Search for periodic signals in the dielectron and diphoton invariant mass spectra using 139 fb−1 of pp collisions at s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document} = 13 TeV with the ATLAS detector

A search for physics beyond the Standard Model inducing periodic signals in the dielectron and diphoton invariant mass spectra is presented using 139 fb−1 of s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document} = 13 TeV pp collision data collected by the ATLAS experiment at the LHC. Novel search techniques based on continuous wavelet transforms are used to infer the frequency of periodic signals from the invariant mass spectra and neural network classifiers are used to enhance the sensitivity to periodic resonances. In the absence of a signal, exclusion limits are placed at the 95% confidence level in the two-dimensional parameter space of the clockwork gravity model. Model-independent searches for deviations from the background-only hypothesis are also performed.


Introduction
techniques. The two previous analyses also estimated the background from simulation and found excellent compatibility with the estimations from data. Therefore, this analysis takes advantage of the precise datadriven background estimates while also benefitting from the availability of a simulation-based background estimates. The background estimates from simulation include a complete set of systematic uncertainties in both of the analysis channels, which is important for this analysis because the "spurious-signal" uncertainty derived in Refs. [10,12] is applicable only to searches which consider a single narrow resonance. The simulation-based uncertainties from previous searches are used instead and their effect on the background estimation is incorporated using the methodology developed in a previous ATLAS search for non-resonant signatures in the dilepton channel [11]. Finally, this analysis also relies on transformational and statistical methods developed in Ref. [15] in order to deal with data featuring periodic structures.
A brief description of the ATLAS detector is given in Section 3. The data and simulated event samples are discussed in Section 4 and an overview of the event selection in each analysis channel is described in Section 5. The signal modelling is presented in Section 6 and the background estimation methods are discussed briefly in Section 7. The sources of systematic uncertainties are discussed in Section 8, while the uncertainty estimation process is described in Section 9.
This analysis performs a search for generic periodic features in the dielectron and diphoton invariant mass spectra, in the same mass ranges considered in Refs. [10,12]. The dielectron and diphoton channels are analysed separately due to potential overlaps in their event selections. This search uses a continuous wavelet transform (CWT) to analyse these mass spectra in the frequency domain. 1 The output of the CWT is a two-dimensional image of the wavelet amplitude in the frequency versus mass space, referred to as a "scalogram." The potential periodicity of a signal can be revealed in the image, for example as a local "blob" around a small range of frequencies or masses. While conventional resonant and non-resonant searches may be suboptimal, particularly in the case of small CW/LD-like signals, the CWT may still provide an enhancement of a new signal's semi-periodic nature. This enhancement provides a clear separation of a signal from the background and suggests that the search should be conducted in the CWT space rather than in the mass space alone. The CWT method is described briefly in Section 10.
In the limit of infinite statistical precision, the local features of a potential signal in the CWT images can be clearly visible and separable from the background-only case. When accounting for realistic statistical fluctuations due to the finite-size distributions in the mass space, the signal features wash out partially in the CWT images and the separation power becomes significantly smaller. Therefore, machine-learning techniques are applied to distinguish periodic contributions to the diphoton and dielectron CWT images made using the invariant mass spectra. Model-independent results are provided using an autoencoder-based anomaly detection procedure to search for generic periodic deviations in the scalograms. The CW/LD model is used as a benchmark model for a model-specific search using the CWT images. A neural network binary classifier is trained on the CWT images made from background-only and signal-plus background mass distributions to provide a test statistic for discovery of potential periodic resonances with specific and 5 values. In both the model-independent and model-specific searches, the test-statistic used during the statistical inference portion of the analysis is based on the machine-learning outputs themselves. As the CW/LD signal may introduce a non-resonant enhancement above the SM background in the mass space, the statistical inference in the two searches can be dominated by the non-resonant signal features instead of the semi-periodic contribution. Therefore, the two searches are performed with and without specific thresholds imposed to reduce potential non-resonant contributions and focus the sensitivity on the periodic features. The statistical analysis of the two methods and the related thresholds are discussed in

Data and simulated event samples
The data used for the search consists of the collision data recorded by the ATLAS experiment at √ = 13 TeV during the 2015 to 2018 LHC data-taking period. After requiring stable beam conditions and data quality selections with all ATLAS subsystems operational [21], the data sample corresponds to an integrated luminosity of 139.0 ± 2.4 fb −1 [22]. The LUCID-2 detector [23] was used for the primary luminosity measurements.
The events used in this analysis were recorded using a set of diphoton and dielectron triggers. Events in the diphoton channel were recorded using a diphoton trigger that required at least two energy clusters in the EM calorimeter with transverse energies ( T ) greater than 35 and 25 GeV for the T -ordered leading and subleading photon candidates, respectively. Both of the clusters were required to satisfy photon identification criteria based on the shower shapes in the EM calorimeters. The triggers used in 2015 and 2 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 upward. 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).
2016 required both of the photons to satisfy the loose identification requirement [24]. In 2017 and 2018, due to the greater instantaneous luminosity, the diphoton trigger requirements were tightened and both of the photons were required to satisfy the medium identification requirement. The efficiency of the diphoton trigger relative to the event selection given in Section 5 is over 99% for the 2015-2016 data and above 98% for the 2017-2018 data [25].
Events in the dielectron channel were collected using several dielectron triggers. The trigger used in 2015 required both of the electrons to satisfy the loose identification criteria and T thresholds of 12 GeV. In 2016, the T thresholds were increased to 17 GeV for both electrons and the electrons were required to satisfy the very loose identification criteria [24]. In 2017 and 2018, the identification criteria were left unchanged and the T thresholds were increased to 24 GeV for both of the electrons [25].
Although the background in this analysis is estimated by using data-driven methods, simulated Monte Carlo (MC) events are used to optimise the analysis selections, determine appropriate fit functions for the data, estimate background compositions, and evaluate signal acceptances and efficiencies. As the KK modes in the CW/LD model are on-shell, the interference effects between the resonant signals and all background processes are neglected in both the diphoton and dielectron channels.
For the diphoton channel, the largest background comes from the production of two prompt photons which represents the irreducible background in this search channel. Smaller background contributions come from events containing a photon and a jet and events with two jets, where the jets are misidentified as photons. These smaller backgrounds are estimated by using a data-driven technique, the two-dimensional sideband method, described in Ref. [26].
Events with two prompt photons were simulated using the S 2.2.4 [27,28] event generator. Matrix elements were calculated with up to one additional parton at next-to-leading-order (NLO) and including two or three additional partons at leading-order (LO) in perturbative QCD (pQCD). These matrix elements were merged with the S parton-shower simulation using the ME+PS@NLO prescription [29-32]. The NNPDF3.0 parton distribution function (PDF) set [33] was used and paired with a dedicated parton-shower tune in the S generator.
For the dielectron channel, the main prompt backgrounds arise from Drell-Yan (DY), top-quark pair (¯), single-top-quark, and diboson production. The background contribution from non-prompt electrons from multĳet and + jets processes is estimated by using a data-driven technique, the matrix method, as described in Ref. [34].
The DY [35] sample was generated using the P B v1 [36][37][38][39] event generator with the CT10 PDF set [40] and interfaced with the P 8.186 [41] parton shower program. Next-to-next-to-leading-order (NNLO) corrections in pQCD and NLO corrections in electroweak (EW) theory were calculated and applied to the simulated DY events. The pQCD corrections were computed with VRAP v0.9 [42] and the CT14 NNLO PDF set [43]. The EW corrections were computed with MCSANC [44] which accounts for quantum electrodynamic effects due to initial-state radiation, interference between initial and final-state radiation, and Sudakov logarithm single-loop corrections.
The diboson [45] processes with fully leptonic and semileptonic final states were simulated using S 2.2.1 with the CT10 PDF set. Matrix elements were calculated at NLO accuracy in QCD for up to one additional parton and at LO accuracy for up to three additional parton emissions. The matrix element calculations were matched and merged with the S parton shower based on Catani-Seymour dipole factorisation [46,47] using the ME+PS@NLO prescription. The diboson and DY backgrounds were generated in slices of dilepton mass in order to enhance the MC statistical precision in the high-mass region.
The¯and single-top-quark samples were generated with P B v2 [36][37][38][48][49][50][51] at NLO using the NNPDF3.0 PDF [33] in the matrix element and interfaced to P 8.230 [52] in order to model the parton shower, hadronisation, and underlying event, with parameters set according to the A14 tune [53] and using the NNPDF2.3 set of PDFs [54]. These samples were normalised to the theoretical cross-sections calculated at NNLO in pQCD and include resummation of the next-to-next-to-leading logarithmic soft gluon terms as provided by T ++ 2.0 [55].
Resonant single-graviton MC samples were simulated using a Randall-Sundrum model [9]. These MC samples were generated at LO in pQCD using P 8 with the NNPDF2.3 PDF set and the A14 tune. In these samples, only the lightest KK graviton excitation was generated. For the diphoton decay channel, the samples were generated with a KK graviton mass * in the range of 150-5000 GeV. A fixed coupling value of / Pl = 0.01, where Pl = Pl / √ 8 is the reduced Planck scale, was used to ensure a sufficiently narrow-width signal. For the dielectron decay channel, samples were generated with masses in the range of 200-6000 GeV and with a fixed coupling value of / Pl = 0.1. These samples are used to determine the acceptance and efficiency of selecting the CW/LD signal.
Additional MC samples incorporating a series of RS graviton resonances were generated at LO in pQCD using P 8.244 with the NNPDF2.3 PDF set and the A14 tune. In these samples, only the direct graviton decays into dielectron and diphoton final-states were simulated. Samples were generated for values of 300 GeV, 500 GeV, 1 TeV, and 2 TeV with 5 fixed to a value of 6 TeV. These samples are used to validate the analytic signal templates of the CW/LD signals, which are discussed in detail in Section 6.
The effects of multiple interactions in the same bunch crossing as that of the hard scatter plus those from adjacent bunch crossings (pile-up) are included in all simulated samples. Pile-up collisions were generated with P 8.186 using the NNPDF2.3 PDF set and the ATLAS A3 set of tuned parameters [56]. Simulated event samples were weighted to reproduce the distribution of the average number of interactions per bunch crossing observed in the data [57].
The spin-2 diphoton simulated signal events, and DY,¯, single-top-quark, and diboson background events were processed using a detailed simulation of the ATLAS detector [58] based on G 4 [59]. The irreducible prompt background and spin-2 dielectron signals were processed using a fast simulation of the ATLAS detector [60], where the full simulation of the calorimeter is replaced with a fast parameterisation of the calorimeter response. All simulated events were reconstructed with the same reconstruction algorithms as those used for data.
Generator-level-only MC samples of the NLO DY background are used for the modelling studies described in Section 7. These samples could not be produced with the ATLAS detector simulation due to the large number of events required [10].
The event selection in the diphoton channel requires at least two reconstructed photon candidates with T > 25 GeV and | | < 2.37, excluding candidates in the transition region 1.37 < | | < 1.52 between the barrel and endcap EM calorimeters. The two highest-T photons are used to form the diphoton candidate and are used with additional information from the tracking detectors to identify the diphoton primary vertex [61]. After the diphoton primary vertex is identified, the leading and subleading photons are required to have T / > 0.35 and 0.25, respectively. The diphoton invariant mass, , is required to be greater than 150 GeV. To reduce the background from jets, photon candidates must satisfy the tight identification criteria based on shower shapes in the EM calorimeter [24].
Events in the dielectron channel are selected by requiring at least one pair of reconstructed electron candidates. Each event is required to contain at least one reconstructed interaction vertex, where the primary vertex is defined as the vertex with the highest sum of track transverse momenta squared. Electron candidates are reconstructed from ID tracks matched to energy clusters deposited in the EM calorimeter [24,62]. All selected electrons are required to have T > 30 GeV and | | < 2.47. As in the diphoton channel, electron candidates in the transition region (1.37 < | | < 1.52) are not considered. The final selection requires that the electrons satisfy the medium identification working point.
Electron candidate tracks are required to satisfy | 0 / ( 0 )| < 5 and | 0 sin | < 0.5 mm, where 0 and 0 are the transverse and longitudinal impact parameters defined relative to the primary vertex position and ( 0 ) is the uncertainty on 0 . This selection ensures that the electron candidate track is consistent with the primary vertex of the event.
If there are more than two electrons in the same event, the two electrons with the highest T are selected to form the dielectron pair. An opposite-sign electron charge requirement is not applied due to a high probability of charge misidentification for high-T electrons. The reconstructed invariant mass of the dielectron pair, , is required to be above 225 GeV to avoid the region dominated by decays of the boson which cannot be described using the same background parameterisation as the high-mass region.
To suppress the backgrounds from misidentified jets, the photon and electron candidates are required to satisfy calorimeter-and track-based isolation criteria. The photons are required to satisfy the tight isolation which is 90-95% efficient for the analysis selection [24]. Electrons are required to satisfy the gradient isolation which is 99% efficient for the analysis selection [24].

Signal modelling
The CW/LD signals are modelled using analytic invariant mass templates constructed from a range of inputs including PDF information, branching ratio, and detector resolution modelling derived from the previous ATLAS diphoton and dielectron analyses.
For each reconstructed mass bin, the expected number of signal events is determined by computing the contribution from the entire KK graviton tower. The estimate of the expected number of signal events for a given reconstructed mass bin is given by where the index runs over the KK graviton modes whose masses are within the search range. In Eq. (1), int is the integrated luminosity, is the production cross-section, B denotes the branching ratio, and ( × ) is the acceptance times efficiency for a given mass, , of the KK graviton. The ( | ) term represents the probability to reconstruct an event in invariant mass bin for events with an input true mass of that satisfy the selection requirements and is referred to as the transfer function (TF). The TF provides a smooth transformation between the true and reconstructed invariant mass spectrum by modelling the detector effects. The convolution becomes a product as the true masses are discrete and the natural widths can be neglected as the resonances are described by the narrow-width-approximation (NWA) over the range of true masses considered in this analysis.
The cross-sections and branching ratios for the gravitons in the CW/LD signal are taken from Ref. [2], where the effects of heavy graviton cascade decays into lighter graviton pairs are included in the total width when calculating the diphoton and dielectron branching ratios. The additional contributions to the signal from these cascade decays into diphoton and dielectron final states are not considered.
The TF and acceptance times efficiency terms are derived following the methodologies of the previous ATLAS dielectron and diphoton resonance searches [10,12] and are briefly outlined below. These terms are derived using the DY MC samples and the resonant single-graviton MC samples for the dielectron and diphoton channels, respectively.
For the dielectron channel, the detector response is defined in the TF with respect to the relative dilepton mass resolution ( ℓℓ − true ℓℓ )/ true ℓℓ , where true ℓℓ is the generated dilepton mass at Born level before final-state radiation. The mass resolution is parameterised as the sum of a Gaussian distribution and a Crystal Ball function [63,64]. The Gaussian component in this function describes the central peak of the detector response and the Crystal Ball component is used to model the effects of bremsstrahlung in the dielectron system. The parameterisation of the relative dilepton mass resolution is determined using a simultaneous fit to the resolution function to the DY MC events. The DY MC sample is separated into 200 bins of equal size in true ℓℓ based on a logarithmic scale in the dielectron mass range from 130 GeV to 6000 GeV. The mass resolution fit is repeated to evaluate the uncertainty in the fit parameters by shifting individually the lepton energy and momentum scale and resolutions by their uncertainties. Although the resolution function is derived on spin-1 events, the results are found to be compatible with the true response coming from the spin-2 gravitons.
For the diphoton channel, the TF describes the detector resolution of the invariant mass using a double-sided Crystal Ball (DSCB) function [63,65] with the parameters expressed as a function of the resonance mass, . The DSCB parameters are extended to lower masses relative to Ref.
[12] as the CW/LD signals considered in this analysis can contain graviton masses below 500 GeV, which were not considered in the previous analysis.
Analytic templates allow for signal shapes to be created for any choice of and 5 . These templates are confirmed to agree with the MC samples generated at the specific points in the -5 plane discussed in Section 4. Examples of the expected signal shapes in each analysis channel are shown in Figure 1 for = 1200 GeV, 5 = 3000 GeV and for = 2000 GeV, 5 = 8500 GeV.

Background modelling
The diphoton and dielectron search channels in this analysis use the same data-driven background estimation technique based on a functional form fit to the data. Each analysis channel uses a single functional form to model the background across the full invariant mass spectrum. The analytic form for the background in each channel is determined from a fit to the background expectation derived from MC simulation and control-region data. For the dielectron channel, the functional form is taken from the analysis in Ref. [10], where the fit was performed on a background template with contributions from the MC estimate of the DY,¯, diboson, and single-top processes and the data-driven estimate of the reducible backgrounds arising from the multĳet and +jet processes. The function is parameterised as a function of ℓℓ and is given by: and BW, ( ℓℓ ) is a non-relativistic Breit-Wigner function with = 91.1876 GeV and Γ = 2.4952 GeV [66]. The and parameters are fixed to the values from the fit to data in Ref. [10] and the (1 − ) term ensures that the background fit tends to zero as → 1, consistent with the expectation from the collision energy of the LHC. The background function is normalised such that its integral corresponds to the total number of events in the signal region.
For the diphoton channel, the largest background component arises from the non-resonant production of photon pairs ( events). Smaller contributions to the total background come from events containing a photon and a jet misidentified as a photon ( events) and from events with two jets where both of the jets are misidentified as photons ( events). The relative contribution from each of these processes is determined using a two-dimensional sideband method described in Ref. [26]; the overall purity increases as a function of from around 89% at 150-200 GeV to around 97% above 400 GeV. The relative uncertainty in the purity ranges from 0.5%-3% and the remainder of the background is dominated by events.
The analytic form for the diphoton channel is chosen from the family of functions used in previous analyses to describe the diphoton invariant mass spectrum [67], following the procedure described in Ref. [12]. The background function is described by the following form where , 0 , 1 are free parameters and = / √ .

Uncertainties
The previous dilepton [10] and diphoton [12] data-driven analyses extracted full MC-based uncertainties, which are used in this search to construct systematically-varied distributions as discussed in Section 9. The variations considered in both the dielectron and diphoton channels come from theoretical and experimental uncertainties in the simulated backgrounds, uncertainties in the detector response and signal yield, and uncertainties induced by the data-driven background estimates.
The following theoretical uncertainties are considered for variations in the DY and components: the variations of the nominal PDF set, variations of the factorisation and renormalisation scales, and the effect of choosing different PDF sets. The variations in the nominal PDF set for the DY background are determined using 90% CL CT14NNLO PDF error set, while the PDF variations for the background are constructed from all of the replicas within the NNPDF3.0 NNLO set. The factorisation and renormalisation scales are varied independently by a factor of two to estimate the perturbative uncertainties for these background MC samples. Theoretical uncertainties in the DY component from the variation of the strong coupling, electroweak corrections, and photon-induced corrections [68] are also considered. An additional uncertainty is applied in the dielectron channel to account for the theoretical cross-section uncertainties of the¯process.
The experimental uncertainties considered in the diphoton (dielectron) analysis channel include uncertainties on the luminosity determination, pile-up profile, trigger efficiency, photon (electron) energy scale and resolution, and the efficiency of photon (electron) identification and isolation requirements. More details of the determination of the experimental systematic uncertainties can be found in Refs. [10,12].
The systematic uncertainties in the detector response are determined by varying the energy scale and energy resolution of the electrons and photons. The effect of the experimental uncertainties in the signal yield, based on the variations in the acceptance times efficiency, are also considered. An additional uncertainty is also derived for the signal based on the internal variations of the PDF set.
The largest data-driven uncertainty in the diphoton channel is a shape uncertainty that corresponds to the uncertainty in the relative contribution of the reducible background component. This uncertainty is estimated by varying the measured fraction of the background according to its uncertainty from the two-dimensional sideband method mentioned in Section 7. A data-driven uncertainty in the normalisation of the reducible background component (referred to as the "fake" background) in the dielectron channel is also derived, following the methodology of Ref. [34].
In this analysis, each systematic uncertainty is either naturally two-sided or is symmetrised to provide a two-sided systematic uncertainty. The variations from systematic uncertainties in both analysis channels are smoothed using a sliding window method [10]

Generation of pseudo-experiments
The analysis uses pseudo-datasets as inputs for the CWT, where one scalogram is generated for each pseudo-dataset. These scalograms are the basic ingredient for the statistical procedures used in this analysis and are also used for estimating the systematic uncertainties. The pseudo-datasets (or "toys") are generated with background-only models based on the functional forms from Section 7 and with signal-plus-background models using the signal models discussed in Section 6 and the background-only models from Section 7. These toys are produced with finely-binned (1 GeV) mass spectra separately for the diphoton and dielectron final states. For the toys containing a signal, a specific choice of CW/LD model parameters is used. The generated pseudo-datasets are hereafter called statistical-only toys (stat-toys) or systematic toys (syst-toys) depending on whether systematic variations are considered when constructing the toy.
To make the stat-toys, the analytic shapes are constructed containing either a background-only shape or a signal-plus-background shape. The background component is given a normalisation corresponding to the data normalisation in the signal region of each analysis channel. Next, a Poisson-fluctuated toy distribution is generated from the analytic shape chosen. Each of these toys is then treated as a proxy for the real data in the subsequent statistical analysis discussed in Section 11.
To make the syst-toys, a similar procedure is followed as with the stat-toys, but starting from an alternative description of the background-only and signal-plus-background shapes. This analysis follows the procedure introduced in Ref. [11], which is described briefly below.
To make one syst-toy for the background-only case, the alternative background shape (called hereafter an uncertainty template) is built from the systematic variations that arise from the sources of both experimental and theoretical uncertainties as discussed in Section 8. In the background-only case, these systematic variations are given in the invariant mass space as shape variations around the background description from simulation, as obtained in Refs. [10,12]. A relative variation is defined as the relative difference between the up or down variation and the background description from simulation. To obtain the absolute systematic difference for the nominal background description from data defined in Sections 6 and 7, the relative variation is multiplied by the nominal background description from data. Each uncertainty template is constructed from the nominal background description from data, summed with the weighted absolute systematic differences. The weights are randomly sampled from a Gaussian distribution, with a mean of zero and standard deviation of one. The number of such weights in each uncertainty template is equal to the number of systematic variations considered.
To replicate the procedures given in Refs. [10,12], the resulting uncertainty template is Poisson-fluctuated to generate one uncertainty pseudo-dataset. This uncertainty pseudo-dataset is then fit with the background model from either Eq. (2) or Eq. (3), depending on the analysis channel. This procedure results in one smooth alternative background description, replacing the fits done in Refs. [10,12]. This alternative background description is then used to generate one respective background-only uncertainty pseudo-dataset.
To make one syst-toy in the signal-plus-background case, a similar procedure is followed with minor modifications. The procedure starts from the signal systematic variations discussed in Section 8. The nominal signal shape is described by Eq. (1). The absolute systematic difference for each variation is calculated directly for the nominal signal shape. The signal uncertainty template is constructed from the nominal signal shape, summed with the Gaussian-weighted absolute systematic differences. Each signal-plus-background uncertainty template is constructed as the sum of one smooth alternative background description, as discussed above, and one signal uncertainty template. The result is Poisson-fluctuated to generate one respective signal-plus-background uncertainty pseudo-dataset.
These procedures are repeated to generate background-only and signal-plus-background syst-toys ensembles, where each toy in these ensembles reflects a different (random) combination of all uncertainties. The syst-toys ensembles are used in the subsequent statistical analysis discussed in Section 11.

Continuous wavelet transforms
Continuous wavelet transforms are used in this search to convert the diphoton and dielectron invariant mass spectra into mass versus frequency information in the form of scalograms before applying the machine learning techniques discussed in Section 11. An overview of the CWT method used in this search is detailed below.
The CWT is a measure of similarity between a chosen wavelet and a signal. The wavelet is used to scan the signal for different frequencies of the wavelet throughout the invariant mass spectrum.
The wavelet ( ) is a basis function localised in both the mass and frequency space and the CWT of a signal ( ) at a scale 3 and translational parameter is given by a projection on ( ) for different and values [15]: A scalogram can be produced by taking the norm of the coefficient ( , ) for all values of and . Therefore, the CWT in this analysis defines how much of a certain frequency is in the signal at a given invariant mass bin.
The Morlet wavelet [69] is used in this analysis as the choice of ( ) in Eq. (4). The Morlet wavelet is chosen because it is Gaussian-shaped in the frequency domain, which minimises possible edge effects that could be interpreted as signal, as opposed to wavelets with sharp boundaries that can introduce edge effects [70]. It consists of a localised wave packet and is given by: where and are constants which are chosen to be 2 and 1 respectively. This choice of ensures that the CWT of a signal reaches a maximum when the signal wavelength approximately equals the scale . The Morlet Wavelet is shown in Figure 2. signal-island is shifted horizontally to higher masses because the signal turn-on point in mass is roughly equal to the value of . The signal-island is also shifted vertically towards higher scales because the spacing in the KK tower increases with . Fixing and changing 5 only determines the prominence of the island, that is, it becomes more distinct with decreasing 5 because of the inverse relationship between 5 and the signal cross-section. Changing the mass resolution due to detector effects (within the uncertainties) only affects the | ( , )| values of the signal-island and this change is effectively equivalent to changing 5 .

Data analysis
The diphoton and dielectron invariant mass distributions from the toys detailed in Section 9 are transformed into images using the CWT method described in Section 10 before being used to train a neural network (NN). The diphoton and dielectron images are treated as independent channels and are trained separately. Two types of convolutional NNs are used in this search, a classifier NN and an autoencoder (AE), which are discussed in Section 11.1 and Section 11.2, respectively. In both of the cases, the training is done using stat-toys, while the prediction is performed using syst-toys to evaluate the impact of the systematic uncertainties in the NN responses. As discussed in Section 9, the syst-toys already include a proper statistical representation of the data.
The classifier NN is used to probe for the periodic signals of the CW/LD model specifically (denoted here as model-dependent). On the other hand, the AE NN is used to search for more general anomalies in the data (denoted here as model-independent). Both of the neural networks used in this analysis are based on the setups given in Ref.
[15]. The neural networks are implemented in Keras [71] using the TensorFlow backend [72]. The ADAM [73] optimiser is used to minimise the loss functions in each NN setup. The search is performed using a general form of a test statistic derived from the NN output itself.
The loss functions associated with each of the NNs play the role of the likelihood function in a typical analysis. The loss function used in the classifier NN is the Binary Cross-Entropy (BCE) [74], while the AE NN uses the Mean Squared Error (MSE) loss function. The MSE loss function is further discussed in Section 11.2. The training and validation loss are recorded as a function of the number of epochs and are inspected to verify that each NN gives good agreement in the output of the loss function when comparing the training and validation datasets.
In this analysis, the NN output in the classifier setup and the value of the loss function in the AE setup are used as test statistics. To set the exclusion limits on the model-dependent parameter space, the modified frequentist method, commonly known as the CL method [75][76][77], is used. This choice results in more conservative limits than those obtained by the standard -value procedure [75,76]. The dielectron and diphoton channels are analysed separately due to potential overlaps in their event selections.

Classifier NN
One possible way to discover a specific signal in a scalogram is with a classifier NN. In this analysis, the convolutional NN classifier setup from Ref.
[15] is implemented for the model-dependent search. The CWT scalogram of the mass spectrum for each stat-toy, syst-toy, and the signal region data is produced and rebinned to ensure the inputs to the NN remain manageable. The mass for the NN inputs is rebinned into 40 GeV wide bins, while the scale is binned logarithmically. The stat-toy scalograms for the background-only and signal-plus-background cases are used for training the binary classifier convolutional NN, where the output of the network is between 0 (background-like) and 1 (signal-plus-background like).
In addition to the signal periodicity of the CW/LD model, these models can introduce a non-resonant tail above the SM background at high dielectron or diphoton invariant masses. For these high-mass ranges, the periodicity may no longer be as apparent due to the widening of the resonances with mass. This signal widening occurs because the experimental mass resolution worsens, which causes the merging of the signal mass peaks. This analysis focuses on the periodic features of the signal by adjusting the procedure such that the subsequent inference does not consider the effects of the non-resonant tail.
To achieve this adjustment, a cutoff is applied to the dielectron or diphoton mass distribution to remove events above a certain point in mass prior to training the classifier. This mass threshold is determined by calculating a signal's local significance, / , where is the signal yield and is the local per-bin uncertainty. This uncertainty includes the statistical uncertainty and the systematic uncertainties described in Section 8, added in quadrature, of the background. The bin in the invariant mass distribution on a local signal peak where the per-bin-significance falls below 50% of its maximal value is chosen as the threshold. While the cutoff is independent of 5 by construction, it does rise monotonically with . Therefore, the cutoff values are calculated for a few points in and are then fit with a second-order polynomial to allow for interpolation between the points. As the long-range signal shape falls rapidly after reaching its maximum, alternative choices close to the nominal choice of 50% of the maximum local significance have little impact on the result. A lower significance choice, for example / ≈ 10% of the maximal value, effectively introduces no cutoff and the subsequent inference remains affected by the presence of the non-resonant tail feature. A higher significance choice, e.g., / ≈ 90% of the maximum value, removes the non-resonant tail effectively but also removes a large mass range where the periodicity is still apparent and reduces the search sensitivity to the intended feature. For the dielectron channel, the 50% threshold mass value ranges from 1400 to 6000 GeV for in the range of 200 GeV to 5850 GeV. For the diphoton channel, the mass threshold ranges from 2000 to 5000 GeV for in the range of 150 GeV to 4900 GeV.
For completeness, the classifier search without mass thresholds is also performed. In this case, the non-resonant feature of the signal may be present, while the periodicity may be unresolvable in the high mass region. The CWT and NN procedure can still provide some discrimination power between the signal-plus-background and background-only distributions due to the non-resonant feature and additional oscillations available to describe the signal. Therefore, the sensitivity for low-( 1000 GeV) signals is expected to be higher than for the case with mass thresholds. For high-values, ( 1000 GeV), the sensitivities of the two methods are expected to be almost identical.

Autoencoder NN
A model-independent approach searching for anomalies in scalograms is performed using AEs. This technique has previously been applied to jet images in Refs. [78,79] and was implemented and suggested for this type of search in Ref. [15]. The AE compresses a scalogram to a smaller set of parameters, which are then used to reconstruct the input scalogram. Instead of using the original background-only scalograms, each bin in the scalogram is remapped monotonically to uniformly distributed values in the range of 0 to 1. The negative logarithm of the remapped bin value is used to build a new two-dimensional input. This procedure is identical to the procedure in Ref. [15] and ensures that the loss function is not dominated by the region with high statistical precision. The AE is trained on background-only, stat-toy remapped scalograms to reproduce the original remapped scalograms (y) by minimising the MSE loss function [79], MSE , as defined below: where is the number of bins used in the scalogram andŷ is the output remapped scalogram from the AE.
After the AE model is trained, the AE should be able to approximately reproduce the original remapped scalogram if applied to a typical background-only syst-toy scalogram. Conversely, the AE may fail to reproduce the scalogram if applied to a data scalogram that contains a signal. This metric, MSE , is used as a test statistic. In this analysis, the AE is particularly sensitive to signals with periodic structures.
The MSE is also calculated per scalogram bin. For each bin, an ensemble of the MSE values for the background predictions is generated. From these ensembles, a local -value is calculated for every scalogram bin. The statistically significant bins are then visible with -values close to 0.
As discussed in Section 11.1, to focus the search on the periodicity features and avoid possible inference that is based only on non-resonant signals, two sets of selections are introduced. Unlike in Section 11.1, these selections cannot be based on a specific signal and should be characterised only based on information pertaining to the SM backgrounds. In the first method, the non-resonant features are removed by using the statistical error at the tail of the background template shape, = 1/ √︁ ( min ), where ( min ) is the integral of the background template shape above some minimum invariant mass min . Different min thresholds are derived for different values, where for each, the tail of the distribution above the threshold is removed before calculating the CWT. The thresholds and the respective invariant mass ranges for each channel are found in Table 1. Table 1: Different precision-driven thresholds considered for each analysis channel and the corresponding mass ranges probed in the model-independent analysis. For the no threshold scenario, the upper limit in mass is chosen such that the range goes 300 GeV beyond the last observed data point in the dielectron and diphoton mass spectra. This procedure also prevents large statistical fluctuations in the high-mass tail of the background distribution from affecting the MSE of the AE predictions.
In the second approach, the scale range of the scalograms is instead modified and the mass range remains unrestricted. This approach, referred to as scale thresholding, avoids the large-scale region of the scalogram that may include continuum contributions from a signal. In this method, the bin content in the scalogram is replaced with zero if the scale in that bin satisfies > /4, where denotes the mass value of that bin. In a similar way, to exclude the scales that are short and cannot be resolved experimentally, the bin content in the scalogram is replaced with zero if the bin satisfies < /100. The results from both methods are given in Section 12.
Possible deviations in the data, arising from SM or beyond the SM effects, from the smoothly falling background model assumed in each analysis channel could be captured in the AE analysis. However, the sensitivity to any new signal is lessened as this approach is designed to be most sensitive for deviations which appear periodic.

Results
The scalograms for the data events passing all selections for the diphoton and dielectron channels are shown in Figure 5. The model-independent results from the AE-based search with < 50% are presented as the negative logarithm of the local -values corresponding to these data scalograms and are shown in Figure 6.
To obtain the significance for the model-independent results, the MSE is calculated for each original and predicted scalogram pair, with the sum in Eq. (6) running over all of the scalogram bins. An ensemble of MSE values for the background prediction is then generated from many pairs of original and predicted scalograms. From this ensemble, the significance is calculated for the data scalogram.
The significances for the model-independent results in each analysis channel are given in Table 2. No significant deviations from the background-only hypothesis are observed. The largest deviation from the background-only (SM) hypothesis is found in the dielectron channel with the 50% threshold and the scale threshold, with an excess corresponding to a significance of 1.5 .
For the AE-based model-independent result, an alternative approach is also tested where the training is performed using syst-toys rather than stat-toys. In this approach, all steps of the AE-based search after the training match the procedure described in Section 11.2. The significance values in this alternative approach are found to be comparable to the results given in Table 2.  The model-dependent results are also presented for each analysis channel. No significant deviation from the background-only hypothesis is seen in either of the analysis channels. Test statistic distributions assuming both a signal model, with = 1033 GeV and 5 = 9000 GeV, and the Standard Model are shown in Figure 7 and they are compared with the test statistic from the observed data.
In the absence of a clear signal, limits at 95% CL are set on the CW/LD model in the -5 plane. The limits for the case with mass thresholding, as discussed in Section 11.1, are shown in Figure 8. Additional exclusions are shown for the case without mass thresholds in Figure 9. As expected, the sensitivity and corresponding limits are stronger for the case without mass thresholds in the region of 1000 GeV.
For the case with mass thresholds, the maximum excluded 5   The systematic uncertainties mainly impact the classifier result in the range of 500 < < 1500 GeV for both of the analysis channels. The statistical uncertainties dominate in the higher ranges. The sensitivity of the classifier with systematic uncertainties is at most ∼ 1 TeV weaker in 5 exclusion than the limits evaluated without including the systematic uncertainties. The most dominant uncertainty contribution in both of the channels is due to the theoretical uncertainties in the background modelling.
It is worthwhile to mention that the NNs are initially trained on a lattice of points in the -5 plane, where a dedicated training is performed for each lattice point. The results are verified to be similarly effective for the models in-between those lattice points.

Conclusion
Searches for periodic new phenomena in the dielectron and diphoton invariant mass spectra with the ATLAS experiment at the LHC are presented. The searches are conducted with 139 fb −1 of collision data at √ = 13 TeV. The model-dependent result is optimised for spin-2 periodic resonances (KK gravitons) in the clockwork/linear-dilaton model. Model-independent results sensitive to more general periodic contributions are also presented.
The data observed in the and invariant mass spectra are found to be consistent with the SM expectations. Without a significant excess, the 95% CL exclusion limits are set in the parameter space of the CW/LD model. This result excludes values of 5 in the range 11 TeV to 1 TeV for values of in the range 100 GeV to 5 TeV.