Measurements of Absolute Branching Fractions of Fourteen Exclusive Hadronic $D$ Decays to $\eta$

Using $2.93\,\rm fb^{-1}$ of $e^+e^-$ collision data taken at a center-of-mass energy of 3.773\,GeV with the BESIII detector, we report the first measurements of the absolute branching fractions of fourteen hadronic $D^{0(+)}$ decays to exclusive final states with an $\eta$, e.g., $D^0\to K^-\pi^+\eta$, $K^0_S\pi^0\eta$, $K^+K^-\eta$, $K^0_SK^0_S\eta$, $K^-\pi^+\pi^0\eta$, $K^0_S\pi^+\pi^-\eta$, $K^0_S\pi^0\pi^0\eta$, and $\pi^+\pi^-\pi^0\eta$; $D^+\to K^0_S\pi^+\eta$, $K^0_SK^+\eta$, $K^-\pi^+\pi^+\eta$, $K^0_S\pi^+\pi^0\eta$, $\pi^+\pi^+\pi^-\eta$, and $\pi^+\pi^0\pi^0\eta$. Among these decays, the $D^0\to K^-\pi^+\eta$ and $D^+\to K^0_S\pi^+\eta$ decays have the largest branching fractions, which are $\mathcal{B} (D^0\to K^-\pi^+\eta )=(1.853\pm0.025_{\rm stat}\pm0.031_{\rm syst})\%$ and $\mathcal{B}(D^+\to K^0_S\pi^+\eta)=(1.309\pm0.037_{\rm stat}\pm0.031_{\rm syst})\%$, respectively. We also determine the $CP$ asymmetries for the six decays with highest event yields. No evidence of $CP$ violation is found.

Studies of CP violation in the weak decays of hadrons are powerful tools for understanding physics within the SM and searches for physics beyond it.The CP violation in D decays is predicted to be up to a few times 10 −3 [23][24][25][26][27][28][29] and has been recently observed to be (1.54 ± 0.29) × 10 −3 in D 0 → K + K − and π + π − decays by LHCb [30].However, knowledge of CP violation in D decays is still very limited.Searching for CP asymmetries in hadronic D decays, which have been much less explored than (semi-)leptonic decays, allows for a more comprehensive understanding of CP violation in the D sector.
This Letter reports the first measurements of the absolute BFs for the decays Throughout this Letter, the charge conjugate processes are implied unless stated otherwise.In addition, the CP asymmetries are determined for the six decays with the highest yields.To avoid double-counting previously measured decays, the narrow peaks for the K 0 S , η, ω, η ′ , and φ are removed from the mass spectra of the π +(0) π −(0) , The data sample was collected with the BESIII detector at a center-of-mass energy √ s = 3.773 GeV and has an integrated luminosity of 2.93 fb −1 [31].Details about the design and performance of the BESIII detector are given in Ref. [32].The Monte Carlo (MC) simulated events are produced with a geant4based [33] detector simulation software package.An inclusive MC sample, including D 0 D0 , D + D − , and non-D D decays of the ψ(3770), initial state radiation (ISR) production of the ψ(3686) and J/ψ, and the processes e + e − → q q (q = u, d, s) and e + e − → (γ)ℓ + ℓ − (ℓ = e, µ, τ ), is produced to determine the detection efficiencies and to estimate any potential backgrounds.The production of the charmonium states is simulated by the MC generator kkmc [34].The measured decay modes of the charmonium states are generated using evtgen [35] with BFs from the Particle Data Group [11], and the remaining unknown decay modes are generated by lundcharm [36].
The BFs of the hadronic D (D 0 or D + ) decays are measured via the reaction chain e + e − → ψ(3770) → D D. If a D meson is fully reconstructed, it is called a singletag (ST) D meson.The ST D − mesons are reconstructed via the decays , and K + K − π − , while the ST D0 mesons are reconstructed using the decays D0 → K + π − , K + π − π 0 , and K + π − π − π + .If a signal decay is fully reconstructed in the system recoiling against an ST D meson, the candidate event is called a double-tag (DT) event.The BF of the signal decay is given by where N ST = i N i ST and N DT are the total ST and DT yields in data, respectively, and /N ST is the effective efficiency for detecting the signal decay, averaged over tag mode i, where ǫ ST and ǫ DT are the efficiencies for detecting ST and DT candidates, respectively.
To determine the DT yields in the data (N fit DT ), a two-dimensional (2D) unbinned maximum likelihood fit is performed on the M tag BC vs. M sig BC distribution of the accepted DT candidates (See Fig. 1 of the supplemental material [44] for an example).Signal events concentrate around M tag BC = M sig BC = M D , where M D is the nominal D mass [11].Background events are divided into three categories.The first one (named BKGI) is from events with correctly reconstructed D ( D) and incorrectly reconstructed D (D).They are spread along the lines around M tag BC or M sig BC = M D .The second one (named BKGII) is from events smeared along the diagonal, which are mainly from the e + e − → q q processes.The third one (named BKGIII) comes from events with uncorrelated and incorrectly reconstructed D and D.
In the 2D fit, the probability density functions (PDFs) of the backgrounds are constructed as • BKGII: Here, x = M sig BC , y = M tag BC , z = (x + y)/ √ 2, and k = (x − y)/ √ 2. The PDFs for signal, a(x, y), b(x), and b(y), are described by the corresponding MC-simulated shapes. ) , where f denotes x, y, or z; E b is fixed at 1.8865 GeV/c 2 ; A f is a normalization factor; and ξ f is a fit parameter.g(k; 0, σ k ) is a Gaussian function with mean of zero and standard deviation , where σ 0 and p are two free parameters.For the decays S π 0 π 0 η, and D + → K 0 S π + π 0 η, the yields and shapes of the peaking backgrounds mentioned above are fixed based on MC simulations.All other parameters are left free.
Combinatorial π + π − pairs can also satisfy the K 0 S selection criteria and form peaking backgrounds around the D mass in the M sig BC distribution.This kind of background is estimated by the data events in the K 0 S sideband region.For one-dimensional (1D) signal and sideband regions are defined as M π + π − ∈ (0.486, 0.510) GeV/c 2 and M π + π − ∈ (0.454, 0.478) ∪ (0.518, 0.542) GeV/c 2 , respectively.For D 0 → K 0 S K 0 S η, 2D signal and sideband regions are defined.The 2D sideband 1 (2) regions are defined as the boxes in which one (two) of the two π + π − combinations lie in the K 0 S sideband regions and the rest are located in the K 0 S signal regions.See Fig. 2 of the supplemental material [44] as an example.
For the decays involving K 0 S , the net DT yields are obtained by , where N fit DT and N fit sid-i are the fitted DT yields in the K 0 S signal region and sideband i region, respectively.For the other decays, the net DT yields are N fit DT .For each signal decay mode, the statistical significance is calculated by −2ln(L 0 /L max ), where L max and L 0 are the maximum likelihoods with and without the signal component in the fits, respectively.The effect of combinatorial π + π − backgrounds in the K 0 S signal regions has been considered for the decays involving K 0 S .The statistical significances of the four decays with lowest yields, , and K 0 S π + π 0 η, are 5.5σ, 2.8σ, 5.7σ, and 8.4σ, respectively; while those for the other decays are all greater than 10σ.
To determine the signal efficiencies (ǫ sig ), the D → Kπη decays are simulated with a modified data-driven generator BODY3 [35], which was developed to simulate different intermediate states in data for a given threebody final state.The Dalitz plot of M 2 Kπ vs. M 2 πη found in data, corrected for backgrounds and efficiencies, is as input for the BODY3 generator.The efficiencies across the kinematic space are obtained with MC samples generated with the PHSP generator.Intermediate states in the D 0 → K + K − η, K 0 S K 0 S η, K 0 S π 0 π 0 η, and D + → K 0 S K + η, K 0 S π + π 0 η decays cannot be determined due to limited statistics; these decays are therefore simulated with the PHSP generator.Each of the other decays is simulated with a mixed signal MC sample.Here, the decays generated with PHSP generator and the decays containing K * (892), ρ(770), and a 0 (980) 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 in the e + e − rest frame) of the daughter particles, and the invariant masses of each of the twoand three-body particle combinations, agree with the MC simulations.
The values for N DT , ǫ sig , and the BFs of the signal decays are summarized in Table 1.The BF upper limit for D 0 → K 0 S K 0 S η at 90% confidence level is determined to be < 2.4 × 10 −4 , using the Bayesian approach after incorporating the systematic uncertainty [46].
The systematic uncertainties arise from the sources discussed below and are estimated relative to the measured BFs.The uncertainties in the total ST yields come from the M tag BC fits to the ST D candidates, which Table 1.Requirements on ∆Esig, net DT yields in data (NDT), detection efficiencies (ǫsig, including the BFs of K 0 S , η, and π 0 as well as correction factors described later), and the obtained BFs (Bsig).Numbers in the first and second brackets are last two effective digits of statistical and systematic uncertainties, respectively, for Bsig.The uncertainty is statistical only for NDT. were determined as 0.5% for both neutral and charged D [37][38][39].The systematic uncertainties of the tracking efficiencies are found to be (0.2-0.5)% per K ± or π ± , while those for PID efficiencies are taken as (0.2-0.3)% per K ± or π ± , by using DT D D hadronic events.The systematic uncertainty in K 0 S reconstruction is estimated to be 1.6% per K 0 S by using the J/ψ → K * (892) ∓ K ± and J/ψ → φK 0 S K ± π ∓ candidates [47].The systematic uncertainty of the π 0 reconstruction is assigned as (0.7-0.8)% per π 0 from studies of DT D D hadronic decay samples of 37,38].The systematic uncertainty for η reconstruction is taken to be the same as that for π 0 .The uncertainties of the quoted BFs of K 0 S → π + π − , η → γγ, and π 0 → γγ decays are 0.07%, 0.5%, and 0.03% [11], respectively.
To estimate the systematic uncertainty in 2D fit, we repeat the fits by varying the signal shape and the endpoint of the ARGUS function.The systematic uncertainty of the D D opening angle requirement is assigned as 0.4% by using the DT events of D 0 → K − π + π 0 .The systematic uncertainty due to the ∆E sig requirement is assigned to be 0.3%, which is the largest efficiency difference with and without smearing the data-MC Gaussian resolution of ∆E sig for signal MC events.Here, the parameters of the Gaussian are obtained by using the DT samples of D 0 → K 0 S π 0 , K − π + π 0 , K − π + π 0 π 0 , and D + → K − π + π + π 0 .The systematic uncertainties due to the choice of the K 0 S sideband and the K 0 S /ω/η (′) /φ rejection windows are assigned by examining the changes of the BFs when varying the nominal K 0 S sideband and rejection windows by ±5 MeV/c 2 .The uncertainties due to limited MC statistics (0.3-1.1)% are considered as a source of systematic uncertainty.
The systematic uncertainties in MC modeling are categorized into three cases.For the D → Kπη decays, the differences between the DT efficiencies obtained with the BODY3 and PHSP generators are assigned as the uncertainties.For the decays whose efficiencies are estimated with PHSP generator, the uncertainties are assigned by referring to the largest change of the efficiencies among D 0 → K − π + η, K 0 S π 0 η, and D + → K 0 S π + η.For the decays whose efficiencies are estimated with the mixed signal MC events, the systematic uncertainties are assigned as the change of the DT efficiency after removing the smallest component.
The D 0 D0 pairs are produced coherently at the ψ(3770).For the decays and π + π − π 0 η, the measured BFs are affected by various CP components due to quantum-correlation (QC) effects.The fractions of CP + components in these decays are examined by the CP + tag of D 0 → K + K − and CP − tag of D 0 → K 0 S π 0 , with the same method described in Ref. [48], and the necessary parameters are taken from Refs.[49][50][51].The obtained impact of QC effects on the BFs (f QC ) is shown in Table 1 of the supplemental material [44].The signal efficiencies are corrected by the corresponding f QC factors, the residual statistical errors of f QC are assigned as the systematic uncertainties.
For each signal decay, the total systematic uncertainty is obtained by adding the above effects in quadrature.
The systematic uncertainties for the various signal decays are given in Table 2 of the supplemental material [44].
For the six decay modes with the highest yields, the BFs of D and D decays, B + sig and B − sig , are measured separately.Their asymmetry is determined by The obtained BFs and asymmetries are summarized in Table 2.
No CP violation is found within the current statistics.Several systematic uncertainties cancel in the asymmetry: the tracking and PID of π + π − /K + K − pair, K 0 S reconstruction, π 0 /η reconstruction, quoted BFs, K 0 S sideband choice, K 0 S /ω/η (′) /φ rejection windows, MC modeling, and strong phase of D 0 decays.The other systematic uncertainties are estimated separately as above.In summary, with 2.93 fb −1 of data taken at √ s = 3.773 GeV with the BESIII detector, we report the first measurements of the absolute BFs of fourteen exclusive D 0(+) decays to η. Summing over the BFs measured in this work, and using the world averaged values of other known decays [11], the total BFs of all the exclusive D 0 and D + decays to η are determined to be (8.62 ± 0.35)% and (4.68 ± 0.18)%, respectively.Here, the systematic uncertainties of N ST , K ± /π ± tracking and PID, K 0 S and η reconstruction, and the quoted BFs are correlated.They are consistent with the corresponding inclusive rates (9.5 ± 0.9)% and (6.5 ± 0.7)% within 0.9σ and 2.5σ, respectively, leaving little room for other exclusive decays involving η.The reported BFs provide key inputs for accurate background estimations in LFU tests with semileptonic B decays, which are crucial to explore possible new physics beyond the SM.The obtained B(D 0 → K − π + η) agrees with the recent Belle result [11,12] within 1.3σ, with precision improved twofold.Our B(D 0 → K 0 S π 0 η) is greater than CLEO's result [11,13] by 3.7σ.Combining the measured B(D 0 → K 0 S π 0 η) with the fit fraction B(D 0 → K * (892) 0 η, K * (892) 0 → K 0 S π 0 )/B(D 0 → K 0 S π 0 η) from CLEO [13], we find B(D 0 → K * (892) 0 η) = (1.77± 0.44)%, where the uncertainty is dominated by the fit fraction.This deviates from various theoretical calculations [18,19,23] by 1.9-2.9σ.Future amplitude analyses of these decays at BESIII [54] and Belle II [55] will open a window to extract more two-body hadronic D decays, which are important to understand quark U-spin and SU(3)-flavor symmetry breaking effects, and will be beneficial for the predictions of D 0 D0 mixing and CP violation in D decays [18,19,23].In addition, we determine the asymmetries of the charge-conjugated BFs for the six D decays with highest yields, and no CP violation is found with current statistics.
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.Figure 2 shows the definitions of 1D and 2D K 0 S signal and sideband regions.Table 1 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 vs. M sig BC distributions of the accepted DT candidates, and the QC factors obtained with the same method as described in Ref. [48] and the necessary parameters quoted from Refs.[49][50][51].No DT event are observed from the D 0 → K + K − η and K 0 S K 0 S η decays.The systematic uncertainties arising from QC effects are directly assigned as the averaged strong-phase factor C f by the flavor tag yields.
Table 2 summarizes the systematic uncertainties for various sources in the measurements of BFs, which are assigned relative to the measured BFs.For each signal decay, the total uncertainty is obtained by quadratically adding all errors.

g
Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China h Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People's Republic of China i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA j Currently at: Institute of Physics and Technology, Peace Ave.54B, Ulaanbaatar 13330, Mongolia k Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China l School of Physics and Electronics, Hunan University, Changsha 410082, China 2 .Tagged D (signal D) mesons are identified by two variables, the energy difference ∆E tag (sig) ≡ E tag (sig) −E b and the beam-constrained mass M tag (sig) BC ≡ E 2 b − | p tag (sig) | 2 , where tag (sig) represents the tagged D (signal D), E b the beam energy, and p tag (sig) and E tag (sig) the momentum and energy of the D (D) candidate in the e + e − rest frame.For each tag (signal) mode, if there are multiple combinations, only the one with the minimum |∆E tag (sig) | is kept for further analysis.The D tags are required to satisfy ∆E tag ∈ (−55, 40) MeV for the modes containing π 0 in the final states and ∆E tag ∈ (−25, 25) MeV for the other modes.

Fig. 1 .
Fig. 1.Projections on M tag BC and M sig BC of the 2D fits to the DT candidate events.Data are shown as dots with error bars.Blue solid, red-dotted, blue dot-dashed, black dot-long-dashed, green long-dashed, and pink dashed curves denote the overall fit results, signal, BKGI, BKGII, BKGIII, and peaking background components (see text), respectively.

Figure 1 Fig. 1 .
Figure1shows the M tag BC vs. M sig BC distribution of the accepted DT candidate events.

Table 1 .
Summary of the ST yields of CP ± tags (S ± measured ), the DT yields tagged by CP ± tags (M ∓ measured ), and the QC factor (fQC).The errors are statistical only.

Table 2 .
Systematic uncertainties (%) in the measurements of the BFs.