Measurement of the Higgs boson mass in the $H\rightarrow ZZ^* \rightarrow 4\ell$ and $H \rightarrow \gamma\gamma$ channels with $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector

The mass of the Higgs boson is measured in the $H\rightarrow ZZ^* \rightarrow 4\ell$ and in the $H\rightarrow \gamma\gamma$ decay channels with 36.1 fb$^{-1}$ of proton-proton collision data from the Large Hadron Collider at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector in 2015 and 2016. The measured value in the $H\rightarrow ZZ^* \rightarrow 4\ell$ channel is $m_{H}^{ZZ^{*}} = 124.79 \pm 0.37$ GeV, while the measured value in the $H\rightarrow \gamma\gamma$ channel is $m_{H}^{\gamma \gamma} = 124.93 \pm 0.40$ GeV. Combining these results with the ATLAS measurement based on 7 TeV and 8 TeV proton-proton collision data yields a Higgs boson mass of $m_H=124.97 \pm 0.24$ GeV.


Introduction
The observation of a Higgs boson, H, by the ATLAS and CMS experiments [1,2] with the Large Hadron Collider (LHC) Run 1 proton-proton (pp) collision data at centre-of-mass energies of √ s = 7 TeV and 8 TeV was a major step towards understanding the mechanism of electroweak (EW) symmetry breaking [3][4][5].The mass of the Higgs boson was measured to be 125.09± 0.24 GeV [6] through a combination of the individual ATLAS [7] and CMS [8] mass measurements with Run 1 data.Recently, the CMS Collaboration measured the Higgs boson mass in the H → Z Z * → 4 channel using 35.9 fb −1 of 13 TeV pp collision data [9].The measured value of the mass is 125.26 GeV, with an observed (expected) uncertainty of 0.21 (0.26) GeV.This Letter presents a measurement of the Higgs boson mass, m H , with 36.1 fb −1 of √ s = 13 TeV pp collision data recorded with the ATLAS detector.The measurement is derived from a combined fit to the four-lepton and diphoton invariant mass spectra in the decay channels H → Z Z * → 4 ( = e, µ) and H → γγ, based on the current understanding of the reconstruction, identification, and calibration of muons, electrons, and photons in the ATLAS detector.A combination with the ATLAS Run 1 data is also presented.

ATLAS detector
The ATLAS experiment [10] at the LHC is a multi-purpose particle detector with nearly 4π coverage in solid angle.1It consists of an inner tracking detector (ID) surrounded by a 2 T superconducting solenoid, electromagnetic (EM) and hadronic calorimeters, and a muon spectrometer (MS) incorporating three large superconducting toroidal magnets.The ID provides tracking for charged particles for |η| < 2.5.The calorimeter system covers the pseudorapidity range |η| < 4.9.Its electromagnetic part is segmented into three shower-depth layers for |η| < 2.5 and includes a presampler for |η| < 1.8.The MS includes high-precision tracking chambers (|η| < 2.7) and fast trigger chambers (|η| < 2.4).Online event selection is performed by a first-level trigger with a maximum rate of 100 kHz, implemented in custom electronics, followed by a software-based high-level trigger with a maximum rate of 1 kHz.

Data and simulated samples
This measurement uses data from pp collisions with a centre-of-mass energy of 13 TeV collected during 2015 and 2016 using single-lepton, dilepton, trilepton and diphoton triggers, with looser identification, isolation and transverse momentum (p T ) requirements than those applied offline.The combined efficiency of the lepton triggers is about 98% for the H → Z Z * → 4 events (assuming m H = 125 GeV) passing the offline selection.The diphoton trigger efficiency is higher than 99% for selected H → γγ events (assuming m H = 125 GeV).After trigger and data-quality requirements, the integrated luminosity of the data sample is 36.1 fb −1 .The mean number of proton-proton interactions per bunch crossing is 14 in the 2015 data set and 25 in the 2016 data set.
Monte Carlo (MC) simulation is used in the analysis to model the detector response for signal and background processes.For the H → Z Z * → 4 measurement, a detailed list and description of the MC-simulated samples used can be found in Ref. [11] and only a few differences specific to the mass analysis are mentioned here.For the gluon-gluon fusion (ggF) signal, the NNLOPS sample generated at next-to-next-to-leading order (NNLO) in QCD [12] with m H = 125 GeV and the PDF4LHC NLO parton distribution function (PDF) set [13] was used.Additional samples generated at different m H values (120,122,124,125,126,128,130 GeV) at next-to-leading order (NLO) were also used.The NLO ggF simulation was performed with P -B v2 [14] interfaced to P 8 [15] for parton showering and hadronisation, and to E G [16] for the simulation of b-hadron decays.The CT10NLO [17] PDF set was used for the hard process and the CTEQ6L1 [18] set for the parton shower.The non-perturbative effects were modelled using the AZNLO set of tuned parameters [19].
The Z Z * continuum background from quark-antiquark annihilation was modelled at NLO in QCD using P -B v2 and interfaced to P 8 for parton showering and hadronisation, and to E G for b-hadron decays.The PDF set used is the same as for the NLO ggF signal.NNLO QCD [20,21] and NLO EW corrections [22,23] were applied as a function of the invariant mass of the Z Z * system (m Z Z * ). 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe.The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upwards.Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the z-axis.
The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2).Angular distance is measured in units of ∆R ≡ (∆η For the H → γγ measurement, the same H → γγ signal (generated for m H = 125 GeV) and background simulated events used for the measurements of the Higgs boson couplings and fiducial cross-sections in the diphoton final state [24] were used.In addition, signal samples with alternative m H values (110, 122, 123, 124, 126, 127, 130, 140 GeV) were produced, with the same generators and settings as the m H = 125 GeV samples, but only for the four Higgs boson production modes with largest cross-section: gluon-gluon fusion, vector-boson fusion (VBF), and associated production with a vector boson V = W, Z (V H), for q q → V H and gg → Z H.For rarer processes, such as associated production of the Higgs boson with a top-quark pair (t tH) or a single top-quark (tH), contributing to less than 2% of the total cross-section, only samples at m H = 125 GeV were used.
Except for the γγ background sample, whose modelling requires a large MC sample obtained through a fast parametric simulation of the calorimeter response [25], the generated events for all processes were passed through a G 4 [26] simulation of the response of the ATLAS detector [25].For both detector emulation methods, events were reconstructed with the same algorithms as the data.Additional protonproton interactions (pile-up) were included in both the parametric and the G 4 simulations, matching the average number of interactions per LHC bunch crossing to the spectrum observed in the data.
The Standard Model (SM) expectations for the Higgs boson production cross-section times branching ratio, in the various production modes and final states under study and at each value of m H , were taken from Refs.[27][28][29][30] and used to normalise the simulated samples, as described in Refs.[11,24].

Muon reconstruction, identification and calibration
Muon track reconstruction is first performed independently in the ID and the MS.Hit information from the individual subdetectors is then used in a combined muon reconstruction, which includes information from the calorimeters.
Corrections to the reconstructed momentum are applied in order to match the simulation to data precisely.These corrections to the simulated momentum resolution and momentum scale are parameterised as a power expansion in the muon p T , with each coefficient measured separately for the ID and MS, as a function of η and φ, from large data samples of J/ψ → µ + µ − and Z → µ + µ − decays.The scale corrections range from 0.1% to 0.5% for the p T of muons originating from J/ψ → µ + µ − and Z → µ + µ − decays and account for inaccurate measurement of the energy lost in the traversed material, local magnetic field inaccuracies and geometrical distortions.The corrections to the muon momentum resolution for muons from J/ψ → µ + µ − and Z → µ + µ − are at the percent level.After detector alignment, there are residual local misalignments that bias the muon track sagitta, leaving the track χ 2 invariant [31,32], and introduce a small charge-dependent resolution degradation.The bias in the measured momentum of each muon is corrected by an iterative procedure derived from Z → µ + µ − decays and checked against the E/p ratio measured in Z → e + e − decays.The residual effect after correction is reduced to the per mille level at the scale of the Z boson mass.This correction improves the resolution of the dimuon invariant mass in Z boson decays by 1% to 5%, depending on η and φ of the muon.The systematic uncertainty associated with this correction is estimated for each muon using simulation and is found to be about 0.4 × 10 −3 for the average momentum of muons from Z → µ + µ − decays.
The momentum scale is determined to a precision of 0.05% for muons with |η| < 2, and about 0.2% for muons with |η| ≥ 2. The resolution is known with a precision ranging from 1% to 2% for muons with |η| < 2 and around 10% for muons with |η| ≥ 2 [33].Both the momentum scale and momentum resolution uncertainties in the corrections to simulation are taken as fully correlated between the Run 1 and Run 2 measurements.
In total, five independent sources are used to describe the uncertainties affecting the muon momentum scale and resolution corrections in data and simulation.

Photon and electron reconstruction, identification and calibration
Photon and electron candidates are reconstructed from clusters of electromagnetic calorimeter cells [34].Clusters without a matching track or reconstructed conversion vertex in the inner detector are classified as unconverted photons.Those with a matching reconstructed conversion vertex or a matching track, consistent with originating from a photon conversion, are classified as converted photons [35].Clusters matched to a track consistent with originating from an electron produced in the beam interaction region are considered electron candidates.
The energy measurement for reconstructed electrons and photons is performed by summing the energies measured in the EM calorimeter cells belonging to the candidate cluster.The energy is measured from a cluster size of ∆η ×∆φ = 0.075×0.175 in the barrel region of the calorimeter and ∆η ×∆φ = 0.125×0.125 in the calorimeter endcaps.The procedure for the energy measurement of electrons and photons closely follows that used in Run 1 [36], with updates to reflect the 2015 and 2016 data-taking conditions: • The different layers of the electromagnetic calorimeter are intercalibrated by applying methods similar to those described in Ref. [36].The first and second calorimeter layers are intercalibrated using the energy deposited by muons from Z → µ + µ − decays, with a typical uncertainty of 0.7% to 1.5% (1.5% to 2.5%) as a function of η in the barrel (endcap) calorimeter, for |η| < 2.4.This uncertainty is added in quadrature to the uncertainty in the modelling of the muon ionisation in the simulation (1% to 1.5% depending on η).The energy scale of the presampler is estimated using electrons from Z boson decays, after correcting the simulation on the basis of the correlations between the amount of detector material and the ratio of the energies deposited in the first and second layers of the calorimeter.The uncertainty in the presampler energy scale varies between 1.5% and 3% depending on η.
• The cluster energy is corrected for energy loss in the inactive materials in front of the calorimeter, the fraction of energy deposited outside the area of the cluster in the η-φ plane, the amount of energy lost behind the electromagnetic calorimeter, and to account for the variation of the energy response as a function of the impact point in the calorimeter.The calibration coefficients used to apply these corrections are obtained from a detailed simulation of the detector response to electrons and photons, and are optimised with a boosted decision tree (BDT) [37].The response is calibrated separately for electron candidates, converted photon candidates and unconverted photon candidates.In data, small corrections are applied for the φ-dependent energy loss in the gaps between the barrel calorimeter modules (corrections up to 2%, in about 5% of the calorimeter acceptance) and for inhomogeneities due to sectors operated at non-nominal high voltage (corrections between 1% and 7%, in about 2% of the calorimeter acceptance).
• The global calorimeter energy scale is determined in situ with a large sample of Z → e + e − events.The energy response in data and simulation is equalised by applying η-dependent correction factors to match the invariant mass distributions of Z → e + e − events.The uncertainty in these energy scale correction factors ranges from 0.02% to 0.1% as a function of η, except for the barrel-endcap transition region (1.37 < |η| < 1.52), where it reaches a few per mille.In this procedure, the simulated width of the reconstructed Z boson mass distribution is matched to the width observed in data by adding in the simulation a contribution to the constant term c of the electron energy resolution, σ E E = a E ⊕ b E ⊕ c.This constant term varies between 0.7% and 2% for |η| < 2.4 with an uncertainty of 0.03%-0.3%,except for the barrel-endcap transition region, where the constant term is slightly higher (2.5%-2.9%)with an uncertainty reaching 0.6%.
The main sources of systematic uncertainties in the calibration procedure discussed in Ref. [36] have been revisited.These sources include uncertainties in the method used to extract the energy scale correction factors, as well as uncertainties due to the extrapolation of the energy scale from Z → e + e − events to photons, and also to electrons with energies different from those produced in Z → e + e − decays.The latter arise from the uncertainties in the linearity of the response due to the relative calibration of the different gains used in the calorimeter readout, in the knowledge of the material in front of the calorimeter (inside and outside of the ID, referred to as ID and non-ID material in the following), in the intercalibration of the different calorimeter layers, in the modelling of the lateral shower shapes and in the reconstruction of photon conversions.The total calibration uncertainty for photons with transverse energy (E T ) around 60 GeV is 0.2%-0.3% in the barrel and 0.45%-0.8% in the endcap.These uncertainties are close to those quoted in Ref. [36], but typically about 10% larger.The small increase in the uncertainty arises mostly from a larger uncertainty in the relative calibration of the first and second calorimeter layers with muons because of a worse ratio of signal to pile-up noise in Run 2 data.In the case of electrons with E T around 40 GeV, the total uncertainty ranges between 0.03% and 0.2% in most of the detector acceptance.For electrons with E T around 10 GeV the uncertainty ranges between 0.3% and 0.8%.A total of 64 independent uncertainty sources are used to describe the overall uncertainty affecting the energy calibration of electron and photons.
The accuracy of the energy calibration for low-energy electrons (5-20 GeV) is checked by computing residual energy calibration corrections (after applying the corrections extracted from the Z → e + e − sample) for an independent sample of J/ψ → e + e − events.These residual correction factors are found to be compatible with one within uncertainties.A similar check is performed by computing residual corrections for photons in a sample of radiative Z boson decays.They are found to be compatible with one within uncertainties which are given by the combination of the statistical uncertainty of the radiative Z boson decays sample and of the systematic uncertainty from the extrapolation of the energy scale from electrons to photons.Systematic uncertainties in the calorimeter energy resolution arise from uncertainties in the modelling of the sampling term a/ √ E and in the measurement of the constant term in Z boson decays, in the amount of material in front of the calorimeter, which affects electrons and photons differently, and in the modelling of the contribution to the resolution from fluctuations in the pile-up from additional protonproton interactions in the same or neighbouring bunch crossings.The uncertainty of the energy resolution for electrons and photons with transverse energy between 30 and 60 GeV varies between 5% and 10%.
The identification of photons and the rejection of background from hadrons is based primarily on shower shapes in the calorimeter.The two levels of selection, loose and tight, are described in Ref. [35].To further reduce the background from jets, two complementary isolation selection criteria are used, based on topological clusters of energy deposits in the calorimeter and on reconstructed tracks in a direction close to that of the photon candidate, as described in Ref. [24].
Electrons are identified using a likelihood-based method combining information from the electromagnetic calorimeter and the ID.As in the case of photons, electrons are required to be isolated using both the calorimeter-based and track-based isolation variables as described in Ref. [38].

Statistical methods
The mass measurement is based on the maximisation of the profile likelihood ratio [39,40] where the vectors θ and mH denote the unconditional-maximum likelihood estimates of the parameters of the likelihood function L, while θ is the conditional maximum-likelihood estimate of the parameters θ for a fixed value of the parameter m H . Systematic uncertainties and their correlations are modelled by introducing nuisance parameters θ described by likelihood functions associated with the estimate of the corresponding effect [6].
The statistical uncertainty of m H is estimated by fixing all nuisance parameters to their best-fit values, all remaining parameters are thus left unconstrained.This approach yields a lower bound on the statistical uncertainty, when the combination of the different event categories discussed in the next sections is performed neglecting the different impact of the systematic uncertainties in each category.The upper bound on the total systematic uncertainty is estimated by subtracting in quadrature the statistical uncertainty from the total uncertainty.
Alternatively, the decomposition of the uncertainty into statistical and systematic components is performed using the BLUE method [41][42][43].The two approaches may lead to different results from the decomposition of the uncertainty for a combination of measurements with significant and uncorrelated systematic uncertainties.
7 Mass measurement in the H → Z Z * → 4 channel

Event selection
Events are required to contain at least four isolated leptons ( = e, µ) that emerge from a common vertex, form two pairs of oppositely charged same-flavour leptons.Electrons are required to be within the full pseudorapidity range of the inner tracking detector (|η| < 2.47) and have transverse energy E T > 7 GeV, while muons are required to be within the pseudorapidity range of the muon spectrometer (|η| < 2.7) and have transverse momentum p T > 5 GeV.A detailed description of the event selection can be found in Ref. [11,44].
The lepton pair with an invariant mass closest to the Z boson mass in each quadruplet is referred to as the leading dilepton pair, while the remaining pair is referred to as the subleading dilepton pair.The selected events are split according to the flavour of the leading and subleading pairs; ordered according to the expected selection efficiency, they are 4µ, 2e2µ, 2µ2e, 4e.Reconstructed photon candidates passing final-state radiation selections are searched for in all the events [45].Such photons are found in 4% of the events and their energy is included in the mass computation.In addition, a kinematic fit is performed to constrain the invariant mass of the leading lepton pair to the Z boson mass, improving the m 4 resolution by about 15% [7].The improvement brought by the correction of the local tracker misalignments, as discussed in Section 4, is at the percent level for the m 4 resolution of signal events.After event selection, the m 4 resolution for the signal (at m H = 125 GeV), estimated with a Gaussian fit around the peak, is expected to be about 1.6 GeV, 2.4 GeV, 2.2 GeV and 1.8 GeV for the 4µ, 4e, 2µ2e and 2e2µ channels respectively.In the fit range of 110 < m 4 < 135 GeV, 123 candidate events are observed.The yield is in agreement with an expectation of 107 ± 6 events, 53% of which are expected to be from the signal, assuming m H = 125 GeV.
The dominant contribution to the background is non-resonant Z Z * production (about 84% of the total background yield).Events with hadrons, or hadron decay products, misidentified as prompt leptons also contribute (about 15%).Events originating from t t+Z, Z Z Z, W Z Z, and WW Z production are estimated to contribute less than 1% of the total background.The residual combinatorial background, originating from events with additional prompt leptons, was found to be negligibly small [44].
The precision of the mass measurement is further improved by categorising events with a multivariate discriminant which distinguishes the signal from the Z Z * background.The BDT described in Ref.
[7], based on the same input variables, is trained on simulated signal events with different mass values simultaneously (124, 125 and 126 GeV) and Z Z * background events that pass the event selection.For each final state, four equal-size exclusive bins in the BDT response are used.This improves the precision of the m H measurement in the 4 decay channel by about 6%.

Signal and background model
The invariant mass in each category is described by the sum of a signal and a background distribution.
Non-resonant Z Z * production is estimated using simulation normalised to the most accurate predictions and validated in the sidebands of the selected 4 mass range.Smaller contributions to the background from t t+Z, Z Z Z, W Z Z and WW Z production are also estimated using simulation while the contributions from Z+jets, W Z, and t t production where one or more hadrons, or hadron decay products, are misidentified as a prompt lepton are estimated from data using minimal input from simulation following the methodology described in Ref. [11].For each contribution to the background, the probability density function (pdf) is estimated with the kernel density estimation.
For the determination of the signal distribution, an approach based on the event-by-event response of the detector is employed.The measured m 4 signal distribution is modelled as the convolution of a relativistic Breit-Wigner distribution, of 4.1 MeV width [27][28][29][30] and a peak at m H , with a four-lepton invariant mass response distribution which is derived event-by-event from the expected response distributions of the individual leptons.The lepton energy response distributions are derived from simulation as a function of the lepton energy and detector region.The lepton energy response is modelled as a weighted sum of three Gaussian distributions.For an observed event, the m 4 pdf is derived from the convolution of the response distributions of the four measured leptons.The direct convolution of the four leptons distributions, leading to 3 4 = 81 Gaussian distributions, is simplified to a weighted sum of four Gaussian pdfs following an iterative merging procedure as performed with the Gaussian-sum filter procedure [46,47].An additional correction is applied to remove the residual differences which arise from the correlation between the lepton energy measurements introduced by the kinematic constrained fit on the leading dilepton pair and the BDT categorisation of events.
Finally, the mass of the Higgs boson m H is determined by a simultaneous unbinned fit of signal-plusbackground distributions to data over the sixteen categories. 2 The per-event component of the signal pdf is added to the background distribution which is integrated over all kinematic configurations of the four final state leptons.In each of the four BDT categories, the signal yield is factorised by a floating normalisation modifier independent for each BDT category.The measured Higgs boson mass depends on the lepton energy resolution and the lepton energy scale.Uncertainties in these quantities are accounted for in the fit by Gaussian-distributed penalty terms whose widths are obtained from auxiliary data or simulation control samples.The expected uncertainty, with m H = 125 GeV and production rates predicted by the SM, for a data sample of the size of the experimental set, evaluated using simulation-based pseudo-experiments, is ±0.35 GeV.
A validation with data is performed with Z → 4 events to test the performance of the method on a known resonance with similar topology.In this test, the peak and width of the relativistic Breit-Wigner function are set to those of the Z boson.The measured Z boson mass was found to be 91.62 ± 0.35 GeV including statistical and systematic uncertainty.The observed uncertainty is in agreement with the expectation of ±0.34 GeV, as evaluated from simulation.The measured value is in agreement with the world average of 91.1876 ± 0.0021 GeV [48].As an independent check, the template method [7] is also used to measure m H .The expected statistical uncertainty of m H obtained with the per-event method from a sample equal in size to the experimental data set is, on average, 3% smaller than the statistical uncertainty obtained with the template method.Both methods are found to be unbiased within the statistical uncertainty of the simulated samples used of about 8 MeV on m H .

Results
The estimate of m H for the per-event and template methods is extracted with a simultaneous profile likelihood fit to the sixteen categories.The free parameters of the fit are m H , the normalisation modifiers of each BDT category, and the 51 nuisance parameters associated with systematic uncertainties.
The measured value of m H from the per-event method is found to be m Z Z * H = 124.79± 0.36 (stat) ± 0.05 (syst) GeV = 124.79± 0.37 GeV.
The total uncertainty is in agreement with the expectation and is dominated by the statistical component.
The root-mean-square of the expected uncertainty due to statistical fluctuations in the event yields of each category was estimated to be 40 MeV.The p-value of the uncertainty being as high or higher than the observed value, estimated with pseudo-experiments, is found to be 0.47.The total systematic uncertainty is 50 MeV, the leading sources being the muon momentum scale (40 MeV) and the electron energy scale (26 MeV), with other sources (background modelling and simulation statistics) being smaller than 10 MeV.
For the template method, the total uncertainty is found to be +0.41 −0.39 GeV, larger by 35 MeV than for the perevent method.The observed difference for the m H estimates of the two methods is found to be 0.16 GeV, which is compatible with the expected variance estimated with pseudo-experiments and corresponds to a one sided p-value of 0.19. Figure 1(a) shows the m 4 distribution of the data together with the result of the fit to the H → Z Z * → 4 candidates when using the per-event method.The fit is also performed independently for each decay channel, fitting all BDT categories simultaneously; the resulting likelihood profile is compared with the combined fit in Figure 1(b).The combined measured value of m H is found to be compatible with the value measured independently for each channel, with the largest deviation being 1.4σ for the 2µ2e channel and the others being within 1σ.
The Higgs boson mass in the four-lepton channel is also measured by using a profile likelihood ratio to combine the information from the Run 1 analysis [6], where m H = 124.51± 0.52 GeV, and the Run 2 analysis, keeping each individual signal normalisation parameter independent.The systematic uncertainties taken to be correlated between the two runs are the muon momentum and electron energy scales, while all other systematic uncertainties are considered uncorrelated.The combined Run 1 and Run 2 result is m Z Z * H = 124.71± 0.30 (stat) ± 0.05 (syst) GeV = 124.71± 0.30 GeV.The difference between the measured values of m H in the four-lepton channel in the two runs is ∆m Z Z * H = 0.28 ± 0.63 GeV, with the two results being compatible, with a p-value of 0.84.

Mass measurement in the H → γγ channel
In the diphoton channel, the Higgs boson mass is measured from the position of the narrow resonant peak in the m γγ distribution due to the Higgs boson decay to two photons.Such a peak is observed over a large, monotonically decreasing, m γγ distribution from continuum background events.The diphoton invariant mass is computed from the measured photon energies and from their directions relative to the diphoton production vertex, chosen among all reconstructed primary vertex candidates using a neural-network algorithm based on track and primary vertex information, as well as the directions of the two photons measured in the calorimeter and inner detector [49].
Events are selected and divided into categories with different mass resolutions and signal-to-background ratios, optimised for the measurement of simplified template cross-sections [30,50] and of production mode signal strengths of the Higgs boson in the diphoton decay channel.The event selection and classification are described in Ref. [24].A potential reduction of the total expected uncertainty by 4% could have been obtained using the same event categories chosen for the mass measurement with the Run 1 data [7].Given the small expected improvement, a choice was made to use the same categorisation for the measurement of the mass and of the production mode signal strengths.

Event selection and categorisation
After an initial preselection, described in Ref. [24], requiring the presence of at least two loosely identified photon candidates with |η| < 1.37 or 1.52 < |η| < 2.37, events are selected if the leading and the subleading photon candidates have E T /m γγ > 0.35 and 0.25 respectively, and satisfy the tight identification criteria and isolation criteria based on calorimeter and tracking information.Only events with invariant mass of the leading and subleading photon in the range 105 GeV < m γγ < 160 GeV are kept.
The events passing the previous selection are then classified, according to the properties of the two selected photons and of jets, electrons, muons and missing transverse momentum, into 31 mutually exclusive categories [24].The most populated class, targeting gluon-gluon fusion production without reconstructed jets, is split into two categories of events with very different energy resolution: the first ("ggH 0J Cen") requires both photons to have |η| < 0.95, while the second ("ggH 0J Fwd") retains the remaining events.

Signal and background models
For each category, the shape of the diphoton invariant mass distribution of the signal is modelled with a double-sided Crystal Ball function [51], i.e. a Gaussian function in the peak region with power-law functions in both tails.The dependence of the parameters on the Higgs boson mass m H is described by first-order polynomials, whose parameters are fixed by fitting simultaneously all the simulated signal samples generated for different values of m H .The quantity σ 68 , defined as half of the smallest range containing 68% of the expected signal events, is an estimate of the signal m γγ resolution and for m H = 125 GeV it ranges between 1.41 GeV and 2.10 GeV depending on the category, while for the inclusive case its value is 1.84 GeV. Figure 2(a) shows an example of the signal model for a category with one of the best invariant mass resolutions and for a category with one of the worst resolutions.
The expected signal yield is expressed as the product of integrated luminosity, production cross-section, diphoton branching ratio, acceptance and efficiency.The cross-section is parameterised as a function of m H separately for each production mode.Similarly, the branching ratio is parameterised as a function of m H .The product of acceptance and efficiency is evaluated separately for each production mode using only the samples with m H = 125 GeV.Its dependence on the mass is weak (relative variation below 1% when varying the Higgs boson mass by ±1 GeV) and is thus neglected.The cross-sections are fixed to the SM values multiplied by a signal modifier for each production mode: µ ggF , µ VBF , µ V H and µ t t H .The expected yield for m H = 125 GeV varies between about one event in categories sensitive to rare production modes (t tH, tH) to almost 500 events in the most populated event category ("ggH 0J Fwd").The background invariant mass distribution of each category is parameterised with an empirical continuous function of the diphoton system invariant mass value.The parameters of these functions are fitted directly to data.The functional form used to describe the background in each category is chosen among several alternatives according to the three criteria described in Ref. [24]: (i) the fitted signal yield in a test sample representative of the data background, built by combining simulation and control regions in data, must be minimised; (ii) the χ 2 probability for the fit of this background control sample must be larger than a certain threshold; (iii) the quality of the fit to data sidebands must not improve significantly when adding an extra degree of freedom to the model.The models selected by this procedure are exponential or power-law functions with one degree of freedom for the categories with few events, while exponential functions of a second-order polynomial are used for the others.
From the extrapolation of a background-only fit to the sidebands of the m γγ distribution in data, excluding events with 121 GeV < m γγ < 129 GeV, the expected signal-to-background ratio in a m γγ window containing 90% of the signal distribution for m H = 125 GeV varies between 2% in the "ggH 0J Fwd" category and 100% in a high-purity, low-yield (about 12 events) category targeting H+2jet, VBF-like events with low transverse momentum of the H+2jet system.

Systematic uncertainties
The main sources of systematic uncertainty in the measured Higgs boson mass in the diphoton channel are the uncertainties in the photon energy scale (PES), the uncertainty arising from the background model, and the uncertainty in the selection of the diphoton production vertex.They are described in detail in Ref. [24].
For each source of uncertainty in the PES described in Section 5, the diphoton invariant mass distribution for each category is recomputed after varying the photon energy by its uncertainty and is then compared with the nominal distribution.The sum in quadrature of the positive or negative shifts of the m γγ peak position due to such variations ranges from ±260 MeV in the "ggH 0J Cen" category to ±470 MeV in the "jet BSM" category, which requires at least one jet with p T > 200 GeV.All the PES effects are considered as fully correlated across categories.
The uncertainty due to the background modelling is evaluated following the procedure described in Ref. [7].The expected signal contribution as predicted by the signal model is added to the background control sample.The bias in the estimated Higgs boson mass from a signal-plus-background fit to the test sample relative to the injected mass is considered as a systematic uncertainty due to the background modelling.Its value is around ±60 MeV for the most relevant categories for the mass measurement.In the other categories it can assume larger values, which are compatible with statistical fluctuations of the background control sample.For this reason this systematic uncertainty is ignored in the poorly populated t tH categories, which give a negligible contribution to the mass measurement.This systematic uncertainty is assumed to be uncorrelated between different categories.
The systematic uncertainty related to the selection of the diphoton production vertex is evaluated using Z → ee events, as described in Ref. [7].An expected uncertainty of ±40 MeV in m H is used for all the categories and assumed to be fully correlated across different categories.
Systematic uncertainties in the diphoton mass resolution due to uncertainties in the photon energy resolution vary between ±6% (for the "ggH 0J Cen" category) and 11% (for the "jet BSM" category), and are expected to have a negligible impact on the mass measurement.
Systematic uncertainties in the yield and in the migration of events between categories described in Ref. [24] have a negligible impact on the mass measurement.
The uncertainty due to the signal modelling is evaluated similarly to that due to the background modelling.
A sample is built using the expected background distribution and the simulated signal events at m H = 125 GeV.The bias in the fitted Higgs boson mass is considered as a systematic uncertainty and is assumed to be correlated between different categories.The relative bias is below 10 −4 in most of the categories, and at most a few times 10 −4 in the other categories.

Results
The Higgs boson mass in the diphoton channel is estimated with a simultaneous binned maximumlikelihood fit to the m γγ distributions of the selected event categories.In each category, the distribution is modelled with a sum of the background and signal models.The free parameters of the fit are m H , the four signal strengths, the number of background events (31 sources) and the parameters (37 sources) describing the shape of the background invariant mass distribution in each category, and all the nuisance parameters (206) associated with systematic uncertainties.Figure 2(b) shows the distribution of the data overlaid with the result of the simultaneous fit.All event categories are included.For illustration purposes, events in each category are weighted by a factor ln(1 + S/B), where S and B are the fitted signal and background yields in a m γγ interval containing 90% of the signal.
The measured mass of the Higgs boson in the diphoton channel is m γγ H = 124.93± 0.21 (stat) ± 0.34 (syst) GeV = 124.93± 0.40 GeV where the first error is the statistical uncertainty while the second is the total systematic uncertainty, dominated by the photon energy scale uncertainty.
Assuming signal strengths as in the SM and the signal model determined from the simulation, the expected statistical uncertainty is 0.25 GeV and the expected total uncertainty is 0.41 GeV, with a root-mean-square, estimated from pseudo-experiments, of about 40 MeV.Compared to the expectation, the slightly larger systematic uncertainty and smaller statistical uncertainty observed in data are due to a lower than expected signal yield in some categories with large expected yield and small photon energy scale uncertainty, and to the fitted resolution in data being a few percent better than in the simulation (but still agreeing with it within one standard deviation).
To check if the measurement is sensitive to the assumption about the splitting of the production modes, the measurement is repeated using one common signal strength for all the processes.A small shift of the measured m H by 20 MeV is observed.
Other checks targeting possible miscalibration due to detector effects for some specific category of photons are performed by partitioning the entire data sample into detector-oriented categories, different from those used for the nominal result, and determining the probability that m H measured in one of these categories is compatible with the average m H from the other categories.A first categorisation is based on whether the photons are reconstructed as converted or not, a second is based on the photons' impact points in the calorimeter (either in the barrel region, |η| < 1.37, or in the endcap region, |η| > 1.52), and a third is based on the number of interactions per bunch crossing.For each of these categories a new background model, a new signal model and new systematic uncertainty values are computed.For each category the compatibility of its m H value with the combined m H value is tested by considering as an additional likelihood parameter the quantity ∆ i equal to the difference between that category's m H value and the combined value.No value of ∆ i significantly different from zero is found.A similar test is performed to assess the global compatibility of all the different categories with a common value of m H .In the three categorisations considered the smallest global p-value is 12%.The same procedure is applied to the categories used in the analysis: the smallest p-value computed on single categories is 7% while the global p-value is 94%.
A combination of the Higgs boson mass measured in the diphoton channel by ATLAS in Run 1, 126.02 ± 0.51 GeV [6], and in Run 2 is performed using a profile likelihood ratio.The signal strengths are treated as independent parameters.The systematic uncertainties considered correlated between the two LHC run periods are most of the photon energy scale and resolution uncertainties and those in the pile-up modelling, while all the other systematic uncertainties are considered uncorrelated.The photon energy calibration uncertainties that are treated as uncorrelated between the two LHC data-taking periods are a few uncertainties included only in the Run 2 measurement, the uncertainty in the photon energy leakage outside the reconstructed cluster, whose measurement is limited by the statistical accuracy of Z → γ, and the uncertainty in the electromagnetic calorimeter response non-linearity, which is estimated with different procedures in the two LHC run periods.The result is m

Combined mass measurement
The Higgs boson mass is measured by combining information from both the H → Z Z * → 4 and H → γγ channels.The correlations between the systematic uncertainties in the two channels are accounted for in the profile likelihood function.The main sources of correlated systematic uncertainty include the calibrations of electrons and photons, the pile-up modelling, and the luminosity.Signal yield normalisations are treated as independent free parameters in the fit to minimise model-dependent assumptions in the measurement of the Higgs boson mass.
The combined value of the mass measured using Run 2 data is m H = 124.86± 0.27 GeV.Assuming statistical uncertainties only, the uncertainty in the combined value is ±0.18GeV.The corresponding profile likelihood, for the two channels and for their combination, is shown in Figure 3(a).This result is in good agreement with the ATLAS+CMS Run 1 measurement [6], m H = 125.09± 0.24 GeV.
The combined mass measurement from the ATLAS Run 1 (m H = 125.36± 0.41 GeV) and Run 2 results is m H = 124.97± 0.24 GeV.Assuming statistical uncertainties only, the measurement uncertainty amounts to 0.16 GeV. Figure 3(b) shows the value of −2 ln Λ as a function of m H for the two channels combined, separately for the ATLAS Run 1 and Run 2 data sets, as well as for their combination.The contributions of the main sources of systematic uncertainty to the combined mass measurement, using both ATLAS Run 1 and Run 2 data, are summarised in Table 1.The impact of each source of systematic uncertainty is evaluated starting from the contribution of each individual nuisance parameter to the total uncertainty.This contribution is defined as the mass shift δm H observed when re-evaluating the profile likelihood ratio after fixing the nuisance parameter in question to its best-fit value increased or decreased by one standard deviation, while all remainder nuisance parameters remain free to float.The sum in quadrature of groups of nuisance parameter variations gives the impact of each category of systematic uncertainties.The nuisance parameter values from the unconditional maximum-likelihood fit are consistent with the pre-fit values within one standard deviation.The probability that the m H results from the four measurements (in the 4 and γγ final states, using Run 1 or Run 2 ATLAS data) are compatible is 12.3%.The results from each of the four individual measurements, as well as various combinations, along with the LHC Run 1 result, are summarised in Figure 4.
The combination of the four ATLAS measurements using the BLUE approach as an alternative method, assuming two uncorrelated channels,3 is found to be m H = 124.97± 0.

Conclusion
The mass of the Higgs boson has been measured from a combined fit to the invariant mass spectra of the decay channels H → Z Z * → 4 and H → γγ.The results are obtained from a Run 2 pp collision data sample recorded by the ATLAS experiment at the CERN Large Hadron Collider at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 36.1 fb −1 .The measurements are based on the latest calibrations of muons, electrons, and photons, and on improvements to the analysis techniques used to obtain the previous results from ATLAS Run  From the combination of these two channels, the mass is measured to be m H = 124.86± 0.27 GeV.This result is in good agreement with the average of the ATLAS and CMS Run 1 measurements.The combination of the ATLAS Run 1 and Run 2 measurements yields m H = 124.97± 0.24 GeV.
[4] P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys.Rev. Lett.13 (1964)  u Also at Georgian Technical University (GTU),Tbilisi; Georgia.v Also at Giresun University, Faculty of Engineering; Turkey.w Also at Graduate School of Science, Osaka University, Osaka; Japan.
x Also at Hellenic Open University, Patras; Greece.y Also at Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest; Romania.z Also at II.Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen; Germany.aa Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona; Spain.ab Also at Institut de Física d'Altes Energies (IFAE), Barcelona Institute of Science and Technology, Barcelona; Spain.

Figure 1 :
Figure 1: (a) Invariant mass distribution for the data (points with error bars) shown together with the simultaneous fit result to H → Z Z * → 4 candidates (continuous line).The background component of the fit is also shown (filled area).The signal probability density function is evaluated per-event and averaged over the observed data.(b) Value of −2 ln Λ as a function of m H for the combined fit to all H → Z Z * → 4 categories.The intersection of the −2 ln Λ curve with the horizontal lines labelled 1σ and 2σ provide the 68.3% and 95.5% confidence intervals.

Figure 2 :
Figure2: (a) Invariant mass distributions (circles) of simulated H → γγ events reconstructed in two categories with one of the best ("ggH 0J Cen": open circles) and one of the worst ("ggH 0J Fwd": solid circles) experimental resolutions.The signal model derived from a fit of the simulated events is superimposed (solid lines).(b) Diphoton invariant mass distribution of all selected data events, overlaid with the result of the fit (solid red line).Both for data and for the fit, each category is weighted by a factor ln(1 + S/B), where S and B are the fitted signal and background yields in a m γγ interval containing 90% of the expected signal.The dotted line describes the background component of the model.The bottom inset shows the difference between the sum of weights and the background component of the fitted model (dots), compared with the signal model (black line).

Figure 3 :
Figure 3: The value of −2 ln Λ as a function of m H for (a) H → γγ, H → Z Z * → 4 channels and their combination (red, blue and black, respectively) using Run 2 data only and for (b) Run 1, Run 2 and their combination (red, blue and black, respectively).The dashed lines show the mass measurement uncertainties assuming statistical uncertainties only.

Figure 4 :
Figure 4: Summary of the Higgs boson mass measurements from the individual and combined analyses performed here, compared with the combined Run 1 measurement by ATLAS and CMS [6].The statistical-only (horizontal yellow-shaded bands) and total (black error bars) uncertainties are indicated.The (red) vertical line and corresponding (grey) shaded column indicate the central value and the total uncertainty of the combined ATLAS Run 1 + 2 measurement, respectively.
1 data.The measured values of the Higgs boson mass for the H → Z Z * → 4 and H → γγ channels are m H = 124.79± 0.37 GeV, m H = 124.93± 0.40 GeV.

Table 1 :
Main sources of systematic uncertainty in the Higgs boson mass m H measured with the 4 and γγ final states using Run 1 and Run 2 data.
23GeV = 124.97± 0.19 (stat) ± 0.13 (syst) GeV.The splitting of the errors takes into account the relative weight of the two channels in the combined measurement.
508.[5] G. Guralnik, C. Hagen and T. Kibble, Global conservation laws and massless particles, Phys.Rev.Lett.13(1964)585.[6] ATLAS and CMS Collaborations, Combined Measurement of the Higgs Boson Mass in pp Collisions at √ s = 7 and 8 TeV with the ATLAS and CMS Experiments, Phys.Rev. Lett.114 (2015) 191803, arXiv: 1503.07589[hep-ex].Measurement of the Higgs boson mass from the H → γγ and H → Z Z * → 4 channels in pp collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector, Phys.Rev. D 90 (2014) 052004, arXiv: 1406.3827[hep-ex].[8] CMS Collaboration, Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8 TeV, Eur.Phys.J. C 75 (2015) 212, arXiv: 1412.8662[hep-ex].[9] CMS Collaboration, Measurements of properties of the Higgs boson decaying into the four-lepton final state in pp collisions at √ s = 13 TeV, JHEP 11 (2017) 047, arXiv: 1706.09936[hep-ex].[10] ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider, JINST 3 (2008) S08003.[11] ATLAS Collaboration, Measurement of inclusive and differential cross sections in the H → Z Z * → 4 decay channel in pp collisions at √ s=13 TeV with the ATLAS detector, JHEP 10 (2017) 132, arXiv: 1708.02810[hep-ex].Also at Centre for High Performance Computing, CSIR Campus, Rosebank, Cape Town; South Africa.c Also at CERN, Geneva; Switzerland.d Also at CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille; France.e Also at Département de Physique Nucléaire et Corpusculaire, Université de Genève, Genève; Switzerland.f Also at Departament de Fisica de la Universitat Autonoma de Barcelona, Barcelona; Spain.g Also at Departamento de Física Teorica y del Cosmos, Universidad de Granada, Granada (Spain); Also at Department of Financial and Management Engineering, University of the Aegean, Chios; Also at Department of Physics and Astronomy, University of Louisville, Louisville, KY; United States of America.j Also at Department of Physics and Astronomy, University of Sheffield, Sheffield; United Kingdom.k Also at Department of Physics, California State University, Fresno CA; United States of America.l Also at Department of Physics, California State University, Sacramento CA; United States of America.m Also at Department of Physics, King's College London, London; United Kingdom.n Also at Department of Physics, Nanjing University, Nanjing; China.o Also at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg; Russia.p Also at Department of Physics, Stanford University, Stanford CA; United States of America.q Also at Department of Physics, University of Fribourg, Fribourg; Switzerland.r Also at Department of Physics, University of Michigan, Ann Arbor MI; United States of America.s Also at Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa; Italy.t Also at Faculty of Physics, M.V.Lomonosov Moscow State University, Moscow; Russia.
b h i