Measurements of the branching fractions of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ and $\chi_{cJ(J=0,1,2)}\rightarrow\Lambda\bar{\Lambda}$

Since the final state of interest is $\gamma p\bar{p}\pi^{+}\pi^{-},$ we require each $\psi(3686)$ candidate to contain four charged tracks with zero net charge, and at least one photon. Each charged track, detected in the MDC is required to satisfy $|\rm{cos\theta}|<0.93$, where $\theta$ is defined with respect to the $z$-axis, which is the symmetry axis of the MDC. The distance of closest approach to the interaction point (IP) must be less than 30 cm along the $z$-axis, and less than 10 cm in the transverse plane. Pions and protons are identified by the magnitude of their momentum, and charged tracks with momentum larger than 0.7 $\mathrm{GeV}/c$ in the lab frame are identified as protons. Other tracks are identified as pions. An isolated cluster in the EMC is considered to be a photon if the following requirements are satisfied: 1) the deposited energy of each shower must be more than 25 MeV in the barrel region ($|\!\cos\theta|<0.80$) and more than 50 MeV in the end cap region ($0.86<|\!\cos\theta|<0.92$); 2) to suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within (0, 700) ns; 3) to exclude showers that originate from charged tracks, the angle between the position of each shower in the EMC and the closest extrapolated charged track of $p,\pi^{+},$ or $\pi^{-}$ must be greater than 10 degrees, and greater than 20 degrees for the $\bar{p}$ track.

The $\Lambda(\bar{\Lambda})$ candidate is reconstructed with any $p\pi^{-}$ $(\bar{p}\pi^{+})$ combination satisfying a secondary vertex fit secondvertex . The secondary vertex fit is required to be successful, but no additional requirements are placed on the fit $\chi^{2}$. To improve the momentum and energy resolution and to reduce background contributions, a six-constraint (6C) kinematic fit is applied to the event candidates with constraints on the total four-momentum and the invariant masses of the $\Lambda$ and $\bar{\Lambda}$ candidates. The $\chi_{6C}^{2}$ of the kinematic fit is required to be less than 25.

To further suppress background, we require: 1) the $\chi_{6C}^{2}$ for the $\gamma p\bar{p}\pi^{+}\pi^{-}$ hypothesis is less than those for any $\gamma\gamma p\bar{p}\pi^{+}\pi^{-}$ or $p\bar{p}\pi^{+}\pi^{-}$ hypothesis: $\chi_{\gamma p\bar{p}\pi^{+}\pi^{-}}^{2}<\chi_{\gamma\gamma p\bar{p}\pi^{+}\pi^{-}}^{2}$, $\chi_{\gamma p\bar{p}\pi^{+}\pi^{-}}^{2}<\chi_{p\bar{p}\pi^{+}\pi^{-}}^{2}$; 2) the $\Lambda$($\bar{\Lambda})$ lifetime must satisfy $L_{\Lambda(\bar{\Lambda})}/\sigma>2$ where $L$ and $\sigma$ are the decay length and its uncertainty obtained from the secondary vertex fit; 3) the invariant mass of the $\Lambda\bar{\Lambda}$, $M_{\Lambda\bar{\Lambda}}$ is required to be greater than 3.48 $\mathrm{GeV}/c^{2}$ in order to suppress the $\psi(3686)\rightarrow\gamma\chi_{c0},~{}\chi_{c0}\rightarrow\Lambda\bar{\Lambda}$ background; 4) $\rm M_{\gamma\Lambda}>1.15$ $\mathrm{GeV}/c^{2}$ and $\rm M_{\gamma\bar{\Lambda}}>1.15$ $\mathrm{GeV}/c^{2}$ are required to suppress background from $\psi(3686)\rightarrow\Lambda\bar{\Lambda}$.

After imposing the above requirements, Fig. 1 shows the scatter plots of $\rm M_{\gamma\bar{\Lambda}}$ versus $\rm M_{\gamma\Lambda}$ for data, inclusive MC samples, and signal MC samples of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda$ and $\psi(3686)\rightarrow\Sigma^{0}\bar{\Lambda}$ processes. The $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ signals are clearly visible in Fig. 1 (a). The two sloped bands are backgrounds from $\psi(3686)\rightarrow\gamma\chi_{c1,2},\chi_{c1,2}\rightarrow\Lambda\bar{\Lambda}$, which are well simulated by the inclusive MC samples as shown in Fig. 1 (b). The inclusive MC indicates that the only peaking background is from $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Sigma^{0}$. Figures 1 (c) and (d) are the signal shapes of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda$ and $\psi(3686)\rightarrow\Sigma^{0}\bar{\Lambda}$, which are vertical and horizonal bands in the scatter plot distributions, respectively.

We determine the signal yields by an unbinned maximum likelihood fit to the two-dimensional distributions of the $\gamma\Lambda$ and $\gamma\bar{\Lambda}$ invariant masses. The signal shapes are determined from signal MC simulation for the $\Sigma^{0}\bar{\Lambda}$ and $\bar{\Sigma}^{0}\Lambda$ processes. The background shape includes five items: $\psi(3686)\rightarrow\Sigma^{0}\bar{\Sigma}^{0}$, $\psi(3686)\rightarrow\gamma\chi_{cJ}(\chi_{cJ}\rightarrow\Lambda\bar{\Lambda})$ with $J$ = 0, 1, 2, and other background contributions. The shapes of the first four items are determined from MC simulation while the last one is described by a two-variable first-order polynomial function $f(m_{\gamma\Lambda},m_{\gamma\bar{\Lambda}})=am_{\gamma\Lambda}+bm_{\gamma\bar{\Lambda}}+c$ where $a$, $b$, $c$ are constant parameters that are determined in the fit. The background yields are floated in the fit except the peaking background $\psi(3686)\to\Sigma^{0}\bar{\Sigma}^{0}$ which is included with its magnitude determined from previous measurements a2 . The $\chi_{cJ}$ background yields are consistent with expectation after considering branching fractions PDG and efficiencies. Figure 2 shows the projections of the two-dimensional fitting results. The numbers of signal events are determined to be $N^{\textrm{sig}}_{\gamma\bar{\Lambda}}=26.1\pm 6.6$, $N^{\textrm{sig}}_{\gamma\Lambda}=37.2\pm 7.7$ from the fit.

The contribution from the continuum process, $i.e.$ $e^{+}e^{-}\rightarrow\gamma^{*}\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ is estimated from the collision data at 3.773 GeV with integrated luminosity of 2931.8 pb^-1 taken during 2010 and 2011. The same event selection criteria as for the $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ decay is applied. In addition, $|\rm M_{\Lambda\bar{\Lambda}}-3.686$ GeV/$c^{2}|>0.01$ GeV/$c^{2}$ is required to suppress background from the $e^{+}e^{-}\rightarrow\gamma_{ISR}$ $\psi(3686)$, $\psi(3686)\rightarrow\Lambda\bar{\Lambda}$ process. An unbinned one dimensional maximum likelihood fit is done to determine signal yields, where the peaking background $e^{+}e^{-}\rightarrow\bar{\Sigma}^{0}\Sigma^{0}$ has been considered with its shape from MC simulation and magnitude from previous measurements a2 . The other backgrounds are described with a second order polynomial function. To account for the difference of the integrated luminosity and cross sections between the two energy points 3.686 GeV and 3.773 GeV, a scaling factor $f=0.24$ is applied. The continuum contributions at 3.686 GeV are determined to be: $N_{\gamma\bar{\Lambda}}^{\rm{cont}}=6.2~{}\pm~{}1.2$ and $N_{\gamma\Lambda}^{\rm{cont}}=3.9~{}\pm~{}1.0$ in the $\Sigma^{0}\bar{\Lambda}$ and $\bar{\Sigma}^{0}\Lambda$ processes, respectively, where the contributions from $e^{+}e^{-}\rightarrow\psi(3770)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ decay have been ignored due to its low production.

The branching fraction of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda$ is calculated by

Here, $N_{\psi(3686)}$ is the total number of $\psi(3686)$ events totalnumber , $Br~{}=~{}\mathcal{B}\left(\bar{\Lambda}\rightarrow\bar{p}\pi^{+}\right)\cdot\mathcal{B}\left(\Lambda\rightarrow p\pi^{-}\right)\cdot\mathcal{B}\left(\bar{\Sigma}^{0}\rightarrow\gamma\bar{\Lambda}\right)$ PDG , and the efficiency $\epsilon_{\bar{\Sigma}^{0}\Lambda}=16.52\%$ is determined from simulation. The branching fraction of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda$ is determined to be $(0.66\pm 0.22)\times 10^{-6}$. Similarly, the branching fraction of $\psi(3686)\rightarrow\Sigma^{0}\bar{\Lambda}$ is calculated to be $(0.94\pm 0.22)\times 10^{-6}$, with $\epsilon_{\Sigma^{0}\bar{\Lambda}}=19.44\%$. The clear difference between $\epsilon_{\Sigma^{0}\bar{\Lambda}}$ and $\epsilon_{\bar{\Sigma}^{0}\Lambda}$ comes from the different selection criteria on the open angle between photon and (anti-)proton. The combined branching fraction of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ is $\mathcal{B}\left(\psi(3686)\!\rightarrow\!\bar{\Sigma}^{0}\Lambda\!+\!c.c.\right)\!=(1.60\pm 0.31)\times 10^{-6}$, where the uncertainty is statistical only.

The systematic uncertainties on the branching-fraction measurement include those from track and photon reconstruction efficiencies, kinematic fit, angle requirement, $\Lambda(\bar{\Lambda})$ reconstruction efficiency, signal and background shapes, and the branching fraction of $\Lambda(\bar{\Lambda})$ decay.

The uncertainty due to photon detection efficiency is $1\%$ per photon, which is determined from a study of the control sample $J/\psi\rightarrow\rho\pi$ photonEfficiency .

The efficiency of $\Lambda(\bar{\Lambda})$ reconstruction is studied using the control sample of $\psi(3686)\rightarrow$ $\Lambda\bar{\Lambda}$ decays, and a correction factor $0.980\pm 0.011$ trackEfficiency is applied to the efficiencies obtained from MC simulation. The uncertainty of the correction factor, 1.1% already includes the uncertainties of MDC tracking and $\Lambda(\bar{\Lambda})$ reconstruction, and 1.1% is taken as the uncertainty of the efficiency of $\Lambda(\bar{\Lambda})$ reconstruction.

To study the uncertainty caused by the requirement on the angle between the position of each shower in the EMC and the closest extrapolated charged track, we utilize the processes $\psi(3686)\rightarrow\gamma\chi_{cJ},\chi_{cJ}\rightarrow\Lambda\bar{\Lambda}$ due to their large statistics. Two sets of branching fractions are obtained. One is with the nominal requirement, i.e. the angle of each shower is at least $10^{\circ}$ away from $p,\pi^{+},\pi^{-}$ tracks and $20^{\circ}$ away from $\bar{p}$ track; the other one requires the angle of each shower is at least $20^{\circ}$ away from $p,\pi^{+},\pi^{-}$ tracks and $30^{\circ}$ away from $\bar{p}$ track. The difference of the branching fractions obtained with the nominal and modified requirements is $1.6\%$, which is taken as the associated systematic uncertainty.

To study the uncertainty associated with the kinematic fit, the track helix parameters are corrected in the MC simulation kfitUncer . The resulting $0.3\%$ efficiency difference before and after the correction is taken as the systematic uncertainty related to the kinematic fit.

The systematic uncertainty associated with the signal shape is mainly due to the resolution difference between data and MC simulation. It is estimated by smearing the signal shape with a resolution of $5\%$ of the one determined from MC simulation according to the study using the control sample of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Sigma^{0}$. The difference is 0.4% and is taken as the corresponding systematic uncertainty due to the signal shape.

For the peaking background, we fixed the shape and number of events in the fitting. We vary the number of background events by its uncertainty. A difference of 1.9% is assigned as systematic uncertainty.

The uncertainty associated with the $\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$ backgrounds is estimated by fixing their contributions to the world average values PDG instead of floating them in the nominal fit. The differences between fixing and floating are 0.7%, 0.4%, and 2.1% for $\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$, respectively, and are taken as the systematic uncertainties.

The systematic uncertainty for the description of the other background contributions is estimated by changing from a two-variable first-order polynomial to a two-variable second-order polynomial to describe the shape of the other backgrounds. The systematic uncertainty is determined to be 0.1%.

In the signal MC sample, $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ is simulated with an uniform distribution over phase space. However, the angular distribution should be described as $\mathrm{d}N/\mathrm{d}\cos\theta\propto 1+\alpha\cos^{2}\theta$ physicsmode where $\theta$ is the polar angle of the (anti) baryon. Since the observed number of events does not allow the determination of the angular distribution in our analysis, we generate MC samples with $\alpha=-1.0$ and $\alpha=1.0$, the two extreme scenarios. The efficiency difference between them is divided by $\sqrt{12}$ under the assumption that the prior distribution of $\alpha$ is uniform, and the result $6.9\%$ is taken as the uncertainty associated to the angular distribution.

The uncertainty on the total number of $\psi(3686)$ events is $0.6\%$ totalnumber . The uncertainty of the $\Lambda$ decay branching fraction is taken from the world average value PDG . Table 1 lists all sources and values of systematic uncertainties, and the total systematic uncertainty is determined by adding them in quadrature.

So far, we have not considered possible interference between the $\psi(3686)$ decay and the continuum process, which is described by

where $A_{\psi^{\prime}}$ and $A_{\textrm{cont}}$ are the amplitudes of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ and the continuum contribution, respectively. The difference between $\theta=0^{\circ}$ and $\theta=180^{\circ}$, corresponding to the extreme constructive and destructive cases respectively, is adopted as the uncertainty associated with the interference. This difference is divided by $\sqrt{12}$, since the prior distribution of the interference angle is assumed to be uniform. Finally, the branching fractions are $\mathcal{B}\left(\psi(3686)\rightarrow\Sigma^{0}\bar{\Lambda}\right)=(0.94~{}\pm~{}0.39)\times 10^{-6}$ and $\mathcal{B}\left(\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda\right)=(0.66~{}\pm~{}0.49)\times 10^{-6}$, where the uncertainties are only the systematic arising from interference. The combined branching fraction of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ is $\mathcal{B}\left(\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.\right)=(1.60~{}\pm~{}0.31~{}\pm~{}0.13~{}\pm~{}0.58)\times 10^{-6}$, where the first uncertainty is statistic, the second is systematic, and the third is the uncertainty due to interference with the continuum.

The initial selection criteria for charged tracks and photons and the $\Lambda(\bar{\Lambda})$ reconstruction are the same as those described above for $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$. Additional selection criteria are 1.) the $\chi^{2}$ from the $6\mathrm{C}$ kinematic fit is required to be less than 50, and 2.) to veto $\psi(3686)\rightarrow\Sigma^{0}\bar{\Sigma}^{0}$ background, the $\gamma\Lambda(\gamma\bar{\Lambda})$ combination is required to be outside of the $\Sigma^{0}\left(\bar{\Sigma}^{0}\right)$ region, that is defined as within $12$ MeV to the $\Sigma^{0}$ nominal mass PDG . The $\chi_{c0}$ veto is also removed.

After all selection requirements have been applied, Fig. 3 shows all background contributions according to the inclusive MC sample, which include two parts: non-flat $\Sigma^{0}\bar{\Sigma}^{0}$ background contributions that tend to accumulate in the $\chi_{c2}$ region and flat non- $\Sigma^{0}$ background contributions. Both background levels are quite low compared with the signal.

To determine the $\chi_{cJ}$ signal yields, we fit the $\Lambda\bar{\Lambda}$ invariant mass distribution with an unbinned maximum likelihood fit. Each $\chi_{cJ}$ signal shape is described by a Breit-Wigner function convolved with a Gaussian function, and the parameters of the Breit-Wigner functions are fixed to the world average values PDG . The Gaussian function represents the resolutions, whose parameters are floated in the fit but shared with all three $\chi_{cJ}$ resonances. The background shape is composed of two parts: a MC simulation of $\psi(3686)\rightarrow\Sigma^{0}\bar{\Sigma}^{0}$ events with both shape and number fixed and a second-order polynomial with floating parameters to describe other background contributions. The results are shown in Fig. 4. The amount of other backgrounds from the fit is obvious larger than that simulated in inclusive MC as shown in Fig. 3. It indicates the inclusive MC simulation is not perfect yet. The fitted $\chi_{cJ}$ signal yields are $N_{\chi_{c0}}=1486\pm 42$, $N_{\chi_{c1}}=528\pm 24$, $N_{\chi_{c2}}=670\pm 27$.

The product branching fractions of $\mathcal{B}\left(\chi_{cJ}\rightarrow\Lambda\bar{\Lambda}\right)\cdot\mathcal{B}\left(\psi(3686)\rightarrow\gamma\chi_{cJ}\right)$ are determined by

where $\epsilon$ is the efficiency. We calculate the branching fractions of $\chi_{cJ}\rightarrow\Lambda\bar{\Lambda}$ decays based on world averaged values of $\mathcal{B}\left(\psi(3686)\rightarrow\gamma\chi_{cJ}\right)$ PDG . The results are listed in Table 3.

The uncertainties in the branching-fraction measurement include photon and $\Lambda(\bar{\Lambda})$ reconstruction efficiencies, kinematic fit, signal and background shapes, fitting range, and the branching fraction of $\Lambda(\bar{\Lambda})$ decay.

The uncertainties due to the photon detection and $\bar{\Lambda}$ reconstruction efficiencies, the requirement on the angle between the position of each shower in the EMC and the closest extrapolated charged tracks, the $\Lambda$ branching fraction, and the number of $\psi(3686)$ events are the same as in the study of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$, while that due to the kinematic fit is calculated in the same manner.

For the signal shapes, the single Gaussian is changed to a double Gaussian, and the differences in the yields of signal events, $0.6\%,0.3\%,$ and $0.1\%$ for $\chi_{c0},\chi_{c1},$ and $\chi_{c2},$ respectively, are taken as the systematic uncertainties associated with the signal shapes. We study the uncertainty associated with other backgrounds shape by changing description from the second-order polynomial function to the third-order polynomial function. It turns out the difference is negligible.

The uncertainty associated with the fitting range is estimated by varying it from [3.30, 3.60] $\mathrm{GeV}$ to [3.32, 3.62] $\mathrm{GeV}.$ The differences 0.6%, 1.8%, and 0.7% for $\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$ are taken as the uncertainty due to the fitting range.

The angular distributions of $\psi(3686)\rightarrow\gamma\chi_{c1,2}$ are known. However, knowledge of the $\chi_{c1,2}\rightarrow\Lambda\bar{\Lambda}$ angular distributions is still limited. We generate signal MC samples with an uniform distribution over phase space. To estimate the uncertainty caused by the angular distribution, we adopt the method used in Ref. Beschicj . We regenerate the signal MC samples according to the helicity amplitudes $B_{\lambda_{3},\bar{\lambda}_{3}}$ defined in Ref. angleUncer , where $\lambda_{3}(\bar{\lambda}_{3})$ is the helicity of $\Lambda(\bar{\Lambda})$ in the rest frame of $\chi_{cJ}$. The amplitudes $B_{\frac{1}{2},\frac{1}{2}}$ and $B_{\frac{1}{2},-\frac{1}{2}}$ are both set to be $1.0$ to obtain the efficiency again, and the differences of $3.0\%$ and $8.7\%$ between these two models are taken as the associated systematic uncertainties.

Table 2 lists all sources of systematic uncertainty and the values for each decay channel. The total systematic uncertainties are determined by adding each contribution in quadrature.

The branching fractions, compared with the world averaged values, are listed in Table 3, where the third uncertainties for $\mathcal{B}\left(\chi_{cJ}\rightarrow\Lambda\bar{\Lambda}\right)$ are the uncertainties due to the branching fractions of $\psi(3686)\rightarrow\gamma\chi_{cJ}$.