Determination of the resonant parameters of excited vector strangenia with data *

: We determine the resonant parameters of the vector states and by performing a combined fit to the cross sections from the threshold to measured by the BaBar, Belle, BESIII, and CMD-3 experiments. The mass and width are obtained for , and the mass and width are obtained for . The statistical significance of is . Depending on the interference between , , and a non-resonant amplitude in the nominal fit, we obtain four solutions and , , or , and , , or . We also search for the production of , and the significance is only . We then determine the upper limit of at the 90% confidence level.

Hadronic transitions with or η emittance have contributed significantly to discoveries of quarkonium(like) states, such as in via initial-state radiation (ISR) by the BaBar experiment [1].While searching for an version of , BaBar discovered (now called ' ') in via ISR [2], which was later confirmed by Belle [3].While searching for in the hadronic transition with η, BaBar studied the process via ISR using a data sample and found an excess with a mass of (tens of lower than the world average value of [4]) and a width of [5,6].Hereinafter, the first quoted uncertainties are statistical and the second ones are systematic.Belle measured this process with considerably larger statistics in a data sample but did not find this excess, and the statistical significance of was only [7].and found it to be dominated by the contribution [8].CMD-3 then calculated the contribution to the anomalous magnetic moment of the muon: , .With a data sample taken at 22 CM energy points in the range of 2.00 to , BESIII measured the Born cross sections of [9] and [10].BESIII reported the observation of in the final state and determined its resonant parameters to be and [9], for which the width was considerably narrower than the world average value of approximately MeV [4].BESIII also observed a resonance near in the final state with a statistical significance exceeding [10].Assuming it is , one can infer the ratio , which is smaller than the prediction of hybrid models by several orders of magnitude.
It is puzzling that is not significant in the η transition compared with the transition, and the measurement of in the final state is still poor.
σ(e + e − → ηϕ) σ(e The lineshape of is considerably different from that of [2,3], which may help us understand the difference between and .In the sector, at the peak of , whereas at the peak of and at the peak of in the sector.In a recent lattice quantum chromodynamics (QCD) calculation [11], the properties of the two lowest states were found to comply with those of ϕ and but had no obvious correspondence to .

Besides
, there is one more state, known as ' ,' that is a candidate of the quarkonium.The observation of in and is sometimes cited as evidence that this state is an quarkonium, as the radial excitation of ϕ.However, it was argued that the one true evidence for ϕ as an state should be the large branching fractions to hidden strangeness modes such as [12].The FOCUS experiment reported a highstatistics study of the diffraction photo-production of and observed with a mass of and a width of [13].Meanwhile, FOCUS observed a slight enhancement below the region but no obvious signal in the final state.e + e − K + K − q q (q = u, d)

If
and are the same state, the mass measured in collisions and photoproduction experiments typically has a difference of , with dominance in collisions and dominance in photo-production.This may constitute evidence for two distinct states, although interference with vectors may complicate a comparison of these two processes.This issue can be addressed by studying channels in which interference with vectors is expected to be unimportant, notably, .With a sample of 4.48 million events, BESIII performed the first partial wave analysis of and simultaneously observed and in the mass spectrum [14], which indicates that is distinct from .Meanwhile, BESIII determined to be a resonance. .In this study, we combine the measured from the BaBar, Belle, BESIII, and CMD-3 experiments to gain greater lineshape precision, which is helpful for the study of the anomalous magnetic moment of the muon.Then, we perform combined fits to these measured cross sections for the resonant parameters of and and estimate the production of in the final state.
• We show comparisons of the latest results from Belle and the previous measurements from BaBar, BE-SIII, and CMD-3 in Fig. 2. The comparisons reveal good agreement between the four experiments.
• In the BaBar measurements, measured in the mode is slightly lower than that measured in the mode; however, both have similar lineshapes, including a small bump around .The expected signal according to the world average value of [4] is not clear in the BaBar measurements.• BESIII reported the Born cross section of .We calculate the dressed cross section of with the vacuum polarization and Born cross sections from Ref. [9], as shown in Fig. 1(f).The measurements of the dressed cross section of from the four experiments are consistent with each other.Therefore, we combine these measurements to obtain the best precision of .A precise is helpful for studying the anomalous magnetic moment of the muon [8] and may provide hints of or .The calculation for the combination uses where is the value of the ith ( ) experimental measurement of the cross section at the energy point ( ) illustrated in Fig. 1, and and are their related uncertainties.The average of takes into account the difference in in the data taking of the BE-SIII or CMD-3 experiment and the average reported by BaBar and Belle using ISR technology.The uncertainties of in the BESIII and CMD-3 experiments are of the level, and the two experiments have no overlap in the region.We take half of the bin width in the BaBar and Belle measurements as the uncertainty ( ).However, there are correlations between the measurements, such as the branching fraction of ϕ or η decay.We revisit the estimation of Eq. ( 3) according to Ref. [15] and construct the matrices of the statistical uncertainties and uncorrelated systematic uncertainties as , where and are the statistical and uncorrelated systematic relative uncertainties of .We then construct the matrix of the correlated systematic uncertainties as . We obtain the effective global covariance matrix σ According to Ref. [15], we calculate the error of using We show the results of the combination in Fig. 3  , , and may exist in the process.We perform combined fits to measured by the BaBar, Belle, BESIII and CMD-3 experiments and shown in Fig. 1.The fit range is from the threshold to .Assuming there are , , and components and a nonresonant contribution in the final state, we take the parameterization of similar to that used in the BaBar analysis [5]: where is the phase space of the final state, the non-resonant amplitude takes the form , and , , and are the amplitudes of , , and , respectively.
For and , we describe the form with a Breit-Wigner (BW) function, where X is or , the resonant parameters , , and are the mass, total width, and partial width to , respectively, is the branching fraction of decay, and is the relative phase.
The BaBar measurement [5] shows that and are two major decays of and , where is the branching fraction of the decay.We also take the same form as in Ref. [5]: Here, is the phase space of the decay.The other decays of are neglected, and their phase space dependence are correspondingly ignored.Because both the and final states contain a vector meson (V) and pseudoscalar meson (P), the phase space takes the form

X(1750)
We take the same form of as in the BESIII measurement [14]: where [ ] is the momentum of a daughter particle in the rest frame of the resonance with energy (mass ), and l is the orbital angular momentum of the daughter particle.

ϕ(2170) → ηϕ
We describe the amplitude of the decay as in Ref. [7]: We perform several combined fits to measured by BaBar, Belle, BESIII, and CMD-3.These are fits with 1) only , 2) and the non-resonant component, 3) , , and the non-resonant component, and 4) , , , and the non-resonant component.The input data are and the related uncertainties shown in Fig. 1.Based on the fit results, which are described below, we obtain the nominal fit results from the third case.
The input data of the combined fits are the values of measured by the BaBar, Belle, BESIII, and CMD-3 experiments, and a least method with MINU-IT [16] is used.According to Ref. [15], we define of the kth energy point as ) is the difference between the measured value from the ith (jth) data sample and the fitted value of , and the effective global covariance matrix C is described in Eq. ( 6).The total is the sum of over all energy points.
In the measurement of one experiment, there may be a correlation between the two modes of η decays in one bin, or a correlation between all bins.For the first correlation, we calculate for the ith bin.Here, we also use the relative uncertainty between the two modes of η decays to calculate the elements of the correlation matrix.We obtain for the sum of all the bins in one experiment.Similarly, we calculate for the second correlation as Here, the matrix element is for the correlation between the ith and jth bins.Note that refers to different correlated systematic uncertainties in the calculations of and .
We add and to the total for the constraints owing to the two types of correlations in the combined fits.Fitting to measured by the four experiments with only the component in Eq. ( 8), we obtain reasonably good results with a quality of , as illustrated in Fig. 4. Here, is the number of all fitted data points minus the number of free parameters.We obtain the resonant parameters of as , , and . The world average values of the mass and width of are and [4], respectively.We can see that the mass and width from this fit are considerably different from the world average values, which is due to the absence of several components in our fit, such as the nonresonant contribution and .We also notice that the world average value of the width has a large uncertainty.Fitting to with only and the non-resonant contribution in Eq. ( 8), we obtain two solutions of equivalent quality with , as illustrated in Fig. 5 and Table 2. Hereinafter, we use all the data from the four experiments as input for the combined fits but show only the combined from Fig. 3 represent the data in the plots.We obtain the same resonant parameters and while eV or from the two solutions.The two resonant parameters have good agreement with the world aver- σ(e + e − → ηϕ) 2.17 GeV ϕ(2170) age values, and the precision is effectively improved.Meanwhile, the branching fraction of is or , which is close to the value that can be calculated according to the BaBar measurement [6].We can see that most of the measured from the four experiments are above the fit curve around in Fig. 5, which indicates the requirement of .With and but no in Eq. ( 8), we obtain four solutions of equivalent quality with 284/247 from the nominal combined fit, as illustrated in Fig. 6 and Table 2.The four solutions have the same resonant parameters, ,   at the world average values [4], we obtain the fit results listed in Table 3, with curves similar to those in Fig. 6.
We then estimate the statistical significance of to 7.4σ be .We describe the systematic uncertainties in the fit results in Sec.V.
As discussed in Ref. [17]  fit with n components in the amplitude.They have the same goodness of fit and the same mass and width of a resonance.Unfortunately, we cannot find a proliferation of the solutions as in several previous measurements [3, 18−20].

V. SYSTEMATIC UNCERTAINTIES
We characterize the following systematic uncertainties for the nominal fit results and estimate the uncertainty of the parameterization in Eq. ( 9) with two different parameterization methods, which have the forms and .
[1.6, 2.9] GeV/c 2 A n.r.ηϕ (s) = a 0 /s By changing the fit range to , we find that the systematic uncertainty due to the fit range is negligible.We use to estimate the model dependence of the non-resonant contribution.We obtain the uncertainty in by varying according to the previous measurement [5].To estimate the uncertainty due to the possible contribution from , we take [3] and modify Eq. (10) to in the combined fits.Here, is the phase space of the decay.Assuming all these sources are independent and summing them in quadrature, the total systematic uncertainties are listed in Table 5. which are consistent with the world average values [4].The mass and width of are and , with a good precision compared with the world average values [4].The branching fraction of the decay is approximately 20%, with uncertainties of less than 6%.We also determine and from the fits.

VI. SUMMARY
e + e − → ηϕ X(1750) 2.0σ X(1750) e + e − → ηϕ Assuming its existence in the process, the statistical significance of is only .We determine the UL of in at the 90% C.L.

Fig. 1 .
Fig. 1. measured in (a) the mode at Belle, (b) the mode at Belle, (c) the mode at BaBar, (d) the mode at BaBar, (e) the mode at CMD-3, and (f) the mode at BESIII.

Fig. 2 .
Fig. 2. (color online) Measurements of from the BaBar, BESIII, and CMD-3 experiments compared with the latest measurements from the Belle experiment.Plots (a) and (b) show the comparison between BaBar and Belle in the mode and mode, respectively; plots (c) and (d) show the comparison between CMD-3 and Belle and between BESIII and Belle, respectively, where the Belle measurement has the and modes combined.

Fig. 3 .
Fig. 3. Cross section of from the combination of the measurements by the BaBar, Belle, BESIII, and CMD-3 experiments.

Fig. 4 .
Fig. 4.(color online) Results of fitting to measured by the BaBar, Belle, BESIII, and CMD-3 experiments with only .The blue solid line shows the fit results, and the red dashed line shows the component.

Fig. 5 .
Fig. 5. (color online) Fitting to measured by the BaBar, Belle, BESIII, and CMD-3 experiments, including and the non-resonant contribution.The blue solid lines show the fit results, and the red, green, gray dashed lines show the and non-resonant components.The interference between the non-resonant component and is not shown.e + e − → ηϕ Determination of the resonant parameters of excited vector strangenia with the data Chin.Phys.C 47, 113003 (2023)

Table 1 .
and

Table 1 .
Cross section of versus calculated with the measurements from the BaBar, Belle, BESIII, and CMD-3 experiments.The first errors are statistical and the second ones are systematic.

Table 2 .
Results of fitting to measured by the BaBar, Belle, BESIII, and CMD-3 experiments with the non-resonant

Table 3 .
[4]ults of fitting to measured by the BaBar, Belle, BESIII, and CMD-3 experiments with the , , and non-resonant components.The mass and width of are fixed at the world average values[4].

Table 5 .
Systematic uncertainties of the resonance parameters for and .M, Γ, , and are the mass with units of , total width with units of , the production of the branching fraction and the partial width to with units of , and the branching fraction (%), respectively.