Coherent photoproduction of $J/\psi$ in nucleus-nucleus collisions in the color dipole approach

TL;DR

This paper models the exclusive production of J/ψ mesons in ultraperipheral heavy ion collisions using the color dipole approach, comparing theoretical predictions with experimental data to understand nuclear shadowing effects.

Contribution

It applies the color dipole model with Glauber-Gribov theory to calculate nuclear diffractive amplitudes and compares results with recent collider data.

Findings

Good agreement with data at certain energies

Indicates potential for additional shadowing corrections at high energies

Highlights the importance of real part contributions in the amplitude

Abstract

We investigate the exclusive photoproduction of $J/\psi$-mesons in ultraperipheral heavy ion collisions in the color dipole approach. We first test a number of dipole cross sections fitted to inclusive $F_2$-data against the total cross section of exclusive $J/\psi$-production on the free nucleon. We then use the color-dipole formulation of Glauber-Gribov theory to calculate the diffractive amplitude on the nuclear target. The real part of the free nucleon amplitude is taken into account consistent with the rules of Glauber theory. We compare our results to recent published and preliminary data on exclusive $J/\psi$ corrections in ultraperipheral lead-lead collisions at $\sqrt{s_{NN}}=2.76 \, \rm{TeV}$ and $\sqrt{s_{NN}} = 5.02 \, \rm{TeV}$. Especially at high $\gamma A$ energies there is room for additional shadowing corrections, corresponding to triple-Pomeron terms or shadowing from large mass diffraction.

Tables

Table 1. Table 1: BGK fit with fitted valence quarks for σ r subscript 𝜎 𝑟 \sigma_{r} for H1ZEUS-NC data in the range Q 2 ≥ 3.5 superscript 𝑄 2 3.5 Q^{2}\geq 3.5  GeV 2 and x ≤ 0.01 𝑥 0.01 x\leq 0.01 . NLO fit. Soft gluon . m u ​ d ​ s = 0.14 , m c = 1.3 formulae-sequence subscript 𝑚 𝑢 𝑑 𝑠 0.14 subscript 𝑚 𝑐 1.3 m_{uds}=0.14,m_{c}=1.3 GeV. Q 0 2 = 1.9 superscript subscript 𝑄 0 2 1.9 Q_{0}^{2}=1.9 GeV 2 .

$Q_{min}^2$ [GeV2]	${\sigma }_0\left[\mathrm{mb}\right]$	$A_g$	${\lambda }_g$	$C_g$	$N_{df}$	${\chi }^2$	${\chi }^2/N_{df}$
$3.5$	89.99$\pm$	2.44$\pm$	-0.079$\pm$	7.24$\pm$	530	540.35	1.02
	9.2	0.145	0.099	0.61

Table 2. Table 2: Subenergies W ± subscript 𝑊 plus-or-minus W_{\pm} and Bjorken- x 𝑥 x values x ± subscript 𝑥 plus-or-minus x_{\pm} for s N ​ N = 2.76 ​ TeV subscript 𝑠 𝑁 𝑁 2.76 TeV \sqrt{s_{NN}}=2.76\,\rm TeV for a given rapidity y 𝑦 y . Also shown are photon fluxes n ​ ( ω ± ) 𝑛 subscript 𝜔 plus-or-minus n(\omega_{\pm}) and the photoproduction cross sections on Pb 208 superscript Pb 208 {}^{208}{\rm Pb} at energies W ± subscript 𝑊 plus-or-minus W_{\pm} for the IIM-dipole cross section.

$\sqrt{s_{NN}}=2.76\mathrm{TeV}$
$y$	$W_{+}\left[\mathrm{GeV}\right]$	$W_{-}\left[\mathrm{GeV}\right]$	$x_{+}$	$x_{-}$	$n\left({\omega }_{+}\right)$	$n\left({\omega }_{-}\right)$	$\sigma \left(W_{+}\right)\left[\mu b\right]$	$\sigma \left(W_{-}\right)\left[\mu b\right]$
0.0	92.5	92.5	$1.12\cdot {10}^{-3}$	$1.12\cdot {10}^{-3}$	69.4	69.4	27.4	27.4
1.0	152	56.1	$4.13\cdot {10}^{-4}$	$3.05\cdot {10}^{-3}$	39.5	100	37.1	19.5
2.0	251	34.0	$1.52\cdot {10}^{-4}$	$8.29\cdot {10}^{-3}$	14.5	132	48.6	13.0
3.0	414	20.6	$5.59\cdot {10}^{-5}$	$2.25\cdot {10}^{-2}$	1.68	163	62.2	7.18
3.8	618	13.8	$2.51\cdot {10}^{-5}$	$5.02\cdot {10}^{-2}$	0.03	188	74.6	2.81

Table 3. Table 3: Subenergies W ± subscript 𝑊 plus-or-minus W_{\pm} and Bjorken- x 𝑥 x values x ± subscript 𝑥 plus-or-minus x_{\pm} for s N ​ N = 5.02 ​ TeV subscript 𝑠 𝑁 𝑁 5.02 TeV \sqrt{s_{NN}}=5.02\,\rm TeV for a given rapidity y 𝑦 y . Also shown are photon fluxes n ​ ( ω ± ) 𝑛 subscript 𝜔 plus-or-minus n(\omega_{\pm}) and the photoproduction cross sections on Pb 208 superscript Pb 208 {}^{208}{\rm Pb} at energies W ± subscript 𝑊 plus-or-minus W_{\pm} for the IIM-dipole cross section.

$\sqrt{s_{NN}}=5.02\mathrm{TeV}$
$y$	$W_{+}\left[\mathrm{GeV}\right]$	$W_{-}\left[\mathrm{GeV}\right]$	$x_{+}$	$x_{-}$	$n\left({\omega }_{+}\right)$	$n\left({\omega }_{-}\right)$	$\sigma \left(W_{+}\right)\left[\mu b\right]$	$\sigma \left(W_{-}\right)\left[\mu b\right]$
0.0	125	125	$6.17\cdot {10}^{-4}$	$6.17\cdot {10}^{-4}$	87.9	87.9	32.9	32.9
1.0	206	75.6	$2.27\cdot {10}^{-4}$	$1.68\cdot {10}^{-3}$	57.2	119	43.8	24.0
2.0	339	45.9	$8.35\cdot {10}^{-5}$	$4.56\cdot {10}^{-3}$	28.5	150	56.5	16.8
3.0	559	27.8	$3.07\cdot {10}^{-5}$	$1.24\cdot {10}^{-2}$	7.5	181	71.3	10.6
4.0	921	16.9	$1.13\cdot {10}^{-5}$	$3.37\cdot {10}^{-2}$	0.35	213	88.6	4.98
4.8	1370	11.3	$5.08\cdot {10}^{-6}$	$7.50\cdot {10}^{-2}$	0.001	238	103	1.22

Equations

$$\displaystyle\mathcal{A}(\gamma N\rightarrow VN;W,\mbox{\boldmath$q$})$$

$$\displaystyle{d\sigma(\gamma N\to VN;W)\over dt}={d\sigma(\gamma N\to VN;W)\over d\mbox{\boldmath$q$}^{2}}={1\over 16\pi}\Big{|}\mathcal{A}(\gamma^{*}N\rightarrow VN;W,\mbox{\boldmath$q$})\Big{|}^{2}\,.$$

$$\displaystyle\rho_{V\leftarrow\gamma}(\mbox{\boldmath$r$},\mbox{\boldmath$q$})=\int_{0}^{1}dz\Psi_{V}(z,\mbox{\boldmath$r$})\Psi_{\gamma}(z,\mbox{\boldmath$r$})\exp[i(1-2z)\mbox{\boldmath$r$}\mbox{\boldmath$q$}/2]\,.$$

$$\displaystyle\sigma(x,\mbox{\boldmath$r$},\mbox{\boldmath$q$})=\sigma(x,r)\exp[-B\mbox{\boldmath$q$}^{2}/2]\,.$$

$$\displaystyle\Psi^{*}_{V}(z,\mbox{\boldmath$r$})\Psi_{\gamma}(z,\mbox{\boldmath$r$})$$

$$\displaystyle\Delta_{\bf I\!P}={\partial\log\Big{(}\langle{V}|\sigma(x,r)|{\gamma}\rangle\Big{)}\over\partial\log(1/x)}\,,$$

$$\displaystyle\rho_{N}=\tan\Big{(}{\pi\Delta_{\bf I\!P}\over 2}\Big{)}\,.$$

$$R_{\mathrm{skewed}}=\frac{2^{2{\Delta }_{\mathbf{I}\mathbf{P}}+3}}{\sqrt{\pi }}\cdot \frac{\Gamma ({\Delta }_{\mathbf{I}\mathbf{P}}+5/2)}{\Gamma ({\Delta }_{\mathbf{I}\mathbf{P}}+4)}.$$

$$\displaystyle\Gamma_{A}(x,\mbox{\boldmath$b$},\mbox{\boldmath$r$})=1-\exp[-{1\over 2}\sigma(x,r)T_{A}(\mbox{\boldmath$b$})]\,.$$

$$\displaystyle\Gamma_{A}(x,\mbox{\boldmath$b$},\mbox{\boldmath$r$})$$

$$\displaystyle T_{A}(\mbox{\boldmath$b$})=\int_{-\infty}^{\infty}dz\,n_{A}(\vec{r})\,;\,\vec{r}=(\mbox{\boldmath$b$},z),\,\int d^{2}\mbox{\boldmath$b$}\,T_{A}(\mbox{\boldmath$b$})=A.$$

$$\displaystyle\mathcal{A}(\gamma A\rightarrow VA;W,\mbox{\boldmath$b$})$$

$$\displaystyle\sigma(\gamma A\to VA;W)={1\over 4}\int d^{2}\mbox{\boldmath$b$}\,\Big{|}\mathcal{A}(\gamma A\rightarrow VA;W,\mbox{\boldmath$b$})\Big{|}^{2}\,.$$

$$\displaystyle\sigma_{T,L}(\gamma^{*}p;x,Q^{2})=\int d^{2}\mbox{\boldmath$r$}\int_{0}^{1}dz\Big{|}\Psi^{\gamma^{*}}_{T,L}(z,\mbox{\boldmath$r$},Q^{2})\Big{|}^{2}\,\sigma(x,r)\,,$$

$$\sigma (x,r)={\sigma }_0\left(1-\exp \left[-\frac{Q_s^2(x)r^2}{4}\right]\right),$$

$$Q_{s}^{2}(x)=Q_{0}^{2}\cdot\Big{(}{x_{0}\over x}\Big{)}^{\lambda}\,.$$

$$\sigma (x,r)={\sigma }_0\left(1-\exp \left[-\frac{{\pi }^2{r}^2{\alpha }_s({\mu }^2)xg(x,{\mu }^2)}{3{\sigma }_0}\right]\right).$$

$$xg(x,{\mu }_0^2)=A_g{x}^{-{\lambda }_g}(1-x{)}^{C_g},$$

$$\sigma (x,r)=2\pi R_p^2\{\begin{array}{ll}{N}_0\exp [-2\gamma L-\frac{L^2}{\kappa \lambda Y}]& \text{if}L\ge 0,\\ 1-\exp [-a(L-L_0{)}^2]& \text{else},\end{array}$$

$$\displaystyle L=\log\Big{(}{2\over rQ_{s}}\Big{)}\,,\,Q_{s}^{2}=\Big{(}{x_{0}\over x}\Big{)}^{\lambda}{\rm GeV}^{2},\,Y=\log\Big{(}{1\over x}\Big{)}\,.$$

$$\displaystyle L_{0}={1-N_{0}\over\gamma N_{0}}\,\log\Big{(}{1\over 1-N_{0}}\Big{)}\,,a={1\over L_{0}^{2}}\log\Big{(}{1\over 1-N_{0}}\Big{)}.$$

$$N_0=0.7,R_p=3.44{\mathrm{GeV}}^{-1},\gamma =0.7376,\kappa =9.9,\lambda =0.2197,x_0=1.632\cdot 10^{-5}.$$

$$\sigma (\gamma p\to J/\psi p;W)=\frac{1+{\rho }_N^2}{16\pi B}{R}_{\mathrm{skewed}}^2|⟨V|\sigma (x,r)|\gamma ⟩{|}^2.$$

$$\sigma (x,r)={\sigma }_0\left(1-\exp \left[-\frac{{\pi }^2{r}^2{\alpha }_s({\mu }^2)R_{\mathrm{skewed}}xg(x,{\mu }^2)}{3{\sigma }_0}\right]\right),$$

$$\frac{d\sigma (AA\to AAJ/\psi ;\sqrt{s_{NN}})}{dy}=n({\omega }_{+})\sigma (\gamma A\to J/\psi A;W_{+})+n({\omega }_{-})\sigma (\gamma A\to J/\psi A;W_{-}).$$

$$\displaystyle n(\omega)={2Z^{2}\alpha_{\rm em}\over\pi}\Big{[}\xi K_{0}(\xi)K_{1}(\xi)-{\xi^{2}\over 2}(K_{1}^{2}(\xi)-K_{0}^{2}(\xi))\Big{]}.$$

Full text

I Introduction

Following the early theoretical work Klein:1999qj and the recent measurements Abelev:2012ba ; Abbas:2013oua ; Khachatryan:2016qhq ; Kryshen:2017jfz ; LHCb:2018ofh (see also the review Contreras:2015dqa ) there has been recently much interest in the coherent exclusive production of vector mesons in ultraperipheral heavy-ion collisions at the LHC. The production takes place via the diffractive photoproduction process, where one of the ions serves as a source of quasireal photons. The second ion plays the role of the hadronic target on which the diffractive photoproduction proceeds.

The production of vector mesons composed of heavy quarks, such as the $J/\psi$ is of special interest and the exclusive production of $J/\psi$ in ultraperipheral heavy-ion collisions has been investigated using a number of different theoretical approaches Klein:2003vd ; Goncalves:2005yr ; AyalaFilho:2008zr ; Cisek:2012yt ; Lappi:2013am ; Guzey:2013xba ; Santos:2014zna ; Goncalves:2017wgg ; Guzey:2016piu ; Xie:2016ino ; Kopp:2018xvu . In this case the heavy quark mass provides a hard scale which ensures a dominant contribution from short distances, so that a perturbative QCD approach becomes applicable. The diffractive photoproduction then becomes a sensitive probe of the gluon structure of the target.

Much attention has been paid in the past on diffractive photo- and electroproduction of vector mesons on the proton. A large body of data has been accumulated at the DESY-HERA facility. For a review of experimental data and of the theoretical approaches, see Ivanov:2004ax . Here we will use the color-dipole approach, which allows us to take into account nuclear effects once the dipole cross section on a free nucleon has been fixed. To this end we take advantage of available data on exclusive $J/\psi$ production to check a variety of dipole cross sections against them. In view of the later application to ultraperipheral heavy-ion collisions the HERA energy range is the most relevant to us.

Here we discuss the coherent diffractive photoproduction in the same approach which we used earlier for the incoherent photoproduction of $J/\psi$ Luszczak:2017dwf . This work is organized as follows: in Section II we review the formalism and main formulas for diffractive vector meson production on nucleons and nuclei in the color dipole approach. In Section III we review different parametrizations/fits of the dipole cross section. Then, in Section IV we compare the results of our numerical results to available published and preliminary experimental data. We summarize our findings in Section V.

II Coherent photoproduction in the color dipole approach

II.1 Free nucleon target

Let us start with a brief review of the formalism for production of a vector mesons $V$ of mass $M_{V}$ (in this work we concentrate on $V=J/\psi$) at high enough energies, so that the coherence length $l_{c}=2\omega/M_{V}^{2}$ is much larger than the size of the proton $l_{c}\gg R_{N}$, where $\omega$ is the photon energy. In such a situtation the $J/\psi$ photoproduction can be described as a elastic scattering of a $c\bar{c}$ of size $r$ conserved during the interaction (see e.g. Nikolaev:1992si ). The $\gamma\to c\bar{c}$ transition and projection of the $c\bar{c}$ pair on the bound state are encoded in the relevant light-cone wave functions, which depend also on the fraction $z$ of the photon’s light-front momentum carried by the quark. The coherent diffractive amplitude on the free nucleon then takes a form

[TABLE]

Here $x=M_{V}^{2}/W^{2}$, where $W$ is the $\gamma p$-cms energy. Our amplitude is normalized such that the differential cross section is obtained from

[TABLE]

The overlap of light-front wave functions of photon and the vector meson is

[TABLE]

Here a sum over quark and antiquark helicities is implicit. The overlap depends also on photon and vector meson helicities and in general gives rise to nonzero helicity flip transitions. We concentrate on the helicity conserving amplitude, as helicity flip transitions for heavy vector mesons are suppressed. In this case one can also neglect the $q$-dependent phase factor. 111An form of the phase which doesn’t vanish at $z=1/2$ is common in the literature. Note that our phase is consistent with the Feynman-diagram calculations in momentum space Ivanov:2004ax . All the dependence on transverse momentum transfer $q$ is contained then in the off-forward dipole cross section, for which we assume a factorized form

[TABLE]

The diffraction slope $B$ depends on energy, for the explicit parametrization used, see section IV.1 below. Explicitly, the overlap of vector meson and photon light-cone wave function, obtained from the $\gamma_{\mu}$-vertex for the $Q\bar{Q}\to V$ vertex is given by Nemchik:1994fp ; Nemchik:1996cw

[TABLE]

For the radial wave function $\psi(z,r)$, we choose the so-called “boosted Gaussian” wave function Nemchik:1994fp ; Nemchik:1996cw as parametrized in Kowalski:2006hc for the $J/\psi$ meson.

The real part of the amplitude is restored from analyticity from the $x$-dependent effective intercept

[TABLE]

so that

[TABLE]

As we discuss below, the color dipole cross section has been obtained from a fit of the total photoabsorption cross section on the nucleon, i.e. a fit to the absorptive part of the forward, (Mandelstam $t=0$), Compton amplitude. In vector meson production, even at \mbox{\boldmath\Delta}=0 the $t=0$ limit is not reached at finite energy and there is always a finite $t_{\rm min}$ due to the vector meson mass. Consequently, gluons exchanged in the amplitude carry different longitudinal momenta, at small $x=M_{V}^{2}/W^{2}$ we have typically, say $x_{1}\sim x,x_{2}\ll x_{1}$. In such a situation, the corresponding correction which multiplies the amplitude is Shuvaev’s Shuvaev:1999ce factor

[TABLE]

This correction has been studied with some rigour only for the two-gluon ladder, where it accounts for the “skewedness” of gluon momentum fractions. It is to be applied only at small-$x$.

II.2 Nuclear Target

When it comes to nuclear targets one should realise that color dipoles can be regarded as eigenstates of the interaction and one can apply the standard rules of Glauber theory Glauber for each of the eigenstates. We now require that the coherence length be much larger than the nuclear size, $l_{c}\gg R_{A}$. Then we can obtain the Glauber form of the dipole scattering amplitude:

[TABLE]

Notice that being states of fixed size color dipoles are not eigenstates of a mass operator, and the rescattering of dipoles (see Fig.1) corresponds to the diffractive transitions $M_{i}^{2}\to M_{j}^{2}$ in the individual scatterings. The dipole rescattering therefore is a particular realization of Gribov’s generalization of Glauber theory Wilkin . The inelastic shadowing corrections will include masses which are different from $M_{V}^{2}$, but not much larger. That is to say that inelastic shadowing corresponding to explicit triple Pomeron terms are not included.

The dipole amplitude of eq.9 corresponds to a rescattering of the dipole in a purely absorptive medium. The real part of the dipole-nucleon amplitude is often neglected. It induces the refractive effects Glauber , and instead of eq.9 we should take

[TABLE]

Note that the real part of the dipole amplitude $\Gamma_{A}$ contributes to the imaginary (”absorptive”) part of the diffractive amplitude, while the imaginary part of $\Gamma_{A}$ yields the real (”dispersive”) part of the diffractive amplitude. We adopt the standard assumption of the nucleus being a dilute gas of uncorrelated nucleons. The optical thickness T_{A}(\mbox{\boldmathb}) is calculated from a Wood-Saxon distribution $n_{A}(\vec{r})$:

[TABLE]

The diffractive amplitude in $b$-space is

[TABLE]

We denote by $W$ the per-nucleon cms-energy in the $\gamma A$-collision. The nuclear form factor ${\cal F}_{A}(q)=\exp[-R^{2}_{\rm ch}q^{2}/6]$ depends on the finite longitudinal momentum transfer $q_{z}=xm_{N}$. It serves to cut-off the diffractive contribution at low energies (large $x$) where the coherence condition is not satisfied. The total cross section for the $\gamma A\to VA$ reaction is finally obtained as

[TABLE]

III Dipole models

In the dipole picture the deep inelastic scattering is viewed as a two stage process; first the virtual photon fluctuates into a dipole, which consists of a quark-antiquark pair (or a $q\bar{q}g$ or $q\bar{q}gg$ … system) and in the second stage the dipole interacts with the proton. Dipole denotes a quasi-stable quantum mechanical state, which has a very long life time ($\approx 1/m_{p}x\;$) and a size $r$, which remains unchanged during scattering. The wave function \Psi^{\gamma^{*}}_{T,L}(z,\mbox{\boldmathr},Q^{2}) determines the probability amplitude to find a dipole of size $r$ within a photon. This probability depends on the value of external $Q^{2}$ and the fraction of the photon momentum carried by the quarks forming the dipole, $z$.

The scattering amplitude is a product of the virtual photon wave function, $\Psi$, with the dipole cross section, $\sigma(x,r)$, which determines a probability of the dipole-proton scattering. Thus, within the dipole formulation of the $\gamma^{*}p$ scattering Nikolaev:1990ja

[TABLE]

where $T,L$ denotes the virtual photon polarization and $\sigma_{T,L}^{\gamma^{*}p}$ the total inclusive DIS cross section. It is worth to remember that besides a contribution of dipoles of sizes $r^{2}\sim 1/Q^{2}$, the total photoabsorption also gets a scaling contribution from large dipole sizes $r^{2}\sim 1/m_{f}^{2}$, where $m_{f}$ is a mass of the quark of flavour $f$.

Numerous models for the dipole cross section have been developed to test various aspects of the data In the following we will shortly review some of them, which have been obtained from fitting data on the inclusive proton structure function and which we will test against the $J/\psi$ photoproduction data before we proceed to the calculation of the nuclear observables.

III.1 GBW model

The dipole model became a popular tool in investigations of deep-inelastic scattering following the observation of Golec-Biernat and Wüsthoff (GBW) GolecBiernat:1998js ; GolecBiernat:1999qd , that a simple ansatz for the dipole cross section was able to describe simultaneously the total inclusive and diffractive cross sections measured at HERA.

In the GBW model the dipole-proton cross section $\sigma_{\text{dip}}$ is given by

[TABLE]

where $Q_{s}^{2}(x)$ is the $x$ dependent saturation scale. It is parametrized in the form

[TABLE]

The free fitted parameters are: the cross-section normalisation, $\sigma_{0}$, as well as $x_{0}$ and $\lambda$. For dipole sizes which are large in comparison to the saturation radius, $R\sim 1/Q_{s}$, the dipole cross section saturates by approaching a constant value $\sigma_{0}$, i.e. saturation damps the growth of the gluon densitIn this model saturation is taken into account in the eikonal approximation and the saturation radius can be related to the gluon density in the transverse plane. y at low $x$.

The GBW model provided a good description of data from medium $Q^{2}$ values ($\approx 30$ GeV2) down to low $Q^{2}$ ($\approx 0.1$ GeV2). Despite its success and its appealing simplicity the model has some shortcomings; in particular it describes the QCD evolution by a simple $x$ dependence, $\sim(1/x)^{\lambda}$, i.e the $Q^{2}$ dependence of the cross section evolution is solely induced by the saturation effects. Therefore, it does not match with DGLAP QCD evolution, which is known to describe data very well from $Q^{2}\approx 4$ GeV2 to very large $Q^{2}\approx 10000$ GeV2. However, we have to remember, that for the case of $J/\psi$ production the hard scale is just at the lower range of the perturbative regime: $Q^{2}\approx M_{J/\psi}^{2}/4\sim 2.5\,\rm GeV^{2}$. Therefore one may expect that the DGLAP evolution effects are not very strong.

In this work we use a new fit of the GBW-form of the dipole cross section obtained by Golec-Biernat and Sapeta in Golec-Biernat:2017lfv . We take the parameters which they obtained by fitting HERA data for $Q^{2}<5\,\rm{GeV}^{2}$, and which read $\sigma_{0}=28.18\,\rm{mb}$, $\lambda=0.237$ and $x_{0}=0.31\cdot 10^{-4}$, with $Q_{0}^{2}=1\,\rm{GeV}^{2}$.

III.2 BGK model

The evolution ansatz of the GBW model was improved in the model proposed by Bartels, Golec-Biernat and Kowalski, (BGK) Bartels:2002cj , by taking into account the DGLAP evolution of the gluon density in an explicit way. The model preserves the GBW eikonal approximation to saturation and thus the dipole cross section is given by

[TABLE]

The evolution scale $\mu^{2}$ is connected to the size of the dipole by $\mu^{2}=C/r^{2}+\mu^{2}_{0}$. This assumption allows to treat consistently the contributions of large without making the strong coupling constant, $\alpha_{s}(\mu^{2}$), unphysically large.

The gluon density, which is parametrized at the starting scale $\mu_{0}^{2}$, is evolved to larger scales, $\mu^{2}$, using LO or NLO DGLAP evolution. For the initial condition, we consider here the soft ansatz, as used in the original BGK model

[TABLE]

The free parameters for this model are $\sigma_{0}$, $\mu^{2}_{0}$ and the parameters for gluon $A_{g}$, $\lambda_{g}$, $C_{g}$. Their values have been obtained by a fit to the data using the xFitter framework xfitter in Ref.Luszczak:2016bxd . The fit results were found to be independent on the parameter $C$, which was therefore fixed as $C=4$ GeV2, in agreement with the original BGK fits. For convenience we show the parameters in Table 1.

III.3 IIM model

Another parametrization of the dipole cross section which gives the latter in a simple analytic form is the IIM model Iancu:2003ge . It is also meant to take into account the saturation effects. While the GBW and BGK models use for saturation the eikonal approximation, the IIM model uses a simplified version of the Balitsky-Kovchegov equation Balitsky:1995ub ; Kovchegov:1999yj . Here we use a parametrization obtained by Soyez which includes heavy quarks into the fit Soyez:2007kg . The model was compared with the recent H1 data in H1FL , where it was shown that it provides a good data description in the lower $Q^{2}$ range, $0.2<Q^{2}<40$ GeV2. As this model also applies to the range of moderately large $Q^{2}$ it is an appropriate choice for our problem. The dipole cross section is parametrized as

[TABLE]

where

[TABLE]

and

[TABLE]

We take the numerical values found in the xFitter code:

[TABLE]

IV Numerical results

IV.1 Predictions for $J/\psi$ production on the proton target

Let us now turn to the numerical results we obtained for the total exclusive photoproduction cross section of $J/\psi$ on the proton target. For the GBW and IIM dipole cross sections, we calculate the total cross section from

[TABLE]

The diffraction slope $B$ is taken as $B=B_{0}+4\alpha^{\prime}\,\log(W/W_{0})$, with $B_{0}=4.88\,\rm{GeV}^{-2}$, $\alpha^{\prime}=0.164\,\rm{GeV}^{-2}$, and $W_{0}=90\,\rm{GeV}$. For the BGK type of parametrizations, it proves to be more stable numerically to substitute the “skewed glue” in the exponent:

[TABLE]

where the exponent $\Delta_{\bf I\!P}$ which enters the Shuvaev-factor is calculated from the relevant gluon distribution. This avoids taking tedious derivatives in the numerical grid for the dipole cross section.

Our results are shown if Fig.2, where we compare the results from the three different parametrizations of the dipole cross section against the data from Refs. Binkley:1981kv ; Denby:1983az ; Frabetti:1993ux ; Alexa:2013xxa ; Aktas:2005xu ; Chekanov:2002xi ; Aaij:2014iea ; Aaij:2018arx ; TheALICE:2014dwa We observe that the range of 30\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}W\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}300\rm GeV is reasonably well described by all dipole cross sections. The very high-energy domain is covered by data which have been extracted from the $pp\to ppJ/\psi$ reaction by the LHCb collaboration Aaij:2014iea ; Aaij:2018arx . Here none of the models does a good job. While one could certainly try to obtain a dipole cross section which also fits the high-energy data, possibly by including vector meson data into the fit, this is not necessary for the purpose of this paper. Namely it is the HERA energy-range which will be most crucial for the calculations in ultraperipheral heavy-ion collisions later on. In that respect, the description of free-nucleon data for the chosen dipole cross sections is satisfactory for our purposes.

IV.2 Results for photoproduction in ultraperipheral collisions

We now turn to our results for ultraperipheral heavy-ion collisions. We obtain the rapidity-dependent cross section for exclusive $J/\psi$ production from the Weizsäcker-Williams fluxes of quasi-real photons $n(\omega)$ as

[TABLE]

Here the two terms correspond to the contributions where the left-moving ion serves as the photon source and the right-moving one as the target and vice-versa (see Fig. 3). Note that we neglected the interference between the two processes. This interference is concentrated at very small transverse momenta Klein:2003vd ; Hencken:2005hb ; Schafer:2007mm . It introduces the azimuthal correlation between the outgoing ions, and in the absence of absorptive corrections it would vanish after the angular integration Schafer:2007mm .

We use the standard form of the Weizsäcker-Williams flux (see e.g. the reviews Bertulani:2005ru ; Baur:2001jj ) for the ion moving with boost $\gamma$:

[TABLE]

Here $\omega$ is the photon energy, and $\xi=2R_{A}\omega/\gamma$. This flux was obtained by imposing the constraint on the impact parameter of the collision $b>2R_{A}$, where we use $R_{A}=7\,\rm{fm}$. This means that configurations where nuclei touch each other are excluded, as otherwise inelastic processes would destroy the rapidity gaps in the event. The photon energies corresponding to the two contributions are $\omega_{\pm}=m_{V}\exp[\pm y]/2$, the corresponding cms-energies for the $\gamma A\to J/\psi A$ subprocesses are $W_{\pm}=2\sqrt{s_{NN}}\omega_{\pm}$.

In order to understand the kinematics a bit better, in Tables 2 and 3 we show the values of $W_{\pm}$, the associated Bjorken-$x$ values $x_{\pm}$ as well as photon fluxes $n(\omega_{\pm})$. For convenience we have also included the values of the photoproduction cross section $\sigma(\gamma A\to J/\psi A:W_{\pm})$ for the example of the IIM dipole cross section, and for the ${}^{208}\rm Pb$ nucleus. Table 2 is for the energy $\sqrt{s_{NN}}=2.76\,\rm TeV$ and Table 3 for $\sqrt{s_{NN}}=5.02\,\rm TeV$. At midrapidity of course $W_{\pm}$ coincide, and we are always well in the energy range that has been explored at HERA for the free nucleon target. If we move out to larger rapidities of the two processes it is the low-energy reaction which tends to dominate. This has its explanation in the rather quick drop of the nuclear photon fluxes at high photon energies and the modest rise of the nuclear photoproduction cross section.

In figure 4 we show the cross section as a function of $J/\psi$-rapidity, for $\sqrt{s_{NN}}=2.76\,\rm TeV$ for the three different dipole cross sections introduced previously. We compare them to the data obtained by the ALICE Abelev:2012ba ; Abbas:2013oua and CMS Khachatryan:2016qhq collaborations. We see that at large rapidities we obtain a fair description of the data, while all of the dipole cross sections overestimate the data at mid-rapidity.

In figure 5 the rapidity dependent cross section is shown at $\sqrt{s_{NN}}=5.02\,\rm TeV$ and compared to preliminary data from the ALICE collaboration Kryshen:2017jfz . Figure 6 also is calculated at $\sqrt{s_{NN}}=5.02\,\rm TeV$ but compares to the preliminary data by the LHCb collaboration LHCb:2018ofh . We see that again we get a reasonable desription of the preliminary data. We should point out that we did not include a skewedness correction to our results on nuclear targets. It is not completely clear how the skewedness correction should be applied in the case of nuclear rescatterings. Obviously the assumption of a two-gluon exchange does not apply to our mechanism, and the necessary longitudinal momentum transfer can be shared by many gluons. As the diffractive amplitude has a “form-factor”-like (perhaps exponential) behaviour an equipartition of longitudinal momentum transfers looks more likely than having one “large” momentum transfer and many small ones. In addition, the data at large rapidities have a large contribution from not very small $x$ where the skewedness correction may not be justified. We therefore follow the authors of Goncalves:2017wgg and omit the skewedness correction.

The overall picture suggests that the Glauber-Gribov formalism in the color dipole approach works reasonably well at not too high energies (or not too small $x$), while at higher energies (smaller $x$) there is room for additional nuclear suppression. In fact in our calculations we included only the rescattering of the $c\bar{c}$ pair. Due to the “scanning-radius” property of vector meson photoproduction, in rescatterings of the small $c\bar{c}$ pair are in fact higher twist effects. This is different than the case of the inclusive structure function $F_{2}$, where large dipoles contribute to a scaling (up to logarithms) nuclear shadowing Nikolaev:1990ja .

It is well understood that at small-$x$ the coherency condition can be also satisfied by higher Fock $c\bar{c}g$, $c\bar{c}gg\dots$ states. In fact these Fock states (in a configuration of strongly ordered transverse sizes) are responsible for the DGLAP evolution of structure functions, while configurations strongly ordered in gluon longitudinal momenta correspond to the BFKL/BK limit. We believe that the missing higher Fock-states are the main culprit behind the overprediction of the ALICE data at midrapidity. Whether their effect can eventually be absorbed into a leading twist shadowing correction to the DGLAP evolving nuclear glue is an open issue at the moment.

One may ask finally, if the light-cone wave function can be the scapegoat. Indeed in a careful analysis of some theoretical uncertainties Santos:2014zna (see also e.g. Xie:2016ino ) it was shown that there can be a sizeable dependence on the meson light-cone wave-function. However here we take the point of view that the succesful description of free nucleon HERA data in the important for us energy range “fixes” the wave-function overlap. Indeed with the dipole cross sections we employed we would rather spoil the agreement with HERA data if we substitute another of the popular parametrizations. From this point of view attempts to obtain light cone wave functions from other sources Chen:2016dlk are interesting.

V Summary and outlook

In this paper we have presented calculations using the Glauber-Gribov theory for coherent exclusive photoproduction of $J/\psi$-mesons on heavy nuclei within the color dipole approach. The dipole cross sections which we utilize have all been obtained from fitting inclusive deep-inelastic structure function data from HERA. We first calculated the total elastic photoproduction of $J/\psi$ on the free nucleon comparing to the data available from fixed-target epxeriments, from the H1 and ZEUS collaborations at HERA as well as to data extracted from $pp$ or $pA$ collisions by the LHCB and ALICE collaborations. All the three dipole cross sections used in this work give a reasonable description of the data, up to and including the HERA energy range, when used together with the so-called “boosted Gaussian” parametrization of the $J/\psi$ wave function. The higher energy data extracted mainly by the LHCb collaboration from exclusive $pp$ collisions are not well described.

We have applied our results to the exclusive $J/\psi$ production in heavy-ion (lead-lead) collisions at the energies $\sqrt{s_{NN}}=2.76\,\rm GeV$ and $\sqrt{s_{NN}}=5.02\,\rm GeV$,

The color dipoles play the role of the eigenstates of the scattering matrix and take into account the inelastic shadowing corrections. We have taken into account the rescattering of a $c\bar{c}$ dipole in the nucleus taking into account the real part of the free nucleon amplitude consistent with the rules of Glauber theory.

Although there is substantial uncertainty as to how to include the skewedness correction in to the nuclear amplitude, the description of published and preliminary data can be regarded satisfactory. However the data point taken by ALICE at midrapidity for $\sqrt{s_{NN}}=2.76\,\rm{TeV}$ is overpredicted. This seems to point to the fact that rescattering of the $c\bar{c}$ dipole is insufficient at energies $W_{\gamma A}\sim 100\,\rm{GeV}$ or $x\sim 0.001$.

We believe that explicit account of higher Fock-states is necessary in this kinematic region. This is consistent with an analysis of nuclear shadowing and deep inelastic diffraction on nuclei in Nikolaev:2006mh . There it is shown, that $q\bar{q}g$ states are important for nuclear shadowing at x\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}0.005.

Whether the correct approach is a resummation of their effect in a BFKL/BK framework, or whether they can be absorbed into a leading twist shadowing of the collinear nuclear glue is an open issue. It stands to reason that this issue can hardly be resolved by only one observable, measured essentially at one hard scale, and that future measurements at an electron-ion collider will be crucial for a deeper understanding of the nuclear glue.

Acknowledgements

This work is partially supported by the Polish National Science Center grant DEC-2014/15/B/ST2/02528.