Measurement of the fragmentation fraction ratio $f_{s}/f_{d}$ and its dependence on $B$ meson kinematics

The relative production rate of $B^{0}_{s}$ and $B^{0}$ mesons is determined with the hadronic decays $B^{0}_{s} \rightarrow D^{-}_{s}\pi^{+}$ and $B^0 \rightarrow D^{-}K^{+}$. The measurement uses data corresponding to 1.0 fb$^{-1}$ of $pp$ collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV recorded in the forward region with the LHCb experiment. The ratio of production rates, $f_{s}/f_{d}$, is measured to be $0.238 \pm 0.004 \pm 0.015 \pm 0.021 $, where the first uncertainty is statistical, the second systematic, and the third theoretical. This is combined with a previous LHCb measurement to obtain $f_{s}/f_{d} = 0.256 \pm 0.020$. The dependence of $f_{s}/f_{d}$ on the transverse momentum and pseudorapidity of the $B$ meson is determined using the decays $B^{0}_{s} \rightarrow D^{-}_{s}\pi^{+}$ and $B^{0} \rightarrow D^{-}\pi^{+}$. There is evidence for a decrease with increasing transverse momentum, whereas the ratio remains constant as a function of pseudorapidity. In addition, the ratio of branching fractions of the decays $B^{0} \rightarrow D^{-}K^{+}$ and $B^{0} \rightarrow D^{-}\pi^{+}$ is measured to be $0.0822 \pm 0.0011 (\textrm{stat}) \pm 0.0025 (\textrm{syst})$.


Introduction
The ratio of fragmentation fractions f s /f d quantifies the relative production rate of B 0 s mesons with respect to B 0 mesons. Knowledge of this quantity is essential when determining any B 0 s branching fraction at the LHC. The measurement of the branching fraction of the rare decay B 0 s → µ + µ − [1] is the prime example where a precise measurement of f s /f d is crucial for reaching the highest sensitivity in the search for physics beyond the Standard Model. The branching fractions of a large number of B 0 and B + decays have been measured to high precision at the B factories [2], but no B 0 s branching fraction is yet known with sufficiently high precision to be used as a normalisation channel.
The relative production rates of b hadrons are determined by the fragmentation fractions f u , f d , f s , f c and f Λ , which describe the probability that a b quark will hadronize into a B q meson (where q = u, d, s, c), or a b baryon, respectively 1 . The ratio of fragmentation fractions f s /f d has been previously measured at LHCb with hadronic [3] and semileptonic decays [4], and the resulting values were combined [4].
In this paper, the ratio of fragmentation fractions f s /f d is determined using B 0 s → D − s π + and B 0 → D − K + decays collected in pp collisions at a centre-of-mass energy of √ s = 7 TeV, with data corresponding to an integrated luminosity of 1.0 fb −1 recorded with the LHCb detector. Since the ratio of branching fractions of the two decay channels is theoretically well understood [5], their relative decay rates can be used to determine the ratio of fragmentation fractions for B 0 s and B 0 mesons through where N corresponds to a signal yield, corresponds to a total efficiency, τ B 0 s /τ B 0 = 0.984 ± 0.011 [6] corresponds to the ratio of lifetimes and B(D − → K + π − π − ) = (9.14 ± 0.20)% [7] and B(D − s → K + K − π − ) = (5.50 ± 0.27)% [8] correspond to the D − (s) meson branching fractions. The factor N a = 1.00 ± 0.02 accounts for the ratio of non-factorizable corrections [9], N F = 1.092 ± 0.093 for the ratio of B 0 (s) → D − (s) form factors [10], and Φ PS = 0.971 for the difference in phase space due to the mass differences of the initial and final state particles. The numerical values used for the CKM matrix elements are |V us | = 0.2252, |V ud | = 0.97425, and for the decay constants are f π = 130.41 MeV, f K = 156.1 MeV, with negligible uncertainties, below 1% [2]. The measurement is not statistically limited by the size of the B 0 → D − K + sample , and therefore the theoretically less clean B 0 → D − π + decays, where exchange diagrams contribute to the total amplitude, do not contribute to the knowledge of f s /f d .
The ratio of fragmentation fractions can depend on the centre-of-mass energy, as well as on the kinematics of the B 0 (s) meson, as was studied previously at LHCb with partially reconstructed B decays [4]. The dependence of the ratio of fragmentation fractions on 1 Charge conjugation is implied throughout this paper. the transverse momentum p T and pseudorapidity η of the B 0 (s) meson is determined using fully reconstructed B 0 → D − π + and B 0 s → D − s π + decays. Since it is only the dependence that is of interest here, the more abundant B 0 → D − π + decay is used rather than the B 0 → D − K + decay. The B 0 → D − K + and B 0 → D − π + decays are also used to determine their ratio of branching fractions, which can be used to quantify non-factorizable effects in such heavy-to-light decays [9].
The paper is organised as follows: the detector is described in Sec. 2, followed by the event selection and the relative selection efficiencies in Sec. 3. The fit to the mass distributions and the determination of the signal yields are discussed in Sec. 4. The systematic uncertainties are presented in Sec. 5, and the final results are given in Sec. 6.

Detector and software
The LHCb detector [11] is a single-arm forward spectrometer covering the pseudorapidity 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. Data are taken with both magnet polarities. The combined tracking system has momentum resolution ∆p/p that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c, and impact parameter 2 resolution of 20 µm for tracks with high transverse momentum. Charged hadrons are identified using two ring-imaging Cherenkov detectors.
The trigger [12] 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. The events used in this analysis are selected at the hardware stage by requiring a cluster in the calorimeters with transverse energy larger than 3.6 GeV. The software stage requires a two-, three-or four-track secondary vertex with a high sum of the p T of the tracks and a significant displacement from the primary pp interaction vertices (PVs). At least one track should have p T greater than 1.7 GeV/c, track fit χ 2 over the number of degrees of freedom less than two, and IP χ 2 with respect to the associated primary interaction greater than sixteen. The IP χ 2 is defined as the difference between the χ 2 from the vertex fit of the associated PV reconstructed with and without the considered track. A multivariate algorithm is used for the identification of the secondary vertices consistent with the decay of a b hadron.
In the simulation, pp collisions are generated using Pythia 6.4 [13] with a specific LHCb configuration [14]. Decays of hadronic particles are described by EvtGen [15], whilst final state radiation is generated using Photos [16]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [17] as described in Ref. [18].

Event selection
The three decay modes, B 0 → D − π + , B 0 → D − K + and B 0 s → D − s π + , are topologically very similar and can therefore be selected using the same event selection criteria, thus minimizing efficiency differences between the modes. The B 0 (s) candidates are reconstructed from a D − (s) candidate and an additional pion or kaon (the "bachelor" particle), with the After the trigger selection, a loose preselection is made using the B 0 (s) and D − (s) masses, lifetimes and vertex qualities. A boosted decision tree (BDT) [19] is used to further separate signal from background. The BDT is trained on half the B 0 s → D − s π + data sample. The most discriminating variables are the B 0 (s) impact parameter χ 2 , the pointing angle of the B 0 (s) candidate to the primary vertex and the p T of the tracks. A cut value for the BDT output variable was chosen to optimally reduce the number of combinatorial background events, retaining approximately 84% of the signal events.
The D − (s) candidates are identified by requiring the invariant mass under the K + π − π − (K + K − π − ) hypothesis to fall within the selection window 1844 -1890 (1944 -1990) MeV/c 2 . The relative efficiency of the selection procedure is evaluated for all decay modes using simulated events, generated with the appropriate Dalitz plot structures [20,21]. Since the analysis is only sensitive to relative efficiencies, the impact of any discrepancy between data and simulation is small.
The final B 0 s → D − s π + and B 0 → D − π + event samples are obtained after a particle identification (PID) criterion, based on the difference in log-likelihood between the kaon and pion hypotheses (DLL). A cut on the bachelor particle is placed at DLL(K − π)< 0. The B 0 → D − K + sample is selected by requiring DLL(K − π)> 5 for the bachelor particle.
decays by imposing DLL(K − π)> 5 on the kaon candidate with the same charge as the D meson, whilst the DLL criteria for the π − and K + are identical between D − and D − s and are used to discriminate D − (s) decays from background. The PID performance as a function of p T and η of the track is estimated from data using a calibration sample of approximately 27 million D * − → D 0 (K + π − )π − decays, which are selected using kinematic criteria only.

Event yields
The relative yields of the three decay modes are determined from unbinned extended maximum likelihood fits to the mass distributions of the reconstructed B 0 (s) candidates as shown in Fig. 1. In order to achieve the highest sensitivity, the sample is separated according to the two magnet polarities, allowing for possible differences in PID performance and in running conditions. A simultaneous fit to the two magnet polarities is performed for each decay mode, with the peak position and width of each signal shared between the two.
The signal mass shape is described by a Gaussian distribution with power-law tails on either side to model the radiative tail and non-Gaussian detector effects. It consists of a  Table 1. For illustration purposes the figures include events from both magnet polarities, although they are fitted separately as described in the text. Crystal Ball function [22] f left (m, α, n, µ, σ) = N · and a second, similar but mirrored, function to describe the right tail, resulting in the signal mass shape f 2CB (m) = f left (m) + f right (m). The parameters of the tails are obtained from simulated events. The mean µ and the width σ of the Gaussian distribution are equal in both Crystal Ball functions, and are allowed to vary in the fit. The parameter N is a normalisation factor. Three classes of background are considered in the fit: fully reconstructed decays where at least one track is misidentified, partially reconstructed decays with or without misidentified tracks and combinatorial background. The shapes of the invariant mass distributions for the partially reconstructed decays are taken from large samples of simulated events. The main sources are B 0 → D − ρ + and B 0 → D * − π + (K + ) for the B 0 → D − π + (K + ) sample, and B 0 s → D − s ρ + and B 0 s → D * − s π + for the B 0 s → D − s π + sample. The invariant mass distributions of the misidentified decays are affected by the PID criteria. The shapes are obtained from simulated events, with the appropriate mass hypothesis applied. The distribution is then reweighted in a data-driven way, according to the particle identification cut efficiency obtained from the calibration sample, which is strongly dependent on the momentum of the particle.
Despite the small π → K misidentification probability of 2.8%, the largest misidentified background in the B 0 → D − K + sample originates from Cabibbo-favoured B 0 → D − π + decays where the bachelor pion is misidentified as a kaon. The shape of this particular misidentified decay is determined from data using a high purity sample of B 0 → D − π + decays (see Fig. 1(a)), obtained by selecting events in a narrow mass window 5200-5340 MeV/c 2 . The yield of this prominent peaking background is allowed to vary in the fit and is found to be consistent with the expected yield based on the B 0 → D − π + signal yield and the misidentification probability. The contamination of B 0 → D − π + events in the B 0 s → D − s π + sample can be caused by the misidentification of either pion from the D − decay. The misidentification probability is 2.0% (3.2%) for the higher (lower) p T pion. After selecting the D − s candidate within the mass window around the nominal D − s mass [2], the number of misidentified pions is reduced to 0.75% (0.02%). The yield of this background is constrained in the fit, based on the B 0 → D − π + signal yield, the misidentification probability and their associated uncertainties.
The yield of Λ 0 b → Λ − c π + decays is allowed to vary in the fit. The cross-feeds from B 0 → D − K + and B 0 s → D − s π + events in the B 0 → D − π + signal is small, and are constrained to their respective predicted yields. In addition, a contribution from the rare B 0 → D − s π + decay is expected with a yield of 3.3% compared to the B 0 s → D − s π + signal, and is accounted for accordingly. Signal The combinatorial background consists of events with random pions and kaons, forming a fake D − or D − s candidate, as well as real D − or D − s mesons, that combine with a random pion or kaon. The combinatorial background is modelled with an exponential shape.
The results of the fits are presented in Fig. 1, and the corresponding signal yields are listed in Table 1. The total yields of the decays B 0 → D − π + and B 0 → D − K + are used to determine the ratio of their branching fractions, while the event yields of the decays B 0 s → D − s π + and B 0 → D − K + are used to measure the average ratio of fragmentation fractions.
The dependence of the relative b-hadron production fractions as a function of the transverse momentum and pseudorapidity of the B 0 (s) meson is studied in the ranges 2.0 < η < 5.0 and 1.5 < p T < 40 GeV/c, using B 0 → D − π + and B 0 s → D − s π + decays. The event sample is subdivided in 20 bins in p T and 10 bins in η, with the bin sizes chosen to obtain approximately equal number of events per bin. The fitting model for each bin is the same as that for the integrated samples, apart from the treatment of the exponent of the combinatorial background distribution, which is fixed to the value obtained from the fits to the integrated sample.

Systematic uncertainties
The systematic uncertainties on the measurement of the relative event yields of the B 0 → D − π + , B 0 → D − K + and B 0 s → D − s π + decay modes are related to trigger and offline selection efficiency corrections, particle identification calibration and the fit model.
The response to charged pions and kaons of the hadronic calorimeter used at the hardware trigger level has been investigated. As the hardware trigger mostly triggers on the high-p T bachelor, a systematic uncertainty of 2% is assigned to the ratio of trigger efficiencies for the decays B 0 → D − K + and B 0 → D − π + , estimated from dedicated studies with D * − → D 0 (K + π − )π − decays. This uncertainty is assumed to be uncorrelated between the individual bins in the binned analysis.
The relative selection efficiencies from simulation are studied by varying the BDT criterion, changing the signal yields by about ±25%. The variation of the relative efficiency is 1.0% which is assigned as systematic uncertainty.
The uncertainty on the PID efficiencies is estimated by comparing, in simulated events, the results obtained using the D * − calibration sample to the true simulated PID performance on the signal decays. The corresponding uncertainty ranges from 1.0% to 1.5% for the different measurements.
The exponent of the combinatorial background distribution is allowed to vary in the fits to the B 0 → D − π + and B 0 s → D − s π + mass distributions. By studying D − π − and D − K − combinations, it is suggested that the value of the exponent is smaller for the B 0 → D − K + decays than for the B 0 → D − π + decays, and therefore in the fit to the B 0 → D − K + candidates the exponent is fixed to half the value found in the fit to the B 0 → D − π + sample. The uncertainty on the signal yields due to the shape of the combinatorial background is estimated by reducing the exponent to half its value in the fits to the B 0 → D − π + and B 0 s → D − s π + mass distributions, and by taking a flat background for the fit to the B 0 → D − K + mass distribution. An uncertainty of 1.0% (0.7%) is assigned to the relative B 0 → D − K + and B 0 s → D − s π + (B 0 → D − π + ) yields. The tails of the signal distributions are fixed from simulation due to the presence of large amounts of partially reconstructed decays in the lower sidebands. The uncertainty on the signal yield is estimated by varying the parameters that describe the tails by 10%. The uncertainty from the shape of the central peak is taken from a fit allowing for two different widths for the Crystal Ball functions in Eq. 2, leading to a 1.0% (0.8%) uncertainty on the relative B 0 → D − K + and B 0 s → D − s π + (B 0 → D − π + ) yields. The contribution of charmless B decays without an intermediate D meson is ignored in the fit. To evaluate the systematic uncertainty due to these decays, the B mass spectra for candidates in the sidebands of the D mass distribution are examined. A contribution Table 2: Systematic uncertainties for the measurement of the corrected ratio of event yields used for the measurements of f s /f d and the relative branching fraction of B 0 → D − K + . The systematic uncertainty in p T and η bins is shown as a range in the last column, and the total systematic uncertainty is the quadratic sum of the uncorrelated uncertainties. The systematic uncertainties on the ratio of B 0 → D − π + and B 0 s → D − s π + yields that are correlated among the bins do not affect the dependence on p T or η, and are not accounted for in the total systematic uncertainty. of 0.4% relative to the signal yield is found in the B 0 → D − π + decay mode, and no contribution is seen in the other modes. For the B 0 → D − π + decay mode no correction is applied and the full size is taken as an uncertainty. No systematic uncertainty is assigned for the other decay modes. The various sources of the systematic uncertainty that contribute to the uncertainties on the ratios of signal yields are listed in Table 2. No uncertainty is associated to the Λ 0 b → Λ − c π + background, as the yield is allowed to vary in the fit. Other cross checks, like varying the show a negligible effect on the signal yields.
All systematic variations are also performed in bins, and the corresponding relative changes in the ratio of yields have been quantified. Variations showing correlated behaviour do not affect the slope and are therefore not considered further.

Results
The relative signal yields of the decays B 0 → D − π + , B 0 → D − K + and B 0 s → D − s π + are used to determine the branching fraction of the decay B 0 → D − K + , and the ratio of fragmentation fractions f s /f d .
The efficiency corrected ratio of B 0 → D − K + and B 0 → D − π + signal yields results in the ratio of branching fractions = 0.0822 ± 0.0011 (stat) ± 0.0025 (syst). This is combined with the world average branching fraction B (B 0 → D − π + ) = (26.8 ± 1.3) × 10 −4 [2], to give B B 0 → D − K + = (2.20 ± 0.03 ± 0.07 ± 0.11) × 10 −4 , where the first uncertainty is statistical, the second is systematic and the last is due to the uncertainty on the B 0 → D − π + branching fraction. The ratio of fragmentation fractions is determined from the efficiency corrected event yields. The ratio of efficiencies is 0.913 ± 0.027. This results in f s f d = (0.261 ± 0.004 ± 0.017) × 1 N a N F = 0.238 ± 0.004 ± 0.015 ± 0.021 , where the first uncertainty is statistical, the second is systematic containing the sources listed in Table 2 as well as errors from external measurements, and the third is theoretical, due to the knowledge of N a and N F . The last source is dominated by the uncertainty on the form factor ratio.
This measurement supersedes and is in agreement with the previous determination with hadronic decays [3]. It also agrees with the previous measurement based on semileptonic

Conclusions
The relative production rate of B 0 s and B 0 mesons is determined using the hadronic decays B 0 s → D − s π + and B 0 → D − K + resulting in f s /f d = 0.238 ± 0.004(stat) ± 0.015(syst) ± 0.021(theo). This value is consistent with a previous LHCb measurement based on semileptonic decays, with which it is averaged to obtain f s /f d = 0.256 ± 0.020. The ratio of fragmentation fractions f s /f d is determined as a function of the transverse momentum and pseudorapidity of the B 0 (s) meson, and a variation consistent with a linear dependence on the transverse momentum of the the B 0 (s) meson is observed, with a significance of three standard deviations. In addition, the ratio of branching fractions of the decays B 0 → D − K + and B 0 → D − π + is measured to be 0.0822 ± 0.0011 (stat) ± 0.0025 (syst).