# Disentangling coherent and incoherent quasielastic \(J/\psi \) photoproduction on nuclei by neutron tagging in ultraperipheral ion collisions at the LHC

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## Abstract

We consider \(J/\psi \) photoproduction in ion–ion ultraperipheral collisions (UPCs) at the LHC and RHIC in the coherent and incoherent quasielastic channels with and without accompanying forward neutron emission and analyze the role of nuclear gluon shadowing at small \(x\), \(x=10^{-4}{-}10^{-2}\), in these processes. We find that despite the good agreement between large nuclear gluon shadowing and the ALICE data in the coherent channel, in the incoherent channel, the leading twist approximation predicts the amount of nuclear suppression which is by approximately a factor of \(1.5\) exceeds that seen in the data. We hypothesize that part of the discrepancy can be accounted for by the incoherent inelastic process of \(J/\psi \) photoproduction with nucleon dissociation. To separate the high-photon-energy and low-photon-energy contributions to the \(\mathrm{d}\sigma _{AA\rightarrow AAJ/\psi }(y)/\mathrm{d}y\) cross section, we consider ion–ion UPCs accompanied by neutron emission due to electromagnetic excitation of one or both colliding nuclei. We describe the corresponding PHENIX data and make predictions for the LHC kinematics. In addition, in the incoherent quasielastic case, we show that the separation between the low-photon-energy and high-photon-energy contributions can be efficiently performed by measuring the correlation between the directions of \(J/\psi \) and the emitted neutrons.

## Keywords

Neutron Emission ALICE Data Photoproduction Cross Section Dipole Cross Section Vector Meson Dominance Model## 1 Introduction

Recently coherent and incoherent photoproduction of \(J/\psi \) in ultraperipheral collisions (UPCs) of nuclei was measured by the ALICE collaboration at the LHC [1, 2]. In the coherent channel, a large reduction of the coherent cross section—approximately by a factor of three—as compared to the impulse approximation has been reported. Such a magnitude of the suppression was found to be in the reasonable agreement with the expectations of the approaches predicting significant nuclear gluon shadowing at \(x \approx 10^{-3}\) (\(x\) is the fraction of the nucleus momentum carried by gluons), notably with predictions of the leading twist approach to nuclear shadowing [3, 4] and with the results of the EPS09 global QCD fit to nuclear parton distributions [5]. Thus, charmonium photoproduction on nuclear targets is a useful tool to study nuclear gluon shadowing at small \(x\).

The aim of this paper is twofold. First, we extend application of the leading twist approach to nuclear shadowing [6] to incoherent quasielastic photoproduction of \(J/\psi \) on nuclei and show that the suppression of both coherent and incoherent \(J/\psi \) photoproduction in ion–ion UPCs can be described in the same framework. A comparison of the resulting theoretical prediction for the cross section of incoherent \(J/\psi \) photoproduction in Pb–Pb UPCs at the LHC to the ALICE data, which is also characterized to correspond to an incoherent quasielastic process [1], shows that the expected suppression due to nuclear shadowing is larger than that seen in the data. We argue that this does not only place additional constraints on models of nuclear shadowing down to \(x \approx 10^{-4}\) but also indicates that additional processes can contribute to the ALICE data. In particular, on top of incoherent \(J/\psi \) photoproduction on nuclei resulting from the target nucleus excitation, the \(\gamma +A \rightarrow J/\psi +Y+(A-1)^{*}\) process driven by the \(\gamma +N \rightarrow J/\psi +Y\) nucleon dissociation (\(Y\) denotes products of the nucleon dissociation) accompanied by the nucleus breakup into the \((A-1)^{*}\) system consisting of nucleus debris or nucleons also leads to the inelastic final state. The calculation of the \(\gamma +A \rightarrow J/\psi +Y+(A-1)^{*}\) contribution is rather involved, reflecting different mechanisms of the elementary reaction at small and large \(|t|\) considered in [7], and it will be addressed in a separate publication.

Second, we discuss specifics of and make predictions for coherent and incoherent charmonium production in nucleus–nucleus UPCs accompanied by forward neutron emission which can be studied at the LHC with the ALICE, CMS, and ATLAS detectors equipped by zero degree calorimeters (ZDC). The following channels can be studied: (i) one of the nuclei emits at least one neutron while its partner does not—(0nXn); (ii) both nuclei emit neutrons in opposite directions—(XnXn), (iii) neither of the nuclei emits neutrons—(0n0n). We show that selection of a specific channel can strongly influence the ratio of the cross sections of incoherent to coherent \(J/\psi \) photoproduction in Pb–Pb UPCs at the LHC. In particular, we argue that the study of incoherent production of charmonium in ion–ion UPCs with the nucleus breakup allows one to separate the low-photon-energy and high-photon-energy contributions to nuclear \(J/\psi \) photoproduction and, hence, to provide additional information on the dynamics of nuclear shadowing of the gluon distribution in nuclei which is complementary to that obtained from coherent onium production.

This paper is organized as follows. In Sect. 2, we discuss the suppression of the coherent and incoherent nuclear \(J/\psi \) photoproduction cross sections due to nuclear shadowing. We briefly recapitulate main results of the vector meson dominance and the color dipole models for these processes and present the derivation of the coherent \(\sigma _{\gamma A \rightarrow J/\psi A}\) and the incoherent \(\sigma _{\gamma A \rightarrow J/\psi A^{\prime }}\) cross sections in the leading twist approximation. In Sect. 3, using the results of Sect. 2, we make predictions for the coherent and incoherent cross section of \(J/\psi \) photoproduction in ion–ion UPCs without and with neutron emission and analyze the obtained results. Section 4 presents a brief summary of the obtained results.

## 2 Nuclear gluon shadowing in coherent and incoherent \(J/\psi \) photoproduction on nuclei

### 2.1 The coherent nuclear \(J/\psi \) photoproduction cross section

The main issue with Eqs. (1) and (2) is what value of the elementary \(\sigma _{J/\psi N}^\mathrm{tot}\) cross section to use. It has been well known for a long time that if one tries to determine \(\sigma _{J/\psi N}^\mathrm{tot}\) using the vector meson dominance model and the data on the elementary \(\gamma p\rightarrow J/\psi p\) process, the obtained value of \(\sigma _{J/\psi N}^\mathrm{tot}\) is small, namely, \(\sigma _{J/\psi N}^\mathrm{tot}(W_{\gamma p}=5\, \mathrm{GeV}) \approx 1\) mb and \(\sigma _{J/\psi N}^\mathrm{tot}(W_{\gamma p}=100\, \mathrm{GeV})\approx 3\) mb; see, e.g., [11]. As a result, the effect of nuclear shadowing in the \(\sigma _{\gamma A\rightarrow J/\psi A}^\mathrm{VMD}\) cross section predicted using Eqs. (1) and (2) turns out to be small in the small-\(x\) region, which contradicts the ALICE data. Also, the smallness of \(\sigma _{J/\psi N}^\mathrm{tot}\) serves as an indication that in the strong interaction, \(J/\psi \) reveals properties of a small-size dipole built from a heavy quark–antiquark pair.

In a more general case [16], (i.e., beyond the \(k_t^2 \ll m_c^2\) limit for the \(J/\psi \) wave function), there exists a theoretical uncertainty in the value of \(\mu ^2\) in Eq. (4) which means that one could use a reasonable range of values, e.g., \(\mu ^2=2.4{-}3.4~\hbox {GeV}^2\). For example, the suitable value of \(\mu ^2\) can be determined phenomenologically [4] comparing predictions of Eq. (4) for the proton target with the data.

The leading twist theory of nuclear shadowing [6] is based on the space-time picture of the strong interaction at high energies, the generalization of the Gribov–Glauber theory of nuclear shadowing in soft hadron–nucleus scattering [10, 21] to hard processes with nuclei, and the QCD collinear factorization theorems for the total and diffractive cross sections of deep inelastic scattering (DIS). The approach allows one to make predictions for the leading twist shadowing correction to nuclear parton distributions (nPDFs), structure functions and cross sections, which are given as a series in the number of simultaneous interactions with the target nucleons (the multiple scattering series). The structure of each term in the series is unambiguously given by the Gribov–Glauber theory supplemented by Abramovsky–Gribov–Kancheli (AGK) cutting rules [22, 23] and the QCD factorization theorems.

The first term in Eq. (6) describes photoproduction of \(J/\psi \) on a single nucleon and, hence, is proportional to the number of nucleons \(A\); it is the impulse approximation corresponding to graph \(a\) in Fig. 2.

For the interaction with \(N \ge 4\) nucleons (not shown in Fig. 2), we assume that the effect of cross section fluctuations is the same as for the \(N=3\) term, i.e., \(\langle \sigma ^N \rangle =\langle \sigma ^2 \rangle \sigma _3^{N-2}=\langle \sigma \rangle \sigma _2 \sigma _3^{N-2}\) for \(N \ge 3\).

It is important to note that unlike the case of the color dipole formalism, the shadowing correction in Eq. (10) (and also in \(R\)) is a leading twist quantity determined by the elementary hard diffraction in lepton–proton DIS. In the low nuclear density limit, when the interaction with \(N \ge 3\) nucleons can be neglected, the shadowing correction is driven by the leading twist \(\sigma _2\) cross section. At the low values of \(x\), the \(N \ge 3\) terms also become important; their contributions are also leading twist quantities, which can be summed using the \(\sigma _3\) cross section.

### 2.2 The incoherent nuclear \(J/\psi \) photoproduction cross section

Equation (13) has the straightforward and well-known interpretation: the probability of incoherent (quasielastic) photoproduction of a vector meson on a nucleus is given by the product of the probability of elastic scattering on a single nucleon times the probability for the produced vector meson to survive the passage through the nucleus on its way out. Since \(\sigma _{J/\psi N}^\mathrm{in} \approx \sigma _{J/\psi N}\) is small, the nuclear suppression of \(\sigma _{\gamma A\rightarrow J/\psi A^{\prime }}^\mathrm{VMD}\) due to nuclear shadowing is also small.

In the color dipole formalism, the coherent and incoherent nuclear \(J/\psi \) photoproduction cross sections can be calculated on the same footing. The only difference is the order of averaging over the dipole sizes \(d_t\): in the coherent case, one first averages the \(\gamma A \rightarrow J/\psi A\) amplitude over \(d_t\) and then squares the result, while in the incoherent case, one first squares the appropriate scattering amplitude and then averages the result over \(d_t\), see, e.g., Ref. [19]. Since the relevant dipole cross section is small, similarly to the coherent case considered above, nuclear suppression of incoherent nuclear \(J/\psi \) photoproduction cross sections is small [19]. At small \(x\) typical for the LHC kinematics, \(x \sim 10^{-3}\) and below, the dipole formalism predictions are subject to rather significant theoretical uncertainties due to the choice of the model for the dipole cross section and for the \(J/\psi \) wave function. Nevertheless, one can still make the observation that the shadowing suppression of the incoherent cross section of \(J/\psi \) photoproduction on nuclei appears to be larger than that for the coherent case. At the same time, in the RHIC kinematics corresponding to much larger values of \(x\), \(x \approx 10^{-2}\), where the effect of nuclear shadowing is small, both the dipole framework and the leading twist approach provide the good description of the PHENIX UPC data [28]. Note also that at \(x \sim 10^{-2}\), numerous versions of the dipole model correspond to a similar color dipole cross section because the models have been fitted to the same data, see, e.g., Refs. [29, 30].

The physical interpretation of Eq. (15) is similar to that of Eqs. (9) and (11): the nuclear suppression factor in the square brackets arises from multiple interactions of the produced diffractive state with nucleons of the target, which are driven by the \(\sigma _2\) and \(\sigma _3\) cross sections.

A comparison of Eqs. (15) and (10) shows that in the leading twist approximation (LTA), nuclear suppression in both coherent and incoherent photoproduction is determined by the same quantities: \(\sigma _{2}\) and \(\sigma _{3}\) [see Eqs. (11) and (16)]. The \(\sigma _{2}\) cross section is a model-independent quantity whose magnitude and \(x\) dependence are fixed by the experimentally measured diffractive parton distributions, inclusive gluon distributions and DGLAP evolution equations; see Eq. (7). The \(\sigma _{3}\) cross section is a model-dependent quantity of the LTA approach, whose value is constrained using the formalism of cross section fluctuations. In general, \(\sigma _{3}\ge \sigma _{2}\) [see Eq. (8)]; the lower limit on the value of \(\sigma _{3}\), \(\sigma _{3}=\sigma _{2}\), corresponds to the upper limit on the predicted nuclear shadowing.

One should note that since both suppression factors of \(S^\mathrm{LTA}_\mathrm{coh}\) (11) and \(S_\mathrm{incoh}^\mathrm{LTA}\) (16) are determined by the essentially soft physics, we expect them to be numerically of a similar magnitude, with \(S_\mathrm{incoh}^\mathrm{LTA}\) being somewhat smaller than \((S^\mathrm{LTA}_\mathrm{coh})^2\). Indeed, at \(x=10^{-3}\) corresponding to \(y \approx 0\) for Pb–Pb UPCs at \(\sqrt{s_{NN}}=2.76~\hbox {GeV}\), we obtain \(S_\mathrm{incoh}^\mathrm{LTA}=0.16{-}0.35\) from Fig. 3 and \((S_\mathrm{coh}^\mathrm{LTA})^2=0.35{-}0.43\) from Fig. 3 of Ref. [4].

## 3 Photoproduction of \(J/\psi \) in Pb–Pb UPCs at the LHC

### 3.1 Coherent and incoherent cases

The presence of two terms in Eq. (17) reflects the fact that each nucleus can radiate a photon as well as can serve as a target. In the case of symmetric UPCs (e.g., in the case of Pb–Pb UPCs at the LHC and Au–Au UPCs at RHIC), at a fixed value of the \(J/\psi \) rapidity \(y \ne 0\), Eq. (17) contains two contributions: one corresponding to the interaction of high-energy photons with a nucleus and another corresponding to the interaction of low-energy photons with a nucleus.

In the coherent case, the separation of these overlapping contributions is not an easy problem since a priory one cannot say in the interaction with which of the two nuclei \(J/\psi \) was produced. As a result, the photoproduction cross section \(\sigma _{\gamma A\rightarrow J/\psi A}\) can be unambiguously extracted from the measured \(\mathrm{d}\sigma _{AA\rightarrow AA J/\psi }(y)/\mathrm{d}y\) cross section only in the following two cases: (i) at \(y=0\), where the two photon energies are equal and, hence, both terms in Eq. (17) contribute equally, and (ii) in the rapidity range where one of the contributions strongly dominates. Exactly this situation is realized in the ALICE experiment [1, 2]: the \(\mathrm{d}\sigma _{AA\rightarrow AA J/\psi }(y)/\mathrm{d}y\) cross section was measured (i) for \(-1<y<1\), which allowed one to extract \(\sigma _{\gamma A\rightarrow J/\psi A}(W_{\gamma p})\) at \(W_{\gamma p}\approx 100\) GeV corresponding to the gluon momentum fraction of \(x\approx 10^{-3}\) (\(W_{\gamma p}\) is the \(\gamma \)–nucleus center-of-mass energy per nucleon) and (ii) for \(-4<y<-2.5\), where the low-energy photon contribution dominates (more than 95 %), which probes the nuclear gluon density at \(x\approx 10^{-2}\).

In the same figure, the LTA predictions for the incoherent \(\mathrm{d}\sigma _{AA\rightarrow AA^{\prime }J/\psi }(y)/\mathrm{d}y\) cross section made using Eqs. (15) and (17) are compared to the ALICE data point at \(|y|\approx 0\) [1]. One can see from the comparison that the LTA predicts the magnitude of suppression due to nuclear gluon shadowing exceeding the one seen in the data by approximately a factor of \(1.5\).

The shaded bands in Fig. 4 represent the dominant theoretical uncertainty of the LTA predictions associated with the uncertainty in the value of the \(\sigma _3\) cross section, which in turn results in the uncertainty of the LTA predictions for nuclear parton distributions [6]. The uncertainty associated with the value of the hard scale \(\mu ^2\), which was studied in [4], is much smaller and has been safely neglected.

Note that in our calculations, we consider quasielastic scattering and do not take into account the incoherent contribution associated with the nucleon dissociation \(\gamma +N \rightarrow J/\psi +Y\) [33]. We explained in the Introduction that this process could potentially contribute to the inelastic final state and, thus, affect the ALICE extraction of the incoherent \(\mathrm{d}\sigma _{AA\rightarrow AA^{\prime }J/\psi }(y)/\mathrm{d}y\) cross section [1] due to the fact that the ALICE detector does not cover the full range of the rapidity \(y\). While the calculation of the \(\gamma +A \rightarrow J/\psi +Y+(A-1)^{*}\) contribution requires a a separate publication, one can still make several qualitative observations. First, this contribution is expected to have approximately the same \(A\) dependence as that in Eq. (15) (it is proportional to \(A\) in the impulse approximation). Second, the magnitude of this contribution is expected to be sizable: \((\mathrm{d}\sigma _{\gamma N \rightarrow J/\psi Y}/\mathrm{d}t)/(\mathrm{d}\sigma _{\gamma N \rightarrow J/\psi N}/\mathrm{d}t)\approx 0.15\) at \(t \approx 0\) and increases with an increase of \(|t|\) when the \(\mathrm{d}\sigma _{\gamma N \rightarrow J/\psi Y}/\mathrm{d}t\) cross section becomes progressively more important and eventually exceeds that of the elastic \(\gamma +N\rightarrow J/\psi +N\) process; \(\sigma _{\gamma N\rightarrow J/\psi Y}/\sigma _{\gamma N \rightarrow J/\psi N} \approx 0.8\) for the \(t\)-integrated cross sections and for \(M_Y < 10\) GeV (\(M_Y\) is the invariant mass of the proton-dissociative system \(Y\)) [34]. It would be desirable to perform an additional analysis of the ALICE data [1] by assuming that the \(\gamma +N \rightarrow J/\psi +N\) and \(\gamma +N \rightarrow J/\psi +Y\) contributions to incoherent nuclear \(J/\psi \) photoproduction have different slopes of the \(t\) dependence, which would enable one to experimentally estimate the contribution of the nucleon dissociation and, thus, will enable a direct comparison of the data with predictions of Eq. (15). In addition, it is likely that due to the interaction of the system \(Y\) with the nucleus, nucleon dissociation will lead to a larger number of neutrons originating from the nucleus dissociation. Finally, the study of neutron production in the quasielastic \(\gamma A \rightarrow J/\psi A^{\prime }\) process at \(|t| \ge 1~\hbox {GeV}^2\), where the \(\gamma +N \rightarrow J/\psi +Y\) contribution dominates, may be interesting for understanding of the formation time in diffraction.

### 3.2 UPCs accompanied by neutron emission

Besides ALICE, the ATLAS and CMS detectors at the LHC are capable to measure UPC production of \(J/\psi \) in the \(-2.5<y<2.5\) range of rapidity. While for central rapidities, the interpretation of the corresponding measurements is unambiguous, it is difficult to disentangle the high-photon-energy and low-photon-energy contributions to \(\mathrm{d}\sigma _{AA\rightarrow AAJ/\psi }(y)/\mathrm{d}y\) for non-central values of \(y\) and, thus, to access the small-\(x\) region that we are interested in. In particular, according to the estimates of [3, 4], for \(1.5<|y|<2.5\) in Pb–Pb UPCs at 2.76 TeV, the \(\mathrm{d}\sigma _{AA\rightarrow AAJ/\psi }(y)/\mathrm{d}y\) rapidity distribution in the coherent case is exceedingly dominated (by the factor of four) by the low-photon-energy contribution corresponding to \(10^{-2} > x > 5 \times 10^{-3}\) of the probed nuclear gluons. The high-photon-energy contribution is suppressed by the much lower photon flux and the larger nuclear gluon shadowing. Hence, it is rather difficult to extract the high-energy nuclear coherent \(J/\psi \) photoproduction cross section from the UPC data and, hence, to probe the nuclear gluon distribution around \(x\approx 10^{-4}\).

The method to overcome this difficulty was suggested in [35]. It is based on the observation of [36] that coherent photoproduction of vector mesons in heavy ion UPCs can be accompanied by additional photon exchanges which lead to electromagnetic excitation of one or both nuclei with the subsequent neutron emission. These neutrons will have the energy close to that of the colliding beams and can be detected by zero degree calorimeters placed at large distances on both sides of the detectors. With the additional requirement to have in the final state only two muons from the \(J/\psi \) decay (in addition to the neutrons) and the large rapidity gap, i.e., by requiring the absence of any other charged particle in the whole range of \(y\) covered by the detector system, the strong interaction of the colliding nuclei should be suppressed.

Production of forward neutrons in quasielastic incoherent photoproduction of \(J/\psi \) in heavy ion UPCs with the nuclear breakup has been considered in [40]. Since in this case the momentum transfer in elastic \(J/\psi \) photoproduction on the nucleon can be as large as \(|t|=1~\hbox {GeV}^2\), this target nucleon escaping from the nucleus participates in additional quasielastic rescattering. The average excitation energy of a heavy nucleus in the one-nucleon removal process is about \(20{-}25\) MeV, which is much higher than the separation energy of \(7{-}8\) MeV of one neutron. It was shown in [40] that in incoherent \(J/\psi \) photoproduction in heavy ion UPCs, the residual nucleus will decay emitting one or more neutrons with the probability of about 85 %. Therefore, imposing the constraint that no neutrons are emitted, i.e., considering the 0n0n-channel, one can almost completely (at the level of 10–15 %) suppress the incoherent contribution.

As we mentioned in the end of Sect. 3.1, the contribution of nucleon dissociation becomes important/dominant with an increase of the transverse momentum of \(J/\psi \). This process should lead to at least as many neutrons as the quasielastic process. Therefore, the \(\gamma +N \rightarrow J/\psi +N\) and \(\gamma +N \rightarrow J/\psi +Y\) contributions should be either separated experimentally or the latter should be included in the theoretical calculation of the \(\gamma A \rightarrow J/\psi A^{\prime }\) cross section. The procedure for the extraction of the high-photon-energy contribution that we discuss below involves the use of the different \(p_t\) dependences of the \(\gamma +N \rightarrow J/\psi +N\) and \(\gamma +N \rightarrow J/\psi +Y\) cross sections, which allows one to separate their contributions. We also note in passing that the study of the neutron multiplicity at \(p_t \ge 0.8\) GeV, where the process of nucleon dissociation dominates, would produce for the first time information about the space-time formation of hadrons in the diffractive processes like \(\gamma +N \rightarrow J/\psi +Y\).

It is worth noting here that for the discussed kinematics, the results for the dipole–nucleon cross section obtained in different dipole models are rather close since they are well constrained by the DIS data for these energies [29, 30]. The theoretical uncertainty is much smaller than the PHENIX experimental errors and, hence, it is not shown in Fig. 5. Note also that in the discussed model, the nuclear shadowing effect is driven by the \(\sigma _{c{\bar{c}} N}\) dipole cross section and, hence, shadowing is suppressed (it is a higher twist effect) for the dipoles of such a small size; see the discussion above.

Selection of events with two-side neutron detection means that in the XnXn-channel, coherent production should take into account the mutual electromagnetic dissociation that requires at least two additional photon exchanges. In contrast, according to the predictions of [40], incoherent production is associated with excitation and neutron decay of only the target nucleus. Hence, in this case, detection of the XnXn-channel requires only one photon exchange to excite the nucleus, which serves as a source of the photon flux in the process of incoherent production.

Figure 5 presents a comparison of the PHENIX results for the XnXn-channel with our theoretical calculations using the simple dipole described above. Despite large experimental errors, it seems that the agreement between our calculations and the PHENIX data demonstrated in Fig. 5 justifies our approach to neutron production in UPCs. Coupling this with the good description of coherent \(J/\psi \) photoproduction at the LHC makes our approach reasonable for the prediction of coherent and incoherent UPCs accompanied by neutron emission at the LHC.

Each panel contains two sets of curves: the upper shaded area is a sum of the two terms in Eq. (17) and the lower shaded area represents the contribution of \(J/\psi \) production by the photon emitted by the nucleus moving with the momentum in the direction of positive \(J/\psi \) rapidity (the dashed curves labeled “one side”).

We can draw several conclusions from Fig. 6. First, one can see from the two left panels that using the data on the 0nXn and XnXn-channels, one can try to extract the high-photon-energy contribution using Eq. (20). In the range of rapidities of \(1.5<y<2.5\), this will allow one to determine nuclear gluon shadowing in Pb down to \(x\approx 10^{-4}\) and at the scale of \(\mu ^2\approx 3~\hbox {GeV}^2\) from coherent \(J/\psi \) photoproduction in UPCs.

Second, a comparison of the corresponding upper shaded areas in Fig. 6 shows that we predict that at central rapidities, (i) in the 0nXn-channel, the coherent and incoherent contributions will be practically of the same magnitude and (ii) in the XnXn-channel, the coherent contribution exceeds the incoherent one by approximately a factor of two.

Third, besides the coherent channel, the incoherent cross section can also be used to study nuclear gluon shadowing at small \(x\). Based on the dominance of the strong interaction mechanism of neutron production in incoherent photoproduction, we predict that there is a good opportunity to separate the low-photon-energy and high-photon-energy contributions in the 0nXn-channel (see the upper right panel in Fig. 6). Indeed, since in this case neutrons are emitted by the target nucleus, there should by a correlation between the direction of the produced \(J/\psi \) and the direction of neutrons in UPC events. In the kinematics of UPCs, the direction of charmonium produced by a high-energy photon and, hence, by low-\(x\) gluons from the target, is opposite to the direction of the target nucleus and, correspondingly, to the direction of neutrons. Conversely, in the low-photon-energy production, the direction of charmonium coincides with the direction of the target and neutrons.

The standard procedure of the separation of coherent and incoherent events consists in the analysis of momentum transfer distributions. In coherent photoproduction, this distribution is dictated by the nuclear form factor squared and presents several distinct peaks at small \(p_t <200\) MeV/c, where \(p_t\) is the transverse momentum of produced \(J/\psi \) and \(t=-p_t^2\). One can see such first three diffractive peaks in Fig. 7. The \(t\) dependence of the incoherent nuclear cross section is the same as in the elementary process, i.e., \(\exp [-B_{J/\psi } |t|]\). Our predictions for the sum of the coherent and incoherent cross sections of \(J/\psi \) photoproduction in Pb–Pb UPCs accompanied by neutron emission in the 0nXn-channel as a function of \(p_t\) in the rapidity range of \(1.5<y<2.5\) is shown in the upper panel of Fig. 7. As in Fig. 6, our theoretical predictions are based on the LTA; the shaded bands represent the LTA uncertainty in the predicted nuclear gluon distribution. In the figure, the peaks at small \(p_t\) correspond to the coherent signal. Therefore, the \(p_t <200\) MeV/c cut will effectively reject incoherent events with good accuracy.

These predictions are demonstrated in the middle panel of Fig. 7, where the upper band corresponds to the situation when the direction of \(J/\psi \) coincides with that of the neutrons and the lower band is for the opposite directions of \(J/\psi \) and the neutrons. The bottom panel shows the ratio of the two curves from the middle panel. Since we identify the events, where the directions of \(J/\psi \) and neutrons are opposite, with high-energy photoproduction, there could be two sources of suppression of the corresponding cross section: (i) the falloff of the flux of high-energy photons emitted by the nucleus and (ii) stronger nuclear gluon shadowing of the small-\(x\) gluons at \(x \approx 10^{-4}\), which we are mainly interested in.

## 4 Conclusions

We considered \(J/\psi \) photoproduction in ion–ion UPCs at the LHC and RHIC in the coherent and incoherent channels with and without accompanying forward neutron emission and analyzed the role of nuclear gluon shadowing at small \(x\), \(x=10^{-4}{-}10^{-2}\), in these processes. We extended the formalism of leading twist nuclear shadowing characterized by large nuclear gluon shadowing to the incoherent \(\sigma _{\gamma A \rightarrow J/\psi A^{\prime }}\) cross section. We found that despite good agreement between the approaches predicting large nuclear gluon shadowing at \(x \approx 10^{-3}\) and the large nuclear suppression factor extracted from the ALICE data on coherent \(J/\psi \) photoproduction in Pb–Pb UPCs at the LHC [1, 2], in the incoherent channel, the leading twist approximation predicts the amount of nuclear suppression due to gluon shadowing which exceeds that seen in the data by approximately a factor of \(1.5\) (Fig. 4). We hypothesize that one source of the discrepancy could be the contribution of incoherent nucleon dissociation, \(\gamma N \rightarrow J/\psi Y\), which could potentially contribute to the ALICE data [1] and which is not taken into account in our theoretical analysis.

In coherent \(J/\psi \) photoproduction in ion–ion UPCs, it is problematic (except for \(y \approx 0\) and large \(|y|\)) to separate the contributions of high-energy and low-energy photons to the \(\mathrm{d} \sigma _{AA\rightarrow AAJ/\psi }(y)/\mathrm{d}y\) cross section, which reduces the range of \(x\) for the studies of small-\(x\) nuclear gluon shadowing. This problem can be circumvented by considering \(J/\psi \) photoproduction in ion–ion UPCs accompanied by neutron emission due to electromagnetic excitation of one or both colliding nuclei.

Using the leading twist approximation for nuclear gluon shadowing, we made predictions for coherent and incoherent nuclear \(J/\psi \) photoproduction in Pb–Pb UPCs accompanied by neutron emission in various channels at the LHC (Fig. 6). In particular, we discuss the strategy allowing one to separate the low-photon-energy and the high-photon-energy contributions to coherent \(J/\psi \) photoproduction performing a joint analysis of the data in the 0nXn and XnXn-channels. This gives an opportunity to shift the study of nuclear gluon shadowing to the lower \(x\) region of \(x \approx 10^{-4}\).

In addition, in the incoherent case accompanied by neutron emission, we show that the separation between the low-photon-energy and high-photon-energy contributions can be efficiently performed by measuring the correlation between the directions of \(J/\psi \) and the emitted neutrons (Fig. 7).

In the kinematics where nuclear shadowing is small, we showed (Fig. 5) that theoretical predictions based on the dipole model agree with the PHENIX data on \(J/\psi \) photoproduction in Au–Au UPCs at \(\sqrt{s_{NN}}=200\) GeV in the XnXn-channel (both colliding nuclei emit neutrons detected in the ZDCs).

## Notes

### Acknowledgments

We would like to thank L. Frankfurt for useful discussions. M. Strikman’s research was supported by DOE Grant No. DE-FG02-93ER40771.

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