Observation of the Decay $D^0\to \rho^-\mu^+\nu_\mu$

By analyzing an $e^+e^-$ annihilation data sample corresponding to an integrated luminosity of $2.93~\mathrm{fb}^{-1}$ collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, we measure the branching fraction of the $D^0\to \rho^- \mu^+\nu_\mu$ decay for the first time. We obtain ${\mathcal B}_{D^0\to \rho^- \mu^+\nu_\mu}=(1.35\pm0.09_{\rm stat}\pm0.09_{\rm syst})\times 10^{-3}$. Using the world average of ${\mathcal B}_{D^0\to \rho^- e^+\nu_e}$, we find a branching fraction ratio of ${\mathcal B}_{D^0\to \rho^- \mu^+\nu_\mu}/{\mathcal B}_{D^0\to \rho^- e^+\nu_e}=0.90\pm0.11$, which agrees with the theoretical expectation of lepton flavor universality within the uncertainty. Combining the world average of ${\mathcal B}_{D^+\to \rho^0 \mu^+ \nu_\mu}$ and the lifetimes of $D^{0(+)}$, we obtain a partial decay width ratio of ${\Gamma}_{D^0\to \rho^- \mu^+ \nu_{\mu}}/(2{\Gamma}_{D^+\to \rho^0 \mu^+ \nu_{\mu}}) = 0.71\pm0.14$, which is consistent with the isospin symmetry expectation of one within $2.1\sigma$. For the reported values of ${\mathcal B}_{D^0\to \rho^- \mu^+\nu_\mu}/{\mathcal B}_{D^0\to \rho^- e^+\nu_e}$ and ${\Gamma}_{D^0\to \rho^- \mu^+ \nu_{\mu}}/2{\Gamma}_{D^+\to \rho^0 \mu^+ \nu_{\mu}}$, the uncertainty is the quadratic sum of the statistical and systematic uncertainties.

For each decay, the difference between the measured branching fraction ratio (R X µ/e = B D→Xµ + νµ /B D→Xe + νe ) and the corresponding SM prediction is less than 1.7σ. The decay D 0 → ρ − µ + ν µ , calculated using the quark potential model in 1989 [23], has not yet been measured. Observation of this decay and verification of the SM prediction for R ρ − µ/e offer a crucial LFU test.
Under the assumption of isospin symmetry, the partial width ratio R ρ,ℓ is expected to be unity. Here, τ D 0(+) is the lifetime of the D 0(+) meson. Using the world average values [35], one obtains R ρ,e IS = 0.87 ± 0.13, which agrees with unity within the uncertainty. A measurement of the branching fraction of the decay D 0 → ρ − µ + ν µ allows a determination of R ρ,µ IS which tests isospin symmetry in D 0(+) → ρ −(0) µ + ν µ decays.
Using a data sample corresponding to an integrated luminosity of 2.93 fb −1 [36] taken at a center-of-mass energy of 3.773 GeV with the BESIII detector, we report the first observation and a branching fraction measurement of D 0 → ρ − µ + ν µ , a determination of |V cd | and tests of both LFU with D 0 → ρ − ℓ + ν ℓ decays and isospin symmetry in D 0(+) → ρ −(0) µ + ν µ decays. Throughout this Letter, charge conjugate channels are always implied and ρ denotes the ρ(770).
Details about the design and performance of the BESIII detector are given in Ref. [37].
Monte Carlo (MC) simulated data samples, produced with a geant4-based [38] software package including the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate the backgrounds.
The simulation includes the beam-energy spread and initialstate radiation in the e + e − annihilations modeled with the generator kkmc [39]. The inclusive MC sample consists of the production of DD pairs with consideration of quantum coherence for all neutral D modes, the non-DD decays of the ψ(3770), the initial-state radiation production of the J/ψ and ψ(3686) states, and the continuum processes. The known decay modes are modeled with evtgen [40] using the branching fractions taken from the Particle Data Group [35], and the remaining unknown decays from the charmonium states are modeled with lundcharm [41]. Final state radiation from charged final state particles is incorporated with the photos package [42]. This analysis assumes that the same form factors are applicable even in the presence of LFU violation. The vector hadronic form factors of the semileptonic decay D 0 → ρ − µ + ν µ are simulated with those of the D 0 → ρ − e + ν e decay [43], which give good data/MC consistency.
At the center-of-mass energy of 3.773 GeV, D 0 and D 0 mesons are produced in pairs without additional hadrons. This feature results in an ideal environment to study D 0 decays with the double-tag (DT) method. At first, the single-tag (ST)D 0 meson is reconstructed using the hadronic decaysD 0 → K + π − , K + π − π 0 , and K + π − π − π + . Then, the DT candidate events, in which a D 0 → ρ − µ + ν µ decay candidate is found in the system recoiling against an STD 0 meson, are selected. The branching fraction of the D 0 → ρ − µ + ν µ decay is determined by where N tot ST and N DT are the yields of the ST and DT candidates in data, respectively. Here, is the effective signal efficiency of finding D 0 → ρ − µ + ν µ in the presence of the STD 0 meson, where ε ST and ε DT are the detection efficiencies of the ST and DT candidates, respectively, and i labels the ST modes.
For theD 0 → K + π − tag mode, backgrounds related to cosmic rays and Bhabha scattering events are vetoed by using the requirements described in Ref. [51]. To distinguish the STD 0 mesons from combinatorial backgrounds, we define the energy difference ∆E ≡ ED0 − E beam and the beam-constrained where E beam is the beam energy, and ED0 and pD0 are the total energy and momentum of the STD 0 candidate in the e + e − centerof-mass frame, respectively. When multiple combinations for an ST mode are present in an event, the combination with the smallest |∆E| per tag mode per charge is retained for further analysis. The ST candidates are required to be within ∆E ∈ (−0.055, 0.040) GeV for D 0 → K + π − π 0 and ∆E ∈ (−0.025, 0.025) GeV for Figure 1 shows the M BC distributions of the accepted STD 0 candidates. For each tag mode, the yield of STD 0 mesons is obtained from a maximum likelihood fit to the M BC distribution of the accepted candidates. In the fit, the signal and background are described by the signal shape from MC simulation and an ARGUS function [52], respectively. To compensate for offsets in calibration and resolution differences between data and MC simulation, the signal shape is convolved with a double-Gaussian function. The means, widths and relative fractions of the Gaussian components are free parameters in the fit. The resulting fits to the M BC distributions are also shown in Fig. 1 In the presence of the STD 0 mesons, candidates for D 0 → ρ − µ + ν µ are selected from the tracks and showers which have not been used in the tag reconstruction. The ρ − candidates are reconstructed via the ρ − → π − π 0 decay. The selection criteria of π − and π 0 candidates are the same as those used in the ST selection. The invariant mass of the π − π 0 candidate is required to be within (0.625, 0.925) GeV/c 2 . To suppress the background from hadronic D 0(+) decays, it is required that there is no additional charged track or π 0 except for those used to form the signal and ST candidates.
The combined information from the specific energy loss in the drift chamber, the time-of-flight system, and the electromagnetic calorimeter (EMC) is used to identify the muon candidates. The combined confidence levels for various particle hypotheses (CL e , CL µ , CL π , and CL K ) are calculated. Charged tracks satisfying CL µ > 0.001, CL µ > CL e , and CL µ > CL K are identified as muons. In muon identification, no requirement of CL µ > CL π is applied because of inefficient separation between muon and pion due to their very close masses. Also, no muon counter information is used because most of muons in D 0 → ρ − µ + ν µ have momenta lower than 0.6 GeV/c, which are too low to leave effective information in muon counter. To reduce misidentification of hadrons as muons, the deposited energy in the EMC of the muon candidate (E µ,EMC ) is required to be in the range (0.125, 0.275) GeV. This requirement suppresses about 40% of total background.
The signal yield of the D 0 → ρ − µ + ν µ decay is determined by a kinematic quantity defined as M 2 miss ≡ E 2 miss /c 4 − | p miss | 2 /c 2 , which is expected to peak around zero for correctly reconstructed signal events. Here, E miss ≡ E beam − E ρ − − E µ + and p miss ≡ p D 0 − p ρ − − p µ + are the missing energy and momentum of the DT event in the e + e − center-of-mass frame, in which E ρ − (µ + ) and p ρ − (µ + ) are the energy and momentum of the ρ − (µ + ) candidates. The M 2 miss resolution is improved using whereˆ pD0 is the unit vector in the momentum direction of the STD 0 and m D 0 is the D 0 nominal mass [35].
To suppress the background from D 0 → K * (892) − (→ K − π 0 )µ + ν µ , the candidate events are further required not to be within the range |M 2 miss π − →K − | < 0.05 GeV 2 /c 4 , where M 2 miss π − →K − is the M 2 miss value calculated by replacing the mass of the charged pion candidate with the kaon mass in the calculation of M 2 miss . Figure 2 shows the M 2 miss distribution of the accepted DT events in data. The semileptonic decay yield is obtained from an unbinned maximum likelihood fit to the M 2 miss distribution. In the fit, the semileptonic signal is modeled by the MC-simulated shape convolved with a Gaussian function describing differences in resolution and calibration between data and MC simulation. The parameters of this Gaussian function are fixed to the values obtained from a similar fit to D 0 → ρ − e + ν e candidate events which have much cleaner environment and comparable momentum resolution. The peaking background of D 0 → π + π − π 0 π 0 is modeled by the M 2 miss shape derived from the D 0 → π + π − π 0 π 0 control sample in data, in which one π 0 is removed and the π + mass is replaced by the µ + mass. The non-peaking backgrounds, including the contribution from wrongly reconstructed ST candidates, are described by the MC-simulated shape obtained from the inclusive MC sample. The yields of the signal, peaking background, and non-peaking backgrounds are free parameters in the fit. The fit result is also shown in Fig. 2. From the fit, we obtain the signal yield of D 0 → ρ − µ + ν µ to be N DT = 570 ± 40 stat and the yield of the peaking background of D 0 → π + π − π 0 π 0 to be 373 ± 36. The statistical significance, calculated by −2ln(L 0 /L max ), is greater than 10σ. Here, L max and L 0 are the maximum likelihoods of the fits with and without the signal component, respectively, and the difference in the number of fit parameters is one.
Inserting N DT , ε D 0 →ρ − µ + νµ , and N tot ST into Eq. (1), we obtain where the first uncertainty is statistical and the second is systematic as discussed below.
In the branching fraction measurement with the DT method, most uncertainties related to the ST selection cancel. Systematic uncertainties arise from the following sources. The uncertainty in the total yield of STD 0 mesons has been studied in Refs. [19,20,44] and is 0.5%. The systematic uncertainties originating from the tracking and PID efficiencies of π ± are 0.3% and 0.2% per pion, respectively, based on an analysis of DT DD hadronic events [53]. The muon tracking and PID efficiencies are studied by analyzing e + e − → γµ + µ − events. Here, the muon identification efficiencies include the E µ,EMC requirement. Using this control sample, data-MC differences are studied in the twodimensional momentum versus cos θ plane. We re-weight using the obtained data-MC differences, accounting for the different distribution of events in momentum versus cos θ for the D 0 → ρ − µ + ν µ signal decays. Systematic uncertainties are obtained as the integral over the re-weighted two-dimensional distribution, giving 0.2% and 0.2% per muon for the muon tracking and PID efficiencies, respectively. The uncertainty of the π 0 reconstruction is studied with DT DD hadronic decays of D 0 → K − π + , K − π + π + π − versusD 0 → K + π − π 0 , K 0 S π 0 [19,44] and is found to be 0.6%. The uncertainty of the combined E max extra γ and N extra π 0 requirements is  Fig. 3. Comparison of five kinematic variables [54,55] of the D 0 → π − π 0 µ + νµ candidates between data (points with error bars) and MC simulation (histograms): the invariant mass of the π − π 0 system, M π − π 0 ; the invariant mass squared of the µ + νµ system, q 2 ; the angle between the momentum of the µ + (π − ) in the µ + νµ (π − π 0 ) rest frame and the momentum of the µ + νµ (π − π 0 ) system in the D 0 rest frame, θµ (θπ); and the angle between the normals of the decay planes defined in the D 0 rest frame by the π − π 0 pair and the µ + νµ pair, χ. Pink and blue histograms denote the peaking BKG and CBKG components, respectively. Except for M π − π 0 to be shown, events have been imposed with all requirements described in text and |M 2 miss | < 0.025 GeV 2 /c 4 . In the M π − π 0 distribution, pair of red arrows indicate the ρ − mass window. estimated to be 1.3% by analyzing the DT candidate events of D 0 → π − π 0 e + ν e .
The uncertainty of the M 2 miss fit is found to be 6.6% by examining the branching fraction changes with an alternative signal shape without Gaussian smearing of the MC-simulated signal shape (0.9%), an MC-simulated shape of the peaking background of D 0 → π + π − π 0 π 0 (5.3%), and combinatorial background shapes after varying the quoted branching fractions by ±1σ for the two main combinatorial components of D 0 → K 0 S π + π − π 0 and D 0 → K * (892) − µ + ν µ (3.8%). The uncertainty arising from the finite MC statistics used to determine the efficiencies is 0.7%. The uncertainty due to the signal MC model is 0.3%, determined by the difference between our nominal DT efficiency and that determined by varying the input form factors by ±1σ. Systematic uncertainties from other selection criteria are found to be negligible. Adding these uncertainties in quadrature yields a total systematic uncertainty of 6.8%.