Measurement of the branching fraction ratio $\mathcal{B}(B_c^+ \rightarrow \psi(2S)\pi^+)/\mathcal{B}(B_c^+ \rightarrow J/\psi \pi^+)$

Using $pp$ collision data collected by LHCb at center-of-mass energies $\sqrt{s}$ = 7 TeV and 8 TeV, corresponding to an integrated luminosity of 3 fb$^{-1}$, the ratio of the branching fraction of the $B_c^+ \rightarrow \psi(2S)\pi^+$ decay relative to that of the $B_c^+ \rightarrow J/\psi\pi^+$ decay is measured to be 0.268 $\pm$ 0.032 (stat) $\pm$ 0.007 (syst) $\pm$ 0.006 (BF). The first uncertainty is statistical, the second is systematic, and the third is due to the uncertainties on the branching fractions of the $J/\psi \rightarrow \mu^+\mu^-$ and $\psi(2S) \rightarrow \mu^+\mu^-$ decays. This measurement is consistent with the previous LHCb result, and the statistical uncertainty is halved.

was measured to be 0.250 ± 0.068 (stat) ± 0.014 (syst) ± 0.006 (BF).The first uncertainty is statistical, the second is systematic, and the third is due to the uncertainties on the branching fractions of the J/ψ → µ + µ − and ψ(2S) → µ + µ − decays.The statistical uncertainty is dominant.Several theoretical predictions for R B based on different effective models [14][15][16][17][18][19] exist, and vary between 0.07 and 0.29.The analysis presented here updates the previous LHCb measurement of R B [8], using the full pp collision data collected by LHCb in 2011 and 2012 at √ s = 7 TeV and 8 TeV respectively, corresponding to an integrated luminosity of 3 fb −1 .Due to the increased data sample and an improved analysis method, the statistical uncertainty is reduced by half, allowing a more powerful test of the theories.
The LHCb detector [20,21] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, and is 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 [22], 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 [23] placed downstream of the magnet.The tracking system provides a measurement of 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 vertex (PV), the impact parameter, is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of the track momentum transverse to the beam, in GeV/c.Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [24].Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter.Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [25].The online event selection is After the full selection, the background in the B + c → J/ψ π + sample consists of three categories: combinatorial background; partially reconstructed background, mainly from B + c → J/ψ ρ + decays with ρ + → π + π 0 , where the π 0 is not reconstructed; and contamination from the Cabibbo-suppressed decay, B + c → J/ψ K + , with the kaon misidentified as a pion.The background in the B + c → ψ(2S)π + sample consists of a combinatorial background and a partially reconstructed background.The contribution from B + c → ψ(2S)K + is negligible.
The signal yields are extracted from unbinned extended maximum likelihood fits to the invariant mass distributions of J/ψ π + or ψ(2S)π + in the range [6027, 6527] MeV/c 2 , as shown in Figs. 1 and 2 for 2011 and 2012 data, and are summarized in Tables 1 and 2. To improve the B + c mass resolution, the masses of J/ψ and ψ(2S) candidates are constrained to their known values [27].For the B + c → J/ψ π + channel, the signal probability density function is modelled by the sum of two double-sided Crystal Ball functions [36], with the same mean value and tail parameters determined from simulation; the combinatorial background is described with an exponential function; and the partially reconstructed background is modelled with the distribution of the B + c invariant mass obtained from a simulated B + c → J/ψ ρ + sample using a kernel estimation [37].This last shape is convolved with a Gaussian distribution to take into account a difference in mass resolution between data and simulation.For the B + c → J/ψ K + background, the shape of the B + c mass distribution is modelled by a double-sided Crystal Ball function with parameters determined from simulation.For the B + c → ψ(2S)π + channel, due to the limited statistics, the signal shape is modelled by a single double-sided Crystal Ball function with the tail parameters determined from simulation; the combinatorial and partially reconstructed backgrounds are described with the same models as used for the B + c → J/ψ π + channel.The total selection efficiency is the product of the detector geometrical acceptance, the trigger efficiency, the reconstruction and selection efficiency, the PID efficiency, and the BDT classifier efficiency.All efficiencies are determined using simulated samples.To account for any discrepancy between data and simulation, the PID efficiencies are calibrated using a π + sample from D * -tagged D 0 → K − π + decays.The BDT classifier efficiencies of BDT1 and BDT2 are slightly different.After correcting for the BDT classifier efficiencies the signal yields of the sub-samples are consistent within the statistical uncertainties.The BDT classifier efficiency, ε BDT , and the product of all other efficiencies, ε * , are listed in   Tables 1 and 2.
Several sources of systematic uncertainty on the R B measurement are studied and are summarized in Table 3.To account for the uncertainty due to the signal shape modelling, the data are refitted with an alternative shape.The B + c invariant mass distributions are modelled by a kernel estimation convolved with a Gaussian function, as determined from simulation.A difference of 0.6% from the nominal result is observed and is taken as a systematic uncertainty.
The modelling of the partially reconstructed background can also introduce a systematic uncertainty.This is estimated by reducing the fit range to [6164, 6527] MeV/c 2 to exclude its contribution.A change of 2.4% in the result is observed.In the nominal fits, the parameters for B + c → J/ψ K + and the partially reconstructed background are fixed; the results change by less than 1% when these parameters are allowed to vary.The systematic uncertainty due to background modelling is estimated to be 2.4%.
Systematic uncertainties on the R B measurement can be introduced by the BDT  classifier efficiency if the simulation fails to describe the data.The distributions of all training variables from simulation and background-subtracted data are compared, where the background subtraction is performed using the sPlot technique, taking the B + c invariant mass as the discriminating variable [38].They are generally in agreement within statistical fluctuations.Only one variable, which describes the consistency between the pion track and the PV, indicates small differences between simulation and data.Therefore, the simulated sample is reweighed to match the data, and the BDT efficiencies are recalculated with the reweighed simulated sample.The result obtained with these BDT efficiencies is different from the nominal value by 0.2%, which is taken as the uncertainty from the BDT classifier.
The efficiences determined from simulated samples have uncertainties due to the limited statistics.This leads to an uncertainty of 0.3%.An uncertainty of 1.1% is assigned due to imperfect simulation of the trigger, which is determined using data driven methods [39,40].The B + c lifetime of simulated samples is set according to the latest LHCb measurement [4].To estimate the systematic uncertainty due to this, the B + c lifetime is varied within the uncertainty of this measurement, and the change in the result, 0.1%, is taken as a systematic uncertainty.The total systematic uncertainty is 2.7%.
The ratio of the branching fractions with J/ψ and ψ(2S) mesons decaying to dimuons, denoted as is calculated as where N cor 2011(2012) are the signal yields from 2011(2012) after efficiency correction.The ratio is measured to be R = 0.0354 ± 0.0042 (stat) ± 0.0010 (syst).
The ratio of the branching fractions of B + c → ψ(2S)π + and B + c → J/ψ π + is calculated as Assuming electroweak universality, the J/ψ → µ + µ − and ψ(2S) → µ + µ − branching fractions can be substituted with the more precisely measured ones in the e + e − channel [27].Using these values, the ratio R B is measured to be: R B = 0.268 ± 0.032 (stat) ± 0.007 (syst) ± 0.006 (BF), where the first uncertainty is statistical, the second is systematic, and the last term is due to the uncertainty on B(J/ψ → e + e − )/B(ψ(2S) → e + e − ).This result is in agreement with the previous LHCb result [8].Our measurement is consistent with the predictions of non-relativistic QCD at next-to-leading order (0.26 +0.05 −0.06 ) [18] and perturbative QCD based on k T factorization (0.29 +0.17 −0.11 ) [19].The result disfavors the theoretical calculations based on the relativistic quark model [14], the quark potential model [15], the relativistic constituent quark model [16] and the QCD relativistic potential model [17].

Figure 1 :
Figure 1: Fit to the reconstructed B + c mass distribution for B + c → J/ψ π + using (top) 2011 and (bottom) 2012 data samples.The plots on the left (right) correspond to the data selected with BDT1 (BDT2).Black points with error bars represent the data, and the various components are indicated in the keys.

Figure 2 :
Figure 2: Fit to the reconstructed B + c mass distribution for B + c → ψ(2S)π + using (top) 2011 and (bottom) 2012 data samples.The plots on the left (right) correspond to the data selected with BDT1 (BDT2).Black points with error bars represent the data, and the various components are indicated in the keys.

Table 1 :
Summary of the signal yields and efficiencies for the B + c → J/ψ π + decay.

Table 2 :
Summary of the signal yields and efficiencies for the B + c → ψ(2S)π + decay.