Exclusive dielectron production in ultraperipheral Pb+Pb collisions at $\sqrt{s_{_\text{NN}}} = 5.02$ TeV with ATLAS

Exclusive production of dielectron pairs, $\gamma\gamma\rightarrow e^+e^-$, is studied using $\mathcal{L}_\mathrm{int}=1.72\; \mathrm{nb^{-1}}$ of data from ultraperipheral collisions of lead nuclei at $\sqrt{s_{_{\text{NN}}}} = 5.02$ TeV recorded by the ATLAS detector at the LHC. The process of interest proceeds via photon-photon interaction in the strong electromagnetic fields of relativistic lead nuclei. Dielectron production is measured in the fiducial region defined by following requirements: electron transverse momentum, $p_{\textrm{T}}^{e}>2.5$ GeV, absolute electron pseudorapidity, $|\eta^{e}|<2.5$, invariant mass of the dielectron system, $m_{ee}>5$ GeV, and transverse momentum of the dielecton pair, $p_{\textrm{T}}^{ee}<2$ GeV. Differential cross-sections are measured as a function of $m_{ee}$, average $p_{\textrm{T}}^{e}$, absolute rapidity of the dielectron system, $|y_{ee}|$, and scattering angle in the dielectron rest frame, $|\cos\theta^*|$ in the inclusive sample, and also under the requirement of no activity in the forward direction. The total integrated fiducial cross-section is measured to be $215 \pm 1 \text{(stat.)} ^{+23}_{-20} \text{(syst.)} \pm 4 \text{(lumi.)}\; \mu$b. Within experimental uncertainties the measured integrated cross-section is in good agreement with the QED predictions from the Monte Carlo programs Starlight and SuperChic, confirming the broad features of the initial photon fluxes. The differential cross-sections show systematic differences with these predictions which are more pronounced at high $|y_{ee}|$ and $|\cos\theta^*|$ values.

→ + − process or via photonuclear reactions, occurring when an additional photon is exchanged. [6,7]. A Feynman diagram of the leading-order → + − reaction is shown in Figure 1. Even for large invariant masses, the relatively large cross-section associated with this process allows precise differential measurements to be made. Thus, this process is a particularly effective tool for studying the modelling of photon fluxes and elementary production cross-sections, as well as for studying the effects of nuclear break-up induced by additional photon exchanges, whose probability is strongly correlated with the internuclear impact parameter [2]. Nuclear break-up gives rise to forward neutron production, and the fraction of events with such activity is larger at smaller impact parameters.
Exclusive dilepton production, via both electron-pair and muon-pair final states, has been measured by ATLAS and CMS in collisions at √ = 7 TeV [8-10] and √ = 13 TeV [11][12][13]. The ALICE Collaboration has measured exclusive production of electron pairs in Pb+Pb collisions at √ NN = 2.76 TeV [14] over a limited kinematic range. The STAR and PHENIX experiments at RHIC have measured exclusive dileptons at lower invariant masses at √ NN = 200 GeV for both Au+Au and U+U collisions [15][16][17]. With the higher centre-of-mass energy of √ NN = 5.02 TeV, ATLAS has performed differential measurements of → + − production in UPC Pb+Pb collisions [18]. Both STAR and ATLAS have observed substantial broadening of angular distributions for exclusive dileptons from interactions in events where the nuclei overlapped and interacted hadronically [15,19]. Finally, CMS has observed that angular correlations in exclusive dimuon events are broadened significantly as a function of the impact parameter [20], as inferred by the amount of forward neutron production.
Exclusive dielectron production is an important reference process for other measurements. References [21] and [22] propose using it in the context of → + − production, in order to reduce the impact of correlated systematic uncertainties for the measurement of the -lepton anomalous magnetic moment. It is also an important background for light-by-light scattering, which proceeds via loop diagrams and thus has a much lower cross-section. This has been assessed in several publications on light-by-light scattering by ATLAS [23][24][25] and CMS [26]. This paper presents a measurement of the exclusive production of dielectrons with the ATLAS detector at the LHC. It uses Pb+Pb data collected in 2018, which have an integrated luminosity three times larger than the sample used in the ATLAS dimuon measurement [18]. Dielectron production is measured in the fiducial region defined by the following requirements: electron transverse momentum T > 2.5 GeV, electron pseudorapidity | | < 2.5, dielectron invariant mass > 5 GeV, and dielectron transverse momentum T < 2 GeV. Compared to the dimuon measurement discussed in Ref. [18], this fiducial region has wider coverage in lepton T and dilepton invariant mass, with the minimum values lowered by 1.5 GeV and 5 GeV, respectively. The backgrounds originating from single-dissociative processes, Υ( ) production, and exclusive -lepton pair production, → + − , are estimated and subtracted. Differential cross-sections are measured as a function of , average electron transverse momentum T , absolute dielectron rapidity | |, and scattering angle in the dielectron rest frame, | cos * |. The cross-sections are extracted both inclusively in forward neutron activity and exclusively in → + − events without activity in the forward direction. The latter is a unique feature of this paper. 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the -axis along the beam pipe. The -axis points from the IP to the centre of the LHC ring, and the -axis points upwards. Cylindrical coordinates ( , ) are used in the transverse plane, being the azimuthal angle around the -axis. The pseudorapidity is defined in terms of the polar angle as = − ln tan( /2). Angular distance is measured in units of Δ ≡ √︁ (Δ ) 2 + (Δ ) 2 . The photon (electron) transverse energy is T = /cosh( ), where is its energy.
ZDC detects individual neutrons originating from the incoming nuclei. The ZDC calibration is performed in each set of four modules using photonuclear processes that deposit one or more neutrons on one side, and a single neutron, carrying the full per-nucleon beam energy, on the other. Time-dependent weights are determined for each module in short time intervals to minimise the variance around the nominal per-nucleon beam energy. Energy resolutions achieved are typically around Δ / ≈ 16%.
The ATLAS trigger system [29, 30] consists of a first-level (L1) trigger implemented using a combination of dedicated electronics and programmable logic, and a software-based high-level trigger (HLT). An extensive software suite [31] is used in the reconstruction and analysis of real and simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.

Data and Monte Carlo simulation samples
The data used in this measurement are from Pb+Pb collisions with a centre-of-mass energy of √ NN = 5.02 TeV, recorded in 2018 with the ATLAS detector at the LHC. The full data set corresponds to an integrated luminosity of 1.72 nb −1 . Only high-quality data [32] with all detectors operating normally are analysed.
Monte Carlo (MC) simulated events for the → + − signal process were generated at leading order (LO) using S v3.13 [33]. In this approach, the cross-section is computed by convolving the Pb+Pb photon flux with the LO calculation of the elementary → + − process. The photon spectrum is calculated in impact parameter space by integrating the photon number density over all impact parameters while assuming that the beam projectiles do not interact hadronically. This is done by utilising a simple Glauber model [34] which provides an impact-parameter-dependent probability of inelastic processes. S also requires that the dilepton pairs are not formed within either nucleus. Several signal samples were produced for exclusive intervals within the range 4.5 < < 200 GeV. An alternative sample for the signal → + − process uses the S C v3.05 [35] program. The difference between the nominal and alternative signal prediction is mainly in the implementation of the initial photon flux. The S generator relies on flux from point-like sources restricted to impact parameters larger than nuclear radius ( > ), while S C generator implements flux calculations down to = 0 and takes the nuclear form factor into account.
Incoherent emission of photons participating in → + − process contribute to background in the dielectron production sample. In such a process either one (single dissociation) or both (double dissociation) nuclei interact inelastically and dissociate. Simulation of this process is not available in any MC generator, however a similar process in collisions is modelled using S C v4.0 (SC4) [37]. The sample generated with SC4 was interfaced to P 8 for showering and hadronisation. A data-driven approach, discussed in detail in Section 5, is used in the analysis to utilise this sample in Pb+Pb collisions.
Apart from the alternative signal sample, all generated events are processed with a detector simulation [38] based on G 4 [39] and are reconstructed with the standard ATLAS reconstruction software [31]. The alternative signal sample is used for comparisons with the measured differential cross-sections discussed in Section 8.

Signal selection and detector corrections
Candidate dielectron events were recorded using a dedicated trigger for events with moderate activity in the calorimeter but little additional activity in the entire detector. A logical OR of two L1 trigger conditions was required: (1) at least one EM cluster with T > 1 GeV in coincidence with a total T of 4-200 GeV registered in the calorimeter, or (2) at least two EM clusters with T > 1 GeV and a total T below 50 GeV registered in the calorimeter. At the HLT, the total T on each side of the FCal detector was required to be below 3 GeV. Additionally, a veto condition on the maximum activity in the Pixel detector, hereafter referred to as the Pixel-veto, had to be satisfied at the HLT. The number of hits was required to be at most 15 to be compatible with low-multiplicity UPC events.
Electrons are reconstructed from EM clusters in the calorimeter and tracking information provided by the ID [40]. Selection requirements are applied to remove EM clusters with a large amount of energy from poorly functioning calorimeter cells, and a timing requirement is made to reject out-of-time candidates. An energy calibration specifically optimised for electrons and photons [40] is applied to the candidates to account for upstream energy loss and both lateral and longitudinal shower leakage. The calibration is derived for nominal collisions with dedicated factors applied to account for the much lower contribution from multiple Pb+Pb collisions in the same bunch crossing.
The electron identification in this analysis is based on a 'loose' cut-based working point [40] which is defined using selections on the shower-shape and tracking variables. Only electrons with T > 2.5 GeV and | | < 2.47, excluding the calorimeter transition region 1.37 < | | < 1.52, are considered. The minimum T requirement is driven by the electron reconstruction efficiency, which drops below 20% for T values below this threshold. Preselected events are required to have exactly two opposite-charge electrons satisfying the above selection criteria, with a dielectron invariant mass, , greater than 5 GeV. To suppress non-exclusive backgrounds, only two charged-particle tracks [41, 42] each with T > 100 MeV, | | < 2.5, at least seven hits in the Pixel and SCT detectors in total and at most two silicon sensors without a hit, and associated with the dielectron are allowed. To reject non-collision backgrounds such as cosmic-ray muons, the event must not have a track in the MS. Finally, the total T of the dielectron, T , is required to be less than 2 GeV. Low T values are a key feature of the purely EM process, which involves initial-state photons with very low T .
Each of the events satisfying the → + − criteria can be further classified into one of three categories based on the observed activity in the ZDC detector: 1) no neutron is registered in either ZDC ('0n0n'), 2) one or more forward neutrons registered in one ZDC and none in the other ('Xn0n'), and 3) one or more forward neutrons detected in both ZDC arms ('XnXn'). The observed fractions of events falling into these categories are: 0n0n = (62.9 ± 0.3)%, Xn0n = (29.7 ± 0.3)%, and XnXn = (7.4 ± 0.2)%. Due to the relatively large instantaneous luminosity of Pb+Pb collisions, which peaked around 7 × 10 27 cm −2 s −1 , additional neutrons might be generated per bunch crossing by single and mutual dissociation processes and detected in one or both arms of the ZDC, but they are not associated with the exclusive dielectron process. This leads to an outflow of events from the 0n0n and Xn0n categories to both the Xn0n and XnXn categories. This effect is accounted for using the method established in Ref. [18]. A matrix equation with two fundamental parameters representing probabilities for single and mutual dissociation is built. The corrected fractions are measured in four bins of , with boundaries at 5, 10, 20, and 40 GeV, and three bins of | |, with boundaries at 0, 0.8, 1.6, and 2.4, and also in the sample integrated over and | |. On average, in the 0n0n category, they are about 13% larger than the observed fractions. Figure 2 shows the fractions of events in the 0n0n category as a function of in three bins of | |, corrected for the presence of additional neutrons. These fractions tend to drop with increasing mass, and are in general larger for higher | | values. For the rapidity range of | | < 0.8, which has the largest number of events, the 0n0n values drop from about 78% in the lowest mass bin to about 57% in the highest mass bin. The systematic uncertainties in the fractions of events in the 0n0n category originate from several sources: uncertainties in the exclusive single and double EM dissociation cross-sections measured by the ALICE Collaboration [43], and their extrapolation from √ NN = 2.76 TeV to 5.02 TeV as evaluated in Ref.
[18]; the uncertainty in the dissociative background contribution as discussed in Section 5; and the uncertainty in the ZDC efficiency.  The efficiency of the primary physics trigger ( T ) is determined as T = L1 · PixVeto · FCal , where L1 is the efficiency of the L1 EM trigger to register the moderate calorimeter activity characteristic of the signal process, PixVeto is the efficiency of the trigger to reject events with large numbers of Pixel detector hits, and FCal is the efficiency of the FCal selection to reject events with large energy depositions on either side. Individual efficiencies are evaluated in a sample of → + − events collected with a set of dedicated supporting triggers that do not use the condition under study in the primary physics trigger to reject any events. The L1 value rises with the sum of the transverse energies of the two electron clusters and reaches 100% for Σ T > 8 GeV. The Pixel-veto efficiency and its uncertainties are measured as a function of the dielectron rapidity in the dedicated dielectron sample collected by the supporting trigger requiring at least two tracks with T above 1 GeV and without veto requirement on the number of Pixel hits. The efficiency is evaluated as a ratio of events passing the Pixel-veto requirement to all selected events. It is just over 80% for | | ∼ 0 and falls to about 50% for | | > 2. The dependence on | | originates from the growing number of Pixel-detector layers in the forward direction that a dielectron pair has to pass through. The average number of Pixel hits for the signal process is 10 in the central region, and it increases to 14 in the forward direction. The systematic uncertainties are evaluated by repeating the efficiency measurement on the dielectron sample selected by varying requirements on T and , and acoplanarity ( = 1 − |Δ |/ , where Δ is the azimuthal angle between the two electrons). They are at the level of 0.5%, while the statistical uncertainty ranges between 1 and 2%. Finally, the FCal veto efficiency is measured to be (99.1 ± 0.6)%, and it is constant for the entire range in Σ T . The total uncertainty in the trigger efficiency is determined by increasing and decreasing all of the individual components by their respective total uncertainties. They amount to about 3%-4% for the primary calorimeter pair trigger, driven mainly by the limited number of dielectron events collected by an independent ZDC-based trigger used to measure L1 , and less than a percent for the other contributions.
The total electron efficiency is the product of the electron reconstruction efficiency and the 'loose' electron identification efficiency [40]. This is determined in data using a sample of events triggered by the presence of EM clusters, limited total T , and a maximal number of Pixel hits, on which a tag-and-probe procedure is performed. The tag is a well-reconstructed, high-purity electron candidate with T > 2.5 GeV, and the probe is an opposite-charge track built from at least three hits in the Pixel detector (referred to as a 'Pixel-track'). Pixel tracks are required to have T > 50 MeV. The invariant mass of the tag-and-probe system must exceed 5 GeV and the acoplanarity has to be less than 0.1. The extracted mass distribution is found to agree well with a reconstructed sample of S events. The reconstruction efficiency is defined as the fraction of probes which are reconstructed electrons, while the identification efficiency is the fraction of reconstructed electrons which are identified as 'loose' electrons. The reconstruction efficiency has large variations with both T and Pixel-track , and ranges from about 30% at T = 2.5 GeV to 95% above 15 GeV. The identification efficiency is found to vary more weakly with Pixel-track , ranging between 80% and 90%. Then, the overall reconstruction scale factors are extracted as the ratio of efficiencies measured in data and MC simulation. They vary between 0.9 and 1.2, with the largest deviations from unity being in the forward direction for Pixel-track | | > 1.1. Systematic uncertainties in the scale factors are evaluated using tighter selection criteria for the tag and probe candidates, as well as reducing a potential contribution from background processes by limiting the measurement to the 0n0n category or to a narrow acoplanarity region, < 0.01. In particular the requirement on maximum reduces the contribution from the exclusive + − production to 0.25% level. The total systematic uncertainty is at the level of 5% for central Pixel-tracks with | | < 1, and grows to 10% in the forward direction. In the forward region, the statistical and systematic uncertainties are of similar size.

Background contributions
There are three primary sources of background considered in this analysis, presented in order of decreasing contributions: dissociative → + − production; Υ-meson production; and exclusive -lepton pair ( + − ) production.
The largest background originates from → + − production with nuclear dissociation. In this process one (or both) of the initial photons originates from the substructure of the nucleon, rather than from the exterior EM field of the nuclei as a whole. The photon interaction that produces the + − pair is thus accompanied by the dissociation of the emitting nucleus, whose remnants are produced in the forward direction and are typically captured by the ZDC detector.
The contribution from dissociative events is estimated using a template-fitting approach applied to the acoplanarity distribution. The signal template is simulated with S + P 8 and it is peaked at ≈ 0, with some contribution in the tail originating from events with FSR. The background template shape is taken from the single-dissociative events simulated in collisions with S C v4.0 interfaced with P 8. These events have a much wider distribution than the signal. The acoplanarity shape is strongly correlated with the transverse momentum of the system, which is driven by the transverse momenta of the initial photons. For the photons emitted coherently from the nucleus, the transverse momentum is of order ℏ / ≈ 30 MeV, while typical T scale for dissociative events is of order GeV. In the case of dilepton production in collisions, the typical initial T scale is about 200 MeV. The convolution of photon fluxes originating from either proton or ion with photons emitted from nucleon substructure is always dominated by the harder spectrum of the latter. Therefore, the shape of the acoplanarity distribution for dissociative dielectron production in Pb+Pb collisions can be described by the simulation of this process in collisions. The fit to the data is performed in the same intervals of and | | as in the study of the fractions of events in the 0n0n, Xn0n and XnXn categories. In each bin, a binned maximum-likelihood fitting procedure is performed separately in three ZDC categories. The normalisation of the relative background contribution, bkg , is taken to be a free parameter of the fit. The signal fraction is thus (1 − bkg ). For the inclusive sample, bkg is a weighted sum of the results for the 0n0n, Xn0n, and XnXn categories. The bkg fraction accounts for contributions from dissociative production and exclusive + − production, → + − . The latter may contribute to the electron background, especially when both -leptons decay in the electron channel. The + − contribution is estimated using a dedicated MC sample from S . The resulting background fraction of → + − events in the full data sample amounts to 0.1%. It is found that the shape of the distribution for the exclusive + − events is similar to the distribution for the pure dissociative component. However, the origin of this shape in + − events is due to the presence of the neutrino in -lepton decay. The dissociative contribution, diss , is therefore determined as the background fraction obtained from the fitting procedure, then reduced by the + − background fraction.
The results of the fitting procedure for the data from the 10 < < 20 GeV and | | < 0.8 interval are presented in Figure 3 for three ZDC categories as well as for the inclusive sample. The bkg fraction amounts to (0.3 ± 0.2)%, (9.9 ± 0.6)%, (13 ± 1)% and (4.3 ± 0.3)% for the 0n0n, Xn0n, XnXn categories, and the inclusive sample, respectively and increases with and | |.
The contribution from Υ-meson production is estimated using the dedicated S + P 8 samples. Three Υ states, Υ(1 ), Υ(2 ), and Υ(3 ) are considered. This background is significant only for below 14 GeV and amounts to 2.4% of events satisfying the selection criteria for the → + − process. The distribution of the dielectron candidates from Υ decays is peaked at zero, similarly to the signal shape. It does not contribute to bkg because the resulting fraction is not sensitive to a 2.4% change in the acoplanarity peak. Hence, the Υ contribution is subtracted separately from the data.
A background contribution originating from photonuclear processes occurring in ultraperipheral heavy-ion collisions is largely suppressed by the trigger requirement limiting the maximum T deposited in the FCal to 3 GeV per side. The validity of this assumption was tested by examining the multiplicity distribution of Pixel-tracks. The fraction of events that have more than two Pixel-tracks is 1.3% and consistent with simulations of → + − events. This also confirms that the pileup contribution from additional low-T → + − interactions, which should be below 1%, can be neglected in the selected data sample.  general, good agreement between data and the sum of the predictions for signal and background processes is found. On average, the observed discrepancies are at the level of 10%-15% with some exceptions which are discussed further. In the distribution the data excess is more strongly pronounced for between 10 and 20 GeV, where the difference between data and MC simulation is 10%-20%. The data-to-MC ratio drops below unity for masses above 40 GeV. The same features are observed in the T distribution, with the largest deviations from unity in the range 5-10 GeV. In the distribution, the data excess is smaller, up to 10%, in the range from −1.2 to 1.2, with larger discrepancies for higher | | values. The data-to-MC ratio in the | cos * | distribution drops slowly from 1.2 for | cos * | = 0 to unity at | cos * | = 0.75, and then falls more steeply, to 0.5 for the largest values of | cos * |. In the distribution, a difference in the overall shape is observed in the full range. This can be explained by a sensitivity of the results to the T spectrum assumed by S , since this spectrum determines the width of the distribution. In general, all these discrepancies tend to be consistent with the observations made in the ATLAS → + − measurement [18], where the S predictions were found to underestimate the measured integrated fiducial cross-sections by about 10%. , and (bottom) for the inclusive sample in data and the MC predictions for signal and background processes. The lower panels show the ratio of data to MC simulation. Error bars represent statistical uncertainties. The shaded area represents the overall uncertainty of the total MC prediction. In the , | cos * |, and distributions, the Υ and + − contributions are shown together. The dissociative contribution is scaled to constitute the diss fraction from the data fit.

Analysis
The integrated fiducial cross-section for exclusive dielectron production is calculated using the following formula: where: • data and bkg refer to the number of events in data after event selection and the expected number of background events in this selected sample, respectively; • is a correction factor accounting for detector inefficiencies (including the trigger), calculated as fid,cut MC,reco / fid,cut MC,gen where fid,cut MC,gen is the number of generated events passing fiducial requirements of the analysis, while fid,cut MC,reco is the number of simulated signal events that also pass the reconstructionlevel selection; • is the acceptance correction, used to correct the result for the exclusion of the calorimeter transition region and extrapolation from | | < 2.47 to | | < 2.5; and is calculated as fid,cut MC,gen / fid MC,gen , where fid MC,gen is the number of generated events passing all fiducial requirements of the analysis, except the requirement to exclude the calorimeter transition region; • is the total integrated luminosity.
Both fid,cut MC,gen and fid MC,gen are extracted with respect to the generator-level electrons before FSR. The fiducial region is defined by the following requirements: T > 2.5 GeV, | | < 2.5, > 5 GeV, and T < 2 GeV. The number of events passing the fiducial selection is data = 30456. The dissociative and + − background fraction obtained from the fit amounts to 4.5%. The Υ background amounts to 2.4% of all events satisfying the selection criteria.
The selected data sample is corrected in a few subsequent steps in order to compare it with the theoretical predictions. In the first step the backgrounds are subtracted. Distributions in the data are reweighted event-by-event by the factor (1 − bkg ) where bkg is the fraction of background (inclusive in ZDC) from the fit in a given and | | range. For masses above 40 GeV, the fraction obtained from the fit in the full | | < 2.4 range is taken. For events with below 40 GeV, the fraction as a function of | | is used. If | | exceeds 2.4, the fraction from the 1.6 < | | < 2.4 bin is applied. In the next step, the background expected from Υ( ) decays is subtracted.
For differential cross-section measurement, the data are corrected with fiducial correction factors defined as the fraction of events in each bin which fall into the fiducial region at generator level. These factors correct for the events that are reconstructed within the fiducial region, but fall outside it at the generator level. They are parameterised using the reconstructed kinematic variables. They deviate from unity at the subpercent level, so their impact is marginal. After this, the reconstructed data are unfolded using a Bayesian-inspired iterative procedure [44] with one iteration for all distributions implemented in the RooUnfold package [45], using response matrices derived from signal MC samples. The number of iterations is chosen to minimise the resulting statistical uncertainty and at the same time provide good closure. A closure test based on the signal MC samples is performed to validate the unfolding procedure. The signal sample is split into two parts. The first part is used to fill the response matrices, while the second one is unfolded. The ratio of the unfolded yields to the generated yields deviates from unity by 1% at most.
Finally, the distributions are divided by the luminosity as well as the product of correction factors, × , which account for detector inefficiencies as well as acceptance losses. They are determined for each bin of the unfolded distribution as the fraction of events that pass the fiducial requirements at reconstruction level, in events that pass them at generator level. The × factors are parameterised using generator-level kinematics, but then weighted by trigger and reconstruction efficiency scale factors, evaluated at reconstruction level. The average and factors amount to 0.087 and 0.878, respectively.

Systematic uncertainties
The following systematic uncertainties are considered in the cross-section measurement. The total scale factors for the electron reconstruction and identification efficiency [40] are varied upwards and downwards coherently over the full kinematic range, as a conservative estimate. The data-driven trigger efficiency, which is the product of the L1 efficiency, the Pixel-veto efficiency, and the forward transverse energy requirement efficiency, is increased and decreased by its total uncertainty. To assess known uncertainties in the EM energy scale and energy resolution, the calibrations are varied by factors determined in 13 TeV collisions [40]. The background contributions are increased and decreased by their total uncertainties. The dissociative backgrounds are dominated by their statistical uncertainties from the fit. The systematic uncertainties are also evaluated, and the largest contribution is related to the shape of the signal template. This is estimated using data from the 0n0n category as a signal distribution for the Xn0n and XnXn categories. The background template shape uncertainty is estimated by adding the double dissociative component. The uncertainty in the expected Υ yields is dominated by both the efficiency scale factors and the EM energy scale. Given the small contribution from Υ production, the theoretical uncertainties of its cross-section are considered to have negligible impact on the final measurement.
For the differential cross-sections, additional systematic uncertainties are related to the unfolding procedures. The MC sample is split in two, with one subsample used to determine the response matrix and the other treated as a simulated data set. The differences between the generated and unfolded yields are treated as a systematic uncertainty. Similarly, the sensitivity to the Bayesian prior is tested by reweighting the simulated data set to agree with the reconstructed data. Again, the differences between simulated and reconstructed yields in this closure test are applied as an uncertainty. While the primary unfolding is evaluated in one dimension rather than two, a cross-check is performed using the response in two dimensions. For each of the unfolded variables, every other one is used in the second dimension, but with the number of bins reduced to four (three) for and T (| | and | cos * |) to compensate for the limited number of events in the MC sample. The three resulting two-dimensional cross-sections are projected to one dimension and compared with the nominal results for each unfolded variable. The largest variations in each bin are included as an uncertainty. Finally, the spectra are evaluated in the 0n0n category, using the fractions determined in Section 4, but evaluated for generator-level and | | (as opposed to the reconstructed values). The differences are found to be within 1%-2%.
The uncertainty in the integrated luminosity of the data sample is 2.0%. It is derived from the calibration of the luminosity scale using -beam-separation scans, following a methodology similar to that detailed in Ref. [46], and using the LUCID-2 detector for the baseline luminosity measurements [47].
A summary of the systematic uncertainties as a function of and | | is shown in Figure 5. The dominant source of uncertainty arises from the uncertainties in the electron scale factors. They are at the level of 10%-11% in the whole range of , and rise from 9% at | | ≈ 0 to about 15% for | | close to 2. The systematic uncertainty from the trigger efficiency is approximately 2% for above 10 GeV and | | below 1.6. It rises to 4% for smaller , and to 6% for the highest | | values. The uncertainty related to the energy scale and resolution is below 1% in the whole range of | |, but exceeds this value in some bins, reaching approximately 5% for the lowest values. The background uncertainties are within 1%-3% and increase slightly with increasing and | |. Uncertainties related to unfolding procedures do not exhibit such clear dependencies in and | |. They are mostly within the 2%-3% range but exceed this value, up to 5%, at intermediate and | |.  Figure 5: Breakdown of relative systematic uncertainties in the differential cross-section as a function of (left) and | | (right).

Results
The total integrated fiducial cross-section is measured to be 215 ± 1(stat.) +23 −20 (syst.) ± 4(lumi.) b. The S prediction for the total integrated fiducial cross-section is 196.9 b, while the S C prediction is 235.1 b. Both predictions are statistically compatible with the measurement.
The differential cross-sections for exclusive dielectron production are presented as a function of , T , | |, and | cos * | in Figure 6. The cross-sections are measured inclusively in the ZDC categories. The results are corrected for detector inefficiency and resolution effects, and are compared with S v3.13 and S C v3.05 predictions for the signal → + − process. The bottom panel in each plot shows the ratio of the unfolded data to MC predictions. On average the S predictions underestimate the data by about 10%-15%, while S C predictions are higher by about the same amount. The S and S C predictions tend to have very similar shapes. The difference in the absolute normalisation of the two predictions is due to different approaches in the calculation of the initial photon flux. The predictions describe the shape of the data well, except at high | | and high | cos * |. The differences are more pronounced for between 10 and 20 GeV, and for T between 5 and 10 GeV. The ratio of data to S rises from about 1.1 to 1.2 as | | increases from 0 to 2.5. For | cos * | close to 0, the data-to-S ratio reaches its largest value, around 1.15, and then slowly decreases to about 1.05 for | cos * | = 0.8. The ratio falls more steeply in the last two bins of | cos * | and drops below unity to 0.75 and 0.65 for S and S C respectively. The measured cross-section in the highest | cos * | bin is 1.8 (2.7) standard deviations below the theory prediction from S (S C ). There is a plausible proposal that higher-order scattering processes (involving more than two photons in the initial state) are relevant and would tend to reduce the predicted cross-sections by the observed discrepancies [48].  Figure 6: Fully corrected differential cross-sections measured inclusively in ZDC categories for exclusive dielectron production, → + − , as a function of , T , | | and | cos * | for data (dots) and MC predictions from S (solid blue) and S C (dashed red). Bottom panels present the ratios of data to MC predictions. The shaded area represents the total uncertainty of the data, excluding the 2% luminosity uncertainty.
The differential cross-sections as a function of , T , | | and | cos * | for the 0n0n category are presented in Figure 7. They are compared with the MC predictions from S v3.13 and S C v3.05. Both simulated samples were produced inclusively and reweighted to the 0n0n category using the measured fractions presented in Fig. 2. The reweighting was performed event-by-event based on the generator level values of and | |. Each theory prediction is represented by two curves reflecting the systematic variations of the measured 0n0n fractions. S can also generate a prediction conditional on the presence of neutron emission in one or both directions. These dedicated predictions from S for the 0n0n category are shown in the same plots. That prediction agrees well with the shape of the inclusive S prediction corrected for the measured 0n0n fractions, but is systematically lower by 2%-3% for | | < 1.4. The general conclusions from this comparison between MC predictions and data are consistent with the inclusive case. Agreement between data and MC events is generally better for lower | | and | cos * | values, i.e. involving lower-energy initial-state photons. 10 [GeV] ee m 2 − for the 0n0n category (dashed-dotted black) is shown. The bottom panels show the ratios of data to predictions. The shaded area represents the total uncertainty of the data, excluding the 2% luminosity uncertainty.

Conclusions
A measurement of the cross-section for exclusive dielectron production, → + − , is performed using L int = 1.72 nb −1 of ultraperipheral Pb+Pb collision data at √ NN = 5.02 TeV recorded by the ATLAS detector at the LHC. The cross-section is corrected for detector efficiency, acceptance losses, and background contributions. The backgrounds from dissociative processes, Υ decays, and + − production are subtracted, with the first contribution estimated using a template fit to the acoplanarity distribution. After all corrections, the integrated cross-section for the → + − process in the fiducial region, defined by T > 2.5 GeV, | | < 2.5, > 5 GeV, and T < 2 GeV requirements, is measured to be 215 ± 1(stat.) +23 −20 (syst.) ± 4(lumi.) b. Within experimental uncertainties the data are in good agreement with the QED predictions from S v3.13 and S C v3.05. The differential cross-sections are presented as a function of , T , | | and | cos * |, both with and without requirements on forward neutron activity. The differential results are compared with the predictions from S v3.13, which are generally lower than measured cross-sections, and from S C v3.05, which tend to be higher. In general, the shapes of the distributions agree well, but some systematic differences are observed. In particular, the discrepancy between data and the S prediction rises with higher | |, similarly to what ATLAS observed previously in → + − production. For | cos * | ≈ 1, the measured cross-section is 1.8 (2.7) standard deviations below the theory prediction from S (S C ). Agreement between data and MC simulation is generally better for lower | | and | cos * | values.  [49] ATLAS Collaboration, ATLAS Computing Acknowledgements, ATL-SOFT-PUB-2021-003, 2021, : https://cds.cern.ch/record/2776662.