Measurement of Born cross section of $e^{+}e^{-}\rightarrow\Sigma^{+}\bar\Sigma^{-}$ at center-of-mass energies between 3.510 and 4.951 GeV

Using 24.1 fb$^{-1}$ of $e^{+}e^{-}$ collision data collected with the BESIII detector at the BEPCII collider, the Born cross sections and effective form factors of the $e^{+}e^{-}\rightarrow\Sigma^{+}\bar\Sigma^{-}$ reaction are measured. The measurements are performed at center-of-mass energies ranging from 3.510 to 4.951 GeV. No significant evidence for the decay of the charmonium(-like) states, $\psi(3770)$, $\psi(4040)$, $\psi(4160)$, $Y(4230)$, $Y(4360)$, $\psi(4415)$, and $Y(4660)$, into a $\Sigma^{+}\bar\Sigma^{-}$ final state is observed. Consequently, upper limits for the products of the branching fractions and the electronic partial widths at the 90% confidence level are reported for these decays.

The complex situation reflects our limited understanding of the strong interaction, particularly in its non-perturbative aspects.In order to address this challenging problem, it is imperative to make additional experimental measurements, and the study of ψ/Y → B B decays holds great promise.These decays exhibit a straightforward topology in terms of the final states, and the underlying interaction mechanism is assumed to be dominated by three-gluon or one-photon processes.Additionally, investigations into the electromagnetic form factors or effective form factors of B B pairs have the potential to provide insight into the internal composition of charmonium(-like) states.Although many experimental studies [8,[24][25][26][27][28][29][30][31] of B B pair production in this energy region have been performed by the BESIII and Belle experiments, except for two evidences of ψ(3770) → Λ Λ and Ξ − Ξ+ , no significant indication for B B decay of other vector charmonium(-like) states has been found.Thus, more precise measurements of exclusive cross sections for B B final states above the open-charm threshold are crucial.This paper reports the measurements of the Born cross section and the effective form factor for the process of e + e − → Σ + Σ− using the data corresponding to a total integrated luminosity of 24.1 fb −1 collected at center-of-mass (CM) energies ( √ s) between 3.510 and 4.951 GeV with the BESIII detector [32] at the BEPCII collider [33].In addition, potential resonances are searched for by fitting the dressed cross section of the e + e − → Σ + Σ− reaction.

BESIII Detector and Monte Carlo simulation
The BESIII detector [32] records symmetric e + e − collisions provided by the BEPCII storage ring [33] in the range of √ s from 2.0 to 4.95 GeV, with a peak luminosity of 1 × 10 33 cm −2 s −1 achieved at √ s = 3.77 GeV.BESIII has collected large data samples in this energy region [34][35][36].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 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 was 110 ps.The end cap TOF system was upgraded in 2015 using multigap resistive plate chamber technology, providing a time resolution of 60 ps [37][38][39] and benefiting 82% of the data used in this analysis.
To evaluate detection efficiencies and estimate backgrounds, simulated data samples are produced using geant4-based Monte Carlo (MC) software [40], which incorporates the geometric description of the BESIII detector [41] as well as the detector response.The simulation of the e + e − → Σ + Σ− production process models the beam energy spread in the e + e − annihilation process, employing kkmc [42].For each of the 41 CM energy points ranging from 3.510 to 4.951 GeV, a sample of 100,000 events is simulated with a uniform phase space (PHSP) distribution.The Σ + Σ− baryon pair and their subsequent decays are simulated using evtgen [43,44] with a PHSP model.

Event selection
Due to the large background in the selection of e + e − → Σ + Σ− events, both the Σ + and Σ− are required to be reconstructed via the decay modes Σ + → pπ 0 and Σ− → pπ 0 with the subsequent decay π 0 → γγ.
Tracks of charged particles detected in the MDC are required to lie within the angular coverage of the MDC |cos θ| < 0.93, where θ is the angle between the charged track and the z axis, which is the symmetry axis of the MDC.At least one positively charged and one negatively charged track are required to be reconstructed in the MDC with good Kalman fits.Because the proton and anti-proton can be separated according to their momenta in a Σ + Σ− decay, a charged particle with momentum greater than 0.5 GeV/c is identified as a proton or anti-proton.
For π 0 reconstruction, the energies of photons are required to be greater than 25 MeV in the EMC barrel region (|cos θ| < 0.8) and greater than 50 MeV in the EMC end cap (0.86 < |cos θ| < 0.92).To suppress electronic noise and energy deposits unrelated to the events, the EMC shower time measured with respect to the collision signal, is required to satisfy 0 < t < 700 ns.After these selections, at least four photons are required.
The best candidate of all combinations of ppγγγγ within an event is determined by a six-constraint (6C) kinematic fit, which imposes energy and momentum conservation and constrains the masses of photon pairs to the known mass of π 0 [45].The ppπ 0 π 0 combination with the smallest fit χ 2 is chosen.For different p(p) and π 0 combinations, the Σ + and Σ− pair with the minimum of (M pπ 0 − m Σ + ) 2 + (M pπ 0 − mΣ−) 2 , is selected.
Here, M pπ 0 (pπ 0 ) is the invariant mass of the pπ 0 (pπ 0 ) combination, and m Σ + ( Σ− ) is the known mass of Σ + ( Σ− ) from the Particle Data Group (PDG) [45].Figure 1 shows the distributions of M pπ 0 versus M pπ 0 for each energy point and the sum of all energy points.M pπ 0 (pπ 0 ) is required to be within the range of [m Σ + − 4σ, m Σ + + 3σ], which is labeled by S in Fig. 1.The resolution σ and the signal region are determined by a fit with the Crystal-Ball function [46].Due to the longer tail of the photon energy deposition at the low energy side in the EMC, the signal region is asymmetric.

Determination of signal yields
After applying the event selection criteria on data, the remaining background mainly comes from non-Σ + Σ− events, such as e + e − → π 0 π 0 J/ψ → π 0 π 0 pp.To estimate the Sum M pπ 0 (GeV/c 2 ) . Distributions of M pπ 0 versus M pπ 0 for data at each energy point between 3.510 and 4.951 GeV and the sum of all energy points (bottom right) from data.The red boxes represent the signal regions and the green boxes represent the selected sideband regions.
background yield in the signal region, four sideband regions B i (where i = 1, 2, 3, 4) are utilized.These sideband regions, shown in figure 1, have the same area as the signal region, and the exact ranges are defined by The signal yield N obs for the e + e − → Σ + Σ− reaction at each energy point is determined by subtracting the number of events in the sideband regions from the signal region, i.e., N obs = N S − N bkg , where N S is the number of events from the signal region and N bkg = 1 4 4 i=1 N B i .Statistical uncertainties are calculated based on the TRolke method [49], and the statistical significance is evaluated based on the observed p-value [48].The results are listed in table 1.For the energy points with statistical significance less than 3σ, the upper limit at the 90% confidence level (C.L.) is also estimated based on the TRolke method, which takes into account systematic uncertainties.

Determination of Born cross section and effective form factor
The Born cross section for the e + e − → Σ + Σ− process at a given CM energy is calculated by where N obs is the number of observed signal events, L is the integrated luminosity, (1 + δ) is the ISR correction factor, 1 |1−Π| 2 is the vacuum polarization (VP) correction factor, ϵ is the detection efficiency, and B Σ + →pπ 0 and B π 0 →γγ are the PDG branching fractions [45].The ISR correction factor is obtained using the QED calculation as described in ref. [50], and the VP correction factor is calculated according to ref. [51].The efficiency and ISR correction factor are obtained through an iterative process.Initially, the cross section is measured without any correction factors.Using this initial line shape, signal MC samples are regenerated, and their efficiencies and ISR correction factors are recalculated.Subsequently, the Born cross section of e + e − → Σ + Σ− is updated and used as input for the next iteration.To expedite the iteration procedure, an iterative weighting method, as proposed in ref. [52], is employed.The procedure is iteratively performed until the difference of ϵ • (1 + δ) with the last iteration falls below 0.5%.The results of the measured Born cross sections and the Σ + effective form factors G eff (s) for different energy points are listed in table 2. G eff (s) is defined as [26] where s is the square of the CM energy, α = 1 137 is the fine structure constant, the variable , and the Coulomb factor C [53,54] parameterizes the electromagnetic interaction between the outgoing baryon and anti-baryon.For neutral baryons, the Coulomb factor is unity, while for point-like 1−e − πα β [55][56][57][58].Figure 2 displays the energy dependence of the Born cross section and Σ + effective form factor G eff (s) measurements, as well as a comparison of the Born cross sections and the effective form factors with the CLEO-c results [59] at √ s = 3.770 and 4.160 GeV.3.0 +2.6 −1.9 (< 8.0) 1.9 ), the ISR correction factor and the detection efficiency (ϵ • (1 + δ)), the signal yield (N obs ), the upper limit of signal yield at the 90% C.L. (N UL ), the Born cross section (σ B ), the effective from factor (|G eff (s)|) and the statistical significance (S).The first and second uncertainties for σ B and |G eff (s)| are statistical and systematic, respectively.

Systematic uncertainty
The systematic uncertainties on the Born cross section measurements mainly originate from the integrated luminosity, Σ + ( Σ− ) reconstruction, background, angular distribution, branching fractions, and input line shape.

Luminosity
The luminosity at all energy points is measured using Bhabha events with the uncertainties of 1.0% [60] below 4.0 GeV, 0.7% [61] from 4.0 to 4.6 GeV and 0.5% [62] above 4.6 GeV, which are taken as the systematic uncertainties due to the luminosity measurement.
The efficiency difference correlated with the angular distribution (cos θ) between data and MC is taken as the systematic uncertainty.

Background
The systematic uncertainty associated with the background, which is estimated based on the sidebands, is determined by shifting the sideband region inward or outward by 1σ.Since the number of events for each energy point is limited, all energy points are combined in the estimation.The maximum difference before and after moving the sideband region is taken as the systematic uncertainty.

Angular distribution
Since there are not enough events to determine the angular distribution for each energy point separately, a control sample with large statistics and a DIY model [44] with customizable angular distribution are used.With a large sample of ψ(3770) events as the control sample, the angular distribution is obtained by a maximum likelihood fit to the helicity amplitude [73][74][75].The DIY sample is initially generated with the central value of the fitting, and its efficiency is considered as the nominal result.Subsequently, two additional DIY samples are generated using the upper and lower limits of the fit uncertainty.The maximum difference in efficiency between them and the nominal one is taken as the systematic uncertainty.

Branching fractions
The uncertainty from the branching fraction of Σ + → pπ 0 is 0.58% from the PDG [45], and the uncertainty of the branching fraction of π 0 → γγ is 0.03%.Combining with the branching fraction of the opposite side Σ− decay, the systematic uncertainty from the branching fractions is 0.8%.

Input line shape
The systematic uncertainty of the input line shape is estimated by varying the central value of the nominal input line shape within ±1σ of the statistical uncertainty, and the ϵ • (1 + δ) value for each energy point is recalculated.This process is repeated 200 times, after which a Gaussian function is used to fit the ϵ • (1 + δ) distribution.The width of the Gaussian function is taken as the corresponding systematic uncertainty.

Total systematic uncertainty
The various systematic uncertainties on the Born cross section measurement are summarized in table 3. Assuming all sources are independent, the total systematic uncertainty on the cross section measurement is determined by adding them in quadrature.

Fit to the dressed cross section
The potential resonances in the line shape of the cross section for the e + e − → Σ + Σ− reaction are searched for by fitting the dressed cross section, σ dressed = σ B /|1 − Π| 2 (including the VP effect), using the least χ 2 method: This is done considering the covariance matrix V and the vector of residuals ∆X between the measured and fitted cross sections.The covariance matrix incorporates both the correlated and uncorrelated uncertainties among different energy points.The systematic uncertainties associated with the luminosity, Σ + ( Σ− ) reconstruction, and branching fraction are assumed to be fully correlated among the CM energies, while the other systematic uncertainties are assumed to be uncorrelated.
Assuming the cross section of e + e − → Σ + Σ− includes a resonance plus a continuum contribution, a fit to the dressed cross section with the coherent sum of a power-law (PL) function plus a Breit-Wigner (BW) function is applied.Here ϕ is the relative phase between the BW function and the PL function where c 0 and n are free fit parameters, P ( √ s) is the two-body PHSP factor, the mass M and total width Γ are fixed to the assumed resonance with the PDG values [45], and Γ ee B is the products of the electronic partial width and the branching fraction for the resonance decaying into the Σ + Σ− final state.Note that due to limited statistics, we only take into account the interference between the continuum contribution and each resonance, and no longer consider the interference between resonances.The parameters without a resonance are fitted to be (c 0 = 2.3 ± 0.8, n = 8.5 ± 0.3) with the goodness-of-fit χ 2 /n.d.f = 31.2/(41− 2), and the parameters including a resonance are summarized in table 4. Considering systematic uncertainties, the significance for each resonance is calculated by comparing the change of χ 2 /n.d.f with and without the resonance.Charmonium(-like) states, ψ(3770), ψ(4040), ψ(4160), Y (4230), Y (4360), ψ(4415), Y (4660), are all fitted separately by eq.(6.2), but no significant resonance is found.Thus, upper limits of the products of branching fraction and two-electronic partial width for these charmonium(-like) states decaying into the Σ + Σ− final state including the systematic uncertainty are determined at the 90% C.L. using a Bayesian approach [76].Figure 3 shows the fit to the dressed cross section including a resonance [i.e.ψ(3770), ψ(4040), ψ(4160), Y (4230), Y (4360), ψ(4415) and Y (4660)] and without a resonance.Due to the quadratic form of the cross section like eq. (6.2), there are multiple solutions [77], which can be determined by scanning the parameters ϕ and Γ ee B, similar to the method used in ref. [25].The fit results and their multiple solutions are summarized in table 4.

Summary
Using a total of 24.1 fb −1 of e + e − collision data above open-charm threshold collected with the BESIII detector at the BEPCII collider, the process e + e − → Σ + Σ− is studied.The Born cross sections and effective form factors are measured at 41 CM energies that range from 3.510 to 4.951 GeV.A fit to the dressed cross section of the e + e − → Σ + Σ− reaction is performed, in which the line shape is described by a series of resonance hypotheses plus a continuum contribution, or only a continuum contribution.However, no obvious signal of ψ(3770), ψ(4040), ψ(4160), Y (4230), Y (4360), ψ(4415), or Y (4660) is found, and upper limits for the products of branching fraction and di-electronic partial width at the 90% C.L. for these charmonium(-like) states decaying into the Σ + Σ− final state are determined.

Figure 2 .
Figure 2. The measured Born cross section (top) and Σ + effective form factor (bottom) for e + e − → Σ + Σ− versus CM energy, where the uncertainties include both the statistical and systematic ones.

Figure 3 .
Figure 3. Fits to the dressed cross section at the CM energy from 3.510 to 4.951 GeV with the assumptions of a PL function only (upper left) and a power-law function plus a resonance (ψ(3770), ψ(4040), ψ(4160), Y (4230), Y (4360), ψ(4415), and Y (4660)).Dots with error bars are the dressed cross sections, and the solid lines show the fit results.The error bars represent the statistical and systematic uncertainties summed in quadrature.

Table 1 .
Number of events: N S is from the signal region, N bkg is the number of background events, N obs is the number of events by subtracting the backgrounds, S(σ) is the statistical significance, and N UL is the upper limit for an energy point with statistical significance less than 3σ.

Table 3 .
Systematic uncertainties (in %) and their sources for each energy point on the Born cross section measurement.Here, AD denotes angular distribution, B denotes branching fraction, and ILS denotes input line shape.

Table 4 .
The fitted resonance parameters to the dressed cross section for the e + e − → Σ + Σ− process with two solutions.The fit procedure includes both statistical and systematic uncertainties except for the CM energy calibration.The relative phase is given by ϕ.