On the challenge of estimating diphoton backgrounds at large invariant mass
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Abstract
We examine, using the analyses of the 750 GeV diphoton resonance as a case study, the methodology for estimating the dominant backgrounds to diphoton resonance searches. We show that close to the high energy tails of the distributions, where background estimates rely on functional extrapolations or Monte Carlo predictions, large uncertainties are introduced, in particular by the challenging photon–jet background. Analyses with loose photon and low photon \(p_T\) cuts and those susceptible to high photon rapidity regions are especially affected. Given that diphotonbased searches beyond 1 TeV are highly motivated as discovery modes, these considerations are relevant for future analyses. We first consider a physicsdriven deformation of the photon–jet spectrum by nexttoleading order effects and a phase space dependent fake rate and show that this reduces the local significance of the excess. Using a simple but more general ansatz, we demonstrate that the originally reported local significances of the 750 GeV excess could have been overestimated by more than one standard deviation. We furthermore crosscheck our analysis by comparing fit results based on the 2015 and 2016 LHC data sets. Finally we employ our methodology on the available 13 TeV LHC data set assessing the systematics involved in the current diphoton searches beyond the TeV region.
Keywords
Invariant Mass Invariant Mass Distribution Standard Model Background Photon Candidate High Invariant Mass1 Introduction
Searches for new physics at the energy frontier often look for new phenomena at the edge of distributions. In this kinematical region the knowledge of the standard model (SM) background is typically limited and the challenge is to look for a new resonance where only partial knowledge on the SM background is available. In this paper we focus in particular on new physics probes based on the high diphoton invariant mass spectrum. We examine, using the analyses of the 750 GeV diphoton resonance as a case study, the strategy currently used by the experimental collaborations in estimating the dominant SM backgrounds. We employ our methodology on the 13 TeV LHC data set to asses the systematics involved in the current diphoton searches beyond the TeV region.
 (i)
this excess is a result of a rare statistical fluctuation;
 (ii)
this excess implies a discovery of nonStandard Model dynamics.
As both conclusions are quite extraordinary (certainly the second one), they motivate an investigation into their robustness. In particular, we raise a third option, to be considered in conjunction with (i), namely, we ask how unlikely is the possibility that
 (iii)
the significance of the excess is overestimated due to underestimating fakebased backgrounds.
While our conclusion is independent of the 750 GeV resonance we use it as an example case to scrutinize the hypothesis of the underestimated background and its implications. First, the main rationale behind our hypothesis is presented in Sect. 2, followed by a detailed description of our approach to background estimation (Sect. 3) and the statistical treatment of the data (Sect. 4). The comparison with the full 2016 data set is presented in Sect. 5. Our main conclusions are summarized in Sect. 6. For other relevant work, see Refs. [5, 6].
2 The rationale
Superficially, the experimental situation related to the diphoton excess was fairly straightforward. The experiments had reported a relatively narrow “bump”, \(\Gamma /m\lesssim 6\%\ll 1\). Such a bump implies a rise in the differential distribution while, due to the rapidly falling parton luminosity functions, it is expected that any reasonable backgroundrelated distribution should be a monotonically decreasing function of the invariant mass. Consequently, the presence of a nonStandard Model feature seemed to have been indicated by the measurements. While this was qualitatively correct the challenge is to quantify the significance of the excess. To endow the bump with a significance, one needs to control and quantify the background.
The following approaches can be used to constrain the form of the background:
I. Datadriven approach Assuming \(\Gamma /m\ll 1\) and a featureless monotonic background, a robust way to constrain it is through interpolation via a twosided side band analysis. However, this requires one to have enough measured events at invariant masses both below the resonance and above it. In the case of the 750 GeV excess, there were less than 40 events in all of the analyses measured with invariant masses above 850 GeV. Such a small number of events does not allow one to use this method reliably.
II. “Firstprinciple”/MonteCarlo approach There is a rather narrow class of observables for which the theory has reached an advanced enough level such that we can fully trust our ability to correctly predict the shape of the background distributions. We believe that the invariant mass distribution of experimentally measured diphoton events does not (yet) belong to this selected class of observables. Namely, the continuous diphoton distribution consist of an admixture of two dominant components: (i) The first is made of two real isolated hard photons. This diphoton distribution is currently known to nexttonexttoleading order (NNLO) accuracy [7, 8] in perturbative QCD and imposing cuts similar to the ATLAS spin0 analysis suggests an overall uncertainty of about 5% for the invariant mass distribution [8]. (ii) An additional important background component is due to fakes coming mostly from processes involving a hard photon and a jet that passes the various photon quality and isolation cuts [9]. In addition, depending on these cuts, also the dijet background could play an important role. The prompt photon–jet cross section is currently known at nexttoleading order (NLO) in QCD, and several codes are available to produce the relevant distributions, including JetPhox [10] and PeTeR [11]. In addition, QCD threshold resummation at nexttonexttonexttoleading logarithmic (N\({}^3\)LL) order [12, 13] as well as electroweak Sudakov effects are being included [14], resulting in theory uncertainties of about 10–20% [9, 14, 15]. However, a comparison with the 8 TeV ATLAS measurement [9] shows that at low photon \(p_T\sim 50\,\)GeV the data exhibits some level of deviations from the theoretical predictions (a larger uncertainty is found for the invariant mass distribution; see [16]). In addition, it is important to note that the fake rate strongly depends on the quark/gluon “flavor” of the tagged jet (for some discussion of jet flavor definitions, see [17, 18, 19, 20]): intuitively one can understand the difference through the quark and gluon fragmentation functions to pions. At large x, as required to be able to pass photon isolation criteria, gluon fragmentation to few pions is much more suppressed (see e.g. Chapter 20 in Ref. [21]). Accordingly, a dedicated ATLAS study [22] found that there is a probability of about \(1:2\times 10^3\) for a quark jet to fake a photon, and only \(1:2\times 10^4\) for a gluon jet to fake a photon, for jets with \(E_T>40\,\)GeV. Applying this to the photon–jet background, we also note that subleading jets might become an important source of fakes if the leading jet is predominantly gluoninitiated.
In order to theoretically predict the purity of the diphoton mass distribution, an appropriate admixture of the diphoton and the photon–jet(s) components needs to be constructed [23]. Furthermore, for the latter component, one is required to convolve the photon–jet distribution with the relevant fragmentation functions or at least tag the flavor of the jet(s). It is also important to note that the purity is a highly phase space dependent quantity. Not only does it depend on the ratio of the differential jet–photon and photon–photon production but also on the jettophoton fake rate. The fake rate may exhibit a strong dependence on the differential quantities such as \(p_T\) and (pseudo)rapidity \(\eta \). For instance, as discussed below, in the CMS analyses purity is estimated to be better than 90% in the (centralcentral) EBEB event category but only better than 80% in the (forwardcentral) EBEE one. Both experiments consider the purity in an inclusive way. However, in the relevant kinematical region the data is not sufficient to constrain possibly large deviations from the inclusive purity estimation (see Fig. 4).
III. Functionalfit approach Given the present practical limitations of the methods I and II, one is lead to a more phenomenological approach in which the background estimate is obtained by fitting an universal function to control regions in the data and then extrapolating into the signal regions using the fitted functional form. This allows one to predict the background at relatively high invariant masses in a straightforward manner. Consequently, both experiments are essentially following this approach in most of their analyses,^{2} although the functional forms used by ATLAS in the spin0 analysis and by CMS are slightly different. Thus, the significance of the excess is mostly determined by comparing measured events to a background estimate predicted by a fitting function.
While method III is very transparent and makes the search for bumps easy to analyze, it is also rather susceptible to systematic effects, in particular a lack of understanding of the physics modifying the tails of the distributions, as we argue below. The fitting functions used by ATLAS and CMS are well suited for describing rapidly falling distributions and are fitted to the available data. With the amount of data in the 2015 data sets, the differentially measured number of events is abundant in the low invariant mass region and is spare in the high mass region. The extraction of the functions’ parameters is thus dominantly controlled by the low \(m_{\gamma \gamma }\) region and hardly affected by modifications of the invariant mass distribution at diphoton masses of above roughly 500 GeV. However, the significance of the excess with respect to the fitting function is very much affected by such deformations. As it is hard to directly test or predict the correct form of the diphoton mass distribution, this raises the following questions:
\({\mathcal {I}}\). Is the experimental signal over background estimation robust against the presence of deviations from the fitting function predictions at large invariant masses?
\(\mathcal{II}\). If this is not the case, can one produce smokinggun predictions to show that indeed the significance of the excess is being overestimated?
Let us first focus on point \(\mathcal{I}\). To examine the sensitivity of the significance of the excess to the variation of the tails of the distributions. We consider a family of background shapes that are formed by an admixture of the diphoton and photon–jet distributions. We keep the overall inclusive purity of the samples at 90 and 80%, respectively, in accordance with the measured data at low invariant masses. More specifically, we use two classes of deformations. The first is derived from a modification of the photon–jet spectrum due to NLO and showering effects combined with an increased fake rate for larger transverse momenta and pseudorapidities of the jets.
We then consider a simpler ansatz where we allow the distribution of the \(p p \rightarrow \gamma j\) component to be reweighted at invariant masses above 500 GeV such that the purity of events with large invariant masses is reduced leading to a controlled deviation from the functional fit. In the following section we provide a detailed description of our approach. We also provide some tests of our procedure to check that our method complies with public data (below and above the resonance region) and is passing the relevant statistical tests. We then report how the significance is affected by the amount of rescaling of the distributions of fakes. Finally we can use our ansatz to address item \(\mathcal{II}\) and provide smoking guns to test our hypothesis on overestimating the excess significance. With the full statistics of the 2016 data sets at hand it would be fairly easy to eliminate our hypothesis.
3 Reducible and irreducible backgrounds
Cuts of the analyses where the subscript refers to the hardest and second hardest photon candidate, the cross section of the \(p p \rightarrow \gamma \gamma \) sample passing these cuts (calculated at NNLO with MCFM) and of the \(p p \rightarrow \gamma j\) sample at hadron level (before applying any photon mistag rate), calculated at NLO with MadGraph5_aMC@NLO, showered with Pythia [29] and the jets clustered with anti\(k_T\), \(R=0.4\) algorithm using FastJet [30]. In the last line the ratios of the two distributions in the invariant mass region above 500 GeV are given
Analysis  ATLAS spin0  ATLAS spin2  CMS EBEB  CMS EBEE 

\(m_{\gamma \gamma }\)  \({>}150\,\,\text {GeV}\)  \({>}200\,\,\text {GeV}\)  \({>}230\,\,\text {GeV}\)  \({>}330\,\,\text {GeV}\) 
\(p_{T,1}\)  \({>}0.4\,m_{\gamma \gamma }\)  \({>}55\,\,\text {GeV}\)  \({>}75\,\,\text {GeV}\)  \({>}75\,\,\text {GeV}\) 
\(p_{T,2}\)  \({>}0.3\,m_{\gamma \gamma }\)  \({>}55\,\,\text {GeV}\)  \({>}75\,\,\text {GeV}\)  \({>}75\,\,\text {GeV}\) 
\(\left \eta _1\right \)  \({<}2.37\)  \({<}2.37\)  \({<}1.44\)  \({<}1.44\) 
\(\left \eta _2\right \)  \({<}2.37\)  \({<}2.37\)  \({<}1.44\)  \(1.57<\eta _2<2.5\) 
\(\left \eta \right \) excluded  \(1.37<\eta _{1,2}<1.52\)  \(1.37<\eta _{1,2}<1.52\)  n.a.  n.a. 
\(\sigma _{\gamma \gamma }\) [pb] (NNLO)  2.7  1.9  0.52  0.23 
\(\sigma _{\gamma j}\) [pb] (NLO)  1400  1000  250  130 
\(\left. {\sigma _{\gamma j}}{}/{\sigma _{\gamma \gamma }}{}\right _{m>500\,\,\text {GeV}_{}}\)  510  670  470  640 
3.1 QCD and jetfake dependence of the diphoton shape
3.2 Effective shape deformation
In the left panel of Fig. 2, we show the normalized differential \(p p \rightarrow \gamma j\) cross section for the ATLAS spin0 analysis as a function of the invariant mass for several choices of \(\mathcal {R}\). In the right panel, the \(\mathcal {R}\)dependent reweighting factor of Eq. (3.4) is shown. Since the spin0 analysis applies the strongest cuts on the transverse momenta of the photon candidates (\(0.4\,m_{\gamma \gamma }\) and \(0.3\,m_{\gamma \gamma }\), respectively) its distribution is the steepest. Thus the reweighting factor of this analysis is the largest being almost 7 above 770 GeV. The maximal reweighting factors for the other analyses are just above 6. We verified that increasing the flat region by 20 GeV has only a small impact on the reported results.
While the local purity is within the error band in most of the considered mass range (even for \(\mathcal {R}=1\)), it does decrease for large invariant masses and our ansatz predicts a deviation from the experimental value. Given the low statistics in this range, we consider this as a way to falsify our proposal in the future rather than a contradiction with the currently available data.
4 Statistical treatment
In order to see how the significance of the 750 GeV excess changes with our ansatz, we fit the distribution \(w_{\text {mix}}\), defined in Eq. (3.2), once with \(w_{\gamma j}^{\text {MG}}\) corresponding to \(\mathcal R=0\), and then with \(w_{\gamma j}^{}\) as estimated at NLO with showering and hadronization, including fakes and finally with \(\mathcal {R}\) as a free fit parameter (as well as the appropriate fit function f(x)) to the measured data. As an additional template, one could extend the fit function f(x) by a modification similar to the one described in Eq. (3.4), which we will, however, not do for the sake of simplicity. The fits are performed with two methods, which yield similar results.
Results of the fits to the data of all four analyses. In the first block from the top the signal region is defined and the number of measured events in this region \(N_m\) is given. The results of a likelihood and a \(\chi ^2\) fit of the fit function (value of the maximal likelihood and minimal \(\chi ^2\), respectively, and the local significance of the \(750\,\,\text {GeV}\) excess) are given in the second block. Finally, the third block contains the results of a likelihood and \(\chi ^2\) fit of the background distributions described in Sect. 3 to the data. When \(\mathcal {R}\) was fitted its bestfit value and the corresponding local significance of the excess are given, otherwise just the significance. In the last line the result of the Ftest, testing whether \(\mathcal {R}\) should be used as fit parameter, is given. For the minimized \(\chi ^2\) the parameter n is the difference of number of bins and fit parameters. The errors indicate the 1\(\sigma \) interval of the systematic uncertainty of the fit. Note that the results of the spin0 analysis with \(3.2\,\text {fb}^{1}\) are based on the analysis with looser photon identification as described in [1]
Analysis  ATLAS spin0  ATLAS spin2  CMS EBEB  CMS EBEE  

Measurement  
SR  730–770 GeV  720–760 GeV  720–780 GeV  710–770 GeV  710–770 GeV  
\(\int \mathcal {L}\,\text {d}t\)  \(3.2\,\text {fb}^{1}\)  \(15.4\,\text {fb}^{1}\)  \(3.2\,\text {fb}^{1}\)  \(2.7\,\text {fb}^{1}\)  \(12.9\,\text {fb}^{1}\)  \(2.7\,\text {fb}^{1}\)  \(12.9\,\text {fb}^{1}\) 
\(N_m\)  15  33  40  12  24  21  53 
Fitfunction  
\(2\log L\)  270  –  330  200  –  200  – 
\(\sigma \)  3.4  –  2.9  1.9  –  1.7  – 
\(\chi ^2/n\)  1.6  0.75  1.2  0.80  1.0  1.2  0.98 
\(\sigma \)  \(3.1_{0.2}^{+ 0.2}\)  \(1.2^{+0.2}_{0.2}\)  \(2.9_{0.3}^{+ 0.3}\)  \(1.8_{0.2}^{+ 0.3}\)  \(1.5^{+0.2}_{0.3}\)  \(1.6_{0.3}^{+ 0.3}\)  \(1.3^{+0.2}_{0.2}\) 
Distribution  
\(\mathcal {R}\)  
\(2\log L\)  270  360  330  210  310  210  300 
\(\mathcal {R}\)  0.86  0.11  0.97  −0.048  −0.7  0.24  −0.12 
\(\sigma \)  2.2  0.0  2.0  1.5  −1.2  1.4  −0.23 
MG  
\(\chi ^2/n\)  1.5  0.76  1.3  0.78  1.6  1.1  1.1 
\(\sigma \)  \(3.0_{0.0}^{+ 0.0}\)  \(0.2^{+0.0}_{0.0}\)  \(3.4_{0.1}^{+ 0.1}\)  \(1.4_{0.1}^{+0.1}\)  \(2.6^{+0.1}_{0.1}\)  \(1.9_{0.2}^{+ 0.2}\)  \(0.86^{+0.14}_{0.14}\) 
NLO  
\(\chi ^2/n\)  1.5  0.76  1.2  0.80  1.6  1.2  1.0 
\(\sigma \)  \(3.0_{0.0}^{+0.0}\)  \(0.3^{+0.0}_{0.0}\)  \(3.2_{0.1}^{+0.1}\)  \(1.4_{0.1}^{+0.1}\)  \(2.6^{+0.1}_{0.1}\)  \(1.7_{0.2}^{+0.2}\)  \(1.2^{+0.1}_{0.1}\) 
NLO\(\times \)fakes  
\(\chi ^2/n\)  1.5  1.4  1.1  0.92  2.0  1.2  1.2 
\(\sigma \)  \(2.7_{0.0}^{+0.0}\)  \(0.3^{+0.0}_{0.0}\)  \(2.9_{0.1}^{+0.1}\)  \(1.2_{0.1}^{+0.1}\)  \(3.0^{+0.1}_{0.1}\)  \(1.4_{0.2}^{+0.2}\)  \(1.7^{+0.1}_{0.1}\) 
\(\mathcal {R}\)  
\(\chi ^2/n \)  1.2  0.75  1.0  0.81  1.3  1.1  1.1 
\(\mathcal {R}\)  \(1.2_{0.5}^{+ 0.6}\)  \(0.2^{+0.2}_{0.2}\)  \(1.1_{0.4}^{+ 0.4}\)  \(0.15_{0.39}^{+ 0.51}\)  \(0.6^{+0.2}_{0.2}\)  \(0.30_{0.22}^{+ 0.29}\)  \(0.091^{+0.084}_{0.074}\) 
\(\sigma \)  \(2.0_{0.4}^{+ 0.4}\)  \(0.2^{+0.4}_{0.4}\)  \(1.9_{0.5}^{+ 0.5}\)  \(1.5_{0.4}^{+ 0.4}\)  \(1.3^{+0.4}_{0.4}\)  \(1.2_{0.4}^{+ 0.4}\)  \(0.40^{+0.42}_{0.42}\) 
\(p_\text {Ftest}\)  0.021  0.20  0.0027  0.73  0.014  0.19  0.30 
We find that, for the ATLAS searches, the Ftest suggests that \(\mathcal {R}\) should be included as a fitting parameter. The probability of an accidental improvement due to \(\mathcal {R}>0\) is only \(2.1\%\) (ATLAS spin0) and \(0.27\%\) (ATLAS spin2). On the other hand, the CMS categories do not prefer a significant nonzero \(\mathcal R\); see Table 2. Furthermore, as a consistency check, we apply the Ftest on the ATLAS fitting function for spin0, Eq. (4.1), and find that adding a \(k=1\) component to the function does not pass the test. Hence, as mentioned in [1] only the leading term of the function with \(k=0\) is retained. The above in conjunction with the results collected in Table 2 suggest that it is possible that the basis of functions used in Eq. (4.1) is not sufficient to accommodate the deformation of the distribution proposed by us (or at least not the first term in the functional form).
Since \(\mathcal {R}>0\) flattens the distribution one might worry that the reduction in the local significance is obtained by overshooting the measured distribution in the high invariant mass region. By verifying that both the minimal \(\chi ^2\) and the maximal likelihood hardly change between the functional and the distribution fit we show that this is not the case.
As a final exercise, we try to obtain a “combined” significance from the analyses of the 2015 data set. Clearly a proper statistical combination cannot be done, since we neglect correlations between the various analyses and also fit for a single universal value of \(\mathcal {R}\). Realistically, \(\mathcal R\) is expected to be somewhat different for the different analyses since they cover different regions of phase space. Nevertheless, since the naive combination in Eq. (1.1) suffers from similar issues we set them aside and proceed as follows. We sum the \(\chi ^2\) of the analyses included in the combination and fit for a common \(\mathcal {R}\) while keeping the normalizations as separate variables. By combining the two CMS analyses we obtain \(\sigma =2.4\) (1.9) for \(w_{\gamma j}^{\text {MG}}\) (with \(w_{\gamma j}(\mathcal {R})\), bestfit \(\mathcal {R}=0.22\)) and \(\sigma =1.9\) with \(w_{\gamma j}^{\text {NLO}\times \text {fakes}}\). A combination of the ATLAS analyses is impossible since they are not independent. However, we can combine each of them with the two CMS analyses and obtain for ATLAS spin0 combined with CMS \(\sigma =3.6\) (2.6 with \(w_{\gamma j}(\mathcal {R})\), bestfit \(\mathcal {R}=0.46\); 3.1 with \(w_{\gamma j}^{\text {NLO}\times \text {fakes}}\)) and for ATLAS spin2 combined with CMS \(\sigma =4.2\) (2.8 with \(w_{\gamma j}(\mathcal {R})\), bestfit \(\mathcal {R}=0.53\); 3.4 with \(w_{\gamma j}^{\text {NLO}\times \text {fakes}}\)), where the significance numbers before the brackets are obtained for \(w_{\gamma j}^{\text {MG}}\).
5 The new energy frontier: searches beyond 1 TeV
Around ICHEP 2016, ATLAS and CMS updated their analyses, now based on 15.4 \(\text {fb}^{1}\) and 12.9 \(\text {fb}^{1}\), respectively. In the updated ATLAS spin0 analysis [3] and the CMS EBEB and EBEE analyses [4] the large excess around \(750\,\,\text {GeV}\) vanished and no other significant excesses were found. An update of the ATLAS spin2 analysis has not been presented. While CMS processed the data exactly as before, ATLAS made some adjustments, perhaps most importantly, using a tighter photon isolation. We repeat the fits and the statistical treatment of the reported results with the same methods as described above and report the results for the larger data set in Table 2. Note that there is a downwards fluctuation in the signal region in the full CMS data set which even leads to a slightly negative significance.
Comparing the new fit functions to the ones based on the previous small data sets presented at Moriond 2016 we find a steeper functional fit in all three analyses; see Fig. 5. While the new bestfit parameters are within one standard deviation for the two CMS fits, the ones for the ATLAS fit deviate by almost two standard deviations after marginalizing over the normalization; see Fig. 6. This might, however, be an effect of the changed photon isolation as the fit to the \(3.2\,\text {fb}^{1}\) data set with the updated photon identification also deviates by more than one standard deviation from the previous bestfit point. A better understanding of the effect of the fake photons could be obtained by investigating the result of changing the isolation criteria with the full \(15.4\,\text {fb}^{1}\) data set. The tighter isolation criteria are also reflected in the better agreement between the fitted distributions and the MCFM generated digamma spectrum.
In order to show the changes in the fits, the ratio of the normalized fit functions for the ATLAS spin0 analysis is shown in the lower left plot of Fig. 5. A direct comparison of the data and the unnormalized fit functions, even for the fits to the two different \(3.2\,\text {fb}^{1}\) sets, is difficult since the binning of data has changed. The large change in the fit parameters is reflected in the deviation of more than 5% for the comparison of the fits to the two \(3.2\,\text {fb}^{1}\) data sets and the even greater deviation compared with the fit function to the full \(15.4\,\text {fb}^{1}\) data set. While in the previous signal region near 750 GeV the change is of the order of 10% it is greater than 30% near 1.6 TeV. This shows that the actual shape of the digamma spectrum at high invariant masses is hard to predict precisely by an extrapolation and is therefore very much subject to systematic uncertainties.
Finally in Fig. 7 the ratios of several normalized distributions and fit functions to the normalized distribution with \(\mathcal {R}=0\) are shown. These include the distributions with the bestfit value for \(\mathcal {R}\) based on the Moriond 2016 data set and the ICHEP 2016 data set and also the NLO distributions and the fit functions to the old and new data sets. In the case of the ATLAS spin0 analysis also the distribution and fit function to the 2015 data set with the new photon isolation is shown. By comparing the curves we find that a sizable systematic uncertainty can be inferred from the differences between the fit functions.
6 Conclusions
This paper deals with a problem that often arises in searches for new physics at the energy frontier. In this context the challenge is to look for a new resonance at the upper end of a distribution where only limited knowledge on the SM background is available. As a case study we focus on the 750 GeV anomaly where we examine in particular the implications of the possibility that the excess in the 2015 data set is not only due to a (malicious) statistical fluctuation but also a result of a physical effect. We discuss possible issues with the background: how much photon–jet contamination is still allowed in the region of interest? How could it affect the significance of the excess?
We study these questions using currently available theoretical tools for computing the photon–jet mass distributions and apply them to the small set of publicly available data. However, this approach is limited by our ability to thoroughly disentangle the effects of the \((p_T,\eta )\)dependent jetfake rate and the theoretical uncertainty of the shape of the photon–jets background. We therefore choose to model these combined effects by an \(m_{\gamma j}\) dependent reweighting of the invariant mass distribution, keeping the overall purity within the quoted ranges. We first study a physicsdriven reweighting procedure: we convolve a mass dependent Kfactor with a rapidity and transverse momentum dependent photon fake rate for the jets. The Kfactor is extracted comparing the NLO leadingjet–photon to the LO quark–photon spectrum, and the phase space dependence of the fake rate is estimated from the experimental literature [22]. Both correction factors are approximate, based on incomplete information, and should be taken with a grain of salt. Motivated by this result, we then consider a more phenomenological deformation of the \(p p \rightarrow \gamma j\) spectrum. It allows us to study the sensitivity of the significances on a single continuous quantity \(\mathcal {R}\) (see Eq. (3.4)) which parametrizes an effective deformation.

For the ATLAS spin0 analysis, the significance of the excess can be reduced by \(\Delta \sigma \simeq 1.1 \) when comparing the fitting function defined in (4.1) with our best fit to the \(\mathcal R\)modified distribution. A comparable reduction is found for the ATLAS spin2 measurement. Here however, it is less straightforward to determine the reduction, since the estimation of the background shape in our ansatz differs from that of the ATLAS analysis, which is not reproducible since the required data is not publicly available. Strictly speaking, \(w_{\gamma j}^{\text {MG}}\) does therefore not correspond to the ATLAS approach but is the best approximation we can get. Since, however, ATLAS claims to find comparable results with the corresponding fitting function defined in (4.2), we can reduce the significance with the \(\mathcal {R}\)modified distribution with respect to the fitting function by \(\Delta \sigma \simeq 1.0\) as well as with respect to the distribution with \(w_{\gamma j}^{\text {MG}}\) by \(\Delta \sigma \simeq 1.5\).

The effect is smaller for the CMS 13 TeV analyses with \(\Delta \sigma \simeq 0.30.4\), depending on the category.

In a “combined fit” to independent ATLAS and CMS data sets, the significance can be reduced by as much as \(\Delta \sigma \approx 1.0 \,(1.4)\) for the ATLAS spin0 (spin2) combined with CMS.

For the ATLAS spin0 analysis, we find that the new data prefers \(\mathcal {R}\) in the range \(0.2\pm 0.2\), eliminating the remaining significance of \(1.2\,\sigma \) in the full data set. As for the spin2 case no data is currently available.

The updated CMS analyses based on \(12.9\,\text {fb}^{1}\) even have a downwards fluctuation with respect to the fit function near 750 GeV leading to negative significances. Correspondingly the bestfit values for \(\mathcal {R}\) are negative and ameliorate the situation.

A variation of about 30% in the extrapolated background near \(m_{\gamma \gamma }=1.6\,\,\text {TeV}\) is obtained.
Footnotes
 1.
We do not discuss here the global significance as it strongly depends on the lower value of \(m_{\gamma \gamma }\) defined for the search region. ATLAS (CMS) chose it to be about 200 GeV (400 GeV). Furthermore, as discussed below, the region below 500 GeV is dominating the fit to the functional form which is used to estimate the background. Thus, it is not clear whether one should consider this region as a control region or as the region of interest for the search itself.
 2.
An exception is the ATLAS spin2 analysis which employs a Monte Carlo approach (II) with a datadriven estimate of the photon–jet and jetjet background; see Sect. 4.
Notes
Acknowledgements
We would like to thank Rikkert Frederix for useful discussions. J.F.K. would like to thank CERN for hospitality while this work was being completed and acknowledges the financial support from the Slovenian Research Agency (research core funding No. P10035). The work of GP is supported by grants from the BSF, ISF and ERC and the WeizmannUK Making Connections Programme. AW is supported by the DFG cluster of excellence “Origin and Structure of the Universe” and the European Commission (AMVA4NewPhysics, 2020MSCAITN2015).
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