Measurement of the branching fraction of $\Lambda_c^+ \to p \omega$ decay at Belle

Using 980.6 $\rm fb^{-1}$ of data collected with the Belle detector operating at the KEKB asymmetric-energy $e^+e^-$ collider, we present a measurement of the branching fraction of the singly Cabibbo-suppressed decay $\Lambda_c^+ \to p \omega$. A clear $\Lambda_c^+$ signal is observed for $\Lambda_c^+ \to p \omega$ with a statistical significance of 9.1 standard deviations, and we measure the ratio of branching fractions ${\cal B}(\Lambda_c^+ \to p \omega)/{\cal B}(\Lambda_c^+ \to p K^- \pi^+) = (1.32 \pm 0.12 (\rm stat) \pm 0.10 (\rm syst))\times 10^{-2}$, from which we infer the branching fraction ${\cal B}(\Lambda_c^+ \to p \omega) = (8.27 \pm 0.75 (\rm stat) \pm 0.62 (\rm syst) \pm 0.42 (\rm ref))\times 10^{-4}$. The first quoted uncertainty is statistical, the second systematic, and the third from the reference mode $\Lambda_c^+ \to p K^- \pi^+$.


I. INTRODUCTION
Charmed mesons and baryons are copiously produced in the B-factory experiment, providing an excellent arena for understanding Quantum Chromodynamics (QCD) with transitions involving charm quark.SU(3) F flavor symmetry [1,2] and QCD dynamical models [3][4][5] provide theoretical estimates of charmed baryon decays.The former relies on experimental results as the input; the latter models often make different predictions for unknown baryon wave functions and non-factorizable contributions, which makes it difficult to perform definitive tests between theoretical models.
Experimentally, the investigation of charmed baryon decays is more difficult than for charmed mesons due to their smaller production rate.Only the lowest-lying charmed baryon Λ + c decays weakly.Since it was first discovered [6], many hadronic weak decays, mostly Cabibbofavored, have been observed [7].In contrast, the knowledge of Cabibbo-suppressed decays has been limited.Both measurements and theoretical models point to nonfactorizable contributions, such as W -exchange, having a sizable impact on individual decay rates as well as the total widths [8][9][10][11].
In this analysis, we measure the branching fraction of the Λ + c → pω(→ π + π − π 0 ) channel for the first time at Belle, taking advantage of the large value of B(ω → π + π − π 0 ) [7].To improve the measurement precision, we measure the ratio of the branching fractions of this SCS process with respect to the Λ + c → pK − π + reference decay mode: where N data and ǫ MC are the number of fitted Λ + c events in data and the detection efficiency, respectively; the subscript "ref" refers to the reference mode and "sig" to the signal mode; and B ′ = B(ω → π + π − π 0 )×B(π 0 → γγ) [7].

II. THE DATA SAMPLE AND THE BELLE DETECTOR
Measurement of the branching fraction of Λ + c → pω is based on a data sample taken at or near the Υ(1S), Υ(2S), Υ(3S), Υ(4S), and Υ(5S) resonances collected with the Belle detector at the KEKB asymmetric-energy e + e − collider [16], corresponding to an integrated luminosity of 980.6 fb −1 .The Belle detector is a large-solidangle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons (KLM).The detector is described in detail elsewhere [17].
A signal Monte Carlo (MC) sample of e + e − → cc; cc → Λ + c X with X denoting anything; Λ + c → pω with ω → π + π − π 0 , π 0 → γγ is used to optimize the selection criteria and estimate the reconstruction and selection efficiency.Events are generated with pythia [18] and EvtGen [19], and decay products are propagated by geant3 [20] to simulate the detector performance.Charge-conjugate modes are also implied unless otherwise stated throughout this paper.

III. EVENT SELECTION
The Λ + c candidates are reconstructed in two decay modes, Λ + c → pK − π + and Λ + c → pω with ω → π + π − π 0 , π 0 → γγ, corresponding to the reference and signal modes, respectively.Final-state charged particles, p, K, and π, are selected using the likelihood information derived from the charged-hadron identification systems (ACC, TOF, CDC) into a combined likelihood, R(h|h ′ ) = L(h)/(L(h) + L(h ′ )) where h and h ′ are π, K, and p as appropriate [22].The protons are required to have R(p|π) > 0.9 and R(p|K) > 0.9, charged kaons to have R(K|p) > 0.4 and R(K|π) > 0.9, and charged pions to have R(π|p) > 0.4 and R(π|K) > 0.4.A likelihood ratio for e and h identification, R(e), is formed from ACC, CDC, and ECL information [23], and is required to be less than 0.9 for all charged tracks to suppress electrons.For the typical momentum range of our signal decay, the identification efficiencies of p, K, and π are 82%, 70%, and 97%, respectively.Probabilities of misidentifying h as h ′ , P (h → h ′ ), are estimated to be 3% [P (p → π)], 7% [P (p → K)], 10% [P (K → π)], 2% [P (K → p)], 5% [P (π → K)], and 1% [P (π → p)].Furthermore, for each charged-particle track, the distance of closest approach with respect to the interaction point along the direction opposite the e + beam (z axis) and in the transverse rφ plane is required to be less than 2.0 cm and 0.1 cm, respectively.In addition, at least one SVD hit for each track is required.
For Λ + c → pK − π + , a common vertex fit is performed on Λ + c candidates and the corresponding χ 2 vtx value is required to be less than 40 to reject the combinatorial background.We require a scaled momentum of x p > 0.53 to suppress the background, especially from B-meson decays, where , and E cm is the center-of-mass (CM) energy, p * and M are the momentum and invariant mass, respectively, of the Λ + c candidates in the CM frame.All of these optimized selection criteria are the same as those in our previous publication [25].
An ECL cluster not matching any track is identified as a photon candidate.To reject neutral hadrons, the sum of the energy deposited in the central 3 × 3 square of ECL cells is required to be at least 90% of the total energy deposited in the enclosing 5 × 5 square of cells for each photon candidate.Moreover, the energy of photon candidates must exceed 50 MeV and 70 MeV in the barrel (−0.63 < cos θ < 0.85) and endcap (−0.91 < cos θ < −0.63 or 0.85 < cos θ < 0.98) regions of the ECL, respectively, where θ is the inclination angle with respect to the z axis.A π 0 candidate is reconstructed by two photons and 0.08 < M (γγ) < 0.18 GeV/c 2 is required.We perform a mass-constrained (1C) fit on the two photons to require their mass at the π 0 nominal mass [7] and the corresponding χ 2 1C value must be less than 10.For ω → π + π − π 0 , we place a requirement on the momentum of ω candidates in the CM frame: P * (ω) > 0.9 GeV/c.An ω candidate and a proton candidate are combined to form a Λ + c candidate.A common vertex fit is performed for the three charged tracks, p and π ± , and the requirement of χ 2 vtx < 15 is set to suppress background events without a common vertex, especially due to longlived particles such as K 0 S and Σ + .Again, x p > 0.53 is required for Λ + c → pω candidates.With the above requirements, ∼8% of events have multiple Λ + c candidates.We select the best Λ + c candidate based on the minimum χ 2 1C value; the efficiency for this best candidate selection is around 70%.All the above selection criteria are based on an optimization with a maximum figure-ofmerit S/ √ S + B, where S and B are the numbers of signal and background events, respectively, expected in the Λ + c signal region [(2.25, 2.32) GeV/c 2 , corresponding to ±2.5σ around the nominal Λ + c mass [7]].S is estimated via , where B(Λ + c → pω) is assumed to be 9.4 × 10 −4 [12], while the other parameters have been introduced in Eq. (1).Likewise, B is the number of background events obtained from inclusive MC samples normalized to the signal region.
From the study of inclusive MC samples [21], there are several peaking backgrounds from the decays Λ which have the same final-state topology as the signal.However, owing to the long lifetime of K 0 S , Σ + , and Λ, many of the decay vertices of these particles are displaced by several centimeters from the main vertex.Therefore, the χ 2 vtx requirement suppresses most of these background events, subsequently leaving no K 0 S nor Σ + peaks in the M (π + π − ) and M (pπ 0 ) distributions, respectively.In the M (pπ − ) spectrum, a Λ signal is seen and an optimized requirement of |M (pπ − ) − m(Λ)| > 2.756 MeV/c 2 (≈ 3σ) is placed, where m(Λ) is the nominal mass of Λ [7].There is a small ∆ ++ signal observed in the M (pπ + ) distribution.Due to the broad width of the ∆ ++ (∼118 MeV) [7], no requirement on M (pπ + ) is imposed.Since such a background can be described by the ω sidebands, a simultaneous fit to the M (pω) distributions from the selected events in the ω signal region and the normalized ω sidebands is used to handle the ∆ ++ background in extracting the Λ + c signal events, as introduced in the following section.

IV. EFFICIENCY ESTIMATION AND FIT RESULTS
To measure the ratio of the branching fractions, B(Λ + c → pω)/B(Λ + c → pK − π + ), we first determine the yields of Λ + c → pK − π + and Λ + c → pω by fitting the corresponding invariant mass distributions.Figure 1 shows the M (pK − π + ) distribution overlaid with the fit result.A clear Λ + c signal is seen and we fit the M (pK − π + ) distribution using a binned maximum likelihood fit with a bin width of 3 MeV/c 2 .A sum of two Gaussian functions with a common mean value is used to model the signal events and a second-order polynomial is used to model the background events.The parameters of the signal and background shapes are free in the fit.The reduced χ 2 value of the fit is χ 2 /ndf = 87/82 = 1.06 and the fitted number of signal events is 1476200 ± 1560, where ndf is the number of degrees of freedom and the uncertainty is statistical only.The signal efficiency for this reference mode is estimated to be (14.06 ± 0.01)% via a Dalitz-plot method [26]; the details can be found in Ref. [25].Since the decay Λ + c → pη with η → γγ has been well measured [25], the same transition Λ + c → pη, followed by the decay η → π + π − π 0 , having the same final-state topology as our signal mode, is taken as a control channel to validate the event selection criteria.With the final selection criteria, a clear η signal is observed in the M (π + π − π 0 ) distribution and the η signal region is defined as 0.535 < M (π + π − π 0 ) < 0.561 GeV/c 2 .In the M (pη) distribution, a significant Λ + c signal is observed and a one-dimensional fit is performed on the M (pη) distribution using an unbinned maximumlikelihood method.A sum of two Gaussian functions with the same mean value is used to model the Λ + c signal and a second-order polynomial function is used to model the background, with all parameters floated in the fit.The determined number of Λ + c signal events is 819.9 ± 78.6 and the signal efficiency is (1.48 ± 0.01)%, as determined from a signal MC sample.Therefore, the branching ratio of Λ + c → pη with respect to the reference mode Λ With the final selection criteria applied, the π + π − π 0 invariant mass distribution is displayed in Fig. 2.There is a clear ω signal and a fit to the sum of a polynomial and a signal function is performed using an unbinned maximum-likelihood method.The ω signal is described by a Breit-Wigner (BW) function convolved with a double Gaussian function to represent the detector resolution.The mass and width of the BW function are set to the ω world average value [7], the means are constrained to be the same for the double Gaussian function, and the remaining parameters are free.A third-order polynomial function is used to model the combinatorial background.The fit result is shown in Fig. 2, along with the pulls (N data − N fit )/σ data , where σ data is the error on N data .The ω signal region is determined to be 0.75 to 0.81 GeV/c 2 in the M (π + π − π 0 ) spectrum, corresponding to a 92% selection efficiency, and the sideband regions of ω are set to be (0.64, 0.70) GeV/c 2 and (0.86, 0.92) GeV/c 2 .
The M (pω) distribution for events in the ω signal region and the normalized ω sideband regions are shown in Fig. 3.There is a clear Λ + c signal observed and we perform a simultaneous extended unbinned maximumlikelihood fit to extract the Λ + c signal yield.The function for an event in the ω signal region (SR) is described as and that for an event in the ω sidebands (SB) is where P s and P b are probability density functions (PDFs) of the Λ + c signal and background for the M (pω) distribution with the events in SR, respectively; P s and P b sb are the Λ + c signal and background PDFs for the M (pω) distribution with the events in SB; n s , n b , n s sb , and n b sb are the corresponding numbers of the fitted events; f norm = S sb /S sr = 0.428 is the normalization factor determined by fitting the M (π + π − π 0 ) distribution (S sb and S sr are the numbers of the fitted background events in defined ω sidebands and signal region, respectively).The extended likelihood function is where sb , and N sr and N sb are the number of events in SR and SB.The P s and P s sb are both a sum of two Gaussian functions with the same mean value.The parameters of P s and P s sb are kept the same and floated.The P b and P b sb are described by second-order and third-order polynomial functions, respectively.All parameters of the background functions are free.The fit result and pulls are shown in Fig. 3.After fitting, n s = 1829 ± 168 and n s sb = 39 ± 14 are obtained.The χ 2 /ndf for the fit is 44/41 = 1.07 for the fit.The statistical significance is evaluated with −2ln(L 0 /L max ), where L 0 is the maximized likelihood value with the number of signal events set to zero, and L max is the nominal maximized likelihood value.We obtain 9.1σ as the statistical significance.With all event selections, the M (pω) distribution from signal MC sample is obtained and signal events of Λ + c are determined by fitting the M (pω) distribution.We use a sum of two Gaussian functions with the same mean value to model the signal and a second-order polynomial  FIG.3: A simultaneous fit to the pω invariant mass distribution in the ω signal region, and the normalized ω sideband regions is shown.The black dots with error bars represent the data; the red solid line represents the total fitted result; the blue dashed line represents the signal shape; the magenta long-dashed line represents the fitted sideband line shape, and the green filled region is from the normalized sideband regions.
function to model the background.All parameters of the signal and background functions are free.The efficiency of our signal decay is obtained by the ratio of the number of fitted signal events in the M (pω) distribution to that of generated events from signal MC sample, which is (1.50 ± 0.01)%, where the uncertainty is statistical only.The branching ratio is thus B(Λ + c → pω)/B(Λ + c → pK − π + ) = (1.32 ± 0.12) × 10 −2 , where the uncertainty is statistical.

V. SYSTEMATIC UNCERTAINTIES
Since the branching fraction is obtained from a ratio of quantities in Eq. ( 1), some systematic uncertainties cancel.The sources of systematic uncertainties include the fits of the reference and signal modes, particle identification (PID), photon efficiency, the uncertainties of branching fractions for the ω → π + π − π 0 and π 0 → γγ decays, and the statistics of the signal MC sample.
The systematic uncertainty from the fit of the M (pK − π + ) spectrum is estimated by modifying the signal and background functions, bin width, and the fit range.To evaluate the uncertainty from the signal function, the signal shape is fixed to that from the fit to the MC sample.The uncertainty from the background shape is assessed by using a first-order polynomial.Furthermore, the bin width is varied from 2 MeV/c 2 to 4 MeV/c 2 , and the fit range of the invariant mass spectrum adjusted to estimate the uncertainties from binning and fit range.The fractional difference in measured branching ratios, 2.1%, is taken as the uncertainty.The system-atic uncertainty from the fit of the M (pω) distribution is estimated by changing the signal and background line shapes, the fit range, and the fit method.The signal shape is changed from the double Gaussian function to a single Gaussian function, and the background line shape is changed from the second-order polynomial function to a third-order polynomial function, as well as enlarging the fit range.In addition, a two-dimensional unbinned maximum-likelihood fit of the (M (pω), M (π + π − π 0 )) distribution is performed, to evaluate the the fit method uncertainty, and the fractional difference in the branching ratio, 5.2%, is taken as the systematic uncertainty.Systematic uncertainties from PID efficiencies of the p and π + cancel approximately, resulting in negligible amount of systematic uncertainty in the ratio.Systematic uncertainties of 1.6% and 1.3% are assigned for the K − and π − identification efficiencies, respectively, calculated using a D * + → D 0 π + with D 0 → K − π + sample.The total systematic uncertainty from PID is 2.9%.The systematic uncertainty due to tracking efficiency cancels in the ratio.Based on a study of radiative Bhabha events, a systematic uncertainty of 2.0% is assigned to the photon efficiency for each photon, and the total systematic uncertainty from photon reconstruction is thus 4.0%.Since the signal efficiency is independent of the decay angular distribution of proton in the Λ + c rest frame, the model-dependent uncertainty has negligible effect on efficiency.The systematic uncertainty from B(ω → π + π − π 0 )×B(π 0 → γγ) is 0.7% [7], and that from the size of the signal MC sample is estimated to be 0.8% for Λ + c → pω.These systematic uncertainties are summarized in Table I, where a total systematic uncertainty of 7.6% is obtained by assuming all uncertainties are independent and adding them in quadrature.We measure the ratio of branching fractions B(Λ + c → pω) B(Λ + c → pK − π + ) = (1.32 ± 0.12 ± 0.10) × 10 −2 .(5) Using B(Λ + c → pK − π + ) = (6.28 ± 0.32) × 10 −2 [7], we obtain the branching fraction: B(Λ + c → pω) = (8.27± 0.75 ± 0.62 ± 0.42) × 10 −4 , ( where the first uncertainty is statistical, the second systematic, and the third from the reference mode Λ + c → pK − π + .This result is consistent with the LHCb result (9.4 ± 3.9) × 10 −4 [12], and agrees with the theoretical predictions of (11.4 ± 5.4) × 10 −4 [13] and (6.3 ± 3.4) × 10 −4 [14] within uncertainties based on the SU(3) F flavor symmetry.However, our result contradicts the QCD dynamical model prediction of (3.4 − 3.8) × 10 −4 [15].

FIG. 1 :
FIG.1: Fit to the invariant mass distribution of pK − π + from data.Black dots with error bars represent the data; the pink dashed line, the blue dash-dotted line, the green long-dashed line, and the red solid line represent the background contribution, the core Gaussian, tail Gaussian, and the total fit, respectively.

2 FIG. 2 :
FIG.2:A fit to the π + π − π 0 invariant mass distribution is shown.The black dots with error bars represent the data; the red solid line represents the total fitted result; the blue dashed line represents the signal shape; and the magenta dasheddotted line represents the fitted background.The region between the two violet vertical lines is regarded as the signal region and the two regions between the pairs of green vertical lines are regarded as the ω sideband regions.