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Re-examining the significance of the 750 GeV diphoton excess at ATLAS Bradley J. Kavanagh∗ LPTHE, CNRS, UMR 7589, 4 Place Jussieu, F-75252, Paris, France and Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France Theexcessseeninthediphotonchannelataround750GeVbybothATLASandCMShascaused agreatdealofexcitementintheparticlephysicscommunity. However,therehasrecentlybeenmuch discussion about uncertainties in the significance of the peak seen by the ATLAS experiment. In this note, we aim to estimate this significance using a range of possible parametrisations for the smooth diphoton background. We obtain a local significance close to that reported by ATLAS and further demonstrate that the significance of the excess is not substantially reduced when more complicated background functions are considered. In particular, the background contribution is strongly constrained by the small numbers of events at large diphoton invariant mass. Future data releases will improve constraints on the diphoton background, as well as clarifying the true nature 6 of the 750 GeV excess. 1 0 2 I. INTRODUCTION invariant mass spectrum to estimate the significance of b the excess under different background assumptions. In e In the first results from Run-II of the LHC, both particular, weperformaPoissonianlikelihoodfitover40 F the ATLAS and CMS collaborations reported an excess bins in the diphoton invariant mass mγγ, making this 9 above the expected background in the diphoton channel analysis distinct from that of Davis et al. The code used 1 [1, 2] at a diphoton invariant mass of m ∼ 750 GeV. to perform this analysis is publicly available online [7]. γγ The local significance of this excess was reported as 3.9σ The intention of this note is not to improve upon the ] h (2.3σ global)byATLASand2.6σ (1.2σ global)byCMS. ATLAS analysis or to report definitive values for the p The possibility that this excess could be due to a new significance of the 750 GeV excess. Indeed, this is not - bosonic particle has caused a great deal of excitement in possiblewiththeinformationwhichispubliclyavailable. p e theparticlephysicscommunityandthelistofpapersex- Instead, we aim to explore the robustness of the excess h ploring the implications for New Physics is of course too to variations in background fitting, as well as to clarify [ long to mention. certain aspects of the fits performed by ATLAS. Since the announcement of these results [3], a number 3 The different functional forms we consider for the of authors have performed fits to the Run-II diphoton v diphoton background are given in Eq. 2, while the local 0 data from a theory perspective in order to estimate the significances we obtain for the 750 GeV diphoton excess 3 mass, width and cross section times branching ratio of a (using each functional form) are given in Table III. We 3 possible new particle (see e.g. Ref. [4]). However, these describe in detail the fitting procedure used in this work 7 analyses typically assume a fixed form for the diphoton 0 inSec.II,followedbytheresultsofbackground-onlyand background whose parameters are not included in the . signal + background fits in Sec. III and Sec. IV. Finally, 1 fits. This procedure is not strictly correct, as parameter in Sec. V we discuss the implications of these results for 0 values extracted from the data should take into account the significance of the 750 GeV excess. 6 uncertainties in the background. A full likelihood analy- 1 sis of both the ATLAS and CMS binned diphoton data : v wasperformedinRef.[5],assumingthesamebackground i models used by the respective collaborations. The lo- X cal significances reported therein are in broad agreement II. FITTING PROCEDURE r a with those reported by ATLAS and CMS. However,arecentpaperbyDavisetal.[6]exploredthe Within a frequentist framework, the significance of a possibility of allowing more freedom in the modelling of possible excess in the diphoton spectrum can be esti- theATLAScontinuumdiphotonbackground. Theircon- mated by calculating the maximum likelihood obtained clusionwasthatthebackgroundcouldbefitbyanumber assuming that only backgrounds contribute to the ob- ofdifferentfunctionalformsandthatsomeoftheseforms served events (background-only) and that obtained as- lead to a lower significance for the diphoton excess than suming that both signal and background have a contri- that reported by ATLAS, perhaps as low as 2.0σ. Moti- butiontotheobservedeventrate(signal+background). vated by this discrepancy between the results of the AT- By comparing the maximum likelihood obtained under LAS collaboration and those of Davis et al., we perform these two assumptions, we can then calculate the signif- an independent analysis of the binned ATLAS diphoton icance of a possible signal in the data. ThelikelihoodL(θ)issimplytheprobabilityofobtain- ingtheobserveddataassumingaparticularsetofparam- ∗ [email protected] eters θ. For binned data, this has the Poisson form: 2 Parameter Prior range L(θ)=n(cid:89)bins (Nei)Noi exp(−Ni), (1) lbo,ga100,Na1, a2 [[--2255,, 2255]] Ni! e m [700, 800] GeV i=1 o γγ α [1, 10] % where Ni = Ni(θ) is the expected number of events in NS [0.01, 100] e e theith bin(foragivensetofparametersundertheback- groundorsignal+backgroundhypothesis)andNi isthe TABLEI.Priorrangesforthebackgroundandsignalparam- observed number of events in the ith bin. In partoicular, etersusedinthisanalysis. Ineachcase,thepriorisconstant overthespecifiedrange. WenotethatbecauseweuseMCMC we do not include any additional terms in the likelihood onlytofindthemaximumlikelihoodpoint,theexactformof due to uncertainties on the diphoton invariant mass res- the prior does not affect the results presented. olution (as in Ref. [1]). We use a total of n = 40 bins in the m invari- bins γγ ant mass, each with width 40 GeV, spanning the range m ∈[150,1750] GeV. We obtain the number of events background in their analysis, though for comparison we γγ observed in each bin from Ref. [1] (by digitising Fig. 1 willconsidercaseswhereN issetto1(‘fixed-N’)aswell therein using a publicly available plot digitiser [8]). This as where N is allowed to vary (‘free-N’). proceduremayintroducesomedigitisationerrorintothe The expected number of signal events coming from a analysis,particularlyforbinswhichcontainalargenum- possible new resonance near m ∼750 GeV is obtained ber of events. However, we have explicitly checked the γγ using either the narrow width approximation (NWA) or impact of digitisation errors on our results and, as de- by allowing the width of the resonance to vary freely scribed in Sec. V, these do not affect the conclusions we (free-width). UndertheNWA,thesignalismodelledasa report. Gaussian centred on m =m with standard deviation Theexpectednumberofbackgroundeventsisobtained γγ B σ given by the diphoton invariant mass resolution. This by integrating the background event distribution over resolution is taken to increase linearly from σ = 2 GeV each bin in m , for a given choice of functional form γγ at m = 200 GeV to σ = 13 GeV at m = 2 TeV [1], and for a given set of background parameters θ . In γγ γγ b giving a value of around σ = 5.4 GeV near the putative Ref. [1], the ATLAS collaboration discuss a set of pos- resonanceat750GeV.Thisisarathercrudeapproxima- sible empirical functions for the continuum background. tion to the double-sided Crystal Ball function (DCSB) These have been adapted from functions used in multi- used by the ATLAS collaboration in their NWA anal- jet searches for New Physics [9] and have been validated ysis. However, given that the bin widths used in this against both Monte Carlo and data samples. In Ref. [6], analysis are much larger than the mass resolution, we do Davisetal.introducedanotherpossibleparameterisation not expect the precise shape of the narrow resonance to for the continuum diphoton background (also validated strongly affect our results. Indeed, we are still able to with a Monte Carlo study) which we discuss further in recover a significance close to that reported in Ref. [1]. Sec. V. In this work, however, we focus on those back- The signal is normalised so as to contribute a total of groundfunctionsusedbytheATLAScollaboration,given N events, meaning that for the NWA we have two free explicitly by S parameters describing the signal: θ =(m ,N ). s B S In the free-width analysis, we model the signal as a fk(x)=N(1−x1/3)bx(cid:80)kj=0aj(logx)j (2) Breit-Wigner distribution with width Γ = αmB. We do √ not include the effects of finite diphoton invariant mass where x=mγγ/ s and f(x) is the differential distribu- resolution in the free-width analysis. However, as in the tionofexpectedbackgroundevents(inunitsofevents/40 NWA analysis, we can still recover a significance close to GeV).1Wewillconsiderk =0,1,2inthiswork,forwhich that reported in Ref. [1]. In the free-width analysis, we the parameters b, a0, a1 and a2 determine the shape of have three free parameters: θs =(mB,NS,α). the background and are allowed to vary in the analysis. In order to explore the signal and background param- The parameter N which controls the overall normalisa- eter spaces, we use the publicly available Markov Chain tion of the background is typically included when using Monte Carlo (MCMC) software emcee [11], which uses this class of functions [9]. While it is not explicitly de- Affine Invariant MCMC Ensemble sampling [12]. We scribed in Ref. [1], the free normalisation parameter has performatleast20000likelihoodevaluationsforeachfit. beenincludedinpreviousanalysesofthediphotonchan- The prior ranges for the parameters used in this work nel [10] and its inclusion will allow us to better match are listed in Table I, though we emphasise that emcee the results reported by ATLAS. In light of this, we will wasonlyusedtofindthemaximumlikelihoodpointsand assumethatATLASallowforafreenormalisationofthe therefore that the precise form of the priors should have little impact on the analysis. We quantify the significance of a possible excess by 1 Notethatweusethenotationlogforthenaturallogarithmloge. constructing the test statistic q0 [13]: 3 103 k=0  −2log L(NS =0,ˆˆθb) Nˆ ≥0 k=1 q0 = L(NˆS,mB,α,θˆb) S (3) 102 FFrixeeed n noormrm.. 0 NˆS <0, eV G 0 whereˆˆθb isthesetofbackgroundparameterswhichmax- s / 4 101 imise the likelihood under the background-only hypoth- nt esis and θˆ is the set of parameters which maximise the ve E full likelihood under the signal + background hypothe- 100 sis for a given value of m and (if we are considering B the free-width analysis) α. Here, we restrict to positive Background-only fit parameter values for the number of signal events, so the nbins=40 best fit value Nˆ will always be positive. 10-1 S 200 400 600 800 1000 1200 1400 1600 The local p-value for the background-only hypothesis m [GeV] γγ is obtained from the observed value of the test statistic q using, 0,obs FIG. 1. Background-only fits to the ATLAS diphoton invariant mass spectrum. The observed numbers of events in (cid:90) ∞ eachbinareshownasblackcircles,whilethecurvesshowthe p0 = f(q0)dq0 (4) background distribution for the k=0 (red) and k=1 (blue) q0,obs empiricalfunctionsdefinedinEq.2. Weshowbackgroundfits with both free normalisation (solid) and with fixed normali- where f(q0) is the probability density function of q0 un- sation (dashed). The background parameters used are those derthebackground-onlyhypothesis. AccordingtoWilks’ which maximise the likelihood for the background-only hy- Theorem [14], the log-likelihood ratio (appearing in the pothesis. Note that the k=2 background fits are not shown top line of Eq. 3) is asymptotically χ2 distributed if the as they lie close to the k =1, free normalisation curve (solid m background-only hypothesis is correct. The number of blue). degrees of freedom m is given by the difference between the number of parameters in the background-only and thesignal+backgroundhypotheses. Forafixedvalueof free parameter. When adding an additional parameter m (and α), there is only one free parameter (the num- to the background function, either using k =0 with free B ber of signal events N ) so we have m = 1. As detailed normalisation (solid red) or k = 1 with fixed normalisa- S in Ref. [13], in this case q follows a ‘half chi-square’ dis- tion (dashed blue), the fits tend to prefer a higher back- 0 tributionandthelocalsignalsignificanceissimplygiven ground at high mass in order to alleviate this possible √ by Z = q . The maximum value of the local signif- tension. The background contribution is also increased 0,obs icance can then be obtained by maximising over m (or in the region of the 750 GeV excess, suggesting that the B m and α in the free-width analysis). significance of the peak will be reduced when including B these additional parameters. We note that both the k = 0, free-N background and III. BACKGROUND-ONLY FITS the k = 1, fixed-N background appear to match closely the background-only fit reported in Fig. 1 of Ref. [1]. WebeginbyexamininginFig.1thefitstotheATLAS The ATLAS collaboration state that they use the k =0 data (black points) which are obtained when no signal background function in Ref. [1] and we therefore assume contribution is included. We show the best fit (maxi- thattheyalsofitthenormalisationN ofthebackground, mum likelihood) background curves for k = 0 (red) and in order to match the results shown here in Fig. 1. It k = 1 (blue), both using a free normalisation N (solid therefore appears then that the background curve used lines) and when setting N = 1 (dashed lines). We note by ATLAS fits the data well, apart from in the region of that the error bars on the data points are for illustrative the 750 GeV excess and above mγγ ∼ 1600 GeV (where purposes and denote the 1σ confidence intervals on the ATLAS report an excess with 2.8σ local significance). mean number of expected events in each bin given the FollowingRef.[6], wehavecalculatedtheBayesianIn- number of observed events. formation Crierion (BIC) [15], a model selection crite- As pointed out by Davis et al. [6], the k = 0 fixed- rion which be used to compare the fit to data obtained normalisation background curve (dashed red) appears to usingdifferentmodels, penalisingmodelswhichhavead- underestimate the background above m > 1000 GeV. ditional free parameters. The Akaike Information Cri- γγ It is perhaps not surprising that this parametrisation is terion (AIC) [16] is a related model selection criterion, notabletogiveagoodfittothebackgroundovertheen- though in general the BIC penalises the addition of ex- tire invariant mass range, given that it includes only two tra parameters more strongly. The BIC is defined as: 4 Background function Np logLˆ BIC 103 Fixed normalisation k=0 k=0 2 -87.9 183.2 k=1 k=1 3 -82.4 175.9 Free norm. k=2 4 -80.4 175.6 102 Fixed norm. Free normalisation V e k=0† 3 -81.9 174.9 G 0 kk==12 45 --8800..90 117768..64 s / 4 101 nt e v TABLE II. Number of parameters Np, maximum log- E likelihoodlogLândBayesianInformationCriterion(BIC)ob- 100 tainedinbackground-onlyfitstotheATLASdiphotoninvari- ant mass spectrum using the background functions in Eq. 2. Signal+Background fit TheBICisdefinedinEq.5. Thebackgroundfunctionusedby nbins=40 theATLAScollaborationinRef.[1]ismarkedwithadagger. 10-1 200 400 600 800 1000 1200 1400 1600 m [GeV] γγ FIG. 2. Signal + background fits to the ATLAS diphoton BIC=−2logLˆ+Nplognbins, (5) invariantmassspectrum,allowingthewidthoftheresonance near750GeVtovaryfreely. Thecolourschemematchesthat whereLîsthemaximumlikelihoodandNpisthenumber of Fig. 1. The parameters used are those which maximise of free parameters in the model. The lower the BIC, the likelihood for the signal + background hypothesis. As in the stronger the evidence for the model in question. A Fig. 1, we do not show the fits for k=2, as these lie close to difference of around 2 in the BIC between two different the k=1, free normalisation curve (solid blue). models is considered positive evidence that the model with the higher BIC should be rejected in favour of the one with the lower. A difference in BIC greater than 6 is contribution to the rate at either mγγ ∼ 750 GeV or considered strong evidence [17]. mγγ ∼ 1600 GeV by going to more complex smooth We show in Table II the maximum log-likelihood and background parametrisations. the value of the BIC obtained in background-only fits usingthethreeparametrisationsk =0,1,2withbothfree andfixednormalisation. Wefindthatthek =0,fixed-N IV. SIGNAL + BACKGROUND FITS backgroundgivesthelargestBIC,indicatingthatthereis strongevidencetorejectthisbackgroundinfavourofthe We now examine the signal + background fits to the alternative parametrisations. Note that the remaining ATLAS diphoton spectrum and determine the signifi- BIC values are all rather close in value, indicating that canceofthediphotonexcessunderdifferentassumptions thereshouldbenostrongpreferenceamongstthem. This for the background. In Fig. 2, we show the maximum matches the conclusion of the ATLAS collaboration [1], likelihood fits to the data for the signal + background which used both a Fisher test and a ‘spurious signal’ hypothesis using the free-width analysis for the possible analysis2 to determine that no background models more new resonance near 750 GeV. As before, we include fits complex than k = 0, free-N were necessary to fit the for k =0,1 (see Eq. 2) using both free and fixed normal- data. isation, with the colour scheme matching that of Fig. 1. We finally note that increasing the number of param- Adding a new resonance near 750 GeV clearly allows eters further, with the k = 2 functional form, does not the data to be better fit with all background parametri- substantiallyimprovethefits, asreflectedbytheslightly sations. As in the background-only fits, the k = 0, free- larger BIC value for k = 2, free-N. We do not show N and k = 1, fixed-N backgrounds produce similar re- these functional forms in Fig. 1, because they lie very sults. In all cases, the expected rate at high m is re- γγ close to the solid blue k = 1, free-N curve. The smooth ducedcomparedwiththebackground-onlyfits. Withthe backgroundathighm cannotbeincreasedfurtherdue 750GeVexcesssaturatedbythesignalcontribution, the γγ to the tension with a number of empty bins. It there- backgroundabove1000GeVcanbereducedtobetterfit fore seems difficult to further increase the background the large number of empty bins. InTableIII,wereportthemaximumlocalsignificance of the ATLAS 750 GeV excess obtained in this work, us- ingeachofthefunctionalformsintroducedinEq.2. For 2 The‘spurioussignal’analysisrequiresthat(foragivenfunctional reference,thesignificancesreportedbyATLASinRef.[1] form for the background) the bias on the fitted signal yield is significantlysmallerthanthestatisticaluncertaintyonthesignal are3.6σ (NWA)and3.9σ (free-width). Thesignificances yield,asdeterminedfromMonteCarlosamples. SeeRefs.[1,10] obtained in this work, using the same background func- forfurtherdetails. tion used by ATLAS, are 3.4σ (NWA) and 3.6σ (free- 5 Background function NWA Free-width see no events. Any smooth background is constrained Fixed normalisation not to overshoot these bins. k=0 4.2σ 4.9σ Davis et al. introduce a different possible parametri- k=1 3.4σ 3.7σ sation for the background (which was also validated by k=2 3.4σ 3.7σ a Monte Carlo study) and find that the significance of Free normalisation the excess is further reduced with respect to the k = 1, k=0† 3.4σ 3.6σ fixed-N case. However,theemptybinsathighm were γγ k=1 3.5σ 3.8σ not included in that analysis, leading to a background k=2 3.4σ 3.6σ fit which overestimates the high m event rate. In- ATLAS reported 3.6σ 3.9σ γγ deed, using the Davis et al. background parametrisation (with free normalisation) in this analysis gives a local TABLE III. Estimated local significance of the ATLAS 750 significanceof3.8σ forafree-widthresonance. Thisdoes GeV diphoton excess obtained in this work using each of the not discount the possibility that exploring a wider range background functions described in Eq. 2, assuming a freely varying resonance width (free-width) and under the narrow of possible background functions may impact the signifi- width approximation (NWA). The background function used cance of the 750 GeV excess, but the correct constraints bytheATLAScollaborationinRef.[1]ismarkedwithadag- from the entire range of m should be taken into ac- γγ ger. For comparison, we also give the local significance re- count. ported by the ATLAS collaboration. It is of course necessary to point out that the significances we report are only estimates and care must be taken when comparing with the official ATLAS analy- width). We discuss in Sec. V the possible sources of this sis. In particular, the results reported by ATLAS use small discrepancy. the full unbinned data set, while we consider here only Forthesimplestbackgroundfunction(k =0,fixed-N) binned data. This has the largest impact on our NWA thesignificanceoftheexcessissignificantlylarger.3 How- results. The diphoton invariant mass resolution (∼5.4 ever, in Sec. III, we found (in agreement with ATLAS) GeV) is substantially smaller than the bin width (40 thatthissimplefunctionfitsthebackgroundsignificantly GeV), meaning that a narrow resonance cannot be re- worsethanthemorecomplexones. Theremainingback- solved in the binned data. The fitted signal resonance is groundparametrisationsleadtoverysimilarsignificances effectively widened by integration over a single bin, im- comparedwiththek =0,free-N function. Thus,thesig- provingthefittotherelativelywideobservedexcess(∼3 nificance of the excess appears to be robust against the bins). Thismeansthatthesignificanceweobtainforthe choice of background. NWAislargerthanitwouldbeusingtheunbinneddata. However,theATLASanalysisincludesanadditionalnui- sance parameter for the invariant mass resolution, which V. DISCUSSION we have omitted. This mass resolution parameter can be increased to improve the fit to the data, leading to a As initially pointed out by Davis et al. [6], the signif- significance similar to that obtained here. icance of the 750 GeV diphoton excess reported by AT- We also emphasise that the binned data was obtained LASishigherwhenthebackgroundisfitwiththek =0, by digitising the results released in Ref. [1]. However, fixed-normalisationparametrisationandisreducedwhen we have investigated the possible impact of digitisation anextraparameterisaddedtothebackgroundfit. How- error on our analysis. In order to do this, we added ever, this appears to be consistent with the results re- random noise to the first 10 bins in m (distributed γγ ported by ATLAS. Assuming that the overall normalisa- uniformlybetween−3%and+3%ofthenumberofevents tionofthek =0backgroundisalsoincludedinthefit,we ineachbin)inordertosimulatedigitisationerrors.4 Ten recover the background-only fit presented in Ref. [1] and such ‘randomised’ datasets were generated and the peak obtain significances close to (but slightly smaller than) significance for each was calculated assuming the k = 0, those reported by the ATLAS collaboration. free-N background. Theresultinglocalsignificanceswere We also note that the data show no preference for in the range: an increase in complexity of the background function (as demonstrated by the Bayesian Information Criterion study in Table II). Furthermore, we find that adding ad- NWA: 3.3σ–3.5σ ditional parameters to the background fit does not have (6) Free-width: 3.5σ–3.8σ. any impact on the significance of the excess. This is because of a number of bins above m ∼1100 GeV which γγ 4 Atlargervaluesofmγγ thesmallnumberofeventsmeansthat 3 Wenotethatwefindanevenlargersignificanceusingthisback- integer numbers of events can accurately be read off the figure. ground than that reported by Davis et al. [6], though given the However at small values of mγγ digitisation error could induce differentstatisticalapproaches,thisisperhapsnotsurprising. variationsoforder10-100events. 6 These tests indicate that digitisation could have induced to ensure the robustness of the significance reported by an error of order 0.2σ in the analysis, and may also ex- ATLAS, these are likely to be highly constrained by the plain some of the discrepancy between our results and lack of observed events at high diphoton invariant mass. those reported by ATLAS. We further note that in this work we have used only approximatefunctionalformsforthesignaldistributions and while we approximately recover the significances re- ACKNOWLEDGMENTS ported by the ATLAS collaboration, our analysis does notcapturemanyoftheimportantdetailsinvolvedinfit- ting the signal and background (for example, uncertain- The author would like to thank the authors of Ref. [6] ties in the diphoton mass reconstruction). Furthermore, for stimulating and helpful discussions. The help of we have taken an empirical approach and allowed a wide AdamFalkowskiwasinvaluableinthecompletionofthis range of values for the background parameters N, b, a , work and the preparation of this manuscript. The or- 0 a , a . This may not accurately reflect the background ganisers and participants of the Magic Journal Club are 1 2 distributions seen in MCMC and data samples. How- also thanked for interesting (if long) discussions on the ever, restricting the possible ranges of the background 750 GeV excess. The author is supported by the John parameters is likely only to increase the significance of a TempletonFoundationGrant48222andbytheEuropean possible excess. Research Council (Erc) under the EU Seventh Frame- In spite of these simplifications, the broad message of work Programme (FP7/2007-2013)/Erc Starting Grant this note still holds. The significance of the 750 GeV (agreement n. 278234 — ‘NewDark’ project). The au- excess does not appear to be strongly affected by differ- thor also acknowledges the hospitality of the Institut ent choices of smooth background model. While more d’Astrophysique de Paris, where part of this work was complicated background distributions could be explored completed. 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