A ug 2 01 9 Improved measurements of the absolute branching fractions of the inclusive decays D + ( 0 ) → φX

By analyzing 2.93 fb$^{-1}$ of $e^+e^-$ annihilation data taken at the center-of-mass energy $\sqrt s=$ 3.773 GeV with the BESIII detector, we determine the branching fractions of the inclusive decays $D^+\to\phi X$ and $D^0\to\phi X$ to be $(1.135\pm0.034\pm0.031)\%$ and $(1.091\pm0.027\pm0.035)\%$, respectively, where $X$ denotes any possible particle combination. The first uncertainties are statistical and the second systematic. We also determine the branching fractions of the decays $D\to\phi X$ and their charge conjugate modes $\bar{D}\to\phi \bar{X}$ separately for the first time, and no significant CP asymmetry is observed.

By analyzing 2.93 fb −1 of e þ e − annihilation data taken at the center-of-mass energy ffiffi ffi s p ¼ 3.773 GeV with the BESIII detector, we determine the branching fractions of the inclusive decays D þ → ϕX and D 0 → ϕX to be ð1.135 AE 0.034 AE 0.031Þ% and ð1.091 AE 0.027 AE 0.035Þ%, respectively, where X denotes any possible particle combination. The first uncertainties are statistical, and the second are systematic. We also determine the branching fractions of the decays D → ϕX and their charge conjugate modesD → ϕX separately for the first time, and no significant CP asymmetry is observed. DOI

I. INTRODUCTION
Experimental studies of the inclusive D → ϕX decays, where X denotes any possible particle combination, are important for charm physics due to the following reasons. First, precise measurements of their branching fractions offer an independent check on the existence of unmeasured or overestimated exclusive decays that include a ϕ meson. A measurable difference between the inclusive and exclusive decay branching fractions would indicate the size of as yet unmeasured decays or would imply that some decays are overestimated, requiring complementary or more precise measurements. Previous measurements of the branching fractions for inclusive D þ → ϕX and D 0 → ϕX decays were made by BES and CLEO [1,2] with 22.3 and 281 pb −1 of e þ e − annihilation data samples taken at the center-of-mass energies ffiffi ffi s p ¼ 4.03 and 3.774 GeV, respectively. Table I summarizes the branching fractions of the reported exclusive D decays to ϕ, where the branching fractions of D þ → ϕπ þ , D 0 → ϕπ 0 , and D 0 → ϕη are quoted from the recent BESIII measurements [3]; the branching fraction of D þ → ϕK þ is from the LHCb measurements [4,5]; while the others are quoted from the particle data group [6]. In this paper, we report improved measurements of the branching fractions of these inclusive decays by using 2.93 fb −1 of e þ e − annihilation data taken at ffiffi ffi s p ¼ 3.773 GeV with the BESIII detector. Throughout this paper, the charged conjugate modes are implied unless stated explicitly. Second, charge-parity (CP) violation plays an important role in interpreting the matter-antimatter asymmetry in the Universe and in searching for new physics beyond the standard model (SM). It has been well established in the K and B meson systems. In the SM, however, CP violation in charm decays is expected to be much smaller [7][8][9]. Searching for CP violation in D meson decays is important for exploring physics beyond the SM. Recently, CP violation in the charm sector was observed for the first time in the charm hadrons decays at the LHCb [10]. In this paper, we search for CP violation in the inclusive D → ϕX andD → ϕX decays.

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector is a magnetic spectrometer [11] located at the Beijing Electron Positron Collider [12]. The cylindrical core of the BESIII detector consists of a heliumbased multilayer drift chamber, a plastic scintillator timeof-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over a 4π solid angle. The chargedparticle momentum resolution at 1 GeV=c is 0.5%, and the dE=dx resolution is 6% for the electrons from Bhabha scattering. The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end cap) region. The time resolution of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps. The end cap TOF system was upgraded in 2015 with multigap resistive plate chamber technology, providing a time resolution of 60 ps [13]. More details about the design and performance of the detector are given in Ref. [11].
Simulated samples of events produced with the GEANT4based [14] Monte Carlo (MC) package, which includes 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 initial state radiation (ISR) in the e þ e − annihilations modeled with the generator KKMC [15,16]. The inclusive MC samples consist of the production of DD pairs with the consideration of quantum coherence for all neutral D modes, the non-DD decays of the ψð3770Þ, the ISR production of the J=ψ and ψð3686Þ states, and the continuum processes incorporated in KKMC [15,16]. The known decay modes are modeled with EVTGEN [17,18] using branching fractions taken from the Particle Data Group [6], and the remaining unknown charmonium decays are modeled by LUNDCHARM [19]. Final state radiation from charged final state particles is incorporated with the PHOTOS package [20].

III. ANALYSIS METHOD
As the ψð3770Þ resonance peak lies just above the DD threshold, it decays predominately into DD meson pairs.  This advantage is leveraged by using a double-tag method, which was first developed by the MARKIII Collaboration [21,22], to determined absolute branching fractions. If aD (D − orD 0 ) meson is found in an event, the event is identified as a "single-tag (ST) event." If the partner D (D þ or D 0 ) is reconstructed in the rest of the event, the event is identified as a "double-tag (DT) event." In this analysis, the ST D − mesons are reconstructed by using K þ π − π − , K þ π − π − π 0 , K 0 S π − , K 0 S π − π 0 , and K 0 S π − π − π þ , and the STD 0 mesons are reconstructed by using K þ π − , K þ π − π 0 , and K þ π − π − π þ . The signal D þ and D 0 mesons are reconstructed by using ϕX, ϕ → K þ K − . The branching fraction for D → ϕX decay is given by where i is the ith ST mode, N i DT and N i ST are the yield of the DT and ST events, ϵ i ST is the efficiency for reconstructing the tag candidate, and ϵ i DT is the efficiency for simultaneously reconstructing theD decay to tag mode i and D decay to ϕX. N DT and N ST are the total yields of the DTand STevents, and is the average efficiency of finding the signal decay, weighted by the yields of tag modes in data. Here, f i QC is a factor to take into account the quantum-correlation (QC) effect in D 0D0 pairs, called QC correction factor. The f i QC is taken as unity for charged D tags, but determined for neutral D tags following Refs. [23,24] (see the Appendix for more details).

IV. SELECTION AND YIELD OF STD MESONS
All charged tracks, except those originating from K 0 S decays, are required to originate in the interaction region, which is defined as V xy < 1 cm, jV z j < 10 cm, j cos θj < 0.93, where V xy and jV z j denote the distances of the closest approach of the reconstructed track to the interaction point perpendicular to and parallel to the beam direction, respectively, and θ is the polar angle with respect to the beam axis. Charged tracks are identified using confidence levels for the kaon (pion) hypothesis CL KðπÞ [11], calculated with both dE=dx and TOF information. The kaon (pion) candidates are required to satisfy CL KðπÞ > CL πðKÞ and CL KðπÞ > 0. The K 0 S candidates are formed from two oppositely charged tracks with jV z j < 20 cm and j cos θj < 0.93. The two charged tracks are assumed to be a π þ π − pair without particle identification (PID), and the π þ π − invariant mass must be within ð0.487; 0.511Þ GeV=c 2 . The photon candidates are selected from isolated EMC clusters. To suppress electronics noise and beam backgrounds, the clusters are required to have a start time within 700 ns after the event start time and have an opening angle greater than 10°with respect to the nearest extrapolated charged track. The energy of each EMC cluster is required to be larger than 25 MeV in the barrel region (j cos θj < 0.8) or 50 MeV in the end cap region (0.86 < j cos θj < 0.92). To select π 0 meson candidates, the γγ invariant mass is required to be within ð0.115; 0.150Þ GeV=c 2 . The momentum resolution of π 0 candidates is improved with a kinematic fit that constrains the γγ invariant mass to the π 0 nominal mass [6]. ForD 0 → K þ π − candidates, backgrounds arising from cosmic rays and Bhabha scattering events are rejected with the same requirements as those described in Ref. [25].
Two variables, the energy difference ΔE≡ ED − E beam and the beam-energy-constrained mass , are used to identify the STD candidates. Here, E beam is the beam energy, and EDðpDÞ is the reconstructed energy (momentum) of the STD candidates in the center-of-mass frame of the e þ e − system. For a given tag mode, if there are multiple candidates per charm per event, the one with the smallest value of jΔEj is retained. Combinatorial backgrounds are suppressed by mode dependent ΔE requirements, as shown in Table II.     Table II.

V. SELECTION AND YIELD OF D → ϕX
DT events containing a ϕ meson are selected by investigating the system recoiling against the ST D − ðD 0 Þ. Candidate DT events are required to have at least two good charged tracks with opposite charges. The ϕ candidates are reconstructed through ϕ → K þ K − decays. The selection and identification criteria of the charged kaons are identical to those for the tag side.
The K þ K − invariant mass (M K þ K − ) spectra of the accepted candidates for D → ϕX in the M BC signal region are shown in the top row of Fig. 2. The events in the M BC sideband region, ð1.844; 1.860Þ GeV=c 2 for D þ and ð1.840; 1.856Þ GeV=c 2 for D 0 , are used to estimate the peaking backgrounds in the M K þ K − spectra, as shown in the bottom row of Fig. 2. For each case, the yield of DT events containing D → ϕX signals is obtained by fitting these spectra. A MC-simulated signal shape convolved with a Gaussian function is used to model the ϕ signal, and the combinatorial backgrounds are modeled by a reversed ARGUS background function [26]. The sideband contributions are normalized to the same background areas in the M BC signal region. The fit results are also shown in Fig. 2. The fitted DT yields in the M BC signal and sideband regions in the data, N sig DT and N sid DT , are given in Table III. The background-subtracted DT yields are calculated by   background area in the M BC signal region over that in the M BC sideband region and is determined to be 0.82 for the D þ decay and 0.92 for the D 0 decay. These results have been verified by analyzing the inclusive MC sample.

VI. BRANCHING FRACTION
The detection efficiencies are estimated by analyzing exclusive signal MC samples with the same procedure as for analyzing data. For the ST side, all possible subresonances have been included in the MC simulations. For the signal side, all known D meson decays involving ϕ have been included in the MC simulations. Especially, to obtain better data/MC agreement, we have readjusted the branching fraction of D þ → ϕπ þ π 0 , which is dominated by D þ → ϕρ þ , to be 0.6% in the MC simulations. The efficiencies have been corrected by the small differences in K AE tracking and PID between the data and MC simulation. To verify the reliability of the detection efficiencies, we compare the cos θ and momentum distributions for ϕ, K þ , and K − for the selected candidate events in data and MC simulations, as shown in Figs. 3 and 4. Good data-MC agreement is observed. The detection efficiencies and the measured branching fractions for D → ϕX are given in Table III.
Most of the systematic uncertainties originating from the ST selection criteria cancel when using the DT method. The systematic uncertainties in these measurements are assigned relative to the measured branching fractions and are discussed below.
The uncertainties due to the M BC fits are estimated by using alternative signal shapes, varying the bin sizes, varying the fit ranges, and shifting the end point of the ARGUS background function. We obtain 0.5% as the total systematic uncertainty due to the M BC fits.
The tracking and PID efficiencies for K AE are studied by using DT DD hadronic events. In each case, the efficiency to reconstruct a kaon is determined by using the missing mass recoiling against the rest of the event and determining the fraction of events for which the missing kaon can be reconstructed. The differences in the momentum weighted efficiencies between the data and MC simulations (called the data-MC difference) due to tracking and PID are determined to be ð4.2 AE 0.5Þ% and ð0.5 AE 0.5Þ% per K AE . After correcting the detection efficiencies obtained by MC simulations by these differences, the uncertainties of the data-MC differences are assigned as the systematic uncertainties for the K AE tracking and PID efficiencies. This gives a systematic uncertainty for the K AE tracking or PID efficiency of 0.5% per track.
The systematic uncertainties arising from the fit range in the M K þ K − fits are estimated by a series of fits with alternative intervals. The maximum deviations in the resulting branching fractions are assigned as the associated systematic uncertainties, which are 0.4% and 1.3% for D þ → ϕX and D 0 → ϕX, respectively. To estimate the systematic uncertainties due to the signal shape in the M K þ K − fits, we use a Breit-Wigner function to describe the ϕ signal. The maximum deviations in the resulting branching fractions are assigned as the associated systematic uncertainties, which are 1.6% and 1.8% for D þ → ϕX and D 0 → ϕX, respectively. To estimate the systematic uncertainties due to the background shape in the M K þ K − fits, we use an  alternative background shape, to describe the background. The maximum deviations in the resulting branching fractions are assigned as the associated systematic uncertainties, which are 0.2% and 1.6% for D þ → ϕX and D 0 → ϕX, respectively. We assume that systematic uncertainties arising from the fit range, signal, and background shape are independent and add them in quadrature to obtain the systematic uncertainty of the M K þ K − fit.
In our nominal analysis, the measured branching fraction of D 0 → ϕX has been corrected by an averaged QC factor f QC defined in Sec. VI. After this correction, we take the residual uncertainty of f QC , 0.5%, as the systematic uncertainty due to the QC effect. The uncertainties due to limited MC samples are 0.8% and 0.7% for D þ and D 0 decays, respectively. The uncertainty in the quoted branching fraction of ϕ → K þ K − is 1.0% [6].
Assuming all the sources are independent, the quadratic sum of these uncertainties gives the total systematic uncertainty in the measurement of the branching fraction for each decay. Table IV summarizes the systematic uncertainties in the branching fraction measurements.

VII. ASYMMETRY OF BðD → ϕXÞ AND BðD → ϕXÞ
We determine the branching fractions of D → ϕX and D → ϕX separately. In this section, charge conjugated modes are not implied. Table III summarizes the ST yields, the DT yields in the M BC signal and sideband regions, detection efficiencies, and the measured branching fractions. The asymmetry of the branching fractions of D → ϕX andD → ϕX is determined by The asymmetries for charged and neutral D → ϕX decays are determined to be ð−0.7 AE 2.8 AE 0.7Þ% and ð−0.4AE 2.5 AE 0.7Þ%, where the uncertainties due to the M BC fit, K AE tracking, K AE PID, M K þ K − fit, QC effect, and quoted branching fractions in the measurements of BðD → ϕXÞ and BðD → ϕXÞ cancel. No CP violation is found at the current statistical and systematic precision.

VIII. CONCLUSIONS
By analyzing 2.93 fb −1 of e þ e − annihilation data taken with the BESIII detector at ffiffi ffi s p ¼ 3.773 GeV, the branching fractions of D þ → ϕX and D 0 → ϕX decays are measured to be ð1.135 AE 0.034 AE 0.031Þ% and ð1.091 AE 0.027 AE 0.035Þ%, respectively, where the first uncertainties are statistical and the second are systematic.
Comparisons of our results with the previous measurements by CLEO [2] and BES [1] are shown in Table V. Our results are consistent with previous measurements, but with much better precision. These results indicate that the nominal values of the branching fractions for some known exclusive decays of the D þ meson, e.g., D þ → ϕπ þ π 0 , may be overestimated. Precision measurements of some exclusive ϕX decays of D þ and D 0 mesons are required to further understand the discrepancy. We also determine CP asymmetries in the branching fractions of D → ϕX andD → ϕX decays for the first time, but no CP violation is found.

APPENDIX: QC CORRECTION FACTOR
At ψð3770Þ, the D 0D0 pairs are produced coherently. The impact of the QC effect on the measurement of the branching fraction of D 0 → ϕX is considered by two aspects: the strong-phase parameters of the tag modes and the CPþ fraction of the D 0 → ϕX decay.

Formulas
Due to the QC effect, the yield of the ith ST candidates can be written as [23,24] and the yield of the DT candidates, i.e., CPAE eigenstate decay vs the ith tag, can be written as where N D 0D0 is the total number of D 0D0 pairs produced in data; ϵ i STðDTÞ is the efficiency of reconstructing the ST (DT) candidates; B i ST and B i sig are the branching fractions of the ST and signal decays, respectively; R i WS;f is the ratio of the Cabibbo-suppressed and Cabibbo-favored rates; r i f is defined as and δ i f is the strong-phase difference between these two amplitudes.
In this analysis, R i WS;f is taken to be r 2 i , where r i is the ratio of the Cabibbo-suppressed and Cabibbo-favored amplitudes for D 0D0 decays to same final state. Then, we have where R i is the coherence factor, 0 < R i ≤ 1, that quantifies the dilution due to integrating over the phase space (for D → K AE π ∓ , R ¼ 1.00) [27,28]. According to Eqs. (A3) and (A4), the absolute branching fraction for the signal decay is calculated by where C i f is the strong-phase factor, which can be calculated by The amplitude of the neutral D decays can be decomposed as mixture of the CPþ and CP− components. This gives F sig þ ¼ 1 − F sig − , where F sig þ and F sig − are the CPþ and CP− fractions of the decay, respectively. The yield of the DT candidates tagged by the Cabibbo-favored tag mode i can be written as According to Eqs. (A3) and (A7), the branching fraction of the signal decay can be calculated by Here, þ −1Þ is the QC correction factor to be determined.
2. Strong-phase factor C i f Based on Eq. (A6) and quoted parameters of r i , R i , and δ i f , we obtain the strong-phase factor C i f for the different ST modes. The quoted parameters of r i , R i and δ i f as well as the obtained C i f are listed in Table VI.

CP + fraction of the signal decay
According to Ref. [30], the CPþ fraction for the signal decay is determined by in which N AE is the ratio of the DT and ST yields with CP ∓ tags and is obtained by where M AE is the DT yields for D 0 → ϕX vs CP ∓ tags and S AE is the corrected ST yields for the CPAE decay modes. Here, η AE ¼ AE1 for CPAE decay modes, and y D is the D 0D0 mixing parameter from the heavy flavor averaging group (HFAG) average [6].
To extract F sig þ of the D 0 → ϕX decay, we use the CPþ tag of D → K þ K − and the CP− tag of D → K 0 S π 0 . Figures 5 and 6 show the fits to the M BC distributions of the ST candidates and the M K þ K − distributions of the DT candidates. From the fits, we obtain the measured ST and DT yields (S AE measured and M AE measured ), as summarized in Table VII. Inserting these numbers in Eqs. (A9) and (A10), we obtain F sig þ ¼ 0.64 AE 0.05.

Impact on the measured branching fraction
Inserting the C i f and F sig þ obtained above in Eqs. (A6) and (A9), we obtain the QC correction factors for the D → K AE π ∓ , D → K AE π ∓ π 0 , and D → K AE π ∓ π ∓ π AE ST decays to be ð96.9 AE 0.3 AE 1.1Þ%, ð98.1 AE 0.3 AE 0.7Þ%, and ð99.2 AE 0.7 AE 0.3Þ%, where the first and second uncertainties are from C i f and F sig þ , respectively.