Measurements of + and-Time-like Electromagnetic Form Factors for center-of-mass energies from 2.3864 to 3.0200 GeV

The Born cross sections of the ee → ΣΣ̄ and ee → ΣΣ̄ processes are determined for center-of-mass energy from 2.3864 to 3.0200 GeV with the BESIII detector. The cross section lineshapes can be described properly by a pQCD function and the resulting ratio of effective form factors for the Σ and Σ is consistent with 3. In addition, ratios of the Σ electric and magnetic form factors, |GE/GM |, are obtained at three center-of-mass energies through an analysis of the angular distributions. These measurements, which are studied for the first time in the off-resonance region, provide precision experimental input for understanding baryonic structure. The observed new features of the Σ form factors require more theoretical discussions for the hyperons.


Introduction
Nucleons, as the lightest baryons, are the largest component of the observable matter in the universe, and were shown to be non-pointlike particles in the middle of last century [1,2]. However, nucleon properties, such as their radii and the sources of their spin, are still not well understood [3]. The hyperons are the SU (3)flavour-octet partners of the nucleons that contain one or more strange quarks, and offer crucial additional dimensions to the study of nucleon structures [4,5]. Treating the heavier strange quarks as spectators, hyperons can provide valuable insight into the behaviour of the lighter up and down quarks in different environments. Electromagnetic form factors (EMFFs) are fundamental observables of baryons that are intimately related to their internal structure and dynamics [6][7][8]. Despite the fact that much work has been done on the EM structures of protons in both the space-like and time-like regions [9][10][11][12][13][14], experimental information regarding the EMFFs of hyperons remains limited [15][16][17][18]. Moreover, the few existing measurements of timelike neutron FFs [19,20] differ from each other and lead to conflicting conclusions when compared to those for the proton [21,22]. A Σ + hyperon is formed by replacing the proton's down quark with a strange quark; likewise a Σ − is formed by replacing the neutron's up quark with a strange quark. The corresponding ratio of FFs between the Σ + and Σ − hyperons could provide guidance for the nucleons. Therefore, experimental measurements for Σ hyperons, especially the Σ − , which has never been measured in the time-like region, provide essential tests of various theoretical models [22][23][24] and produce important input for the understanding of baryonic structures.
The differential, one-photon exchange cross section for the e + e − → BB process, where B is a spin-1/2 baryon, can be expressed in terms of the electric and magnetic FFs G E and G M as [25]: where α is the fine-structure constant, s is the square of center-of-mass (c.m.) energy, β = , m B is the baryon mass, and θ is its c.m. production angle. The Coulomb correction factor C [26,27] accounts for the electromagnetic interaction of charged point-like fermion pairs in the final state. It reads C = y/(1 − e −y ) with y = πα(1 + β 2 )/β for a charged point-like fermion pair and C = 1 for a neutral point-like fermion pair. For charged point-like fermion pairs, the cross section at threshold is non-zero, σ(4m 2 B ) = π 2 α 3 /2m 2 B = 848(m p /m B ) 2 pb, where m p is the proton mass [28], and then grows with increasing β. Experimentally, a rapid rise of the e + e − → pp cross section near threshold followed by a plateau is observed [12,13]. The crosss section of plateau near threshold is consistent with the 848 pb expectation for a point-like charged particle. However, in this case, the pp is produced by a virtual photon with Q 2 = 4m 2 p = 3.53 GeV 2 , which corresponds to a Compton wavelength of ∼0.1 fm, a scale at which the proton is definitely not point-like. A similar feature of the cross section for e + e − → Λ + cΛ − c is observed by the BESIII experiment [29], where the cross section of plateau near threshold is around 240 pb. This is 1.6 times the predicted value for point-like charged particles. These unexpected threshold effects have been widely discussed in the literature where they are interpreted as final state interactions [30], bound states or near-threshold meson resonances [31], or an attractive Coulomb interaction [32]. To understand the nature of these threshold effects, experimental measurements of the near threshold charged pair production of other hyperons will be of critical importance.

Detector and data sample
In this Letter, we present precision measurements of e + e − → Σ +Σ− and e + e − → Σ −Σ+ with a data sample of 329.7 pb −1 collected at BESIII with c.m. energies between 2.3864 and 3.0200 GeV [33]. The threshold energies for Σ +Σ− and Σ −Σ+ pair production are 2.3787 GeV and 2.3949 GeV, respectively. The BESIII detector is described in detail in Ref. [34]. The critical elements for the measurements reported here are: the main drift chamber (MDC), which measures the momenta of charged particles with 0.5% resolution for 1 GeV/c tracks and the dE/dx for chargedparticle identification (PID); a barrel array of scintillation counters that measures charged particles' time of flight for additional PID information; and an electromagnetic calorimeter (EMC) comprising an array of CsI(Tl) crystals that measures photon energies with a resolution of 2.5% at 1 GeV.
Simulated event samples produced with a GEANT4based [35] Monte Carlo (MC) package that includes the geometric description of the BESIII detector and its response, are used to determine the detection efficiency and to estimate the backgrounds. The signal processes e + e − → Σ ±Σ∓ are generated according to the differential amplitude presented in Ref. [36]. Initial state radiation (ISR) is simulated with CONEXC [37] and the corresponding correction factors are calculated for higher order processes. Background from the QED processes e + e − → l + l − (l = e, µ) and e + e − → γγ are investigated with BABAYAGA [38], while for e + e − →hadrons and two-photon processes we use LUNDARLW [39] and BESTWOGAM [40], respectively.

Data Analysis
In the process e + e − → Σ +Σ− , there are four dominant final state topologies which account for more than 99% of its total decay width: pπ 0p π 0 , nπ +p π 0 , pπ 0n π − and nπ +n π − , All four configurations are selected in this analysis, significantly improving the statistics. At BESIII, charged particles are efficiently detected and identified by the MDC and PID systems and π 0 mesons are reconstructed in the EMC via their π 0 → γγ decay mode. The selection criteria for charged tracks, PID, and photon candidates are the same as those used in Ref. [41]. Most of the anti-neutrons (n) annihilate in the EMC and produce several secondary particles with a total energy deposition that can be as high as 2 GeV; the position of then interaction and, from this, then direction can be inferred from the weighted center-of-energy of the shower [17]. Neutron (n) detection is not done because of its low interaction efficiency and small energy deposition.
The pπ 0p π 0 and nπ +p π 0 final-state configurations, classified as category A, can be analyzed by a partial reconstruction technique in which only the detection ofΣ − →pπ 0 is required. Candidate events are required to have at least one charged track that is identified as ap by the PID system and at least two good photons that are consistent with originating from π 0 → γγ. The mass spectrum of γγ is required to be from 0.127 < M γγ < 0.139 GeV/c 2 to 0.123 < M γγ < 0.14 GeV/c 2 , depending on c.m. energies. Thē Σ − is reconstructed using all combinations of the selectedpγγ. The two-body process exploits two variables that are based on energy and momentum conservation: the energy difference ∆E ≡ E − E beam and the beam-constrained mass M bc ≡ E 2 beam − p 2 . Here, E(p) is the total measurement energy (momentum) of thepγγ combinations in the c.m. system, and E beam is the beam energy. Candidates are accepted with optimized ∆E requirements of −16 < ∆E < 7 MeV to −24 < ∆E < 13 MeV, depending on c.m. energies, and with M bc > 1.15 GeV/c 2 .
The pπ 0n π − and nπ +n π − final states, classified as category B, are reconstructed by requiring two good charged tracks with one identified as a π − and the other identified as either a π + or p, and the most energetic shower in these events is assigned as then candidate. To discriminaten-initiated showers from those produced by photons, three variables are retained for further selection based on c.m. energy-dependent requirements: the total energy in then-assigned EMC shower, the second moment of the shower [17], and the number of crystals with above-threshold signals within a 40 • cone around the shower. After that, kinematic fits that include then direction are performed to identify signal events. Since then shower does not provide a good measure of its total energy, En, this is left as a free parameter in the kinematic fits. If a π + is identified, the fit imposes the nnπ + π − hypothesis with a missing n. If a p is identified, the fit imposes the pnπ − π 0 hypothsis with a missing π 0 . In both fits, total energy-momentum conservation is constrained and Mn π − is also constrained to the mass of theΣ − . The pπ − invariant mass is required to be |M (pπ − ) − m(Λ)| > 0.005 GeV/c 2 to eliminate background from e + e − → ΛΛ → pπ −n π 0 . Furthermore, the χ 2 value from the kinematic fit is required to be less than 20.
The reconstruction of e + e − → Σ −Σ+ is similar to that for nπ +n π − in the e + e − → Σ +Σ− analysis since they have the same final states. The only difference is that Mn π + is constrained to the mass of theΣ + in the kinematic fit. Figure 1 shows the distributions of M bc for category A and the recoil mass ofnπ − , M rec nπ − , for category B using selected e + e − → Σ +Σ− candidates, where significant signals in both categories are observed in data at √ s = 2.3864 and 2.3960 GeV. Backgrounds are studied with MC samples and only hadronic final states survive the selection criteria. In category A, the backgrounds are from e + e − annihilation events with the same final states as the signal process, with one or more additional π 0 , and with an additional γ-ray. In category B, the backgrounds are from annihilation events with the same final states as the signal process, multi-π processes such as π + π − π 0 π 0 and processes with one more π 0 in the final states. These background processes are mainly from contributions including intermediate states such as ∆, Λ and Σ baryons, but none of them produce peaks in the signal regions as shown by the histograms of Fig. 1. Figure 2 shows distributions of M nπ − for e + e − → Σ −Σ+ candidate events at √ s = 2.3960 and 2.6444 GeV, respectively, where significant signals in data are observed. In the background study, no peaking background is observed in the nπ − mass spectrum.  The Born cross section for e + e − → Σ +Σ− is determined from the relation: where N is the signal yield extracted from the fits; L is the integrated luminosity; 1 + δ r is the ISR correction factor incorporating the input cross section from this analysis iteratively; 1 |1−Π| 2 is the vacuum polarization factor [42]; ε is the detection efficiency determined from signal MC events. The factor δ data/MC is a correction factor for efficiency differences between data and MC simulation, determined from studies of high statistics, low-background control samples of J/ψ → Σ +Σ− and J/ψ → ΛΣ − π + , respectively. The decay branching fraction B accounts for the intermediate states in theΣ − decay (51.57% forΣ − →pπ 0 and 48.31% for Σ − →nπ − ).
To determine the signal yields, un-binned maximum likelihood fits are performed to the M bc and M nπ + distributions for categories A and B, respectively. The probability density function (PDF) for the signal is described with a MC-simulated shape convolved with a Gaussian function to account for mass resolution differences between data and MC simulation. The background PDF for category A is described by an Argus function [43]; for category B by a second order polynomial. In the fit, the two categories are constrained by the same Born cross section σ Born , and the expected signal yields are calculated from N i = σ Born · L · ε i · (1 + δ) · δ data/MC i · B i . The fit results at √ s = 2.3864 and √ s = 2.3960 GeV are shown in Fig. 1. Similarly, the signal yield of e + e − → Σ −Σ+ is determined by fitting the nπ − mass spectrum, where the signal is described with the MC simulated shape convolved with a Gaussian function and the background is described with a 2nd-order polynomial. Fit results at √ s = 2.3960 and √ s = 2.6444 GeV are shown in Fig. 2.
The quantities used in the cross section calculations for e + e − → Σ +Σ− and e + e − → Σ −Σ+ are summarized in Table 1 and Table 2, respectively. It should be noted that, due to limited statistics, data at c.m energies 2.7000 and 2.8000 GeV are combined; data at 2.9500, 2.9810, 3.0000 and 3.0200 GeV are combined. Currently, individual measurements on |G E | and |G M | at each energy point are not possible due to statistics. Therefore, the effective FFs of Σ ± , defined as |G eff | 2 ≡ (|G E | 2 + 2τ |G M | 2 )/(2τ + 1) [44], are reported here and shown in Table 1, 2. Table 1: Summary of the calculated cross section for e + e − → Σ +Σ− and effective FFs of Σ + at each c.m. energy and the quantities used in the calculation, ǫ = ε(1 + δ r ) 1 |1−Π| 2 δ data/MC , defined in the text. The energy points with asterisks are combined data samples with c.m energies weighted by the luminosities of the subsamples. The 2.7500 GeV is a combined data set of 2.7000 and 2.8000 GeV, and 2.9884 GeV is a combined data set of 2.9500, 2.9810, 3.0000 and 3.0200 GeV. The last column shows the results of Systematic uncertainties associated with the cross section measurements include event selection, cross section line-shape, angular distribution, fitting method, energy scale, and luminosity. In the nominal results, the differences of data and MC efficiencies are corrected with control samples. We vary the data/MC correc-tion factors within their ±1σ uncertainty and the resulting differences in the cross sections are taken as the uncertainty from the event selection. The uncertainty associated with the cross section line-shape is 1.0%, which includes both the theoretical uncertainty and the parameter uncertainty in the line-shape fit. The uncertainty from the angular distribution is evaluated by varying |G E /G M | ratios within ±1σ at the three energy points with the highest statistics. For the energy points with unknown |G E /G M | values, two extreme cases G E = 0 and G M = 0 are considered and the difference in the efficiencies divided by a factor of √ 12 is taken as the uncertainty [45]. Alternative fits are performed to study the uncertainty from the fit procedure. These include varying the fitting range, varying the signal shape by fixing the resolution of the convolved Gaussian to be ±1σ different from its nominal value, and changing the background PDF from a second order to a third order polynomial. The effects of the c.m. energy and energy resolution uncertainties are studied for energy points near threshold. The difference of the cross sections in e + e − → Σ +Σ− is very small and the corresponding uncertainty on the cross sections can be neglected. The uncertainty on the effective FFs are 4.9% and 2.8% at √ s = 2.3864 and 2.396 GeV due to the change of Coulomb correction factors. For the e + e − → Σ −Σ+ process, the variation of c.m energy and energy resolution introduce uncertainties of 12.0% and 14.2% in the cross section and effective FF, respectively, at √ s = 2.396 GeV. The integrated luminosity is determined with large angle Bhabha events with an uncertainty of 1.0% [33]. All sources of systematic uncertainties are treated as uncorrelated and summed in quadrature; they are in the range between 3.5% and 13.0% of the cross sections, depending on the c.m. energy.

Line shape analysis
The measured cross section line-shapes of e + e − → Σ ±Σ∓ from √ s = 2.3864 to 3.0200 GeV are shown in Fig. 3. The near threshold cross sections for e + e − → Σ +Σ− and e + e − → Σ −Σ+ are measured to be 58.2 ± 5.9 +2.8 −2.6 and 2.3 ± 0.5 ± 0.3 pb, respectively, both are inconsistent with the value of 520 pb expected for point-like charged baryons. Instead, a new feature is observed in which the cross sections for e + e − → Σ −Σ+ are consistently smaller than those for e + e − → Σ +Σ− . A perturbative QCD-motivated energy power function [46,47], given by (3) is used to fit the line-shapes, where c 0 is the normalization, c 1 is the mean effect of a set of intermediate states that mediates the coupling between the virtual photon [48] and is regarded as common for the two processes, and Λ QCD is the QCD scale, fixed to 0.3 GeV. The fit results are shown in Fig. 3 with a fit quality of χ 2 /ndof = 9.7/12, where ndof is number of degrees of freedom. The cross section ratio between e + e − → Σ +Σ− and e + e − → Σ −Σ+ is obtained from c 0 to be 9.7±1.3, and c 1 is 2.0±0.2 GeV 2 . Since the effective FF is proportional to the square root of the Born cross section, the ratio of the effective Σ + and Σ − FFs is consistent with 3, which is the ratio of the incoherent sum of the squared charges of the Σ + and Σ − valence quarks, q∈B Q 2 q . The results are in disagreement with the prediction from octet baryon wave functions [22], where the typical SU (3)-symmetry breaking effects for hyperon FFs are about 10 ∼ 30%. In the di-quark model, the Σ + FFs should be comparable to that of Λ [23]. The Σ ± FFs are also predicted in Ref. [24] from Unitary and Analytic model. We notice that a recent prediction for the non-resonant cross section of e + e − → Σ ±Σ∓ at the J/ψ mass [49], based on an effective Lagrangian density, is consistent with our result when extrapolated to √ s = 3.097 GeV using Eq.

Extraction of |G
The value of |G E /G M | can be obtained by fitting the differential angular distribution according to Eq. (1). The statistics at √ s = 2.3960, 2.6444, 2.6464 and 2.9000 GeV for e + e − → Σ +Σ− allow us to perform a study of the polar angle of Σ + in the c.m. frame. The angular distributions for categories A and B at √ s = 2.3960 GeV are shown in Fig. 4. These angular distributions have been corrected for the detection efficiency and ISR, which are obtained from signal MC simulation. Additional bin-by-bin corrections due to the data/MC detection differences, for categories A and B, respectively, have also been applied. Simultaneous fits to the two data sets to the expression in Eq. (1) sharing a common value for |G E /G M | are performed. The result of |G E /G M | = 1.83 ± 0.26 is significantly higher than 1. Using the normalized number of events, |G M | is determined to be (9.14 ± 1.42) × 10 −2 and (9.30 ± 1.53) × 10 −2 for category A and B, respectively. Similar angular distribution fits are performed for the combined √ s = 2.6444 and 2.6464 GeV data sets, denoted as 2.6454 GeV, and √ s = 2.90 GeV and the results are listed in Table 1. The systematic uncertainties on |G E /G M | considered here are the difference between data and MC efficiency, the bin size, and the fit range. For the Σ − , on the other hand, the statistics only allow for the determination of |G eff |; they are not sufficient to extract |G E /G M |.

Summary
In summary, the data collected by BESIII at c.m. energies between 2.3864 and 3.0200 GeV, are exploited to perform measurements of e + e − → Σ ±Σ∓ . This is the first time that cross sections of e + e − → Σ ±Σ∓ in the off-resonance region are presented. The precision has been significantly improved by reconstructing all dominant decay modes of the Σ. Cross sections near threshold are observed for e + e − → Σ +Σ− and e + e − → Σ −Σ+ to be 58.2 ± 5.9 +2.8 −2.6 and 2.3 ± 0.5 ± 0.3 pb, respectively. The values disagree with the point-like expectations near threshold, 848(m p /m B ) 2 pb, as has been seen for the proton [12,13]. The cross section line-shapes for e + e − → Σ +Σ− and e + e − → Σ −Σ+ are well-described by pQCD-motivated functions. The ratio of the σ Born (e + e − → Σ +Σ− ) to σ Born (e + e − → Σ −Σ+ ) is determined to be 9.7±1.3, which is inconsistent with predictions from various models [22][23][24]. The EMFF ratio |G E /G M | of the Σ + is determined from its production angle dependence at three high-statistics energy points. The |G E /G M | of the Σ + shows similar features to those of the proton [12,14], Λ [18], and Λ c [29], that is larger than 1 within uncertainties near threshold and consistent with 1 at higher c.m. energies.