Measurement of the Branching Fraction For the Semi-Leptonic Decays $D^{0(+)}\to \pi^{-(0)}\mu^+\nu_\mu$ and Test of Lepton Flavor Universality

Using a data sample corresponding to an integrated luminosity of $2.93\,\rm fb^{-1}$ taken at a center-of-mass energy of 3.773\,GeV with the BESIII detector operated at the BEPCII collider, we perform an analysis of the semi-leptonic decays $D^{0(+)}\to \pi^{-(0)}\mu^+\nu_\mu$. The branching fractions of $D^0\to \pi^-\mu^+\nu_\mu$ and $D^+\to \pi^0\mu^+\nu_\mu$ are measured to be $(0.272 \pm 0.008_{\rm stat.} \pm 0.006_{\rm syst.})\%$ and $(0.350 \pm 0.011_{\rm stat.} \pm 0.010_{\rm syst.})\%$, respectively, where the former is of much improved precision compared to previous results and the latter is determined for the first time. Using these results along with previous BESIII measurements of $D^{0(+)}\to \pi^{-(0)}e^+\nu_e$, we calculate the branching fraction ratios to be ${\mathcal R}^0\equiv {\mathcal B}_{D^{0}\to \pi^{-}\mu^+\nu_\mu}/{\mathcal B}_{D^{0}\to \pi^{-}e^+\nu_e}=0.922\pm 0.030_{\rm stat.}\pm0.022_{\rm syst.}$ and ${\mathcal R}^+\equiv {\mathcal B}_{D^{+}\to \pi^{0}\mu^+\nu_\mu}/{\mathcal B}_{D^{+}\to \pi^{0}e^+\nu_e}=0.964\pm 0.037_{\rm stat.}\pm0.026_{\rm syst.}$, which are compatible with the theoretical expectation of lepton flavor universality within $1.7\sigma$ and $0.5\sigma$, respectively. We also examine the branching fraction ratios in different four-momentum transfer square regions, and find no significant deviations from the standard model predictions.

This Letter presents tests of LFU in D 0(+) → π −(0) + ν decays [14] at BESIII. Recently, the Cabibbofavored decays D 0(+) →K + ν were precisely studied at BESIII, and the measured BF ratios (BFRs) B D→Kµ + νµ /B D→Ke + νe are compatible with the SM expectations [15][16][17][18]. Nevertheless, tension between previous measurement and the SM prediction for the Cabibbosuppressed decays D 0 → π − + ν is found. In the SM, the BFRs R 0(+) LFU = B D 0(+) →π −(0) µ + νµ /B D 0(+) →π −(0) e + νe are expected to be 0.985 ± 0.002 [19], which deviates from unity due to different phase space available to the two processes. With the world-average values of B D 0 →π − µ + νµ and B D 0 →π − e + νe [20], R 0 LFU is 17% lower than the SM prediction, corresponding to 2.1 standard deviations. Currently, the most precise measurements of B D 0(+) →π −(0) e + νe have reached an accuracy better than 3% [15,16]. However, the world-average value of B D 0 →π − µ + νµ has a large relative uncertainty of 10% [20][21][22], and the decay D + → π 0 µ + ν µ has not been measured. To clarify this tension, it is crucial to precisely measure The analysis is performed by using a data sample corresponding to an integrated luminosity of 2.93 fb −1 [23] taken at a center-of-mass energy of √ s = 3.773 GeV with the BESIII detector. Details about the design and performance of the BESIII detector are given in Ref. [24]. A geant4-based [25] Monte Carlo (MC) simulation software package, which includes a description of the detector geometry and its response, is used to determine the detection efficiency and to estimate potential backgrounds. An 'inclusive' MC sample corresponding to about 10 times the luminosity of data is produced at √ s = 3.773 GeV. It includes the D 0D0 , D + D − , and non-DD decays of ψ(3770), the initial state radiation (ISR) production of ψ(3686) and J/ψ, and the qq (q = u, d, s) continuum process, along with Bhabha scattering, µ + µ − and τ + τ − events. The production of ψ(3770) is simulated by the MC generator kkmc [26]. The measured decay modes of the charmoniums are generated using EvtGen [27] with the BFs reported in Ref. [28], and the remaining decay modes are generated using LundCharm [29]. The signal D 0(+) → π −(0) µ + ν µ decays are simulated incorporating the modified pole model [30], where the parameters of vector and scalar hadronic form factors (HFFs) are taken from Refs. [15,16,31]. The ISR effects [32] and final state radiation (FSR) effects of all particles [33] have been included in the event generation.

0(+) ST
and N 0(+) DT are the ST and DT yields in data, πµν is the signal efficiency of finding D 0(+) → π −(0) µ + ν µ events in the presence of a STD meson. Here, are the ST yield and the ST[DT] efficiency of the k th tag mode, respectively.
All charged tracks are required to be within a polarangle range of |cosθ| < 0.93. Except for those from K 0 S decays, the good charged tracks are required to come from the interaction region defined by V xy < 1 cm and |V z | < 10 cm, where V xy and |V z | are the distances of closest approach of the reconstructed track to the interaction point (IP) in the xy plane and the z direction (along the beam), respectively. Charged particle identification (PID) is performed by combining the time-offlight information with the specific ionization energy loss measured in the main drift chamber. The information of the electromagnetic calorimeter (EMC) is also included to identify muon candidates. Combined confidence levels for electron, muon, pion and kaon hypotheses (CL e , CL µ , CL π and CL K ) are calculated individually. The kaon and pion are required to satisfy CL K > CL π and CL π > CL K , respectively, while muon candidates are selected with CL µ > 0.001, CL µ > CL e and CL µ > CL K . Additionally, muon candidates are required to deposit an energy in the EMC within the range (0.1, 0.3) GeV and to satisfy a polar angle and momentum dependent hit depth criterion in the muon counter (MUC) [34]; these criteria suppress the number of pions misidentified as muons. The K 0 S candidate is reconstructed from two oppositely charged tracks with |V z | < 20 cm. These two charged tracks are assumed to be pions (without PID), constrained to a common vertex and are required to have an invariant mass satisfying [20]. A selected K 0 S candidate must have a decay length larger than two times of the vertex resolution away from the IP. Photon candidates are selected from the shower clusters in the EMC that are not associated with a charged track. The shower time is required to be within 700 ns of the event start time, its energy is required to be greater than 25 (50) MeV in the EMC barrel (endcap) region [24]. The opening angle between the shower and any charged tracks must be greater than 10 • . A π 0 candidate is reconstructed from a γγ pair with an invariant mass M γγ within (0.115, 0.150) GeV/c 2 . A kinematic fit constrain-ing M γγ to the π 0 nominal mass [20] is imposed to improve its momentum resolution.
The STD mesons are identified by the energy difference ∆E ≡ ED − E beam and the beam-constrained mass Here, E beam is the beam energy, pD and ED are the momentum and energy of thē D candidate in the e + e − rest frame. For each ST mode, if there are multiple candidates in an event, only the one with the smallest |∆E| is kept. The ST candidates are required to have ∆E ∈ (−55, 40) MeV and (−25, 25) MeV for the modes with and without a π 0 in the final states, respectively. For the ST candidates ofD 0 → K + π − , the backgrounds from cosmic rays and Bhabha events are further rejected using the requirements described in Ref. [35]. After the above selection criteria, the ST yields are obtained by performing maximum likelihood fits to the M BC distributions for individual ST modes, as shown in Fig. 1. In the fits, theD signal is modeled by a MCsimulated shape convolved with a double Gaussian function that describes any resolution difference between data and MC simulation. For individual tags, the peaks and resolutions of the convolved Gaussian functions fall in the regions of (−0.3, 0.3) MeV/c 2 and (0.7, 3.2) MeV/c 2 , respectively. The combinatorial background is described by an ARGUS function [36]. The candidates in the M BC signal regions, defined as (1.859, 1.873) GeV/c 2 and (1.863, 1.877) GeV/c 2 forD 0 and D − , respectively, are kept for further analysis. In the part of the event recoiling against the STD meson, the SL decay candidate is selected from the remaining tracks that have not been used for tag reconstruction. Events containing a muon candidate, with opposite charge to the STD candidate, and a π −(0) candidate are considered as SL D 0(+) decays. We require there are no additional charged tracks in the event. The potential backgrounds from D 0 → K − π + , D 0(+) → π −(0) π + and D 0(+) → π −(0) π + π 0 /η/K 0 are suppressed by the optimized requirements of M π −(0) µ + < 1.7 GeV/c 2 and E extra γ max < 0.07 GeV, where M π −(0) µ + is the π −(0) µ + invariant mass and E extra γ max is the maximum energy of any additional photon candidates unused in the DT reconstruction. The relative efficiencies of the requirements on M π −(0) µ + and E extra γ max are approximately 99% and 70%, respectively. To further reject the peaking backgrounds of D 0 → K 0 S (π + π − )π 0 and D + →K 0 π + for D 0 → π − µ + ν µ and D + → π 0 µ + ν µ , we require M π − µ + and M rec D − µ + (D − µ + recoil mass) to be outside the ranges (0.46, 0.50) GeV/c 2 and (0.45, 0.55) GeV/c 2 , respectively. The undetected neutrino is inferred from the variable M 2 miss ≡ E 2 miss /c 4 − | p miss | 2 /c 2 , which peaks at zero for signal events. Here E miss and | p miss | are the missing energy and momentum calculated by and p π −(0) ( p µ + ) are the energy and momentum of π −(0) (µ + ) in the rest frame of e + e − system. Furthermore, p D ≡ (−pD) E 2 beam /c 2 − M 2 D c 2 is the momentum of D meson, wherepD is the momentum direction of the STD meson and M D is the D nominal mass [20]. Figure 2 shows the M 2 miss distributions of the selected DT candidates for D 0 → π − µ + ν µ and D + → π 0 µ + ν µ . Both the candidate events contain two peaks corresponding to the D 0(+) → π −(0) µ + ν µ signals and the D 0(+) → π −(0) π +K 0 backgrounds (named BKGI) at zero and 0.25 GeV 2 /c 4 , respectively. MC studies indicate that the small peaking backgrounds from decays D 0 → K − π + , D 0(+) → π −(0) π + and D 0(+) → π −(0) π + π 0 (named BKGII) peak around 0.02 GeV 2 /c 4 , under the right side of signal. The DT signal yields are determined by performing unbinned maximum likelihood fits on the M 2 miss distributions. In the fits, the signals, the peaking backgrounds of BKGI and BKGII and other non-peaking backgrounds (named BKGIII) are described by the corresponding MC-simulated shapes. The signal, BKGI and BKGII shapes are smeared with Gaussian functions with free parameters to take into account the resolution difference between data and MC simulation. The parameters of the Gaussian function for BKGII are the same as those for the signal, while those for BKGI can be different. All but one of the BKGII peaking background yields are fixed to the values from MC simulation; the exception is the D 0 → π + π − π 0 background to the D 0 → π − µ + ν µ signal, which is determined from data due to its good separation from the signal. All the other background component yields are floated in the fit.
The ST and DT yields, the detection efficiencies and the obtained BFs are shown in Table I. In BF measurements using the DT method, the uncertainties from the ST selection mostly cancel. The relative systematic uncertainties from the different sources considered are shown in Table 2. The uncertainty from the ST yield is taken as 0.5% by examining its relative change between data and MC simulation by varying the fit range, signal shape and endpoint of ARGUS function. The efficiencies of µ + and π − tracking (PID) and π 0 reconstruction are verified using e + e − → γµ + µ − events and DT DD hadronic events, respectively. We assign the uncertainties of π − tracking (PID), µ + tracking (PID) and π 0 reconstruction to be 0.5% (0.5%), 0.5% (0.5%) and 1.0%, respectively. The uncertainty related to the choice of the E extra γ max requirement is assigned by analyzing the control sample D 0(+) → π −(0) e + ν e ; it is 1.2% (1.7%) for the D 0(+) decay. The uncertainty associated with the M πµ + requirement is investigated by using the alternative requirements of 1.65 GeV/c 2 or 1.75 GeV/c 2 . The uncertainty due to the K 0 S veto is estimated by varying the M π − µ + (M rec D − µ + ) requirement by ±0.01 GeV/c 2 . The changes to the measured BFs with the different requirements are taken as the systematic uncertainties. The uncertainties related to the M 2 miss fits are investigated by varying the fit ranges by ±0.025 (0.050) GeV 2 /c 4 for D 0(+) decays, and with different parameterizations of signals, combinatorial and peaking backgrounds. The effects due to signal shapes are estimated with different requirements on the MC-truth matched signal shapes. The relative magnitudes of the dominant combinatorial background components in BKGIII are varied by ±20%. The fixed magnitudes of the dominant peaking backgrounds in BKGII are changed according to the BF uncertainties [20], the limited MC statistics of background channels, and the data-MC differences of the rates of misidentifying K − as π − and π + as µ + . The maximum changes of BFs are taken as their respective uncertainties. The uncertainties due to limited MC statistics are 0.3% for both decays. The uncertainty related to MC generator assumptions is estimated to be 0.3% via comparing the DT efficiencies by varying the quoted vector HFF param-eters by ±1 standard deviation and replacing the nominal scalar HFF model with the simple pole model [30]. The uncertainty due to FSR effect is assigned as 0.3%, which is obtained by comparing the nominal DT efficiency to that when the FSR photon probability is changed by ±20%. The total systematic uncertainty is the quadratic sum of the individual contributions.   Combining the B D 0(+) →π −(0) µ + νµ measured in this work with previous BESIII measurements [15,16] B D 0 →π − e + νe = (0.295 ± 0.004 stat. ± 0.003 syst. )% and B D + →π 0 e + νe = (0.363 ± 0.008 stat. ± 0.005 syst. )%, we obtain R 0 LFU = 0.922 ± 0.030 stat. ± 0.022 syst. and R + LFU = 0.964±0.037 stat. ±0.026 syst. . Here, the systematic uncertainties in ST yields, π − tracking and PID, and π 0 reconstruction cancel, and an additional uncertainty of 0.5% is included to take into account different FSR effects for electron and muon. The measured values of R 0(+) LFU coincide with the SM expectation 0.985 ± 0.002 [19] within 1.7σ (0.5σ).
The BFRs R 0(+) LFU are obtained in the full q 2 (fourmomentum transfer square of µ + ν µ ) region. To investigate the q 2 dependence of R 0(+) LFU , we examine BFRs in different q 2 ranges. Using the method described in Refs. [15,16], the partial width of D 0(+) → π −(0) µ + ν µ in the i th q 2 bin is calculated by where τ D 0(+) is the lifetime of the D 0(+) meson, and N 0(+) i is the produced DT yield in the i th q 2 bin, calculated by N Here M 0(+) j is the observed DT yield in the j th q 2 bin, 0(+) is the efficiency matrix and ( 0(+) ) ij are the elements of a matrix that describes the efficiency and smearing across q 2 bins. See Supplemental Material [37] for the observed and produced DT yields, efficiency matrices as well as the partial widths for D 0(+) → π −(0) µ + ν µ . Combining with the measured partial widths for D 0(+) → π −(0) e + ν e in the same q 2 bins [15,16], we obtain R 0(+) LFU in various q 2 bins. Figure 3 shows ∆Γ  /∆q 2 of D 0(+) → π −(0) e + νe are quoted from Refs. [15,16]. Data are shown as dots with error bars, where the uncertainties are combined from statistical and systematic errors, and the uncertainties in R 0(+) LFU are dominated by the statistical uncertainties of semi-muonic modes. The blue, green and black curves with bands show the LQCD predictions with uncertainties, using the equations and HFF parameters described in Refs. [19,38], where the theoretical uncertainties in R 0(+) LFU are tiny due to strong correlation of the form factors.
In summary, using 2.93 fb −1 e + e − collision data collected at √ s = 3.773 GeV with the BESIII detector, we have measured the BFs of D 0 → π − µ + ν µ and D + → π 0 µ + ν µ . The value of B D 0 →π − µ + νµ is consistent with the world-average value [20] and has much improved precision; B D + →π 0 µ + νµ is determined for the first time. Combining the previous BESIII measurements of D 0(+) → π −(0) e + ν e , we calculate the q 2 -integrated and q 2 -dependent BFRs, and find no significant evidence of LFU violation.
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong sup-port. Authors thank S. Simula, L. Riggio Table I shows tag dependent ST yields and efficiencies, DT efficiencies and signal efficiencies of D 0(+) → π −(0) µ + ν µ . Table II and Table III present the efficiency matrices ( 0(+) ) ij , the range of each q 2 bin, the number of the observed DT events M j , the number of produced DT events N i , and the partial decay rate ∆Γ i in each q 2 bin for D 0 → π − µ + ν µ and D + → π 0 µ + ν µ decays, respectively.  Table II: Summary of the efficiency matrix ( 0(+) )ij (in %), the range of each q 2 bin (in GeV 2 /c 4 ), the number of the observed DT events Mj, the number of produced DT events Ni and the partial decay rate ∆Γi (in ns −1 ) in each q 2 bin for D 0 → π − µ + νµ. The column of ( 0(+) )ij gives the true q 2 bin j, while the row gives the reconstructed q 2 bin i. Uncertainties are statistical only. For ∆Γi, the uncertainty is combined from the statistical and systematic uncertainties, in which the statistical uncertainty for muon mode is dominated.  Table III: Summary of the efficiency matrix ( 0(+) )ij (in %), the range of each q 2 bin (in GeV 2 /c 4 ), the number of the observed DT events Mj, the number of produced DT events Ni and the partial decay rate ∆Γi (in ns −1 ) in each q 2 bin for D + → π 0 µ + νµ. The column of ( 0(+) )ij gives the true q 2 bin j, while the row gives the reconstructed q 2 bin i. Uncertainties are statistical only. For ∆Γi, the uncertainty is combined from the statistical and systematic uncertainties, in which the statistical uncertainty for muon mode is dominated.