Combination of Run-1 exotic searches in diboson final states at the LHC

We perform a statistical combination of the ATLAS and CMS results for the search of a heavy resonance decaying to a pair of vector bosons with the s=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s}=8 $$\end{document} TeV datasets collected at the LHC. We take into account six searches in hadronic and semileptonic final states carried out by the two collaborations. We consider only public information provided by ATLAS and CMS in the HEPDATA database and in papers published in refereed journals. We interpret the combined results within the context of a few benchmark new physics models, such as models predicting the existence of a W′ or a bulk Randall-Sundrum spin-2 resonance, for which we present exclusion limits, significances, p-values and best-fit cross sections. A heavy diboson resonance with a production cross section of ∼4-5 fb and mass between 1.9 and 2.0 TeV is the exotic scenario most consistent with the experimental results. Models in which a heavy resonance decays preferentially to a WW final state are disfavoured.


Introduction
Searches for new heavy resonances are one of the major components of the ATLAS and CMS physics programmes at the Large Hadron Collider (LHC) at CERN.Of particular interest is the coupling of new resonances to pairs of vector bosons.Models with Vectorial heavy resonances (i.e.W -like and Z -like bosons) are commonly considered as possible extensions of the SM, either in weakly coupled (see [1][2][3]) or strongly coupled versions, the so-called composite Higgs scenarios [4,5].In these scenarios, the existence of new resonances is introduced to alleviate the hierarchy problem in the SM.Another common SM extension is the Warped Extra Dimensions or Randall-Sundrum (RS) model [6], which is an example of a class of models predicting neutral spin-2 resonances as Kaluza-Klein (KK) excitations of the graviton field (G * ).Two types of models are usually considered: the original version, in which only gravity is allowed to propagate into the extra-dimensional bulk ("RS1" models, see Ref. [7]) and variants of the original model, in which the SM fields are also allowed to propagate into the extra dimensional bulk ("bulk RS" models, see for example Ref. [8]).RS1 models favour the decay of G * to q q, + − and γγ final states, whereas in bulk RS models its decay to vector bosons.
After a number of direct and indirect bounds from previous experiments, and in particular, the stringent constraints from the electro-weak precision measurements carried out at LEP [9] 1 , nowadays searches for heavy exotic resonances decaying to pairs of vector bosons typically focus on resonance masses above 1 TeV.When produced and decayed at the LHC, these particles would generate vector bosons with O(1 TeV) transverse momenta, requiring special reconstruction strategies.In particular, the quarks from a hadronically-decaying vector boson are very close to each other in the η −φ space.In their showering and hadronisation process they produce highly overlapping jets, in a so-called boosted topology.ATLAS and CMS handle this experimental signature by reconstructing the two partially overlapping jets as a single massive (or "fat") jet, noted in this paper as "J".One then exploits the jet mass m J and the momentum flow around the jet axis to distinguish these special jets from those originating from quark or gluon production [12][13][14][15][16][17].A typical boosted longitudinally polarised and hadronically-decaying V boson2 can be identified by a tagger with an efficiency of ∼ 50% and with a false-positive rate for light quarks or gluons of 2% [18,19].
The ATLAS and CMS collaborations have employed hadronic boson taggers in searches for heavy resonances in diboson final states with the proton-proton collision data collected in 2012 at a centre-of-mass energy of 8 TeV.In particular, the ATLAS search in the fully hadronic final state [20] has generated significant interest due to an excess of diboson events with invariant mass mass around 1.9 TeV.Small deviations in the same mass region are observed in other channels as well, e.g. the CMS search in the Z( + − ) V (q q) channel with = e, µ [21], and the CMS search in the fully hadronic V (q q) V (q q) final state [22].Other analyses, e.g. the ATLAS and CMS searches in the W ( ν) V (q q) channel see no evidence of a deviation, indicating a possible tension between these experimental results in the scenario of a heavy exotic resonance.Additional results with potentially interesting deviations in the same mass region include a moderate excess (≈ 1 − 2σ of local significance) reported in the ATLAS [23,24] and CMS [25,26] searches in the dijet channel, as well as in the CMS search in the dilepton channel [27].In addition, a search for right-handed W (and heavy neutrinos) [28] by CMS has reported a small excess in the electron channel [29] (however, this excess is not confirmed by a similar ATLAS analysis [30]).Finally, a CMS search for W( ν) H(b b) resonances reported an excess of ≈ 2σ, originating from a stronger excess in the electron channel and no evidence of a deviation in the muon channel [31].At the same time, the CMS searches for WH or ZH resonances in the fully hadronic channel were inconclusive, with a mild upward fluctuation around 1.8 TeV and a lack of events around 2 TeV [32].The dedicated searches for Z(q q) H(τ + τ − ) and H(b b) H(τ + τ − ), H(b b) H(b b) final states showed no excess [33,34].
A natural next step would be to carry out a systematic comparison of the results reported by ATLAS and CMS in various channels, and examine if the apparent deviations work in a synergistic way towards a coherent picture.In particular, the goal is to quantify the level of agreement among the different results, and by using an exotic signal hypothesis for the interpretation of these deviations, to calculate the corresponding production cross section.We hereby present the first step in addressing this question, starting with the statistical combination of the results of the ATLAS and CMS Run-1 searches for vector boson pair resonances.The exotic models considered by the experiments are usually connected with the electroweak sector, with the predicted resonances mainly coupling to longitudinally polarised vector bosons V L .We consider the experimental results of the searches for heavy resonances decaying to three final states: Z L Z L , W L W L and W L Z L .We combine the results and interpret the derived exclusion limits in the context of a (W -like) spin-1 charged particle decaying to a W L Z L boson pair, and a neutral spin-2 particle (G bulk ).For the latter case, we only consider bulk RS scenarios, namely particles decaying to the Z L Z L , W L W L final states 3 .
The paper is organised as follows: in Section 2 we present a general overview of the methodology used to emulate the ATLAS and CMS analyses; Sections 3 and 4 discuss the emulation of the hadronic and semileptonic analyses, respectively.Each section covers the individual searches by ATLAS and CMS, and their combination; in Section 5 we combine the Run-1 results provided by the two collaborations and discuss their interpretation in a few benchmark models considered in this study; we present the summary of the findings, along with the conclusions in Section 6.A brief note on the compatibility of the findings of this study with the preliminary Run-2 search results reported by ATLAS and CMS in December 2015 has been added in v2 of this paper and is presented after the conclusions.Additional information on the determination of the background and signal modelling for the ATLAS search in the fully hadronic channel is given in Appendices A, B.

General methodology
All exotic searches considered in this paper are looking for a diboson mass peak emerging on top of a falling background spectrum.In order to evaluate the significance of a deviation observed in the data, we need as input the shapes of the signal and background distributions, the total number of expected background events, the signal efficiency, and the experimentally measured distribution (data).
This study is based exclusively on the public information provided by the two experimental collaborations in the HEPDATA database [69] and the cited papers (published in refereed journals).In particular, we employ the expected backgrounds with their corresponding uncertainties, as they have been estimated directly by ATLAS and CMS, wherever possible.The modelled signal distributions (namely, shapes and signal efficiencies for a few benchmark models and mass values) are also taken from the information publicly provided by the experiments, when available 4 .In order to emulate signal distributions for additional mass values, we carry out linear interpolations of the available models within the benchmark mass points.We derive exclusion limits on hypothetical signals by performing binned templated fits of the data distributions with linear combinations of the signal and background distributions.These calculations are carried out with the open-source statistical framework THETA [70] which uses the asymptotic approximation [71] of the CLs method [72,73].
In a few cases, the information published by ATLAS and CMS is not sufficient for this simple approach to produce satisfactory results.For example, uncertainty correlations that affect the background determination, or the mass-dependence of an important systematic uncertainty are not always properly documented.In these cases, we fit the data distributions to the functional form documented in the published analysis, e.g. the function used in the hadronic searches or an exponential function for the leptonic channels.Details about these fits are given in the corresponding sections of the paper, where we also discuss the agreement achieved in the background modelling.When it is necessary to model a signal distribution ourselves, we either use a Gaussian approximation with a resolution inferred from the relevant experimental paper, or we generate Monte Carlo (MC) samples using the Madgraph5 matrix-element event generator [74], matched to Pythia8 [75] for the hadronisation process.For the G bulk signal we use the Madgraph5 model files as presented in Ref. [76], while for the spin-1 signal W the ones described in Ref. [77].
These approximations are mainly motivated by our familiarity with the diboson and similar searches by ATLAS and CMS.The described procedure is validated using the nominal published results as benchmarks, as well as the comparison of our own calculations of the per-experiment combinations against the official combination of diboson searches [21,78].We are able to reproduce the exclusion limits of each analysis individually and their combinations with an agreement of better than 20% in the region of interest for all channels, with the exception of the fully hadronic search in ATLAS (see Appendix A).Our methodology can be used as a set of guidelines for model builders in the absence of official combined results published by the two experiments.
All diboson final states considered in this study contain at least one vector boson (W or Z) decaying hadronically.Because of the limited hadronic detector resolution, it is not possible to distinguish between hadronic W and hadronic Z jets.When interpreting an experimental result, special care is needed to account for possible cross-channel contamination of the final state under consideration.For example, a neutral heavy resonance decaying to a pair of vector bosons is expected to decay to both WW and ZZ final states.We consider models in which the relative branching fractions of neutral particle decays to WW and ZZ can vary, in order to study the relative importance of the different bosonic sub-channels to the combined result.We quantify this dependence by introducing as a free parameter the ratio r of the corresponding branching fractions: with r = 2 being the default ratio in the baseline bulk RS scenario.
The full list of channels that we consider in this study is as follows: the fully hadronic searches X → V (q q) V (q q) (labelled "JJ"), searches including a W decaying leptonically X → W( ν) V (q q) (labelled " νJ"), and searches including a Z decaying leptonically X → Z( ) V (q q) (labelled " J").Table 1 summarises the methods that have been used to emulate each of the analyses considered.Details of the individual analyses are given in the sections that follow.3 Fully hadronic searches: VV → JJ In this Section we discuss the analysis of the ATLAS and CMS searches in the VV → JJ channel.We first present the results of our analysis for the two searches separately, followed by their combination and a summary of the findings.

Description of the ATLAS analysis
The ATLAS fully hadronic search analyses calorimetric dijet events.The main irreducible background is dijet production in QCD, which is dominated by 2 → 2 t-channel processes involving quarks and gluons.The contribution of these processes is minimised by restricting the jet acceptance to |η| < 2.0 and the rapidity difference between those two jets to |∆η| < 1.2.The events are required to have low missing transverse momentum and a rather symmetric dijet topology (similar p T for the two leading jets) to reduce the detector noise.After this selection, the efficiency is approximately 70-80% for a heavy vector boson signal, and above 80% for a G bulk signal.
To further reduce the multijet background, two fat jets are reconstructed using the Cambridge-Aachen algorithm [81,82] with radius parameter R = 1.2.The mass-drop filtering algorithm [12] is applied to each of these jets for the identification of the subjets and grooming.Events are kept if each of the two leading jets satisfies the following conditions: have two sub-jets with similar transverse momentum, have less than 30 tracks matched to it, and have a pruned mass within a ±13 GeV window either around 82.4 GeV (for W tagging) or around 92.8 GeV (for Z tagging).The selection efficiency of the grooming algorithm for fat jets from a W resonance is between 30% and 40%.
The events are subsequently classified into three non-mutually-exclusive categories, based on the jet-mass values: WW, WZ and ZZ.The overall product of the geometric acceptance with the signal efficiency for this analysis is typically 10-20%.

Statistical analysis
The analysis uses the smoothness test ("bump search") approach: the background is approximated by a steeply falling function, while the signal template is taken from simulation.The sum of the two components is then fitted to the data.The background function used by the ATLAS collaboration is: where p 0 , p 1 and p 2 are free parameters and m JJ is the dijet invariant mass; ATLAS has also made the signal templates used in the analysis public.We employ the same function for the background description, but recalculate the background uncertainties in order to better account for the large scale correlations in m JJ .To this end, we refit the data in each of the three categories above using the aforementioned background parametrisation.We diagonalise the uncertainty matrix and obtain three uncertainty eigenvectors (σ λ i , with i = 0, 1, 2).Our fit result produces a background estimate which agrees with the nominal background within 10%, which is well within the uncertainties (see Appendix A).This background is subsequently used together with the associated uncertainties in our statistical analysis (see Fig. 1).
We consider the following systematic uncertainties, treated as fully correlated across m JJ histogram bins: • Background uncertainty, obtained as described above.
• Signal normalisation uncertainty, which is separated into two further sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.8%), and an additional term applicable to the JJ channel that covers V-tagging uncertainties as well as jet systematics.
• Signal jet energy scale uncertainty, which includes jet transverse momentum and mass uncertainties (with a ±2% and ±5% impact on m JJ , respectively).An additional jet energy resolution uncertainty is known to have a negligible effect on the signal shape and is ignored in this study.
Our statistical analysis produces expected exclusion limits that are typically 50% more stringent than the ones publicly provided by ATLAS.This discrepancy, discussed in detail in Appendix A, is corrected for with the introduction of a fudge factor, defined as the ratio of the ATLAS expected exclusion limits and the ones from this study obtained with the THETA statistical framework (see Fig. 2).With this correction, our calculated exclusion limits are in good agreement with the public ATLAS results (see Fig. 3).

Results with WW, WZ and ZZ signal hypotheses
As discussed above, due to the finite detector resolution, the V-tagging tool is not capable to differentiate between fat jets originating from W or Z bosons.However, there is a significant performance difference between W and Z tagging efficiencies of up to ≈ 30%, mainly as

ATLAS JJ
Z L Z L Figure 3. ATLAS hadronic search: Observed exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with this study, with (black) and without (red) the correction discussed in the text ("fudge"), and comparison with the official ATLAS results (grey) for G bulk → W L W L (left), W → W L Z L (middle) and G bulk → Z L Z L (right) signal hypotheses and tagging selections.The green and yellow bands represent the one and two sigma variations around the median expected limits (dashed lines) calculated with the same fudge factor.
a result of the different boson masses.By using the mass distribution of longitudinal Vjets, as documented in Fig. 1 of Ref. [20], and by taking into account the different W and Z efficiencies, we can calculate the efficiency of tagging selections for different signal hypotheses (WW, WZ, ZZ).The comparison of the tagging selection efficiencies can be found in Table 2.
The effect of applying the different tagging selections to the WW, WZ and ZZ signal hypotheses as a function of the resonance mass is shown in Fig. 4. We assume that the m JJ spectrum is not affected by the mass window difference in the tagging selections, i.e. that the same distribution describes the three tagging categories WW, WZ and ZZ.Since the three categories have common events, they cannot be combined as if they were statistically independent.Instead, for each theoretical model under consideration we choose the tagging category that gives the best expected exclusion limits.For the W model the WZ tagging selection gives the best result, whereas for the G bulk graviton model in the W L W L and Z L Z L final states the ZZ tagging selection has the best performance.

Description of the CMS analysis
The jet acceptance is restricted to |η| < 2.5 and |∆η| < 1.3 in order to reduce the contamination from multijet events.The detector noise is removed by requiring tight quality criteria on the jets.The pruning algorithm [13] is used to clean up the jet from soft and large-angle radiation.The mass of the resulting fat jet is constrained in the70 < m J < 100 GeV range.Finally, the signal-to-background ratio is enhanced by exploiting the jet N-subjettiness [14][15][16] variable τ N .This variable is used to quantify how well the jet constituents can be arranged into N sub-jets, i.e. in a consistency check with the hadronic V boson hypothesis.The ratio τ 12 = τ 2 /τ 1 is built with the two leading jets: the smaller the ratio, the larger the probability that the jet consists of two sub-jets.The analysis considers two categories: the high purity (HP) one, defined by requiring τ 12 < 0.5 for both jets, and the low purity (LP) one, defined by requiring one jet with τ 12 < 0.5 and the other one with 0.5 < τ 12 < 0.75.The HP category is characterised by a smaller background contamination.The LP category captures signal events with asymmetric decays of the vector-boson candidates in the laboratory frame.Dividing the event sample into the LP and HP categories improves the sensitivity of the analysis in the mass range between 1 TeV and 2 TeV, while avoiding the inefficiency of a tight τ 12 selection at large jet momenta.
The product of the geometrical acceptance with the signal efficiency is similar to the one in the ATLAS search, ranging between 10% and 20%.

Statistical analysis
The CMS collaboration provides the binned data and background distributions with the associated uncertainties in the HEPDATA database (see Fig. 5), as well as the signal distributions for three different models along with their efficiencies [22]: W → W L Z L and G bulk decaying exclusively to Z L Z L or W L W L .We consider the following systematic uncertainties: • Background uncertainty, provided by CMS (in HEPDATA) and considered as fully correlated across the bins of the m JJ distribution.
• Signal normalisation uncertainty, which is separated further into two sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.2%), and an additional term applicable to the JJ channel that covers V-tagging uncertainties, such as p T , pile-up and PDF dependencies (13%).The τ 12 uncertainties are treated separately in the category below.
• Signal purity category migration uncertainty, which covers the effects of events "migrating" from the HP to the LP category, or vice-versa.This uncertainty amounts to 7.5% and 54 %, respectively.
• Signal jet energy scale uncertainty, propagates to ±1% of uncertainty on m JJ ; It is treated in the same way as in the ATLAS case.
All systematic uncertainties are treated as fully correlated across different m JJ bins.They are also considered as fully correlated between the LP and the HP categories, with the exception of the "purity category migration" uncertainty, which is treated as fully anticorrelated.
Our statistical analysis for W → W L Z L , G bulk → W L W L and G bulk → Z L Z L models produces exclusion limits that are in very good agreement with the ones publicly provided by CMS.An example of this agreement can been seen in the left plot of Fig. 6.The exclusion limits calculated in a few benchmark models can be seen in the right plot of Fig. 6.The most stringent limits are obtained for the G bulk → Z L Z L hypothesis, thanks to the higher V-tagging efficiency for Z bosons.

CMS JJ
. CMS hadronic search.Left: Expected (dashed lines) and observed (continuous lines) exclusion limits on W → W L Z L production cross sections as a function of the resonance mass m X obtained with this study (black), and comparison with the official CMS results (red).The green and yellow bands (dashed lines) represent the one and two sigma variations around the median expected limits calculated in this study (by CMS).Right: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with this study for

Combined LHC results of hadronic searches
This section describes the combination of the ATLAS and CMS searches in the fully hadronic channel JJ and the interpretation of the results under different signal hypotheses.
As a first step we note that ATLAS assumes a wide resonance in its JJ searches, whereas CMS assumes a narrow one.To ensure a consistent treatment of the search in the hadronic channel between the two experiments we introduce a +10% scale factor in the ATLAS selection efficiency.A description of the derivation of the scale factor and its impact on the search sensitivity is discussed in Appendix B. For every signal hypothesis under consideration we use the optimal mass selection windows as defined by ATLAS.
We proceed by combining the THETA data cards of the individual ATLAS and CMS searches.The results of the statistical combination for the W L Z L , W L W L , and Z L Z L signal hypotheses can be seen in Fig. 7.In the 1.7 < m X < 2.2 TeV region we observe the largest discrepancy between expected and observed exclusion limits due to the presence of the excess in the m JJ spectrum.The excess is much smaller in the CMS analysis, which forces the combined results to lie between the ATLAS and the CMS curves.The sensitivity of the combined search as we move away from the deviation region is driven by the CMS analysis.
The impact of the individual experimental results on the combination can be seen in the distribution of p-values (obtained using Wilks' theorem) depicted in Fig. 8.The CMS z-value or significance5 in the excess region is of the order of 1σ, independently of the considered model and corresponding selections.The ATLAS significance ranges from less than 3σ for the W L W L selection to nearly 4σ for the Z L Z L selection, as a result of the different W and Z mass selection windows.The statistical significance of the combined result is very close to the one obtained with the ATLAS result alone, although slightly reduced.In fact, the ATLAS and CMS results are not contradictory: due to the small CMS excess observed in the same mass region, the CMS result cannot exclude the larger ATLAS excess.
In order to further characterise the interplay between the ATLAS and the CMS results in the combination, we show in Fig. 9 the best-fit exotic signal cross section as a function of the resonance mass m X value for a few benchmark models and corresponding selections: The best-fitted cross section values are shown separately for the emulation of ATLAS and CMS searches, and their combination.The largest excess for the W L Z L and W L W L signal hypotheses is observed in the 1.9 < m X < 2.1 TeV mass range, while the excess extends down to m X = 1.8 TeV for the Z L Z L signal hypothesis.In these mass ranges, the ATLAS data suggests a production cross section of ≈ 10 fb, whereas the CMS data favours smaller values (≈ 3 fb) and is more consistent with the no-signal hypothesis.The m X profile of the fitted exotic signal cross section is essential identical to the one obtained from the ATLAS search emulation.
Further tests of the compatibility between the ATLAS and CMS results can be seen in Fig. 10, showing scans of the profiled likelihood as a function of the exotic production cross section for m X = 2 TeV (mass value of largest excess).Due to the large uncertainties of the fit, the best-fit cross-section values by ATLAS and CMS are compatible within ±1σ for the W L Z L and W L W L hypotheses.The compatibility of the results from the two experiments is slightly reduced in the Z L Z L scenario.The dependence of these results on r ≡ B(X → W L W L )/B(X → Z L Z L ) can be seen in Fig. 11.The conclusions discussed above remain mostly unchanged.
In summary, in the combination of fully hadronic results the small CMS excess results in a slight reduction of the larger ATLAS excess.However, the combined-search statistical significance stays well above 3σ for the W L Z L and Z L Z L hypotheses and close to 3σ for the W L W L hypotheses.The preferred mass range for a hypothetical exotic signal is 1.9 < m X < 2.0 TeV, with the corresponding production cross section in the 8-12 fb region.
Combination of hadronic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses.The green and yellow bands represent the one and two sigma variations around the median expected limits.The results include the 10% scale factor discussed in the text.The dashed black curve corresponds to the combined search without the 10% scale factor discussed in the text.4 Semi-leptonic searches: WV → νJ and ZV → J

ATLAS+CMS
In this Section we discuss the analysis of the ATLAS and CMS searches in the WV → νJ and ZV → J channels.We follow the discussion pattern of the fully hadronic section: we first present the results of our analysis for the two searches separately, followed by their combination and a summary of our findings.The ATLAS semileptonic search considers both the case in which the two quarks from the vector boson decay are reconstructed as a single merged jet (boosted regime), and the case in which they are reconstructed as two distinct jets (resolved regime).In this study, we focus on resonances heavier than 1.5 TeV, for which the merged regime largely drives the sensitivity.Thus we consider only the Merged Region (MR) categories of Refs.[79,80].

ATLAS+CMS JJ
In both ZV → J and WV → νJ searches, the boosted jet is identified using the mass-drop filtering algorithm (as in the VV → JJ search).In addition, two same-flavour opposite-sign leptons, or one charged lepton and missing transverse energy (MET) are required.The events are selected online by single-or double-lepton based triggers.The detector coverage includes the tracker volume (|η| < 2.5) and the fiducial region of the electromagnetic calorimeter (for electrons) or the muon detector.The typical p T threshold for the charged leptons and for MET is 25 GeV.The main backgrounds are inclusive V production (i.e.Z +jets for the J channel and W +jets for the νJ channel), as well as t t production.

Statistical analysis
We build the likelihood for the ATLAS semileptonic searches using the information documented in the HEPDATA database.The ATLAS collaboration estimates the background uncertainties separately for each lepton category.The electron p T resolution is better than that of the muon in the high-p T region.The systematic uncertainties associated with different background sources (t t and electroweak components) are also treated separately.Nevertheless, the background distributions documented in the HEPDATA database (see Fig. 12) are presented jointly for electrons and muons.We model the signal distributions in the diboson mass spectrum with a Gaussian function, centred at the assumed resonance mass and with a width reflecting the experimental resolution.We assume a fixed value of 4% resolution in the J channel for all mass values 6 .Similarly, we assume a fixed value of 10% resolution in the νJ channel for all mass values7 (see Fig. 1 in Ref. [79]).
The signal distributions are normalised to the expected yield, as calculated from the theoretical cross section and the selection efficiency provided by the ATLAS collaboration.
We consider the following systematic uncertainties, treated as fully correlated across m JJ histogram bins: • Background uncertainty, provided by the ATLAS experiment (in HEPDATA).
• Signal normalisation uncertainty, which is separated into two further sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.8%), and an additional term accounting for all types of scale and efficiency systematic effects (10%).The latter is treated as uncorrelated between the J and νJ channels.
Given the approximations that we have introduced to model the signal, we do not expect our statistical analysis to produce results matching with high accuracy the public ATLAS results.Similarly to the procedure followed for the emulation of the fully hadronic ATLAS search, we introduce a fudge factor to reduce this discrepancy.The value of the fudge factor is chosen such that the expected exclusion limits produced by this study agree with the official limits by ATLAS.It is found to be between 0.8 and 1.2 in the resonance mass range of interest, slowly decreasing for larger mass values (Fig. 13).With this correction, our calculated exclusion limits are in good agreement with the public ATLAS results (Fig. 14).

ATLAS
V L V L Figure 13.ATLAS semileptonic searches: Fudge factor as a function of the mass m X of the exotic resonance, calculated via the ratio of observed exclusion limits obtained with this study to the ones of the official ATLAS result, for the W → W L Z L (red) and G bulk → Z L Z L (black) signal hypotheses in the J channel, and for the W → W L Z L (magenta) and G bulk → W L W L (orange) signal hypotheses in the νJ channel.

Emulation of CMS search 4.2.1 Description of the CMS analysis
The CMS semileptonic analyses [21] are performed with data collected by single-lepton triggers for the νJ channel and double-lepton triggers for the J channel.Jets are identified as boosted vector bosons using the same algorithm employed for the fully hadronic search (see Section 3).Similarly to the strategy developed in the fully hadronic search, LP and HP categories are introduced, based on the value of τ 21 , to increase the analysis sensitivity.
The analysis is performed by using a G bulk graviton as the benchmark signal model.In order to facilitate the interpretation of the search results in other theoretical models, the CMS collaboration provides the reconstruction efficiencies of leptonic and hadronic W L and Z L in the HP category, as function of the boson's p T and η.Those 2D efficiency maps include the effects of the pruned jet mass and τ 21 selections, as well as the resonance mass reconstruction.

ATLAS lνJ
W L Z L Figure 14.ATLAS semileptonic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross sections as a function of the resonance mass m X obtained with this study (black), and comparison with the official CMS results (red) for signal hypotheses in the J (top) and νJ (bottom) channels.The green and yellow bands represent the one and two sigma variations around the median expected limits calculated in this study, with all the corrections described in the text included.

Statistical analysis
The background model is extracted by fitting the m VV data distributions for each lepton flavour with a levelled exponential where N , k and σ are free parameters.This function saturates in the high m VV region, and is meant to describe events where m VV was significantly mismeasured.For example, this may happen if a high p T muon leaves a nearly straight track barely bent by the magnetic field, or if the calculation of the neutrino momentum fails.In practice, this function is used in Ref. [21] to model the HP category with k as a free parameter, whereas for the LP category k can be set to 0. In the J channel we focus on the m VV > 700 GeV region, and we merge the contents of the (publicly available) 50 GeV wide bins to obtain a uniform, 100-GeV-wide binning for the m VV distribution.We use the diagonalised uncertainties from the fit (σ λ i , with i = 0, 1, 2) as background uncertainties.Figs. 15 and 16 show the comparison between the fits produced in this study and the official CMS fits on the data distributions.
We model the signal distributions in the diboson mass spectrum with a Gaussian function.The HP signal yield is calculated from the theoretical cross section and the selection efficiency obtained from the algorithm described in Ref. [21].The first step in this process is the generation of signal samples with the Madgraph5 generator as described in Sec. 2. We then apply acceptance selections on the leptons and generator-level jets, and use the 2D efficiency maps to emulate the V-boson reconstruction and tagging processes.Finally, we apply a 90% correction to account for b-jet veto inefficiencies.Considering the approximations made, this procedure is expected to reproduce the official CMS results within a 10% accuracy.The HP-category efficiencies that we obtain are consistent with the nominal G bulk → W L W L efficiencies for m X = 1.2 TeV within 6%.
The LP category signal efficiencies are generally not provided, but examples of the LP/HP efficiency ratios are given for a G bulk signal with m X = 1.2 TeV.The ratio is 0.47 (0.25) for the J ( νJ) channel.The reason for the efficiency difference between the two cases lies in the different boosted jet selection applied in the two channels.We make the assumption that we can use the same LP/HP ratio for all mass points under consideration in this study, and use the values above to estimate the expected signal yields in the LP category.Finally, the τ 21 categorisation is not sensitive to the nature of the resonance8 , therefore we use the same LP/HP ratio also for the W signal hypothesis.
We consider the following systematic uncertainties, treated as fully correlated across m JJ histogram bins: • Background uncertainty, extracted from our fit to the data distributions.
• Signal normalisation uncertainty, which is separated into two further sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.2%), and an additional uncertainty covering all lepton-related uncertainties (3.7% for electrons, 3% for muons), applied separately for the J and νJ channels.
• Signal purity category migration uncertainty, which covers the effects of events "migrating" from the HP to the LP category, or vice-versa.This uncertainty amounts to 9% and 24%, respectively.
As already discussed in previous sections, we apply a fudge factor to account for differences between our background description and the one from the public CMS result, as well as for the approximations introduced in the signal modelling (Fig. 17 our calculated exclusion limits are in good agreement with the public CMS results (Fig. 18).The statistical uncertainties (one-and two-sigma coverage bands) are ≈ 50% smaller than expected, as they have been calculated with the asymptotic CLs method, which is known to underestimate uncertainties in tests with small statistics.
We use the same procedure to recast the results in the context of a W → W L Z L signal search, with the results presented in Fig. 19.The jet mass selection for the J channel is 70 < m J < 110 GeV, to be compared with 65 < m J < 105 GeV for the νJ analysis.This choice was made in order to optimise the search for a neutral resonance (at the expense of the search for a charged one).Since the νJ channel mass window is shifted to a region with more background, the signal sensitivity for the νJ channel is reduced.

Combined LHC results of semi-leptonic searches
Here we discuss the combination of the ATLAS and CMS searches in the semileptonic channels ( νJ and J) and the interpretation of the results under different signal hypothe- ses, with final states including a leptonic W (→ ν) or Z (→ ) decay.The results are summarised in Fig. 20.
Under the hypothesis of a Z L Z L benchmark model, only the J searches are relevant.In this channel, CMS observes a small excess (≈ 1σ) between 1.7 and 1.9 TeV, while ATLAS a < 1σ excess between 1.9 and 2.0 TeV, driven by the presence of one event in the highest bin of the merged analysis distribution.The combination of the two channels results in a more stringent limit and a moderate excess of the order of 1σ around 1.9 TeV.Above 2 TeV, ATLAS has not published their search results and the limit considered here is the one provided by CMS.While the significance of the observed deviation is too small to cause any excitement, the sensitivity of this analysis is strongly reduced.This has implications for the combination result discussed in Sec. 5.
On the contrary, under the hypothesis of a W L W L benchmark model, only the νJ searches are relevant.An observed upward fluctuation around m VV = 1.8 TeV in the CMS data spectrum is compensated by a downward fluctuation in the same region for the ATLAS data.The two deviations effectively cancel each other, resulting into observed

CMS
V L V L Figure 17.CMS semileptonic searches: Fudge factor as a function of the mass m X of the exotic resonance, calculated via the ratio of observed exclusion limits obtained with this study to the ones of the official CMS result for the G bulk → W L W L (red) and G bulk → Z L Z L (black) semileptonic analyses.

CMS llJ
Z L Z L Figure 18.CMS semileptonic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross sections as a function of the resonance mass m X obtained with this study (black), and comparison with the official CMS results (red) for the G bulk → W L W L search in the νJ channel (left) and the G bulk → Z L Z L search in the J channel (right).The green and yellow bands represent the one and two sigma variations around the median expected limits calculated in this study, with all the corrections described in the text included.

CMS llJ
V L V L Figure 19.CMS semileptonic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with this study for the G bulk → W L W L (red) and W (black) signal hypotheses in the νJ channel (left) and for the G bulk → Z L Z L (red) and W (black) signal hypotheses in the J channel (right).
exclusion limits which are consistent with the experimental sensitivity and the backgroundonly hypothesis expectations.
For the W L Z L benchmark model, we are able to combine the experimental results in the J and νJ channels.The sensitivity and the relative weight of the J channel is larger than those of the νJ channel in the combination.Similar to the interpretation of the search results in the Z L Z L signal hypothesis, we observe here that the combined results give a small excess (≈ 1σ) around m VV = 1.9 TeV.

Combination of hadronic and semi-leptonic channels
This section is dedicated to the combination of both hadronic and semileptonic channels by ATLAS and CMS under different signal hypotheses.The searches in the JJ and νJ channels contribute to constrain a hypothetical G bulk → W L W L production; searches in the JJ and J channels enter the combination for the interpretation of the results in a G bulk → Z L Z L signal scenario.Finally, all six searches (i.e.results in three channels by two experiments) enter the combination in the W → W L Z L signal hypothesis.
The exclusion limits on production cross section, likelihood-ratio p-values, and best-fit cross sections as a function of a hypothetical resonance mass are summarised in Fig. 21.Scans of the profile likelihood as a function of the exotic production cross section for m X = 1.9 and 2.0 TeV (mass values of largest excesses for the benchmark models considered) are given in Fig. 22.The sensitivity of the search in the G bulk → Z L Z L signal hypothesis is dominated by the semileptonic analyses below 1.9 -2.0 TeV and the fully hadronic searches at higher mass ranges.The largest deviation is observed at m X = 1.9 TeV, driven by the ATLAS excess in the VV → JJ channel.The overall significance remains above 3σ.The preferred cross section for a hypothetical G bulk → Z L Z L signal as calculated in the J channel is ≈ 2 fb and increases to ≈ 9 fb for the JJ channel.When combined, the estimated cross section is 5 fb.The combination of the two channels reduces the exotic cross section favoured by the JJ results, and alleviates the potential disagreement between different channels, without reducing the overall significance of the excess.In other words, the combination of the two channels leads to a more coherent picture of the results by the two experiments.This is also evident from the profile likelihood scans shown in Fig. 21: given the uncertainty on the best-fit exotic production cross section, and contrary to what one might expect by considering the individual exclusion limits, the results obtained in different final states are not in tension with each other.In addition, the combination pushes the excess to mass values below 2 TeV.
The picture is quite different in the G bulk → W L W L signal interpretation.The lack of a significant excess in the νJ channels is strong enough to reduce the significance of the JJ excess below the 1σ threshold.The combination of the ATLAS and CMS results disfavours the hypothesis of a resonance decaying exclusively to WW (an interpretation which in any case would be difficult to justify phenomenologically).
Finally, the interpretation of the results in the context of a W signal hypothesis lies between the G bulk → Z L Z L and G bulk → W L W L scenarios: the νJ analyses are more sensitive than the fully hadronic ones, but their contribution is not as dominant as in the G bulk → W L W L case.Nevertheless, the excess survives above the 3σ threshold, thanks to the presence of a moderate excess in the J search around the same mass region.Overall, the estimated cross section of a hypothetical exotic signal is strongly reduced: the best-fit value changes from ≈ 10 fb (when using the JJ channel results only) to ≈ 5 fb (when combining the JJ, νJ and J channels).At this smaller cross section value, the outcome of the searches in the different channels is quite coherent, as shown in the profile likelihood scans depicted in Fig. 21.The mitigating effect of the J result is evident if one compares the νJ-and-J combined likelihood scan for the W combination to the likelihood scans in the semileptonic searches.The W L Z L curve is much more similar to the Z L Z L curve in the J channel than to the W L W L curve in the νJ channel.
In conclusion, a resonance with a production cross section of ∼5 fb and mass between 1.9 and 2.0 TeV is the scenario most consistent with the experimental results out of all benchmark models considered in this study, as long as it does not decay exclusively to a W L W L final state.
An example of the model independent combination of the Z L Z L and W L W L channels is shown in Fig. 23.In this case, one considers a resonance that can decay to both W L W L and Z L Z L , with the relative branching fraction determined by the r parameter introduced in Eq.(2.1).For r → 0 one recovers the G bulk → Z L Z L case, while for r → ∞ one recovers the G bulk → W L W L limit.It should be noted that for this combination we use a common mass window for the ATLAS analyses, namely the one that corresponds to the ZZ search, giving the best overall sensitivity (see Sec. 3).Therefore, the results obtained here on the W L W L exclusion limits and p-values are somewhat different than the ones presented in Fig. 21.
The results obtained for generic values of r are similar to the Z L Z L case, i.e. they point to an overall excess.The size of the excess is reduced to 2σ, with a best-fit exotic production cross section around 4 fb.Particularly interesting is the r = 2 case, corresponding to a resonance with universal couplings to the pseudo-Goldstone bosons.In this case, despite the fact that B(X → W L W L ) = 2B(X → Z L Z L ), the combined deviation is found to have a ≈ 2.4σ significance for a cross section of ≈ 4 fb.
It may be interesting to comment here on how the statistical methods that we have employed in this study compare with the simplified practices used by the theoretical community.A standard technique employed in many theoretical papers is to assume Gaussian likelihoods for the cross section of hypothetical signals, taking as central value the difference between the observed and expected limits, and as standard deviation the expected (95% C.L.) limit divided by 1.96.Then, one can use the cross sections and uncertainties as estimated in the various channels, and calculate a weighted average.This method should, in principle, work well for cases in which the fitted cross section comes with a relatively small uncertainty (which is, typically, not the case in most searches) and the systematic uncertainties can be considered as uncorrelated among channels and experiments (which may, or may not be the case).As an example, we note that the simplified combination of the search results in the G bulk → W L W L interpretation yields a best-fit cross section of 2.5 ± 1.6 fb (2.5 ± 1.4 fb) at m X = 1.9 TeV (2.0 TeV), to be compared with our result of 0.75 +1.67 −0.75 fb (1.1 +1.4 −1.1 fb).Similarly, the simplified combination in the G bulk → Z L Z L interpretation yields a best-fit cross section of 4.7 ± 1.9 fb (4.4 ± 1.8 fb) at m X = 1.9 TeV (2.0 TeV), to be compared with our result of 5.2 +2.1 −1.6 fb (4.2 +1.9 −1.2 fb).While more data is needed to clarify the situation, the results from the analysis of the diboson searches is unquestionably one of the most interesting outcomes of the ATLAS and CMS exotic programmes during the first LHC run.The situation is even more intriguing if one adds to the picture the ≈ 2σ excess at 1.8-1.9TeV observed by CMS in a WH resonance search.The W results shown in Fig. 21 emerge as the most promising hint in the quest for a new heavy resonance in the ATLAS and CMS data, as already pointed out in Ref. [39].

Conclusions
We have performed a combination of the ATLAS and CMS searches for a heavy resonance decaying to a diboson final state, derived from the public information available for the six relevant analyses [20-22, 79, 80].We have developed a methodology for the combination procedure, which begins with the work to emulate the public results by ATLAS and CMS for each individual analysis.This process is adjusted when necessary with correction factors to account for unknown uncertainties, and has been validated by reproducing the official results by the two experiments.We have presented combinations of the ATLAS and CMS searches for individual decay modes in various simplified models.At each step, the 95% CL limits, the likelihood ratio p-values, the profile likelihood scans, and the maximum likelihood fits of the production cross section as function of the resonance mass m X are provided.
The combination is obtained in three scenarios: W → W L Z L , G bulk → W L W L , and G bulk → Z L Z L .We also obtain the full combination results for a G bulk resonance with generic W L W L and Z L Z L branching fractions.Out of all benchmark models considered, the combination favours the hypothesis of a resonance with mass 1.9-2.0TeV and a production cross section ≈ 5 fb, as long as the resonance does not decay exclusively to W L W L final states.Depending on the details of the resonance model, a signal significance between 2.4 and 3.4σ is obtained for notable benchmark scenarios (see Table 3).In particular, the possibility of a W resonance, suggested by other searches in different final states, is corroborated by the diboson searches, with a significance of ≈ 3σ for a resonance mass of 1.9 TeV.

Note added in v2 of the paper
While preparing this manuscript for submission, ATLAS and CMS presented preliminary results in searches for diboson resonances with the first √ s = 13 TeV pp collision data.They include results in the W ( ν) V (q q) [83], Z( + − ) V (q q) [84], V (q q) V (q q) [85] and Z(ν ν) V (q q) [86] channels by ATLAS, and the W ( ν) V (q q) and V (q q) V (q q) channels by CMS [87].No significant excess above the SM expectations is observed, however the experimental sensitivity is, in most cases, not comparable with the one from Run-1 yet.The notable exception is the newly added Z(ν ν) V (q q) channel.The most stringent exclusion limits in the preliminary analysis of Run-2 data are obtained in the following channels: • (HVT) W → W L Z L : 25 fb (20 fb) for m X = 1.9 TeV (2.0 TeV) in the W (q q) Z(ν ν) channel by ATLAS, and the combination of the two channels considered by CMS.
In assessing the compatibility of the Run-2 exclusion limits with the results obtained in this study (summarised in Table 3) we use parton luminosity ratio values of 13 (15) for m X = 1.9 TeV (2.0 TeV) for gg production (G bulk → W L W L and G bulk → Z L Z L channels) and 8 (8.5) for m X = 1.9 TeV (2.0 TeV) for q q production (W → W L Z L channels) [88] to calculate the increase in the exotic signal production cross section from 8 to 13 TeV.We observe that the absence of a significant deviation in the Run-2 data • creates a ∼ 2 − 3σ tension with the best-fit cross section derived in this paper in the • is consistent (within 1σ) with the (consistent-with-zero) result we obtain in the G bulk → W L W L channel, and • is also consistent (within 1σ) with the best-fit cross section that we have derived in the W → W L Z L channel.
We, therefore, conclude that the preliminary analysis of the Run-2 data by ATLAS and CMS does not rule out the small deviation reported in the W → W L Z L channel of the Run-1 diboson searches.It is widely expected that a clear picture will emerge with the analysis of the larger 13 TeV datasets.

ATLAS+CMS
no fudge    Combination of all ATLAS and CMS resonance searches: Scans of the profile likelihood as a function of the production cross section for a m X = 2.0 (1.9) TeV signal shown with continuous (dashed) lines in the hadronic (red) and semileptonic (blue) channels and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses.obtained in this study on the r ≡ B(X → WW)/B(X → ZZ) parameter for a neutral bulk RS-like spin-2 particle hypothesis, and as a function of the resonance mass m X .Left: expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section.Middle: likelihood-ratio p-values.Right: best fitted exotic production cross section.

A Comparison of different approaches to emulate ATLAS VV → JJ analysis
The expected limits obtained in the emulation of the ATLAS VV → JJ channel show a 40% discrepancy with respect to the official results (see Sec. 3).This is the largest discrepancy observed among all the channels considered in this study.We have considered alternative approaches in our strategy and carried out several cross-checks, which are summarised here: • Nominal background: ATLAS publishes a background description with a total background uncertainty.This information can be used directly as an input to our analysis.The disadvantage of this approach is that it combines all systematic uncertainties into a single contribution, implying a correlation model that may not reflect the accuracy of the fit performed by the ATLAS collaboration.
• Pure fitting: We have repeated the fit on the data distribution provided by the ATLAS collaboration.The fitting procedure naturally yields a covariance matrix for the shape parameters, which allows to adopt a more realistic correlation model.
• Rescaling: This is a mixed approach in which the fit is performed over the data distribution to obtain the covariance matrix of the fitting function parameters, but the resulting background prediction and the corresponding uncertainties are then rescaled to match those provided by ATLAS.In this approach, the official ATLAS background prediction is used and our fit is only used to model the uncertainties and their correlations.
• Sidebands: In this case we repeat the fit procedure described above, after excluding the region of the largest deviation (1700-2300 GeV) from the fit range, in order to exclude the possibility that it could bias the fit.
Fig. 24 shows the ratio of the observed exclusion limits to the ones from the official ATLAS results for the different approaches summarised above.In all cases the differences are very small, which suggests that the explanation for the observed discrepancy should be attributed to a factor other than the background determination procedure.The discrepancy is absorbed in the fudge factor which, when tuned to deliver the official expected exclusion limits, remarkably removes (to a large extent) the differences in the observed limits.One should note that the decision to employ these correction factors in our analysis (for this and other channels) does not change qualitatively the conclusions of this study.This can be seen, for example, in the middle plot of Fig. 21, where it is shown that the two different approaches yield significances that differer typically by 0.5σ.

B Narrow width approximation
The CMS collaboration assumes a signal with negligible width, whereas the ATLAS collaboration simulates signal distributions with a model-dependent width of ≈ 7% of the resonance mass (see Table 1 of Ref. [20]).In this appendix we estimate the effect of this ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

ATLAS JJ
▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ difference in the final exclusion limits and provide a recipe for obtaining the ATLAS results in the narrow-width approximation.
The large width hypothesis used by the ATLAS collaboration impacts the limits through the modification of the signal shapes.In the JJ channel it widens the core for the signal distribution and creates a large left tail due to the interplay between proton PDFs [89] and the natural width of the resonance, as one can see in the left plot of Fig. 25.In practice, for a given total cross section we have events leaking outside the ±10% window around m X .This value corresponds typically to the experimental resolution of this channel.The amount of this leakage, f l is provided in Ref. [20] and corresponds typically to 15% in the region under study in this paper.
We expect the events in the left tail to have no significant impact on the exclusion limits.A test was performed by truncating the signal to m X ± 200 GeV and repeating the JJ limit-setting procedure for the W hypothesis.As one can see in the right plot of Fig. 25, the difference in the expected exclusion limits does not exceed 2%.To map the ATLAS limits into a narrow width hypothesis we make the following approximation: The main difference between the wide and narrow resonances is the presence of leaking events in the right tail or under the peak.Consequently, by multiplying the signal efficiency of ATLAS by 1/f l we recover most of the properties of the narrow signal.In conclusion, we approximate the narrow signal hypothesis for ATLAS analyses by scaling the fully hadronic and semi-leptonic signals by a factor of 1.1 (i.e. by increasing the signal yield by 10%).Right: Ratio of the expected (dashed lines) and observed (continuous lines) exclusion limits when constraining the signal width to 10% of the resonance mass over those obtained with the default shape.

Figure 1 .
Figure 1.ATLAS hadronic search: Comparison between the official ATLAS fit (blue line) and the fit of this study with uncertainties as described in the text (coloured bands), with the overlaid data of the m JJ spectrum for the WW (left), WZ (middle) and ZZ (right) tagging selections.

Figure 4 .
Figure 4. ATLAS hadronic search: Expected exclusion limits for different tagging and masswindow selections, as a function of the mass m X of the exotic resonance for G bulk → W L W L (left), W → W L Z L (middle) and G bulk → Z L Z L (right) signal hypotheses.The results have been obtained with the correction discussed in the text.

Figure 5 .
Figure 5. CMS hadronic search: m JJ data distribution overlaid with the background fit employed in this study with uncertainties for High (left) and Low (right) Purity samples.See text for details.

Figure 8 .
Figure 8. Combination of hadronic searches: likelihood ratio p-values as a function of the exotic resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (continuous black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections.The dashed black curve corresponds to the combined search without the 10% scale factor discussed in the text.

Figure 9 .Figure 10 .
Figure 9. Combination of hadronic searches: Best fitted exotic production cross section as a function of the resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses.The green and yellow bands represent the one and two sigma variations around the median values.The results include the 10% scale factor discussed in the text.

Figure 11 .
Figure 11.Combination of hadronic searches, and dependence of results obtained in this study on the r ≡ B(X → WW)/B(X → ZZ) parameter for a neutral bulk RS-like spin-2 particle hypothesis, and as a function of the resonance mass m X .Left: expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section.Middle: likelihood-ratio p-values.Right: best fitted exotic production cross section.

Figure 12 .
Figure 12.ATLAS ZV → J (left) and WV → νJ (right) searches: Comparison between the official ATLAS background (blue line) and its uncertainties (purple band) with the overlaid data of the m JJ spectrum for the Merged Region (of the vector boson hadronic reconstruction) category.

Figure 15 .
Figure 15.CMS WV → νJ search: Comparison between the official CMS background (blue line) and the background modelling with uncertainties employed by this study (coloured bands), with the overlaid data of the m JJ spectrum for the HP (left-hand side) and LP (right-hand side) categories, plotted separately for the electron (top) and the muon (bottom) channels.

Figure 16 .
Figure 16.CMS ZV → J search: Comparison between the official CMS background (blue line) and the background modelling with uncertainties employed by this study (coloured bands), with the overlaid data of the m JJ spectrum for the HP (left-hand side) and LP (right-hand side) categories, plotted separately for the electron (top) and the muon (bottom) channels.

Figure 20 .
Figure 20.Combination of semileptonic searches forG bulk → Z L Z L (top), G bulk → W L W L (middle) and W → W L Z L (bottom)selections and signal hypotheses, and as a function of the resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black).Left: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section.The green and yellow bands represent the one and two sigma variations around the median expected limits.The results include the correction factors discussed in the text.Right: Likelihood ratio p-values.The dashed black curve corresponds to the combined search without the corrections discussed in the text.

Figure 21 .Figure 22 .
Figure 21.Combination of all ATLAS and CMS resonance searches forG bulk → Z L Z L (top), G bulk → W L W L (middle) and W → W L Z L (bottom)selections and signal hypotheses, and as a function of the resonance mass m X carried out in the hadronic (red) and semileptonic (blue) channels and their combination (black).The results include all correction factors discussed in the text.Left: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section.The green and yellow bands represent the one and two sigma variations around the median expected limits.Middle: Likelihood ratio p-values.The dashed black curve corresponds to the combined search without the corrections discussed in the text.Right: Best fitted exotic production cross section.The green and yellow bands represent the one and two sigma variations around the median values.

Figure 23 .
Figure23.Combination of all ATLAS and CMS resonance searches, and dependence of results obtained in this study on the r ≡ B(X → WW)/B(X → ZZ) parameter for a neutral bulk RS-like spin-2 particle hypothesis, and as a function of the resonance mass m X .Left: expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section.Middle: likelihood-ratio p-values.Right: best fitted exotic production cross section.

Figure 24 .
Figure24.Emulation of ATLAS VV → JJ search and comparison of the alternative approaches for the background prediction considered: Fudge factors as a function of the resonance mass m X , determined via the ratio of the expected limits obtained with different background estimation techniques (black: "pure fitting", red: "nominal background", blue: "rescaling", magenta: "sidebands") over those in the official ATLAS result for the W L W L (left), W L Z L (middle) and Z L Z L channels (right).See text for details.

Figure 25 .
Figure 25.Narrow-width approximation.Left: Signal distribution in the diboson invariant mass for a 2 TeV W signal.The hatched ±200 GeV region around the signal represents the narrowwidth approximation.Right: Ratio of the expected (dashed lines) and observed (continuous lines) exclusion limits when constraining the signal width to 10% of the resonance mass over those obtained with the default shape.

Table 1 .
Summary of the methods used and the corresponding uncertainties for the signal and background modelling per channel and experiment.

0.2 0.4 0.6 0.8 1.0 m X (TeV) our fit / official result ATLAS JJ V L V L
Figure2.ATLAS hadronic search: Ratio of observed exclusion limits obtained with this study to the ones of the official ATLAS result, as a function of the mass m X of the exotic resonance for the WW (black), ZZ (red) and WZ (magenta) tagging selections.

Table 2 .
Relative efficiencies for WW, WZ, ZZ signal hypotheses for tagging selection using different mass windows.

Table 3 .
Summary of results obtained in this study: significance, p-values and best-fit cross sections for different model interpretations at m X = 1.9 and m X = 2.0 TeV, i.e. the mass values where the largest excesses have been observed for different models.Our main results contain corrections that have been introduced to account for unknown uncertainties in the official results.(Additional results calculated without these correction factors are given inside the parentheses.)