Absolute Measurements of Branching Fractions of Cabibbo-Suppressed Hadronic $D^{0(+)}$ Decays Involving Multiple Pions

By analyzing $e^+e^-$ annihilation data with an integrated luminosity of $2.93~\rm fb^{-1}$ collected at the center-of-mass energy $\sqrt s=$ 3.773 GeV with the BESIII detector, we present the first absolute measurements of the branching fractions of twenty Cabibbo-suppressed hadronic $D^{0(+)}$ decays involving multiple pions. The largest four branching fractions obtained are $\mathcal{B}(D^0\to\pi^+\pi^-\pi^0)= (1.343\pm0.013_{\rm stat}\pm0.016_{\rm syst})\%$, $\mathcal{B}(D^0\to\pi^+\pi^-2\pi^0)= (0.998\pm0.019_{\rm stat}\pm0.024_{\rm syst})\%$, $\mathcal{B}(D^+\to 2\pi^+\pi^-\pi^0)= (1.174\pm0.021_{\rm stat}\pm0.021_{\rm syst})\%$, and $\mathcal{B}(D^+\to 2\pi^+\pi^-2\pi^0)=(1.074\pm0.040_{\rm stat}\pm0.030_{\rm syst})\%$. The $CP$ asymmetries for the six decays with highest event yields are also determined.

Investigations of hadronic D 0(+) decays are of general importance for both charm and bottom physics. For example, Ref. [1] suggests that hadronic D 0(+) decays involving three charged pions are crucial backgrounds for the tests of lepton flavor universality (LFU) in semileptonic B decays. However, many Cabibbo-suppressed hadronic D 0(+) decays with three charged pions are unexplored mainly due to low detection efficiencies and high background contamination. Precision and comprehensive measurements of the absolute branching fractions (BFs) of these decays provide necessary inputs to unravel the hints of LFU violation observed in semileptonic B decays.
To date, only the BFs of seven of these decay modes have been measured relative to reference modes and only the CP asymmetries in D 0 (D 0 ) → π + π − π 0 and D ± → π + π − π ± have been measured by various experiments [13]. Throughout this Letter, chargeconjugated processes are implied except when discussing CP asymmetries.
The BESIII detector is a magnetic spectrometer [14] located at the Beijing Electron Positron Collider (BEPCII) [15]. Simulated samples produced with a geant4-based [16] Monte Carlo (MC) package including the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiencies and to estimate backgrounds. The simulation of e + e − annihilations modeled with the generator kkmc [17] includes the beam-energy spread and initialstate radiation. The inclusive MC samples consist 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 continuum processes. Known decay modes are modeled with evtgen [18] using the BFs taken from Ref. [13], and the remaining unknown decays of the charmonium states are modeled with lundcharm [19,20]. Final-state radiation from charged final-state particles is incorporated using photos [21].
At √ s = 3.773 GeV, D 0D0 or D + D − pairs are produced without accompanying hadron(s), thereby offering a clean environment to investigate hadronic D decays with double-tag (DT) method [22]. The singletag (ST)D candidates are selected by reconstructing aD 0 or D − in the hadronic decay modes:D 0 → K + π − , K + π − π 0 , and K + π − π − π + , and D − → K + π − π − , K 0 S π − , K + π − π − π 0 , K 0 S π − π 0 , K 0 S π + π − π − , and K + K − π − . Events in which a signal candidate is reconstructed in the presence of an STD meson are referred to as DT events. The BF of the signal decay is determined by [23] where N tot ST = i N i ST and N DT are the total yields of the ST and DT candidates in data, respectively. The ST yield for the tag mode i is N i ST , and the efficiency ǫ sig for detecting the signal D decay is averaged over the tag modes i.
TaggedD mesons are identified using two variables: is the beam energy, p tag and E tag are the momentum and energy ofD in the rest frame of e + e − system, respectively. The ∆E tag of STD candidates must be in the range (−55, 40) MeV for the tag modes involving π 0 and (−25, 25) MeV for the other tag modes, due to differing resolutions. For each tag mode, if there are multiple candidates in an event, only the one yielding the smallest |∆E tag | is accepted.
To extract the yields of STD candidates for individual tag modes, binned maximum-likelihood fits are performed to the corresponding M tag BC distributions of the accepted ST candidates following Ref. [24]. TheD signal is modeled by an MC-simulated shape convolved with a double-Gaussian function describing the resolution difference between the data and MC simulation. The combinatorial background shape is described by an ARGUS function [26]. The total yields of the STD 0 and D − candidates in data are (232.8 ± 0.2 stat ) × 10 4 and (155.8 ± 0.2 stat ) × 10 4 , respectively.
The signal D decays are selected from the remaining tracks and showers recoiling against the tagged D candidates.
To reject the main backgrounds from Cabibbo-suppressed decays containing K 0 S , the π + π − and π 0 π 0 combinations are required to not fall in the mass windows (0.468, 0.528) GeV/c 2 and (0.428, 0.548) GeV/c 2 , respectively. These mass windows correspond to at least 4σ of resolution.
Signal D mesons are identified using the energy difference ∆E sig and the beam-constrained mass M sig BC , calculated similarly to the ST side. For each signal mode, if there are multiple candidates in an event, only the one with the minimum |∆E sig | is chosen. Signal decays are required to satisfy the ∆E sig requirements shown in Table 1.
For each signal decay mode, the signal yield (N DT ) is obtained from a two-dimensional (2D) unbinned maximum-likelihood fit [27]  . Events smeared along the diagonal (BKGII) are mainly from the e + e − → qq processes. Events with uncorrelated and incorrectly reconstructed D andD (BKGIII) disperse across the whole allowed kinematic region.
In the fit, the probability density functions (PDFs) for signal, BKGI, BKGII, and BKGIII contributions are constructed as a(x, y), b x (x) · c y (y; M end BC , ξ y ) + b y (y) · c x (x; M end BC , ξ x ), c z (z; √ 2M end BC , ξ z ) · g(k; 0, σ k ), and The PDFs of signal, a(x, y), b x (x), and b y (y), are described by the MC-simulated shapes smeared with individual Gaussian resolution function with parameters derived from the corresponding one-dimensional M BC fits, to consider resolution difference between data and MC simulation. c f (f ; M end BC , ξ f ) is an ARGUS function [26] (f denotes x, y, or z), where the endpoint M end BC = 1.8865 GeV/c 2 is fixed in the fit. The Gaussian function g(k; 0, σ k ) has a mean of zero and a standard deviation parameterized by σ k = σ 0 · ( √ 2M end BC − z) p , where σ 0 and p are fit parameters. In addition, the yields and shapes of the peaking background (PBKG) components, which are mainly from D decays into the same final state as a signal mode but involve K 0 S → π + π − or π 0 π 0 and K − → π − π 0 decay, are fixed based on MC simulations and the known BFs of various PBKG components [13,29]. Relative to the corresponding signal yields, the PBKG components are 9.5%, 18.2%, and 36.2% for D + → π + 4π 0 , D 0 → 2π + 2π − π 0 , and D 0 → π + π − 3π 0 , respectively, and range from 0.1% to 6.3% for the other signal decays.
The M tag BC and M sig BC projections of the 2D fits of the DT candidates reconstructed from data are shown in Fig. 1 and the fitted DT yields are summarized in Table 1.
To determine the signal efficiencies (ǫ sig ), the threebody decays are simulated with a modified data-driven generator BODY3 [18], which was developed to simulate different intermediate states in data for a given threebody final state. The Dalitz plot from data, corrected for backgrounds and efficiencies, is taken as input for the BODY3 generator. The efficiencies across phase space are obtained with MC samples generated according to a phase space distribution. Each of the four-body and five-body decays is simulated with a mixed signal MC sample. These decays generated with phase space models including contributions from η, ω, η ′ , ρ(770), a 0 (980), a 1 (1260), b 1 (1235) and φ intermediate states are mixed with fractions obtained by examining the corresponding invariant mass spectra. The data distributions for momenta and cos θ (where θ is the polar angle of particle in the e + e − rest frame) of the daughter particles, and the invariant masses of each of the two-and multi-body particle combinations agree with the MC simulations. The results for N DT , ǫ sig , and the extracted BFs are summarized in Table 1. The smallest statistical significance is 6.8 standard deviations for D + → π + 4π 0 mode. The signal efficiencies have been corrected by the data-MC differences in the selection efficiencies of π ± tracking, particle identification (PID) procedures and the reconstruction of π 0 or η.
The systematic uncertainties relative to the obtained BFs are discussed below. In the BF determinations using Eq. (1), all uncertainties from selecting the ST D candidates are cancelled in the ratio. Systematic uncertainties in the total yields of STD mesons related to the M tag BC fits to the STD candidates, were previously estimated to be 0.5% for both neutral and chargedD [30][31][32].
The tracking and PID efficiencies of π ± are investigated using other DT DD hadronic events. The differences of efficiencies between data and MC simulations are weighted by the corresponding π ± momentum spectra of signal MC events. The systematic uncertainties due to tracking and PID are assigned to be (0.2-0.4)% per π ± , based on the residual statistical uncertainties of the measured data-MC differences.
The systematic uncertainty of the π 0 reconstruction is assigned as (0.4-0.9)% per π 0 from studies of DT DD hadronic decay samples of [30,31]. Due to limited η statistics, the systematic uncertainty for η reconstruction is assigned by referring to that of π 0 .
The systematic uncertainty in the 2D fit to the M tag BC versus M sig BC distribution is examined by varying the smeared Gaussian function (±1σ), the endpoint of the ARGUS function (±0.2 MeV/c 2 ), and the fixed PBKG yields (±1σ of the quoted BF). Adding the changes from these three sources in quadrature yields the corresponding systematic uncertainties.
The systematic uncertainty due to the ∆E sig requirement ranges from (0.1-1.3)% depending on the signal mode. They are evaluated from the efficiency differences obtained with and without smearing the ∆E sig distributions for signal MC events with parameters derived from D 0 → π + π − π 0 , D 0 → π + π − 2π 0 , D + → π + 2π 0 , and D + → 2π + π − π 0 to get the data-MC differences.
The systematic uncertainty due to the BODY3 generator is considered by varying the number of bins by ±20% and the systematic uncertainty in the mixed MC model is assigned by varying the fractions of various components by ±1σ of the quoted BF, when available. Unmeasured components are varied by ±25%, beyond which comparisons with observed mass spectra are unsatisfactory. The largest change of the signal efficiencies, (0.2-5.7)% for various signal modes, are assigned as the corresponding systematic uncertainties.
The systematic uncertainties due to the mass window applied to reject K 0 S events are crosschecked by examining the changes of the BFs by varying the corresponding boundaries of the window by ±5 MeV/c 2 . If the difference of the BF is larger than 1 time the statistical uncertainty on the difference (taking the correlated samples into account), it is assigned as the corresponding systematic uncertainty. Otherwise, it is neglected.
The measurements of the BFs of the neutral D decays are affected by quantum correlation effect [33]. To take this effect into account, the CP -even fractions in various decays are needed. The D 0 → 4π 0 and D 0 → 3π 0 η final states are both CP -even eigenstates. For D 0 → π + π − π 0 , its CP -even fraction has been determined to be 0.973 ± 0.017 [5]. For D 0 → π + π − 2π 0 , D 0 → 2π + 2π − π 0 , D 0 → π + π − 3π 0 , and D 0 → 2π + 2π − 2π 0 , the CP -even fractions are estimated by the CP -even tag D 0 → K + K − and the CP -odd tag D 0 → K 0 S π 0 . Using the same method as described in Ref. [1] and the necessary parameters quoted from Refs. [2][3][4], we obtain the correction factors to account for the quantum correlation effect on the measured BFs; the results are summarized in Table 3 of the Supplemental Material [28]. After correcting the signal efficiencies by the individual factors, the residual uncertainties are assigned as systematic uncertainties.
Adding all individual effects for each signal decay quadratically yields the total systematic uncertainties to be (1.2-11.9)% depending on the signal mode. The detailed systematic uncertainties are given in Table 5 of the Supplemental Material [28].
For the six decay modes with the highest yields, the BFs of D andD decays, B + sig and B − sig , are measured separately. Their asymmetry is determined The obtained BFs and asymmetries are summarized in Table 2. We find no statistically significant CP violation. Several systematic uncertainties cancel in the asymmetry: the tracking and PID of π + π − pairs, π 0 and η reconstruction, daughter BFs, K 0 S rejection windows, MC modeling, and strong phase of D 0 decays. The other systematic uncertainties are estimated separately as above.
To summarize, by analyzing 2.93 fb −1 of e + e − annihilation data recorded at √ s = 3.773 GeV with the BESIII detector, we present the first absolute measurements of the BFs of twenty Cabibbo-suppressed hadronic D 0(+) decays involving multiple pions. For D 0 → π + π − π 0 , π + π − 2π 0 , 2π + 2π − π 0 , and D + → 2π + π − , π + 2π 0 , 2π + π − π 0 , 3π + 2π − , the BF precisions are improved by factors of 1.2-2.9 compared to the world average values based on relative measurements. For the other 13 decay modes, the BFs are measured for the first time. The reported BFs offer important input for reliable estimations of potential background sources in the precision measurements of B and D decays, especially to properly evaluate the tensions found in the LFU tests with semileptonic B decays. Amplitude analyses of these multi-body decays with larger data samples available in the near future [39,40] will open Table 1. Requirements of ∆Esig, DT yields in data (NDT), detection efficiencies (ǫsig, including the BFs of η, and π 0 as well as correction factors described later), and the obtained BFs (Bsig). The first nine modes are D 0 decays and the others are D + decays. For Bsig, numbers in the first and second parentheses are last two digits of the statistical and systematic uncertainties, respectively. For NDT, uncertainties are statistical only.
134.8 ± 1.8 133.3 ± 1.8 +0.6 ± 0.9 ± 0.4 π + π − 2π 0 97.1 ± 2.6 102.3 ± 2.7 −2.6 ± 1.9 ± 0.7 2π + π − 33.5 ± 1.0 32.7 ± 1.0 +1.2 ± 2.1 ± 0.6 π + 2π 0 48.9 ± 1.8 43.4 ± 1.7 +6.0 ± 2.7 ± 0.5 2π + π − π 0 117.7 ± 3.0 116.8 ± 3.0 +0.4 ± 1.8 ± 0.8 2π + π − 2π 0 102.7 ± 5.6 111.6 ± 5.8 −4.2 ± 3.8 ± 1.3 an opportunity to precisely extract more quasi-two-body hadronic D 0(+) decay rates, e.g. D + → ρ + π 0 . Detailed knowledge of these hadronic D 0(+) decays is essential to deeply explore quark U-spin and SU (3) Figure 2 shows the M tag BC versus M sig BC distribution of the accepted DT candidates in data. Detailed selection criteria can be found in context of the main paper.   11 show comparisons between data and MC simulations for the distributions of invariant mass spectra of two-, three-, four-or five-body particle combinations, momenta and cos θ of daughter particles for the signal DT candidates with more than 100 signal events. The candidates must satisfy additional requirements of |M tag(sig) BC −M D | < 0.006 GeV/c 2 and multiple possible combinations of daughter particles are all plotted when relevant (e.g., all four π + π − combinations for D 0 → 2π + 2π − π 0 , etc.) Table 3 summarizes the ST yields of CP ± tags from the fits to the M tag BC distributions of the accepted ST candidates, the DT yields tagged by CP ± tags from the 2D fits to the M tag BC versus M sig BC distributions of the accepted DT candidates, and the quantum correlation (QC) factors obtained with the same method as described in Ref. [1] and the necessary parameters quoted from Refs. [2][3][4]. Table 4 summarizes the statistical significances of the decay modes. Table 5 summarizes the systematic uncertainties for various sources in the measurements of BFs, which are assigned relative to the measured BFs. They are from the ST yield (N tag ), π ± tracking efficiency, π ± PID efficiency, π 0 and η reconstruction efficiency, daughter BFs, ∆E requirement, K 0 S rejection, MC statistics, MC generator, 2D fit, and strong phase. For each signal decay, the total uncertainty is obtained by quadratically adding all uncertainties.  Fig. 3. Comparisons of the distributions of invariant masses of two-body particle combinations, momenta and cos θ of daughter particles for the D 0 → π + π − π 0 candidates between data (points with error bars) and the BODY3 signal MC events (black solid line histograms) plus the MC-simulated backgrounds from the inclusive MC sample (yellow filled histograms). Table 3. Summary of the ST yields of CP ∓ tags (S ± measured ), the DT yields tagged by CP ∓ tags (M ± measured ), the CP + fraction (fCP +), and the QC factor (fQC). The uncertainties are statistical only. A "/" denotes unmeasured quantites, occuring for one mode with a high-precision extrernal result and for the two CP -eigenstates. [5] 93.5 ± 0.5 0.5 D 0 → π + π − 2π 0 65.7 ± 11.1 169.8 ± 13.9 0.682 ± 0.077 97.4 ± 0.7 0.7