NLO QCD predictions for \(Z+\gamma \) + jets production with Sherpa
Abstract
We present precise predictions for prompt photon production in association with a Z boson and jets. They are obtained within the Sherpa framework as a consistently merged inclusive sample. Leptonic decays of the Z boson are fully included in the calculation with all offshell effects. Virtual matrix elements are provided by OpenLoops and partonshower effects are simulated with a dipole parton shower. Thanks to the NLO QCD corrections included not only for inclusive \(Z\gamma \) production but also for the \(Z\gamma \) + 1jet process we find significantly reduced systematic uncertainties and very good agreement with experimental measurements at \(\sqrt{s}=8\,\mathrm {TeV}\). Predictions at \(\sqrt{s}=13\,\mathrm {TeV}\) are displayed including a study of theoretical uncertainties. In view of an application of these simulations within LHC experiments, we discuss in detail the necessary combination with a simulation of the Z + jets final state. In addition to a corresponding prescription we introduce recommended cross checks to avoid common pitfalls during the overlap removal between the two samples.
1 Introduction
The production of a Z boson is one of the standard candle processes at hadron colliders like the LHC. The massive boson is often produced in association with photons, which are typically lowenergetic or collinear with charged final state particles. Cases where one of the photons happens to be wellisolated and highenergetic can be regarded as an individual important final state, \(Z\gamma \) production.
\(Z\gamma \) production plays an important role both as a signal and as a background process at the LHC. The absence of couplings of the photon to the uncharged Z boson in the Standard Model can be probed by measurements in this channel, resulting in differential cross sections and limits on anomalous couplings from LEP experiments [1, 2, 3, 4], Tevatron experiments [5, 6, 7], and by ATLAS [8, 9, 10] and CMS [11, 12, 13, 14].
The \(Z\gamma \) process also constitutes an irreducible background in the search for the Higgs boson [15] or new massive gauge bosons decaying to \(Z\gamma \) [16, 17] or in more inclusive searches in final states containing a photon and missing transverse momentum [18, 19, 20, 21].
Theoretical predictions for \(Z\gamma \) production can be divided into onshell and offshell calculations. Results for onshell \(Z\gamma \) production leave out the decays of the Z boson or include them only in a narrowwidth or pole approximation. Beyond the leadingorder results [22], the first onshell higherorder calculations included NLO QCD [23, 24, 25] and, more recently, also NNLO QCD corrections [26]. In reality, the Z boson is unstable and is thus never produced as an onshell finalstate particle. Recent calculations take this finite width into account and provide predictions for the offshell \(\ell \ell \gamma \) final state. The most accurate offshell predictions contain NNLO QCD [27, 28] and NLO EW corrections [29].
Experimental analyses at the LHC rely heavily on theoretical predictions for their signal and background processes. Fixedorder predictions as listed above can provide such an input only to some extent. While they describe the dominant features of the given final state objects at the parton level, they are not made to simulate a realistic behaviour of the hadronic final state. MonteCarlo event generator programs on the other hand combine fixedorder predictions and an approximate allorder resummation of QCD corrections to enable a full simulation at the hadron level. Different approaches and programs are available, but the simulation currently in use in LHC experiments for \(Z\gamma \) production is generated with leadingorder multileg generators using approaches like CKKW(L) or MLM merging [30, 31, 32, 33, 34, 35, 36]. For the related process of \(W\gamma \) production, an implementation of a NLO QCD calculation of the inclusive process matched to a parton shower exists within the Powheg framework [37].
2 Methods
2.1 Matching and merging with Sherpa at NLO
To obtain NLOaccurate multijetmerged predictions the “MEPS@NLO” formalism [38] is applied to \(Z\gamma \) production within the Sherpa framework. It combines two essential ingredients, NLO + partonshower matching and multijet merging, which are briefly summarised in the following.
For the combination of NLOaccurate matrixelement calculations with a parton shower (PS), a matching procedure to avoid double counting of the QCD emission effects at \(\mathcal {O}(\alpha _s)\) is needed. Here, the prescription proposed in [39, 40] is applied, both to \(pp\rightarrow Z\gamma \) and \(pp\rightarrow Z\gamma \) + jet simulations. It is based on the original MC@NLO method [41] but extends it to a fully colourcorrect formulation also in the limit of soft emissions.
Both simulations and further PS emissions are then consistently merged into one inclusive \(pp\rightarrow Z\gamma +0\),1j@NLO sample using the “MEPS@NLO” method [38]. Shower emissions above a predefined separation criterion, \(Q_\text {CUT}\), are vetoed in the lower multiplicity contribution, and Sudakov factors are applied where appropriate such as to make this contribution exclusive and allow the combination with a higher jet multiplicity. This applies not only to the emissions from the parton shower but to all contributions of the NLOmatched emission, including the hard remainder (“\(\mathbb {H}\) events”). In the application of the Sudakov factors care has to be taken to remove the \(\mathcal {O}(\alpha _s)\) contribution which is already present in the NLO(matched) emission.
Beyond the processes simulated at NLO accuracy, highermultiplicity processes can be added to the simulation at LO accuracy to improve the modelling of high jet multiplicities beyond the partonshower approximation.
Similar to merging methods at LO, an appropriate scale choice for the evaluation of multijet configurations is obtained by statistically identifying a partonshower history in the matrixelement final state. To that end, the parton shower is run in reverse mode, i.e. the closest parton pair is identified according to the shower splitting probabilities and then recombined using the kinematical properties of the shower. When applied recursively, this clustering results in a core process, e.g. \(pp\rightarrow Z\gamma \), and in an ordered history of shower emissions. The factorisation scale is then determined by a typical momentum transfer within the core process (core scale), and the renormalisation scale is calculated from the n identified branchings to resemble \(\alpha _s^n(\mu _R)=\prod _{i=1}^n\alpha _s(k_{\perp ,i})\), where \(k_{\perp ,i}\) is a suitably scaled relative transverse momentum encountered in the ith splitting. If the clustering algorithm is restricted to splitting functions active in the shower (“exclusive clustering”) it stops if no possible ordered shower history can be reconstructed. Thus, for events with very hard QCD emissions, the core process can also contain jets and contribute to the renormalisation scale accordingly.
By contrast, the clustering applied by default (“inclusive clustering”) may include electroweak combinations to preserve the ordering in the history, even though electroweak splitting functions are not enabled in the shower.
An example configuration which demonstrates the different treatment in the inclusive and the exclusive clustering algorithm is shown in Figs. 1 and 2. The possible core processes will become relevant for the discussion of different scale choices later in this work.
2.2 Soft photon resummation with YFS
It is worth noting that in a Monte Carlo code photons radiated by YFS are—in contrast to e.g. QED showers—unordered.
2.3 Isolated photons
3 NLOaccurate multijet predictions for \(Z\gamma \) production
3.1 Setup
All results in this publication are obtained with the Monte Carlo event generator Sherpa [45] using merged calculations. Version 2.2.2 of the program was upgraded in the context of this work to enable the \(Z\gamma \) process and to generically improve the stability of the onthefly assessment of systematic uncertainties.
 MEPS@NLO

\(pp \rightarrow e^+ e^ \gamma \,+\, 0,1\mathrm {jets@NLO} + 2,3\mathrm {jets@LO} \),
 MEPS@LO

\(pp \rightarrow e^+ e^ \gamma + 0,1,2,3\mathrm {jets@LO}\).
The matrix elements are calculated by the internal matrixelement generators Amegic++ [46] and Comix [47]. Virtual diagrams are calculated by OpenLoops 1.3.1 [48], using CutTools [49] and OneLoop [50]. In all MEPS@NLO setups Amegic++ is used only for Bornlike processes. All realsubtracted contributions and the leadingorder diagrams of higher multiplicity are calculated by Comix. The merging cut \(Q_\mathrm {CUT}\) is set to 30\(\,\mathrm {GeV}\).
As parton distribution functions NNPDF3.0 [54] sets are used, taking the NLO set for MEPS@LO calculations and the NNLO set for MEPS@NLO. The running of \(\alpha _\mathrm {s}\) and its value at \(M_\mathrm {Z}\) are thereby set according to these PDF sets, resulting in \(\alpha _\mathrm {s}(M_\mathrm {Z})=0.118\) and a running at two(three) loop order when using the NLO(NNLO) sets. YFS is set active and includes matrixelement corrections for further photon emissions.
For the event generation, all partonlevel cuts are set to be more inclusive than the respective analysis cuts. As described in Sect. 2.3, it is not possible to use the experimental isolation criterion; instead the smooth cone criterion is used and validated for two different sets of parameters.
When comparing to experimental data the simulation is performed including the default multiple interactions [55, 56] and hadronisation models [57]. The final state analyses are done within the Rivet framework [58].
3.2 Predictions for \(\sqrt{s}=\)13\(\,\mathrm {TeV}\)
3.2.1 Merging cut variation
Using the ME + PS merging method defined in Sect. 2.1 a new parameter is introduced, the merging cut \(Q_{\mathrm {CUT}}\). It separates the different phase space regions for the parton shower and highermultiplicity matrix elements. Since this parameter is unphysical, physical observables should be independent of its exact value as long it is chosen in a reasonable range.
An interesting observable for checking this behaviour are the splitting scales as defined by the \(k_\perp \)algorithm [59]. All final state partons^{2} are clustered to jets according to this algorithm. The splitting scale \(d_\mathrm {(n1)(n)}\) is then defined as the jet measure which describes the cluster step of a nparton final state to a \((n1)\) final state. Following this, \(d_{01}\) gives the \(p_\perp \) of the hardest jet and \(d_{12}\) either describes the production of a second jet or the second splitting of the first jet in the case there is only one.
In the context of matching and merging, jets can emerge either from highermultiplicity matrix elements or from the parton shower. Since the \(k_\perp \) algorithm uses a jet measure which is very similar to the jet criterion used by the merging algorithm, its splitting scales are very sensitive observables to study the interplay between LO/NLO matrix elements and partonic showers.
In Fig. 3, these differential jet rates are evaluated for the first two splittings while varying the merging cut between 20 and 40\(\,\mathrm {GeV}\). Figure 3a shows the hardest splitting scale. This jet rate is sensitive to the transition from the zerojet NLO matrix element^{3} with a shower emission at \(Q_\mathrm {emission} < Q_\mathrm {CUT}\) to a onejet NLO matrix element with \(Q_\mathrm {emission} > Q_\mathrm {CUT}\).
By contrast, in Fig. 3b the subleading splitting scale is shown. This splitting probes the transition from the onejet NLO matrix element with an unresolved emission to a LO matrix element with two resolved jets.
3.2.2 QCD core scale choice
This table summarizes all cuts which define the differential cross section analysis for 13\(\,\mathrm {TeV}\). The leading photon has to be isolated from all other particles by fulfilling the requirement \(\sum _{\mathrm {\Delta }R<0.4} E < \epsilon _{\mathrm {iso}}E^{\gamma }\) where the sum includes all particles which have an angular distance of 0.4 or less to the photon axis. In addition, the photon is required to not come from a hadron decay. The leptons are dressed with all photons not coming from a hadron decay and within \(\mathrm {\Delta }R < 0.1\)
Lepton  \(p_\perp >25\) \(\,\mathrm {GeV}\), \(\eta <2.5\) 
Jet  \(E_\perp >30\) \(\,\mathrm {GeV}\), \(\eta <4.4\), \(\mathrm {\Delta }R(\mathrm {jet}\), \(e/\gamma ) > 0.3\) 
Boson  \(M_{e^+, e^}>40\) \(\,\mathrm {GeV}\) 
Photon  \(E_\perp >15\) \(\,\mathrm {GeV}\), \(\eta <2.5\) 
Isolation  \(\mathrm {\Delta }R(\gamma , e^\pm )>0.4\), \(\epsilon _{\mathrm {iso}}=0.5\) 
Since this is a merged calculation it is not possible to use these scale definitions directly. However, the STRICT_METS scale setter also allows one to use custom scales for the core process. In order to use this possibility, the clustering has to be restricted such that it results in a \(Z\gamma \) core process. This can be enforced by using the exclusive cluster mode which exactly reconstructs a possible shower history using QCD splittings only (see Sect. 2.1).
Altogether, four different scale choices are compared. The first one is inclusive, it uses the default settings of Sherpa as described in Sect. 3.1. By contrast, the three remaining setups make use of the exclusive cluster mode and use different core scales as defined in Fig. 4.
In Fig. 4 the differential \(E_\perp ^\gamma \) distribution and the \(p_\perp \) spectrum of the leading jet are compared for the four different scale schemes.
In the \(E_\perp ^\gamma \) spectrum all scale choices are in good agreement and differ by not more than 10%. Switching from the default to the exclusive clustering algorithm does not make a large difference for this observable. This also holds for all other observables which where measured by ATLAS and are discussed later on in Sect. 3.3, those measurements do not allow us to discriminate between the different scale choices.
By contrast, the difference is much higher when looking at the \(p_\perp \) spectrum of the leading jet. Here, all scale and cluster choices are in good agreement for low \(p_\perp \) but differ as soon as \(p_\perp \) exceeds 100\(\,\mathrm {GeV}\). At large values at order of 1000\(\,\mathrm {GeV}\) the difference reaches almost a factor of 2. There, the highest cross section corresponds to the core scale defined in Eq. (8) in combination with the exclusive cluster model, whereas the lowest cross section is given by the default settings. This is not surprising since such a high \(p_\perp \) region probes configurations where the jet is harder than the typical scale of this process, e.g. the Z mass. Such configurations are unordered in terms of the partonshower evolution variable and are thus very sensitive to the clustering definition. If here a \(pp \rightarrow Z \gamma + n ~\mathrm {partons}\) core process is determined but the core scale is evaluated solely based on Z and \(\gamma \), this scale will underestimate the physical scale and thus overestimate the strong couplings, resulting in a larger cross section.
3.2.3 Scale and PDF variation uncertainties
In addition to the core scale variations studied above, independent variations of the \(\mu _\mathrm {R}\) and \(\mu _\mathrm {F}\) scales are performed. All replicas of the NNPDF set are used to estimate the PDF uncertainty. These variations are performed using onthefly weights [62] and include only contributions from the matrix elements but not from the shower.
All figures are structured as follows. Each figure has three ratio plots. The main plot and the first ratio plot point out the difference between MEPS@NLO and MEPS@LO. Here, the MEPS@NLO prediction is chosen as reference. By contrast, the additional subplots show the size of all performed scale variations for each method as a ratio with respect to the corresponding nominal prediction.
Figures 5 and 6 show predictions for \(E_\perp ^\gamma \) and \(M_{ll\gamma }\). The corrections between MEPS@LO and MEPS@NLO are almost flat at a level of 20%. As expected, the scale uncertainties are reduced when moving from MEPS@LO to MEPS@NLO. The factorisation scale dependency vanishes almost completely for \(p_\perp ^\gamma \lesssim 60\,\mathrm {GeV}\) and \(m_{ll\gamma }\) close to the Z peak, compared to 10% in the MEPS@LO case. The renormalisation scale uncertainty is reduced by roughly a factor of 2 for both observables, it shrinks from 20 to 10% for high \(E_\perp ^\gamma \).
Here both the factorisation and the renormalisation scale uncertainties are reduced significantly for small \(H_\perp \) but have almost the same size if \(H_\perp \) exceeds \(200\,\mathrm {GeV}\). The leading jet \(p_\perp \) is depicted in Fig. 8 and shows a very similar behaviour. Again the renormalisation scale uncertainty is reduced from 15 to 5% for low \(p_\perp ^\mathrm {jet}\) but does not change for values from 200\(\,\mathrm {GeV}\) onwards. Finally, Fig. 9 shows the jet multiplicity. In the zerojet bin the factorisation scale dominates when using MEPS@LO, this uncertainty vanishes almost completely when moving to MEPS@NLO. However, the onejet bin is dominated by the renormalisation scale uncertainty. This uncertainty is reduced from 20 to 10% when moving to MEPS@NLO.
All of these observables only show minor improvements in the multijet regions. This is not surprising since observables which are sensitive to a high number of hard jets are hardly improved by the MEPS@NLO applied here. Only the zero and onejet matrix elements are calculated at nexttoleading order but the two and threejet calculations still have leadingorder accuracy. Both \(H_\perp \) and \(p_\perp ^\mathrm {jet}\) are dominated at large values by multijet configurations and are thus described at leadingorder accuracy only. This effect is reflected by the ratio between the MEPS@NLO and the MEPS@LO method, too. At low values of \(H_\perp \) and \(p_\perp ^\mathrm {jet}\) it amounts to 0.7 but increases to one for larger values.
3.3 Comparison with \(\sqrt{s}=\)8\(\,\mathrm {TeV}\) measurements
Both, the stability and the reduction of the perturbative uncertainties have been demonstrated in the 13 \(\,\mathrm {TeV}\) results in the last section. Now, the focus is on the comparison with recent experimental data and a study of the interplay between the smooth isolation criterion with the experimental one.
This section relies on a measurement of the ATLAS collaboration at 8\(\,\mathrm {TeV}\)[10], using 20.3 fb\(^{1}\) of data. In this measurement, final states with \(ll\gamma \) and up to three jets are studied. Here, the focus lies on the \(e^+ e^ \gamma + \mathrm {jets}\) final state. All cuts which define the extended, differential fiducial cross section are summarised in Table 2.
This table summarizes all cuts which define the extended, fiducial cross section in the measurement [10]. Leptons are dressed with all photons having an angular distance of \(\mathrm {\Delta }R<0.1\)
Lepton  \(p_\perp >25\) \(\,\mathrm {GeV}\), \(\eta <2.47\) 
Jet  \(E_\perp >30\) \(\,\mathrm {GeV}\), \(\eta <4.45\), \(\mathrm {\Delta } R(\mathrm {jet}\), \(e/\gamma ) > 0.3\) 
Boson  \(M_{e^+, e^}>40\) \(\,\mathrm {GeV}\) 
Photon  \(E_\perp >15\) \(\,\mathrm {GeV}\), \(\eta <2.37\) 
Isolation  \(\mathrm {\Delta }R(\gamma , e^\pm )>0.7\), \(\epsilon _{\mathrm {max}}=0.5\) 
Even if one assumes that both the final jet \(E_\perp \) and its direction are in perfect agreement with the closest parton \(E_\perp \) at matrixelement level, this criterion differs from the one used in our calculation and described in Sect. 2.3 when going to lower angular distances.
The second parameter set is thus chosen more inclusively in R, here \(R=0.1\), \(n=2\) and \(\epsilon =0.1\). Such a setup also reflects the requirement that experiments want to generate event samples to be as universal as possible, usable not only as signal process but also as background for many other measurements.
In Fig. 10 both these parameter sets are compared with the \(E_\perp ^\gamma \) spectrum measured by ATLAS. Both predictions are in good agreement with the data but the more inclusive set gives a slightly higher cross section. This is most obvious in a \(p_\perp ^\gamma \) region of around 70\(\,\mathrm {GeV}\), there the difference reaches almost 10%. Although it is expected that the first set with \(R=0.4\) may miss some contributions due to the smoothing of the cone, it is not guaranteed that the second set gives a more accurate prediction. A more inclusive partonlevel isolation always allows configurations which come closer to the collinear, nonperturbative region. This region cannot be described without fragmentation functions or QED partonshower matching [63]. However, this is not expected to happen if the angular distance is large enough and thus the inclusive parameter set is used in the following.
In almost all observables the MEPS@NLO prediction is in excellent agreement with the data. A small deviation is found in the invariant mass prediction with zero jets in a region around 250 GeV. The MEPS@NLO differential cross sections are about 20% larger with respect to the MEPS@LO results at small scales. At large scales the difference gets smaller since the contribution of the additional LO jets is increasing.
As already seen in the 13 \(\,\mathrm {TeV}\) section, the uncertainties estimated by the scale variations are reduced when moving from MEPS@LO to MEPS@NLO. At MEPS@LO, for lower values of \(E_\perp ^\gamma \) and \(m_{ee\gamma }\) the factorisation scale is the dominant source of uncertainty and reaches up to 10%. This is reflected in the zero jet bin of Fig. 13, too. By contrast, in the MEPS@NLO case this uncertainty is removed almost completely. At higher values of \(E_\perp ^\gamma \) or \(m_{ee\gamma }\) the renormalisation scale uncertainty takes over in all inclusive observables and reaches values of 10–20 % in the MEPS@LO case. This uncertainty is reduced for MEPS@NLO to 5–10% in both large \(E_\perp ^\gamma \) and the lower jet multiplicity bins. Both, the size of the corrections and their uncertainties behave very similarly between 8 and 13 \(\,\mathrm {TeV}\).
In contrast to the 13 \(\,\mathrm {TeV}\) section, here an additional comparison with an inclusive \(Z\gamma \) NLO calculation matched to the parton shower is performed. The MC@NLO method [41] as implemented in Sherpa [39] is used with the same settings as for MEPS@NLO but the NLO PDF set. The most significant difference between the multilegmerged and NLOmatched methods can be seen in the jet multiplicity distribution, shown in Fig. 13. In the MC@NLO simulation the two and threejet bins are only generated by the parton shower which cannot describe hard jet production properly. The same situation holds for large \(E_\perp ^\gamma \), as a hard photon is likely to be produced in conjunction with several hard jets.
4 Interplay with Z+jets production and QED final state radiation
4.1 Motivation
When predictions for \(V\gamma \) production are used in experimental searches to determine background contributions, they have to be combined with predictions for V + jets production in several cases. A jet from the V + jets sample can be misidentified as a photon at the detector level and thus contribute to the \(V\gamma \) event selection. Another example is a selection requiring multiple leptons, if the photon is misidentified as an electron.
At the same time the two types of MC samples are not exactly complementary: the simulation of QED final state radiation (FSR) from the leptons in the V + jets sample generates a fragmentation contribution also contained in the FSRlike diagrams of the \(V\gamma \) process. It is obvious that this overlap has to be removed before the samples can be used for background estimation.
This combination of V and \(V\gamma \) samples can be achieved by means of a QED merging as introduced in [63]. However, due to the smallness of the corresponding QED Sudakov suppression, it is not necessary to use an elaborate implementation of such a QED merging within MEPS@NLO samples. Instead, it suffices to define a more simple overlap removal prescription for the case discussed in this publication.
This overlap removal is a conceptually straightforward requirement, which is complicated by two facts. The QED FSR photons in the V+jets simulation are produced at the hadron level and can thus not simply be subjected to partonlevel cuts matching the ones in the \(V\gamma \) simulation. Furthermore the photon cuts in a multijet merged sample of \(V\gamma \)+jets require an isolation of the photon with respect to partons from the multijet matrix elements. This constraint has to be respected when defining the complementary cuts for the V+jets sample.
An implementation of such an overlap removal at the event generation level is discussed in this section using the example of \(V=Z\).
4.2 Implementation of overlap removal
In order to combine Z and \(Z\gamma \) events^{4} the phase space is split into two regions. The \(Z\gamma \) process includes photons directly in the matrix elements. The phase space of this region should be as large as possible but is limited since the matrix elements diverge when the photon is soft or collinear either to a massless lepton or quark.^{5} By contrast, photons generated by YFS in Z events do not have these limitations, but cannot describe initial state radiation which usually gives most of the contribution to hard photons.
In principle, the phase space slicing is defined by three components. First, a \(p_\perp ^{\gamma }\) cut, secondly a lepton photon isolation and finally a photon–hadron isolation. These cuts exclude a region where collinear or soft divergences are present and no fixedorder calculation in QCD is possible. Thus, an \(Z\gamma \) event has to pass all these cuts while a Z event has to fail at least one of them.
In case of \(Z\gamma \) events, there are already cuts at matrixelement level present and it would be desirable to use them directly for the overlap removal. Unfortunately, this is not possible since the generation of additional final state photons via YFS happens technically after the parton shower. The shower can shift the kinematics of all particles, thus cutting once before and once after the shower would result in a mismatch. As a consequence, the slicing cuts are applied to both Z and \(Z\gamma \) events at hadron level.
Special care has to be taken when selecting the photon and leptons which take part in the slicing procedure. Nonprompt leptons and photons can easily be produced by the decay of hadrons and there is no requirement that these additional particles are softer than the particles coming directly from the hard interaction or the YFS algorithm. However, it has to be guaranteed that all divergences which are present at matrixelement level are covered by the slicing cuts since otherwise the result would still depend on the matrixelement level cuts.
This table summarizes all cuts which define the phase space region which is used for testing the introduced overlap removal procedure
Region i, inclusive Z control region  

Lepton  \(p_\perp >25\) \(\,\mathrm {GeV}\), \(\eta <2.5\) 
Jet  \(E_\perp >30\) \(\,\mathrm {GeV}\), \(\eta <4.4\), \(\mathrm {\Delta }R(\mathrm {jet}\), \(e/\gamma ) > 0.3\) 
Boson  \(M_{e^+, e^}>40\) \(\,\mathrm {GeV}\) 
Photon  \(E_\perp >15\) \(\,\mathrm {GeV}\), \(\eta <2.5\) 
Isolation  \(\mathrm {\Delta }R(\gamma , e^\pm )>0.4\), \(\epsilon _{\mathrm {iso}}=0.5\) 
Region ii, inclusive Z control region  
Leptons  \(p_\perp >25\) \(\,\mathrm {GeV}\), \(\eta <3.5\), opposite charge 
Z  \(65\,\mathrm {GeV}< M^{ll} <115\) \(\,\mathrm {GeV}\) 
Region iii, overlap removal test region  
Leptons  \(p_\perp >15\) \(\,\mathrm {GeV}\), \(\eta <2.5\), opposite charge 
Z  \(30\,\mathrm {GeV}< M^{ll} <87\) \(\,\mathrm {GeV}\) 
Photon  \(E_\perp >5\) \(\,\mathrm {GeV}\), \(\eta <2.5\) 
It should be noted that it cannot be guaranteed that this photon selection indeed chooses the direct photon from the matrix element and not any further photon from final state radiation. The latter ones are not ordered in \(p_\perp \) and selecting one of them would introduce a dependency on the generation cut. This behaviour was studied in the \(p_\perp ^\gamma \) spectrum of \(Z\gamma \) events including additional final state radiation. Two kinds of photon selections are compared. In the first approach always the hardest photon is chosen, no matter whether it stems from the direct production or from additional FSR. The second method uses \(\mathrm {\Delta }R\) matched photons to identify the final state photon which is closest to the original matrixelement photon in angular space. Both spectra are shown in Fig. 14 and agree at the percent level, thus any bias on the overlapremoved result due to the photon choice will be negligible.
In the following, isolated photons are required to fulfill \(p_\perp >10\,\mathrm {GeV}\), a photon–lepton isolation of \(\mathrm {\Delta }R > 0.4\) and the hadronic isolation using a smooth cone isolation with \(R=0.4\), \(n=1\) and \(\epsilon =0.5\).
4.3 Results
The validation of the overlap removal algorithm proceeds with analyses in three different phase space regions. Both the \(Z\gamma \) and the Z prediction should not be altered by the overlap removal in their regions of validity. The first condition is checked using the \(Z\gamma \) analysis introduced in Sect. 3.2. It covers the explicit \(Z\gamma \) phase space and defines region i. By contrast, the Z phase space is probed by the default inclusive Z analysis provided by Rivet. Here, a reconstructed Z boson with an invariant mass between 65 and 115\(\,\mathrm {GeV}\) is required. The leptons are dressed with all photons having an angular distance of 0.2 or smaller. These cuts define the phase space region ii.
Slicing parameters which are used for validation of the overlap removal procedure. These cuts are applied according to the procedure defined in Sect. 4.2
Hardest photon  \(p_\perp >10\) \(\,\mathrm {GeV}\) 
Photon–lepton isolation  \( \mathrm {\Delta }R (\mathrm {leptons}, \gamma ) > 0.4 \) 
Photon hadron isolation  \(R=0.4\), \(n=1\), \(\epsilon =0.5\) 
A region dominated by final state radiation is defined by requiring the lepton pair to have an invariant mass below the Z peak, \(30\,\mathrm {GeV}< m^{ll} < 87\,\mathrm {GeV}\). An event is accepted, if both leading leptons have electron flavour but opposite charge. As photon candidate the leading photon (\(p_\perp ^\gamma > 5\,\mathrm {GeV}\)) is chosen, it has to be isolated from the selected leptons by requiring \(\mathrm {\Delta }R^{\gamma , e^\pm } > 0.05 \). In addition, the total energy of all remaining particles (excluding the selected leptons) in a cone with \(\mathrm {\Delta }R<0.4\) around the photon axis has to be less than \(0.5 \cdot E_\perp ^\gamma \). Details of all analyses are summarised in Table 3. All cuts and observables are implemented as a user module using the Rivet framework.
A comparison is performed using four sets of separately generated event samples; pure \(Z\gamma \), pure Z, sliced Z and sliced \(Z\gamma \). The latter two are summed up to give the total prediction after overlap removal. YFS is set active for all samples.
For event generation, the matrixelement level cuts are selected to be more inclusive than the analysis cuts. For the generation of the sliced direct part, the matrix level cuts are additionally chosen to be more inclusive than the phase space slicing parameters. All events are generated with \(Q_\mathrm {CUT}=30\) \(\,\mathrm {GeV}\) and up to three jets at leadingorder accuracy.^{6} The slicing parameters which are used for this test are summarised in Table 4.
In Fig. 15, the inclusive jet multiplicity and the inclusive \(E_\perp ^\gamma \) spectrum for the \(Z\gamma \) phase space (region i) are shown. In these and all further plots the inclusive Z, the direct \(Z\gamma \) and the summed overlapremoved predictions are shown, together with the corresponding components in the overlap removal. For better readability the statistical uncertainties of the latter have been omitted.
Here, the overlap removal result is in very good agreement with the pure \(Z\gamma \) prediction in both plots. The dominating contribution is the \(Z\gamma \) component, giving about 90% of the cross section for low \(p_\perp \) and almost 100% if \(p_\perp \) exceeds 80\(\,\mathrm {GeV}\).
By contrast, the inclusive Z phase space (region ii) is dominated by the Z component. The Z mass and Z \(p_\perp \) distributions are shown in Fig. 16. The overlap removal result is in excellent agreement with the pure Z prediction. The only region of phase space where the direct component of the overlap removal is sizeable is the low mass region of less than 85\(\,\mathrm {GeV}\), there the direct component gives around 10% of the cross section.
Despite the good agreement between overlap removal and the respective reference, one might want to construct the overlap removal to require only the \(Z\gamma \) component when looking at \(Z\gamma \) analyses. This would require one to take into account the analysis cuts for slicing the phase space and is thus not possible in a generic sample. An overlapremoved contribution with inclusive Z production allows more flexibility as needed in general purpose experiments.
Finally, in Figs. 17 and 18 some observables of the FSR dominated phase space (region iii) are shown. In this region neither the pure Z nor the pure \(Z\gamma \) predictions are guaranteed to give an accurate result. The former one includes only final state radiation and will therefore miss contributions especially in the high \(E_\perp \) region. By contrast, the latter one includes all contributions at a fixed order but may fail to describe the region where the photon is soft or very close to the lepton. Thus, both these predictions can only be interpreted as lower and upper bounds for the overlap removal in the context of this validation.
In Fig. 17, different \(E_\perp ^\gamma \) spectra are shown. The first subplot covers the whole phase space, while the two remaining ones cover only the regions where the photon is either very close to (\(0.05<\mathrm {\Delta }R<0.5\)) or separated from (\(0.5< \mathrm {\Delta }R<3\)) the closest lepton. While in the inclusive plot the two components of the overlap removal procedure give a very similar contribution if \(E_\perp ^\gamma \) exceeds the slicing cut, the two remaining plots reveal the nature of the overlap removal procedure much better. Whereas the low \(\mathrm {\Delta }R\) region is entirely dominated by the Z component, the high \(\mathrm {\Delta }R\) region is dominated by the \(Z\gamma \) component as soon as the cutoff is exceeded.
Figure 18 shows the azimuthal angle between the photon and its closest lepton and the invariant mass of the photon and both leptons. Both observables show an interesting behaviour. As expected, the cross section of the combined overlap removal result always interpolates between the cross section of the pure \(Z\gamma \) and the pure Z sample. When looking at the azimuthal distance, the Z component dominates at lower and the \(Z\gamma \) component at higher values, while both components are equal in a large region of phase space. The invariant mass spectrum is dominated at lower values (\(<105\) \(\,\mathrm {GeV}\)) by the Z component and at higher values by \(Z\gamma \).
5 Conclusions
Precise Standard Model predictions for \(Z\gamma \) + jets production are crucial for the search for new particles or anomalous couplings in measurements of this final state at the LHC.
With the presented simulation within the Sherpa framework using the MEPS@NLO algorithm we provide a simulation which is at the same time precise and realistic: NLO QCD corrections for the \(Z\gamma \) and \(Z\gamma \) + jet processes are included and reduce the uncertainties in relevant observables significantly. At the same time, the matching and merging with the parton shower allows a realistic simulation of the full final state at the hadron level, and the inclusion of all offshell effects allows one to place realistic experimental cuts on the prompt leptons without approximations.
Comparing to data from experimental measurements at \(\sqrt{s}=8\,\mathrm {TeV}\) we find very good agreement. On that basis we make predictions at \(\sqrt{s}=13\,\mathrm {TeV}\) and identify the dominant theoretical uncertainties and the phase space regions affected by them.
To further the application of these precise \(Z\gamma \)+jets predictions in experiments we also demonstrate how they can be combined with event generator predictions for Z+jets including finalstate photon radiation. As a validation we introduce a number of cross checks based on different phase space regions which can be repeated for any specific application of such samples in the experiments.
Footnotes
 1.
Even though the process is denoted with the shorthand \(Z\gamma \), the calculations throughout this paper include the full offshell \(\ell \ell \gamma \) final state.
 2.
In this section the simulation is performed at parton level for better scrutiny, i.e. multiple parton interactions and fragmentation are switched off.
 3.
A njet matrix element refers to a matrix element with n additional, well separated partons in the matrix element.
 4.
Here and in the following \(ll(\gamma ) + \mathrm {jets}\) final states are denoted as \(Z(\gamma )\) for better readability.
 5.
In this publication all leptons and quarks except the top are treated as massless in the matrix elements.
 6.
MEPS@LO is chosen simply for performance reasons. This implies no limitation as long as slicing cuts are IR save since the introduced slicing algorithm is based solely on kinematics.
Notes
Acknowledgements
We are grateful to our colleagues in the Atlas and Sherpa collaborations for useful discussions and support, in particular to Marek Schönherr for his comments on the manuscript. We thank the OpenLoops authors for providing the necessary virtual matrix elements. This research was supported by the German Research Foundation (DFG) under Grant No. SI 2009/11. We thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for generous allocations of computing time.
References
 1.P. Achard et al., L3, study of the \(e^{+} e^{} \rightarrow Z \gamma \) process at LEP and limits on triple neutralgaugeboson couplings. Phys. Lett. B 597 (2004). arXiv:hepex/0407012 [hepex]
 2.J. Abdallah et al., DELPHI, study of triplegaugeboson couplings \(ZZZ\), \(ZZ\gamma \) and \(Z\gamma \gamma \) at LEP. Eur. Phys. J. C 51 (2007). arXiv:0706.2741 [hepex]
 3.G. Abbiendi et al., OPAL, search for trilinear neutral gauge boson couplings in \(Z^\) gamma production at \(\sqrt{s}\) = 189 GeV at LEP. Eur. Phys. J. C 17 (2000). arXiv:hepex/0007016 [hepex]
 4.G. Abbiendi et al., OPAL, constraints on anomalous quartic gauge boson couplings from \(\nu \bar{\nu } \gamma \gamma \) and \(q \bar{q} \gamma \gamma \) events at LEP2. Phys. Rev. D 70 (2004)Google Scholar
 5.V.M. Abazov et al., D0, Measurement of the \(Z \gamma \rightarrow \nu \bar{\nu }\gamma \) cross section and limits on anomalous \(Z Z \gamma \) and \(Z \gamma gamma\) couplings in p antip collisions at \(\sqrt{s}=1.96\) TeV. Phys. Rev. Lett. 102 (2009). arXiv:0902.2157 [hepex]
 6.V.M. Abazov et al., D0, \(Z\gamma \) production and limits on anomalous \(ZZ\gamma \) and \(Z\gamma \gamma \) couplings in \(p\bar{p}\) collisions at \(\sqrt{s}=1.96\) TeV. Phys. Rev. D 85 (2012). arXiv:1111.3684 [hepex]
 7.T. Aaltonen et al., CDF, limits on anomalous trilinear gauge couplings in \(Z\gamma \) events from \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96\) TeV. Phys. Rev. Lett. 107 (2011). arXiv:1103.2990 [hepex]
 8.G. Aad et al., ATLAS, measurements of \(W \gamma \) and \(Z \gamma \) production in \(pp\) collisions at \(\sqrt{s}=7\) TeV with the ATLAS detector at the LHC. Phys. Rev. D 87(11) (2013). arXiv:1302.1283 [hepex]
 9.M. Aaboud et al., ATLAS, studies of \(Z\gamma \) production in association with a highmass dijet system in \(pp\) collisions at \(\sqrt{s}=\) 8 TeV with the ATLAS detector. JHEP 07 (2017). arXiv:1705.01966 [hepex]
 10.G. Aad et al., ATLAS, measurements of \(Z\gamma \) and \(Z\gamma \gamma \) production in \(pp\) collisions at \(\sqrt{s}=\) 8 TeV with the ATLAS detector. Phys. Rev. D 93(11), 112002 (2016). arXiv:1604.05232 [hepex]
 11.S. Chatrchyan et al., CMS, measurement of the production cross section for \(Z\gamma \rightarrow \nu \bar{\nu }\gamma \) in pp collisions at \(\sqrt{s} =\) 7 TeV and limits on \(ZZ\gamma \) and \(Z\gamma \gamma \) triple gauge boson couplings. JHEP 10 (2013). arXiv:1309.1117 [hepex]
 12.S. Chatrchyan et al., CMS, measurement of the \(W\gamma \) and \(Z\gamma \) inclusive cross sections in \(pp\) collisions at \(\sqrt{s}=7\) TeV and limits on anomalous triple gauge boson couplings. Phys. Rev. D 89(9) (2014). arXiv:1308.6832 [hepex]
 13.V. Khachatryan et al., CMS, measurement of the \({{\rm Z}}\gamma \) production cross section in pp collisions at 8 TeV and search for anomalous triple gauge boson couplings. JHEP 04 (2015). arXiv:1502.05664 [hepex]
 14.V. Khachatryan et al., CMS, measurement of the \( {{\rm Z}}\gamma \rightarrow \nu \bar{\nu } \gamma \) production cross section in pp collisions at \(\sqrt{s}=\) 8 TeV and limits on anomalous \( {{\rm ZZ}} \gamma \) and \( {{\rm Z}}\gamma \gamma \) trilinear gauge boson couplings. Phys. Lett. B 760 (2016). arXiv:1602.07152 [hepex]
 15.A. Djouadi, V. Driesen, W. Hollik, A. Kraft, The Higgs photon—Z boson coupling revisited. Eur. Phys. J. C 1 (1998). arXiv:hepph/9701342 [hepph]
 16.M. Aaboud et al., ATLAS, search for heavy resonances decaying to a \(Z\) boson and a photon in \(pp\) collisions at \(\sqrt{s}=13\) TeV with the ATLAS detector. Phys. Lett. B 764 (2017). arXiv:1607.06363 [hepex]
 17.V. Khachatryan et al., CMS, search for highmass Z\(\gamma \) resonances in e\(\mathit{^{+}\rm e }^{}\gamma \) and \( \mu ^{+}\mu ^{}\gamma \) final states in proton–proton collisions at \(\sqrt{s} =\) 8 and 13 TeV. JHEP 01 (2017). arXiv:1610.02960 [hepex]
 18.M. Aaboud et al., ATLAS, search for new phenomena in events with a photon and missing transverse momentum in \(pp\) collisions at \(\sqrt{s}=13\) TeV with the ATLAS detector. JHEP 06 (2016). arXiv:1604.01306 [hepex]
 19.M. Aaboud et al., ATLAS, search for dark matter at \(\sqrt{s}=13\) TeV in final states containing an energetic photon and large missing transverse momentum with the ATLAS detector. Eur. Phys. J. C 77(6) (2017). arXiv:1704.03848 [hepex]
 20.V. Khachatryan et al., CMS, search for supersymmetry in events with photons and missing transverse energy in pp collisions at 13 TeV. Phys. Lett. B 769 (2017). arXiv:1611.06604 [hepex]
 21.A.M. Sirunyan et al., CMS, search for new physics in the monophoton final state in protonproton collisions at sqrt(s) = 13 TeV. arXiv:1706.03794 [hepex]
 22.F.M. Renard, Tests of neutral gauge boson selfcouplings with \(e^+ e^{}\rightarrow \gamma Z\). Nucl. Phys. B 196, 93108 (1982)Google Scholar
 23.J. Ohnemus, Order \(\alpha ^ s\) calculations of hadronic \(W^\pm \gamma \) and \(Z \gamma \) production. Phys. Rev. D 47, 92–54 (1993)Google Scholar
 24.J. Ohnemus, Hadronic \(Z \gamma \) production with QCD corrections and leptonic decays. Phys. Rev. D 51 (1995). arXiv:hepph/9407370 [hepph]
 25.U. Baur, T. Han, J. Ohnemus, QCD corrections and anomalous couplings in \(Z \gamma \) production at hadron colliders. Phys. Rev. D 57 (1998). arXiv:hepph/9710416 [hepph]
 26.M. Grazzini, S. Kallweit, D. Rathlev, A. Torre, \(Z\gamma \) production at hadron colliders in NNLO QCD. Phys. Lett. B 731 (2014). arXiv:1309.7000 [hepph]
 27.M. Grazzini, S. Kallweit, D. Rathlev, \(W\gamma \) and \(Z\gamma \) production at the LHC in NNLO QCD. JHEP 07 (2015). arXiv:1504.01330 [hepph]
 28.J.M. Campbell, T. Neumann, C. Williams, \(Z\gamma \) production at NNLO including anomalous couplings. arXiv:1708.02925 [hepph]
 29.A. Denner, S. Dittmaier, M. Hecht, C. Pasold, NLO QCD and electroweak corrections to \(Z+\gamma \) production with leptonic Zboson decays. JHEP 02 (2016). arXiv:1510.08742 [hepph]
 30.S. Catani, F. Krauss, R. Kuhn, B.R. Webber, QCD matrix elements + parton showers. JHEP 11 (2001). arXiv:hepph/0109231
 31.L. Lönnblad, Correcting the colourdipole cascade model with fixed order matrix elements. JHEP 05 (2002). arXiv:hepph/0112284
 32.F. Krauss, Matrix elements and parton showers in hadronic interactions. JHEP 08 (2002). arXiv:hepph/0205283
 33.M.L. Mangano, M. Moretti, R. Pittau, Multijet matrix elements and shower evolution in hadronic collisions: \(W b\bar{b}+n\)jets as a case study. Nucl. Phys. B 632 (2002). arXiv:hepph/0108069
 34.J. Alwall et al., Comparative study of various algorithms for the merging of parton showers and matrix elements in hadronic collisions. Eur. Phys. J. C 53 (2008). arXiv:0706.2569 [hepph]
 35.K. Hamilton, P. Richardson, J. Tully, A modified CKKW matrix element merging approach to angularordered parton showers. JHEP 11 (2009). arXiv:0905.3072 [hepph]
 36.S. Höche, F. Krauss, S. Schumann, F. Siegert, QCD matrix elements and truncated showers. JHEP 05 (2009). arXiv:0903.1219 [hepph]
 37.L. Barze, M. Chiesa, G. Montagna, P. Nason, O. Nicrosini, F. Piccinini, V. Prosperi, W\(\gamma \) production in hadronic collisions using the POWHEG+MiNLO method. JHEP 12 (2014). arXiv:1408.5766 [hepph]
 38.S. Höche, F. Krauss, M. Schönherr, F. Siegert, QCD matrix elements + parton showers: the NLO case. JHEP 04 (2013). arXiv:1207.5030 [hepph]
 39.S. Höche, F. Krauss, M. Schönherr, F. Siegert, A critical appraisal of NLO+PS matching methods, JHEP 09 (2012), arXiv:1111.1220 [hepph]
 40.S. Höche, F. Krauss, M. Schönherr, F. Siegert, W+njet predictions with MC@NLO in Sherpa. Phys. Rev. Lett. 110 (2013). arXiv:1201.5882 [hepph]
 41.S. Frixione, B.R. Webber, Matching NLO QCD computations and parton shower simulations. JHEP 06 (2002). arXiv:hepph/0204244
 42.D.R. Yennie, S.C. Frautschi, H. Suura, The infrared divergence phenomena and highenergy processes. Ann. Phys. 13, 379–452 (1961)ADSCrossRefGoogle Scholar
 43.M. Schönherr, F. Krauss, Soft photon radiation in particle decays in SHERPA. JHEP 12 (2008). arXiv:0810.5071 [hepph]
 44.S. Frixione, Isolated photons in perturbative QCD. Phys. Lett. B 429 (1998). arXiv:hepph/9801442
 45.T. Gleisberg, S. Höche, F. Krauss, M. Schönherr, S. Schumann, F. Siegert, J. Winter, Event generation with sherpa 1.1. JHEP 02 (2009). arXiv:0811.4622 [hepph]
 46.F. Krauss, R. Kuhn, G. Soff, AMEGIC++ 1.0: a matrix element generator in C++. JHEP 02 (2002). arXiv:hepph/0109036
 47.T. Gleisberg, S. Höche, Comix, a new matrix element generator. JHEP 12 (2008). arXiv:0808.3674 [hepph]
 48.F. Cascioli, P. Maierhöfer, S. Pozzorini, Scattering amplitudes with open loops. Phys. Rev. Lett. 108 (2012). arXiv:1111.5206 [hepph]
 49.G. Ossola, C.G. Papadopoulos, R. Pittau, CutTools: a program implementing the OPP reduction method to compute oneloop amplitudes. JHEP 0803 (2008). arXiv:0711.3596 [hepph]
 50.A. van Hameren, OneLOop: for the evaluation of oneloop scalar functions. Comput. Phys. Commun. 182 (2011). arXiv:1007.4716 [hepph]
 51.J. Butterworth, G. Dissertori, S. Dittmaier, D. de Florian, N. Glover et al., Les Houches 2013: physics at TeV colliders: standard model working group report. arXiv:1405.1067 [hepph]
 52.A. Denner, S. Dittmaier, M. Roth, L.H. Wieders, Electroweak corrections to chargedcurrent e+ e —> 4 fermion processes: technical details and further results. Nucl. Phys. B 724 (2005). arXiv:hepph/0505042 [hepph] [Erratum: Nucl. Phys. B 854, 504 (2012)]
 53.A. Denner, S. Dittmaier, The complexmass scheme for perturbative calculations with unstable particles. Nucl. Phys. Proc. Suppl. 160 (2006). arXiv:hepph/0605312 [hepph]
 54.R.D. Ball et al., NNPDF, parton distributions for the LHC Run II. JHEP 04 (2015). arXiv:1410.8849 [hepph]
 55.T. Sjöstrand, M. van Zijl, A multipleinteraction model for the event structure in hadron collisions. Phys. Rev. D 36, 2019 (1987)ADSCrossRefGoogle Scholar
 56.A. De Roeck, H. Jung (eds.) HERA and the LHC: a workshop on the implications of HERA for LHC physics: proceedings Part A (CERN, Geneva, 2005)Google Scholar
 57.J.C. Winter, F. Krauss, G. Soff, A modified clusterhadronisation model. Eur. Phys. J. C 36 (2004). arXiv:hepph/0311085
 58.A. Buckley, J. Butterworth, L. Lönnblad, D. Grellscheid, H. Hoeth et al., Rivet user manual. Comput. Phys. Commun. 184 (2013). arXiv:1003.0694 [hepph]
 59.S. Catani, Y.L. Dokshitzer, M.H. Seymour, B.R. Webber, Longitudinallyinvariant \(k_\perp \)clustering algorithms for hadron–hadron collisions. Nucl. Phys. B 406, 187–224 (1993)ADSCrossRefGoogle Scholar
 60.L. Dixon, Z. Kunszt, A. Signer, Vector boson pair production in hadronic collisions at \(O({\alpha }_{s})\): lepton correlations and anomalous couplings. Phys. Rev. D 60, 114037 (1999)ADSCrossRefGoogle Scholar
 61.A. Denner, S. Dittmaier, M. Hecht, C. Pasold, NLO QCD and electroweak corrections to \(W \gamma \) production with leptonic \(W\)boson decays. J. High Energy Phys. 2015(4), 18 (2015)CrossRefGoogle Scholar
 62.E. Bothmann, M. Schönherr, S. Schumann, Reweighting QCD matrixelement and partonshower calculations. Eur. Phys. J. C 76(11) (2016). arXiv:1606.08753 [hepph]
 63.S. Höche, S. Schumann, F. Siegert, Hard photon production and matrixelement partonshower merging. Phys. Rev. D 81 (2010). arXiv:0912.3501 [hepph]
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