Estimating nonlinear effects in forward dijet production in ultraperipheral heavy ion collisions at the LHC
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Abstract
Using the framework that interpolates between the leading power limit of the color glass condensate and the high energy (or \(k_{T}\)) factorization we calculate the direct component of the forward dijet production in ultraperipheral \(\mathrm {Pb}\)–\(\mathrm {Pb}\) collisions at CM energy \(5.1\,\mathrm {TeV}\) per nucleon pair. The formalism is applicable when the average transverse momentum of the dijet system \(P_{T}\) is much bigger than the saturation scale \(Q_{s}\), \(P_{T}\gg Q_{s}\), while the imbalance of the dijet system can be arbitrary. The cross section is uniquely sensitive to the Weizsäcker–Williams (WW) unintegrated gluon distribution, which is far less known from experimental data than the most common dipole gluon distribution appearing in inclusive smallx processes. We have calculated cross sections and nuclear modification ratios using WW gluon distribution obtained from the dipole gluon density through the Gaussian approximation. The dipole gluon distribution used to get WW was fitted to the inclusive HERA data with the nonlinear extension of unified BFKL + DGLAP evolution equation. The saturation effects are visible but rather weak for realistic \(p_{T}\) cut on the dijet system, reaching about 20% with the cut as low as \(6\,\mathrm {GeV}\). We find that the LO collinear factorization with nuclear leadingtwist shadowing predicts quite similar effects.
1 Introduction
High energy collisions of heavy ions provide a unique opportunity to investigate the quark–gluon plasma regime of QCD. In addition, they also offer a more direct insight into dense initial nucleus states. Relativistic nuclei are in fact very strong sources of electromagnetic fields, thus when they collide at large impact parameters it is possible to study photon–nucleus interactions. Such ultraperipheral heavy ion collisions (UPC) can be investigated using the current LHC setup, and there are plenty of unique possibilities to explore various aspects of nuclear physics [1] and the smallx regime [2]. In particular, photon–nucleus interactions can shed some light on certain aspects of the smallx physics, principally the saturation phenomenon [3].
In the present work we will be focused on the dijet production in UPC in the kinematic configuration which probes relatively small values of x. The purpose of this analysis is twofold. First, we give predictions within a framework which incorporates a nonlinear gluon saturation phenomenon. For certain differential cross sections we will also give predictions using the collinear factorization with the leadingtwist nuclear shadowing [4, 5]. Second, the dijet configurations in \(\gamma A\) collisions are sensitive to subtle QCD effects related to gluon distributions appearing in saturation formalism; we describe this situation in some more details later below. Thus, the second goal is to check if UPC can shed some light on that subject. A previous study in a similar context for the Electron Ion Collider was done in [6] using a slightly different formulation.
Although in the present work we focus mainly on the nonlinear saturation phenomena, the nuclear effects are not restricted to saturation alone. In fact, before the onset of saturation the nuclear shadowing may be significant. It is a leadingtwist effect since the suppression of gluons is encoded in the collinear PDFs within the collinear factorization approach [5]. Large gluon shadowing consistent with prediction of [5] was reported in the coherent \(J/\Psi \) photoproduction of \(\mathrm {Pb}\). For example the shadowing for \(x=10^{3}\), \(Q^2 \sim 3\, \mathrm {GeV}^2 \) was found to be \(\approx 0.6\), for the recent discussion and extensive list of references see [18] (for \(J/\Psi \) production in CGC see [19, 20, 21]). In the present work we estimate both effects, the leadingtwist nuclear shadowing and the saturation, for selected jet observables. The question whether and how these should be combined remains open and is beyond the scope of this paper.
We are considering the limit when typical transverse momenta are much larger than \(Q_s\). In the limit \(k_t \lesssim Q_s\) production of leading particles/jets may be suppressed very strongly. This effect may be responsible for the large suppression of the forward pion production in \(dAu\) collisions at BNL, for the summary and references see chapter 8 of Ref. [5]. How fast these effects die out at \(k_t > Q_s\) still has to be investigated.
In the smallx literature the gluon distributions with transverse momentum dependence are typically called unintegrated gluon distributions (UGDs). We shall follow this standard terminology, keeping in mind the following issues. First, the name UGD would suggest that integrating it over \(k_T\) one gets a collinear gluon PDF. There is no proof of that statement within QCD. There are theoretical difficulties with such a relation, connected to the proper formulation of factorization in the case with transverse momentum dependence; see for example [22]. Second, the collinear parton densities are obtained from the DGLAP equations (up to NNLO order) whereas the unintegrated gluon distributions are usually obtained from the evolution which includes resummation of large logarithms \(\ln 1/x\). Thus the two types of distributions are typically obtained in the different regimes. In the nucleus case for instance, at present no UGD exists which could lead to the gluon distribution incorporating the leadingtwist shadowing. This, however, does not mean that the latter is inherently incompatible with all approaches using UGDs. In principle, one could construct a framework in which the leadingtwist shadowing gluon distribution could be obtained from the UGDs provided they account for diffractive states and the appropriate resummation of collinear and small x logarithms would be performed in both cases.
From the above review it is clear that the study of dijets in UPC collision is of great importance for better understanding of the WW gluon distribution in the small x regime. One of the goals of this work is to investigate whether the present LHC kinematics can give a restriction of that distribution. This is done, by a direct calculation of the cross sections and nuclear modification factors for various observables taking into account existing information on WW gluon distribution.
The work is organized as follows. In Sect. 2 we describe the framework and ingredients necessary to compute the process of interest within the saturation regime. Next, in Sect. 3 we give detailed description of the unintegrated gluon distribution functions, kinematic cuts and actual implementation of the formalism. The results of numerical simulations are given in Sect. 4. Finally, we give a brief summary in Sect. 5.
2 Factorization formula for the dijet cross section in UPC
We are interested in the \(\gamma A\rightarrow 2\,\text {jet}+X\) process under the following assumptions: (i) The nucleus is probed at sufficiently small longitudinal momentum fraction \(x_{A}\) so that the saturation formalism applies. In reality, as we show later, \(x_A\) can reach values of order \(\sim 10^{3}\) (this would require \(p_T\) of the jet down to \(10 \; \mathrm{GeV}\)) so that we probably venture outside the applicability domain of the saturation formalism. (ii) We focus on the kinematic region where \(x_{A}<x_{\gamma }\), which implies that we look for a forward dijet configuration along the photon direction. A comment is in order here. Unlike for the pA collisions within the hybrid approach [23] this restriction is not absolutely necessary for \(\gamma A\) collisions. The only restriction is that \(x_A\) has to be small enough. Allowing for the full rapidity coverage will, however, unnecessarily make the \(x_A\) spectrum broader towards the larger values (see Fig. 3). Moreover, restricting to forward jets may allow in the future to combine the present results with the resolved component of the cross section. (iii) The average transverse momentum of the dijet system \(P_{T}=\left( p_{T1}+p_{T2}\right) /2\) is much bigger than the saturation scale \(Q_{s}\), \(P_{T}\gg Q_{s}\), and sets the hard scale of the process: \(\mu \sim P_{T}\). iv) The transverse imbalance of the dijets \(k_{T}=\left \mathbf {p}_{T1}+\mathbf {p}_{T2}\right \) can be anything allowed by the kinematics – this implies that the nonleading power corrections have to be taken into account.
Let us now justify the present approach. We shall show that Eq. (5) coincides with known results in two regimes: in the linear regime with large dijet imbalance \(k_{T}\sim P_{T}\gg Q_{s}\) and in the saturation regime with small imbalance \(P_{T}\gg k_{T}\sim Q_{s}\).
Let us start with the first regime. The formula (5) is superficially identical to the high energy factorization (HEF) formalism [25, 27, 28, 29, 30, 31, 32, 33] for heavy quark pair production in inclusive DIS. Namely, in the latter the same phase space and offshell matrix element is used. Consider now the limit \(k_{T}\sim P_{T}\gg Q_{s}\) adequate to the HEF regime of applicability. It can be shown that, for large \(k_T\), both \(xG_{1}\) and \(xG_{2}\) have the same asymptotics [13] (this is in fact true for many other TMD gluon distributions one can define at small x; see [34]). Therefore, in the linear HEF regime there is just one universal UGD (as long as there is no hard scale dependence). This justifies Eq. (5) in the regime \(k_{T}\sim P_{T}\gg Q_{s}\).
To summarize, the factorization formula (5) for the \(P_{T}\gg Q_{s}\) regime coincides with HEF when \(k_{T}\sim P_{T}\) and with the leading power limit of CGC when \(k_{T}\sim Q_{s}\). For \(k_{T}\) between those limiting values it provides a smooth interpolation given by the offshell matrix element and exact kinematics.
Let us note that, if the process under consideration were inclusive single jet production, Eq. (5) would change. In that case, the dipole gluon distribution \(xG_2\) would appear instead of \(xG_1\) [35, 36, 37, 38].
As already mentioned, there is an issue related to Eqs. (5) or (10). Namely for \(P_T\gg k_T\) the hard scale evolution is important and there should be some sort of the Sudakov form factor resumming large logarithms of the form \(\log \left( k_{T}/P_{T}\right) \). In the correlation regime of Eq. (10), it is rather simple – the Sudakov form factor multiplies the r.h.s. This was applied in [6] in the context of dihadron correlations at EIC. On more general ground, in the saturation formalism a comprehensive study was done in [39]. In our study, which, as discussed, goes beyond the leading power, we shall follow a different path. As we describe in the next section, the formalism of Eq. (5) is particularly convenient to implement in a simple Monte Carlo program which is capable of generating complete kinematics, with full final state four momenta, and with unit weight when necessary. Knowing the weights of particular events in a sample one can estimate the Sudakov effect by a suitable modification of the weights applying the Sudakov probability. From that point of view it is independent of the saturation and can be applied for any sample of events. In what follows we shall refer to this approach as the Sudakov resummation model. In short, the model takes a weight \(w_{i}\left( k_{T},P_{T}\right) \) for an event i (we suppress other weight variables for brevity) and modifies it by a surviving probability \(\mathcal {P}\left( k_{T},P_{T}\right) \) of the gluon with transverse momentum \(k_{T}\) which initiated the event. This probability is related to the Sudakov form factor. It is assumed that the gluons that do not survive at the scale \(k_{T}\) appear at the scale \(P_{T}\), so that the model does not change the total cross section. See [40] for details.
3 The framework for the unintegrated gluon distribution function
The complete direct components of the UPC cross section (3) have been implemented in the LxJet Monte Carlo program [41] based on foam algorithm [42]. It was previously used to calculate some jet observables within HEF (and beyond) in pp and pA collisions, including forwardcentral and forward–forward dijets [40, 43], threejet production [44], \(Z_{0}\)+jet production [45], UGD fits [46] and recently forward–forward dijets [47] using the approach of [24]. The Sudakov resummation model is an independent plugin and is applied on the top of the generated and stored events.
where \(\Sigma \left( x,k_{T}\right) \) is the accompanying singlet sea quark distribution and R has the interpretation of the target radius (more precisely, it appears from the integration of the impact parameter dependent gluon distribution assuming the uniform distribution of gluons). The parameter d, \(0<d\le 1\) is set to \(d=1\) for proton and can be varied for nucleus to study theoretical uncertainty. This equation accounts for DGLAP corrections, and a kinematic constraint along the BFKL ladder and running strong coupling constant. Due to all these (formally) subleading corrections this equation has been proven to be useful in modeling more exclusive final states. The actual initial condition \(\mathcal {F}_{0}\) has been fitted to the inclusive DIS HERA [52]. In what follows we shall name this set of UGD KS (Kutak–Sapeta). Also, the parameter R had to be fitted giving \(R\approx 2.4\,\mathrm {GeV}^{1}\). The set for a nucleus (actually the UGD per nucleon) is obtained by changing the proton R parameter according to the simple Woods–Saxon prescription \(R_{A}=A^{1/3}R\) where A is the mass number. The nonlinear term in (12) is enhanced then by \(dA^{1/3}\) resulting in much stronger saturation effects than in the proton case (for \(d=1\)). In [47] except \(d=1\) for nucleus, also the values \(d=\left\{ 0.5,0.75\right\} \) have been used to study the dependence of the results on the strength of the nonlinear term. In the present work, the set with \(d=0.5\), corresponding to weaker saturation will be used. As evident from Eq. (12) the saturation effects will become important whenever the nonlinear term will be of the same order as the linear term. Thus one can characterize the strength of the nonlinear effects by the parameter defined as the ratio of these two terms, which is proportional to the average gluon density per unit area, that is, a saturation scale. We choose \(d=0.5\) to ensure that the saturation scale in the case of scattering off \(\mathrm {Pb}\) is about 3 times larger than in the case of scattering off the proton. This choice is consistent with the ratio of average gluon density in \(\mathrm {Pb}\) and proton for \(x\sim 10^{3} \div 10^{4} \) and with account of the leadingtwist nuclear shadowing, cf. Fig. 100, of [5]. The saturation scale for Pb for this value of x is about \(Q^2_\mathrm{sat}\sim 2 \div 3\, \mathrm {GeV}^2\), to be compared with \(Q^2_\mathrm{sat} \le 1\, \mathrm {GeV}^2\) for the proton case.
The kinematic cuts used in the numerical calculations of the dijet cross section in the ultraperipheral \(\mathrm {Pb}\mathrm {Pb}\) collisions
CM energy  \(\sqrt{S}=5.1\,\mathrm {TeV}\) 

rapidity  \(0<y_{1},y_{2}<5\) 
transverse momentum  \(p_{T1},p_{T2}>p_{T0}\), \(p_{T0}=25,\,10,\,6\,\mathrm {GeV}\) 
Finally, the \(xG_{1}\) distributions for proton and lead were obtained from the KS gluon \(\mathcal {F}\) through Eq. (11), using the method presented in detail in [47]. We note, in particular, that in our procedure of calculating the WW KS distributions, \(xG_{1}\), we used the running coupling and xdependent transverse target area. The x dependence of S(x) was adjusted to ensure that the impact parameter dipole amplitude reaches unity as expected in the black disk limit. The resulting distributions are shown in Fig. 1. Let us note that even for \(x\sim 10^{2}\) the nonlinear effects are still present in that model. This is one of the differences with respect to the leadingtwist shadowing model and will be visible in the physical observables.
4 Numerical results
We start by determining the longitudinal fractions x of the photon and the gluon that can be effectively probed within our cuts. In Fig. 2 we show differential cross sections in the longitudinal fractions for various \(p_{T}\) cuts. We see that for \(p_{T0}=25\,\mathrm {GeV}\) the gluon longitudinal momentum fraction \(x_A\) is probed only slightly below \(10^{2}\), while for \(p_{T0}=10\,\mathrm {GeV}\) the process probes \(x_A\) easily around \(10^{3}\). With the smallest cut tested \(p_{T0}=6\,\mathrm {GeV}\) one can go below \(10^{3}\). We also show the distribution of the \(\gamma A\) CM energy, which reaches \(1.2\,\mathrm {TeV}\). If we did not restrict the calculation to the forward jets, the \(x_A\) fractions probed in the nucleus would have much broader distribution entering the relatively large x, which is an unwanted feature in the present approach. This is illustrated in Fig. 3 for two different \(p_T\) cuts. All the distributions discussed above are shown without the Sudakov effect, as its impact on these spectra is very weak.
In Fig. 4 we present the differential cross sections in the jet \(p_{T}\). In the present formalism the jets in general do not have equal \(p_{T}\) thus we order them, \(p_{T1}>p_{T2}>p_{T0}\), and show separate plots for leading and subleading jet spectra. For comparison we also calculate the same observable from the LO collinear factorization with nuclear PDFs implementing the leadingtwist nuclear shadowing [4]. In the LO collinear factorization both jets have equal \(p_{T}\). Interestingly their spectra are very close to the subleading jet spectrum of the present approach. The error bands are constructed by varying the hard scale by the factor of two with respect to the central value. The two bottom plots in Fig. 4 show the effect of the Sudakov resummation model. It has a significant effect on the subleading jet spectrum making its slope bigger. The error bands are bigger with resummation because the appearing hard scale can be varied as well and the results are sensitive to that scale.
Finally, for completeness, in Fig. 10 we show the suppression as a function of rapidity (these spectra are the same for leading and subleading jets). The curves slowly fall off with increase of the jet rapidity, as one could expect. After applying the Sudakov resummation, the spectra almost do not change, but the error bands become significantly bigger. For the \(25\,\mathrm {GeV}\) \(p_{T}\) cut the spectrum rises, but, again, this is the region that involves quite large x, for which the UGD grids are not trustworthy. It is interesting to note that close to central jet production, i.e. at rather large \(x_A\), there is an initial suppression of around 10% (we note, however, that there is a finite bin width of 0.25 unit, so this statement should be taken in the average sense). This is also clearly visible when we plot the nuclear modification ratio as a function of the longitudinal fraction \(x_A\) probed in the nucleus (Fig. 10 bottom). For definiteness, we plot the result for the \(p_T\) cut of \(10\,\mathrm {GeV}\). We compare the tendency of the saturation formalism used in the present work with the leadingtwist shadowing. The calculation with saturation gives a suppression about 10% over the wide range of x: from \(10^{3}\) up to \(10^{2}\). For larger x (not shown) there are large fluctuations as we approach the edge of the phase space, but the ratio seems to be closer to unity. For the leadingtwist shadowing the ratio approaches the unity much faster (around \(10^{2}\)).
5 Summary
In this work we have investigated potential saturation effects in dijet production in ultraperipheral heavy ion collisions at the LHC, for the \(5.1\,\mathrm {TeV}\) CM energy per nucleon. The quasireal photons are unique probes of the nucleus, as within the saturation formalism the Weizsäcker–Williams (WW) unintegrated gluon distribution is directly involved in the dijet production process. The WW distribution has an interpretation of the gluon number density, unlike other similar quantities that appear at small x.

The formalism has a form of \(k_{T}\)factorization which involves a convolution of unintegrated gluon distribution and offshell matrix element. On phenomenology side, the usage of unintegrated gluon distributions is more convenient than using correlators of Wilson lines. Gluon distributions can be more easily supplemented with additional effects.

It involves full momentum conservation for produced final states, taking into account the transverse momentum of the incoming gluons. This allows for a construction of Monte Carlo generators based on the formalism.

Formalism is simple compared to the full CGC calculation, yet catching its essential features. When using the McLerran–Venugopalan model to obtain the WW gluon distribution, the present formulation should give identical results to CGC for \(\Delta \phi \sim 0\) and \(\Delta \phi \sim \pi \) for large \(p_T\) jets. They could differ in the intermediate region, but taking into account general properties of \(\Delta \phi \) distributions they cannot differ too much. The \(p_T\) spectra should also be similar for large \(p_T\).
The results can be summarized as follows. The suppression due to the saturation effects is around 20% at most, for the smallest \(p_{T}\) cutoff of the dijet transverse momenta. This is because the probed longitudinal fractions x are not very small. In addition, for the \(p_{T}\) spectra, the saturation effects and the leadingtwist shadowing look qualitatively similar. Thus, it would be rather difficult to distinguish between predictions of considered approaches. Nevertheless, it would be promising to focus on two main differences. The first difference is the slope of the \(p_T\) spectra for \(R_{\gamma A}\) in the small \(p_T\) region (Fig. 7). The \(p_T\) spectrum for the leading jet is much steeper for the saturation formalism. The second difference is the slope of the x spectra for \(R_{\gamma A}\), i.e. how fast the nuclear effects vanish with increase of x (see Fig. 10 bottom). For the leadingtwist nuclear shadowing this happens around \(10^{2}\), while for the saturation formalism with the WW gluon distribution used here it is a bigger value, very close to the edge of the phase space, so that we were unable to determine the exact value.
The question whether one should combine both mechanisms (and whether this is possible) remains open. In general, the predicted nuclear effects – regardless of the source – seem to be big enough to be seen in the data. Finally we note that the effect discussed strongly depends on the centrality of the \(\gamma A\) collisions. So the study of the imbalance of jets as a function of centrality appears to be a promising strategy for exploring the effects discussed in this paper.
Notes
Acknowledgements
The authors are thankful to A. van Hameren, C. Marquet, E. Petreska for fruitful discussions.
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