Study of $B_c^+\to J/\psi D_s^+$ and $B_c^+\to J/\psi D_s^{*+}$ decays in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector

A study of $B_c^+\to J/\psi D_s^+$ and $B_c^+\to J/\psi D_s^{*+}$ decays using 139 fb$^{-1}$ of integrated luminosity collected with the ATLAS detector from $\sqrt{s} = 13$ TeV $pp$ collisions at the LHC is presented. The ratios of the branching fractions of the two decays to the branching fraction of the $B_c^+\to J/\psi \pi^+$ decay are measured: $\mathcal B(B_c^+\to J/\psi D_s^+)/\mathcal B(B_c^+\to J/\psi \pi^+) = 2.76\pm 0.47$ and $\mathcal B(B_c^+\to J/\psi D_s^{*+})/\mathcal B(B_c^+\to J/\psi \pi^+) = 5.33\pm 0.96$. The ratio of the branching fractions of the two decays is found to be $\mathcal B(B_c^+\to J/\psi D_s^{*+})/\mathcal B(B_c^+\to J/\psi D_s^+) = 1.93\pm0.26$. For the $B_c^+\to J/\psi D_s^{*+}$ decay, the transverse polarization fraction, $\Gamma_{\pm\pm}/\Gamma$, is measured to be $0.70\pm0.11$. The reported uncertainties include both the statistical and systematic components added in quadrature. The precision of the measurements exceeds that in all previous studies of these decays. These results supersede those obtained in the earlier ATLAS study of the same decays with $\sqrt{s} = 7$ and 8 TeV $pp$ collision data. A comparison with available theoretical predictions for the measured quantities is presented.


Study of 𝑩 +
→ /  +  and  +  → /  * +  decays in   collisions at √  = 13 TeV with the ATLAS detector The ATLAS Collaboration A study of  +  → / +  and  +  → / * +  decays using 139 fb −1 of integrated luminosity collected with the ATLAS detector from √  = 13 TeV   collisions at the LHC is presented.The ratios of the branching fractions of the two decays to the branching fraction of the  +  → / + decay are measured: B ( +  → / +  )/B ( +  → / + ) = 2.76 ± 0.47 and B ( +  → / * +  )/B ( +  → / + ) = 5.33 ± 0.96.The ratio of the branching fractions of the two decays is found to be B ( +  → / * +  )/B ( +  → / +  ) = 1.93 ± 0.26.For the  +  → / * +  decay, the transverse polarization fraction, Γ ±± /Γ, is measured to be 0.70 ± 0.11.The reported uncertainties include both the statistical and systematic components added in quadrature.The precision of the measurements exceeds that in all previous studies of these decays.These results supersede those obtained in the earlier ATLAS study of the same decays with √  = 7 and 8 TeV   collision data.A comparison with available theoretical predictions for the measured quantities is presented.
This paper presents a new study of these decays with the ATLAS detector [13].It is based on   collision data collected at √  = 13 TeV during Run 2 of the LHC, corresponding to an integrated luminosity of 139 fb −1 .The purpose of the new measurement is to improve the precision of measured properties of these decays by using a larger data sample and new analysis methods.In particular, a technique using boosted decision trees is applied to improve the signal decay candidate selection.
Similarly to Ref. [2], the  +  meson is reconstructed via the  +  →  + decay, with the  meson decaying into a pair of charged kaons.The  * +  meson decays into a  +  meson and a soft photon or  0 which is not reconstructed in the analysis.However, the mass difference between the  * +  and  +  mesons is sufficient for the two decay signals to be resolved as two distinct structures in the distribution of the reconstructed mass of the / +  system.The / meson is reconstructed via its decay into a muon pair.The  +  → / + decay is used as a reference to measure the branching fractions.The  +  → / * +  decay, being a transition of a pseudoscalar meson into two vector states, can be described in terms of three helicity amplitudes:  ++ ,  00 and  −− , where the indices correspond to the helicities of the / and  * +  mesons.The contribution of the  ++ and  −− amplitudes, referred to as the  ±± component, corresponds to the / and  * +  transverse polarization.Its fraction, Γ ±± /Γ, is also measured.

The ATLAS detector, data and simulated samples
The ATLAS detector 1 consists of three main components: an inner detector (ID) tracking system immersed in a 2 T axial magnetic field, surrounded by electromagnetic and hadronic calorimeters and by the muon spectrometer (MS).A full description of the detector can be found in Ref. [13], complemented by Ref. [14] for details about the innermost silicon pixel layer that was installed for Run 2. This analysis is based on the full Run 2 dataset of √  = 13 TeV   collisions collected by ATLAS between 2015 and 2018 at the LHC.The data were recorded during stable LHC beam periods.Data quality requirements were imposed, notably on the performance of the MS and ID systems [15].After applying these criteria, the integrated luminosity of the dataset amounts to 139 fb −1 , with an uncertainty of 1.7% [16], obtained using the LUCID-2 detector [17] for the primary luminosity measurements.
The data were collected during periods with different instantaneous luminosities, so several trigger configurations were used in the analysis to maximize the signal yields.Most of the triggers were based on identification of two muons, with various muon transverse momentum ( T ) thresholds (usually 4 and 6 GeV), requiring the oppositely charged muons in the pair to form a good vertex and have an invariant mass compatible with being produced by a / →  +  − decay.Since the event rate from these triggers was too high, prescale factors were applied to reduce it during periods with high instantaneous luminosity.To allow lower thresholds while keeping acceptable rates, other triggers required the presence of a third muon, which may appear in a semileptonic decay of hadrons containing  and c quarks present in an event along with the  +  meson.Another set of triggers fitted a common vertex to two muons and two additional ID tracks, applying an invariant mass selection that assumed the  0  →  +  −  decay topology.The events selected by these triggers require a special treatment described in Section 4.
To model the inelastic   events containing  +  decays, large samples of Monte Carlo (MC) simulated events were prepared using the dedicated generator BCVEGPY 2.2 [18] interfaced to P 8.244 [19] to simulate the parton showers and hadronization, and E G [20] to model heavy-flavour decays.Simulated distributions of  T ( +  ) and |( +  )| were corrected to match the corresponding spectra of  +  mesons in data, following the procedure described in Section 6.The generated events with at least two muons having transverse momenta  T () > 3.5 GeV and pseudorapidities |()| < 2.8 were passed through a full simulation of the detector using the ATLAS simulation [21] based on G 4 [22].A simulation of the triggers and their prescale factors was applied to all MC event samples used in the analysis.

𝑱/𝝍 candidates
The / candidates are built from pairs of oppositely charged muon candidates that are reconstructed using information from the MS and the ID.Muon candidates must satisfy the Loose identification working point [24].The muon track parameters are determined from the ID measurement alone, since the precision of the measured track parameters is dominated by the ID track reconstruction in the  T range of interest for this analysis.Pairs of oppositely charged muon tracks are re-fitted to a common vertex.The quality of the vertex fit is required to satisfy  2 < 10, where  2 is the fit quality (the number of degrees of freedom  dof = 1).This loose selection requirement is chosen to minimize biasing the  2 distribution of further cascade fit.The candidates in the dimuon invariant mass window 2800 MeV < ( +  − ) < 3400 MeV are retained for further analysis.

𝒔 candidates
Candidate  +  mesons from the  +  → / ( * )+  decays 2 are reconstructed using the decay  +  →  + with  →  +  − .Oppositely charged particle tracks are assigned the kaon mass and combined in pairs to form  candidates.Any additional track is assigned the pion mass and combined with the  candidate to form a  +  candidate.Only three-track combinations successfully fitted to a common vertex with  2 / dof < 5 ( dof = 3) are accepted for further analysis.The invariant mass of the  candidate, ( +  − ), is required to be within a ±7 MeV range around the world average  mass,   = 1019.461MeV [25] ) is the transverse distance between the  +  candidate production vertex, which corresponds to the primary vertex of the   interaction (PV), and its decay vertex projected onto the direction of the  +  transverse momentum.As an event can contain several PVs, the actual PV is chosen as the one giving the smallest three-dimensional impact parameter of the  +  candidate.To avoid biasing    , the PV position is recalculated after removing any tracks used in 2 The analysis considers both  +  and  −  meson decays.For simplicity, charge conjugated states are implied from here onwards.
the reconstruction of the  +  meson candidate.To remove poorly reconstructed candidates,    ( +  ) is also required to not exceed 10 mm.To further suppress the combinatorial background, a multivariate classifier based on boosted decision trees (BDTs) as implemented in the TMVA framework [27] is employed.The input variables used to train the classifier are the  T of the  +  meson candidate, the    ( +  ) variable, and four angular variables: • cos  * ( + ), where  * ( + ) is the angle between the pion momentum in the  +  −  + rest frame and the  +  −  + combined momentum in the laboratory frame.The signal distribution of cos  * ( + ) is flat before kinematic selection because the pseudoscalar  +  meson decays isotropically, but it increases as cos  * ( + ) approaches +1 for the background events.
• | cos 3  ( + )|, where  ( + ) is the angle between one of the kaons and the pion in the  +  − rest frame.The decay of the pseudoscalar  +  meson to the  (vector) plus  + (pseudoscalar) final state results in the spin of the  meson being aligned perpendicularly to the direction of motion of the  relative to the  +  .Consequently, the distribution of cos  () follows a cos 2  () shape, implying a uniform distribution for cos 3  ().In contrast, the cos  () distribution of the combinatorial background is approximately uniform and its cos 3  () distribution peaks at zero.
• cos  * ( +  ), where  * ( +  ) is the angle between the  +  momentum in the  +  rest frame and the  +  flight direction in the laboratory frame.The distribution of cos  * ( +  ) is uniform for the decays of pseudoscalar  +  mesons before kinematic selection, while it tends to increase towards negative values for the background.
• cos  ( + ), where  ( + ) is the angle between the / momentum and the pion momentum in the  +  −  + rest frame.Its distribution is nearly uniform for the signal processes but peaks towards −1 and +1 for the background.
The information about the helicity in the  +  → / * +  decay is encoded in the /  +  mass spectrum and in the distribution of | cos  ( + )|, where  ( + ) is the helicity angle defined in the rest frame of the muon pair as the angle between the  + and  +  candidate momenta.It was verified that neither (/ +  ) nor | cos  ( + )| is significantly correlated with the BDT input variables.
The BDT classifier is trained using signal candidates from the simulated samples and background candidates from the sidebands of the / +  system's mass distribution, defined as the union of the [5680, 5900] MeV and [6400, 6800] MeV ranges.No significant difference in the BDT output is observed when it is trained using either the  +  → / +  or  +  → / * +  signal sample, so the classifier used for the following selection is trained using both.
The ranges of the mass sidebands used to train the BDT were varied to ensure that the classifier does not produce a bias in the mass distribution: the training is repeated using either the inner or the outer halves of both sidebands, as well as using only the right sideband.Alternatively, the classifier trained with the nominal sidebands was applied to the  +  → / ( * )+  candidates with  2 ( +  )/ dof > 4, where the signal contribution is expected to be negligible.No significant shaping of the (/ +  ) distribution was found in these tests.
The cutting threshold set on the BDT output is chosen to maximize the significance / √  + , where  and  are the expected yields of signal and background events in the range 5900 MeV < (/ +  ) < 6350 MeV.Following the optimization procedure, a threshold value corresponding to the efficiency of 81% is chosen.
Candidates within the mass range 5680 MeV < (/ +  ) < 6800 MeV are retained for further analysis.If more than one candidate in an event passes the selection (this happens in about 10% of events with at least one passing candidate), all of them are kept.

𝑩 + 𝒄 → 𝑱/𝝍𝝅 + candidates
To reconstruct the  +  → / + candidates, a / candidate is combined with an additional chargedparticle track which is assigned the pion mass.For the pion candidate, tracks identified as muons passing the low- T identification working point [24] are vetoed in order to suppress a substantial background from  +  → / +    decays.The three-track combination is re-fitted to a common vertex, with the / candidate mass constrained to the world average value [25].
In what follows, the  +  invariant mass,  T ( +  ) and ( +  ) are calculated using the track parameters from the  +  candidate vertex fit.The  +  candidates are required to have  T ( +  ) > 15 GeV and |( +  )| < 2.0.The re-fitted muon tracks must have  T > 4 GeV and || < 2.3, while the re-fitted pion track is required to satisfy  T > 3.5 GeV and || < 2.5.To suppress combinatorial background the following requirements are used: where the definitions of all these variables for the  +  → / + candidates are the same as the corresponding ones for the  +  → / ( * )+  selection.
Candidates within the mass range 5800 MeV < (/ + ) < 7100 MeV are retained for further analysis.
If more than one candidate in an event passes the selection, all of them are kept.

Signal parameters extraction
As described in Section 2, various types of trigger selection chains were used to collect the events for the analysis.Most of them are based on identification of muons from / →  +  − decays and hence are equally efficient for  +  → / ( * )+  and  +  → / + decay events.However, one suite of triggers attempts to reconstruct  +  − ( +  − ) candidates; these triggers, originally intended for the collection of  0  →  +  −  events [28], are not efficient for the reference decay events and may produce a bias in measurements using this channel.Nevertheless, they contribute significantly to the  +  → / ( * )+  signal yields and can be used to measure the quantities related exclusively to these decays, namely the ratio of branching fractions   * +  / +  and the transverse polarization fraction Γ ±± /Γ.
To that end, all selected  +  → / ( * )+  candidates are separated into two (non-overlapping) datasets: • Dataset 1: candidates in the events collected by the standard dimuon triggers or by three-muon triggers without requirements on the additional ID tracks.
• Dataset 2: candidates collected only by the dedicated  0  →  +  −  triggers and not by other ones used in the analysis.
The   +  /  + and   * +  /  + ratios are measured using only the  +  → / ( * )+  decay yields in Dataset 1, while the full dataset, i.e. the union of Dataset 1 and Dataset 2, is employed for the   * +  / +  and Γ ±± /Γ measurements.In the  +  → / + signal yield extraction, the events collected by the  0  →  +  −  triggers are not used at all.

𝒔 signal parameters
An extended unbinned maximum-likelihood fit to the two-dimensional distribution of (/ +  ) and | cos  ( + )| is performed to extract the signal yields as well as the transverse polarization fraction in  +  → / * +  decays.The signal and background probability density functions (PDFs) for the fit are assumed to be uncorrelated in (/ +  ) and | cos  ( + )| and can therefore be factorized into mass and angular parts.For the signals, this assumption was validated using the simulated samples.A small correlation (∼ 1%) was observed for the background, manifesting itself in a statistically insignificant difference between the | cos  ( + )| shapes of the left and right mass-sideband events; its effect is accounted for by assigning a systematic uncertainty.
As stated in Section 1, the  +  → / * +  decay can be described in terms of three helicity amplitudes:  ++ ,  00 , and  −− .The (/ +  ) and | cos  ( + )| spectra should be the same for the  ++ and  −− amplitudes, and that is confirmed with the MC simulation.Therefore, the  +  → / * +  signal is described by a model with two helicity components corresponding to the  ±± and  00 contributions.signal are consistent between Dataset 1 and Dataset 2. Efficiency ratios between these signal components are consistent between Datasets 1 and 2 as well.Therefore, the signal shapes and all signal parameters (except the yields) extracted from the fit are treated as being the same in Dataset 1 and Dataset 2.
The (/ +  ) and | cos  ( + )| shapes of the two helicity components of the  +  → / * +  signal are described using templates made from the MC simulated event samples with the adaptive kernel estimation technique [29].The  ±± component's fraction in the total  +  → / * +  reconstructed yield,  ±± , is a free parameter of the fit.
The (/ +  ) PDF for the  +  → / +  signal is parameterized by a modified Gaussian function,  mod [30,31]: where  = |(/ ; their values and uncertainties are calculated from the parameter values and covariance matrix obtained from the fit.The fitted  +  mass is in good agreement with the world average value [25] and the width agrees with the MC simulation.The projections of the two-dimensional fit onto Dataset 1 and Dataset 2 are shown in Figure 2.

Fit of 𝑩 +
→ / + candidates The yield of the  +  → / + decay signal is extracted with an extended unbinned maximum-likelihood fit to the distribution of / + mass.
In the fit, the signal is modelled by a modified Gaussian function, Eq. ( 1), and the combinatorial background is described by a two-parameter exponential function, exp(− 0  × (1 +  1 )).
In addition to the combinatorial background, there is a significant contribution from partially reconstructed  +  decays (PRDs),  +  → / , where a charged-particle track from  can be reconstructed as a pion of the  +  → / + candidate.The contribution of such decays can create structures in the left sideband of the  +  → / + signal peak that could not be described by the smooth shape of the combinatorial background PDF.MC studies show that the dominant contribution to such decays comes from the  +  → /  + decay, with  + decaying into  +  0 , while other decays, if present, exhibit a fairly smooth dependence on (/ + ) that can be absorbed by the functional form used to model the combinatorial background.
The  +  → /  + decay is a transition of a pseudoscalar to two vector states and, similarly to  +  → / * +  , can be described in terms of three helicity amplitudes.The contributions corresponding to the  ±± and  00 amplitudes have a significantly different shape in (/ + ).Therefore, to model this background in the fit, templates for these two contributions are built out of simulated events, using adaptive kernel estimation technique, and added to the fit PDF, leaving their relative fraction a free parameter.The overall normalization of this PRD component is left free as well.
A small peaking background from CKM-suppressed  +  → / + decay events, which behaves similarly to signal, is expected in data.The PDF for this contribution is obtained by applying the adaptive kernel estimation technique to a simulation of the  +  → / + channel.The ratio of the  +  → / + and  +  → / + yields is fixed according to the reconstruction efficiencies (nearly equal for the two modes) and the relative branching fraction [25].
The mass and width of the  +  → / + signal and its yield,   +  → /  + , obtained from the fit are given in Table 2.The fitted  +  mass is in good agreement with the world average value [25] and the width agrees with the MC simulation.Figure 3 shows the measured (/ + ) distribution overlaid with the result of the fit and its signal and individual background components. → / + candidates.Data are shown as points, and the overall result of the fit is given by the red solid curve.The signal contribution is shown by the purple shaded area.The red dotted, blue dashed, and green long-dashed lines correspond to the  +  → /  + ,  +  → / + , and combinatorial background contributions, respectively.The bottom panel shows the pulls, defined as the difference between the data and the fit function divided by the uncertainty of the data point.

Measurement of the decay parameters
The ratio of the branching fractions of  +  → / and the ratio of the branching fractions of  +  → / * +  and  +  → / +  is3 Here  and  denote the yield and the total efficiency of the corresponding mode indicated in the subscripts.The superscript DS1 or DS1&2 defines whether the value corresponds to Dataset 1 or the full dataset, respectively.
The total efficiency is as a product of kinematic acceptance and reconstruction efficiency.The kinematic range of the measurement is defined in terms of the transverse momentum and pseudorapidity of the  +  meson as  T ( +  ) > 15 GeV and |( +  )| < 2.0.The kinematic acceptance is defined as the fraction of the  +  decay events in the kinematic range which pass the same requirements as those placed on the decay's muon and hadron tracks for the signal extraction.It is determined using large generator-level samples without applying a selection on the decay products.The reconstruction efficiency is evaluated using the MC simulation as the ratio of the number of events passing the full analysis selection to the number of events passing the  +  and decay product requirements at generator level.The reconstruction efficiencies for Dataset 1 and for the full dataset are determined by requiring the simulated events to pass any of the triggers qouted in the Dataset 1 definition and any of the triggers used in the analysis, respectively.
For the value of the branching fraction B ( +  → ( +  − ) + ) of the  +  →  + decay followed by  →  +  − , the CLEO measurement [32] of the partial  +  →  +  −  + branching fractions is used, with a kaon-pair mass within various intervals around the nominal  meson mass.An interpolation between the partial branching fractions, measured in ±5 MeV and ±10 MeV intervals by using a relativistic Breit-Wigner distribution for the shape of the resonance, yields the value (1.85 ± 0.11)% for the ±7 MeV interval which is used in the analysis.This is done as in Ref. [2].
The total efficiencies for all decay modes are shown in Table 3.They are different for the  ±± and  00 components of the  +  → / * +  decay, hence the overall efficiency for this mode is given by , which is valid for either Dataset 1 or the full dataset, and  ±± has the value taken from the fit (shown in Table 1).The fraction of transverse polarization in the  +  → / * +  decay, Γ ±± /Γ, is the fraction of the  ±± component and is calculated from the  ±± value by applying a correction to account for the difference between the total efficiencies for the two component contributions:

Systematic uncertainties
Various sources of systematic uncertainty are considered and outlined in this section.The systematic uncertainties can be classified into two categories.The first category relates to the possible differences between the data and MC simulation which affect the signal acceptances and reconstruction efficiencies.The second category includes the uncertainties in the signal extraction procedure, for both the  +  → / ( * )+  and  +  → / + channels.Each source of systematic uncertainty is considered individually by repeating the analysis with a certain systematic change implemented.The deviation from the nominal result is then symmetrized and assigned as the uncertainty.Although some sources can have rather large effects on the individual decay rate measurements, they largely cancel out in the ratios of the branching fractions due to correlations between their effects on the different decay modes.
The systematic uncertainties affecting the acceptances and reconstruction efficiencies are the following.
• MC modelling of  +  production.To correct for the possible difference in  +  kinematics between data and MC simulation, the  T ( +  ) and |( +  )| spectra are extracted using the abundant  +  → / + channel with an sPlot technique [33].A linear fit of data/MC ratio for these spectra is performed and weights are assigned to simulated events according to the slope of the fitted linear function.This slope is varied in a range preserving one standard deviation agreement with data.The slopes are varied independently for the  T ( +  ) and |( +  )| distributions.They affect both the kinematic acceptances and reconstruction efficiencies.The effects are found to be 1.5%-2% for  • Limited knowledge of  +  and  +  mesons lifetimes leads to an additional systematic uncertainty due to different decay time acceptances between the  +  decay modes.The effect is estimated by varying these lifetimes within the uncertainties of their world average values [25] and is found to be below 0.5% for  /  + ratios are found to be about 1%, while those on   * +  / +  and Γ ±± /Γ are statistically insignificant.
• The effect of possible mis-modelling of the efficiency of  2 / dof and impact parameter selection is evaluated by comparing their distributions in simulation and data using the sPlot technique, independently for  +  → / ( * )+  and  +  → / + events.Good agreement is found, and weights are used to vary the simulated distribution, still preserving one standard deviation agreement with data.The effects on the   ( * ) +  /  + ratios are found to be 3.2% and 0.2% for the modelling of  2 / dof and the impact parameters, respectively, with no significant effect on   * +  / +  and Γ ±± /Γ.
• The effect of an uncertainty in the efficiency of the BDT selection is evaluated in the same way, using the distribution of the BDT output variable.Very good agreement is found here as well, and variations preserving one standard deviation agreement with data estimate an effect of 1.3% for   ( * ) +  /  + ratios.
• The possible effect of mis-modelling the trigger efficiency is expected to be small because the same triggers are used to select events from the different decay modes when measuring each ratio of branching fractions and Γ ±± /Γ.It is conservatively estimated using the data-driven MC corrections for dimuon triggers, applying them to all reconstructed events and taking the deviation from the nominal result.The effect amounts to 0.6% for   ( * )+  /  + .For   * +  / +  and Γ ±± /Γ it is estimated at the level below 0.2% and neglected.The fraction of signal decay events with muons not matching those which fire the triggers is found to be about 1.5% and does not affect the measured quantities.
• The muon reconstruction and identification efficiency uncertainty affects the individual channel efficiencies by about 1%-2%.However, these mostly cancel out in the efficiency ratios and the effect is found to be below 0.1% for the measured quantities and is neglected.
• The presence of other  +  decay modes with a  +  −  + final state affects the selection efficiency.This is studied with a dedicated MC simulation.The dominant contribution is found to come from  +  →  0 (980) ( +  − ) + decay, due to the large overlap of its ( +  − ) distribution with the signal one.By considering a range of possible values for the mass and natural width of the  0 (980) meson [25] to quantify that overlap, an uncertainty of 1.6% is assigned to the The following variations are used to estimate uncertainties due to the  +  → / ( * )+  signal fit procedure.
•  +  → / +  signal mass shape modelling effects are tested with alternative models for the  +  → / +  signal (/ +  ) distribution: a double-Gaussian function and a double-sided Crystal Ball function [34][35][36], fixing the tail parameters to the values extracted from simulation.Changes in the  +  → / +  yield are found to be less than 1.8%, with smaller effects on  +  → / * +  and  ±± .• For the  +  → / * +  signal mass shape, variations of the bandwidth scale factor in the kernel templates ( parameter defined in Ref. [29]) in the widest range preserving both smoothness and a sufficient level of detail in the distributions are applied.The (/ +  ) resolution is also varied by ±10%.The largest effect found is 2.7%, for the  ±± value.
• The simulated shapes of the | cos  ( + )| templates for  +  → / ( * )+  signals are varied by similar changes of the kernel bandwidth scale factor parameter.The effect on all measured quantities is below 0.6%.
• For the background mass shape modelling, the two-parameter exponential function and a second-order polynomial function are used as alternatives to the simple exponential shape; the fitted mass range is also reduced in turn by 40 MeV (two bins of the mass distributions in Figure 2) from the low-mass and high-mass ends.These variations represent the largest uncertainty in the  +  → / ( * )+  signal extraction and amount to 6.0%, 9.0%, and 3.2% for   +  /  + ,   * +  /  + , and   * +  / +  , respectively.• The background | cos  ( + )| shape is alternatively parameterized with a third-order polynomial function.A linear combination of the kernel templates made from the left and right data sidebands is also used to describe the background in the mass range of the signals.The latter variation also covers a small deviation from the assumption of factorization of the mass and angular parts of the background PDF.These variations yield a maximum change of about 2% for   * +  / +  and Γ ±± /Γ, and about 1% for   +  /  + and   * +  /  + .• Effect of the  0  →  +  −  triggers: in the nominal fit, the angular and mass shapes of the signals are supposed to be the same in Datasets 1 and 2, and the same value of  ±± is used in both.To estimate the possible effects of deviations from these assumptions, alternative fit models with either of the following changes are tested: different signal mass and | cos  ( + )| templates are used for Datasets 1 and 2 (produced from events selected by the corresponding triggers); the  ±± fraction is set to be different between Datasets 1 and 2 according to slightly different efficiencies in the simulation.
These variations yield an uncertainty of about 4% for Γ ±± /Γ and less than 1.5% for the  ratios.
• The branching fractions of the  * +  decays are varied in simulation within their uncertainties [25] since they affect the  +  → / * +  template shapes.The effect is found to be 0.7% for Γ ±± /Γ and below 0.1% for the other quantities.
The following variations are used to estimate uncertainties due to the  +  → / + signal fit procedure.• For the signal modelling, a double-Gaussian function and a double-sided Crystal Ball function are used as alternatives to the modified Gaussian function,Eq.( 1), fixing the tail parameters to the values extracted from simulation.Changes in the signal yields reach 4.2%.
• Alternative models are used for combinatorial background modelling: a three-parameter exponential function, and second-and third-order polynomial functions are used.These models, providing more freedom at lower (/ + ) values, are able to absorb a part of PRD contribution that may not have been accounted for by using the  +  → /  + templates.Alternatively, an additional hyperbolic tangent function is added to the fit PDF to cover possible mis-modelling of the PRD component.The shape of the  +  → /  + templates is also varied by changing the kernel bandwidth scale factor and (/ + ) resolution.These variations produced changes of up to 5.8% in the signal yield.
The statistical uncertainties of the total efficiency values due to the limited number of MC events are also treated as a separate source of systematic uncertainty.
Contributions from the different sources of systematic uncertainty described above are summed in quadrature.Finally, since the branching fraction of  +  → ( +  − ) + is featured in Eq. ( 2), its uncertainty [32] is propagated to the final values of the relative branching fractions   ( * )+  /  + .This uncertainty is quoted separately.
The systematic uncertainties of the measured quantities are summarized in Table 4.
In the last two quantities, the first uncertainty is statistical and the second is systematic.
These ATLAS results from Run 2 are compared with the results of LHCb [1] and ATLAS Run 1 [2] measurements and with the predictions from various theoretical calculations in Table 5   /  + and Γ ±± /Γ are, respectively, smaller and larger than in the ATLAS Run 1 measurement, although in each case the difference does not exceed 1.5 standard deviations of the Run 1 uncertainty.Since the precision of the new results significantly exceeds that achieved in the ATLAS Run 1 analysis [2], the latter is superseded by this work.Values of the corresponding ratios of branching fractions calculated using Eqs.( 5)- (7) and Eq. ( 8) and of transverse polarization fractions for  + ,  0 , and  0  decays are also shown.No phase-space corrections are applied to the ratios and the quoted uncertainties are the ones propagated from the world average uncertainties of the individual decay branching fractions.For all experimental measurements, quadrature sums of all uncertainties are quoted.Uncertainties in the predictions are quoted only if they are explicitly given in the corresponding references.The ratios of branching fractions are well described by the predictions of a QCD relativistic potential model [3].The predictions of a QCD sum rules calculation [4], the covariant confined quark model [6] 4 and the light-front quark model [8] all underestimate the   +  /  + ratio, although the prediction of the latter model still agrees with data within a large theoretical uncertainty.However, the   * +  /  + ratio is described well by the same predictions, which correspondingly overestimate the   * +  / +  ratio.Nonetheless, for the light-front quark model the predicted value of   * +  / +  is still in agreement within the theoretical uncertainty.
The   +  /  + value predicted in the Bauer-Stech-Wirbel framework [7] is slightly below the data, while the results of calculations within perturbative QCD [9] and the QCD factorization approach [12] are near the upper bound of the experimental uncertainty range; the recent prediction made within the framework of the relativistic independent quark model [11] gives an even higher value.The perturbative QCD prediction [9] for the   * +  / +  ratio is also slightly above the data.The measured value of Γ ±± /Γ clearly agrees with a naive spin-counting expectation of 2/3.It is larger than the values calculated in the covariant confined quark model [6], perturbative QCD [9], and the relativistic  [2], LHCb [1] and theoretical predictions based on a QCD relativistic potential model (QCD PM) [3], QCD sum rules (QCD SR) [4], covariant confined quark model (CCQM) [6], Bauer-Stech-Wirbel relativistic quark model (BSW) [7], light-front quark model (LFQM) [8], perturbative QCD (pQCD) [9], relativistic independent quark model (RIQM) [10,11], and calculations in the QCD factorization approach (FNCM) [12].Hatched areas show the statistical uncertainties of this measurement and yellow bands correspond to the total uncertainties.Quadrature sums of all experimental uncertainties are quoted for the ATLAS Run 1 and LHCb results.The uncertainties of the theoretical predictions are shown only if they are explicitly quoted in the corresponding papers.independent quark model [10], which give values below 0.5.The discrepancies, however, do not exceed 2-3 standard deviations of the experimental uncertainty.
The measured ratios of branching fractions and the transverse polarization fraction can be compared with the corresponding quantities for the other  mesons.Assuming that the colour-favoured spectator diagram (shown in Figure 1(a) for  +  → / ( * )+

𝑠
) dominates the decay amplitudes, the following approximate relations can be established: where  and D * stand for  + and D * 0 ,  0 and  * − , or  0  and  * −  .Similarly, the Γ ±± /Γ value for  +  → / * +  decay can be compared with the same quantity in the corresponding  + or  0  decays.One can also compare the measured value of   * +  / +  with the corresponding ratio for the  0 and  + decays occurring predominantly via the colour-suppressed spectator diagram: where  stands for  + or  0 and  ( * ) is  ( * )+ or  ( * )0 , respectively.The Γ ±± /Γ value, in turn, can be compared with the transverse polarization fraction in the same  → / * decays as well as in  0  → / decays.The estimates obtained using Eqs.( 5)- (7) and Eq. ( 8) with the world average branching fraction values for the  0 ,  + , and  0  decays [25] are also shown in Table 5, along with the Γ ±± /Γ values for the corresponding decays of those mesons [25].Corrections to the above approximations, required due to phase-space differences between the studied   decays and the corresponding  + ,  0 , and  0  decays, are 1%-2% for Eqs. ( 5)- (7) and increase the values from Eq. ( 8) by about 7%.These corrections are not applied in the numbers quoted in Table 5.
The   * +  / +  value agrees with the corresponding ratio calculated with Eq. ( 7) for both the  0 and  + decays and is larger than that obtained from Eq. ( 8) for decays of the same mesons.It supports the assumption that the colour-favoured spectator diagram dominates the  +  → / ( * )+  decay amplitudes.The   +  /  + and   * +  /  + values agree with the calculations made using Eqs.( 5)-( 6) for decays of  0 and  0  , and are larger than those for decays of  + .The measured value of Γ ±± /Γ lies between the transverse polarization fraction values in the  0 →  * −  *  and  0  →  * −   *  decays and is larger than one in the considered  decays occurring via the colour-suppressed spectator diagram.

Summary
A study of  +  → / +  and  +  → / * +  decays has been performed by the ATLAS experiment at the LHC using   collision data corresponding to an integrated luminosity of 139 fb −1 at 13 TeV centre-of-mass energy.The ratios of the branching fractions of these decays to the branching fraction of  +  → / + decay are measured to be   +  /  + = 2.76 ± 0.33 ± 0.29 ± 0.16 and   * +  /  + = 5.33 ± 0.61 ± 0.67 ± 0.32, where the first uncertainty is statistical, the second is systematic, and the third is due to the uncertainty in the branching fraction of  +  → ( +  − ) + decay.The ratio of the two decay branching fractions is also measured, yielding   * +  / +  = 1.93 ± 0.24 ± 0.09.The transverse polarization fraction in the  +  → / * +  decay is found to be Γ ±± /Γ = 0.70 ± 0.10 ± 0.04.The results of this new measurement supersede those of the previous ATLAS measurement using Run 1 data.
All results are consistent with the earlier measurements by ATLAS and LHCb.The precision of the measurement exceeds that of all previous studies of these decays.
Various predictions for the measured quantities are compared with data.A QCD relativistic potential model calculation agrees well with all three ratios of branching fractions.The   * +  /  + ratio also agrees with other calculations made using QCD sum rules and the covariant confined quark model.The   +  /  + and   * +  / +  ratios are, however, poorly described by the same predictions.The measured transverse polarization fraction in the  +  → / * +  decay agrees well with the value of 2/3 expected from naive spin-counting considerations, while the available calculations of this quantity in the covariant confined quark model, perturbative QCD, and the relativistic independent quark model give values below the data, although agreeing within 2-3 standard deviations of experimental uncertainty.
Comparisons of the measured   * +  / +  ratio with the similar ratios for the  0 and  + meson decays support the assumption that the colour-favoured spectator diagram dominates the  +  → / ( * )+  decay amplitudes.

Figure 2 :
Figure 2: Distributions of ((a), (c)) the / +  mass and ((b), (d)) | cos  ( + )| for the selected  +  → / ( * )+  candidates in ((a), (b)) Dataset 1 and ((c), (d)) Dataset 2. Data are shown as points, and the overall result of the fit is given by the red solid curve.The purple dashed line shows the contribution of the  +  → / +  signal, while the orange dotted and blue dashed-dotted lines show the contributions of the  ±± and  00 components of the  +  → / * +  signal, respectively.The background contribution is shown with the green long-dashed line.The bottom panels show the pulls, defined as the difference between the data and the fit function divided by the uncertainty of the data point.

Figure 3 :
Figure3: Distributions of the / + mass for the selected  +  → / + candidates.Data are shown as points, and the overall result of the fit is given by the red solid curve.The signal contribution is shown by the purple shaded area.The red dotted, blue dashed, and green long-dashed lines correspond to the  +  → /  + ,  +  → / + , and combinatorial background contributions, respectively.The bottom panel shows the pulls, defined as the difference between the data and the fit function divided by the uncertainty of the data point.

/
+ ratios.• The tracking efficiency uncertainty is dominated by the uncertainty in the detector material description used in the MC simulation.It is estimated from independent measurements of secondary vertices in the detector volume and results in an effect of about 1% for the   ( * )+  /  + ratios and below 0.1% for   * +  / +  and Γ ±± /Γ.• The effect of uncertainties in the modelling of multiple proton-proton interactions (pile-up) in MC simulation is estimated by correcting the MC pile-up profile to the profile in data.The effects on the   ( * ) +

Figure 4 :
Figure4: Comparison of the results of this measurement with those of ATLAS Run 1[2], LHCb[1] and theoretical predictions based on a QCD relativistic potential model (QCD PM)[3], QCD sum rules (QCD SR)[4], covariant confined quark model (CCQM)[6], Bauer-Stech-Wirbel relativistic quark model (BSW)[7], light-front quark model (LFQM)[8], perturbative QCD (pQCD)[9], relativistic independent quark model (RIQM)[10,11], and calculations in the QCD factorization approach (FNCM)[12].Hatched areas show the statistical uncertainties of this measurement and yellow bands correspond to the total uncertainties.Quadrature sums of all experimental uncertainties are quoted for the ATLAS Run 1 and LHCb results.The uncertainties of the theoretical predictions are shown only if they are explicitly quoted in the corresponding papers.
The re-fitted muon tracks must have  T > 4 GeV and || < 2.3, while the re-fitted hadron tracks are required to satisfy the  T > 1 GeV and || < 2.5 requirements.The following preselection requirements are used to remove combinatorial background from the sample: . The  +  candidate must have an invariant mass, ( +  −  + ), between 1930 MeV and 2010 MeV, which spans about ±2.5 standard deviations of the detector's mass resolution.The  +  → / ( * )+  candidates are built by combining the selected / and  +  candidates.The / meson decays instantly at the same point as the  +  meson does, whereas the  +  meson lives long enough to form another displaced vertex.Following this cascade topology, a two-vertex fit is performed and the  +  candidate's momentum is required to point back to the  +  vertex [26].To improve the mass resolution, the invariant masses of the / and  +  candidates are constrained to the world average measured masses of the / and  +  mesons [25], respectively.In what follows, the  +  invariant mass,  T ( +  ) and ( +  ) are calculated using parameters of the tracks from the  +  cascade vertex fit.The  +  candidates are required to have  T ( +  ) > 15 GeV and |( +  )| < 2.0.
The uncertainties are calculated from the covariance matrix of the PV and the covariance matrix of the  +  pseudo-track extracted from the cascade vertex fit.These two requirements are used to ensure that the  +  candidate points back to the PV.•  T ( +  )/  T (trk) > 0.10, where the sum is taken over all tracks originating from the selected PV, including the tracks of the  +  candidate.Due to the characteristic hard fragmentation of -quarks, this requirement removes a sizeable fraction of the combinatorial background while having almost no effect on the signal.Among various possible exclusive background contributions from  → /  decays, the only significant one was found to arise from the  0  → / process.Combining the tracks from a true  0  → /, / →  +  − ,  →  +  − decay with another, random, track may produce a fake  +  candidate.To suppress background events of this type, the  +  → / ( * )+ Dataset 1 and Dataset 2 are fitted simultaneously.The MC simulation indicates that both the (/ +  ) and | cos  ( + )| distributions of the  +  → / +  signal and of the helicity components of the  +  → / * + +  ) −   +  |/  +  with the mean mass   +  and width   +  being free parameters of the fit.The  +  → / +  signal's angular PDF is also determined from the MC simulated events by using the adaptive kernel estimation technique.Second-order polynomial shapes are used to describe the background | cos  ( + )| PDFs for Dataset 1 and Dataset 2, while the mass PDFs are parameterized by exponential functions.The parameters of these polynomial and exponential shapes are allowed to float independently for Dataset 1 and Dataset 2.The free parameters of the fit also include the ratio of the  + The values of the signal parameters obtained from the fit to the data are shown in Table1.It also shows the  + → / * +  yield to the  +  → / +  yield,   * +  / +  , and the  +  → / +  signal yields in Datasets 1 and 2,  DS1  +  → / +  and  DS2  +  → / +  . → / * +  signal yield in Dataset 1,  DS1  +  → / * +  , and the yields of both signals in the full dataset,

Table 1 :
Parameters of the  +  → / +  and  +  → / * +  signals obtained with the unbinned extended maximumlikelihood fit to the data.Only the statistical uncertainties are included.No acceptance or efficiency corrections are applied to the signal yields.

Table 2 :
Parameters of the  +  → / + signal obtained with the unbinned extended maximum-likelihood fit.Only the statistical uncertainties are included.No efficiency correction is applied to the signal yield.

Table 3 :
Summary of the total efficiencies.Quoted uncertainties correspond to statistical uncertainties of the simulated samples.For the  +  → / + channel, the efficiency   +  → /  + entering the equations for /  + = 2.76 ± 0.33 ± 0.29 ± 0.16 and   * +  /  + = 5.33 ± 0.61 ± 0.67 ± 0.32, where the first uncertainty is statistical, the second is systematic, and the third corresponds to the uncertainty in the branching fraction of the  +  → ( +  − ) + decay.The ratio of the branching fractions for = 1.93 ± 0.24 ± 0.09.

Table 4 :
Relative systematic uncertainties of the measured quantities.
and Figure 4.The measurements of   +  /  + and   * +  / +  agree with the ATLAS Run 1 and LHCb results.The obtained values of   * +

Table 5 :
Comparison of the measured quantities with the results of earlier measurements and theory predictions.