Measurement of e+e− → ωπ+π− cross section at s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document} = 2.000 to 3.080 GeV

A partial wave analysis on the process e+e− → ωπ+π− is performed using 647 pb−1 of data sample collected by using the BESIII detector operating at the BEPCII storage ring at center-of-mass (c.m.) energies from 2.000GeV to 3.080GeV. The Born cross section of the e+e− → ωπ+π− process is measured, with precision improved by a factor of 3 compared to that of previous studies. A structure near 2.25GeV is observed in the energy-dependent cross sections of e+e− → ωπ+π− and ωπ0π0 with a statistical significance of 7.6σ, and its determined mass and width are 2232 ± 19 ± 27MeV/c2 and 93 ± 53 ± 20MeV, respectively, where the first and second uncertainties are statistical and systematic, respectively. By analyzing the cross sections of subprocesses e+e− → ωf0(500), ωf0(980), ωf0(1370), ωf2(1270), and b1(1235)π, a structure, with mass M = 2200 ± 11 ± 17MeV/c2 and width Γ = 74 ± 20 ± 24MeV, is observed with a combined statistical significance of 7.9σ. The measured resonance parameters will help to reveal the nature of vector states around 2.25GeV.


Introduction
The experiment of electron-positron collisions has long been a research topic in elementary particle physics wherein searches for exotic hadronic states have continuously received much attention. Some exciting observations in the e + e − experiments have been recently reported in refs. [1][2][3][4][5][6][7][8][9][10][11][12]. Meanwhile, measurements of the cross sections of exclusive processes are of particular interest for resolving the discrepancy of muon magnetic anomaly a µ = (g µ − 2)/2 between experimental measurements and standard model (SM) predictions. The latest result by the Muon g − 2 Experiment shows 3.3 standard deviation from JHEP01(2023)111 the SM prediction [13], implying a hint for the existence of new physics [14]. However, to confirm this, further improvements in the precision for both experimental measurement and theoretical prediction are necessary. Specifically, the precision measurements on the cross sections of light hadron production in the e + e − annihilation are essential to reduce uncertainties of theoretical calculations on the hadronic vacuum polarization contribution and the muon anomaly a µ [15][16][17].
In this paper, we report the measurement of Born cross sections for the e + e − → ωπ + π − process using the data collected in an energy scan at 19 c.m. energies ( √ s) from 2.000 GeV to 3.080 GeV with a total integrated luminosity of 647 pb −1 . The detailed values of c.m. energies and integrated luminosities of various data sets are presented in section 5.2. Combined with the previous measurements of the e + e − → ωπ 0 π 0 process [52], the cross sections of the e + e − → ωππ process are obtained. In addition, the cross sections of the subprocesses via some intermediate states (e.g., e + e − → ωf 0 (500), ωf 0 (980), ωf 0 (1370), ωf 2 (1270), and b 1 (1235)π) are determined.

BESIII detector and Monte Carlo simulation
The BESIII detector [53] records symmetric e + e − collisions provided by the BEPCII storage ring [54], which operates in the c.m. energy range within 2.000-4.950 GeV. BESIII has collected large data samples in this energy region [55]. The cylindrical core of the BESIII detector covers 93% of the full solid angle and comprises 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 specific ionization energy loss dE/dx resolution is 6% for electrons from Bhabha scattering. The EMC measures JHEP01(2023)111 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) events, including the geometric description of the BESIII detector and the detector response, are produced using GEANT4-based [56] offline software BOSS [53]. Two million of inclusive MC events, e + e − → hadrons, are used to estimate the background contamination. They are generated using a hybrid generator, which integrates PHOKHARA [57,58], ConExc [59,60], and LUARLW [62] models. The PHOKHARA model generates 10 well parameterized and established exclusive channels. The ConExc model simulates a total of 47 exclusive processes according to a homogeneous and isotropic phase space population and then reproduces the measured line shapes of the absolute cross section. The remaining unknown decays of virtual photon are modeled by the LUARLW model.
Signal MC events for the e + e − → ωπ + π − process with subsequent decays ω → π + π − π 0 and π 0 → γγ are generated by using the amplitude model with parameters fixed to the helicity amplitude analysis results, and the significant intermediate states are included with the ISR effects.

Event selection and background analysis
A data sample of the e + e − → π + π − π + π − π 0 process is selected by requiring the net charge of four charged tracks to be zero and at least two photons. Charged tracks detected in the MDC are required to be within a polar angle (θ) range of |cos θ|<0.93, where θ is defined with respect to the z-axis, which is the symmetry axis of the MDC. For charged tracks, the distance of the closest approach to the interaction point (IP) must be less than 10 cm along the z-axis of the MDC, |V z |, and less than 1 cm in the transverse plane, |V xy |. Photon candidates are identified using showers in the EMC. The deposited energy of each shower is more than 25 MeV in the barrel region (|cos θ|<0.80) and more than 50 MeV in the end cap region (0.86<|cos θ|<0.92). 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 should be greater than 10 degrees. To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and event start time must be within (0, 700) ns.
To improve the mass resolution and suppress the background contribution, a fourconstraint (4C) kinematic fit imposing four-momentum conservation is performed under the hypothesis of the e + e − → 2(π + π − )γγ process. When more than two photons exist, the combination with minimum χ 2 4C is retained for further analysis. Based on an optimization of the signal-to-noise ratio for the requirement on χ 2 4C , where N s and N s + N b are the numbers of events in the ω signal region for signal MC sample and data, respectively, candidate events with χ 2 4C < 40 are accepted for further analysis. The ωπ + π − events are reconstructed by selecting the ω candidates from all the four π + π − π 0 combinations with minimal mass difference from the ω known mass [20]. The π 0 candidate is reconstructed by the γγ combination with a mass window of 0.12<M γγ <0.15 GeV/c 2 , and the ω signal region is defined with a requirement of 0.758<M π + π − π 0 <0.808 GeV/c 2 . To estimate the back- The e + e − → π + π − π + π − π 0 process is found to be the dominant background source by analyzing the inclusive MC sample [63]. No peaking background is found in the signal region. The selection of the 3π combination for ω candidate may lead to a wrong π + π − π 0 combination. The ratio due to the mis-combination is examined by matching the MC truth to the reconstructed final state particles using a sample of 0.5 million MC events for the e + e − → 2(π + π − )π 0 process. When a charged pion is assigned to be from the wrong ω candidate, an event will be marked as "mis-combination". The ratio of the number of mis-combination ω to the total number of events is about 1.0% on average. The miscombination yields will be subtracted from the ω candidate events when calculating the e + e − → ωπ + π − Born cross section.
In the amplitude analysis, the intensity for each process is described with the helicity amplitude, which is constructed as the amplitude production of two sequential decays in the helicity frame. Taking process (a) as an example, figure 2 shows the definition of helicity rotation angles, where the R 1 polar angle, θ [123] [12] , is defined as the angle spanned between the R 1 momentum and positron moving direction, and the azimuthal angle φ [123] [12] is the angle between the R 1 production plane and its decay plane. For the R 1 → π + π − decay, the azimuthal angle, φ [12] [1] , is defined as the angle between R 1 production plane and its decay plane, formed by the π + π − momentum. After boosting the two pion momenta to the R 1 rest frame, they remain in the same decay plane. The polar angle θ [12] [1] for π + is defined as the angle between the π + π − and R 1 momentum in the R 1 rest frame. Helicity angles for processes JHEP01(2023)111 Helicity rotation angles defined for the process e + e − → R 1 ω, R 1 → π + π − .
The non-resonant three-body process e + e − → ωπ + π − is also considered, and its amplitude is written as [67][68][69][70]: where µ is the z-component of spin J in the helicity system; m(λ 3 ) is the helicity value for γ * (ω); α, β, γ are the Euler angles as defined in refs. [67][68][69][70]; and F µ,λ 3 is the helicity amplitude. The process conserves the parity and thus leads to F ±,λ 3 = −F ±,−λ 3 and F 0, To match the covariant tensor amplitude, we expand the helicity amplitude in terms of the partial waves for the two-body process in the LS-coupling scheme [68][69][70]. It follows for a spin-J particle decay J → s + σ, and λ and ν are the helicities of two final-state particles s and σ with δ = λ − ν. The symbol g lS is a coupling constant, S is the total intrinsic spin S = s + σ, and l is the orbital angular momentum, r = |r|, where r is the relative momentum between the two daughter particles in their mother rest frame and r 0 corresponds to the invariant mass of the resonance which is equal to its nominal mass. The Blatt-Weisskopf factor B l (r) [68][69][70] up to l = 4 takes the form B 0 (r)/B 0 (r 0 ) = 1, where parameter d is constant fixed to 3 GeV −1 [64].

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The differential cross section is given by where m = ±1 is due to the virtual photon produced from unpolarized e + e − annihilations, λ 3 is the ω helicity value, and dΦ is the element of a standard three-body phase space. The variable Ω( where ε is the ω polarization vector and q 1 (q 2 ) is the momentum vector for π + (π − ) from the ω decay. Here, we factor out the Breit-Wigner function describing the ω line shape into the MC integration when applying the amplitude model fit to the data events.

Simultaneous fit
The relative magnitudes and phases for coupling constants are determined by an unbinned maximum likelihood fit. The joint probability for observing N events in one data set is where p j (j = 1, 2, . . . , 5) denote the four-vector momenta of the final states and P i is a probability to produce event i with the momenta of the final states. The normalized P i is calculated from the differential cross section Here, the normalization factor σ MC is calculated using a signal MC sample with N MC accepted events, which are generated with a uniform distribution in phase space and subjected to the detector simulation, and then are passed through the same event selection criteria as applied to the data events. With a signal MC sample of two million events, the σ MC is approximately evaluated as For technical reasons, rather than maximizing L, the object function, S = − ln L, is minimized using MINUIT [71]. To subtract the background events, the object function is replaced with where L data and L bg are the joint probability densities for data and background, respectively. The background events are obtained by the normalized ω sidebands. We perform a simultaneous fit to an ensemble of the ωπ + π − events with different c.m. energies, and their amplitudes sharing same coupling constants describing the intermediate state processes. The total object function is obtained as the summation of individual ones: for the total N sets of data events.

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After the parameters are determined in the fit, the signal yield of a given subprocess can be estimated by scaling its cross section ratio r i to the number of net events where σ i is the cross section for the i-th subprocess as defined in eq. (4.7), σ tot is the total cross section, N obs is the number of observed events, and N bg is the number of background events. The statistical uncertainty, δN i , associated with the signal yield N i is estimated according to the error propagation formula using the covariance matrix V , obtained from the fit: where X is a vector containing parameters and µ contains the fitted values for all parameters. The sum runs over all N pars parameters.

Intermediate states and significance check
In terms of the π + π − mass spectrum shown in figure 3 for 12 of the 19 samples, the f 0 (980) signals are significantly observed in the data events at c.m. energies below 2.3094 GeV. In the low π + π − energy region, f 0 (500) resonance may have some contributions. In the high energy region, f 0 (1370) and f 2 (1270) resonances are included in the amplitude model. The f 0 (980) line shape is parameterized with the Flattê formula: where ρ(s) = 2k/ √ s; k is the c.m. momentum of the π or K in the resonance rest frame; g 1 and g 2 g 1 are fixed to the measured values of 0.138 ± 0.010 GeV 2 and 4.45 ± 0.25, respectively [72,73]; and m 0 is the mass of the resonance f 0 (980) (m 0 =0.990 GeV/c 2 ).
For the f 0 (500) line shape, many types of energy-dependence-width parameterizations exist in the literature [72,73], among which we choose the E791 parameterizations (used by the E791 Collaboration) in the nominal fit: where m 0 and Γ 0 are the f 0 (500) mass and width, respectively.
For the other resonances (e.g., f 0 (1370), f 2 (1270), b 1 (1235), ρ(1450) ± and ρ(1570) ± ), the width is considered constant, and the line shape is described with the Breit-Weigner function, in which their masses and widths are fixed to the PDG values, as given in table 2.
The partial wave analysis (PWA) fit procedure begins by including all possible intermediate states in the PDG that match J P C conservation in the subsequent two-body decay. Then we examine the statistical significance of the individual amplitudes. Amplitudes with statistical significance less then 5σ are dropped. This procedure is repeated until a baseline solution is obtained with only amplitudes having a statistical significance greater than 5σ.

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Resonance  The statistical significance of each amplitude is evaluated by incorporating the changes in likelihood and number of degrees of freedom with and without the corresponding amplitude being included in the simultaneous fit. The significance for each intermediate state is listed in table 2. We take the subprocess with a significance greater than 5σ in both group A and group B data sets to be the baseline solutions, including the e + e − → ωf 0 (500), ωf 0 (980), ωf 0 (1370), ωf 2 (1270) and b ∓ 1 (1235)π ± processes. JHEP01(2023)111

Fit results
Signal yields for the data sets are calculated using eq. (4.13), and their statistical uncertainties are derived using eq. (4.14), which includes the correlation among parameters. The signal yields are given in table 3

ISR correction factor
In the direct e + e − collision experiments, the observed cross section, σ obs (s), at the c.m. energy √ s for e + e − → ωπ + π − is the Born cross section, σ 0 (s), convolved with the ISR function W (s, x) [60,61]. To unfold the Born cross section, the ISR correction factor should be defined as where Π( √ s) is the vacuum polarization function. We use the calculated results, including the leptonic and hadronic parts both in space-and time-like regions [74][75][76][77][78]. The M th corresponds to the ωπ + π − mass threshold, and x is the ratio of the ISR photon energy to the beam energy.
Calculations of the ISR correction factor and the MC event generation are consistently performed using the generator model "ConExc" [60], and the dressed cross sections σ D 0 (s) are taken from the BaBar experiment ( √ s < 2.000 GeV) [18] and this measurement ( √ s > 2.000 GeV). To achieve stable cross sections in this measurement, the procedure of the ISR correction factor calculation and the MC event generation for the Born cross section calculation are iterated serval times. The iteration is stopped if the updated Born cross section reaches the accuracy within statistical uncertainty. The ISR correction factors for various energy points are given in table 4.
In the MC event generation, the ISR photon is characterized by the soft energy and beam collinear distribution. Its angular dependence is sampled with the Bonneau and Martin formula with an accuracy of up to the m 2 e /s term; where θ is the polar angle of the ISR photon, m e is the electron mass, and E is the beam energy. After the photon emission, the γ * → ωπ + π − events are generated with the amplitude model with parameters fixed to the fitted values so that the measured intermediate states are inclusively simulated. Figure 6 shows an example of the MC simulation at √ s = 2.125 GeV, in which we can observe a good agreement between data and MC simulation in the mass distribution.  Figure 3.
Projections of the fit results for mass spectra m π + π − and m ωπ ± for selected energy points.
Dots with error bars are the data, histograms are the projection of the fit results, and shaded histograms (green) are the background estimated by the ω mass sideband. The subplots are for data sets   The π + π − π 0 invariant mass spectrum of the process e + e − → ωπ + π − at √ s = 2.1250 GeV. The dots with error bars are the data, the red dotted curve is the signal, and the blue dashed line is the background. The blue solid curve is the best fit result. The green histogram is from the MC simulation of the non-resonant contribution from e + e − → π + π − π + π − π 0 . The range between the two red vertical solid lines is regarded as the signal region, and those between the yellow and red vertical lines on each side of the signal peak are regarded as the sideband regions.

Cross section for e + e − → ωπ + π −
The signal yields are determined by an unbinned maximum likelihood fit to the π + π − π 0 invariant mass distribution of the ω candidates. In the fit, the signal shape is modeled with the MC histogram smeared with a Gaussian function for the resolution difference between data and MC simulation. The background is dominated by the process e + e − → 2(π + π − )π 0 , and the corresponding shape is described by a second-order Chebychev polynomial. The parameters of the Gaussian function, the background polynomial and the yields of signal and background are floated. Figure 7 shows the fit results at √ s = 2.125 GeV. The Born cross section is calculated from , (5.4) where N sig is the number of signal events obtained by fitting the π + π − π 0 mass distribution, N mis the mis-combination yield, L the integrated luminosity, and the detection efficiency obtained from the MC simulation according to the PWA result. Br(ω → π + π − π 0 ) and Br(π 0 → γγ) are the branching fractions quoted from the PDG [20], and (1 + δ) represents the correction factors due to the ISR effect and vacuum polarization. The resulting Born cross sections and related variables are summarized in table 4.

Cross sections for the intermediate subprocesses
The  Table 4. Summary of the signal yield (N sig ), number of mis-combination events (N mis ), integrated luminosity L, ISR correction factor (1 + δ), detection efficiency ( ), and total Born cross section of e + e − → ωπ + π − (σ B ) at various energy points √ s. The uncertainties of N sig and N mis are only statistical. The first and second uncertainties of Born cross sections are statistical and systematic, respectively. ratio of the Born cross section for a subprocess R to the total cross section is determined.

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• In the nominal fit, we take the f 0 (500) line shape as that used by the E791 Collaboration. The uncertainty associated with this parametrization is estimated by replacing it with those in ref. [73]. The difference in the signal yield of each intermediate state is considered as the systematic uncertainty.
• The parameters of the f 0 (980) Flattê formula in the nominal fit are fixed to the measurements [72,73]. The uncertainties are estimated by varying the resonance parameters by one standard deviation. The differences in the signal yields are considered as the systematic uncertainties.
• The masses and widths of intermediate states f 0 (1370), f 2 (1270), and b 1 (1235) are fixed to the PDG values [20] in the nominal fit. The associated uncertainties are estimated by varying masses and widths by ±1σ respectively. The differences are taken as the corresponding systematic uncertainties.
• The two resonances ρ(1450) ± and ρ(1570) ± are insignificant and not included in the baseline solution. The differences of the fits with and without the two resonances in the signal yields are assigned as systematic uncertainties.
• To estimate the uncertainties associated with background contamination, alternative fits are performed by changing the number of background events with one standard deviation for various energy points. The differences in the signal yields are considered as the systematic uncertainties.
All these uncertainties are added in quadrature to provide the total systematic uncertainties of intermediate states (table 5).

Uncertainties for the measurements of σ(e + e − → ωπ + π − )
The sources of systematic uncertainties associated with the Born cross section measurements include the tracking efficiency, photon detection, kinematic fit, π 0 mass window requirement, fit procedure, radiative correction, luminosity measurement, and branching fraction of ω → π + π − π 0 . These sources are described as follows: • The uncertainty due to the tracking efficiency is estimated as 1.0% per track [79].
• The uncertainty due to the photon detection is 1.0% per photon [80].
• The uncertainty associated with the kinematic fit originates from the inconsistency between the data and MC simulation of the track helix parameters. Following the procedure described in ref. [81], half of the difference in the detection efficiencies with and without the helix parameter correction is regarded the systematic uncertainty.

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• The uncertainties due to the choices of signal shape, background shape, and fit range are estimated by varying the signal function from the Breit-Wigner to the MC simulated shape convolved with a Gaussian resolution function. Another technique is varying the background function from the second-order to third-order Chebychev polynomial and by extending the fit range from (0.65, 0.89) GeV/c 2 to (0.64, 0.90) GeV/c 2 . The differences in the signal yields are considered as systematic uncertainty.
• The ISR correction factor is calculated with the line shape of Born cross sections, which is smoothed by fitting the cross section distribution with five Gaussian functions. The signal MC samples with the ISR effect are generated according to the input line shape. Using the updated MC samples, we re-estimate the efficiency and ISR correction factor and then update the cross section. Iterations are performed un-

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til the results are stable. Finally, the difference between the last two iterations, 1.0%, is considered the corresponding systematic uncertainty for all the energy points.
• The integrated luminosities of these data sets are measured using large angle Bhabha scattering events, with a uncertainty about 1.0% [82] is propagated to the cross section measurements. The uncertainty in Br(ω → π + π − π 0 ) is quoted as 0.8% from the PDG [20].
• The uncertainty from the MC statistics is estimated by the number of generated events, calculated by where N =1000000 is the number of generated events at each energy points.
• The uncertainty due to the PWA MC model is investigated by smearing masses and widths of intermediate states per event within ±1σ, assuming that they follow Gaussian distribution. The difference in the detection efficiencies is considered the systematic uncertainty, which is about 1%. The effect due to the insignificant intermediate states, such as ρ(1450) and ρ(1570), is investigated by adding them to the nominal fit. Moreover, the difference in the detection efficiencies with and without these two intermediate states is regarded the systematic uncertainty, which varies from 0.2% to 1% depending on the energy point. The total uncertainty due to the MC model is obtained by adding these uncertainties in quadrature.
The systematic uncertainties for all energy points are listed in table 6. These uncertainties are assumed independent and summed in quadrature.
7 Fit to the line shapes 7.1 Line shape of e + e − → ωπ + π − The measured Born cross sections are shown in figure 8, where a clear structure is observed around 2.25 GeV. A minimized χ 2 fit incorporating the correlated and uncorrelated uncertainties is performed for the measured cross section with the following function: where φ is the phase of the resonance structure relative to the continuum components and σ r ( √ s) and σ c ( √ s) are the cross sections for the resonance and continuum components, respectively. The resonant component σ r ( √ s) is parameterized as where m r and Γ r are the mass and width of the resonant structure near 2.25 GeV, respectively. Parameter Γ ee r · Br is the product of the partial width of the resonance decaying to the e + e − and the branching fraction to the ωπ + π − final state. m r , Γ r and Γ ee r · Br are free parameters in the fit. C is a conversion constant that equals to 3.893 × 10 5 nb·GeV 2 [18].
The continuum component is parameterized as where a and b are free parameters and P ( √ s) is the phase factor. For the two-body process, P ( √ s) is taken as p/ √ s, where p is the magnitude of the momentum for one of the two particles. For the three-body final state of ωπ + π − , 4) where m ω and m π + π − are the mass of ω particle and the invariant mass of π + π − , respectively. The results of the fits to the measured cross section are shown in figure 8. The data points of the BaBar experiment are plotted to overlap for comparison. The fit has two solutions with equally good fit quality and identical mass and width of the resonance, but with different phases and Γ ee r · Br. We observe a structure around 2.25 GeV, denoted as X(2230). The statistical significance of X(2230) is 10.3σ, which is determined from the change in the χ 2 value with and without it in the fit versus the change of the number of degrees of freedom. The best fit provides a fit quality of χ 2 /n.d.f = 20.6/13. The fit parameters of the two solutions are listed in table 7. Due to the interference effect, the parameter b is fitted as b = 4.4 ± 0.1 in both two solutions. Note that this is different from the single virtual photon contribution where b = 1.
To estimate the systematic uncertainties for the resonant parameters, an alternative fit is performed by parameterizing the continuum component with an exponential function [83] to determine the uncertainties associated with the continuum component: where p 0 and p 1 are float parameters and u = √ s − (2m π + m ω ), which the masses of m π and m ω take from the PDG [20]. The differences of the results between the alternative and nominal fits are considered the systematic uncertainties for resonant parameters. Further, the systematic uncertainties associated with the signal model are investigated by using a relativistic Breit-Wigner function with the energy-dependent width. Their differences are found to be negligible. Eventually, the mass and width of the resonance are determined to be m r = 2250 ± 25 ± 27 MeV/c 2 and Γ r = 125 ± 43 ± 15 MeV, respectively. In addition, the Γ ee r · Br is 0.9 ± 0.4 ± 0.4 eV or 52.9 ± 17.0 ± 13.1 eV for the two solutions, where the first and second uncertainties are statistical and systematic, respectively.

Cross section of e + e − → ωππ
The Born cross sections for the e + e − → ωπ + π − measured in this work are consistent with that for the e + e − → ωπ 0 π 0 obtained in ref. [52] within a factor of 2 given their uncertainties. Therefore, the pion production in the e + e − → ωππ process complies with the SU(3)-flavor symmetry. The summed cross sections at various energy points are listed in table 8. We fit the e + e − → ωππ cross sections with the same method as in section 7.1 (see figure 9). The resultant parameters are obtained and tabulated in table 9. After considering the systematic uncertainty, the mass and width of X(2230) are obtained as 2232 ± 19 ± 27 MeV/c 2 and 91 ± 53 ± 20 MeV, respectively. The Γ ee r · Br is 0.9 ± 0.5 ± 0.2 eV and 61.1 ± 32.1 ± 15.4 eV for the two solutions. The statistical significance of the structure is 7.6σ.
We also compute the contribution of our cross section σ Born (e + e − → ωππ) to the hadronic contribution of (g − 2) µ , where K(s ) is the kernel function [74]. Our result, a ωππ µ =(2.213 ± 0.106 ± 0.149) × 10 −10 , is the first measurement for the e + e − → ωππ process in this energy region.

X(2230) decays via intermediate states
Using the measured Born cross sections via the intermediate states as tabulated in table 5, we perform a simultaneous fit to the measured cross sections σ exp i with i = 1, 2, . . . , 5 for the modes ωf 0 (500), ωf 0 (980), ωf 0 (1370), ωf 2 (1270), and b 1 (1235)π, respectively. At the energy point √ s, the cross section for the mode i is described by Where, σ r and σ c are given by eqs. ( Figure 9. Fits to the cross sections of e + e − → ωππ. The dots with error bars are the results combined from this work and ref. [52], where the error bars include both statistical and systematic uncertainties. The red solid curves are the total fit results, the blue solid curves represent the continuum components, the green curves denote the resonant components, and the cyan line are the interference of the two components. The two plots are the results of Solutions I (left) and II (right), respectively. five modes, the χ 2 function is minimized, which is defined by where V is a covariance matrix with element V ij = ρ ij σ i σ j and ρ ij is a correlation coefficient and determined by the errors of interference between modes i and j. We take ρ ij = 1 if i = j. The sum over √ s runs over all energy points involved in the fit. Figure 10 shows results of the simultaneous fit to these five intermediate modes.
In the fit, the state X(2230) is assumed as a vector meson, and its mass and width is 2200 ± 11 ± 17 MeV/c 2 and 74 ± 20 ± 24 MeV, respectively. Here, the systematic uncertainties of the mass and width are determined by using the same method as in section 7.1. The statistical significance of this structure is about 7.9σ. The resultant values of Γ ee r · Br are tabulated in table 10. Here, Γ ee r is the partial width of the X(2230) decaying to e + e − , and Br is the branching fraction for the X(2230) decaying to the final state ωππ. Two solutions are found with equally good fit quality. No structure around 2.25 GeV is observed in the non-resonant process, as shown in figure 10(f), indicating that the vector meson favors the intermediate decay.

Conclusion and discussion
In summary, the Born cross sections of the e + e − → ωπ + π − process were measured using 647 pb −1 data samples collected with the BESIII detector at 19 c.m. energies from JHEP01(2023)111

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2.000 GeV to 3.080 GeV. The precision of the measured cross sections are improved by a factor of 3 with respect to the previous measurement [18]. The resonance X(2230) is observed with a significance of 7.6 σ and thereby confirms the structure near 2.25 GeV observed by BaBar [18] and BESIII [52]. The energy dependence of the total cross section for e + e − → ωπ + π − and ωπ 0 π 0 is shown in figure 9. From the fit to the e + e − → ωππ cross sections, we obtained the mass and width of X(2230) as 2232 ± 19 ± 27 MeV/c 2 and 93 ± 53 ± 20 MeV, respectively. The fitted Γ ee r · Br values are 0.9 ± 0.5 ± 0.2 eV and 61.1 ± 32.1 ± 15.4 eV depending on the interference pattern. We also fit the Born cross section of the intermediate processes, such as e + e − → ωf 0 (500), ωf 0 (980), ωf 0 (1370), ωf 2 (1270), and b 1 (1235)π, with the results presented in figure 10. Precision measurement of the cross sections of these intermediate subprocesses will help to reveal the properties and nature of the structure. For the process e + e − → ωππ, we computed its hadronic contribution of the cross section to (g − 2) µ and found that its contribution is small in the energy region between 2.000 GeV and 3.080 GeV.