Class D Amplifier Matlab

Abstract

The branching fraction B(B0 →D∗−τ +ντ) is measured relative to that of the nor- malization mode B0 →D∗−π+π−π+ using hadronic τ + →π+π−π+(π0)ντ decays in proton-proton collision data at a center-of-mass energy of 13 TeV collected by the LHCb experiment, corresponding to an integrated luminosity of 2 fb−1. The measured ratio is B(B0 →D∗−τ +ντ)/B(B0 →D∗−π+π−π+) = 1.79 ± 0.11 ± 0.11,

Systematic

effects.

The

B0 →D∗−π+π−π+ and B0 →D∗−µ+νµ modes, the lepton universality test, R(D∗−) ≡B(B0 →D∗−τ +ντ)/B(B0 →D∗−µ+νµ) is calculated,

R(D∗−) = 0.260 ± 0.015 ± 0.016 ± 0.012 ,

where the third uncertainty is due to the uncertainties on the external branching fractions. This result is consistent with the Standard Model prediction and with previous measurements.

Published In Phys. Rev. D 108 (2023) 1, 012018

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence. †Full author list given at the end of the paper.

Introduction

Measurements of R(D(∗)) ≡B(B0 →D(∗)τ +ντ)/B(B0 →D(∗)ℓ+νℓ), the ratio of branching fractions with ℓ= µ, e, test lepton flavor universality in b →cℓνℓtransitions. First measured by the BaBar Collaboration in 2012 , this ratio has been studied by the Belle and LHCb experiments using hadronic and muonic decay modes of the τ + lepton1. After the latest LHCb Collaboration result using muonic τ + decays , the discrepancy between the world-average values of R(D∗) and R(D) measurements with their theoretical prediction is at the level of 3 standard deviations . The predicted value by the Standard Model (SM) of particle physics is R(D∗) = 0.254 ± 0.005 . Several extensions to the SM can explain this anomaly, e.g., leptoquark models , which typically assume a leptoquark that preferentially couples to third-generation leptons and has a mass below 1 TeV/c2.

The LHCb hadronic R(D∗−) analysis was first performed using proton-proton (pp) collision data collected at center-of-mass energies of √s = 7 and 8 TeV in 2011 and 2012 , corresponding to an integrated luminosity of 3 fb−1. This paper presents a similar measurement of R(D∗−) based on pp collision data taken at 13 TeV in 2015 and 2016, corresponding to an integrated luminosity of 2 fb−1. Despite the lower integrated luminosity, the increase of the bb production cross section with the center-of-mass energy by nearly a factor of 2 and improvements in the LHCb trigger provide about 40% more signal candidates than in the previous analysis.

The analysis strategy in this paper is similar to that detailed in the previous study , and includes changes that improve the signal efficiency. The τ + lepton is reconstructed in the hadronic final state 3π(π0)ντ, where 3π ≡π+π−π+, and the D∗−candidate is reconstructed through the D∗−→π−D0(→K+π−) decay.2 The B0 →D∗−3π decay is chosen as the normalization mode because it has the same visible final state as the signal mode. Many of the systematic uncertainties due to detector and reconstruction effects cancel in the ratio of their branching fractions, defined as

1

B(τ + →3πντ) + B(τ + →3ππ0ντ) .

(1)

Here, Nsig and Nnorm are the yields in the signal and normalization modes, respectively, which are obtained from the data. The efficiencies εsig and εnorm, for the signal and normalization modes, respectively, are determined from the simulation.

The Signal

efficiency εsig is calculated as the average between the τ + →3π¯ντ and the τ + →3ππ0¯ντ decays weighted by their relative branching fractions . Finally, using the known branching fractions B (B0 →D∗−3π) and B (B0 →D∗−µ+νµ) , R(D∗−) is obtained as

R(D∗−) = K(D∗−) B(B0 →D∗−3Π)

B(B0 →D∗−µ+νµ) .

(2)

To avoid biases, the numerical results of this analysis were not examined until the full procedure had been finalized. This paper is organized as follows. The LHCb detector and simulation are described in Sec. 2. Details of the selection criteria used to select the B0 →D∗−τ +ντ and B0 →D∗−3π 1BaBar and Belle used decays with both muons and electrons in the R(D(∗)) denominator, while LHCb has exclusively studied decays with muons so far.

2The inclusion of charge-conjugate decay modes is implied throughout the paper.

1

candidates are presented in Sec. 3, followed by Sec. 4, which describes the study of the double-charm decays of the B meson that form the dominant background for the signal mode. The determination of the signal and normalization yields are provided in Sec. 5, and the systematic uncertainties are discussed in Sec. 6. Finally, the results are given in Sec. 7.

Detector And Simulation

The LHCb detector [12, 13] is a single-arm forward spectrometer covering the pseudo- rapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region , a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of the momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. The minimum distance of a track to a primary pp collision vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/pT) µm, where pT is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors . Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers .

The online event selection is performed by a trigger , which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. At the hardware-trigger stage, events are required to have a muon with high pT or a hadron, photon or electron with high transverse energy in the calorimeters. The hadron can originate from either the decay chain under consideration or the remainder of the event. The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any PV.

At least one charged particle must have a transverse momentum pT > 1.6 GeV/c and be inconsistent with originating from any PV. A multivariate algorithm is used for the identification of secondary vertices consistent with the decay of a b hadron.

Simulation is required to model the effects of the detector acceptance and the imposed selection requirements. In the simulation, pp collisions are generated using Pythia with a specific LHCb configuration . Decays of unstable particles are described by EvtGen , in which final-state radiation is generated using Photos . The τ + decays to 3πντ and 3ππ0ντ are simulated using the resonance chiral Lagrangian model as implemented in the Tauola package tuned according to the results from the BaBar Collaboration . The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit as described in Ref. .

Large samples of simulated events are required to reduce the systematic uncertainty due to the sample size. A fast simulation technique, redecay , is used for this purpose in which the underlying pp interaction is reused multiple times, with an independently generated signal decay for each. These samples have been validated against simulated

2

events using unique underlying interactions.

Event Selection

The reconstruction of the decay kinematics follows the procedure given in Refs. [4, 5]. The signal candidates are formed by combining a D∗−meson that decays as D∗−→π−D0(→K+π−) and a 3π system which is detached from the B0 decay vertex due to the non-negligible lifetime of the τ + lepton. The dominant background contribution is B0 decays in which the 3π system comes promptly from the B0 vertex, called prompt background hereafter. To suppress this background, the distance between the B0 and 3π vertices along the beam direction, ∆z ≡z(3π) −z(B0), is required to be at least twice its uncertainty (σ∆z). Double-charm B →D∗−D(X) decays3 are the next dominant background, with a detached-vertex topology similar to that of the signal decays. The remaining sources of background are suppressed by requiring the 3π system to be consistent with originating from a common vertex.

Three categories of data and simulation are used in this analysis: signal, normalization and control. The control samples are used to study the double-charm background and are

Selected To Enrich The Number Of D+

s decays. The selection process for each set is split into two stages. First, common selection criteria are applied to suppress the candidates originating from random combinations of final-state particles for the signal, normalization and control samples. Second, specific requirements are placed on each set. The selection of control samples is described in Sec. 4.

Common Selection Criteria

The purpose of these common requirements is to suppress the prompt and combinatorial backgrounds. Tracks consistent with a kaon or pion hypothesis and having p > 2 GeV/c and pT > 250 MeV/c are selected to form D0 candidates, which are required to have a mass in the range [1840, 1890] MeV/c2 and pT larger than 1.2 GeV/c. The D0 candidates are then combined with tracks consistent with the pion hypothesis and with pT > 110 MeV/c to form D∗−candidates, where the difference in the masses of the D∗−and D0 candi- dates (∆m) must lie within 143 and 148 MeV/c2. Sideband regions of m(D0) in ranges [1825, 1840] MeV/c2 and [1890, 1905] MeV/c2 and ∆m within [150, 160] MeV/c2 are defined to study the combinatorial background.

The τ + candidates are formed from three tracks, where each track must have pT > 250 MeV/c and be consistent with the pion hypothesis. Additionally, the 3π vertex must be separated from the PV associated with the signal decay by at least 10 times the uncertainty on the distance of separation. The radial distance between the 3π vertex and the beam center in the transverse plane is required to be within [0.2, 5.0] mm to avoid pion triplets coming from secondary interactions or a PV. The combinatorial background is suppressed by requiring that the D0 and τ + candidates are associated with the same PV. For events with B0 mass greater than 5150 MeV/c2, the significance of the IP of the D0 and 3π candidates with respect to the PV associated with the signal decay is required to be larger than 4.

3Throughout the paper, X denotes unreconstructed particles that are known to be present in the decay chain, and (X) stands for those that may or may not be present.

Particle Identification Requirements

The charged pion and kaon tracks must be positively identified using the information provided by the particle identification (PID) system. The requirements are the same as those used in the previous analysis . Contamination due to decays of the type B →D∗−D+(X), with the subsequent decay D+ →K−π+π+(π0), are suppressed by re- quiring that the kaon identification probability is less than 17% for the π−candidate within the 3π system.

Anticombinatorial And Isolation Requirements

The combinatorial background is further suppressed using a boosted decision tree (BDT) classifier that is trained using simulated B0 →D∗−τ +ντ decays as the signal proxy and the data sample with the same-sign charge combination D∗−π−π+π−as the background proxy. The distributions of pT and η of the D∗−and τ + candidates along with variables related to their vertex and IP are used to separate signal and background events. Two

Such Variables Are Χ2

IP, which is defined as the difference in the vertex-fit χ2 of the PV reconstructed with and without the particle under consideration, and vertex χ2, which describes the quality of a vertex. The flight direction information of the B0 and τ + is also utilized. The BDT classifier rejects about 75% of the combinatorial background while preserving 77% of the B0 →D∗−τ +ντ decays.

A closely related challenge is the rejection of partially reconstructed backgrounds with more than six charged tracks. The main source of these candidates is B →D∗−D+

Processes Where The D+

s meson decays into five stable charged particles and B →D∗−D0K+ decays where the D0 meson decays into four stable charged particles.

A Dedicated

algorithm is used to evaluate the isolation of each signal-candidate track from other nonsignal tracks in an event. An isolation BDT classifier is formed from this information for each signal-candidate track, trained using simulated B0 →D∗−τ +ντ decays as the signal proxy and simulated B →D∗−D0K+ decays in which two extraneous charged kaons are present as the background proxy. This classifier removes 82% of background decays with extra charged tracks while retaining 78% of signal decays. The isolation requirements for suppressing background with extra neutral particles in the final state are discussed in Sec. 3.2.2.

Selection Of Signal Mode B0→D∗−Τ +Ντ

The signal-mode B0 →D∗−τ +ντ candidates are identified with the use of a detached-vertex criterion and a targeted suppression of backgrounds where D+

S Decays Mimic Hadronic

decays of τ + leptons.

Vertex Detachment Criteria

In B0 →D∗−τ +ντ decays, the 3π vertex is detached from the B0 vertex.

A Good

approximation for the B0 vertex is the point of closest approach between the D∗−line of flight and the 3π line of flight. A BDT classifier is used to identify this detached topology. The inputs to this BDT classifier are the positions of the D0, B0 and 3π vertices and their related uncertainties, the 3π mass, and the momenta of the tracks forming the B0 candidate. The training of this BDT is performed using simulated B0 →D∗−τ +ντ decays

4

as signal and simulated prompt bb →D∗−3πX production as background samples. The efficiency and rejection performance of this BDT classifier is slightly better than the beam-direction significance used in Refs. ; the rejection rate is 20% higher for the same signal efficiency.

S Backgrounds

The 3π decay of the τ + proceeds predominantly through an a1 resonance with a ρ0π+ intermediate state. There is a major background contribution due to D+

S Decays To 3Π(X),

which primarily proceeds through η and η′ resonances, with only a very small contribution from Ra1 structures, where R designates an η, η′, ω, ϕ or K0 meson . In addition, for the latter three mesons, the phase space is such that the a1 meson must be produced below its on-shell mass, providing further discrimination with respect to τ decays. The π+π−mass is kinematically limited to < 400 MeV/c2 for the decays η →π+π−π0 and η′ →π+π−η, which provides input to a discriminating variable formed by the minimum mass of the two π+π−combinations present in the pion triplet. The 3π system from D+

S

decays is often accompanied with a large number of neutral particles from the same decay. Variables related to the energy from these neutral particles in cones around the 3π system direction are used in a BDT classifier to suppress these backgrounds. This classifier is trained using simulated B0 →D∗−τ +ντ events as the signal training sample and simulated

D+

s events decaying into three pions as the background training sample. The performance

Of This Anti-D+

s BDT is better than that found in the previous analysis with 40%

S Bdt Output Is Used As A

fit variable to estimate the yield of B0 →D∗−τ +ντ decays.

Other Requirements

The invariant mass of the 3π system is required to be below 1600 MeV/c2 to suppress double-charm backgrounds. An upper boundary for the invariant mass of the D∗3π candidates is set at 5100 MeV/c2, consistent with the presence of neutrinos in the final state of the signal decays. After all the selection requirements are applied, only 0.5% of events have multiple candidates, from which one is chosen at random.

Selection Of Normalization Mode B0 →D∗−3Π

The selection of B0 →D∗−3π candidates utilizes the unique characteristics of this mode: the fully reconstructed B0 →D∗−3π decay and the 3π system coming directly from the B0 meson. The B0 candidates are selected in the mass range [5150, 5400] MeV/c2. The selection is kept similar to that of the B0 →D∗−τ +ντ mode, to cancel the majority of systematic bias in the measurement of the ratio K(D∗−). To ensure this, the D0 decay vertex is required to lie further downstream than the 3π vertex, a similar detachment criterion to that for the 3π system from the B0 decay vertex in B0 →D∗−τ +ντ decays.

The Anti-D+

s BDT classifier and m(3π) selection criteria are not applied; however these do not bias K(D∗−), since their efficiency is around 97% for the B0 →D∗−τ +ντ decays.

Simulation Corrections And Efficiencies

The simulation samples used in the analysis are required to match the conditions in data as closely as possible. The pT and η distributions of the B meson, B0 →D∗−τ +ντ form factors, 3π vertex-position uncertainty and the 3π decay dynamics in simulation are calibrated to data. Control samples are used to validate simulation samples and apply corrections where necessary, as described in Sec. 4. The selection efficiencies (ε) for the B0 →D∗−τ +ντ and B0 →D∗−3π modes are estimated from simulation after corrections are applied. The efficiencies used in Eq. (1) are εsig = (1.21 ± 0.01) × 10−4 and εnorm = (3.54 ± 0.04) × 10−4. The uncertainties are due to the limited size of the simulation samples.

Study Of Double-Charm Background

The dominant background category after applying the selection criteria mentioned in Sec. 3 is double-charm decays B →D∗−D(X), where D is a D+

S , D+ Or D0 Meson. Control

samples in data are used to study these backgrounds and evaluate corrections that must be applied to simulated samples which are used to obtain the background probability density functions (PDFs) used in the fit to determine the signal yield. The decays involving

D+

s mesons are analyzed in two stages. First, the corrections to the branching fractions

S Decays With Three Pions In The Final State

are estimated. These corrections are applied to the simulation samples. Second, the

Composition Of Several B →D∗−D+

s (X) decays is determined, serving as constraints on the fractions of these components in the signal-extraction fit.

The Decays Of D+

s mesons and τ + leptons to final states involving three pions are distinct,

S Meson Can Decay To The Ρ0 Meson Via An Η′

resonance, which decays to the ρ0γ final state. Consequently, the ρ0 contribution from

The D+

s decay could be mistakenly attributed to that from a τ + decay. Therefore, it is crucial that contributions from various resonances, especially η′, are correctly normalized in simulation to reflect the data as closely as possible. It is also essential to constrain the relative contributions of certain decay modes, whose branching fractions are not precisely measured.

The Fractions Of Different D+

s →3πX decays are determined from a data sample en- riched in these decays. This sample is selected with the same criteria as for B0 →D∗−τ +ντ decays but with a reverse requirement on the anti-D+

S Bdt Output. A Simultaneous Binned

maximum-likelihood fit is performed to the distribution of four variables: min[m(π+π−)], max[m(π+π−)], m(π+π+), which represents the reconstructed masses of all possible two- pion combinations of the candidate, and m(3π). This last variable allows the exclusive

D+

s →3π decay to be distinguished from candidates where energy is carried by addi-

→3Π X, Where X Escapes

detection.

The Different D+

s decay components are broadly divided into four categories:

• D+

s →ηπ+(π0) decays where charged pions from the η meson are selected,

Min[M(Π+Π−)],

max[m(π+π−)], m(π+π+) and m(3π) in the fit to the control data samples.

• D+

s →η′π+(π0) decays where charged pions from the η′ meson are selected,

S →Ωπ+(Π0) Or D+

s →ϕπ+(π0) decays where charged pions from the ω or ϕ

• D+

s decays where the pions originate either directly from the D+

S Decay Or From

the a1 resonance η3π, ηa1, η′3π, η′a1, ω3π, ωa1, ϕ3π, ϕa1, K03π, K0a1, τ +ντ and nonresonant 3π.

S Decays Are Modeled Using The Inclusive

B →D∗−3πX simulation sample. The fraction of each of the four D+

S Decay Components,

relative fractions of η3π, η′3π and ω3π final states and the total number of D+

S

decays are free parameters in the fit. Compared to the previous analysis , the simulated

S →Ra1 Modes With R ∈{Η, Η′, Ω, Φ, K0},

which enable a more detailed description of the D+

S →3Πx Decay. This Introduces More

fit parameters and hence additional constraints are applied to ensure the stability of the fit. The fit assumes no interference effects, as a full amplitude analysis is beyond the scope of this study. The fit results are given in Fig. 1, which shows the fractions of the four

S Control

sample (see Fig. 1) before applying any bias correction to the mean value of its uncertainty.

0.001 ± 0.001

inclusive categories mentioned above. The relative fraction of the component with the π0 meson with respect to the total fraction of each of the first three categories are Gaussian constrained around the expected value of 2/3 with a standard deviation of 0.2. For each

S →Ra1 + D+

s →R3π) is fixed at 5.5%. This is chosen since it provides the best quality for the signal fit. The fractions of D+

S Decays To Φa1 + Φ3Π, K0A1 + K03Π

and τ +ντ final states are fixed according to their known branching fractions . The χ2 per degree of freedom is evaluated to be 1.5, indicating a reasonable fit quality. Possible effects from mismodeling of the components in the fit are considered as sources of systematic uncertainties, which are described in Sec. 6. The determined fractions of the different modes are given in Table 1. They are used to correct the corresponding modes in the simulation sample. The uncertainties obtained from the fit could be underestimated due to the statistical correlations arising from simultaneously fitting to four one-dimensional distributions. The effect of the correlations is investigated using a set of pseudodata samples built via bootstrapping from simulation, in which events are assigned to each component category randomly. The statistical uncertainty on the fit parameters is found to be underestimated by no more than 40%. The uncertainties of the fit parameters are then corrected according to this estimation. The correction factors are applied to the simulation samples that are used to produce the template PDFs for

B →D∗−D+

s X backgrounds.

Table 2: Decay Fractions For B →D∗−D+

s (X) decays obtained from data control samples. The fractions are normalized relative to that of the B0 →D∗−D∗+

Decay. Εcontrol Is The Efficiency In

the control sample.

S (X) Decays Provides Important Constraints

in the signal fit. These relative fractions are determined from a fit to the data control sample

Enriched In B →D∗−D+

s (→3π)(X) decays. The selection of the control sample differs from the default selection by requiring the 3π mass to be within 20 MeV/c2 of the known

S Mass And Omitting The Anti-D+

s BDT cut, as well as the B0 and τ + mass constraints. This sample comprises modes that can be grouped into the exclusive B0 →D∗−D(∗,∗∗)+

S (X) Decays.4

The q2 ≡(pB0−pD∗−)2 distribution for exclusive modes peaks at the mass of the relevant

S

states, where pH is the momentum of particle H. The inclusive B →D∗−D+

S X

modes often have at least one extra particle, possibly from D

S (X) Final States. These

additional particles carry momentum that contributes to the q2, shifting this distribution to higher values. An extended binned maximum-likelihood fit is performed to the distribution of the difference of the D∗−3π mass and the sum of the reconstructed D0 and 3π masses, i.e., ∆MD∗Ds ≡m(D∗−3π) −m(D0) −m(3π). The total PDF used in the fit is

(3)

where fcomb is the fraction of the combinatorial background and Pcomb is the

Ple;

fi are floating fractions of the different B →D∗−D+

D∗

s0(2317)+ and Ds1(2460)+ states, respectively. The template shape of each component is taken from simulation. The distributions of ∆MD∗Ds, q2, decay time of the τ + candidate (tτ) and anti-D+

S

BDT output are shown in Fig. 2. The fit quality is good with a χ2 per degree of freedom equaling 1.11. The fractions of different decays are given in Table 2, which are used as Gaussian constraints in the signal-extraction fit for the corresponding components after accounting for the efficiency differences between the control and signal samples.

S

are used to refer to any higher-mass excited charm or charm-strange

Mesons That Decay Into The Ground-State D∗−And D+

s meson.

Combinatorial

Figure 2: Distributions of ∆MD∗Ds, q2, tτ and anti-D+

S (X)

components. The results of the fit are overlaid.

B →D∗−(D0, D+)(X) Control Samples

The B →D∗−D+(X) and B →D∗−D0(X) decays are the subleading double-charm back- grounds in the signal sample, where the D0 mesons decay to three charged pions plus extra particles and the D+ mesons decay to the π+K−π+ final state with the kaon misidentified as a pion. Data control samples are used to check the agreement with simulation. A control sample representing the B →D∗−D0X decays is selected using the decay mode D0 →K−3π. The isolation algorithm described in Sec. 3.1.2 searches for extra kaons around the 3π vertex and thus can be used to select such candidates. A B →D∗−D+X control sample is obtained by reversing the PID requirements on the negatively charged pion to form D+ →π+K−π+ decays. In order to retain high statistics, the BDT and B0 and τ + mass requirements are omitted. Other selection criteria remain the same as those for B0 →D∗−τ +ντ decays.

These control samples are used to check the agreement with simulation for the signal

Fit Variables Q2, Tτ And Anti-D+

s BDT output. The q2 distribution shows disagreement between data and simulation in both D0 and D+ modes due to the imperfect modeling of inclusive 3π decays. Therefore, the simulation is corrected to match the data distributions.

Fig. 3 shows the q2 distributions before and after the corrections in the B →D∗−D0(X)

2 Fb

Figure 3: Simulated q2 distributions for B →D∗−D0(X) before and after weighting based on the data control samples. sample. The agreement is good in the case of the other two fit variables, tτ and anti-D+

S

BDT output, and no further correction is necessary.

B0 →D∗−Τ +Ντ Yield

The B0 →D∗−τ +ντ yield is determined from an extended binned maximum-likelihood fit

S Bdt Output. The Binning Scheme Comprises

eight bins in q2 and tτ, and six bins in the BDT output. The chosen binning scheme has maximum sensitivity to the B0 →D∗−τ +ντ yield while still sufficiently populating the bins. The ranges of q2, tτ and BDT distributions are [0, 11] GeV2/c4, [0, 2] ps and [−0.2, 0.5], respectively. The fit model is built as a three-dimensional template. A summary of the components in the fit model is given in Table 3. The fit model assumes that any possible new physics effects on the B0 →D∗−τ +ντ decays are the same as that in B →D

∗∗Τ +Ντ

decays. The templates for the combinatorial components are derived from data in which same-sign combinations of the D∗and τ candidates are selected, whereas the remainder are obtained from simulation.

The fit parameters are itemized below. • Nsig: the number of B0 →D∗−τ +ντ events, which is used as input to K(D∗−). • fτ +→3πντ: the fraction of τ + →3πντ decays relative to the sum of τ + →3πντ and τ + →3ππ0ντ decays. This is estimated and fixed as per the branching fractions and efficiencies of these modes.

• Fd∗∗Τν: The Amount Of B →D

∗∗τ +ντ decays relative to B0 →D∗−τ +ντ decays. This is fixed to the expected value from simulation after correcting for the overestimated branching fractions used to produce them. The correction is done by comparing

11

Table 3: List of components in the signal yield extraction fit and their normalization.

Nfake D∗−

Ref. and branching fractions of B →¯D∗∗µ+νµ . This fraction is determined to be 3.5%, which is significantly lower than that used in Refs. .

• N Same

D0 : the number of B →D∗−D0(X) candidates where all pions in the 3π system originate from the D0 vertex. The yield is estimated from simulation and corrected for data-simulation differences by the ratio of D0 →K−3π decays in both samples.

This value is fixed in the fit.

• F V1V2

D0 : the ratio of the number of B →D∗−D0(X) decays where at least one of the pions comes from the D0 vertex and the other pion(s) from a different vertex relative

• Fd+:

the ratio of the number of B →D∗−D+(X) decays to the number of

B →D∗−D+

s (X) decays.

S : The Yield Of B →D∗−D+

s (X) decays, which have six categories, as described in Sec. 4.2. The fraction parameters are Gaussian constrained to the values obtained from the data control sample given in Table 2 after correcting for efficiency effects.

• NB→D∗−3πX: the yield of B →D∗−3πX events where the three pions come from the B vertex. This value is constrained by using the observed ratio between B0 →D∗−3π exclusive and B →D∗−3πX inclusive decays, corrected for data-simulation differ- ences.

• NB1B2: the yield of combinatorial background events where the D∗−meson and the 3π system come from different B decays. It is fixed to the value obtained in the same-sign data sample with D∗+3π candidates satisfying the criteria of higher mass and nonisolation due to originating from two different b hadrons.

B

Comb.

D

Comb.

D

Comb. Figure 4: Distributions of the fit variables in the B0 →D∗−τ +ντ data sample with the fit result overlaid.

• Nfake D0 and Nfake D∗−: the combinatorial background yields with a fake D0 and D∗−, respectively. These are fixed to the values obtained from a fit to m(K−π+) and m(D∗−) −m(K−π+).

Vary Freely In The Fit. The Fit Results Are

summarized in Table 4 and the distributions of the fit variables are shown in Fig. 4. The fit is performed in two iterations: First, the fractions of D0 are varied freely and the six

D+

s decay modes are Gaussian constrained, and then a second fit is performed by fixing these to their best fit values. This is the same strategy followed in Refs. to determine the statistical uncertainty on the B0 →D∗−τ +ντ yield. Thus the relative statistical precision on the yield changes from 6.2% to 5.9%. The quadratic difference between the statistical uncertainties in the two iterations is treated as a systematic uncertainty from the double-charm decay models. The number of signal events is determined to be 2573 ± 156, where the uncertainty is statistical only. The fit quality is excellent with a χ2 per degree of freedom of 1.0. From studies using pseudoexperiments, the fit is found to be unbiased.

B0 →D∗−3Π Yield

The B0 →D∗−3π yield is estimated from an unbinned maximum-likelihood fit to the D∗−3π± mass distribution. The signal model consists of a Crystal Ball (CB) function

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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