Measurement of the $Z\gamma\rightarrow\nu\bar{\nu}\gamma$ production cross section in $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector and limits on anomalous triple gauge-boson couplings

The production of $Z$ bosons in association with a high-energy photon ($Z\gamma$ production) is studied in the neutrino decay channel of the $Z$ boson using $pp$ collisions at $\sqrt{s}$ = 13 TeV. The analysis uses a data sample with an integrated luminosity of 36.1 fb$^{-1}$ collected by the ATLAS detector at the LHC in 2015 and 2016. Candidate $Z\gamma$ events with invisible decays of the $Z$ boson are selected by requiring significant transverse momentum ($p_{T}$) of the dineutrino system in conjunction with a single isolated photon with large transverse energy ($E_{T}$). The rate of $Z\gamma$ production is measured as a function of photon $E_{T}$, dineutrino system $p_{T}$ and jet multiplicity. Evidence of anomalous triple gauge-boson couplings is sought in $Z\gamma$ production with photon $E_{T}$ greater than 600 GeV. No excess is observed relative to the Standard Model expectation, and upper limits are set on the strength of $ZZ\gamma$ and $Z\gamma\gamma$ couplings.


Introduction
The production of a Z boson in association with a photon in proton-proton (pp) collisions has been studied at the Large Hadron Collider (LHC) since the beginning of its operation in 2010 [1][2][3][4][5]. These studies have been used to test the electroweak sector of the Standard Model (SM) and to search for new physics effects, such as potential couplings of Z bosons to photons. Previous publications from experiments at LEP [6][7][8][9] and the Tevatron [10][11][12] have shown no evidence for anomalous properties of neutral gauge bosons at the LHC. The set of data from the second period of the LHC operation provides the opportunity for more accurate measurements of the diboson production rate in pp collisions, and facilitates higher-precision tests of triple gauge-boson couplings (TGCs). This paper presents a measurement of Zγ production with the Z boson decaying into neutrinos. The analysis uses 36.1 fb −1 of pp collision data collected with the ATLAS detector1 at the LHC, operating at a centre-of-mass energy of 13 TeV. The measurements are made both with no restriction on the system recoiling against the Zγ pair (inclusive events) and by requiring that no jets with |η| < 4.5 and p T > 50 GeV (exclusive events) are present in addition to the Zγ pair. 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 z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the z-axis. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2). The angular distance is measured in units of ∆R ≡ (∆η) 2 + (∆φ) 2 .
The ννγ final state in the SM can be produced by a Z boson decaying into neutrinos in association with photon emission from initial-state quarks or from quark/gluon fragmentation. These processes are illustrated by the leading-order Feynman diagrams shown in Figures 1(a)-(c). An example of an anomalous triple gauge-boson coupling (aTGC) of Z bosons and photons is shown in Figure 1(d). Such couplings are forbidden at tree level in the SM but can arise in theories that extend the SM [13,14]. A study of the Z(νν)γ process has several advantages over processes with Z decay into hadrons or charged leptons. The channel with hadrons in the final state is contaminated by a large multijet background. A higher Z boson branching ratio into neutrinos relative to that into charged leptons provides an opportunity to study the Zγ production in a more energetic (higher E γ T ) region, where the sensitivity of this process to bosonic couplings is higher [5,15]. In addition, the neutrino channel is sensitive to anomalous neutrino dipole moments, although a higher integrated luminosity than that available to this study would be required to significantly improve upon LEP results [16,17].
The measurements of the rate and kinematic properties of the Zγ production from this study are compared with SM predictions obtained from two higher-order perturbative parton-level calculations at next-toleading order (NLO) and next-to-next-to-leading order (NNLO) in the strong coupling constant α S , as well as with a parton shower Monte Carlo (MC) simulation. The measured Zγ production cross section at high values of photon E T is used to search for aTGCs (Z Zγ and Zγγ). For these searches an exclusive selection is used, providing higher sensitivity to the anomalous couplings due to further background suppression.

ATLAS detector and data samples 2.1 ATLAS detector and experimental data set
The ATLAS detector at the LHC is described in detail in Ref. [18]. A short overview is presented here, with an emphasis on the subdetectors needed for a precision measurement of the Z(νν)γ final state. The ATLAS detector covers nearly the entire solid angle surrounding the collision point. Its major components are an inner tracking detector (ID) surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic (ECAL) and hadron (HCAL) calorimeters, and a muon spectrometer (MS). The ID is composed of three subsystems. Two detectors cover the pseudorapidity range |η| < 2.5: the silicon pixel detector and the silicon microstrip tracker (SCT). The outermost system of the ID, with an acceptance of |η| < 2.0, is composed of a transition radiation tracker (TRT). The TRT provides identification information for electrons by the detection of transition radiation. The MS is composed of three large superconducting air-core toroid magnets, a system of three stations of chambers for tracking measurements, with high precision in the range |η| < 2.7, and a muon trigger system covering the range |η| < 2.4.
The ECAL is composed of alternating layers of passive lead absorber interspersed with active liquid-argon gaps. It covers the range of |η| < 3.2 and plays a crucial role in photon identification. For |η| < 2.5 the calorimeter has three longitudinal layers in shower depth, with the first layer having the highest granularity in the η coordinate, and the second layer collecting most of the electromagnetic shower energy for high-p T objects. A thin presampler layer precedes the ECAL over the range |η| < 1.8, and is used to correct for the energy lost by EM particles upstream of the calorimeter. The HCAL, surrounding the ECAL, is based on two different technologies, with scintillator tiles or liquid-argon as the active medium, and with either steel, copper, or tungsten as the absorber material. Photons are identified as narrow, isolated showers in the ECAL with no penetration into the HCAL. The fine segmentation of the ATLAS calorimeter system allows an efficient separation of jets from isolated prompt photons.
Collision events are selected using a hardware-based first-level trigger and a software-based high-level trigger. The resulting recorded event rate from LHC pp collisions at √ s = 13 TeV during the data-taking period in 2015 and 2016 was approximately 1 kHz [19]. After applying criteria to ensure good ATLAS detector operation, the total integrated luminosity useful for data analysis is 36.1 fb −1 . The uncertainty in the combined 2015+2016 integrated luminosity is 2.1%. It is derived, following a methodology similar to that detailed in Ref. [20], and using the LUCID-2 detector for the baseline luminosity measurements [21], from calibration of the luminosity scale using x-y beam-separation scans.

Simulation of signal and backgrounds
Simulated signal and background events were produced with various Monte Carlo event generators, processed through a full ATLAS detector simulation [22] using G 4 [23], and then reconstructed with the same procedure used for data. Additional pp interactions (pile-up), in the same and neighbouring bunch crossings, were overlaid on the hard-scattering process in the MC simulation. The MC events were then reweighted to reproduce the distribution of the number of interactions per bunch crossing observed in data.
For the signal modeling Sherpa 2.2.2 [24] with the NNPDF3.0 NNLO PDF set [25] is used as the baseline event generator. The signal sample was generated with up to three additional final-state partons at leading order (LO) and up to one additional final-state parton at next-to-leading order (NLO). Alternative signal samples, the first generated using S 2. Signal samples with non-zero anomalous triple gauge-boson couplings were also generated using S 2.1.1 with the CT10 PDF set. The values of coupling constants used in the generation are chosen to be equal to the expected limits obtained in a previous ATLAS study [5].
Background events containing Z bosons with associated jets were simulated using S 2.1.1 with the CT10 PDF set, while background events containing W bosons with associated jets were simulated using S 2.2.0 with the NNPDF3.0 NNLO PDF set. For both of these processes the matrix elements were calculated for up to two partons at NLO and four partons at LO. Background events containing a photon with associated jets were simulated using S 2.1.1 with the CT10 PDF set. Matrix elements were calculated with up to four partons at LO. Background events containing a lepton pair and a photon with associated jets were simulated using S 2.2.2 with the NNPDF3.0 NNLO PDF set. Matrix elements including all diagrams with three electroweak couplings were calculated for up to one parton at NLO and up to three partons at LO.

Selection of Z(νν)γ events
The event selection criteria are chosen to provide precise cross-section measurements of Z(νν)γ production and good sensitivity to anomalous gauge-boson couplings between photons and Z bosons. The selection is optimized for obtaining a high signal efficiency together with good background rejection.
Events are required to have been recorded with stable beam conditions and with all relevant detector subsystems operational. Event candidates in both data and MC simulation are selected using the lowest-E T unprescaled single-photon trigger: this requires the presence of at least one cluster of energy deposition in the ECAL with transverse energy E T larger than 140 GeV, satisfying the loose identification criteria described in Ref. [29]. The trigger efficiency is greater than 98% for photons selected for this analysis.

Object selection
Photon candidates are reconstructed [30] from ECAL energy clusters with |η| < 2.37 and E T > 150 GeV. They are classified either as converted (candidates with a matching reconstructed conversion vertex or a matching track consistent with having originated from a photon conversion) or as unconverted (all other candidates). Both kinds of photon candidates are used in the analysis. Electron candidates are reconstructed [31] from ECAL energy clusters with |η| < 2.47 that are associated with a reconstructed track in the ID with transverse momentum p T > 7 GeV. The ECAL cluster of the electron/photon candidate must lie outside the transition region between the barrel and endcap (1.37 < |η| < 1.52). Muon candidates are reconstructed from tracks in the MS that have been matched to a corresponding track in the inner detector, and are referred to as "combined muons". The combined track is required to have p T > 7 GeV and |η| < 2.7.
The shower shapes produced in the ECAL are used to identify photons and electrons. Photons are required to pass all the requirements on shower shape variables which correspond to the tight photon identification criteria [29]. The tight photon identification efficiency ranges from 88% (96%) to 92% (98%) for unconverted (converted) photons with p T > 100 GeV. A sample of "preselected" photons, used for the calculation of missing transverse momentum, are required to satisfy the less restrictive loose identification criteria of Ref. [29]. Electron candidates are required to satisfy loose [31] electron identification criteria, whose efficiency is greater than 84%. Muon candidates are required to satisfy tight identification criteria as described in Ref.
[32], with efficiency greater than 90% for combined muons used in the selection.
Electron and muon candidates are required to originate from the primary vertex2 by demanding that the significance of the transverse impact parameter, defined as the absolute value of the track's transverse impact parameter, d 0 , measured relative to the beam trajectory, divided by its uncertainty, σ d 0 , satisfy |d 0 |/σ d 0 < 3 for muons and |d 0 |/σ d 0 < 5 for electrons. The difference z 0 between the value of the z coordinate of the point on the track at which d 0 is defined, and the longitudinal position of the primary vertex, is required to satisfy |z 0 · sin(θ)| < 0.5 mm for both the muons and electrons.
Photon, electron and muon candidates are required to be isolated from other particles using the following criteria: the total transverse energy in ECAL energy clusters within ∆R = 0.4 of the photon candidate is required to be less than 2.45 GeV + 0.022 · E γ T , and the scalar sum of the transverse momenta of the tracks located within a distance ∆R = 0.2 of the photon candidate is required to be less than 0.05 · p γ T . For preselected photons, isolation criteria are not applied. For muons and electrons, the isolation requirement is based on track information and is tuned to have an efficiency of at least 99% [32].
Jets are reconstructed from topological clusters in the calorimeter [33] using the anti-k t algorithm [34] with a radius parameter of R = 0.4. Events with jets arising from detector noise or other non-collision sources are discarded [35]. A multivariate combination of track-based variables is used to suppress jets originating from pile-up in the ID acceptance [36]. The energy of each jet is calibrated and corrected for detector effects using a combination of simulated events and in situ methods [37] using data collected at √ s = 13 TeV. The selected jets are required to have p T larger than 50 GeV and |η| < 4.5.
The missing transverse momentum is defined as the negative vector sum of the transverse momenta of all reconstructed physics objects in the event [38] (leptons with p T > 7 GeV, preselected photons with p T > 10 GeV and jets with p T > 20 GeV), plus a "soft term" incorporating tracks from the primary vertex that are not associated with any such objects [39]. The resulting vector is denoted ì E miss T since it includes calorimetric energy measurements, and its magnitude E miss T is used as a measure of the total transverse momentum of neutrinos in the event.
To resolve ambiguities in the object reconstruction, jet candidates lying within ∆R = 0.3 of the photon candidates are removed.

Signal region definition
The signal region (SR) is defined to have exactly one tight isolated photon, as described above. In order to reduce the contamination from events that do not contain high-energy neutrinos (mainly γ + jet background with fake E miss T from jet momenta mismeasurements) the selected events are required to have E miss T > 150 GeV. To reduce the number of W( ν)γ and Z( )γ events, a lepton veto is applied: events with any selected electrons or muons are discarded. A requirement of at least 10.5 GeV 1/2 for the E miss T significance, defined as E miss T / Σp jet T + E γ T , further suppresses background contributions with fake 2 Each primary vertex candidate is reconstructed from at least two associated tracks with p T > 0.4 GeV. The primary vertex is selected among the primary vertex candidates as the one with the highest sum of the squared transverse momenta of its associated tracks.

E miss
T . An additional angular separation requirement ∆φ( ì E miss T , γ) > π/2 is made, which suppresses the pp → W(eν) + X background. These object and event selection requirements define the reconstructionlevel fiducial region and are summarized in Table 1.

Photons
Leptons To simplify the interpretation of the results and comparison with theory predictions, the cross section is measured in an extended fiducial region, defined at particle level3 in Table 2. Compared with the fiducial region, the extended fiducial region removes requirements on E miss T significance, ∆φ( ì E miss T , γ), the lepton veto and the transition η region for photons. In the signal event selection at particle level, the E miss T significance and ∆φ( ì E miss T , γ) are given by p νν , respectively. Photon isolation at the particle level is performed using the same requirements and cone sizes as described for the reconstruction-level isolation in Section 3.1.

Background estimation
Backgrounds to the Z(νν)γ signal originate from several sources. The dominant sources (listed in decreasing order of importance) are estimated with data-driven techniques: electroweak processes such as W( ν)γ, where the lepton is not detected; events with prompt photons and mismeasured jet momenta that gives rise to missing transverse momentum; events with real E miss T from neutrinos (such as Z(νν) or W(eν)) and misidentified photons from either electrons or jets. The procedures used to estimate these backgrounds closely follow those of the previous ATLAS measurement [5]. A less important source is γ (mainly ττγ) production, which is estimated from MC simulation and is expected to contribute roughly 1% of the selected event yield. In the following, each source of background is discussed in detail together with the method used for its estimation.
Misidentified events from W( ν)γ production are one of the dominant background contributions. A large fraction (about 60%) of this contamination arises from W(τν)γ events. Photon+jets events form another sizeable background contribution to the signal region. For the estimation of these backgrounds, two control regions (CRs) are defined by selecting events with the same criteria used for the SR but requiring either exactly one charged lepton (e or µ) in the event, or requiring the E miss T significance to be less than 10.5 GeV 1/2 . The first CR is enriched with W( ν)γ events (about 77%) while the second CR is enriched with γ+jets events (about 55%). The use of the 1-lepton (e or µ) control region for the estimation of the W( ν)γ background to the signal region, where can be any of e, µ or τ, relies on the assumption of lepton flavour universality. A simultaneous fit to the background-enriched CRs is performed to allow the CR data to constrain the yield of these main backgrounds, initially estimated with MC simulation, by establishing the normalization factors for the W( ν)γ and γ+jets background contribution as described in Refs. [5,40]. The same background normalization factors are assumed in the CR and SR and the fit uncertainties on these factors accounts for the uncertainty from this assumption. The normalization factor for the W γ background is found to be close to one, while the normalization factor for the γ+jets background is 1.7±0.5, since the pre-fit expectation is computed at LO, for which higher-order corrections would be expected to be considerable. The kinematic distributions of these backgrounds are taken from the MC predictions and the individual bins are allowed to float in the fit within their uncertainties. The variations of the background yields due to each of the experimental and MC modelling uncertainties reported in Section 5.1 are treated as Gaussian-distributed nuisance parameters in the likelihood function fit used to obtain the final background predictions in the SR. The dominant systematic uncertainties in the W( ν)γ process come from MC modelling (mostly due to the QCD scale uncertainty) and from the uncertainty in the electron-photon energy scale. Their contributions are 5.8% and 3.8%, respectively. The systematic uncertainty for γ+jets events is also dominated by the QCD scale component, and amounts to approximately 19%.
Misidentification of electrons as photons also contributes to the background yield in the signal region. The main source of this background is the inclusive W(eν) process, but contributions also arise from the single top-quark and tt production processes. The estimation of the size of these background contributions is done in two steps. The first is the determination of the probability for an electron to be misidentified as a photon using Z(e + e − ) decays reconstructed as e + γ, as described in Refs. [5,40]. The probability of observing an e + γ pair with invariant mass near the Z boson mass is used to determine an electron-tophoton fake factor f e→γ . The fake factor is found to vary between 0.6% to 2.7%, depending on the photon's η and p T . The second step is the construction of a control region by applying the nominal ννγ selection criteria described in Section 3, with the exception that an electron is required instead of the final-state photon, leading to a control region dominated by the W(eν)+jets process. The estimated background is then given by the number of events in the chosen control sample scaled by the electron-to-photon fake factor. The statistical uncertainty is determined by the size of the control sample and does not exceed 5%. The systematic uncertainty for this background varies from 13% to 25%, depending on the photon p T and η, and is dominated by the difference between the fake rates obtained from Z(ee) and W(eν) MC events. The total relative systematic uncertainty of this background estimate is less than 15%, since the main contribution comes from the most populated central pseudorapidity region and has p T < 250 GeV, where the systematics on the fake factor is the smallest.
To estimate the contribution from background due to the misidentification of jets as photons, a twodimensional sideband method is used, as described in Ref. [5]. In this method the Z(νν)γ events are separated into one signal and three control regions. Events in the signal region require the photon to satisfy the nominal photon isolation and tight identification requirements, as described in Section 3. The photon isolation and identification criteria are modified in order to build the control regions, which are disjoint from each other and from the signal region. The modified photon identification criteria requires photons to pass a "non-tight" identification but fail the tight identification. The non-tight selection criteria remove requirements on four out of the nine shower shape variables required for tight photons; the variables that are removed from the list of requirements are those that are least correlated with calorimeter isolation [41]. Two of the control regions are defined by modifying either the photon isolation or photon identification requirement, while for the third control region both the isolation and identification requirements are modified. The number of background events in the signal region can be derived from the number of observed events in the control regions according to the methodology described in Ref. [5]. The statistical uncertainty of the background is established by the event yields in the four regions, while the systematic uncertainty is 29% and is dominated by the size of changes to the background estimate arising from the variation of the control regions' definitions, which leads to changes exceeding the expected size of the statistical fluctuations. This systematic uncertainty also covers possible effects due to the correlation between the isolation and identification criteria.
The resulting signal and background composition is shown in Table 3. Kinematic distributions of the photon transverse energy, missing transverse momentum, and jet multiplicity in the fiducial region for the inclusive selection (N jets ≥ 0) are shown in Figure 2. Kinematic distributions of the photon transverse energy and the missing transverse momentum in the fiducial region for the exclusive selection (N jets = 0) are shown in Figure 3.
Good agreement between data and the SM expectation is observed in the shapes of most of the measured distributions. The discrepancy in the last bin of the inclusive E γ T distribution, after dedicated checks, was found to be consistent with having arisen from a statistical fluctuation of the data. The uncertainties shown in the figures are treated as being uncorrelated among different systematic sources and different backgrounds.

Description of the cross-section measurements
The number of signal events is determined by subtracting the estimated backgrounds from the number of observed events. The signal yield is then corrected for detection efficiencies in the fiducial region, defined 2710 ± 40 ± 130 N data (obs) 3812 2599 Table 3: Summary of observed and expected yields (all backgrounds and signal) for events passing the selection requirements in data for the inclusive (N jets ≥ 0) and exclusive (N jets = 0) selections. The W γ and γ+jet backgrounds are scaled by the normalization factor from the fit, luminosity and cross section. The e → γ and jet → γ backgrounds are estimated using data-driven techniques. The row labelled "N sig (exp)" corresponds to the S NLO prediction. The row labelled "N sig+bkg total " corresponds to the sum of the expected background contributions and expected signal. The first uncertainty is statistical, while the second is systematic.
in Table 1. The integrated cross section in the extended fiducial region, defined in Table 2, is calculated as where N is the number of observed candidate events, B is the expected number of background events and ∫ Ld t is the integrated luminosity corresponding to the analyzed data set. The factors C Zγ and A Zγ correct for detection efficiency and acceptance, respectively: • C Zγ is defined as the number of reconstructed signal events satisfying all selection criteria divided by the number of events that, at particle level, meet the acceptance criteria of the fiducial region; • A Zγ is defined as the number of signal events within the fiducial region divided by the number of signal events within the extended fiducial region, with both numerator and denominator defined at particle level.
The corrections A Zγ and C Zγ are determined using the Zγ signal events generated by S and are summarized in Table 4 along with their uncertainties.

Systematic uncertainties
Systematic uncertainties in the acceptances A Zγ are evaluated by varying the PDF sets, the value of α S , the renormalization and factorization scales (QCD scale uncertainty), and the Monte Carlo parameter tunes for the parton shower (PS) and multi-parton interactions (MPI). In total, 100 error sets are checked for the NNPDF3.0 NNLO PDF variation, leading to a relative uncertainty of 0.76% for the inclusive case and 0.35% for the exclusive case. These numbers fully cover variations arising from the use of alternative PDF sets such as CT14 [42] and MMHT2014 [43]. The uncertainty from α S is estimated by Data Data Data varying it within the range of its world-average value as provided in Ref. [44] and is found to be negligible. The effects of the renormalization and factorization scale uncertainties are assessed by varying these two scales independently by a factor of two from their nominal values, removing combinations where the two variations differ by a factor of four, and taking the envelope of the resulting cross-section variations as the size of the associated systematic uncertainty. Uncertainties from the PS and MPI are evaluated using a series of eigentunes for the P generator with its A14 parameter tune [45]. The size of the uncertainty from the renormalization and factorization scales does not exceed 3.0%, while PS and MPI Data Data   uncertainties cause variations from 1.9% to 2.7% for the inclusive and exclusive cases, respectively. The total uncertainties in the acceptance factors are summarized in Table 4.
Systematic uncertainties affecting the correction factor C Zγ include contributions arising from uncertainties in the efficiencies of the trigger, reconstruction and particle identification, as well as the uncertainties in the energy, momentum scales and resolutions of the final-state objects. Additional systematic uncertainty sources arise from the modelling of particle spectra and pile-up events. Spectrum modelling uncertainties are estimated by varying the PDF set and QCD scales as described above for the case of the acceptance factor A Zγ . Some of these contributions are found to have a non-linear dependence on photon transverse energy, E miss T or jet multiplicity. In these cases, uncertainties estimated as a function of these observables are used in the unfolding process of Section 5.5 when the corresponding kinematic distributions are derived from the signal sample. Table 5 displays the size of the individual contributions to the uncertainties in the C Zγ factor; the total uncertainty is summarized in Table 4 Table 5: Relative systematic uncertainties in the signal correction factor C Zγ for the inclusive and exclusive Zγ measurements.

Integrated extended fiducial cross section
The measurements of the cross sections, along with their uncertainties, are based on the maximization of the profile-likelihood ratio where L represents the likelihood function, σ is the cross section, and θ are the nuisance parameters corresponding to the sources of systematic uncertainty. Theσ andθ terms denote the unconditional maximum-likelihood estimate of the parameters, i.e., the parameters for which the likelihood is maximized for both σ and θ. The termθ(σ) denotes the value of θ that maximizes L for a given value of σ.
The likelihood function is defined as representing the product of the Poisson probability of observing N events, given expectations of S for the signal and B for the background, multiplied by the Gaussian constraints θ on the systematic uncertainties, with central values θ 0 from auxiliary measurements, as described in Section 5.1.
The measured cross sections for Z(νν)γ production in the extended fiducial region are summarized in Table 6, along with the theoretical predictions of the M [46] generator described in Section 5.4. The measured cross sections agree with the SM expectations to within one standard deviation. Systematic uncertainties arise from uncertainties in the acceptances and correction factors, as well as from uncertainties in the background estimates. These two sources contribute roughly equally to the uncertainty in the measured cross sections. Compared with the Zγ measurements at  Table 6: Measured cross sections for Z(νν)γ production within the extended fiducial region for a centre-of-mass energy of √ s = 13 TeV, with corresponding SM expectations obtained from the M [46] generator at next-tonext-to-leading order in the strong coupling constant α S . uncertainty is significantly reduced. This improvement is due primarily to the reduction of systematic uncertainty allowed by the data-driven estimate of the γ+jets and W γ backgrounds.
An overall check of the SM predictions is done with the M generator [47]. Cross sections obtained by M (inclusive case: σ ext.fid. = 78.6 ± 0.4 ± 4.4 fb; exclusive case: σ ext.fid. = 55.8 ± 0.3 ± 3.6 fb, where the uncertainties are statistical and systematic, respectively) are found to be consistent with those from M to within their statistical uncertainty.

Standard Model calculations
The resulting measurement of the rate and kinematic distributions of Zγ production is compared with SM expectations using the parton shower Monte Carlo generator S and the NNLO parton-level generators M and M . The NNPDF3.0 PDF set was used for the S , M and M generation. The values of the renormalization and factorization scales were set to m Zγ for the M and M NNLO generation of the Zγ process.
The photon isolation criterion at the parton level is applied by considering a cone of variable opening angle ∆R (with maximum opening angle ∆R max = 0.1) centred around the photon direction, and requiring that the transverse energy flow inside that cone be always less than a given fraction of the photon p T ; this fraction is set to 0.1 when ∆R = ∆R max , and tends smoothly to zero when ∆R → 0, as described in Ref. [48]. Due to this procedure, the contribution from photon fragmentation to the NNLO calculations of the M and M SM predictions is zero.
Events generated with S , as described in Section 2.2, are also compared with the particle-level measurements. For the NNLO parton-level predictions, parton-to-particle correction factors C * (parton→particle) must be applied in order to obtain the particle-level cross sections. These correction factors are computed as the ratios of the pp → Zγ cross sections predicted by S with hadronization and the underlying event disabled to the cross sections with them enabled. The systematic uncertainty in the correction factors is evaluated by using a signal sample from an alternative generator (MG5_ MC@NLO), taking the resulting change in C * (parton→particle) as the one-sided size of a symmetrized value for the uncertainty. This accounts for uncertainties in both the parton shower modelling and the description of the underlying event.
The value of C * (parton→particle) is found to be 0.87 ± 0.04 for the inclusive predictions and 0.97 ± 0.04 for the exclusive predictions. For the exclusive case, the parton-to-particle correction includes an additional contribution from the jet veto, which compensates for the difference in the photon isolation between the parton and particle levels. The particle-level cross sections are then obtained by multiplying the NNLO parton-level cross-section values by the C * (parton→particle) correction factors, and are displayed in Table 6.
The systematic uncertainty in the expected NNLO SM cross sections arising from uncertainties in the QCD scale is estimated by varying the QCD scales by factors of 0.5 and 2.0 (separately for the renormalization and factorization scales, removing combinations where the two variations differ by a factor of four). The effect of the QCD scale uncertainty on the prediction for the first bin of the various differential cross-section measurements also accounts for uncertainties arising from the incomplete cancellation of divergences associated with soft gluon emission in fixed-order perturbative calculations of Zγ production. This effect is appreciable because of the symmetric E γ T and p νν T thresholds used in defining the SR. The corresponding corrections are estimated conservatively from the cited MC generators by evaluating the degree of compensation of the divergence that arises when the p νν T (E γ T ) requirement is lowered to a value significantly below the value of the E γ T (p νν T ) requirement of 150 GeV. The systematic uncertainty due to the PDF choice is computed using the eigenvectors of the NNPDF 3.0 PDF set [25] and the envelope of the differences between the results obtained with the CT14 [42] and MMHT2014 [43] PDF sets, according to the PDF4LHC recommendations [49]. M predictions do not include the systematic uncertainty due to the PDF choice.

Differential extended fiducial cross section
The measurement of the Zγ production differential cross sections allows a comparison of experimental results with SM expectations for both the absolute rates and the shapes of kinematic distributions. The measurements are performed as a function of several observables that are sensitive to higher-order perturbative QCD corrections [50] and to a possible manifestation of aTGCs [51]: photon transverse energy (E γ T ), the transverse momentum of the neutrino-antineutrino pair (p νν T ), and jet multiplicity (N jets ). The differential cross sections are defined in the extended fiducial region, and are extracted with an unfolding procedure that corrects for measurement inefficiencies and resolution effects that modify the observed distributions. The procedure described in Ref. [5] is followed, using an iterative Bayesian method [52]. For each distribution, events from simulated signal MC samples are used to generate a response matrix that accounts for bin-to-bin migration between the reconstruction-level and particle-level distributions.
The statistical uncertainties of the unfolded distributions are estimated using pseudo-experiments, generated by fluctuating each bin of the observed spectrum according to a Poisson distribution with a mean value equal to the observed yield. The shape uncertainties arising from the limited size of the signal MC sample are also obtained by generating pseudo-experiments. The sources of systematic uncertainty are discussed in Section 5.1, with their impact on the unfolded distribution assessed by varying the response matrix for each of the systematic uncertainty sources by one standard deviation and combining the resulting differences from the nominal values in quadrature.
The differential cross sections as a function of E γ T and p νν T are shown in Figures 4 and 5, respectively, for both the inclusive and exclusive measurements. Figure 6 shows the cross section measured in bins of jet multiplicity. The values of the SM expectations shown in the figures are obtained as described in Section 5.4.
Good agreement with SM expectations is observed in all but the last bin of the E γ T inclusive distribution. This disagreement is a consequence of the corresponding disagreement observed in Figure 2, which was investigated and found to be consistent with having arisen from a statistical fluctuation of the data.

Limits on triple gauge-boson couplings
Vector-boson couplings are completely fixed within the Standard Model by the SU(2) L ×U(1) Y gauge structure. Their measurement is thus a crucial test of the model. Any deviation from the SM prediction is referred to as an anomalous coupling.
Within the framework of the effective vertex function approach [51], anomalous triple gauge-boson coupling contributions to Zγ production can be parameterized by four CP-violating (h V 1 , h V 2 ) and four CPconserving (h V 3 , h V 4 ) complex parameters. Here the V indices are Z and γ, and h Z i and h γ i are the parameters of Z Zγ and the Zγγ vertices, respectively. The h V 3 (h V 1 ) and h V 4 (h V 2 ) parameters correspond to the electric (magnetic) dipole and magnetic (electric) quadrupole transition moments of V, respectively [53].
All of these parameters are zero at tree level in the SM. Since the CP-conserving couplings h V 3,4 do not interfere with the CP-violating couplings h V 1,2 , and their sensitivities to aTGCs are nearly identical [51], the limits from this study are expressed solely in terms of the CP-conserving parameters h V 3,4 .
The yields of Zγ events with high E γ T from the exclusive (zero-jet) selection are used to set limits on h V 3,4 . The exclusive selection is used because it significantly reduces the SM contribution at high E γ T and therefore optimizes the sensitivity to anomalous couplings. The contribution from aTGCs increases with the E T of the photon, and the measurement of Zγ production is found to have the highest sensitivity to aTGCs by restricting the search to the portion of the extended fiducial region with E γ T greater than 600 GeV. [fb/GeV]    Cross-section values modified by the inclusion of aTGCs (σ aTGC Zγ ) are obtained from the M generator. These values are displayed in Table 7 for several combinations of choices of the Z Zγ vertex parameters h Z 3 and h Z 4 . The expected number of Zγ events in the aTGC region (N aTGC The anomalous couplings influence the kinematic properties of the Zγ events and thus the efficiency factor of the event reconstruction (C Zγ ). The maximum variation of C Zγ due to non-zero aTGC parameters within the aTGC limits measured in this paper (about 7%) is adopted as an additional systematic uncertainty. The effect of anomalous couplings on the acceptance factor (A Zγ ) and parton-to-particle factor (C * (parton→particle) ) is an order of magnitude smaller than that on C Zγ , and so is neglected.
Limits on a given aTGC parameters are extracted from a frequentist profile-likelihood test similar to that of Section 5.3. The profile likelihood depends on the observed number of exclusive Zγ candidate events, the amount of expected signal as a function of aTGC given by Eq.
(1), and the estimated number of background events. A point in the aTGC space is accepted (rejected) at the 95% confidence level (CL) if fewer (more) than 95% of randomly generated pseudo-experiments exhibit larger profile-likelihood ratio values than that observed in data. In this context, a pseudo-experiment is a set of randomly generated numbers of events that follow a Poisson distribution with mean equal to the sum of the number of expected signal events and the estimated number of background events. Systematic uncertainties are incorporated into the pseudo-experiments via a set of nuisance parameters with correlated Gaussian constraints. All nuisance parameters are allowed to fluctuate in the pseudo-experiments.
No evidence of anomalous couplings is observed. The allowed 95% CL ranges for the anomalous couplings are shown in Table 8 for Z Zγ (h Z 3 and h Z 4 ) and the Zγγ (h γ 3 and h γ 4 ) vertices. Limits on anomalous couplings imposed by this analysis are 3-7 times more stringent than those from prior studies [5].  Figure 7, which are the only such pairs that are expected to interfere [51].
Allowed ranges are also determined for parameters of the effective field theory (EFT) of Ref. [54], which includes four dimension-8 operators describing aTGC interactions of neutral gauge bosons. The    coefficients of these operators are denoted C BW /Λ 4 , C BW /Λ 4 , C WW /Λ 4 and C BB /Λ 4 , as described in Ref. [55]. The parameter Λ has the dimension of mass and is associated with the energy scale of the new physics described by the EFT. The 95% CL limits on these EFT parameters displayed in Table 9 are derived from the limits of Table 8  C BB /Λ 4 (−0.24, 0.24) (−0.28, 0.27) Table 9: Observed and expected one-dimensional 95% CL limits on the C BW /Λ 4 , C BW /Λ 4 , C WW /Λ 4 and C BB /Λ 4 EFT parameters, assuming that any excess in data over the SM expectation is due solely to a non-zero value of the parameter C BW /Λ 4 , C BW /Λ 4 , C WW /Λ 4 or C BB /Λ 4 . For each row, all parameters other than the one under study are set to 0.

Conclusion
The cross section for the production of a Z boson in association with an isolated high-energy photon is measured using 36.1 fb −1 of pp collisions at √ s = 13 TeV collected with the ATLAS detector at the LHC. The analysis uses the invisible decay mode Z → νν of the Z boson, and is performed in a fiducial phase space closely matching the detector acceptance.
Kinematic distributions are presented in terms of differential cross sections as a function of the transverse energy of the photon, the missing transverse momentum, and the jet multiplicity. Measurements are made for both the inclusive case, with no requirements on the system recoiling against the Zγ pair, and the exclusive case in which no jets with p T > 50 GeV are allowed within |η| < 4.5.
The results are compared with SM expectations derived from a parton shower Monte Carlo generator (S ) and from parton-level perturbative calculations carried out at NNLO (M and M ). Good agreement is observed between the measured and expected total and differential cross sections.
In the absence of significant deviations from SM expectations, the data are used to set limits on anomalous couplings of photons and Z bosons. Limits on aTGCs are determined using a modified SM Lagrangian that includes operators proportional to the h V 3 and h V 4 (V = Z or γ) parameters of the vertex function parameterization of aTGC contributions to Zγ production. The limits are also transformed into limits on the C BW /Λ 4 , C BW /Λ 4 , C WW /Λ 4 and C BB /Λ 4 parameters of an effective field theory formulation of aTGC effects. The limits obtained from the current study are 3-7 times more stringent than those available prior to this study.