Measurement of branching fractions of J/ψ and ψ(3686) decays to Σ+ and Σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\Sigma} $$\end{document}−

Using 1310.6 × 106J/ψ and 448.1 × 106ψ(3686) events collected with the BESIII detector, the branching fractions of J/ψ decays to Σ+Σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\Sigma} $$\end{document}− is measured to be (10.61 ± 0.04 ± 0.36) × 10−4, which is significantly more precise than the current world average. The branching fractions of ψ(3686) decays to Σ+Σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\Sigma} $$\end{document}− is measured to be (2.52 ± 0.04 ± 0.09) × 10−4, which is consistent with the previous measurements. In addition, the ratio of B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B} $$\end{document}(ψ(3686) → Σ+Σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\Sigma} $$\end{document}−)/B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B} $$\end{document}(J/ψ → Σ+Σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\Sigma} $$\end{document}−) is determined to be (23.8 ± 1.1)% which violates the “12% rule”.


Introduction
Measurements on the decays of J/ψ and ψ(3686) (denoted here collectively as Ψ) can be used to study flavor-SU(3) symmetry breaking and test quantum chromodynamics (QCD) in the perturbative energy regime [1][2][3][4]. If we consider J/ψ decays into B 8 B 8 and B 10 B 10 final states, where B 8 and B 10 represent the baryon octet and decuplet states, respectively, and if the electromagnetic contributions are neglected, flavor-SU(3) symmetry gives the same decay amplitudes for all J/ψ decays to baryon anti-baryon pairs. However, broken flavor-SU(3) symmetry can contribute to the differences in branching fractions of different baryonic pairs. Furthermore, the branching fractions are determined not only by strong interaction amplitudes, but also by electromagnetic interactions and interferences between them [5], although these are much smaller than the expected flavor-SU(3) breaking effects. As shown in table 1, a phenomenologically plausible model [6,7] can be made to fit the pattern of branching fractions of J/ψ decays to baryon octet final states well [8]. However the precision on the branching fraction of J/ψ → Σ + Σ − is still relatively poor [9]. The   [8] and phenomenological calculations (B cal ) [6,7] for the branching fractions of J/ψ decays to baryon octet final states, where ∆(σ) is the difference in terms of the total uncertainty. Dash (−) represents no experimental measurement.
According to pQCD, the ratio of Γ h to Γ l , where Γ h is the partial width of J/ψ (ψ(3686)) decay to light hadrons and Γ l is the partial width to leptons, does not depend on the particle wave function [10,11]. The ratio between the branching fractions of J/ψ and ψ(3686) decays to the same final states obeys the so-called "12% rule", Although a large fraction of exclusive decay channels follow the rule approximately, significant violation was first observed in the ρπ channel [12]. The ratio of B(ψ(3686) → ρπ) to B(J/ψ → ρπ) is much smaller than the pQCD prediction, and this is called the "ρπ" puzzle. To understand the "ρπ" puzzle, the theoretical and experimental efforts have been made: amongst the suggested solutions are J/ψ-glueball admixture scheme, Instrinsic-charmcomponent scheme, Squential-fragmentation model, Exponential-form-factor model, S-D wave mixing scheme, Final state interaction scheme and others [13][14][15][16]. But there are no satisfactory explanations for all existing experimental results. Tests of the 12% rule using the baryonic decay modes may be helpful in understanding the ρπ puzzle. With CLEO data [17,18], the branching fraction of ψ(3686) → Σ + Σ − was determined to be

BESIII detector and Monte Carlo simulation
The BESIII detector [19] records symmetric e + e − collisions provided by the BEPCII storage ring [20], which operates with a peak luminosity of 1 × 10 33 cm −2 s −1 in the center-ofmass energy range from 2.0 to 4.9 GeV. BESIII has collected large data samples in this energy region [21]. The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T (0.9 T in 2012) magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel. The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for 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 in the TOF barrel region is 68 ps, while that in the end cap region is 110 ps. Monte Carlo (MC) simulated events are used to determine the detection efficiency, optimize selection criteria, and study possible backgrounds. GEANT4-based [22][23][24] MC simulation software, which includes the geometric and material descriptions of the BESIII detector, the detector response, and digitization models as well as the detector running conditions and performance, is used to generate MC samples. The simulation models the beam energy spread and initial state radiation (ISR) in the e + e − annihilations with the generator kkmc [25,26]. The inclusive MC samples of J/ψ and ψ(3686) includes the production of the J/ψ and ψ(3686) resonances, the ISR production of the J/ψ, and the continuum processes incorporated in kkmc. The known decay modes are modelled with evtgen [27,28] using branching fractions taken from the Particle Data Group [29], and the remaining unknown charmonium decays are modelled with lundcharm [30,31]. Final state radiation (FSR) from charged final state particles is incorporated using the photos package [32][33][34]. To describe the MC simulation of the signal process, the differential cross section is expressed with respect to five observables ξ = (θ Σ + , θ p , φ p , θ p , φ p ) [35]. Here θ Σ + is the angle between the Σ + and electron (e − ) beam in the interaction center-of-mass frame (CM), θ p , φ p and θ p , φ p are the polar and azimuthal angles of the proton and anti-proton measured in the rest frames of their corresponding mother particles. The parameters in the differential cross sections have been determined in ref. [36]. for each hadron hypothesis. The two good charged tracks are identified as proton and anti-proton by requiring L(p) > L(π) and L(p) > L(K).

Selection criteria
Photon candidates are reconstructed from isolated showers in the EMC. Each photon candidate is required to have a minimum energy of 25 MeV in the EMC barrel region or 50 MeV in the end cap region. To improve the reconstruction efficiency and the energy resolution, the energy deposited in the nearby TOF counters is included in the photon reconstruction. 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. The π 0 candidates are reconstructed by requiring the invariant mass of photon pairs to satisfy (M π 0 − 60) < M γγ < (M π 0 + 40) MeV/c 2 , where M π 0 is the nominal mass of π 0 [29]. The asymmetrical mass window is used because the photon energy deposited in the EMC has a tail on the low energy side. A one-constraint (1C) kinematic fit is performed on the photon pairs by constraining their invariant masses to the nominal π 0 mass, and the χ 2 1C is required to be less than 25 to remove fake candidates. Further there must be at least two reconstructed π 0 candidates.
To further remove potential background events and improve the mass resolution, a four-constraint (4C) kinematic fit is performed, constraining the total reconstructed four momentum to that of the initial e + e − state. A requirement on the quality of the 4C kinematic fit of χ 2 4C < 100 is imposed, which is chosen by optimizing the figure-ofmerit, defined as S √ S+B , where S is the number of signal events and B is the number of background events, which are estimated based on MC simulations. If the number of π 0 candidates in an event is greater than two, the ppπ 0 π 0 combination with the lowest χ 2 4C is selected. After kinematic fitting, the Σ + and Σ − candidates are constructed from the proton, anti-proton and neutral-pion candidates, and the combination that minimizes To investigate other possible background processes, inclusive MC samples of 1.2 × 10 9 J/ψ and 5.06 × 10 8 ψ(3686) decays are used and examined TopoAna, a software tool to categorise backgrounds and identify the physics processes of interests from the inclusive MC samples [37]. For J/ψ → Σ + Σ − , the dominant background contributions are found  14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23

Branching fraction measurements
With the above selection criteria, there are significant enhancements close to the Σ + and Σ − nominal masses in the two dimensional distribution of M pπ 0 and M pπ 0 as can be seen in figure 2. To obtain the number of signal events, an unbinned maximum likelihood fit is performed to the M pπ 0 distribution by requiring M pπ 0 to be within the signal region. The signal is described by the MC shape convoluted with a Gaussian function which JHEP11(2021)226 represents the difference between data and MC in the resolution and mean value. The background is described with a second-order polynomial function. The mean and width of the Gaussian function and polynomial function parameters are all floated. Figure 3 shows the fitting of the pπ 0 invariant mass distributions, where the red solid lines are the total fitting functions, the red dashed lines are the signal functions and the blue dotted ones are the background functions. The branching fraction of each channel is calculated according to where N sig is the number of signal events determined by the fit, cor is the corrected detection efficiency, generated according to the decay parameters measured in data but corrected for differences between data and MC simulation, ΠB i is the product of the branching fractions of all the intermediate states in each channel and N Ψ is the number of J/ψ or ψ(3686) events [38,39]. The corresponding numbers of signal events, detection efficiencies and branching fractions are listed in table 2. The initial detection efficiencies are estimated with signal MC simulation. In the calculation of cor , we take into account the difference between data and signal MC, obtained from control samples, which include the differences of detection efficiencies of the proton, anti-proton and π 0 . To study the tracking and PID efficiencies of the proton and anti-proton, the decay processes of Ψ → ppπ + π − are used to select the control samples of the proton or anti-proton. The proton efficiency ratios between MC and data are determined within different proton transverse-momentum and polar-angle regions. The ratios of anti-proton efficiency are also determined using the same method. To study the π 0 reconstruction efficiency, the control samples are selected with the processes of ψ(3686) → π 0 π 0 J/ψ, J/ψ → l + l − and e + e − → ωπ 0 at √ s = 3.773 GeV. The relative difference of the π 0 reconstruction efficiencies between MC and data obtained on the two datasets are consistent with each other and depend on π 0 momentum. The overall correction to the event selection efficiency is the product of correction factors of proton, anti-proton and π 0 in the related kinematic regions.

Systematic uncertainties
The systematic uncertainties of the branching fraction measurements are mainly due to the difference of efficiency between data and simulation. The main sources come from the difference in the detection efficiencies of charged and neutral particles in the final states. In addition, the detector resolution difference between data and MC also affects the efficiency via χ 2 requirement in the kinematic fit. Other sources, such as the fitting method, parameters of the generator and numbers of Ψ events, are also considered. Table 3 summarizes the sources of systematic uncertainties, which are discussed in further detail below.

MC efficiency correction for charged tracks
The tracking and PID efficiency differences between data and simulation for the proton and anti-proton have been studied in bins of transverse momentum and polar angle from control samples Ψ → ppπ + π − . These differences are treated as correction factors to calculate the nominal efficiencies. The systematic uncertainties due to the limited statistics of the control samples are obtained by summing their relative uncertainties in different bins quadratically and are estimated to be 1.6% and 1.5% for J/ψ and ψ(3686), respectively.

π 0 efficiency correction
Based on control samples of ψ(3686) → π 0 π 0 J/ψ, J/ψ → l + l − and e + e − → ωπ 0 at √ s = 3.773 GeV, the relative difference of the π 0 reconstruction efficiencies between data and MC has been obtained. The two datasets are consistent with each other. We studied the relative difference as a function of polar angle and momentum magnitude of π 0 , and found it to decrease linearly, (0.06 − 2.41 × p)%, as a function of momentum p. The detection efficiency differences obtained by varying the correction factor according to its uncertainty, ( 0.76 × p 2 + 1.15 + 0.39 × p)%, are taken as the systematic uncertainties, which are 2.3% and 2.4% for J/ψ and ψ(3686), respectively.

Decay parameters
The signal MC sample is generated according to a set of decay parameters which have been measured through multi-dimensional fitting of angular distributions, where polarization effects and decay asymmetry have been included [36]. Assuming no CP violation (α 0 = −ᾱ 0 ), three decay parameters α Ψ , ∆Φ Ψ and α 0 are used for Ψ decay. We take the mean value and error matrix of these 3 parameters to build a 3 dimensional Gaussian distribution. Based on the 3 dimensional Gaussian distribution, we generate 1000 signal MC sample to evaluate the systematic uncertainty. The distributions of newly obtained efficiencies are fitted using a Gaussian function, and the widths are assigned as the systematic uncertainties, which are 0.6% and 0.7% for J/ψ → Σ + Σ − and ψ(3686) → Σ + Σ − respectively.

Fitting function
To estimate the uncertainties of the fitting function, we use the Crystal Ball function to describe the signal instead of the MC shape convoluted with a Gaussian function, and the JHEP11(2021)226 differences are taken as the systematic uncertainties, 0.4% for J/ψ → Σ + Σ − and 0.6% for ψ(3686) → Σ + Σ − .

Background estimation
There are two kinds of background events: peaking backgrounds and non-peaking backgrounds. For the peaking backgrounds, we neglect this contribution in calculating the branching fractions. Considering this contributions is less than 0.1%, we take 0.1% as conservative estimate for this kind of systematic uncertainties. For the non-peaking background, to estimate the uncertainties due to background modeling, we use the background shape determined by kernel density estimation from the sideband region of M (pπ 0 ) instead of the second-order polynomial function. The differences are taken as the systematic uncertainties, 0.9% for J/ψ → Σ + Σ − and 0.3% for ψ(3686) → Σ + Σ − .

M (pπ 0 ) mass window selection
To select the signal events, we require the 1.

Kinematic fitting
To estimate the systematic uncertainty caused by the χ 2 4C requirement, we obtain the χ 2 4C distributions using the track correction method for the helix parameters [40]. By imposing the requirement of χ 2 4C < 100, the efficiencies are estimated, and compared with the nominal values. The differences, 0.1% for both J/ψ and ψ(3686), are taken as the systematic uncertainties.

Branching fractions and numbers of J/ψ and ψ(3686)
The uncertainties related to the branching fractions of Σ + → pπ 0 and Σ − → pπ 0 are taken as 1.2% according to the PDG [29]. The numbers of J/ψ and ψ(3686) mesons are determined based on inclusive hadronic events, as described in [38,39] with an uncertainty of 0.6% for J/ψ and 0.7% for ψ(3686).

Summary
In summary, with 1310.6 × 10 6 J/ψ and 448.1 × 10 6 ψ(3686) events collected by the BESIII detector, the branching fractions of J/ψ and ψ(3686) decaying to Σ + Σ − are measured to be (10.61 ± 0.04 ± 0.36) × 10 −4 and (2.52 ± 0.04 ± 0.09) × 10 −4 , respectively, and both are in agreement with the previous measurement [9,18]   precision of the branching fraction of J/ψ → Σ + Σ − is improved by a factor of 6.6 relative to the previous best measurement. The branching fraction ratio of the ψ(3686) and J/ψ decays is calculated to be (23.8 ± 1.1)%, where the statistical and systematic uncertainties are combined. The ratio is consistent with the previous measurement in the Σ 0 Σ 0 final states by the BESIII collaboration [41], and both violate the "12% rule".

JHEP11(2021)226
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