Inclusive dijet photoproduction in ultraperipheral heavy-ion collisions at the LHC in next-to-leading order QCD

We compute the cross section of inclusive dijet photoproduction in ultraperipheral Pb-Pb collisions at the LHC using next-to-leading order perturbative QCD. We demonstrate that our theoretical calculations provide a good description of various kinematic distributions measured by the ATLAS collaboration. We find that the calculated dijet photoproduction cross section is sensitive to nuclear modifications of parton distribution functions (PDFs) at the level of 10 to 20%. Hence, this process can be used to reduce uncertainties in the determination of these nuclear PDFs, whose current magnitude is comparable to the size of the calculated nuclear modifications of the dijet photoproduction cross section.


Introduction
Ultraperipheral collisions (UPCs) of relativistic ions correspond to large impact parameters between the nuclei exceeding the sum of their radii, so that short-range strong interactions between the ions are suppressed and reactions proceed rather via the emission of quasireal photons by the colliding ions. Thus, UPCs allow one to study photon-photon and photon-hadron (proton, nucleus) interactions at high energies [1]. During the last decade, UPCs have become an active field of research, driven by experimental results obtained at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) (for a recent experimental review see, e.g., [2]). Notable examples of various UPC processes and their analyses include the two-photon production of dilepton pairs [3,4]; light-by-light scattering γγ → γγ and searches for potential physics beyond the Standard Model [5][6][7]; an electromagnetic double-scattering contribution to dimuon pair production in photonphoton scattering [8]; exclusive photoproduction of charmonia in proton-proton [9,10], proton-nucleus [11] and nucleus-nucleus [12][13][14][15] UPCs and of bottomonia in proton-proton [16] and proton-nucleus UPCs [17]; new constraints on the small-x gluon distribution in the proton [18,19] and heavy nuclei [20,21] and the dynamics of strong interactions at high energies in the color dipole framework [22][23][24]; and exclusive photoproduction of ρ mesons on nuclei [25][26][27][28] as well as tests of models of nuclear shadowing [29,30].
Focusing on UPC studies of nuclear structure in QCD at the LHC, coherent J/ψ photoproduction in Pb-Pb UPCs at √ s N N = 2.76 TeV [12][13][14][15] revealed a significant nuclear suppression of the measured rapidity distributions. In the framework of the leading logarithmic approximation of perturbative QCD [31], it can be interpreted as evidence of large nuclear gluon shadowing, R g = f g/A (x, µ 2 )/[Af g/N (x, µ 2 )] ≈ 0.6 at x = 10 −3 and µ 2 = 3 GeV 2 (f g/A and f g/N are gluon densities in Pb and the proton, respectively). This value of R g agrees with predictions of the leading twist nuclear shadowing model [32], which are characterized by small theoretical uncertainties in this kinematic region. It is also broadly consistent with the EPS09 [33], nCTEQ15 [34], and EPPS16 [35] nuclear parton distribution functions (nPDFs), which however have significant uncertainties in this kinematic regime. Note that in the collinear factorization framework, next-to-leading order (NLO) perturbative QCD corrections to the cross section of J/ψ photoproduction are large [36,37] and the relation between the gluon parton distribution function (PDF) and the gluon generalized parton distribution (GPD) is model-dependent, which makes it challenging to interpret the UPC data on J/ψ photoproduction on nuclei in terms of the NLO gluon nPDF.
The program of UPC measurements continues with Run 2 at the LHC, where besides photoproduction of vector mesons, inclusive dijet photoproduction in Pb-Pb UPCs AA → A + 2jets + X has also recently been measured by the ATLAS collaboration [38] (for leading-order QCD predictions for rates of this process, see [39]). The cross section of this process is sensitive to quark and gluon nPDFs f j/A (x, µ 2 ) in a wide range of the momentum fraction x and the resolution scale µ > O(20) GeV, where one still expects sizable nuclear modifications of the PDFs. In addition, imposing the requirement that the target nucleus stays intact, one can study diffractive dijet photoproduction in UPCs AA → A + 2jets + X + A. Studies of this process may shed some light on the mechanism of QCD factorization breaking in diffractive photoproduction and, for the first time, give access to nuclear diffractive PDFs [40]. While further progress in constraining nPDFs will benefit from studies of high-energy hard processes with nuclei in proton-nucleus (pA) scattering at the LHC [41] and lepton-nucleus (eA) scattering at a future Electron-Ion Collider (EIC) [42] and LHeC [43], UPCs at the LHC present an important and complimentary method of obtaining new constraints on nPDFs in a wide kinematic range already now.
In this work, we make predictions for the cross section of inclusive dijet photoproduction in Pb-Pb UPCs at the LHC using NLO perturbative QCD [44] and nCTEQ15 nPDFs. We show that our approach provides a good description of the shape of various cross section distributions measured by the ATLAS collaboration [38]. While we also semi-quantitatively reproduce the cross section normalization, one has to keep in mind that the preliminary ATLAS data has not yet been corrected for the detector response. Our analysis also shows that the dijet photoproduction cross section in the considered kinematics is sensitive to nuclear modifications of the PDFs. As a function of the momentum fraction x A , the ratio of the cross sections calculated with nPDFs and in the impulse approximation behaves similarly to R g for a given µ and deviates from unity by 10 − 20% for the central nCTEQ15 fit. The calculations using EPPS16 nPDFs and predictions of the leading twist nuclear shadowing model give similar results. This suggests that inclusive dijet photoproduction on nuclei can be used to reduce uncertainties in the determination of nPDFs, which are currently significant and comparable in size to the magnitude of the calculated nuclear modifications of the dijet photoproduction cross section.
The remainder of this paper is structured as follows. In Sec. 2, we outline the formalism of dijet photoproduction in UPCs using NLO perturbative QCD. We present and discuss our results for the LHC in Sec. 3 and draw conclusions in Sec. 4.

Photoproduction of dijets in UPCs in NLO perturbative QCD
Typical leading-order (LO) Feynman diagrams for dijet photoproduction in UPCs of nuclei A and B are shown in Fig. 1, where the graphs (a) and (b) correspond to the direct and resolved photon contributions, respectively. Note that beyond LO, the separation of the direct and resolved photon contributions depends on the factorization scheme and scale.
In the collinear factorization framework, the cross section of the UPC process AB → A + 2jets + X is given by [44] dσ(AB → A + 2jets + X) = a,b ymax y min where a, b are parton flavors; f γ/A (y) is the flux of equivalent photons emitted by ion A, which depends on the photon light-cone momentum fraction y; f a/γ (x γ , µ 2 ) is the PDF of the photon, which depends on the momentum fraction x γ and the factorization scale µ; f b/B (x A , µ 2 ) is the nuclear PDF with x A being the corresponding parton momentum fraction; and dσ(ab → jets) is the elementary cross section for production of two-and three-parton final states emerging as jets in hard scattering of partons a and b. The sum over a involves quarks and gluons for the resolved photon contribution and the photon for the direct photon contribution dominating at x γ ≈ 1. At LO, the direct photon contribution has support exactly only at x γ = 1. The integration limits are determined by the rapidities and transverse momenta of the produced jets, see Sec. 3. Note that Eq. (2.1) is based on the NLO perturbative QCD formalism for jet photoproduction in lepton-proton scattering developed in Refs. [44][45][46][47], which successfully described HERA ep data on dijet photoproduction [48].
In our analysis, we used the following input for Eq. (2.1): the GRV HO [49] photon PDFs f a/γ (x γ , µ 2 ), which we transformed from the DIS γ to the MS factorization scheme; [34]; and the standard expression for the photon flux f γ/A (y) produced by a relativistic point-like charge Z where α e.m. is the fine-structure constant; K 0,1 are modified Bessel functions of the second kind; ζ = ym p b min with m p being the proton mass and b min the minimal distance between two nuclei. For Pb-Pb UPCs, Eq. (2.2) with b min = 14.2 fm reproduces very well the photon flux calculated taking into account the nuclear form factor and the suppression of strong interactions at impact parameters b < b min , see the discussion in [50].
3 Predictions for dijet photoproduction in Pb-Pb collisions at the LHC Using the formalism outlined in Sec. 2, we calculate the cross section of inclusive dijet photoproduction in Pb-Pb UPCs at √ s N N = 5.02 TeV in the kinematics of the ATLAS measurement at the LHC [38]. The ATLAS analysis was performed using the following conditions and selection criteria: • the anti-k T algorithm with the jet radius R = 0.4; • the leading jet has p T,1 > 20 GeV, while the other jets have a different cut on p T,i =1 > 15 GeV as required [51], which corresponds to 35 < H T < 400 GeV, where H T = i p T,i ; • all jets have rapidities |η i | < 4.4; • the combined mass of all reconstructed jets is 35 < m jets < 400 GeV; • the parton momentum fraction on the photon side z γ = yx γ , 10 −4 < z γ < 0.05; • the parton momentum fraction on the nucleus side x A , 5 × 10 −4 < x A < 1.
The ATLAS results are presented as distributions in terms of the total jet transverse momentum H T = i p T,i and the photon z γ and nucleus x A light-cone momentum fractions where In Eqs. (3.2), the index i runs over all measured jets; E i and p i denote the jet energy and momentum, respectively. Note that at LO, the kinematics of 2 → 2 parton scattering and the momentum fractions z γ and x A can be exactly reconstructed from the dijet measurement. At NLO, Eqs.   [38]. In each bin, our predictions are obtained using the central fit of nCTEQ15 nPDFs [34]. The shaded bands quantify the uncertainty of our results ∆σ due to the uncertainty of nCTEQ15 nPDFs. It is calculated by adding in quadrature the individual uncertainties corresponding to each of where σ(f k ) is the cross section calculated using the f k nCTEQ15 error nPDFs. A comparison of our results shown in Figs. 2, 3, 4, and 5 with Figs. 12-15 of Ref. [38] demonstrates that our calculations describe well the shapes of the corresponding distributions. In addition, while we also semi-quantitatively reproduce the normalization of the distributions (judging by eye on a logarithmic scale), a direct comparison should be taken  with a grain of salt because the preliminary ATLAS data has not been corrected (unfolded) for the detector response.
To quantify the magnitude of nuclear modifications of nPDFs, which affect the dijet photoproduction cross section in the considered kinematics, we focus on the x A distribution integrated over H T and z γ . Our results are shown in Fig. 6. The top panel presents separately the resolved (green, dot-dashed) and the direct (blue, dashed) photon contributions to the cross section as well as their sum (red, solid). As can be expected, because of the connection between x A and z γ , see Eq. (3.1), the resolved photon contribution dominates for x A > 0.01. We find that for small x A < 0.01, the two contributions are comparable with x A 0.0002 < z γ < 0.0003 0.0003 < z γ < 0.0006 (×10 -2 ) 0.0006 < z γ < 0.0012 (×10 -4 ) 0.0012 < z γ < 0.0022 (×10 -6 ) 0.0022 < z γ < 0.0042 (×10 -8 ) 0.0042 < z γ < 0.0077 (×10 -10 ) 0.0077 < z γ < 0.0144 (×10 -12 ) 0.0144 < z γ < 0.0269 (×10 -14 ) Figure 5. NLO QCD predictions for the cross section of dijet photoproduction in Pb-Pb UPCs at √ s N N = 5.02 TeV in the ATLAS kinematics as a function of x A for different bins of z γ . the direct contribution being somewhat larger. While this behavior is qualitatively similar to the results of the LO analysis in the framework of PYTHIA 8 with EPPS16 nPDFs [52], the relative contribution of the resolved photon term is larger at NLO, but this statement depends of course on the choice of the photon factorization scheme and scale.
The middle panel of Fig. 6 presents the ratio of the cross section calculated using nCTEQ15 nPDFs in lead to the one calculated in the impulse approximation (IA), where nuclear PDFs are assumed not to include any nuclear modifications and are given by the weighted sum of free proton and neutron PDFs, f IA b/A = Zf b/p + (A − Z)f b/n . One can see from this panel that the cross section ratio as a function of x A behaves similarly to the ratio R g = f g/A (x, µ 2 )/[Af g/N (x, µ 2 )] of the nuclear and nucleon gluon distributions.
It dips below unity for x A < 0.01 due to nuclear shadowing and then becomes enhanced around x A = 0.1 due to the assumed gluon antishadowing. For x A > 0.3, the cross section ratio shows again a suppression due to the EMC effect encoded in the nPDFs. Note that this behavior is similar to the one observed in the case of dijet photoproduction in the kinematics of an Electron-Ion Collider (EIC) [53]. In spite of large values of the resolution scale probed in the considered kinematics, µ > O(20) GeV, one can see that one is still sensitive to nuclear modifications of the PDFs at the 10 − 20% level for the central value of our predictions. One should also note that the uncertainty due to nPDFs, which is given by the shaded band, is significant and comparable to the size of the discussed nuclear modifications. This can be viewed as an opportunity to reduce uncertainties of nPDFs using data on cross section of inclusive dijet photoproduction in nuclei in global QCD fits of nPDFs.
Finally, the bottom panel of Fig. 6 presents the ratio of the dijet cross section calculated using nCTEQ15 nPDFs to the one calculated with the central value of EPPS16 nPDFs. The shaded band quantifies the uncertainty of the nCTEQ15 fit. One can see from the panel that the two parameterizations of nPDFs give similar predictions, which differ by at most 5% for all but one values of x A . We have also explicitly checked that the use of nPDFs calculated in the model of leading twist nuclear shadowing [32] gives similarly close predictions for the dijet photoproduction cross section.
In our calculations, following the standard prescription for setting the hard scale in QCD calculations, we used µ = 2E T,1 in Eq. (2.1). In detail, we performed calculations using µ = (E T,1 /4, E T,1 /2, E T,1 , 2 E T,1 , 4 E T,1 ) both at NLO and LO and found that (i) the integrated cross section of inclusive dijet photoproduction at NLO as a function of µ is approximately constant is the vicinity of µ = 2E T,1 , (ii) while the NLO cross section slightly increases with an increase of µ up to 2E T,1 and then starts to decrease again, the LO cross section steeply decreases monotonically, and (iii) the values of the two cross sections are close around µ = 2E T,1 . Therefore, µ = 2E T,1 in Eq. (2.1) corresponds to the choice, which is most numerically stable against higher-order corrections.
In this work, we used the framework of collinear factorization and NLO perturbative QCD to examine the sensitivity of the dijet photoproduction cross section to nuclear modifications of PDFs. Alternatively, one can use this process to look for signs of the BFKL and gluon saturation dynamics in the high-energy (k T ) factorization approach [54].

Conclusions
In this work, we calculated the cross section of inclusive dijet photoproduction in Pb-Pb UPCs at the LHC using NLO perturbative QCD and nCTEQ15 nPDFs. We showed that our approach provides a good description of various cross section distributions measured by the ATLAS collaboration. We found that the calculated dijet photoproduction cross section is sensitive to nuclear modifications of the PDFs. In particular, as a function of the nucleus momentum fraction x A , the ratio of the cross sections calculated with nPDFs and in the impulse approximation behaves similarly to R g for given µ and deviates from unity by 10 − 20% for the central nCTEQ15 fit. The calculations using EPPS16 nPDFs and predictions of the leading twist nuclear shadowing model give similar results. Therefore, inclusive dijet photoproduction on nuclei has the potential to reduce uncertainties in determination of nPDFs, which are comparable to the magnitude of the calculated nuclear modifications of the dijet photoproduction cross section. Our present analysis is a step in this direction.