Observation of the $W$-Annihilation Decay $D^{+}_{s} \rightarrow \omega \pi^{+}$ and Evidence for $D^{+}_{s} \rightarrow \omega K^{+}$

We report on the observation of the $W$-annihilation decay $D^{+}_{s} \rightarrow \omega \pi^{+}$ and the evidence for $D_{s}^{+} \rightarrow \omega K^{+}$ with a data sample corresponding to an integrated luminosity of 3.19 fb$^{-1}$ collected with the BESIII detector at the center-of-mass energy $\sqrt{s} = 4.178$ GeV. We obtain the branching fractions $\mathcal{B}(D^{+}_{s} \rightarrow \omega \pi^{+}) = (1.77\pm0.32_{{\rm stat.}}\pm0.11_{{\rm sys.}}) \times 10^{-3}$ and $\mathcal{B}(D^{+}_{s} \rightarrow \omega K^{+}) = (0.87\pm0.24_{{\rm stat.}}\pm0.07_{{\rm sys.}}) \times 10^{-3}$, respectively.

We report on the observation of the W -annihilation decay D + s → ωπ + and the evidence for D + s → ωK + with a data sample corresponding to an integrated luminosity of 3.19 fb −1 collected with the BESIII detector at the center-of-mass energy √ s = 4.178 GeV. We obtain the branching fractions B(D + s → ωπ + ) = (1.77±0.32stat. ±0.11sys.)×10 −3 and B(D + s → ωK + ) = (0.87±0.24stat. ± 0.07sys.) × 10 −3 , respectively. Within the Standard Model of particle physics, direct CP violation (CPV) in hadronic decays can only be induced in decays that proceed via at least two distinct decay amplitudes with non-trivial strong and weak phase differences [1][2][3]. In the charm sector, examples for such decays are the single Cabibbo suppressed (SCS) decays involving W -annihilation process [1][2][3][4], for example D + s → ωK + , and other V P final state (V and P refer to the vector and pseudo-scalar mesons, respectively). However, in D decays, the W -annihilation amplitude is dominated by the nonfactorizable long-distance amplitude induced by the final-state interaction (FSI). The corresponding theoretical calculation is very unreliable and results in some ambiguity in the prediction of the branching fractions (BFs) and the CPV of related decays. Instead, the experimental BF measurement of decays through the W -annihilation is used as an input in theoretical calculations [2][3][4][5]. Therefore, the BF of the Cabibbo favored (CF) decay D + s → ωπ + proceeding only via the W -annihilation process provides direct knowledge of the W -annihilation amplitude.
Compared with the SCS decays, the BF of D + s → ωπ + is expected to be larger and may be measured with a higher precision, is thus a more useful experimental input in the W -annihilation amplitude determination. Evidence for this decay was first reported by CLEO in 1997 and a ratio Γ(D + s →ωπ + ) Γ(D + s →ηπ + ) = 0.16 ± 0.04 ± 0.03 was measured based on 4.7 fb −1 data taken at the Υ(4S) peak [6]. Later in 2009, using a data sample corresponding to an integrated luminosity of 0.586 fb −1 taken at center of mass energy √ s = 4.170 GeV, CLEO observed 6.0 ± 2.4 signal events and measured the absolute BF to be (2.1 ± 0.9) × 10 −3 [7].
With the previously measured W -annihilation D → V P decay as input, for D + s → ωK + , when considering the ρ − ω mixing, the BF and CPV are predicted to be 0.07 × 10 −3 and −2.3 × 10 −3 [3], respectively, where this predicted CPV is in the largest order of magnitude in charmed meson decays. However, the corresponding values are predicted to be 0.6 × 10 −3 and −0.6 × 10 −3 [3], respectively, when the ρ − ω mixing is negligible. Therefore, the search for D + s → ωK + can also test the ρ − ω mixing effect and if D + s → ωK + is a good decay mode to search for CPV in charm decays.
In this Letter, we report on measurements of the absolute BFs of the hadronic decays D + s → ωπ + and D + s → ωK + . Charge conjugation is implied throughout this Letter unless explicitly stated. At the center-of-mass energy of √ s = 4.178 GeV, the D + s meson is predominantly produced through the process e + e − → D * + s D − s + c.c., where the D * + s decays to either γD + s or π 0 D + s . As a consequence, any event that contains a D + s meson also contains a D − s meson. This condition enables the usage of a powerful "double tag (DT)" technique [8] to measure absolute BFs. Events with at least one D − s candidate reconstructed, which are referred to as "single tag (ST)" events, provide a sample with a known number of D + s D − s pairs. The absolute BF of the signal mode (B sig ) is determined by forming DT signal events from the tracks and clusters not used to reconstruct the D − s candidate. The value of the BF can then be obtained using where Y sig is the DT signal yield, ǫ i tag,sig is the DT efficiency, and Y i tag and ǫ i tag are the ST yield and the ST efficiency of the ith tag mode, respectively.
A detailed description of the BESIII detector can be found in Ref. [9]. Two endcap time-of-flight systems were upgraded with multi-gap resistive plate chambers [10]. Monte Carlo (MC) simulations of BESIII detector are based on geant4 [11]. The generic MC sample includes all known open charm processes, initial state radiation to the J/ψ or the ψ(3686), and the continuum process. The open charm processes are generated with conexc [12], considering the effects from initial state radiation and final state radiation. The decay modes with known BFs are simulated with evtgen [13]. The generators kkmc [14] and babayaga [15] are used to simulate the continuum. The generic MC sample, corresponding to an effective luminosity of 110.6 fb −1 , is used to determine the ST efficiency and estimate the background. The signal MC sample, two million of events with one D s meson decaying to the signal modes and the other one decaying to anything, as used in generic MC, is generated to estimate the DT efficiency.
ST and DT candidates are constructed from individual π + , K + , K 0 S (π + π − ) and π 0 /η(γγ) candidates in an event. All charged tracks, except the K 0 S daughters, are required to originate from within 10 cm (1 cm) along (perpendicular to) the beam axis with respect to the interaction point (IP). Their polar angles (θ) are required to be within | cos θ| < 0.93. The combination of information about the energy loss in the multi-layer drift chamber and the time-of-flight is used to identify the species of charged particles by calculating a probability P (K + )[P (π + )]) that the track satisfies the hypothesis of being a K + (π + ). The K + and π + candidates are required to satisfy P (K + ) > P (π + ) and P (π + ) > P (K + ), respectively.
For K 0 S candidates, the combination of two oppositely charged tracks whose mass hypotheses are set to the pion, without particle identification (PID) applied and with distances of closest approach to the IP less than 20 cm along the beam axis, is required to have an invariant mass in the interval [0.487, 0.511] GeV/c 2 .
The energy of each photon from the π 0 /η is required to be larger than 25 (50) MeV in the barrel (endcap) region of the electromagnetic calorimeter [9]. The opening angles between the shower and all the charged tracks should be larger than 10 • . The invariant mass of the γγ pair is required to be within the asymmetric intervals [0.115, 0.150] and [0.490, 0.580] GeV/c 2 for π 0 and η. Furthermore, the π 0 /η candidates are constrained to their nominal mass [16] via a kinematic fit to improve their energy and momentum resolution.
The momenta of all pions, except those from the decay of K 0 S , are required to be greater than 0.1 GeV/c in order to reject low momentum pions produced in the D * decay. The ST events are selected by reconstructing a D − s meson in the two highest purity decay modes D − s → K 0 S K − and K + K − π − . For the D s candidate, a recoil mass M rec is defined as where E tot , p Ds , m Ds , p tot and p Ds are the total energy of e + e − , the momentum of the D s candidate, the nominal mass of D s [16], the three-momentum vector of the colliding e + e − system, and the three-momentum vector of the reconstructed D s candidate, respectively. If the selected D − s candidate originates directly from the e + e − annihilation, M rec peaks at the D * s mass (m D * s ). The other D s candidate has a broader distribution around m D * s in the M rec spectrum. The D s invariant mass and M rec of the candidate are required to fall into the ranges  .986] GeV/c 2 for D − s → K + K − π − , respectively. The ST yields determined by the fit for D − s → K 0 S K − and D − s → K + K − π − are 32751 ± 313 and 131862 ± 773, respectively. For the tag mode D − s → K 0 S K − , a small peak in the background is observed in the signal region; this is due to D − → K 0 S π − events with the π − misidentified as a K − . From the generic MC sample, the yield of the D − → K 0 S π − background is estimated to be around 250, corresponding to about 0.2% of the total ST yields, which is considered in the systematic uncertainty. For the tag mode D − s → K + K − π − , a much smaller bump can also be found and the effect is negligible.
The DT events are reconstructed D + s D − s pairs with D − s reconstructed in a tag mode combined with D + s → ωπ + or D + s → ωK + candidates, in which the ω is reconstructed in the π + π − π 0 final state. For the two D s candidates, we require that at least one of them has M rec greater than 2.10 GeV/c 2 . If there is more than one D + s D − s pair candidate, the one with an average invariant mass of the two D s mesons closest to m Ds is chosen.
For D + s → ωK + , the background from the decay D + s → K 0 S K + π 0 is identical to the signal in the M rec distribution and forms a peak around the K * (892) mass in the π + π − π 0 invariant mass (M π + π − π 0 ) spectrum. Consequently, we further perform a K 0 S veto to suppress this background. If the invariant mass of the π + π − (M ππ ) combination in D + s → ωK + signal candidate satisfies |M ππ − m K 0 S | < 0.03 GeV/c 2 and the distance between the decay point and the IP has a significance of more than two standard deviations, the events are vetoed. This veto eliminates about 78% of D + s → K 0 S K + π 0 background, while retaining about 97% of signal events. After the K 0 S veto, this background is found to be negligible according to the generic MC.
A two dimensional (2D) extended unbinned likelihood fit is performed to the M π + π − π 0 and the signal D s invariant mass (M sig ) distributions to extract the signal yield. For D + s → ωπ + candidates, there are two π + π − π 0 combinations formed in each event. In the data sample, there are 5 events with both π + π − π 0 combinations retained in the fit range of [0.60, 0.95] GeV/c 2 , but there is no evidence that these events create a peak. This effect is negligible in the fit.
The D s signal is described by the MC simulated signal shape convolved with a Gaussian (f peak Ds ). Here, the mean and the resolution of the Gaussian are fixed at the values determined from the fit to the sample of D + s → π + π − π 0 π + (K + ) in data. The ω signal is represented by a Breit-Wigner (BW) convolved with a Gaussian (f peak ω ), where the mass and width are fixed to the PDG values [16]. The resolution of the Gaussian is fixed at the value determined from the sample of e + e − → K + K − ω. The combinatorial background in M sig and M π + π − π 0 spectra are parameterized by secondorder Chebychev polynomials (f poly Ds and f poly ω ).
The scatter plots of M sig versus M π + π − π 0 for the two signal decays are shown in Figs. 2(a,b), from which no obvious correlation between M sig and M π + π − π 0 is found, which is also confirmed by signal MC. The 2D fit model is then constructed as following. The signal shape is modeled as the product of f peak Ds and f peak ω . The background that does not peak in both M π + π − π 0 and M sig distributions (BKGI) is modeled as the product of f poly Ds and f poly ω . The background with an ω that peaks in the M π + π − π 0 distribution (BKGII) is modeled as the product of f poly Ds and f peak ω . The background from D + s → π + π − π 0 π + (K + ) that only has a peak in the M sig distribution (BKGIII) is modeled as the product of f peak in the 2D fit are consistent with the results obtained in the individual fits to the M sig and M π + π − π 0 spectra, respectively. Therefore, in the 2D fit, yields of signal and backgrounds are determined by the fit and the other parameters are fixed at the values from the individual fits. From the 2D fits, we obtain 65.0 ± 11.6 D + s → ωπ + signals and 28.5 ± 7.8 D + s → ωK + signals with statistical significances of 6.7σ and 4.4σ, respectively. The fit results are shown in Figs. 2(b,c,e,f). The ((a) and (d)) scatter plots of Msig versus M π + π − π 0 , fit results of ((b) and (e)) M π + π − π 0 , and fit results of ((c) and (f)) Msig for (a-c) D + s → ωπ + and (d-f) D + s → ωK + . In the fits, the dots with error bars are data, the (blue) solid lines describe the total fits, the (red) dashed lines describe the signal shape and the (green) dotted, (cyan) dash-dotted, and (black) long dashed lines describe the BKGI, BKGII, and BKGIII, respectively.
The systematic uncertainties are investigated and are summarized in Table II. For each decay, the total systematic uncertainty is obtained by adding the individual terms in quadrature.
The uncertainties due to the M rec requirement and momentum requirement on pion are estimated with the control sample of D + s → π + π − π + η, with η ′ decays removed by requiring the invariant mass of π + π − η to be greater than 1.0 GeV/c 2 . The uncertainty due to the K 0 S veto is estimated with the control sample of D 0 → K 0 S ω. The differences between the efficiencies caused by corresponding selection criterion obtained from data and MC simulation are taken as the systematic uncertainties.
The uncertainties for charged tracks selection are determined to be 0.5%/track for PID and 1.0%/track for tracking using the control sample of e + e − → K + K − π + π − . The uncertainty of the π 0 reconstruction efficiency is investigated with the control sample of e + e − → K + K − π + π − π 0 , and is determined to be 1.8% (1.9%) for D + s → ωπ + (K + ). The uncertainty due to the MC statistics is 0.6%.
The uncertainty due to the signal shape is estimated by varying the masses and resolutions of the ω and D s within their uncertainties. The uncertainty due to the background shape is investigated by narrowing the fit ranges of M π + π − π 0 and M sig to [0.65, 0.90] GeV/c 2 and [1.91, 2.02] GeV/c 2 , respectively, and replacing the second-order Chebychev polynomial in f poly Ds and f poly ω by a first-order Chebychev polynomial. The uncertainty related to ST yields is estimated using the MC-simulated shape to replace the Gaussian function in the ST yields determination. Here, the effect from the bump under the D − s → K 0 S K − signal region is also taken into account. For each alteration of the fit configuration the measurements are re-performed. The largest change to the BF is taken as the corresponding uncertainty. The uncertainty from the shape and fit ranges effect of background in ST yields is found to be negligible. The uncertainty due to the fit procedure is investigated by studying ten statistically independent samples of generic MC events with the same size as data. With the same method as used in data analysis, the average measured BF is found to have a relative difference of 0.8% with respect to the input value. This difference is taken as the uncertainty from the fit procedure. The uncertainty related to the assumed BFs for ω → π + π − π 0 and π 0 → γγ is taken from the PDG [16].
In summary, we observe the W -annihilation decay D + s → ωπ + with a significance of 6.7σ and measure its BF to be (1.77 ± 0.32 stat. ± 0.12 sys. ) × 10 −3 . This measurement provides critical information to determine the non-perturbative W -annihilation amplitudes. The significantly improved precision benefits the investigations of the underlying dynamics in charmed hadronic decays, and will allow better predictions for the BFs and direct CPV of decays involving W -annihilation [1,[3][4][5]. Among these decays, D + s → ωK + is interest for its possibly large CPV. We find the first evidence for this decay with a significance of 4.4σ. Its BF is measured to be (0.87 ± 0.24 stat. ± 0.07 sys. ) × 10 −3 . According to Ref. [3], our result implies that the ρ − ω mixing is negligible and direct CP asymmetry is expected at the level of −0.6 × 10 −3 .