Measurement of the Higgs boson mass with $H \rightarrow \gamma\gamma$ decays in 140 fb$^{-1}$ of $\sqrt{s}=13$ TeV $pp$ collisions with the ATLAS detector

The mass of the Higgs boson is measured in the $H\to\gamma\gamma$ decay channel, exploiting the high resolution of the invariant mass of photon pairs reconstructed from the decays of Higgs bosons produced in proton-proton collisions at a centre-of-mass energy $\sqrt{s}=13$ TeV. The dataset was collected between 2015 and 2018 by the ATLAS detector at the Large Hadron Collider, and corresponds to an integrated luminosity of 140 fb$^{-1}$. The measured value of the Higgs boson mass is $125.17 \pm 0.11 \mathrm{(stat.)} \pm 0.09 \mathrm{(syst.)}$ GeV and is based on an improved energy scale calibration for photons, whose impact on the measurement is about four times smaller than in the previous publication. A combination with the corresponding measurement using 7 and 8 TeV $pp$ collision ATLAS data results in a Higgs boson mass measurement of $125.22 \pm 0.11 \mathrm{(stat.)} \pm 0.09 \mathrm{(syst.)}$ GeV. With an uncertainty of 1.1 per mille, this is currently the most precise measurement of the mass of the Higgs boson from a single decay channel.


Introduction
After the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) discovered in 2012 [1,2] a particle  with properties consistent with those of the Higgs boson in the Standard Model (SM) of particle physics, the precise determination of its mass became one of the primary goals of their physics programmes.The Higgs boson mass   is a fundamental parameter of the SM and the only unknown parameter of the scalar sector of the Standard Model prior to the Higgs boson discovery.Its measurement is of paramount importance for several reasons.Firstly, its value determines the Higgs boson production rates and decay branching ratios [3].It is also the value assumed by the experimental collaborations when estimating the acceptances, efficiencies and signal models used in their analyses and to report their measured rates.Knowledge of the Higgs boson mass is therefore mandatory for a coherent test of its coupling structure.Secondly, the Higgs boson mass is one of the inputs in global fits to the measurements of electroweak observables [4].Knowing its value therefore plays a key role in verifying the internal consistency of the SM, especially through the interplay between the masses of the top quark and the  and Higgs bosons.Finally, the stability of the electroweak vacuum, and thus the fate of our universe, depends on the value of the Higgs boson mass [5].By measuring   , one can infer whether the universe is in a global, and thus stable, minimum-energy state of the Higgs field potential, or in a local metastable one, from which it could decay to the ground state in the future [6].
Measurements of the Higgs boson mass were performed separately by the ATLAS [7] and CMS [8] collaborations using Higgs boson decays to the high-resolution four-lepton (4ℓ, ℓ = , ) and diphoton () final states reconstructed during the first data-taking phase of the LHC (Run 1).The data consisted of 25 fb −1 of proton-proton ( ) collisions recorded at centre-of-mass energies √  = 7 and 8 TeV in 2011 and 2012.The combination of the ATLAS and CMS results led to a measurement of the Higgs boson mass with an uncertainty of 0.19%,   = 125.09± 0.24 GeV [9].Updated measurements of the Higgs boson mass were performed by both experiments using   collisions collected at √  = 13 TeV between 2015 and 2018 during the second data-taking phase of the LHC (Run 2).Using both  →   * → 4ℓ and  →  decays selected in a partial Run 2 dataset (36 fb −1 of   collisions recorded before 2017), the ATLAS Collaboration measured   = 124.86± 0.27 GeV [10].Combined with the ATLAS Run 1 results from Ref. [7], this study led to a Higgs boson mass measurement of   = 124.97± 0.24 GeV.Using a dataset of equivalent size and both the four-lepton and diphoton final states, the CMS Collaboration found   = 125.46± 0.16 GeV [11], whose combination with the Run 1 results from Ref. [8] led to the most precise determination of   to date, with a 0.11% uncertainty:   = 125.38 ± 0.14 GeV.Recently, the ATLAS Collaboration released an updated measurement of   using  →   * → 4ℓ decays in the full Run 2 dataset [12], consisting of 140 fb −1 of   collisions.The result,   = 124.99±0.19 GeV, combined with that of the corresponding analysis using Run 1 data, yields a single-channel Higgs boson mass measurement with an uncertainty of 0.14%,   = 124.94± 0.18 GeV.This Letter reports a measurement of the Higgs boson mass in the diphoton channel using the full Run 2 dataset.Compared with that in Ref. [10], the analysis presented here profits from a larger data sample, a new photon reconstruction algorithm with better energy resolution [13], an improved estimation of the photon energy scale with reduced uncertainties [14], and an optimised event classification strategy.The selected events are required to contain two energetic photons fulfilling strict identification and isolation criteria.The invariant mass (  ) distribution of the selected photon pairs exhibits a peak near   , arising from resonant Higgs boson decays, over a smoothly falling distribution from background events mainly due to non-resonant diphoton production.The Higgs boson mass is determined from the position of the peak in data through a profile-likelihood fit to the   distribution.Simulated signal and background event samples are used to optimise the analysis criteria, to define the signal and background   models used in the fit, and to estimate the impact of the systematic uncertainties in   .To increase the sensitivity of the measurement, the selected events are classified into mutually exclusive categories with different diphoton invariant mass resolutions and signal-to-background ratios which are analysed simultaneously.Finally, a combination with the ATLAS Run 1 measurement [9] is performed.
The rest of the Letter is organised as follows.The ATLAS detector is described briefly in Section 2. The data and simulated event samples used in the analysis are summarised in Section 3. The photon reconstruction and the event selection and classification are discussed in Section 4. The statistical tools used in the measurement and the methods used to assess the systematic uncertainties are presented in Sections 5 and 6, leading to the results described in Section 7. The conclusions of this study are summarised in Section 8.

The ATLAS detector
The ATLAS experiment [15] at the LHC is a multipurpose particle detector with a forward-backward symmetric cylindrical geometry and a near 4 coverage in solid angle. 1 It consists of an inner tracking detector surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic and hadron calorimeters, and a muon spectrometer.The inner tracking detector covers the pseudorapidity 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 -axis along the beam pipe.The -axis points from the IP to the centre of the LHC ring, and the -axis points upwards.Cylindrical coordinates (, ) are used in the transverse plane,  being the azimuthal angle around the -axis.
range || < 2.5.It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors.Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements with high granularity.A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity range (|| < 1.7).The endcap and forward regions are instrumented with LAr calorimeters for both the EM and hadronic energy measurements up to || = 4.9.The muon spectrometer surrounds the calorimeters and is based on three large superconducting air-core toroidal magnets with eight coils each.The field integral of the toroids ranges between 2.0 and 6.0 T m across most of the detector.The muon spectrometer includes a system of precision chambers for tracking and fast detectors for triggering.
A two-level trigger system is used to select events.The first-level trigger is implemented in hardware and uses a subset of the detector information to accept events at a rate below 100 kHz.This is followed by a software-based trigger that reduces the accepted event rate to 1 kHz on average depending on the data-taking conditions.
An extensive software suite [16] is used in data simulation, in the reconstruction and analysis of real and simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.
3 Data and simulation samples

Data
The measurement uses the full   collision dataset collected at √  = 13 TeV by the ATLAS detector during Run 2 of the LHC.Events were recorded using unprescaled diphoton and single-photon triggers [17].The photon transverse momentum thresholds were 35 GeV and 25 GeV for the diphoton triggers throughout the whole of Run 2, and 120 (140) GeV for single-photon triggers in 2015-2016 (2017-2018).Shower-shape requirements looser than those used in the offline analysis were applied to the photon candidates at the trigger level.The integrated luminosity of the dataset after trigger and data-quality requirements [18] is 140.1 ± 1.2 fb −1 [19,20].The efficiency of the trigger system for signal events passing the full selection is close to 100% [17].

Simulation samples
Monte Carlo (MC) simulated event samples of Higgs bosons produced by   collisions and decaying into diphotons, as well as samples of the main background processes for the same final state, were produced with the set-up described in Ref. [21].Simulated hard-scattering events were overlaid with simulated inelastic proton-proton events generated with Pythia 8.1 [22], to model the effect of multiple 'pile-up' interactions in the same and neighbouring bunch crossings.
Signal samples were produced for the main Higgs boson production modes: gluon-gluon fusion (ggF), vector-boson fusion (VBF), and associated production with a vector boson ( ,  = , ), a top-quark pair ( t), a bottom-quark pair ( b) or a single top quark ().Signal event samples were produced with either the Powheg Box [23] or (for  only) MadGraph5_aMC@NLO [24] event generator, using matrix element calculations at the highest available order of accuracy in the strong coupling constant  s : next-to-next-to-leading order (NNLO) for ggF, next-to-leading order (NLO) for VBF,  ,  q →  ,  t,  b and , and leading order (LO) for  →  .The event generators were interfaced to Pythia 8.2 [25] for the modelling of the parton shower and the underlying event.In the analysis the samples are normalised to the integrated luminosity of the data, using state-of-the-art Standard Model calculations for the Higgs boson production cross-sections and branching ratios at the hypothesised Higgs boson mass [3].The generated signal samples were passed through a detailed simulation of the response of the ATLAS detector [26] based on Geant4 [27].
The nominal signal samples were generated assuming a Higgs boson mass of 125 GeV.The Higgs boson width Γ in all signal samples was set to the SM prediction for   = 125 GeV, Γ = 4.07 MeV, which is much narrower than the experimental resolution.Systematic uncertainties related to the modelling of the parton shower are studied with alternative samples produced with the same matrix-element generator as used for the nominal ones but with the Herwig 7.1.3parton shower algorithm [28].The parameterisations of the expected signal yields and diphoton invariant mass distributions as a function of the Higgs boson mass are obtained by interpolation between results from signal samples with   set to 110,122,123,124,125,126,127,130 or 140 GeV, as described in Section 5.The same set-up as that for the nominal samples was used.
Background events from non-resonant   →  +  parton ( ≥ 0) production were also simulated, using the Sherpa 2.2.4 event generator [29] with NLO-accurate matrix elements for up to one parton, and leading-order (LO) accurate matrix elements for up to three partons.Due to its large size, the   →  sample was processed by a fast simulation of the ATLAS detector [30], based on a parameterisation of the response of the calorimeter.Since the diphoton background is estimated from the sidebands in the diphoton invariant mass distribution in data, the background simulation is only used to select the background model and the systematic uncertainty associated with this choice, and for this the fast simulation was found to be sufficiently accurate.
The effect of interference between resonant signal production and non-resonant background diphoton production is studied using samples of simulated diphoton events including contributions from both processes (produced by either the  or  partonic channels) and their interference.The accuracy of the calculations is NLO for the -interference and LO for the -interference samples.The events were generated using Sherpa 2.2.11 and passed through the Geant4 detector simulation.

Event selection and classification
The event reconstruction and selection closely resemble those used in the latest ATLAS measurement of Higgs boson production cross-sections using the diphoton channel and the full Run 2 dataset [21].The main differences are the use of an updated photon energy calibration [14] with reduced systematic uncertainties, and the classification of events into categories that are optimised to minimise the uncertainty in the measured Higgs boson mass rather than in the Higgs boson production cross-sections.
Compared with the previous mass measurement [10], which used photon candidates reconstructed from a fixed-size cluster of energy deposits in the electromagnetic calorimeter identified by a sliding-window algorithm, this measurement relies on photon candidates that are reconstructed from dynamic, variable-size clusters, called superclusters.The main advantages of this algorithm are improvements in the reconstruction efficiency and energy measurement of converted photons ( →  +  − ), and a lower rate of misclassifying unconverted photons as converted photon candidates [13].
The photon energy is determined from the signals due to energy deposited in the electromagnetic calorimeter, after applying the calibration scheme detailed in Ref. [14].The photon direction is calculated from the positions of the supercluster and the   collision vertex that is chosen from among the reconstructed primary vertex candidates by a neural-network (NN) algorithm [31].The NN inputs are the directions of the two  T -leading photon candidates in the event, determined only from the conversion vertices and the longitudinal sampling of the calorimeter, and vertex candidate information such as the transverse momenta and directions of the associated tracks.
The selection retains events with at least two photon candidates with pseudorapidity in the range || < 1.37 or 1.52 < || < 2.37, meeting tight identification and loose isolation criteria [13], matched to the online photon candidates that passed the trigger selection.Events are kept if the  T -leading and  T -subleading photon candidates have invariant mass   in the range 105-160 GeV and transverse momenta that exceed 0.35 and 0.25 times   , respectively.When more than two photon candidates pass those requirements, only the  T -leading and  T -subleading candidates are considered for further analysis.About 1.2 million events in the data pass the selection.The expected efficiency for the signal for   = 125 GeV is close to 36%, leading to an expected signal yield of about 6200 events.
To increase the precision of the mass measurement, the selected events are classified into 14 categories with different signal-to-background ratios, diphoton invariant mass resolutions and photon energy scale uncertainties.The observables and the thresholds used to define the categories are optimised by minimising the expected total Higgs boson mass uncertainty for   = 125.09GeV using a simplified version of the maximum-likelihood fit described in the next section, including the statistical uncertainties and the dominant systematic uncertainties from the photon energy scale calibration.The final choice of observables and thresholds for the categories used in the measurement is the following: • The number of reconstructed converted photon candidates: events with no photon conversion candidates ('U'-type events) are considered separately from events with one or two  →  +  − candidates ('C'-type events).For each category, the narrowest diphoton invariant mass window, with half-width denoted by   90 , containing 90% of the signal events is listed in Table 1.The expected signal ( 90 ) and background ( 90 ) yields in each category in that interval are also reported, where  90 is determined from the integral of an exponentiated second-order polynomial function fitted to the data   distribution after excluding the range 120 <   < 130 GeV from the interval.The table also indicates the expected fraction of signal events  90 =  90 /( 90 +  90 ), and the signal significance The invariant mass resolution   90 for C-type events is 10%-20% worse than for U-type events, due to asymmetric  →  +  − conversions producing a low-energy electron or positron, and to bremsstrahlung photons emitted by the  +  − pair.In both cases, such soft electrons, positrons or bremsstrahlung photons can escape the supercluster and the calibration procedure may not be fully efficient in recovering the energy of the original photon.The resolution is 6%-20% better in the central-barrel categories than in the corresponding outer-barrel categories, due to the smaller amount of material upstream of the electromagnetic calorimeter in the central region of the detector.Events with high

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Tt have about 15%-20% (30%) better   resolution than events with medium (low)   Tt , due to poorer photon energy resolution at lower photon transverse momentum.The signal fraction  90 also depends on the same quantities: it is larger for U-type events and central-barrel events, and increases with   Tt since the main background process, continuum diphoton production arising mostly from -channel  q →  and  →  scattering, has a softer   Tt spectrum than the signal.The photon energy scale uncertainty is smaller for C-type events and central-barrel events than for U-type events and outer-barrel or endcap events; it increases with   Tt due to uncertainties in the linearity of the response and in the extrapolation to photons in the energy scale calibration, which is mainly determined using electron and positron candidates with relatively low transverse momentum from  →  +  − decays.
The chosen categorisation reduces the expected uncertainty in   from statistical and photon energy scale systematic uncertainties by about 17% compared with a measurement based on the inclusive sample passing the event selection, and by 6% compared with the use of the event classification strategy in Ref. [10].Compared with the 101 event categories developed in Ref. [21] for the measurement of the Higgs boson production-mode cross-sections times branching ratio to diphotons using the same dataset, the 14 categories used in this analysis lead to a small increase (+3%) in the expected statistical uncertainty and to a larger decrease (−14%) in the systematic uncertainty of   , yielding an overall expected decrease (−3%) in the total uncertainty.

Mass measurement procedure
The Higgs boson mass   is measured using the statistical methods described in Ref. [10].The diphoton invariant mass distribution of the data is used to define a likelihood function  depending on   and on additional parameters  describing the signal and background normalisations, their   models, and corresponding systematic uncertainties.The profile likelihood ratio [32,33] is then: where θ and m denote the values of the parameters that maximise the likelihood function , while θ (  ) represent the values of the parameters  that maximise  for a given value of the parameter   .
A numerical fit procedure determines the central value m of the measurement and its 68% confidence interval, defined by all values of   for which −2 ln Λ(  ) < 1.The fit uses the event counts in 25-MeV-wide bins of the   distribution in each category.
The likelihood function  is computed from the product of individual likelihood functions for the observed diphoton invariant mass spectra in each category and distributions representing auxiliary measurements that constrain the nuisance parameters associated with the systematic uncertainties.The model assumes that the observed distribution arises from the sum of a signal and a background component, whose shape and normalisation are inferred from the data themselves, with some input from the simulation.
The   distribution of the simulated signal in each category is found to be properly described, for any value of the Higgs boson mass in the range [110,140] GeV, by a double-sided Crystal Ball [34] probability density function, i.e. a function with a Gaussian core and power-law tails.The parameters ( ± ,  ± ) describing the tails of the model do not depend on   , while the parameters  CB and  CB describing the peak position and the resolution of the core Gaussian component scale linearly with   ,  CB =   +   +   (  − 125 GeV) and  CB =   +   (  − 125 GeV).The nominal values of the parameters   ,   ,   ,   ,  ± and  ± in each category are determined by a simultaneous fit to the simulated diphoton invariant mass spectra in that category for different   hypotheses.As a cross-check, the fit is repeated after removing the   = 125 GeV distribution from the input and new values of the   ,   ,   ,   ,  ± and  ± parameters are obtained.The signal model for   = 125 GeV predicted by this parameterisation is then compared with the model determined from a single fit, with a Crystal-Ball lineshape with floating  CB ,  CB ,  ± and  ± parameters, to the   = 125 GeV simulated signal events.Good agreement is observed.The nominal signal models for the categories with the best and worst expected resolutions for a Higgs boson mass   = 125 GeV are shown in Figure 1.
The normalisations of the signal, one for each category , are free parameters of the fit and are expressed as the product of a per-category signal-strength factor   and the expected number   of Higgs boson events in the same analysis region.The expected signal yield   is determined from the integrated luminosity, the SM values of the Higgs boson cross-section and  →  branching ratio, and the simulation-predicted event selection efficiency in that category.The dependence of   on   in each category  is modelled with a second-order polynomial whose parameters are determined by a fit to the expected yields in that category, calculated for nine discrete values of   between 110 and 140 GeV using the simulated signal samples described in Section 3.
The background   distribution in each category is represented by either an exponential function, a power-law function or an exponentiated second-order polynomial.The background model is chosen in an empirical way [21], using the results obtained by fitting the diphoton invariant mass distribution of a background template with models with free parameters for the signal and background yields.The background template is obtained by summing the   distributions of simulated non-resonant diphoton events and data samples enriched in photon+jet and dĳet events.The photon+jet and dĳet enriched samples are obtained using a selection similar to that in the previous section, the exception being that one or both photon candidates must fail the nominal identification and isolation requirements while passing looser ones.The three distributions are normalised to the yields of the respective contributions estimated in situ.Among all considered background models whose  2 probability is greater than 1% when fitted to the background template, and for which the fitted signal yield (the 'spurious signal') is less than 10% of the expected signal yield, the one with the fewest degrees of freedom is selected.A value of 10% is chosen for the threshold so that the spurious signal, considered as a systematic uncertainty in the signal yield, is small compared to the statistical uncertainty.The background yield in each category and the parameters describing the shape of the background model are free parameters of the likelihood function.
Systematic uncertainties and their correlations are modelled by including, among the parameters , nuisance parameters described by likelihood functions associated with estimates of the corresponding effects.The statistical uncertainty of   is estimated by determining the confidence interval when all nuisance parameters associated with systematic uncertainties are fixed to their best-fit values, and all other parameters are left unconstrained.The total systematic uncertainty is estimated as the square root of the difference of the squares of the total uncertainty and the statistical uncertainty.Similarly, the contribution from a group of uncertainties is determined from the difference of the squares of the total uncertainty and the uncertainty from a fit in which the associated nuisance parameters are fixed to their best-fit values.
The expected statistical uncertainty of 120 MeV in   is a factor of 2.1 less than in the previous measurement [10], due to the present dataset being almost four times larger and improvements in the photon reconstruction algorithm and event classification.

Systematic uncertainties
The main sources of systematic uncertainty in   are the uncertainties in the photon energy scale, the uncertainty from the background modelling and the effect of interference between the signal and the  continuum background.They are described below, together with their expected pre-fit impact on   .
The photon energy scale uncertainty is modelled by using 67 independent components.The effect of every component is evaluated for each analysis category by comparing the nominal signal MC diphoton invariant mass distribution with the one obtained by varying the energy of each photon by the uncertainty under study.The shifts induced in the peak parameters  CB of the Crystal Ball signal models are then included as nuisance parameters in the likelihood function, fully correlated among the categories.The 67 independent sources of uncertainty in the photon energy scale can be classified roughly into four main groups.The first group ( →  +  − calibration) is related to the determination of the -dependent energy scale factors for electrons and positrons from  boson decays, effectively constraining their energy scale for a transverse energy  T ≈ 45 GeV.The second group ( T -dependent electron energy scale) includes uncertainties in the  T -dependence of the energy scale from sources such as the calorimeter readout non-linearity, the calorimeter layer intercalibration, and the amount of material upstream of the calorimeter.The third group ( ± →  extrapolation) includes the uncertainties in the extrapolation of the energy scale from electrons to photons, arising, for instance, from potential mismodelling of differences in lateral shower development between electrons and photons in the calorimeter.Finally, the fourth group (conversion modelling) contains the uncertainties related to the accuracy of the photon conversion modelling in the simulation.Since the simulation-based photon energy calibration [14] is trained and applied separately for unconverted and converted photon candidates, any mismodelling of the conversion reconstruction performance in the simulation may affect the calibrated photon energy scale.
The results presented in this Letter profit from a new auxiliary measurement (linearity fit) of the data-to-MC electron energy scale corrections as a function of the electron  T .In the photon energy scale calibration used in the previous Higgs boson mass measurement [10], only -dependent energy scale factors were derived from 2015-2016 data.These were obtained by comparing the position of the peak of the invariant mass distribution of  +  − pairs from  boson decays with that predicted by the simulation.The possible  T -dependence of the data-to-MC energy scale correction was accounted for as a systematic uncertainty, arising from the various sources belonging to the  T -dependent electron energy scale group, calculated as described in Ref. [35].The new approach used in Ref. [14] exploits the larger sample of  →  +  − decays collected in 2015-2018 to derive residual data-to-MC energy scale factors in bins of electron transverse energy within broad  regions.The measurement of these additional scale factors is used to constrain the  T -dependent electron energy scale systematic uncertainties.The additional constraints and correlation of the systematic uncertainties from the linearity fit are propagated to the Higgs boson mass measurement by implementing in the likelihood function a multivariate Gaussian constraint term whose covariance is that returned by the linearity fit.The impact of the photon energy scale systematic uncertainties, including the effect of the linearity fit, was found to be independent of   for Higgs boson mass values in the range 124-126 GeV, and is expected to be approximately ±83 MeV (with contributions of 60, 43, 30 and 23 MeV, respectively, from the four individual groups of  →  +  − calibration,  T -dependent electron energy scale,  ± →  extrapolation and conversion modelling uncertainties).Compared to the results in Ref. [10], the simulation's more accurate description [13] of material upstream of the EM calorimeter and the lower sensitivity of the new clustering algorithm to the effects of interactions with detector material reduced the associated systematic uncertainty by a factor close to three.In addition, the larger dataset allowed a more precise study of the  ± →  extrapolation procedure [14], and its impact on the expected   uncertainty is reduced by a factor larger than three.Furthermore, the contribution of the  T -dependent electron energy scale uncertainties to the total expected   uncertainty has been reduced by a factor of two thanks to more precise dedicated measurements.Finally, the linearity fit constrains the expected uncertainty in the Higgs boson mass from the group of  T -dependent energy scale uncertainties by a further factor of four.
The effect of interference between the  →  →  signal and the  →  continuum background and between  →  →  and  →  is not included in the simulated event samples used to study the nominal signal model.In the Standard Model, this interference is expected to change the signal cross-section by 1%-2% [36] and to shift the position of the peak in the diphoton invariant mass distribution by a few tens of MeV [37].The size of this effect is treated as a systematic uncertainty, which is quantified by fitting the nominal signal-plus-background model to the sum of the   distributions predicted by the nominal signal and background models and the   distribution arising from such interference in simulated events.The relative differences between the nominal Higgs boson mass and the fitted mass in the various analysis categories are included in the likelihood function as a single nuisance parameter affecting the 14 categories coherently.The impact of the interference term on the determination of   is expected to be approximately ±24 MeV.
The effect of a possible bias in the measured   value, due to mismodelling of the continuum background   distribution, is evaluated by fitting the signal-plus-background model to the sum of the   distribution of the background template described in the previous section and the   distribution predicted by the signal model for   = 125 GeV.The relative differences between the nominal Higgs boson mass and the fitted mass in the various analysis categories are then included in the likelihood function as 14 additional uncorrelated nuisance parameters, one per category.The impact of the background modelling uncertainty on the measurement of   is expected to be approximately ±18 MeV.
The systematic uncertainty related to the selection of the diphoton production vertex is evaluated in a  →  +  − control sample.The directions of the selected electrons and positrons and thus their invariant mass are calculated using either the primary vertex candidate with the largest sum of the squared transverse momenta of the tracks associated to the vertex (which includes also the  ± tracks), or the primary vertex selected by the NN algorithm described in Section 4, where the electron and positron tracks are ignored.The separation between the peak positions of the two  +  − invariant mass distributions is evaluated separately in data and simulation.The maximum difference (5 MeV) between the separation observed in data and that observed in the simulation is taken as an additional systematic uncertainty in   .
Uncertainties from the chosen parameterisation of the nominal signal model are propagated to the final result by including 14 additional uncorrelated nuisance parameters in the likelihood function.They are determined by fitting the nominal signal-plus-background model to the sum of the   distribution of simulated   = 125 GeV signal events and that predicted by the nominal background model.The impact of the signal modelling uncertainty on the measurement of   is expected to be approximately ±5 MeV.
The effect of the photon energy resolution uncertainty is included in the signal model as five nuisance parameters that affect the resolution parameter  CB of the Crystal Ball function and are treated as being correlated among categories.The impact of the five independent sources of photon energy resolution uncertainty is evaluated by comparing the nominal signal invariant mass distribution in each category with the ones obtained by varying the energy resolution of each photon according to its uncertainties.The sum in quadrature of the different components of the photon energy resolution uncertainty ranges from 4.5% for C-type, outer-barrel, low-  Tt events to 17% for U-type, central-barrel, high-  Tt events.The impact of the photon energy resolution uncertainty on   is expected to be approximately ±3 MeV.
Yield uncertainties from the Higgs boson branching ratio to diphotons and the integrated luminosity of the data, and uncertainties in the migrations of events between categories, from various experimental and theoretical sources, are included in the model.Experimental uncertainties in the efficiency of photon conversion reconstruction, photon identification, photon isolation and the photon triggers, as well as in the impact of the simulation's modelling of pile-up are considered.Theoretical uncertainties that are taken into account are those in the signal production cross-sections, in the modelling of the underlying event and parton shower, in the value of the strong coupling constant and in the parton distribution functions of the proton.All the uncertainties described in this paragraph are included in the fit, although their expected impact on the measurement of   was found to be below 1 MeV.
In summary, compared to the previous publication [10] based on 36 fb −1 of ATLAS Run 2 data, the improvements in the photon energy calibration [14] and the optimised event classification lead to an expected fourfold reduction of the dominant systematic uncertainty in   (from the photon energy scale and resolution), from 320 MeV to 80 MeV, and a similar reduction of the total expected systematic uncertainty, from 330 MeV to 90 MeV.

Results
The   distribution of the data, overlaid with the sum of the signal and background models corresponding to the maximum-likelihood estimates of the parameters of the likelihood function, is shown in Figure 2. All event categories are included.For illustration purposes, events in each category are weighted by a factor ln(1 +  obs 90 / obs 90 ), where  obs 90 and  obs 90 are the fitted signal and background yields in the smallest   interval containing 90% of the signal.
The profile likelihood ratio as a function of   is shown in Figure 3(a).The value of the Higgs boson mass determined by fitting the profile likelihood ratio in Eq. ( 1) to the diphoton invariant mass distribution in data is: = 125.17± 0.11(stat.)± 0.09(syst.)GeV = 125.17± 0.14 GeV.
The statistical and systematic uncertainties are in good agreement with the values of 120 MeV and 90 MeV expected for a SM Higgs boson with the observed mass.The main sources of systematic uncertainty and their impact on the measurement are summarised in Table 2.
The signal strength   in each category  is compatible with the SM prediction   = 1, with the largest difference being 2.   results is performed by simultaneously fitting a single   parameter to the two datasets.The nominal model including the 14 signal strengths   of the reconstructed categories is used for the Run 2 dataset, while two separate signal strengths, one each for production processes involving Higgs boson couplings to either fermions or vector bosons, are used for the Run 1 dataset.All 16 signal-strength parameters are profiled in the combined fit for   .Almost all the nuisance parameters, especially the  T -dependent photon energy scale parameters affected by the linearity measurement and the parameters describing the extrapolation of the energy scale from electrons to photons, are assumed to be uncorrelated between the two measurements because of differences in the reconstruction algorithms and in the calibration procedures and control samples.The  →  +  − scale uncertainties and some of the resolution uncertainties, estimated in the same way for the two measurements, are treated as being fully correlated between the two data-taking periods.The combination with the Run 1 ATLAS measurement in the diphoton channel produces a small shift (+50 MeV) of the central value and a slight reduction (<10 MeV) of the statistical uncertainty, leading to: = 125.22± 0.11(stat.)± 0.09(syst.)GeV = 125.22± 0.14 GeV.
The individual ATLAS Run 1 and Run 2 measurements in the diphoton channel and their combination are shown in Figure 3(b), together with the ATLAS+CMS Run 1 measurement [9] using  →  and  → 4ℓ decays.

Conclusion
A measurement of the Higgs boson mass in the diphoton channel has been performed using the Run 2   collision data 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 140 fb −1 .Compared to a previous measurement in the same decay channel, which was based on a four times smaller dataset at the same energy, this measurement has a systematic (total) uncertainty that is reduced by a factor close to four (three).The reduction in systematic uncertainty is mainly due to an improved photon energy scale calibration, with better energy resolution, and smaller  T -dependent and  ± →  extrapolation uncertainties.

• 2 T
The absolute value of the pseudorapidity | 2 | of each of the two energy clusters reconstructed in the electromagnetic calorimeter and associated with the photon candidates.The pseudorapidity  2 is determined from the position of the barycentre of the cluster in the second sampling layer of the calorimeter and from the origin of the ATLAS coordinate system.Both the U-type and C-type events are separated into three subsamples: 'central barrel' (both photons have | 2 | < 0.8), 'outer-barrel' (both photons have | 2 | < 1.37 and at least one of these has | 2 | ≥ 0.8), and 'endcap' (at least one photon has 1.52 ≤ | 2 | < 2.37).• The magnitude   Tt = | ì   T × t| of the component of the diphoton transverse momentum that is orthogonal to the thrust axis, defined as t = ( ì  |.Low (  Tt < 70 GeV), medium (70 GeV <   Tt < 130 GeV) and high (  Tt > 130 GeV)   Tt categories are defined for U-type and C-type central-barrel and outer-barrel events.

Figure 1 :
Figure 1: Invariant mass distributions of simulated  →  events reconstructed in the categories with the best (U-type, central-barrel, high-  Tt : open circles) and the worst (C-type, endcap: open squares) experimental resolutions.The signal model derived from a fit of the simulated events is superimposed (solid lines).

Figure 2 :
Figure2: Diphoton invariant mass distribution of all selected data events (black dots with error bars), overlaid with the result of the fit (solid red line).For both the data and the fit, each category is weighted by a factor ln(1 +  obs 90 / obs 90 ), where  obs 90 and  obs 90 are the fitted signal and background yields in the smallest   interval containing 90% of the expected signal.The dotted line describes the background component of the model.

Figure 3 :
Figure 3: (a) Value of −2 ln Λ as a function of   for the combined fit to all  →  categories.The intersections of the −2 ln Λ curve with the horizontal lines labelled 1 and 2 provide the 68.3% and 95.5% confidence intervals.(b) Summary of the Higgs boson mass measurements from the analysis of  →  decays in ATLAS Run 2 data and combined Run 1 + Run 2 data presented in this Letter, compared with the combined Run 1 ATLAS result in the diphoton channel and with the Run 1 measurement by ATLAS and CMS [9] combining the diphoton and four-lepton channels.The statistical, systematic and total uncertainties are indicated with horizontal yellow-shaded bands, pink-shaded bands and black error bars, respectively.The vertical dashed line and grey shaded band around it indicate the central value and total uncertainty of the  →  ATLAS Run 1 + Run 2 measurement.

Table 1 :
The expected signal ( 90 ) and background ( 90 ) yields, the signal yield as a percentage of the total (  90 ), and the signal significance ( 90 ) in a diphoton invariant mass window whose half-width   90 is chosen in such a way that it is the narrowest interval containing 90% of signal events.All quantities are given for each analysis category and for the inclusive case.

Table 2 :
Estimated impact of the main sources of systematic uncertainty on the   measurement with Run 2 data.