Nuclear geometry at high energy from exclusive vector meson production

Exclusive particle production processes in Deep Inelastic Scattering (DIS) are powerful probes of the structure of protons and nuclei at high energy. The exclusive nature of the process ensures that there is no net color charge transferred from the target, which means that at least two gluons need to be exchanged. This renders the cross section approximately proportional to the square of the gluon distribution at leading order [Ryskin:1992ui] (at next-to-leading order the relation is less direct [Eskola:2022vpi]). Additionally, measuring the total momentum transfer to the target is possible by measuring the produced particle, e.g. a vector meson. As the momentum transfer is the Fourier conjugate to the impact parameter, exclusive processes provide access to the spatial distribution of nuclear matter in protons and nuclei. Indeed, multi dimensional imaging using exclusive photon or vector meson production processes is a central part of the physics programs of future nuclear-DIS facilities, including the EIC [AbdulKhalek:2021gbh, Aschenauer:2017jsk], LHeC/FCC-he [Agostini:2020fmq] and EicC [Anderle:2021wcy].

Before these future facilities are realized, it is also possible to study exclusive vector meson production at high energy in the photoproduction region in Ultra Peripheral Collisions (UPCs) at RHIC and at the LHC [Bertulani:2005ru, Klein:2019qfb]. In UPCs the impact parameter is so large that strong interactions are suppressed, and one of the colliding nuclei acts as a source of quasi real photons, which probe the other nucleus. In particular, ultra peripheral heavy ion collisions provide access to photon-nucleus scattering at collider energies for the first time.

Experiments at both RHIC and at the LHC have performed first measurements of the exclusive $\mathrm{J}/\psi$ photoproduction cross section in heavy ion UPCs [ALICE:2012yye, ALICE:2013wjo, ALICE:2014eof, ALICE:2018oyo, ALICE:2021gpt, ALICE:2019tqa, ALICE:2021tyx, LHCb:2014acg, LHCb:2018rcm, LHCb:2022ahs, LHCb:2021bfl, LHCb:2022ahs, CMS:2016itn, PHENIX:2009xtn]. These measurements have been extensively studied in the context of saturation physics, e.g. in Refs. [Sambasivam:2019gdd, Lappi:2013am, Cepila:2017nef, Mantysaari:2017dwh, Goncalves:2017wgg, Bendova:2020hbb] (see also Refs. [Toll:2012mb, Mantysaari:2019jhh, Lappi:2010dd, Caldwell:2010zza] where vector meson production in photon-nucleus collisions is studied). Very recently first measurements differential in the meson transverse momentum $\mathbf{p}$ or squared momentum transfer $|t|$ have also become available [LHCb:2022ahs, ALICE:2021tyx, star]. These new developments make it possible to study the geometric structure of nuclei, including event-by-event fluctuations [Mantysaari:2020axf], in a so far unexplored kinematical domain down to $x\sim 10^{-5}$. This possibility is the main motivation behind this work.

We calculate within the Color Glass Condensate framework [Iancu:2003xm, Albacete:2014fwa, Blaizot:2016qgz, Gelis:2010nm, Morreale:2021pnn] exclusive $\mathrm{J}/\psi$ production in ultra peripheral lead-lead and gold-gold collisions. In particular we show how the non-linear saturation effects change the nuclear geometry (as measured by the $\mathrm{J}/\psi$ spectra) when one moves from the low-energy region described in terms of nucleon positions following the nuclear density distribution, such as the Woods-Saxon distribution  [Woods:1954zz], to the region of strong color fields in the small momentum fraction $x$ region probed in collider experiments. Compared to our previous study [Mantysaari:2017dwh] we use a full CGC based setup including perturbative small-$x$ evolution calculated by solving the JIMWLK equation (see e.g. [Mueller:2001uk]), which also describes the geometry evolution [Schlichting:2014ipa]. Additionally we take into account the interference effect due to the fact that it is not possible to know which nucleus emitted the photon [Klein:1999gv, Bertulani:2005ru] and the non-zero photon transverse momentum [Xing:2020hwh].

This manuscript is organized as follows. In Sec. II we discuss how ultra peripheral collisions can be considered as photon-nucleus events, and show how the interference effect and photon transverse momentum are taken into account in our calculations. The calculation of exclusive vector meson production from a CGC setup including the small-$x$ evolution is presented in Sec. LABEL:sec:vm_production. Numerical results compared to LHC and RHIC data are presented in Sec. LABEL:sec:results before we present our conclusions in Sec. LABEL:sec:conclusions.

There are two indistinguishable contributions to the exclusive vector meson production in ultra peripheral collisions, as both of the colliding nuclei can act as a photon source. Consequently there is also a quantum mechanical interference contribution which becomes important at small vector meson transverse momentum $|\mathbf{p}|$ [Bertulani:2005ru]. Additionally, although the photons are quasi real with their virtuality limited by the nuclear size $Q^{2}\lesssim 1/R_{A}^{2}$, they carry a non-zero transverse momentum that can have an effect on the vector meson transverse momentum spectra, especially near diffractive minima.

In order to include both the photon transverse momentum $\mathbf{k}$ (which is related to transverse distance between the nuclei $\mathbf{B}$ via Fourier transform) and the interference contribution, we follow Ref. [Xing:2020hwh]. Let us first consider coherent vector meson production, where the target remains intact and one averages the scattering amplitude over the target configurations $\Omega$ [Good:1960ba, Caldwell:2009ke]. The result derived in Ref. [Xing:2020hwh] can be written as (see Appendix LABEL:appendix:amplitude for details)

$\displaystyle\frac{\mathrm{d}\sigma^{A_{1}+A_{2}\to V+A_{1}+A_{2}}}{\mathrm{d}\mathbf{p}^{2}\mathrm{d}y}=\frac{1}{4\pi}\int_{|\mathbf{B}|>B_{\mathrm{min}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mathrm{d}^{2}\mathbf{B}|\langle\mathcal{M}^{j}(y,\mathbf{p},\mathbf{B})\rangle_{\Omega}|^{2}\,,$

(1)

where

	$\displaystyle\mathcal{M}^{j}(y,\mathbf{p},\mathbf{B})=\mathbf{B}^{j}\mathcal{M}_{0}(y,\mathbf{p},\mathbf{B})-\mathcal{M}^{j}_{1}(y,\mathbf{p},\mathbf{B})$
	$\displaystyle-\left[\mathbf{B}^{j}\mathcal{M}_{0}(-y,\mathbf{p},-\mathbf{B})+\mathcal{M}^{j}_{1}(-y,\mathbf{p},-\mathbf{B})\right]e^{-i\mathbf{p}\cdot\mathbf{B}}\,,$		(2)

and

	$\displaystyle\mathcal{M}_{0}(y,\mathbf{p},\mathbf{B})$	$\displaystyle=\int\mathrm{d}^{2}\mathbf{b}e^{-i\mathbf{p}\cdot\mathbf{b}}(-i\widetilde{\mathcal{A}}(y,\mathbf{b}))\widetilde{\mathcal{F}}_{S}(y,\mathbf{b}-\mathbf{B})\,,$
	$\displaystyle\mathcal{M}_{1}^{j}(y,\mathbf{p},\mathbf{B})$	$\displaystyle=\int\mathrm{d}^{2}\mathbf{b}e^{-i\mathbf{p}\cdot\mathbf{b}}(-i\widetilde{\mathcal{A}}(y,\mathbf{b}))\mathbf{b}^{j}\widetilde{\mathcal{F}}_{S}(y,\mathbf{b}-\mathbf{B})\,.$		(3)

An equivalent expression to Eq. (2) is given in Eq. (LABEL:eq:Amplitude_coordinate-space2) where the symmetry in the exchange between photon emitter and target is manifest.

The vector meson production amplitude in photon-target interaction, $\widetilde{A}(y,\mathbf{b})$, is discussed in more detail in Sec. LABEL:sec:vm_production.

The transverse coordinate index is $j=1,2$. Here the vector meson $V$ rapidity is denoted by $y$, and its transverse momentum $\mathbf{p}$ is obtained as a vector sum of the photon transverse momentum $\mathbf{k}$ and the nuclear momentum transfer $\mathbf{\Delta}$. The photon transverse momentum and the momentum transfer are not explicitly visible above as we work in coordinate space, see discussion in Appendix LABEL:appendix:amplitude.

The impact parameter of the photon-nucleus collision is denoted by $\mathbf{b}$. The integral over the transverse separation between the two nuclei, $\mathbf{B}$, is limited from below in ultra peripheral collisions, and we use $B_{\text{min}}=2R_{A}$ where $R_{A}=6.62$ fm for Pb and $R_{A}=6.37$ fm for Au unless stated otherwise.

The function $\mathcal{F}_{S}$ describes the electromagnetic field of the nucleus calculated using an equivalent photon approximation Fourier transformed into coordinate space. As we only need the electromagnetic field at distances $|\mathbf{B}|\geq 2R_{A}$, the nuclear form factor can be replaced by that of a point particle following Gauss’ law (we have confirmed that using a Woods-Saxon form factor has negligible effect on our results). In this case the function $\mathcal{F}_{S}$ reads

$$\widetilde{\mathcal{F}}_{S}(y,\mathbf{B})=\frac{Z{\alpha_{\mathrm{em}}}^{1/2}\omega}{\pi\gamma}\frac{1}{|\mathbf{B}|}K_{1}\left(\frac{\omega|\mathbf{B}|}{\gamma}\right)\,.$$

(4)

The photon energy is $\omega=(M_{V}/2)e^{y}$, $Z$ is the ion charge and $\gamma=A\sqrt{s}/(2M_{A})$ where $M_{A}$ is the mass of the nucleus. The vector meson mass is denoted by $M_{V}$. As discussed above and in Appendix LABEL:appendix:amplitude, the impact parameter $\mathbf{B}$ is related to the photon transverse momentum and as such the size of the nucleus sets the scale for the photon transverse momenta. In this work we use a sharp cutoff $|\mathbf{B}|>B_{\text{min}}$ which potentially has an effect on the photon $k_{T}$ distribution as discussed in Ref. [Klein:2020jom].

The results shown in this work are not highly sensitive to the $B_{\text{min}}$ cut: for example the total coherent $\mathrm{J}/\psi$ production cross section at the LHC discussed in Sec. LABEL:sec:tint_xs changes by $\sim 3\%$ when the minimum distance is changed by $10\%$.

We further note that at midrapidity and for coherent production (using the fact that $\langle-i\widetilde{\mathcal{A}}(y,\mathbf{b})\rangle_{\Omega}$ is real) the amplitude in Eq. (2) averaged over configurations can be cast into a simple form

	$\displaystyle\langle\mathcal{M}^{j}(0,\mathbf{p},\mathbf{B})\rangle_{\Omega}=2ie^{i\mathbf{p}\cdot\mathbf{B}/2}$
	$\displaystyle\times\mathrm{Im}\Big{\{}e^{-i\mathbf{p}\cdot\mathbf{B}/2}\left[\mathbf{B}^{j}\langle\mathcal{M}_{0}(0,\mathbf{p},\mathbf{B})\rangle_{\Omega}-\langle\mathcal{M}^{j}_{1}(0,\mathbf{p},\mathbf{B})\rangle_{\Omega}\right]\Big{\}}\,.$		(5)

Let us next discuss some commonly used approximations. First, as the photon transverse momentum is small, $\mathbf{k}^{2}\lesssim Q^{2}\sim 1/R_{A}^{2}$, it can usually (but not around diffractive minima) be neglected. In coordinate space this corresponds to assuming $|\mathbf{B}|\gg|\mathbf{b}|$. In this case the cross section can be written as