Search for excited $B_c^+$ states

A search is performed in the invariant mass spectrum of the $B_c^{+}\pi^{+}\pi^{-}$ system for the excited $B_c^{+}$ states $B_c(2^{1}S_{0})^+$ and $B_c(2^{3}S_{1})^+$ using a data sample of $pp$ collisions collected by the LHCb experiment at the centre-of-mass energy of $\sqrt{s} = 8 \,{\mathrm{TeV}}$, corresponding to an integrated luminosity of $2 \,{\mathrm{fb^{-1}}}$. No evidence is seen for either state. Upper limits on the ratios of the production cross-sections of the $B_c(2^{1}S_{0})^+$ and $B_c(2^{3}S_{1})^+$ states times the branching fractions of ${B_c(2^{1}S_{0})^+} \to {B_c^{+}\pi^{+}\pi^{-}}$ and ${B_c(2^{3}S_{1})^+} \to {B_c^{*+}\pi^{+}\pi^{-}}$ over the production cross-section of the $B_c^{+}$ state are given as a function of their masses. They are found to be between 0.02 and 0.14 at $95\%$ confidence level for $B_c(2^{1}S_{0})^+$ and $B_c(2^{3}S_{1})^+$ in the mass ranges $[6830, 6890] \,{\mathrm{MeV}}/c^{2}$ and $[6795,6890] \,{\mathrm{MeV}}/c^{2}$, respectively.


Detector and simulation
The LHCb detector [22,23] 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 siliconstrip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector (TT) 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 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 (IP), is measured with a resolution of (15 + 29/p T ) µm, where p T 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 calorimeter 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 stage, events are required to have at least one muon with high p T or a hadron with high transverse energy. At the software stage, two muon tracks or three charged tracks are required to have high p T and to form a secondary vertex with a significant displacement from the interaction point.
In the simulation, pp collisions are generated using Pythia 6 [24] with a specific LHCb configuration [25]. The generator Bcvegpy [19] is used to simulate the production of B c mesons. Decays of hadronic particles are described by EvtGen [26], in which final-state radiation is generated using Photos [27]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [28] as described in Ref. [29]. In the default simulation, the masses of the excited B c states are set as

Event selection
To select B + c → J/ψ π + decays, J/ψ candidates are formed from pairs of opposite-charge tracks. The tracks are required to have p T larger than 0.55 GeV/c and good track-fit quality, to be identified as muons, and to originate from a common vertex. Each J/ψ candidate with an invariant mass between 3.04 GeV/c 2 and 3.14 GeV/c 2 is combined with a charged pion to form a B + c candidate. The pion is required to have p T > 1.0 GeV/c and good track-fit quality. The J/ψ candidate and the charged pion are required to originate from a common vertex, and the B + c candidates must have a decay time larger than 0.2 ps. Each of the particles is associated to the PV that has the smallest χ 2 IP , where χ 2 IP is defined as the difference in the vertex-fit χ 2 of a given PV reconstructed with and without the particle under consideration. The χ 2 IP of the B + c (π + ) candidate is required to be < 25 (> 9) with respect to the associated PV of the B + c candidate. To further suppress background, a requirement on a boosted decision tree (BDT) [30,31] classifier is applied. The BDT classifier uses information from the χ 2 IP of the two muons, the pion, the J/ψ , and the B + c mesons with respect to the associated PV; the p T of both muons, the J/ψ and π + mesons; and the decay length, decay time, and the vertex-fit   [32] to improve the B + c mass resolution. 3 The B + c signal yield is obtained by performing an unbinned extended maximum likelihood fit to the M (J/ψ π + ) mass distribution, as shown in Fig. 1. The signal component is modelled by a Gaussian function with asymmetric power-law tails as determined from simulation. The mean and resolution of the Gaussian function are free parameters in the fit. The combinatorial background is described with an exponential function. The contamination from the Cabibbo-suppressed channel B + c → J/ψ K + , with the kaon misidentified as a pion, is described by a Gaussian function with asymmetric power-law tails. The parameters are also fixed from simulation, with only the Gaussian mean related to the B + c → J/ψ π + signal as a free parameter to account for the possible small mass difference in data and simulation. The signal yield of B + c decays is determined to be 3325 ± 73.
To reconstruct the B c (2S) + mesons are constrained to originate from the associated PV. To optimise the sensitivity of the analysis, a selection based on a multilayer perceptron (MLP) [35] classifier is applied. To distinguish the signal candidates from combinatorial background, the MLP classifier uses information on the angles between the B + c and π + , B + c and π − , and π + and π − candidate momenta projected in the plane transverse to the beam axis; the angles between the B ( * ) c (2S) + momentum and the B + c , π + , and π − momenta in the B Therefore, the combination of the simulated candidates for the decays B c (2S) The differences between the mass resolutions in data and simulation are evaluated with the control decay mode B + c → J/ψ π + π − π + , which has the same final state as the signal and a large yield, and are corrected by applying a scale factor. The obtained mass distributions are consistent with the background-only hypothesis, as determined by the scan described below.

Upper limits
As no significant B where σ is the production cross-section, N the yield, and ε the efficiency of reconstructing and selecting the B + c or B where N obs is the number of observed candidates, N B is the expected background yield, and N S is the expected signal yield. For a given value of the ratio R, N S is determined as The likelihood L is defined as The total statistical test value Q tot is the product of that for each of the four MLP categories. The CL s value is the ratio of CL s+b to CL b , where CL s+b is the probability to find a Q tot value smaller than the Q tot value found in the data sample under the signal-plus-background hypothesis, and CL b is equivalent probability under the background-only hypothesis. The   mass hypotheses. This choice of the search window gives the best sensitivity according to Ref. [37]. The selection efficiencies ε B + c and ε B ( * ) c (2S) + are estimated using simulation. The track reconstruction efficiency is studied in a data control sample of J/ψ → µ + µ − decays using a tag-and-probe technique [38], in which one of the muons is fully reconstructed as the tag track, and the other muon, the probe track, is reconstructed using only information from the TT detector and the muon stations. The track reconstruction efficiency is the fraction of J/ψ candidates whose probe tracks match fully reconstructed tracks. The particle-identification (PID) efficiency of the two opposite-charge pions is determined with a data-driven method, using a π + sample from D * -tagged D 0 → K − π + decays. The total efficiency ε B + c is determined to be 0.0931 ± 0.0005, where the uncertainty is the statistical uncertainty of the simulated sample. The B c π + π − ) distribution of the same-sign sample, which is constructed with B + c π + π + or B + c π − π − combinations. The sources of systematic uncertainties that affect the upper limit calculation are studied and summarised in Table 2. The systematic uncertainty on N B + c comes from the potentially imperfect modelling of the signal, and has been studied using pseudoexperiments. The uncertainty on ε B + c is due to the limited size of the simulated sample. The uncertainty on N B comes both from differences between the combinatorial backgrounds in the opposite-sign and the same-sign data samples and from the potential mismodelling of the background. The former is studied by performing a large set of pseudoexperiments, in which the samples are generated by randomly taking candidates from the data sample, while the candidates in M (B + c π + π − ) ∈ [6785, 6900] MeV/c 2 are taken from the same-sign sample. The M (B + c π + π − ) distributions of the pseudosamples are fit using the same function as in the nominal background modelling. The difference between the mean value of N B obtained from the pseudoexperiments and the nominal value is taken as the systematic uncertainty. The potential mismodelling of the background is estimated by using the Bukin function [39] as an alternative model and the differences to the nominal results are taken as systematic uncertainties. The uncertainties on ε B ( * ) c (2S) + are dominated by the uncertainty due to the finite size of the simulated samples, but also include the systematic uncertainties on the PID and track reconstruction efficiency calibration, which come from the limited size and the binning scheme of the calibration samples. The variations of efficiency with respect to M (B c (2S) + ) and M (B * c (2S) + ) are fitted with linear functions, and the uncertainties of such fits are taken as systematic uncertainties.
No evidence of the B ( * ) c (2S) + signal is observed. The measurement is consistent with the background-only hypothesis for all mass assumptions. The upper limits at 90% and 95% confidence levels (CL) on the ratio R, as functions of the B ( * ) c (2S) + mass states, are shown in Fig. 4. All the upper limits at 95% CL on the ratio R are contained between 0.02 and 0.14. Theoretical models predict that the ratio R has no significant dependence on y and p T of the B + c mesons [19],  Table 3.