Prospects of measuring the branching fraction of the Higgs boson decaying into muon pairs at the International Linear Collider
aa r X i v : . [ h e p - e x ] S e p ILD-PHYS-2020-00209 September 2020
Prospects of measuring the branching fraction of the Higgsboson decaying into muon pairs at the International LinearCollider
Shin-ichi Kawada ∗ , Jenny List ∗ , Mikael Berggren ∗ ∗ DESY, Notkestraße 85, 22607 Hamburg, Germany
Abstract
The prospects for measuring the branching fraction of H → µ + µ − at the InternationalLinear Collider (ILC) have been evaluated based on a full detector simulation of the Interna-tional Large Detector (ILD) concept, considering centre-of-mass energies ( √ s ) of 250 GeVand 500 GeV. For both √ s cases, the two final states e + e − → qqH and e + e − → νν H have been analyzed. For integrated luminosities of 2 ab − at √ s =
250 GeV and 4 ab − at √ s =
500 GeV, the combined precision on the branching fraction of H → µ + µ − is estim-ated to be 17%. The impact of the transverse momentum resolution for this analysis is alsostudied ∗ . ∗ This work was carried out in the framework of the ILD concept group
Introduction
A Standard Model (SM)-like Higgs boson with mass of ∼
125 GeV has been discovered by the ATLASand CMS experiments at the Large Hadron Collider (LHC) [1, 2]. Recently, the decay mode of the Higgsboson to bottom quarks H → bb has been observed at the LHC [3, 4], as well as the ttH productionprocess [5, 6], both being consistent with the SM prediction. However, there are several importantquestions to which the SM does not offer an answer: it neither explains the hierarchy problem, nor doesit address the nature of dark matter, the origin of cosmic inflation, or the baryon-antibaryon asymmetry inthe universe. Most fundamentally, it does not include gravity. The Higgs boson could be a portal towardsthe solution of many of the outstanding questions which are not addressed by the SM. Thus, it is veryimportant to measure the Higgs boson in as many channels as possible. In the SM, the Yukawa couplingbetween matter fermions and the Higgs boson is proportional to the fermion’s mass. If any deviationfrom this proportionality is observed, it is an indication of new physics beyond the SM. The size of thedeviation from the SM depends on the model, but for a large variety of models, it is estimated to beat the level of a few percent [7]. To observe such a small deviation, very precise measurements of theproperties of the Higgs boson are required. The International Linear Collider (ILC) [8–13] is one of thefuture e + e − colliders proposed to deliver this precision with the least possible dependency on models.The interaction of electrons and positrons will provide a cleaner environment than the proton-protoncollisions at the LHC. Besides, the beams would be longitudinally polarised: 80% for electron beamand 30% for positron one. By controlling the polarisation, one can study the chiral structure of the SMinteractions and potentially new physics [14].In this paper, we focus on the channel of the Higgs boson decays to a pair of muons H → µ + µ − atthe ILC. This channel is important because it provides an opportunity to measure the Yukawa couplingbetween the Higgs boson and a second-generation fermion directly. However, this is a very challenginganalysis, because in the SM the branching fraction of H → µ + µ − is predicted to be tiny: 2 . × − forthe mass of the Higgs boson of 125 GeV [15].At the LHC, the H → µ + µ − channel is explored using proton-proton collisions. In ATLAS, an ob-served significance of 2 . σ with respect to the hypothesis of no H → µ + µ − signal was obtained usingthe full Run 2 dataset of 139 fb − [16]. The CMS observed an excess of events in data with a significanceof 3 . σ [17]. The prospects of measuring this channel at the High-Luminosity LHC (HL-LHC) have alsobeen studied. The ATLAS experiment estimates ∼
13% precision on the signal strength with 3 ab − dataassuming the phase-II detector upgrade [18], based on generator-level samples for the main signal andbackground processes. The CMS experiment projects a precision of ∼ σ × BF (cross-section times branching fraction) can be turned into a measurementon the BF itself thanks to the total cross-section σ being accessible via the so-called recoil techniquein a highly model-independent way [20, 21]. Moreover, by combining other measurements at the ILC,absolute couplings of the Higgs boson can be extracted, e.g. based on SM Effective Field Theory [14].In this study, the precision expected for the measurement of the branching fraction BF ( H → µ + µ − ) atthe ILC has been estimated based on a full detector simulation of the International Large Detector (ILD)concept [12, 22] and taking into account all relevant physics and machine-related process. The Higgsproduction cross-section as a function of √ s at the ILC is shown in Fig. 1, together with correspondingFeynman diagrams. The standard running scenario for the ILC has been assumed, which would accumu-late 2 ab − at √ s =
250 GeV and 4 ab − at √ s =
500 GeV with beam polarisation sharing as described inRefs. [23, 24]. Eight different configurations, referred to as analysis channels in the following, have beenconsidered: the two production processes e + e − → qqH → qq µ + µ − and e + e − → νν H → ννµ + µ − , withtwo beam polarisation configurations, and two √ s cases. The case of the electron-positron polarisationcombination P ( e − , e + ) = ( − , + ) will be referred to as the left-handed case (denoted by L)and P ( e − , e + ) = (+ , − ) as the right-handed case (denoted by R). The cross-section of the qqH Introduction
Figure 1: The Higgs production cross-section as a function of √ s . Taken from Ref. [25].Table 1: The expected number of signal events N signal and abbreviations for each channel, where R Ldt isthe integrated luminosity based on the running scenario [23, 24]. √ s process beam pol. abbreviation R Ldt (ab − ) N signal
250 GeV qqH
L qqH250-L 0.9 41.1R qqH250-R 0.9 28.1 νν H L nnH250-L 0.9 15.0R nnH250-R 0.9 8.4500 GeV qqH
L qqH500-L 1.6 24.6R qqH500-R 1.6 16.5 νν H L nnH500-L 1.6 57.5R nnH500-R 1.6 7.9process changes by about 40% between the two beam polarisation configurations, while the νν H processis more significantly affected due to the WW -fusion contribution to the νν final state. At √ s =
250 GeV,the e + e − → ZH process is the dominant production process. Thus, the ZH → qq µ + µ − channel is themost important signal process at this energy due to the large branching fraction of Z → qq . Since WW -fusion is the major production process at √ s =
500 GeV, νν H → ννµ + µ − with left-handed polarisation(including both the WW -fusion as well as ZH with Z → ν ¯ ν ) is the most relevant channel at this energy.As the cross sections of the ZH and WW -fusion processes will be known to very high precision fromother ILC measurements, the separation of the two production modes is not targeted in this paper and itis assumed that the uncertainties on these cross sections will have only a negligible impact on the extrac-tion of the branching fraction. The expected numbers of H → µ + µ − signal events for each channel aresummarised in Table 1, together with the integrated luminosities based on the assumed running scenario.The fourth column introduces the abbreviations to specify the combination of √ s , beam polarisation,and signal production process used throughout this paper. The processes e + e − → ℓ + ℓ − H → ℓ + ℓ − µ + µ − ,where ℓ denotes a lepton ( e , µ , or τ ), are not considered in this paper.The prospects for measuring the H → µ + µ − decay at linear e + e − colliders have been studied previ-ously under various conditions [12, 26–30], but all studies except Ref. [30] have been performed at acentre-of-mass energy of 1 TeV or higher. The studies in Refs. [28] and [30] are based on a mass of theHiggs boson of 120 GeV. In Ref. [30] for example, the signal significance is estimated to be 1 . σ , whichcorresponds to a precision on BF ( H → µ + µ − ) of 91%, based on 250 fb − of data at √ s =
250 GeV from3
The ILD Concept and MC Samples
Figure 2: The transverse momentum resolution for single muon events as a function of the momentumof particles, for tracks with different polar angles. The points show the resolution as obtainedfrom full simulation of the ILD detector, the lines correspond to the design goal of σ / P t = × − ⊕ × − / ( P t sin θ ) for θ = ◦ (green) and θ = ◦ (blue). Taken from Ref. [12].the analysis of the process e + e − → ZH → qq µ + µ − , assuming a Higgs mass of 120 GeV and the Sil-icon Detector (SiD) concept [12, 30] for the ILC. Our study comprehensively evaluates the measurementprecision of H → µ + µ − channel assuming the mass of the Higgs boson of 125 GeV and the runningscenario of the ILC for √ s =
250 GeV and 500 GeV.This paper is structured as follows: in Sec. 2 the ILD concept and the conditions used for producingthe Monte-Carlo (MC) data samples are briefly introduced. The details of the analysis at √ s =
250 GeVand 500 GeV are explained in Sec. 3. The impact of the transverse momentum resolution σ / P t of thecentral ILD tracking system specifically for this analysis is discussed in Sec. 4 before summarising inSec. 5. ILD [12, 22] is one of the proposed detector concepts for the ILC. It is a multi-purpose detector designedfor particle flow analysis based on the reconstruction of hadronic jets. ILD consists of a high precisionvertex detector, a time projection chamber, silicon tracking detectors, a highly granular calorimeter sys-tem and a forward detector system, all placed inside of a solenoid providing a magnetic field of 3 . σ / P t , since the invariant mass of the muon pair will be the final observable for distinguishingthe signal from the background. The goal of the ILD design for the transverse momentum resolution is σ / P t ∼ × − GeV − at high momenta in the central region of the detector [32]. This level of perform-ance ensures that the model-independent selection of e + e − → ZH events from the recoil against leptonic Z → µ + µ − decays is dominated by beam energy spread rather than by the detector effects [12]. Thisgoal is compared to the transverse momentum resolution obtained from the ILD full detector simulationin Fig. 2. The impact of transverse momentum resolution will be discussed in Sec. 4. The performanceof electromagnetic calorimeter will be important for the recovery of final state radiation photons, which,however, is not yet considered in this study. 4 Analysis
Table 2: List of MC samples used in this analysis. √ s =
250 GeV √ s =
500 GeVProcesses e + e − → f e + e − → fe + e − → f e + e − → fe + e − → fe ± γ → f e ± γ → f γγ → f γγ → fe + e − → f f H e + e − → f f H The MC samples have been generated in the context of the ILC Technical Design Report [12] withthe matrix element generator
Whizard [33] (version 1.95). Initial state radiation and the effect ofbeamstrahlung, as simulated with
GuineaPig [34] based on the beam parameters [35], are includedin the event generation.
Pythia [36] (version 6.422) is used for parton shower development, had-ronisation, and to decay short-lived particles, other than leptons. The decays of tau leptons are simu-lated by
Tauola [37–39]. The full detector simulation based on
Geant4 [40] has been performed inthe
Mokka framework [41] with the so-called
ILD_o1_v05 detector model [12]. The pile-up from γγ → low P t hadron events has been generated based on the cross-section model described in Ref. [42].These events have been passed through the same Geant4 -based detector simulation and the resultinghits were overlaid to all MC samples before the reconstruction. Events have been reconstructed using
PandoraPFA [31] in the
Marlin framework [43].To make the analysis as realistic as possible, all relevant SM processes with up to six fermions inthe final state have been included. For the ILC-TDR [12], the SM background samples are groupedwith the number of fermions in the final state. For example, the e + e − → f process comprises the SMprocesses with two fermions † in the final state, i.e. , two quarks or two leptons. Table 2 shows the list ofMC samples used in this analysis. The total number of simulated events are of the order of 10 for eachcentre-of-mass energy. For the production of these MC samples, a significant amount of CPU time wasnecessary. A new MC production of similar size in the context of the recently completed ILD InterimDesign Report [44] required about 320 CPU-years.In the qqH analysis, the process e + e − → qqH with H → µ + µ − is considered as the signal, and allother processes, including other Higgs channels, are considered as background. Similarly, in the νν H analysis, e + e − → νν H → ννµ + µ − is considered as signal, while all other processes are regarded asbackground. The analysis is structured in the same way for all channels: first, a pair of well-measured, prompt, op-positely charged muons consistent with H → µ + µ − is selected by a series of sequential cuts described inSec 3.1. These cuts are “common cuts” for all analysis channels since they only pertain to the propertiesof the H → µ + µ − signal-candidates. Then, the rest of the event is subjected to channel-specific eventreconstruction and event selection as detailed in Sec. 3.2 to 3.4. To perform the final event selection, amultivariate analysis technique is used, as described in Sec. 3.5. Finally, BF ( H → µ + µ − ) is extractedfrom a template fit to the invariant di-muon mass distributions in each channel. A toy MC technique isapplied to estimate the final precision. This technique and its results will be discussed in Sec. 3.6 and 3.7. † In this context, fermions and anti-fermions are counted as fermions. Analysis
Table 3: Requirements for selecting H → µ + µ − candidate in the IsolatedLeptonTagging pro-cessor. The definition of variables is in the text.variables qqH250-L/R nnH250-L/R qqH500-L/R nnH500-L/R E CAL / p track < . < . < . p track (GeV) > > E yoke (GeV) > . | d / σ ( d ) | < | z / σ ( z ) | < > . H → µ + µ − candidate. The definition of variables is in the text. χ / Ndf ( µ ± ) . − . | d ( µ ± ) | (mm) < . | z ( µ − ) − z ( µ + ) | (mm) < . σ ( M µ + µ − ) (GeV) < . < M µ + µ − (GeV) 100 − θ µ + µ − < − . < . H → µ + µ − Candidates
First, the
IsolatedLeptonTagging processor [45] is applied to select the H → µ + µ − candidates.The criteria required for isolated muon candidates are listed in Table 3. Here, E CAL is the total energydeposit in the calorimeter system (apart from the BeamCal), p track is the momentum of the track, E yoke isthe energy deposit in the yoke, d and z are the impact parameters in transverse and longitudinal direc-tions [46], respectively, with their uncertainties σ ( d ) and σ ( z ) as obtained for each individual track fit.A multivariate double-cone method is used to check the isolation of each particle, and a cut on the MVAoutput is applied. In most cases, the default values of the IsolatedLeptonTagging processor areused for isolated muon identification. In the case of the nnh250-L/R channels, the signal rate is ratherlow to start with, while the events hardly contain any other particles than the muons. Therefore, someof the criteria have been relaxed for these channels. The particles passed all the requirements listed inTable 3 are considered as isolated muon candidates. Only events that have exactly one µ + and one µ − are considered for further analysis.The “common cuts” applied to the H → µ + µ − candidates are summarised in Table 4. They have beenchosen by maximising efficiency times purity. Here, χ / Ndf ( µ ± ) is the reduced χ of the muon trackfit, σ ( M µ + µ − ) is the event-by-event mass uncertainty as obtained from error propagation from the trackparameter uncertainties, M µ + µ − is the invariant mass of H → µ + µ − candidate, and θ µ + µ − is the anglebetween the µ + and the µ − . Cut χ / Ndf) serves to select well-measured tracks, followed by twocuts ( d and z ) which ensure prompt tracks and reject muons likely to originate from τ decays. Thecut on σ ( M µ + µ − ) , cut M µ + µ − ensures that the invariant mass of the candidate is well above M Z , while not removing any di-muons witha mass close to M H . Fig. 3 shows the distribution of M µ + µ − before applying cut θ µ + µ − ) requires the di-muons to have a minimum openingangle which is defined by the boost of the produced Higgs boson and thus depends significantly on thecentre-of-mass energy. 6 Analysis (GeV) - µ + µ M E v en t s / G e V All µµ qqH, H-> µµ H/llH, H-> νν ffH, H->other2f µµ
4f qq µ -> τ , ττ
4f qq4f other->2f γγ ->3f γ e ILD simulation
Figure 3: The distribution of M µ + µ − before applying cut Some events include energetic initial state radiation (ISR) photons which, if within the detector accept-ance, will affect the further analysis. Therefore a simple ISR identification procedure is applied afterthe muon identification. First, a candidate photon is selected if its energy E photon is greater than 10 GeV.All charged particle energies in a cone with half-opening angle cos θ cone = .
95 around the photon aresummed up. If this energy sum is less than 5% of the photon energy, the photon is regarded as an ISRphoton. These ISR photons are not subject to jet reconstruction.
For the qqH channels, a jet clustering algorithm is applied to reconstruct Z → qq candidates. Afterthe selection of H → µ + µ − and a possible ISR photon, one can expect that the remaining particlesconsist of Z → qq and some contribution from the overlaid γγ → low P t events. At √ s =
250 GeV, only0 . γγ → low P t hadron events are expected per bunch crossing on average [42]. Thus, no dedicatedattempt is made to remove the overlay and the Durham clustering algorithm [47] is used to force theremaining particles into two jets. However at √ s =
500 GeV, the average number of γγ → low P t hadronevents per bunch crossing increases to 1 . k T clusteringalgorithm [48, 49] is applied to the remaining particles, requesting four jets with a generalised jet radiusof 1.0. The jet radius has been tuned to optimise the reconstruction of the invariant mass spectrum of the Z → qq system. The clustering into four jets has been proven to render the overlay-removal step morerobust in the presence of hard gluon emission [50]. After this process, the Durham algorithm [47] is usedto force the particles contained in the four k T -jets into two final jets. The H → µ + µ − candidates andISR photons are not included in jet clustering. After the general preselection described in Sec. 3.1, channel-specific cuts are applied. Table 5 summar-ises the cuts applied in the qqH channels. Cut H → µ + µ − candidate are expected in the event. For this cut, the IsolatedLeptonTagging processor [45] is applied again to the remaining particles and requires that no isolated leptons (electronsor muons) are found. The cut on the number of charged particles in each jet (
Analysis
Table 5: The channel-specific cuts for the qqH processes. The definition of variables is in the text. µ + µ − pair 02 jet clustering successful yes3 number of charged particles in each jet ≥ M j j (GeV) 50 - 130 50 - 160Table 6: The channel-specific cuts for νν H processes. The definition of variables is in the text. N P t E vis (GeV) 120 - 170 120 - 3503 missing P t (GeV) > | cos θ miss | < . M j j of two jets consistent with Z → qq . Since the di-jetmass resolution is somewhat worse at √ s =
500 GeV, the allowed mass window is wider than in the √ s =
250 GeV case.The channel-specific cuts for the νν H channel are summarised in Table 6. Apart from the H → µ + µ − candidate, there should be no visible particles in the event except for some contribution from γγ → low P t hadron backgrounds which typically have low transverse momentum. Therefore, the number of chargedparticles in an event, excluding the H → µ + µ − candidate, which have a transverse momentum larger than5 GeV, N P t , has to be zero (cut E vis and P t are the energy and the transverse momentum,respectively. The visible four-momentum can be subtracted from the four-momentum of the initial state, ( √ s , √ s · tan θ cross / , , ) where θ cross =
14 mrad is the crossing angle of beam collision, to obtain themissing four-momentum, and in particular its polar angle, θ miss . The cuts E vis , missing P t , andcos θ miss ) use these quantities to select events with neutrinos. The E vis requirement thereby depends onthe centre-of-mass energy.Table 7 shows the number of signal and background events in each channel after the preselection.Overall, the signal selection efficiency is ∼
85% in all channels. The “other Higgs” category includesall events with Higgs bosons with other decay modes than the signal. The category of “irreducible”backgrounds is defined as follows: for the qqH process, e + e − → f → qq µ + µ − and qq τ + τ − with bothtau leptons decaying into µ are defined as irreducible. For the νν H process, e + e − → f → ννµ + µ − , ννµτ with τ decaying into µ , and νντ + τ − with both τ decaying into µ are defined as irreducible. Theseevents have the same or very similar final states to the signal, thus they are difficult to remove. After thepreselection, the irreducible category dominates the background in nearly all analysis channels. In caseof the qqH500-L/R channels, though, the e + e − → f processes (dominated by e + e − → tt ) remain at thesame level as the irreducible background. For the further rejection of background, in particular the “irreducible” one, a multivariate analysis is per-formed based on the gradient boosted decision tree method (BDTG) implemented in the TMVA packagein ROOT [51, 52]. Typically, ∼ MC events remain after the preselection for signal and background,each. In all channels, half of the remaining events after the channel-specific preselection are used fortraining and the other half for testing. The variable M µ + µ − is not used in the BDTG since it will be8 Analysis
Table 7: The number of signal and background events in each channel after the channel-specific cuts,weighted to the beam polarisation and luminosity settings given in Table 1. The definition of“irreducible” is in the text. Numbers in brackets show the signal selection efficiency.channel signal other “irreducible” other SMHiggs backgroundqqH250-L 36 (88%) 143 3 . × . × . × . × . × . × qqH500-R 14 (82%) 64.5 1 . × . × nnH500-L 49 (86%) 4.4 9 . × . × M j j , cos θ j j , E µ + µ − , cos θ µ + µ − , cos θ µ + − cos θ µ − , E sub , cos θ sub nnH250-L/R E vis , E µ + µ − , cos θ µ + − cos θ µ − , M recoil , E sub , cos θ sub qqH500-L/R M j j , cos θ j j , P t , µ + µ − , cos θ µ + µ − , cos θ µ + − cos θ µ − , M recoil , E lead , E sub , cos θ lead , cos θ sub nnH500-L/R E vis , cos θ thrustaxis , E µ + µ − , cos θ µ + µ − , cos θ µ + − cos θ µ − , E lead , E sub , cos θ lead , cos θ sub used later in further analysis. The input variables for the BDTG are summarised in Table 8 for all thechannels. Here, θ j j is the angle between two jets, E µ + µ − is the energy sum of the H → µ + µ − candidate, θ µ + ( µ − ) is the polar angle of µ + ( µ − ) , E lead ( E sub ) is the energy of the higher energy (lower energy) muonof the H → µ + µ − candidate, θ lead ( θ sub ) is the polar angle of the higher energy (lower energy) muonof the H → µ + µ − candidate, M recoil is the recoil mass against the H → µ + µ − candidate (corrected forreconstructed ISR photons), P t , µ + µ − is the transverse momentum of the H → µ + µ − candidate system,and θ thrustaxis is the polar angle of the thrust axis of the visible part of the event. The variable E µ + µ − isused in nnH500-L but not in nnH500-R. For each channel, the minimum number of relevant inputs hasbeen chosen, considering the minimisation of correlations and avoiding overtraining, in particular giventhe finite amount of available MC events. Fig. 4 shows the seven input variables for qqH250-L, whilethe corresponding distribution of the BDTG score for the signal and background events is displayed inFig. 5.For each channel, the final cut value on the BDTG score is chosen such that it optimises the expectedprecision on BF ( H → µ + µ − ) as described in the following section. After applying a cut on the BDTG score, the signal strength is extracted from a template fit to theinvariant di-muon mass M µ + µ − distribution for signal and background with the signal normalisation as a9 Analysis
60 80 100 120 (GeV) jj M N o r m a li z ed / G e V µµ qqH, H-> µµ
4f qqother
ILD simulation (a) M j j − − jj θ cos N o r m a li z ed / . µµ qqH, H-> µµ
4f qqother
ILD simulation (b) cos θ j j
100 120 140 160 (GeV) - µ + µ E N o r m a li z ed / G e V µµ qqH, H-> µµ
4f qqother
ILD simulation (c) E µ + µ − − − − − − − − - µ + µ θ cos µµ qqH, H-> µµ
4f qqother
ILD simulation (d) cos θ µ + µ − − − - µ θ -cos + µ θ cos N o r m a li z ed / . µµ qqH, H-> µµ
4f qqother
ILD simulation (e) cos θ µ + − cos θ µ − (GeV) sub E N o r m a li z ed / G e V µµ qqH, H-> µµ
4f qqother
ILD simulation (f) E sub − − sub θ cos N o r m a li z ed / . µµ qqH, H-> µµ
4f qqother
ILD simulation (g) cos θ sub Figure 4: Example input variables for the BDTG in qqH250-L. All histograms are normalised to anintegral of 1. The signal process is shown in blue, the irreducible backgrounds are contained inthe green histograms, while other backgrounds are shown in magenta.10
Analysis
BDTG score − − N o r m a li z ed / . signalbackground ILD simulation
Figure 5: The distribution of BDTG score (qqH250-L). Both histograms are normalised to an integral of1.free parameter. The final precision on the branching fraction is estimated via a toy MC technique.First, the modeling functions for the M µ + µ − distributions of signal and background have to be defined.These functions are fitted to the M µ + µ − distributions for signal and background as obtained from the fullsimulation analysis described in the previous subsections.Due to the excellent mass reconstruction, the whole template fit is restricted to the range 120 GeV < M µ + µ − <
130 GeV. For the signal, a linear sum of a Crystal Ball function (CB) [53] and a Gaussian, f S ≡ k × CB + ( − k ) × Gaussian ( with 0 < k < ) , (1)is used as modeling function f S . This empirical function models sufficiently well the combined effectof final state radiation photons, which create a tail in the M µ + µ − distribution of the signal process, aswell as effects of the finite detector resolution. It should be noted that no attempt has been made torecover final state radiation photons because the measuring accuracy of the electromagnetic calorimeteris not good enough to improve the mass resolution of events with recovered photons. As we will seein Sec. 4, an excellent invariant mass resolution is a core ingredient to the final performance of theanalysis, and thus recovery of final state radiation is not considered worthwhile in this case. For thesignal modeling, an unbinned fit is performed to avoid effects of the bin width which, due to finite MCstatistics, cannot always be small compared to the width of the mass peak, especially when consideringdifferent P t resolutions in Sec. 4. The Higgs mass itself is assumed to be known very precisely, to about14 MeV, from the recoil analysis [20]. Therefore, the mean value of the CB is fixed to the nominal Higgsmass of 125 GeV in this study. Figure 6(a) illustrates the modeling of the signal M µ + µ − distribution usingthe nnH500-L channel as example. In this example, the parameter k in Eq. (1) is 0.92. The width of thepeak at half its maximum height (FWHM) is 0.23 GeV.The background is modeled by a straight line f B in all channels. An example is given in Fig. 6(b),again based on the nnH500-L channel.The fitted f S and f B are then used as probability density functions for the generation of 2 × pseudo-data sets via a toy MC technique based on RooFit [52, 54]. In each pseudo-experiment, the number ofpseudo-signal(-background) events is drawn from a Poisson distribution with the estimated average num-ber of signal(background) events after all cuts as expectation value. Then, an unbinned fit of the function f ≡ Y S f S + Y B f B to the sum of pseudo-data is performed, where Y S ( Y B ) is the yield of signal(background)events. Thereby, Y B is fixed to the expected average number of backgrounds after all cuts, assumingthat by the time the ILC has collected its full data set, the SM background at a lepton collider can be11 Analysis (GeV) - µ + µ M
120 122 124 126 128 130 E v en t s / . G e V ILD simulation (a) signal with result of f S fit (GeV) - µ + µ M
120 122 124 126 128 130 E v en t s / . G e V ILD simulation (b) background with result of f B fit Figure 6: M µ + µ − distributions for signal and background after all cuts in the nnH500-L channel. Theresult of the f S and f B fits, respectively, is shown as red curves.predicted much more precisely than the statistical uncertainty for rare signal events. Thus, Y S is the onlyfree parameter in the template fit. Fig. 7 shows an example of one pseudo-experiment in the nnH500-Lchannel. The final Y S distribution from 2 × pseudo-experiments is fitted by a Gaussian to extract itsmean and width as shown in Fig. 7(b). The expected relative precision on BF ( H → µ + µ − ) is calculatedas the width of the fitted Gaussian divided by the mean of the fitted Gaussian, which in all channelsagrees with the mean number of signal events expected from the full simulation listed in Table 9.The cut values for the BDTG score have been optimised for each analysis channel by applying the toyMC procedure described above for different values of the cut and selecting the one which gives the bestmeasurement precision. Figs. 6 and 7 correspond to the optimal BDTG score cut case in nnH500-L. Table 9 shows the number of signal and background events in each channel after the optimisation ofthe BDTG score cut. The signal efficiency ranges between 45% and 72%, with an overall average of53%. A notable exception is the nnH250-L channel, which gives an optimal result for a very hard cuton the BDTG score and as a result has a rather low efficiency of only 28%, while the background is stillhigher than in its sister channel nnH250-R. This is an effect of the W + W − contribution to the irreduciblebackground, which has a much higher cross-section in the left-handed polarisation configuration, whilethe corresponding increase for the signal is much smaller since it is — at this energy — dominated by ZH production. At 500 GeV, the effect on the background is even more drastic, but since the now WW -fusion dominated signal profits in the same way from the polarisation, there is no need to optimise foran extremely hard cut on the BDTG score. In all cases, the total background count is strongly dominatedby the irreducible component. This implies that the misidentification of the final-state particles is nota limiting factor in the analysis. In the future it could be investigated, however, whether the eventkinematics could be exploited in a more efficient way to suppress the irreducible component, as will bediscussed in more detail below.The expected precisions on BF ( H → µ + µ − ) obtained in the eight channels are summarised in Table 10.With the √ s =
250 GeV data alone, a precision of 23% can be reached, dominated by the q ¯ qH channels.12 Analysis (GeV) - µ + µ M
120 122 124 126 128 130 E v en t s / . G e V ILD simulation (a) example of one pseudo-experiment S Y − E n t r i e s / ILD simulation (b) distribution of yield of signal events Y S from2 × pseudo-experiments. Figure 7: Signal strength extraction in the nnH500-L channel. (a) Example outcome of one pseudo-experiment, with the pseudo-data shown as black points with error bars, while the solid redcurve shows the result of an unbinned fit of f ≡ Y S f S + Y B f B to the pseudo-data. The dashedred line shows its background component Y B f B . (b) The distribution of the yield of signalevents Y S obtained from 2 × pseudo-experiments, fitted with a Gaussian function. Themean value of the distribution is Y S = . > .
45 29 (72%) 0.1 600 4qqH250-R > .
85 18 (64%) 0 193 3nnH250-L > .
95 4.2 (28%) 0 155 12nnH250-R > .
80 3.7 (45%) 0 105 11qqH500-L > .
60 13 (54%) 4.2 114 9qqH500-R > .
25 10 (61%) 9.6 71 7nnH500-L > .
50 31 (54%) 0 745 48nnH500-R > .
40 3.6 (45%) 0 75 1The √ s =
500 GeV data alone reach 24%, but now the νν H channel in the left-handed data set is themost sensitive. A combination of all data sets improves the expected precision to 17%.These numbers demonstrate a significant improvement with respect to earlier analyses. An extrapol-ation of the result reported by SiD [30] to a luminosity of 0.9 ab − , which corresponds to the size ofthe left-handed data set at √ s =
250 GeV in the present study, yields a precision on BF ( H → µ + µ − ) of ∼ Analysis
Table 10: Expected precisions on BF ( H → µ + µ − ) for √ s =
250 GeV (ILC250), √ s =
500 GeV(ILC500) and their combination (ILC250+500). The luminosities and polarisation sharingcorrespond to the standard ILC running scenario as detailed in Sec. 1. √ s =
250 GeV qqH νν H ILC250 ILC250+500L 34% 113% 23% 17%R 36% 111% √ s =
500 GeV qqH νν H ILC500L 43% 37% 24%R 48% 106%is of relevance here, the ILD momentum resolution is considerably better than that of SiD: in the caseof SiD [12], σ / P t ranges between 3 × − GeV − for P =
100 GeV at θ = ◦ to 2 × − GeV − for P =
40 GeV at θ = ◦ , while in the ILD case the relevant numbers range from 2 × − GeV − for P =
100 GeV at θ = ◦ to 5 × − GeV − for P =
40 GeV at θ = ◦ , as shown in Fig. 2.The combined result of the present study is about 50% larger than the most recent projections for theHL-LHC based on the tracker upgrades for ATLAS and CMS introduced in Sec. 1. Taking into accountthat HL-LHC will provide O ( ) H → µ + µ − events with the full expected integrated luminosity of3 ab − , and thus about 100 times more signal events than ILC with 2 ab − at √ s =
250 GeV and 4 ab − at √ s =
500 GeV together, a difference of only 50% shows the highly efficient use of data possibleat an e + e − collider. In addition, the results of the analysis at ILC1000 presented in Ref. [27] can beextrapolated to the full luminosity of 8 ab − expected at ILC1000 [23] to yield a precision of 14% onBF ( H → µ + µ − ) . Combining this result with our analysis at 250 GeV and 500 GeV yields a precisionof 11%. Thus, from a combination of HL-LHC with the full ILC program a precision of 7% could beexpected, without taking into consideration possible improvements of the analysis.For a better understanding of analysis limitations, one can compare these results with the “theoreticallimit” case, i.e. assuming 100% signal selection efficiency and no backgrounds. In this hypothetical case,the precision would reach 10.4% for ILC250, and 7.1% for the full ILC250+500 data set. The resultspresently achieved in full detector simulation are about a factor of 2 . • If only the signal efficiencies as given in Table 9 are considered, the combined precision at ILC250would be 13.4% and would improve to 9.4% when combined with the 500 GeV data, which is afactor of ∼ . ∼
10% ofthe signal events are lost. In about 4% of the events, the muons are not found, and about 2 .
5% ofevents are lost due to the d and invariant mass requirements, each, c.f.; Table 11. The invariantmass requirement mostly fails due to the presence of FSR, which is not recovered, c.f. discussionin Sec. 3.6. During the rest of the preselection, only a few additional percents are lost, while a ∼
15% reduction occurs via the BDTG score cut. In total, it seems hard to increase the overallsignal efficiency drastically, but some improvement could be achieved by exploiting the variablesdiscussed in the next item. • The irreducible background almost entirely consists of processes with the same final state as thesignal process: e + e − → qq µ + µ − for the qqH process and e + e − → ννµ + µ − for the νν H process,originating from ZZ as well as, in the case of ννµ + µ − , from W + W − production and single- Z / γ ra-diation off a t -channel W . Future upgrades of the analyses could attack these kinds of backgroundsby even better exploitation of all kinematic information, e.g. by testing various intermediate bosonhypotheses in a kinematic fit [55] and/or by evaluating the matrix element probabilities for the14 Impact of the Transverse Momentum Resolution
Table 11: Detailed cut flow of the qqH250-L channel. The numbers in brackets show the signal selectionefficiency. signal other “irreducible” other SMHiggs backgroundno cut 41 (100%) 2 . × . × . × µ ± = . × . × . × common cuts(see Table 4) . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × preselection(see Table 5) . × . × . × . × . × . × . × . × . × . × . × . × BDTG score 30 (73%) 0.2 687 8.5 M µ + µ − >
120 GeV 29 (72%) 0.1 600 4.4signal and various background hypotheses on an event-by-event basis [56]. There are also somebackground events with tau leptons such as νντµ with τ decaying to µ , but this contribution isnegligible compared to the ones with exactly the signal final state. • Last but not least, the di-muon invariant mass resolution is an important ingredient – as soon asthe number of background events at the end of the selection is larger than zero: the sharper thereconstructed Higgs mass peak, the lower the “effective” amount of background under the peak.The invariant mass resolution is dominated by the precision achieved on the transverse momentumof the two muons, while the angular resolutions play a negligible role. Therefore, we will studythe impact of the (inverse) transverse momentum resolution σ / P t in Sec. 4.Finally, it should be noted that in the νν H process, especially at √ s =
500 GeV, two signal processes( ZH process with Z → νν and WW -fusion process) are contributing. The relative contributions of theseproduction modes will be fixed to the percent-level or better from other Higgs decay modes like H → bb and can be used to convert the cross section times branching fraction measurement into a measurementof BF ( H → µ + µ − ) . With the help of the total ZH cross section determined with the recoil method, theabsolute H µµ Yukawa coupling can be extracted. Therefore, the quoted ILC precisions can directly betaken as precision on the branching fraction for H → µ + µ − , or, divided by a factor of two, as precisionon the muon Yukawa coupling. This is qualitatively different from the signal strength determinations atthe (HL-)LHC. The di-muon mass M µ + µ − is the most important observable for this analysis. The uncertainty on M µ + µ − is directly related to the precision of the measurement of the muon momentum, and in particular theresolution on its transverse component, σ / P t , plays a crucial role in this analysis. The transverse mo-mentum resolution of the ILD detector has been shown already in Fig. 2 as a function of the momentumfor different polar angles.Instead of implementing the full p and θ dependency of the resolution, a simplified approach ofsmearing all true muon transverse momenta with the same resolution has been taken here. This is a fully15 Impact of the Transverse Momentum Resolution − − − − ) -1 (GeV t σ B F / B F ( % ) ∆ ILD simulation full (ILC250)smeared (ILC250) theory (10.4%)theory + sig. eff. (13.4%)
Figure 8: Expected precision on BF ( H → µ + µ − ) as a function of transverse momentum resolution σ / P t (triangles), together with full simulation results discussed in Sec. 3.7 (red line) and the theor-etical limits defined in Sec. 3.7 (dashed lines) for 2 ab − collected at √ s =
250 GeV. The redline indicates the typical transverse momentum resolution range at √ s =
250 GeV.justified approach in case of √ s =
500 GeV, since the vast majority of muons have high momenta in theasymptotic regime, and, due to the isotropic decays of the Higgs boson, are mostly at large polar anglesin the centre of the detector. For the case of √ s =
250 GeV, the muons have lower momenta between 40and 100 GeV and the approximation is less precise, but still useful.The dependence of the result on the asymptotic value of the transverse momentum resolution has beenstudied by adding a Gaussian-distributed error to the transverse momentum taken from the MC-truthinformation for all events passing the preselection described in Sec. 3.4. All other quantities in the eventare taken from the full simulation as before. Transverse momentum resolutions between 1 × − to1 × − (GeV − ) have been considered. The background is kept unchanged from the full simulationstudy since its invariant mass distribution after the BDTG score cut does not exhibit any sharp peaks, ascan be seen, e.g. : in Fig. 6, and thus a change in momentum resolution will not affect the distributionsignificantly.Fig. 8 shows the obtained precision on BF ( H → µ + µ − ) as a function of the transverse momentum res-olution σ / P t at √ s =
250 GeV, together with the theoretical limit as defined in Sec. 3.7 shown by dashedlines. The red line indicates the typical value of transverse momentum resolution at √ s =
250 GeV. The“effective” resolution for which the smearing approach gives the same precision on the branching frac-tion as the full simulation is ∼ × − . This result is consistent with Fig. 2, because at this energymuons typically have momenta in the regime of 40 to 100 GeV which corresponds to a resolution ofaround ∼ × − .The following conclusion can be drawn. First of all, with σ / P t = × − GeV − for example, typicalfor LHC experiments, precision would be 36% instead of 23%, i.e. bigger by a factor of 1.6. Therefore,it is very important for this analysis to reach the ILD goal for the transverse momentum resolution. In theother direction, though technologically not realistic, an improvement of σ / P t to a few times 10 − GeV − would allow to nearly reach the “zero-background” scenario, in the sense that although the same amountof (irreducible) background events pass the selection, the Higgs signal peak becomes so narrow that thebackground contribution underneath the peak doesn’t have a significant effect anymore.16 Summary − − − − ) -1 (GeV t σ B F / B F ( % ) ∆ ILD simulation full (ILC500)smeared (ILC500) theory (9.7%)theory + sig. eff. (13.2%)
Figure 9: Expected precision on BF ( H → µ + µ − ) as a function of transverse momentum resolution σ / P t (triangles), together with full simulation results discussed in Sec. 3.7 (star) and the theoreticallimits defined in Sec. 3.7 (dashed lines) for 4 ab − collected at √ s =
500 GeV.Fig. 9 shows the equivalent result at √ s =
500 GeV, while the combined result of both centre-of-massenergies is displayed in Fig. 10. Since at √ s =
500 GeV the momenta of the muons are higher than inthe √ s =
250 GeV case, the transverse momentum resolution is closer to the asymptotic performanceof 2 × − , and thus the “effective” resolution gets closer to the case of 2 × − . Otherwise, theconclusions remain similar to the √ s =
250 GeV case, underlining again the importance to achieve theILD design goal on the transverse momentum resolution.A similar study has also been performed by the Compact LInear Collider (CLIC) [29], based on e + e − → νν H at √ s = . qqh contribution is negligible at 1 . σ / P t = × − , where the precision on BF ( H → µ + µ − ) reaches ∼ σ / P t resolutions compared to the ILD case, the CLICstudy leads to the same conclusion as our analysis, namely that even a large improvement of the muonmomentum resolution would result in only a moderate improvement of the statistical uncertainty of themeasured product of the Higgs production cross-section and the branching fraction for the H → µ + µ − decay. On the other hand, not reaching the design goal for the momentum resolution would lead to asignificant loss of sensitivity. In this study, the prospects for measuring the branching fraction of H → µ + µ − at the ILC have beenevaluated based on full simulation of the ILD detector for the √ s =
250 GeV and 500 GeV data setsas defined by the standard ILC running scenario. Eight channels have been analysed in total, √ s of250 GeV and 500 GeV, two beam polarisation cases, and the two signal processes qqH and νν H . Thecombined precision on BF ( H → µ + µ − ) using all results is estimated to be 17%; the 250 GeV data aloneyield a precision of 23%. These results are about a factor of 2.4 bigger than the “theoretical” limit of zerobackground and 100% efficiency. Compared to most recent HL-LHC prospects based on the full detectorupgrades, the precision is only about 50% larger, despite the fact that 100 times more H → µ + µ − eventsare expected to be produced at HL-LHC. In combination with other ILC measurements, the observed17 eferences − − − − ) -1 (GeV t σ B F / B F ( % ) ∆ ILD simulation full (ILC250+500)smeared (ILC250+500) theory (7.1%)theory + sig. eff. (9.4%)
Figure 10: Expected precision on BF ( H → µ + µ − ) as a function of transverse momentum resolution σ / P t (triangles), together with full simulation results discussed in Sec. 3.7 (star) and the the-oretical limits defined in Sec. 3.7 (dashed lines) for the combination of the 2 ab − collected at √ s =
250 GeV and 4 ab − collected at √ s =
500 GeV data sets.signal strength can be translated into a direct measurement of the branching fraction and thus the muonYukawa coupling. The combination of HL-LHC and ILC full program up to 1 TeV would provide anultimate precision of ∼
7% on BF( H → µ + µ − ). In addition to the full simulation analysis, the impactof the transverse momentum resolution was studied. This study shows the importance to achieve theILD design goal of the transverse momentum resolution, otherwise the precision will be significantlydegraded. The first evaluation of the prospects to measure BF ( H → µ + µ − ) at the lower-energy stages ofthe ILC presented in this paper could be improved in future analyses. Interesting points to study comprisethe application of beam-spot constraint in the track fit of the two muons, a full treatment of events withsignificant FSR and the inclusion of the Z boson decays to charged leptons. Acknowledgements
We would like to thank the LCC generator working group and the ILD software working group forproviding the simulation and reconstruction tools and producing the Monte Carlo samples used in thisstudy. SK would like to thank Junping Tian (The University of Tokyo) for lots of useful comments anddiscussing technical details in the analysis. This work has benefited from computing services providedby the ILC Virtual Organization, supported by the national resource providers of the EGI Federationand the Open Science GRID. We thankfully acknowledge the support by the the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2121“Quantum Universe” 390833306.
References [1] ATLAS Collaboration,
Observation of a new particle in the search for the Standard Model Higgsboson with the ATLAS detector at the LHC , Phys. Lett.
B716 (2012) 1.18 eferences [2] CMS Collaboration,
Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC ,Phys. Lett.
B716 (2012) 30.[3] The ATLAS Collaboration,
Observation of H → bb decays and V H production with the ATLAS detector ,Phys. Lett. B786 (2018) 59.[4] CMS Collaboration,
Observation of Higgs Boson Decay to Bottom Quarks ,Phys. Rev. Lett. (2018) 121801.[5] The ATLAS Collaboration,
Observation of Higgs boson production in association with a topquark pair at the LHC with the ATLAS detector , Phys. Lett.
B784 (2018) 173.[6] CMS Collaboration,
Observation of ttH Production , Phys. Rev. Lett. (2018) 231801.[7] Rick S. Gupta, Heidi Rzehak, James D. Wells,
How well do we need to measure Higgs boson couplings? , Phys. Rev. D (2012) 095001.[8] Ties Behnke et al. , The International Linear Collider Technical Design Report - Volume 1: Executive Summary (2013), arXiv: .[9] Howard Baer et al. , The International Linear Collider Technical Design Report - Volume 2: Physics (2013),arXiv: .[10] Chris Adolphsen et al. , The International Linear Collider Technical Design Report - Volume 3.I:Accelerator R&D in the Technical Design Phase (2013),arXiv: .[11] Chris Adolphsen et al. , The International Linear Collider Technical Design Report - Volume 3.II:Accelerator Baseline Design (2013), arXiv: .[12] Ties Behnke et al. , The International Linear Collider Technical Design Report - Volume 4: Detectors (2013),arXiv: .[13] Philip Bambade et al. , The International Linear Collider A Global Project (2019),arXiv: .[14] Tim Barklow et al. , Improved formalism for precision Higgs coupling fits ,Phys. Rev. D (2018) 053003.[15] LHC Higgs Cross Section Working Group, Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector (2016),arXiv: .[16] The ATLAS Collaboration,
A search for the dimuon decay of the Standard Model Higgs boson with the ATLAS detector (2020), arXiv: .[17] The CMS Collaboration,
Measurement of Higgs boson decay to a pair of muons in proton-protoncollisions at √ s =
13 TeV , tech. rep. CMS PAS HIG-19-006, 2020.[18] The ATLAS Collaboration,
Prospects for the measurement of the rare Higgs boson decayH → µµ with 3000 fb − of pp collisions collected at √ s = TeV by the ATLAS experiment ,tech. rep. ATL-PHYS-PUB-2018-006, Geneva: CERN, 2018.[19] J. Bulter et al. , Technical Proposal for the Phase-II Upgrade of the Compact Muon Solenoid ,tech. rep. CERN-LHCC-2015-010. LHCC-P-008. CMS-TDR-15-02, Geneva, 2015.19 eferences [20] J. Yan et al. , Measurement of the Higgs boson mass and e + e − → ZH cross section usingZ → µ + µ − and Z → e + e − at the ILC , Phys. Rev. D (2016) 113002.[21] M. A. Thomson, Model-independent measurement of the e + e − → HZ cross section at a futuree + e − linear collider using hadronic Z decays , Eur. Phys. J. C76
The ILD detector at the ILC (2019),arXiv: .[23] ILC Parameters Joint Working Group,
ILC Operating Scenarios (2015),arXiv: .[24] LCC Physics Working Group,
Physics Case for the 250 GeV Stage of the International Linear Collider (2017),arXiv: .[25] LCC Physics Working Group,
Physics Case for the International Linear Collider (2015),arXiv: .[26] Constantino Calancha,
Study of H → µ + µ − at √ s = , LC-REP-2013-006, 2013.[27] Michele Faucci Giannelli, Sara Celani, Higgs decay to two muons at ILC ,Proceedings, International Workshop on Future Linear Colliders (LCWS2015): Whistler, B.C.,Canada, November 02-06, 2015, 2016, arXiv: .[28] Christian Grefe, Tomáš Laštoviˇcka, Jan Strube,
Prospects for the measurement of the HiggsYukawa couplings to b and c quarks, and muons at CLIC , Eur. Phys. J.
C73 et al. , Physics potential for the measurement of σ ( H νν ) × BR ( H → µ + µ − ) at the 1.4 TeV CLIC collider , Eur. Phys. J. C75
11 (2015) 515.[30] H. Aihara et al. , SiD Letter of Intent (2009), arXiv: .[31] M. A. Thomson,
Particle flow calorimetry and the PandoraPFA algorithm ,Nucl. Instrum. Meth.
A611
Physics Impact of Detector Performance , Talk presented at LCWS05, 2005.[33] Wolfgang Kilian, Thorsten Ohl, Jürgen Reuter,
WHIZARD—simulating multi-particle processes at LHC and ILC ,Eur. Phys. J.
C71
Study of Electromagnetic and Hadronic Background in the Interaction Region ofthe TESLA Collider , DESY-TESLA-97-08, 1997.[35]
URL : http://ilc-edmsdirect.desy.de/ilc-edmsdirect/item.jsp?edmsid=D00000000925325 .[36] Torbjörn Sjöstrand, Stephen Mrenna, Peter Skands, PYTHIA 6.4 physics and manual ,JHEP
05 (2006) 026.[37] Stanisław Jadach, Johann H. Kühn, Zbigniew Wa¸s,
TAUOLA - a library of Monte Carlo programs to simulate decays of polarized τ leptons ,Comput. Phys. Commun. et al. , The tauola-photos-F environment for the TAUOLA and PHOTOS packages, release II ,Comput. Phys. Commun.
10 (2006) 818.[39] N. Davidson et al. , Universal interface of TAUOLA: Technical and physics documentation ,Comput. Phys. Commun.
Geant4—a simulation toolkit , Nucl. Instrum. Meth.
A506 eferences [41] P. Mora de Freitas, H. Videau,
Detector simulation with MOKKA/Geant4: Present and future ,LC-TOOL-2003-010, 2003.[42] Pisin Chen, Timothy L. Barklow, Michael E. Peskin,
Hadron production in γγ collisions as a background for e + e − linear colliders ,Phys. Rev. D (1994) 3209.[43] F. Gaede, Marlin and LCCD—Software tools for the ILC ,Nucl. Instrum. Meth.
A559
International Large Detector: Interim Design Report , 2020,arXiv: .[45] Junping Tian, Claude Dürig, isolated lepton finder , URL : https://agenda.linearcollider.org/event/6787/contributions/33415/attachments/27509/41775/IsoLep_HLRec2016.pdf .[46] Thomas Krämer, Track Parameters in LCIO , LC-DET-2006-004, 2006.[47] S. Catani et al. , New clustering algorithm for multijet cross sections in e + e − annihilation ,Phys. Lett. B269 et al. , Longitudinally-invariant k ⊥ -clustering algorithms for hadron-hadron collisions ,Nucl. Phys. B406
Successive combination jet algorithm for hadron collisions ,Phys. Rev. D (1993) 3160.[50] M. Chera, Particle Flow: From First Principles to Gaugino Property Determination at the ILC ,PhD thesis, Hamburg: Hamburg U., 2018,
DOI : .[51] P Speckmayer et al. , The Toolkit for Multivariate Data Analysis, TMVA 4 ,J. Phys. Conf. Ser.
ROOT — An object oriented data analysis framework ,Nucl. Instrum. Meth.
A389
Charmonium Spectroscopy from Radiative Decays of the J / ψ and ψ ′ ,PhD thesis, SLAC, 1982.[54] W. Verkerke, D. Kirkby, The RooFit toolkit for data modeling ,eConf
C0303241 (2003) MOLT007, arXiv: physics/0306116 [physics.data-an] .[55] M. Beckmann, B. List, J. List,
Treatment of Photon Radiation in Kinematic Fits at Future e+ e- Colliders ,Nucl. Instrum. Meth.
A624 (2010) 184,
DOI : ,arXiv: .[56] G. Abbiendi et al., Two Higgs doublet model and model independent interpretation of neutral Higgs boson searches ,Eur. Phys. J.
C18 (2001) 425,
DOI : ,arXiv: hep-ex/0007040 [hep-ex]hep-ex/0007040 [hep-ex]