, which just has been experimentally confirmed by the recent STAR measurement of a $\cos 4\phi$ modulation in pure electromagnetic lepton pair production Adam:2019mby . The asymmetries evaluated in the dipole model for $\rho^{0}$ photoproduction at RHIC and LHC energies are shown to be rather sizable. Admittedly, the perturbative treatment for $\rho^{0}$ must be legitimately criticized due to the lack of a hard scale in the problem. However, one might expect that the angular correlation structure is not altered by the non-perturbative effect, for which a more sophisticated phenomenological method is required. Nevertheless, we found that our calculation turns out to be in reasonably good agreement with the $\rho^{0}$ measurement by STAR collaboration. As mentioned in the introduction, a significant $\cos 4\phi$ asymmetry in $\rho^{0}$ production was also observed at RHIC. This observable could potentially give the access to the non-trivial gluon GTMD/Wigner distribution and will be addressed in a future publication.

The obtained transverse momentum dependent $\cos 2\phi$ asymmetries have a distinctive diffractive pattern which undoubtly opens a new window to investigate the coherent and incoherent production of vector meson. As demonstrated by the present study, quasi-real photon beams with linear polarization in heavy ion collisions can be used as a powerful tool to explore novel QCD phenomenology. Meanwhile, as a byproduct of this work, we developed a formalism to compute the joint impact parameter and transverse momentum dependent cross sections that enables us to reliably extract $\Delta_{\perp}$ dependence in UPCs. The Fourier transform of $\Delta_{\perp}$ distribution would provide crucial information on the transverse spatial distribution of gluons inside a nucleus, which is one of the central scientific goals in the forthcoming EIC era.

Note added: After the manuscript was posted on arXiv, a related preprint Zha:2020cst appeared soon, where the similar result for the azimuthal asymmetry was obtained.