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This article documents the performance of the ATLAS muon identification and reconstruction using the LHC dataset recorded at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{s} = 13$$\end{document}s=13 TeV in 2015. 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Introduction
Muons are key to some of the most important physics results published by the ATLAS experiment [1] at the LHC. These results include the discovery of the Higgs boson [2] and the measurement of its properties [3][4][5], the precise measurement of Standard Model processes [6,7], and searches for physics beyond the Standard Model [8][9][10][11].
The performance of the ATLAS muon reconstruction during the LHC run at √ s = 7-8 TeV has been documented in recent publications [12,13]. During the 2013-2015 shutdown, the LHC was upgraded to increase the centre-of-mass energy from 8 to 13 TeV and the ATLAS detector was equipped with additional muon chambers and a new innermost Pixel layer, the Insertable B-Layer, providing measuree-mail: atlas.publications@cern.ch ments closer to the interaction point. Moreover, the muon reconstruction software was updated and improved.
After introducing the ATLAS muon reconstruction and identification algorithms, this article describes the performance of the muon reconstruction in the first dataset collected at √ s = 13 TeV. Measurements of the muon reconstruction and isolation efficiencies and of the momentum scale and resolution are presented. The comparison between data and Monte Carlo (MC) simulation and the determination of the corrections to the simulation used in physics analyses are also discussed. The results are based on the analysis of a large sample of J/ψ → μμ and Z → μμ decays reconstructed in 3.2 fb −1 of pp collisions recorded in 2015.
This article is structured as follows: Sect. 2 describes the ATLAS subdetectors that are most relevant to this work; Sects. 3 and 5 describe the muon reconstruction and identification in ATLAS, respectively; Sect. 4 describes the data samples used in the analysis; the reconstruction and isolation efficiencies are described in Sects. 6 and 7, respectively, while the momentum scale and resolution are described in Sect. 8. Finally, conclusions are given in Sect. 9.

ATLAS detector
A detailed description of the ATLAS detector can be found in Ref. [1]. Information primarily from the inner detector (ID) and the muon spectrometer (MS), supplemented by information from the calorimeters, is used to identify and precisely reconstruct muons produced in pp collisions.
The ID consists of three subdetectors: the silicon pixels (Pixel) and the semiconductor tracker (SCT) with a pseudorapidity 1 coverage up to |η| = 2.5, and the transition radi-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 upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity and the transverse momentum are defined in terms of the polar angle θ as η = − ln tan(θ/2) and p T = ation tracker (TRT) with a pseudorapidity coverage up to |η| = 2.0. The ID measures the muon track close to the interaction point, providing accurate measurements of the track parameters inside an axial magnetic field of 2 T.
The MS is the outermost ATLAS subdetector. It is designed to detect muons in the pseudorapidity region up to |η| = 2.7, and to provide momentum measurements with a relative resolution better than 3 % over a wide p T range and up to 10 % at p T ≈ 1 TeV. The MS consists of one barrel (|η| < 1.05) and two endcap sections (1.05 < |η| < 2.7). A system of three large superconducting air-core toroidal magnets, each with eight coils, provides a magnetic field with a bending integral of about 2.5 Tm in the barrel and up to 6 Tm in the endcaps. Resistive plate chambers (RPC, three doublet layers for |η| < 1.05) and thin gap chambers (TGC, one triplet layer followed by two doublets for 1.0 < |η| < 2.4) provide triggering capability to the detector as well as (η, φ) position measurements with typical spatial resolution of 5-10 mm. A precise momentum measurement for muons with pseudorapidity up to |η| = 2.7 is provided by three layers of monitored drift tube chambers (MDT), with each chamber providing six to eight η measurements along the muon trajectory. For |η| > 2, the inner layer is instrumented with a quadruplet of cathode strip chambers (CSC) instead of MDTs. The single-hit resolution in the bending plane for the MDT and the CSC is about 80 and 60 µm, respectively. The muon chambers are aligned with a precision between 30 and 60 µm.
During the shutdown preceding the LHC Run 2, the MS was completed to its initial design [14] by adding the last missing chambers in the transition region between the barrel and the endcaps (1.0 < |η| < 1.4). Four RPC-equipped MDT chambers were also installed inside two elevator shafts to improve the acceptance in that region compared to Run 1. Some of the new MDT chambers are made of tubes with a smaller radius compared to the ones used in the rest of the detector, allowing the detector to cope with higher rates.
The material between the interaction point (IP) and the MS ranges approximately from 100 to 190 radiation lengths, depending on η, and consists mostly of calorimeters. The lead/liquid-argon electromagnetic calorimeter covers |η| < 3.2. It is surrounded by hadronic calorimeters made of steel and scintillator tiles for |η| < 1.7, and copper or tungsten and liquid argon for |η| > 1.7.

Muon reconstruction
Muon reconstruction is first performed independently in the ID and MS. The information from individual subdetectors is then combined to form the muon tracks that are used in p sin θ, respectively. The η-φ distance between two particles is defined as R = ( η) 2 + ( φ) 2 . physics analyses. In the ID, muons are reconstructed like any other charged particles as described in Refs. [15,16]. This section focuses on the description of the muon reconstruction in the MS (Sect. 3.1) and on the combined muon reconstruction (Sect. 3.2).

Muon reconstruction in the MS
Muon reconstruction in the MS starts with a search for hit patterns inside each muon chamber to form segments. In each MDT chamber and nearby trigger chamber, a Hough transform [17] is used to search for hits aligned on a trajectory in the bending plane of the detector. The MDT segments are reconstructed by performing a straight-line fit to the hits found in each layer. The RPC or TGC hits measure the coordinate orthogonal to the bending plane. Segments in the CSC detectors are built using a separate combinatorial search in the η and φ detector planes. The search algorithm includes a loose requirement on the compatibility of the track with the luminous region.
Muon track candidates are then built by fitting together hits from segments in different layers. The algorithm used for this task performs a segment-seeded combinatorial search that starts by using as seeds the segments generated in the middle layers of the detector where more trigger hits are available. The search is then extended to use the segments from the outer and inner layers as seeds. The segments are selected using criteria based on hit multiplicity and fit quality and are matched using their relative positions and angles. At least two matching segments are required to build a track, except in the barrel-endcap transition region where a single highquality segment with η and φ information can be used to build a track.
The same segment can initially be used to build several track candidates. Later, an overlap removal algorithm selects the best assignment to a single track, or allows for the segment to be shared between two tracks. To ensure high efficiency for close-by muons, all tracks with segments in three different layers of the spectrometer are kept when they are identical in two out of three layers but share no hits in the outermost layer.
The hits associated with each track candidate are fitted using a global χ 2 fit. A track candidate is accepted if the χ 2 of the fit satisfies the selection criteria. Hits providing large contributions to the χ 2 are removed and the track fit is repeated. A hit recovery procedure is also performed looking for additional hits consistent with the candidate trajectory. The track candidate is refit if additional hits are found.

Combined reconstruction
The combined ID-MS muon reconstruction is performed according to various algorithms based on the information provided by the ID, MS, and calorimeters. Four muon types are defined depending on which subdetectors are used in reconstruction: • Combined (CB) muon: track reconstruction is performed independently in the ID and MS, and a combined track is formed with a global refit that uses the hits from both the ID and MS subdetectors. During the global fit procedure, MS hits may be added to or removed from the track to improve the fit quality. Most muons are reconstructed following an outside-in pattern recognition, in which the muons are first reconstructed in the MS and then extrapolated inward and matched to an ID track. An inside-out combined reconstruction, in which ID tracks are extrapolated outward and matched to MS tracks, is used as a complementary approach. • Segment-tagged (ST) muons: a track in the ID is classified as a muon if, once extrapolated to the MS, it is associated with at least one local track segment in the MDT or CSC chambers. ST muons are used when muons cross only one layer of MS chambers, either because of their low p T or because they fall in regions with reduced MS acceptance. • Calorimeter-tagged (CT) muons: a track in the ID is identified as a muon if it can be matched to an energy deposit in the calorimeter compatible with a minimum-ionizing particle. This type has the lowest purity of all the muon types but it recovers acceptance in the region where the ATLAS muon spectrometer is only partially instrumented to allow for cabling and services to the calorimeters and inner detector. The identification criteria for CT muons are optimised for that region (|η| < 0.1) and a momentum range of 15 < p T < 100 GeV. • Extrapolated (ME) muons: the muon trajectory is reconstructed based only on the MS track and a loose requirement on compatibility with originating from the IP. The parameters of the muon track are defined at the interaction point, taking into account the estimated energy loss of the muon in the calorimeters. In general, the muon is required to traverse at least two layers of MS chambers to provide a track measurement, but three layers are required in the forward region. ME muons are mainly used to extend the acceptance for muon reconstruction into the region 2.5 < |η| < 2.7, which is not covered by the ID.
Overlaps between different muon types are resolved before producing the collection of muons used in physics analyses. When two muon types share the same ID track, preference is given to CB muons, then to ST, and finally to CT muons. The overlap with ME muons in the muon system is resolved by analyzing the track hit content and selecting the track with better fit quality and larger number of hits.
The muon reconstruction used in this work evolved from the algorithms defined as Chain 3 in Ref. [12]. These algorithms were improved in several ways. The use of a Hough transform to identify the hit patterns for seeding the segmentfinding algorithm makes the reconstruction faster and more robust against misidentification of hadrons, thus providing better background rejection early in the pattern recognition process. The calculation of the energy loss in the calorimeter was also improved. An analytic parameterization of the average energy loss is derived from a detailed description of the detector geometry. The final estimate of the energy loss is obtained by combining the analytic parameterization with the energy measured in the calorimeter. This method yields a precision on the mean energy loss of about 30 MeV for 50 GeV muons.

Data and Monte Carlo samples
The efficiency measurements presented in this article are obtained from the analysis of 3.2 fb −1 of pp collision data recorded at √ s = 13 TeV at the LHC in 2015 during the data-taking period with 25 ns spacing between bunch crossings. About 1.5 M Z → μμ and 3.5 M J/ψ → μμ events are reconstructed and used for the analysis. For the study of the momentum calibration, 2.7 fb −1 of data were used, rejecting the runs in which the longitudinal position of the beam spot was displaced by about 3 cm with respect to the centre of the detector.
Events are accepted only if the ID, the MS, and the calorimeters were operational and the solenoid and toroid magnet systems were both active. The online event selection was performed by a two-level trigger system derived from the one described in Ref. [18]. The Z → μμ candidates are triggered by the presence of at least one muon candidate with a transverse momentum, p T , of at least 20 GeV. For the reconstruction efficiency and momentum calibration studies, the muon firing the trigger is required to be isolated (see Sect. 7). The J/ψ → μμ candidates used for the momentum calibration are triggered by a dedicated dimuon trigger that requires two opposite-charge muons, each with p T > 4 GeV, compatible with the same vertex, and with a dimuon invariant mass in the range 2.5-4.5 GeV. The J/ψ → μμ sample used for the efficiency measurement is selected using a combination of single-muon triggers and triggers requiring one muon with transverse momentum of at least 4 GeV and an ID track such that the invariant mass of the muon+track pair, under a muon mass hypothesis, is compatible with the mass of the J/ψ. Monte Carlo samples for the process pp → (Z /γ * )X → μμX are generated using the POWHEG BOX [19] interfaced to PYTHIA8 [20] and the CT10 [21] parton distribution functions. The PHOTOS [22] package is used to simulate final-state photon radiation in Z boson decays. Samples of prompt J/ψ → μμ decays are generated using PYTHIA8 complemented with PHOTOS to simulate the effects of final-state radiation. A requirement on the minimum transverse momentum of each muon ( p T > 4 GeV) is applied at the generator level. The samples used for the simulation of the backgrounds to Z → μμ include: Z → τ τ , W → μν, and W → τ ν, generated with POWHEG BOX; W W , Z Z, and W Z generated with SHERPA [23]; tt samples generated with POWHEG BOX + PYTHIA8; and bb and cc samples generated with PYTHIA8.
All the generated samples are passed through the simulation of the ATLAS detector based on GEANT4 [24,25] and are reconstructed with the same programs used for the data. The ID and the MS are simulated with an ideal geometry assuming no misalignment.
The effect of multiple pp interactions per bunch crossing ("pile-up") is modelled by overlaying simulated minimumbias events onto the original hard-scattering event. Monte Carlo events are then reweighted so that the distribution of the average number of interactions per event agrees with the data.

Muon identification
Muon identification is performed by applying quality requirements that suppress background, mainly from pion and kaon decays, while selecting prompt muons with high efficiency and/or guaranteeing a robust momentum measurement.
Muon candidates originating from in-flight decays of charged hadrons in the ID are often characterized by the presence of a distinctive "kink" topology in the reconstructed track. As a consequence, it is expected that the fit quality of the resulting combined track will be poor and that the momentum measured in the ID and MS may not be compatible. Several variables offering good discrimination between prompt muons and background muon candidates are studied in simulated tt events. Muons from W decays are categorized as signal muons while muon candidates from light-hadron decays are categorized as background. For CB tracks, the variables used in muon identification are: • q/p significance, defined as the absolute value of the difference between the ratio of the charge and momentum of the muons measured in the ID and MS divided by the sum in quadrature of the corresponding uncertainties; • ρ , defined as the absolute value of the difference between the transverse momentum measurements in the ID and MS divided by the p T of the combined track; • normalised χ 2 of the combined track fit.
To guarantee a robust momentum measurement, specific requirements on the number of hits in the ID and MS are used. For the ID, the quality cuts require at least one Pixel hit, at least five SCT hits, fewer than three Pixel or SCT holes, and that at least 10 % of the TRT hits originally assigned to the track are included in the final fit; the last requirement is only employed for |η| between 0.1 and 1.9, in the region of full TRT acceptance. A hole is defined as an active sensor traversed by the track but containing no hits. A missing hit is considered a hole only when it falls between hits successfully assigned to a given track. If some inefficiency is expected for a given sensor, the requirements on the number of Pixel and SCT hits are reduced accordingly.
Four muon identification selections (Medium, Loose, Tight, and High-p T ) are provided to address the specific needs of different physics analyses. Loose, Medium, and Tight are inclusive categories in that muons identified with tighter requirements are also included in the looser categories. Medium muons The Medium identification criteria provide the default selection for muons in ATLAS. This selection minimises the systematic uncertainties associated with muon reconstruction and calibration. Only CB and ME tracks are used. The former are required to have ≥3 hits in at least two MDT layers, except for tracks in the |η| < 0.1 region, where tracks with at least one MDT layer but no more than one MDT hole layer are allowed. The latter are required to have at least three MDT/CSC layers, and are employed only in the 2.5 < |η| < 2.7 region to extend the acceptance outside the ID geometrical coverage. A loose selection on the compatibility between ID and MS momentum measurements is applied to suppress the contamination due to hadrons misidentified as muons. Specifically, the q/p significance is required to be less than seven. In the pseudorapidity region |η| < 2.5, about 0.5 % of the muons classified as Medium originate from the inside-out combined reconstruction strategy. Loose muons The Loose identification criteria are designed to maximise the reconstruction efficiency while providing good-quality muon tracks. They are specifically optimised for reconstructing Higgs boson candidates in the four-lepton final state [5]. All muon types are used. All CB and ME muons satisfying the Medium requirements are included in the Loose selection. CT and ST muons are restricted to the |η| < 0.1 region. In the region |η| < 2.5, about 97.5 % of the Loose muons are combined muons, approximately 1.5 % are CT, and the remaining 1 % are reconstructed as ST muons. Tight muons Tight muons are selected to maximise the purity of muons at the cost of some efficiency. Only CB muons with hits in at least two stations of the MS and satisfying the Medium selection criteria are considered. The normalised χ 2 of the combined track fit is required to be <8 to remove pathological tracks. A two-dimensional cut in the ρ and q/p significance variables is performed as a function of the muon p T to ensure stronger background rejection for momenta below 20 GeV where the misidentification probability is higher.

High-p T muons
The High-p T selection aims to maximise the momentum resolution for tracks with transverse momentum above 100 GeV. The selection is optimised for searches for high-mass Z and W resonances [8,9]. CB muons passing the Medium selection and having at least three hits in three MS stations are selected. Specific regions of the MS where the alignment is suboptimal are vetoed as a precaution.
Requiring three MS stations, while reducing the reconstruction efficiency by about 20 %, improves the p T resolution of muons above 1.5 TeV by approximately 30 %. The reconstruction efficiencies for signal and background obtained from tt simulation are reported in Table 1. The results are shown for the four identification selection criteria separating low (4 < p T < 20 GeV) and high (20 < p T < 100 GeV) transverse momentum muon candidates. No isolation requirement is applied in the selection shown in the table. When isolation requirements are applied, the misidentification rates are reduced by more than an order of magnitude. It should be noted that the higher misidentification rate observed for Loose with respect to Medium muons is mainly due to CT muons in the region |η| < 0.1.
The misidentification probability estimated with the MC simulation is validated in data by measuring the probability that pions are reconstructed as muons. An unbiased sample of pions from K 0 S → π + π − decays is collected with calorimeter-based (photon, electron, jet) triggers. Good agreement between data and simulation is observed independent of the p T , η, and impact parameter of the track.

Reconstruction efficiency
As the muon reconstruction in the ID and MS detectors is performed independently, a precise determination of the muon reconstruction efficiency in the region |η| < 2.5 is obtained with the tag-and-probe method, as described in the Sect. 6.1. A different methodology, described in Sect. 6.2, is used in the region 2.5 < |η| < 2.7 where muons are reconstructed using only the MS detector.
6.1 Efficiency measurement in the region |η| < 2.5 The tag-and-probe method is employed to measure the efficiency of the muon identification selections within the acceptance of the ID (|η| < 2.5). The method is based on the selection of an almost pure muon sample from J/ψ → μμ or Z → μμ events, requiring one leg of the decay (tag) to be identified as a Medium muon that fires the trigger and the second leg (probe) to be reconstructed by a system independent of the one being studied. A selection based on the event topology is used to reduce the background contamination.
Three kinds of probes are used to measure muon efficiencies. ID tracks and CT muons both allow a measurement of the efficiency in the MS, while MS tracks are used to determine the complementary efficiency of the muon reconstruction in the ID. Compared to ID tracks, CT muons offer a more powerful rejection of backgrounds, especially at low transverse momenta, and are therefore the preferred probe type for this part of the measurement. ID tracks are used as a cross-check and for measurements not directly accessible to CT muons. A direct measurement of the CT muon reconstruction efficiency is possible using MS tracks.
The efficiency measurement for Medium, Tight, and High-p T muons consists of two stages. First, the efficiency (X|CT) (X = Medium/Tight/High-p T ) of reconstructing these muons assuming a reconstructed ID track is measured using a CT muon as probe. Then, this result is corrected by the efficiency (ID|MS) of the ID track reconstruction, measured using MS probes: (1) A similar approach is used when using ID probe tracks for cross-checks. This approach is valid if two assumptions are satisfied: • the ID track reconstruction efficiency is independent from the muon spectrometer track reconstruction ( (ID) = (ID|MS)). • the use of a CT muon as a probe instead of an ID track does not affect the probability for Medium, Tight, or High-p T reconstruction ( (X|ID) = (X|CT)).
Both assumptions have been tested using generator-level information from simulation and small differences are taken into account in the systematic uncertainties. The muons selected by the Loose identification requirements are decomposed into two samples: CT muons within |η| < 0.1 and all other muons. The CT muon efficiency is measured using MS probe tracks, while the efficiency of other muons is evaluated using CT probe muons in a fashion similar to the Medium, Tight, and High-p T categories.
The level of agreement of the measured efficiency, Data (X), with the efficiency measured with the same method in simulation, MC (X), is expressed as the ratio of these two numbers, called the "efficiency scale factor" (SF): This quantity describes the deviation of the simulation from the real detector behaviour, and is of particular interest to physics analyses, where it is used to correct the simulation.

The tag-and-probe method with Z → μμ events
Events are selected by requiring muon pairs with an invariant mass within 10 GeV of the Z boson mass. The tag muon is required to satisfy the Loose isolation (see Sect. 7.2) and Medium muon identification selections and to have a transverse momentum of at least 24 GeV. Requirements on the significance of the transverse impact parameter d 0 (|d 0 |/σ (d 0 ) < 3.0) and on the longitudinal impact parameter |z 0 | (|z 0 | < 10 mm) of the tag muon are imposed. Finally, the tag muon is required to have triggered the readout of the event.
The probe muon is required to have a transverse momentum of at least 10 GeV and to satisfy the Loose isolation criteria. While this is sufficient to ensure high purity in the case of MS probe tracks, further requirements are applied to both the ID track and CT muon probes. In the case of ID tracks, an isolation requirement is applied which is considerably stricter than the Loose selection in order to suppress backgrounds as much as possible. In addition, the invariant mass window is tightened to 5 GeV around the Z boson mass, rather than the 10 GeV used in the other cases. For CT muon probes, additional requirements on the compatibility of the associated calorimeter energy deposit with a muon signature are applied to further enhance the purity. The ID probe tracks and calorimeter-tagged probe muons must also have transverse and longitudinal impact parameters consistent with being produced in a primary pp interaction, as required for tag muons. A probe is considered successfully reconstructed if a reconstructed muon is found within a cone in the η-φ plane of size R = 0.05 around the probe track.
A small fraction (about 0.1 %) of the selected tag-probe pairs originates from sources other than Z → μμ events. For a precise efficiency measurement, these backgrounds must be estimated and subtracted. Contributions from Z → τ τ and tt decays are estimated using simulation. Additionally, multijet events and W → μν decays in association with jet activity (W +jets) can yield tag-probe pairs through secondary muons from heavy-or light-hadron decays. As these backgrounds are approximately charge-symmetric, they are estimated from the data using same-charge (SC) tag-probe pairs. This leads to the following estimate of the oppositecharge (OC) background, N Bkg , for each region of the kinematic phase-space: where N Z ,tt MC OC is the contribution from Z → τ τ and tt decays, N Data SC is the number of SC pairs measured in data and N Z ,tt MC SC is the estimated contribution of the Z → μμ, Z → τ τ , and tt processes to the SC sample. T is a global transfer factor that takes into account the charge asymmetry of the multijet and W +jets processes, estimated in data using a control sample of events obtained by inverting the probe isolation requirement. For MS (ID) tracks, a value of T = 1.7 (1.1) is obtained, while for calorimeter-tagged muon probes the transfer factor is T = 1.2. The systematic uncertainties in the transfer factor vary between 40 % and 100 % and are included in the systematic error in the reconstruction efficiency described in Sect. 6.1.3.
The efficiency measured in the data is corrected for the background contributions described above by subtracting the predicted probe yields attributed to these sources from the number of observed probes, where N P denotes the total number of probes and N R the number of successfully reconstructed probes. The resulting efficiency can then be compared directly to the result of the simulation.

The tag-and-probe method with J/ψ → μμ events
The reconstruction efficiencies of the Loose, Medium, and Tight muon selections at low p T are measured from a sample of J/ψ → μμ events selected using a combination of single-muon triggers and the dedicated "muon + track" trigger described in Sect. 4. Tag-probe pairs are selected within the invariant mass window of 2.7-3.5 GeV and requiring a transverse momentum of at least 5 GeV for each muon. The tag muon is required to satisfy the Medium muon identification selection and to have triggered the readout of the event. In order to avoid lowmomentum curved tracks sharing the same trigger region, tag and probe muons are required to be R > 0.2 apart when extrapolated to the MS trigger surfaces. Finally, they  | < 5 mm, to suppress background. A probe is considered successfully reconstructed if a selected muon is found within a R = 0.05 cone around the probe track.
The background contamination and the muon reconstruction efficiency are measured with a simultaneous maximumlikelihood fit of two statistically independent distributions of the invariant mass: events in which the probe is or is not successfully matched to the selected muon. The fits are performed in six p T and nine η bins of the probe tracks. The signal is modelled with a Crystal Ball function [26] with a single set of parameters for the two independent samples. Separate first-order polynomial fits are used to describe the background shape for matched and unmatched probes.

Systematic uncertainties
The main contributions to the systematic uncertainty in the measurement of the efficiency SFs with Z → μμ and J/ψ → μμ events are shown in Figs. 1 and 2, as a function of η and p T , respectively.
The uncertainty in the background estimate is evaluated in the Z → μμ analysis by taking the maximum variation of the transfer factor T when estimated with a simulation-based approach as described in Ref.
[12] and when assuming the background to be charge-symmetric. This results in an uncertainty of the efficiency measurement below 0.1 % over a large momentum range, but reaching ∼1 % for low muon momenta where the contribution of the background is most significant. In the J/ψ → μμ analysis, the background uncertainty is estimated by changing the function used in the fit to model the background, replacing the first-order polynomial with an exponential function. An uncertainty due to the signal modelling in the fit, labelled as "Signal" in Figs. 1 and 2, is also estimated using a convolution of exponential and Gaussian  The cone size used for matching selected muons to probe tracks is optimised in terms of efficiency and purity of the matching. The systematic uncertainty deriving from this choice is evaluated by varying the cone size by ±50 %. This yields an uncertainty below 0.1 % in both analyses.
Possible biases in the tag-and-probe method, such as biases due to different kinematic distributions between reconstructed probes and generated muons or correlations between ID and MS efficiencies, are estimated in simulation by comparing the efficiency measured with the tag-and-probe method with the "true" efficiency given by the fraction of generator-level muons that are successfully reconstructed. This uncertainty is labelled as "Truth Closure" in Figs. 1 and 2. In the Z → μμ analysis, agreement better than 0.1 % is observed in the high momentum range. This uncertainty grows at low p T , and differences up to 0.7 % are found in the J/ψ → μμ analysis. A larger effect of up to 1-2 % is measured in both analyses in the region |η| < 0.1. In the extraction of the efficiency scale factors, the difference between the measured and the "true" efficiency cancels to first order. To take into account possible imperfections of the simulation, half of the observed difference is used as an additional systematic uncertainty in the SF.
No significant dependence of the measured SFs with p T is observed in the momentum range considered in the Z → μμ analysis. An upper limit on the SF variation for large muon momenta is extracted from simulation, leading to an additional uncertainty of 2-3 % per TeV for muons with p T > 200 GeV. The efficiency scale factor is observed to be independent of the amount of pile-up. Figure 3 shows the muon reconstruction efficiency as a function of η as measured from Z → μμ events for the different muon selections. The efficiency as measured in data and the corresponding scale factors for the Medium selection are also shown in Fig. 4 as a function of η and φ. The efficiency at low p T is reported in Fig. 5 as measured from J/ψ → μμ events as a function of p T in different η regions.

Results
The efficiencies of the Loose and Medium selections are very similar throughout the detector with the exception of the region |η| < 0.1, where the Loose selection fills the MS acceptance gap using the calorimeter and segment-tagged muons contributions. The efficiency of these selections is observed to be in excess of 98 %, and between 90 and 98 % for the Tight selection, with all efficiencies in very good agreement with those predicted by the simulation. An inefficiency due to a poorly aligned MDT chamber is clearly localised at (η, φ) ∼ (−1.3, 1.6), and is the most significant feature of the comparison between collision data and simulation for these three categories. In addition, a 2 %-level local inefficiency is visible in the region (η, φ) ∼ (1.9, 2.5), traced to temporary failures in the SCT readout system. Further local inefficiencies in the barrel region around φ ∼ −1.1 are also    Within each η region, the efficiency is measured in six p T bins (5-6, 6-7, 7-8, 8-10, 10-12, and 12-15 GeV). The resulting values are plotted as distinct measurements in each η bin with p T increasing from 5 to 15 GeV going from left to right. The error bars on the efficiencies indicate the statistical uncertainty. The panel at the bottom shows the ratio of the measured to predicted efficiencies, with statistical and systematic uncertainties linked to temporary faults during data taking. The efficiency of the High-p T selection is significantly lower, as a consequence of the strict requirements on momentum resolution. Local disagreements between prediction and observation are more severe than in the case of the other muon selections.
Apart from the poorly aligned MDT chamber, they are most prominent in the CSC region. Figure 6 shows the reconstruction efficiencies for the Medium muon selection as a function of transverse momentum, including results from Z → μμ and J/ψ → μμ, for  Fig. 6 Reconstruction efficiency for the Medium muon selection as a function of the p T of the muon, in the region 0.1 < |η| < 2.5 as obtained with Z → μμ and J/ψ → μμ events. The error bars on the efficiencies indicate the statistical uncertainty. The panel at the bottom shows the ratio of the measured to predicted efficiencies, with statistical and systematic uncertainties muons with 0.1 < |η| < 2.5. The efficiency is stable and slightly above 99 % for p T > 6 GeV. Values measured from J/ψ → μμ and Z → μμ events are in agreement in the overlap region between 10 and 20 GeV. The efficiency scale factors are also found to be compatible.
6.2 Muon reconstruction efficiency for |η| > 2.5 As described in the previous sections, the reconstruction of combined muons is limited by the ID acceptance to the pseudorapidity region |η| < 2.5. For |η| > 2.5, the efficiency is recovered by using the ME muons included in the Loose and Medium muon selections. A measurement of the efficiency SF for muons in the region 2.5 < |η| < 2.7 (high-η region) is performed using the method described in Ref. [12]. The number of muons observed in Z → μμ decays in the high-η region is normalised to the number of muons observed in the region 2.2 < |η| < 2.5. This ratio is calculated for both data and simulation, applying all known performance corrections to the region |η| < 2.5. The SFs in the high-η region are defined as the ratio of the aforementioned ratios and are provided in 4 η and 16 φ bins. The values of the SFs measured using the 2015 dataset are close to 0.9 and are determined with a 3-5 % uncertainty.

Isolation
Muons originating from the decay of heavy particles, such as W , Z , or Higgs bosons, are often produced isolated from other particles. Unlike muons from semileptonic decays, which are embedded in jets, these muons are well separated from other particles in the event. The measurement of the detector activity around a muon candidate, referred to as muon isolation, is therefore a powerful tool for background rejection in many physics analyses.

Muon isolation variables
Two variables are defined to assess muon isolation: a trackbased isolation variable and a calorimeter-based isolation variable.
The track-based isolation variable, p varcone30 T , is defined as the scalar sum of the transverse momenta of the tracks with p T > 1 GeV in a cone of size R = min 10 GeV/ p μ T , 0.3 around the muon of transverse momentum p μ T , excluding the muon track itself. The cone size is chosen to be p T -dependent to improve the performance for muons produced in the decay of particles with a large transverse momentum.
The calorimeter-based isolation variable, E topocone20 T , is defined as the sum of the transverse energy of topological clusters [27] in a cone of size R = 0.2 around the muon, after subtracting the contribution from the energy deposit of the muon itself and correcting for pile-up effects. Contributions from pile-up and the underlying event are estimated using the ambient energy-density technique [28] and are corrected on an event-by-event basis.
The isolation selection criteria are determined using the relative isolation variables, which are defined as the ratio of the track-or calorimeter-based isolation variables to the transverse momentum of the muon. The distribution of the relative isolation variables in muons from Z → μμ events is shown in the top panels of Fig. 7. Muons included in the plot satisfy the Medium identification criteria and are well separated from the other muon from the Z boson ( R μμ > 0.3). The bottom panel shows the ratio of data to simulation.

Muon isolation performance
Seven isolation selection criteria (isolation working points) are defined, each optimised for different physics analyses. Table 2 lists the seven isolation working points with the discriminating variables and the criteria used in their definition.
The efficiencies for the seven isolation working points are measured in data and simulation in Z → μμ decays using the tag-and-probe method described in Sect. 6. To avoid probe muons in the vicinity of a jet, the angular separation R between the probe muon and the closest jet, reconstructed using an anti-k t algorithm [29] with radius parameter 0.4 and with a transverse momentum greater than 20 GeV, is required to be greater than 0.4. In addition, the two muons originating from the Z boson decay are required to be separated by R μμ > 0.3. Figure 8 shows the isolation efficiency measured for Medium muons in data and simulation

≥95(99) % efficiency at 25 (60) GeV
as a function of the muon p T for the LooseTrackOnly, Loose, GradientLoose, and FixedCutLoose working points, with the respective data/MC ratios included in the bottom panel. The systematic uncertainties in the SFs are estimated by varying the background contributions within their uncertainties and by varying some of the selection criteria, such as the invariant mass selection window, the isolation of the tag muon, the minimum quality of the probe muon, the opening angle between the two muons, and the R between the probe muon and the closest jet. In Fig. 8, the largest systematic uncertainty contribution over the entire p T region arises from having neglected the η dependence of the SFs, which are usually provided as a function of η and p T . In the lowp T region, other important contributions are due to the background estimation and the mass window variation, while the highp T region is dominated by statistical uncertainties in data and simulation. The total uncertainty is at the per mille level over a wide range of p T and reaches the percent level in the highp T region. The suppression factor for muons from light mesons or b/c semileptonic decays is estimated from simulation and depends on the isolation working point, ranging from a minimum of 15 for LooseTrackOnly to a maximum of 40 for Gradient.

Momentum scale and resolution
The muon momentum scale and resolution are studied using J/ψ → μμ and Z → μμ decays. Although the simulation contains an accurate description of the ATLAS detector, the level of detail is not enough to describe the muon momentum scale to the per mille level and the muon momentum resolution to the percent level. To obtain such a level of agreement between data and simulation, a set of corrections is applied to the simulated muon momentum. The methodology used to extract these corrections is described in Sect. 8.1. In Sect. 8.2, measurements of the muon momentum scale and resolution in data and simulation are presented for various detector regions and for a wide range of p T . To improve the precision of the procedure, the p T and η distributions of the Z and J/ψ resonances in simulation are reweighted to the distributions observed in data.

Muon momentum calibration procedure
In the following, the "muon momentum calibration" is defined as the procedure used to identify the corrections to the simulated muon transverse momenta reconstructed in the ID and MS subdetectors to precisely describe the measure- where p MC,Det T is the uncorrected transverse momentum in simulation, g m are normally distributed random variables with zero mean and unit width, and the terms r Det m (η, φ) and s Det n (η, φ) describe the momentum resolution smearing and the scale corrections applied in a specific (η, φ) detector region, respectively.
The corrections described in Eq. (5) are defined in η-φ detector regions that are homogeneous in terms of detector technology and performance. Both the ID and the MS are divided into 18 pseudorapidity regions. In addition, the MS is divided into two φ bins separating the two types of φ sectors: those that include the magnet coils (small sectors) and those between two coils (large sectors). The small and large MS sectors employ independent alignment techniques and cover detector areas with different material distribution. Therefore, relevant scale and resolution differences exist.
The numerator of Eq. (5) describes the momentum scales. The s Det 1 term corrects for inaccuracy in the description of the magnetic field integral and the dimension of the detector in the direction perpendicular to the magnetic field. The s MS 0 (η, φ) term models the effect on the MS momentum from the inaccuracy in the simulation of the energy loss in the calorimeter and other materials between the interaction point and the MS. As the energy loss between the interaction point and the ID is negligible, s ID 0 (η) is set to zero. The denominator of Eq. (5) describes the momentum smearing that broadens the relative p T resolution in simulation, σ ( p T )/ p T , to properly describe the data. The corrections to the resolution assume that the relative p T resolution can be parameterized as follows: with ⊕ denoting a sum in quadrature. In Eq. (6), the first term accounts mainly for fluctuations of the energy loss in the tra-versed material, the second term accounts mainly for multiple scattering, local magnetic field inhomogeneities and local radial displacements of the hits, and the third term mainly describes intrinsic resolution effects caused by the spatial resolution of the hit measurements and by residual misalignment of the muon spectrometer. The energy loss term is negligible in both the ID and MS measurements, and therefore r ID 0 and r MS 0 are set to zero. The corrected momentum of the combined muons, p Cor,CB T , is obtained by combining the ID and MS corrected momenta using a weighted average: with the weight f derived from the following linear equation which assumes that the relative contribution of the two subdetectors to the combined track remains unchanged before and after momentum corrections.

Determination of the p T calibration constants
The MS and ID correction parameters contained in Eq. (5) are extracted from data using a binned maximum-likelihood fit with templates derived from simulation which compares the invariant mass distributions for J/ψ → μμ and Z → μμ candidates in data and simulation. The exceptions are r ID 0 , r MS 0 , and s ID 0 , which are set to zero, and r MS 2 , which is determined from alignment studies using special runs with the toroidal magnetic field off.
The J/ψ → μμ and Z → μμ candidates are selected by requiring two oppositely charged CB muons satisfying the Medium identification criteria. Both muons must have impact parameters compatible with tracks produced by the primary interaction and pseudorapidity within the acceptance of both the ID and MS detectors (|η| < 2.5). Both muons from J/ψ → μμ (Z → μμ) candidate decays are required to have momenta in the range 5-20 (22-300) GeV and to form an invariant mass in the range 2.65-3.6 (76-106) GeV. Muons from Z boson decays need to be isolated, while no isolation criterion is imposed on muons from J/ψ decays.
The extraction of the correction parameters is performed in η-φ regions of fit (ROFs) defined separately for the ID and the MS. Events are assigned to a specific ROF if at least one muon falls in the corresponding η-φ region.
The ID corrections are extracted using the distributions of the ID dimuon invariant mass, m ID μμ . To enhance the sensitivity to p T -dependent correction effects, the m ID μμ is classified according to the p T of the muons. For J/ψ → μμ (Z → μμ) decays, the fit is performed in two exclusive categories defined requiring the candidates to have p ID T of the sub-leading (leading) muon greater than 5 or 9 (22 or 47) GeV, respectively. Similarly, the MS corrections are extracted using the distributions of the MS-reconstructed dimuon invariant mass, m MS μμ . Since there are more parameters and more ROFs in the MS version of Eq. (5), an additional variable is added to the MS fit. This is defined by the following equation which represents the p T imbalance between the measurement in the ID and in the MS. In Eq. (9), the momentum of the ID, p Cor,ID T , contains the appropriate p T corrections. The variable ρ is used only in Z → μμ candidate events and is binned according to p MS T of the muon with lower bin boundaries of p MS T = 22, 35, 47, 60, 90 GeV. Templates for the m ID μμ , m MS μμ , and ρ are built using J/ψ → μμ and Z → μμ simulated signal samples. In the Z → μμ sample, the small background component (approximately 0.1 %) is extracted from simulation and added to the templates. A much larger (about 15 %) non-resonant background from decays of light and heavy hadrons and from continuum Drell-Yan production is present in the J/ψ → μμ sample. As this background is not easy to simulate, a datadriven approach is used. The dimuon invariant mass distribution in data is fitted in each ROF using a Crystal Ball function added to an exponential background distribution in the ID and MS fits. The background model and its normalisation are then used in the template fit.
The results are shown in Tables 3 and 4, averaged over three η regions. The quoted errors include systematic uncertainties evaluated by varying several parameters of the template fit. The main contributions to the final systematic uncertainty are: • Mass window width for the Z → μμ candidate selection. Non-Gaussian smearing effects are accounted for by varying the m μμ selection by ±5 GeV.  • Background parameterization for the J/ψ fit as well as increased muon p T cut (from 5 to 7 GeV) to reduce the weight of the contribution of lowp T muons.
• Scale parameter for the ID corrections obtained by fitting separately the J/ψ → μμ and Z → μμ samples, to include possible non-linear scale effects. • As r MS 2 is sensitive to the alignment of the MS chambers, its systematic uncertainty is determined from alignment studies performed on special runs where the toroidal magnetic field was turned off.

Dimuon mass scale and resolution after applying momentum corrections
The samples of J/ψ → μμ and Z → μμ decays are used to study the muon momentum scales and resolution in data and simulation and to validate the momentum corrections obtained with the template fit method described in the previous section. The invariant mass distributions for the J/ψ → μμ and Z → μμ candidates are shown in Fig. 9 and compared with uncorrected and corrected simulation. In the uncorrected simulation, it is noticeable that the signal distributions are narrower and slightly shifted with respect to data. After correction, the lineshapes of the two resonances in simulation agree with the data within the systematic uncertainties, demonstrating the overall effectiveness of the p T calibration.
A better demonstration of the effectiveness of the momentum calibration is obtained by comparing, in data and simu-   When the two muons have similar momentum resolution and angular effects are neglected, the relative mass resolution, σ μμ /m μμ , is directly proportional to the relative muon momentum resolution, σ p μ / p μ :

ATLAS
Similarly, the total muon momentum scale, defined as s = ( p meas − p true )/ p true , is directly related to the dimuon mass scale, defined as s μμ = (m meas μμ − m true μμ )/m true μμ : where s μ 1 and s μ 2 are the momentum scales of the two muons. The dimuon mass resolution is obtained by fitting the width of the invariant mass peaks. In J/ψ → μμ decays, the intrinsic width of the resonance is negligible with respect to the experimental resolution. The bulk of the peak is modelled by a Crystal Ball function; a Gaussian distribution centred on the Crystal Ball function is added to the signal description to model the tails of the distribution. The non-resonant background is described by an exponential function. In Z → μμ decays, the fits use a convolution of the true lineshape (modelled by a Breit-Wigner function) with an experimental resolution function (a combination of a Crystal Ball and a Gaussian function). Similarly to the J/ψ, the non-resonant background is described by an exponential function. The peak position and width of the Crystal Ball function are used as estimators for the m μμ and σ (m μμ ) variables in the various η and p T bins. Figure 10 shows the position of the peak of the invariant mass distribution, m μμ , obtained from the fits to the Z boson and J/ψ samples as a function of the pseudorapidity of the highestp T muon for CB pairs. The distributions are shown for data as well as corrected simulation, with the ratio of the two in the lower panel. The simulation is in very good agreement with the data. Minor deviations are contained within the scale systematic uncertainties of 0.05 % in the barrel region, increasing with |η| to 0.1 %(0.3 %) in the region |η| ∼ 2.5 for Z → μμ (J/ψ → μμ) decays. The systematic uncertainties shown in the plots include the effects of the uncertainties in the calibration constants described in Sect. 8.1 and the changes in the fit parameterization. The observed level of agreement demonstrates that the p T calibration for combined muon tracks described above provides a very accurate description of the momentum scale in all η regions, over a wide p T range. Similar levels of data/MC agreement are observed for the ID and MS components of the combined tracks. Figure 11 displays the dimuon mass resolution σ (m μμ ) as a function of the leading-muon η for the two resonances. The dimuon mass resolution is about 1.2 and 1.6 % at small η values for J/ψ and Z bosons, respectively, and increases to 1.6 and 1.9 % in the endcaps. This corresponds to a relative muon p T resolution of 1.7 and 2.3 % in the centre of the detector and 2.3 and 2.9 % in the endcaps for J/ψ and Z boson decays, respectively. After applying the momentum corrections described above, the simulation reproduces the resolution measured in data, well within the systematic  Fig. 11 Dimuon invariant mass resolution for CB muons for Z → μμ (left) and J/ψ → μμ (right) events for data and corrected simulation as a function of the pseudorapidity of the highestp T muon. The upper panels show the fitted resolution value for data and corrected simulation.
The lower panels show the data/MC ratio. The error bars represent the statistical uncertainty; the shaded bands represent the systematic uncertainty in the correction and the systematic uncertainty in the extraction method added in quadrature uncertainties. The systematic uncertainties are estimated following the same procedure described for the determination of the energy scale. Good agreement between the dimuon mass resolution measured in data and simulation is also observed for the ID and MS components of the combined tracks.
The relative dimuon mass resolution σ μμ /m μμ depends approximately on the average momentum of the muons, as shown in Eq. (10). This allows a direct comparison of the momentum resolution function determined with J/ψ and Z boson decays. This is shown in Fig. 12, where the relative dimuon mass resolution from J/ψ → μμ and Z → μμ events is compared to simulation. The J/ψ → μμ and Z → μμ resolutions are in good agreement. For the J/ψ, the average momentum is defined as p T = 1 2 ( p T,1 + p T,2 ) while for the Z boson it is defined as where m Z is the Z boson mass [30], θ 1 and θ 2 are the polar angles of the two muons, and α 12 is the opening angle of the muon pair. This definition, based on angular variables only, removes the correlation between the measurement of the dimuon mass and the average p T .

Conclusions
The performance of the ATLAS muon reconstruction has been measured using 3.

Fig. 12
Dimuon invariant mass resolution divided by the dimuon invariant mass for CB muons measured from J/ψ → μμ and Z → μμ events as a function of the average transverse momentum variables p T and p * T defined in the text. Both muons are required to be in the same |η| range. The error bars represent the statistical uncertainty while the bands show the systematic uncertainties decays and J/ψ → μμ decays allows for a precise measurement of the reconstruction and isolation efficiency as well as of the momentum resolution and scale over a wide p T range.
The muon reconstruction efficiency is close to 99 % over most of the pseudorapidity range of |η| < 2.5 for p T > 5 GeV. The Z → μμ sample enables a measurement of the efficiency with a precision at the 0.2 % level for p T > 20 GeV. The J/ψ → μμ sample provides a measurement of the reconstruction efficiency between 5 and 20 GeV with a precision better than 1 %.
The Z → μμ sample is also used to measure the isolation efficiency for seven isolation working points in the momentum range 10-120 GeV. The isolation efficiency varies between 93 and 100 % depending on the selection and on the momentum of the particle, and is well reproduced in the simulation.
The muon momentum scale and resolution have been studied in detail using J/ψ → μμ and Z → μμ decays. These studies are used to correct the simulation to improve the agreement with data and to minimise the systematic uncertainties in physics analyses. For Z → μμ decays, the uncertainty in the momentum scale varies from a minimum of 0.05 % for |η| < 1 to a maximum of 0.3 % for |η| ∼ 2.5. The dimuon mass resolution is about 1.2 % (1.6 %) at small values of pseudorapidity for J/ψ (Z ) decays, and increases to 1.6 and 1.9 % in the endcaps for J/ψ and Z decays, respectively. This corresponds to a relative muon p T resolution of 1.7 and 2.3 % at small values of pseudorapidity and 2.3 and 2.9 % in the endcaps for J/ψ and Z decays, respectively. After applying momentum corrections, the p T resolution in data and simulation agree to better than 5 % for most of the η range.
Acknowledgments We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia;