Electromagnetic form factors of Σ+and Σ-in the vector-meson dominance model

Zhong-Yi Li and Ju-Jun Xie,3,*

1 Institute of Modern Physics,Chinese Academy of Sciences,Lanzhou 730000,China

2 School of Nuclear Science and Technology,University of Chinese Academy of Sciences,Beijing 101408,China

3 School of Physics and Microelectronics,Zhengzhou University,Zhengzhou,Henan 450001,China

Abstract Based on the recent measurements of theandprocesses by the Beijing Spectrometer III(BESIII)collaboration,the electromagnetic form factors of the hyperon Σ+ and Σ- in the timelike region are investigated using the vector-meson dominance model,where the contributions of the ρ,ω,and φ mesons are taken into account.The model parameters are determined from the BESIII experimental data of the timelike effective form factors|Geff|of the Σ+and Σ-baryons for center-of-mass energies from 2.3864 to 3.02 GeV.It is found that we can provide quantitative descriptions of the available data using as few as one adjustable model parameter.We then progress to an analysis of the electromagnetic form factors in the spacelike region and evaluate the spacelike form factors of the hyperons Σ+and Σ-.The electromagnetic form factors obtained for the Σ+ and Σ- baryons are comparable with those of other model calculations.

Keywords: Electromagnetic form factors,Vector-meson dominance,Hyperon-antihyperon pair production

1.Introduction

The study of the structure of hyperons,which contain strange quarks,via their electromagnetic form factors (EMFFs) is crucial for a deep understanding of the nonperturbative quantum chromodynamics (QCD) effect whereby quarks are bounded by baryons[1-5].As discussed in[6-8],EMFFs are crucial experimental observables of baryons,which are intimately related to their internal structure and dynamics.In the last few decades,there has been great progress in the study of baryon EMFFs,especially for nucleons,in both the timelike[9-20] and spacelike regions [21-31].For example,in [32]the nucleon form factors and the N-Δ(1232) transitions were theoretically investigated using the framework of a constituent quark model,where the effect of the π meson cloud is also considered.However,since hyperons cannot be targets,hyperon EMFFs in the spacelike region have hardly been experimentally measured.In the timelike region,the Beijing Spectrometer III(BESIII)collaboration has investigated the Σ hyperon EMFFs using the reactionand[4,33].The effective form factors|Geff|of Σ+and Σ-and the ratios of the Σ+electric and magnetic form factors |GE/GM| were obtained [4] from the high-precision Born cross-sections determined for the two abovementioned reactions for center-of-mass energies from 2.3864 to 3.02GeV.

Very recently,the EMFFs of hyperons have been investigated in the timelike region throughe+e-→(Y is the hyperon)reaction[5],where thefinal state interactions were taken into account,and the EMFFs in the timelike region were calculated for the Λ,Σ,and Ξ hyperons based on the one-photon exchange approximation of the elementary reaction mechanism.In [2],the timelike EMFFs of hyperons were accurately reproduced within a relativistic quark model.In addition,the Σ hyperon EMFFs have been calculated using lattice QCD [34],the light-cone sum rule (LCSR) [35],and chiral perturbative theory (ChPT) [36].In addition to these,the vector-meson dominance (VMD) model is a very successful approach for the study of nucleon electromagnetic form factors,in both the spacelike and timelike regions[37-39].Within a modified VMD model,in[1]the EMFFs of the Λ hyperon have been studied in the timelike region using thereaction,where the contributions from φ and ω mesons and their excited states were included.It was found that the VMD model was able to simultaneously describe the effective form factor and also the electromagnetic form factor ratio of the Λ hyperon.

In this work,based on the recent measurements of theandreactions,we aim to

2.Theoretical formalism

In this paper,we study the EMFFs of Σ+and Σ-using the VMD model.As in[1],we first introduce the electromagnetic current of the Σ hyperon with a half spin in terms of the Dirac form factor F1(Q2) and the Pauli form factors F2(Q2) as

where F1and F2are functions of the squared momentum transfer Q2=-q2.In the VMD model,the Dirac and Pauli form factors are parameterized into two parts.One is the intrinsic three-quark structure described by the form factor g(Q2) [39],the other is the meson cloud,which is used to describe the interaction between a bare baryon and a photon through the intermediate isovector ρ meson and the isoscalar ω and φ mesons [37].Following [37-39],F1and F2can be decomposed intowhereandare the isoscalar and isovector components of the form factors,respectively.The Dirac and Pauli form factors of Σ+and Σare then easily obtained without considering the total decay widths of the vector mesons,as follows:

where we take the intrinsic form factor g(Q2)as a dipole form

This form is consistent with g=1 at Q2=0,and the free model parameters γ1and γ2are determined by fitting them to the experimental data.

On the other hand,the observed electric and magnetic form factors,GEand GM,can be expressed in terms of the Dirac and Pauli form factors F1and F2by

In equations(2)-(9),βρ,βω,βφ,αφ,and αρrepresent the product of a vector-meson-photon coupling constant and a VΣΣ coupling constant.The VΣΣ coupling constants are obtained through an SU(3) flavor symmetry,as in [42],

where gBBV=gNNρ=3.20 and αBBV=1.15,as used in[42,43].We can now easily get gΣΣω=7.36,gΣΣφ=-5.88,gΣΣρ=7.36,while the tensor couplings are given by determine the parameters of the VMD model by fitting them to the experimental data of |Geff| of Σ+and Σ-.We have included the contributions of the ρ,ω and φ mesons.The ratios |GE/GM| of Σ+are then estimated with the model parameters,which are comparable to those obtained using other theoretical calculations.

This article is organized as follows:the formalism of the Σ hyperon form factors in the VMD model are shown in the following section.In section 3,we introduce effective form factors and the method that analytically continues the expressions of the form factors from the timelike region to the spacelike region.The numerical results of the timelike form factors of the Σ+and Σ-hyperons and the ratios|GE/GM|of Σ+and Σ-are presented in section 4,followed by a short summary in the final section.

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Table 1.Parameters used in this work.

where fNNρ=gNNρκρand κρ=6.1[42,43].Therefore,we can get fΣΣφ=13.80 and fΣΣρ=9.76.

In addition,we calculate the Vγ coupling constants following [44-46],

where αe=e2/(4π)=1/137 is the fine-structure constant andis the three momentum of the electron in the rest frame of the vector meson.is the partial decay width of the vector meson when it decays into a e+e-pair.With the experimental values,we now get 1/fρ=0.200,1/fω=0.059,and 1/fφ=0.075.

Finally,we obtain βρ,βω,βφ,αφand αρthrough

which are summarized in table 1.

We now explain how the large total width of the ρ meson contributions are implemented.4We do not consider the effects of the widths of ω and φ,since they are so small.For this purpose,one needs to replace [37]

where we let Γρ=149.1,and the average value of mπ=138.04 MeV [40].The function α(Q2) is given by

The timelike form factors can be obtained from the spacelike form factors by an appropriate analytic continuation.Within the above ingredients,g (q2) has the form required for an analytical continuation,

Figure 1.The solid curves represent the theoretical results for the |Geff|of the Σ+and Σ-using the fitted parameters.The experimental data for Σ+ and Σ- are taken from [4].

3.Numerical results and discussion

In the timelike region,the EMFFs of the hyperons Σ+and Σare experimentally studied via the electron-position annihilation processes.Under the one-photon exchange approximation,the total cross-section ofe+e-→,where Y is Σ+or Σ-,can be expressed in terms of the electric and magnetic form factors GEand GM,as described in [47]

In general,one can easily obtain the effective form factor Geff(q2) from the total cross-section of the e+e-annihilation process.The effective form factor,Geff(q2),is defined as

We then perform a χ2fit to the experimental data of the effective form factor |Geff| of Σ+and Σ-taken from [4].In the fitting,we only have one free parameter:γ1for Σ+,and γ2for Σ-.The fitted parameters are γ1=0.46±0.01 GeV-2and γ2=1.18±0.13 GeV-2,with χ2/dof=2.0 and 1.1,respectively.The corresponding best-fitting results for the effective form factor|Geff|of Σ+and Σ-in the energy range 2.3864GeV ＜＜2.9884 GeV are shown in figure 1 using solid curves.We also show the theoretical band obtained from the above uncertainties of the fitted parameters.The numerical results show that we can give a good description for the experimental data.

Figure 2.Theoretical results for the ratio of ∣Ge ff∣ Σ + ∣Geff ∣Σcompared with the experimental data taken from [4].

From the fitted results of the effective form factors,we can easily obtain the values ofwhich are shown in figure 2.One can see that the ratio is about three in the energy region of 2.4 ＜＜3.0 GeV.The value of three is just the ratio of the incoherent sum of the squared charges of the Σ+and Σ-valence quarks.

Through the vector-meson dominance model,using the fitted parameter γ,one can also easily calculate the ratio of|GE| and|GM|.This ratio is determined to be one at the mass threshold of a baryon-anti-baryon pair,due to the kinematical restriction.We show our theoretical calculations in figure 3,where the results are obtained with γ1=0.46 and γ2=1.18.It is found that when the reaction energyincreases,the ratio of |GE| and |GM| for Σ+slowly decreases,while for the case of Σ-,it is almost flat.Our results here do not adequately explain the experimental data,which show that the ratio is larger than one within uncertainties close to the threshold.This may indicate that there should be also other contributions in that energy region.For example,the electromagnetic form factors should be significantly influenced by the interaction in the finalsystem[5].However,since the experimental and empirical information about thefinal state interaction is so limited,we leave those contributions to further study when more precise data are available.

We next pay attention to the EMFFs in the spacelike region,which can be straightforwardly calculated with the parameter γ determined by the experimental measurements in the timelike region.Since the parameterization forms shown in equations (2)-(9) are valid in the low Q2regime,we calculate GEand GMbelow Q2=3 GeV2,and compare our numerical results with other calculations.

Figure 3.The results for the ratio of|GE/GM|of the Σ+and Σ-.The data are for Σ+ and taken from [4].

Figure 4.The results of the magnetic form factors of Σ+ and Σ-.The blue dotted line is the result of the lattice QCD calculations[34].The red dashed curves are the results of LCSR calculations [35].

Figure 5.Results of the electric form factor GE of the Σ+ and Σ-.The blue dotted line is the result of lattice QCD[34].The red dashed curves are the results of LCSR [35].The purple dot-and-dashed curves are the result of chiral perturbation theory [36].

The numerical results for the GMand GEobtained with γ1=0.46 and γ2=1.18 are shown in figures 4 and 5,respectively.5To compare our estimations with other calculations,we convert the unit of our results into nucleon magnetons.In figure 4,the predictions of the light-cone sum rules [35] and the lattice QCD calculations [34] are also shown for comparison.Our results for the magnetic form factors of Σ+and Σ-are somewhat quantitatively different from those obtained using other theories.Our results for the magnetic form factor of Σ-are more consistent with other calculations,while for the case of the Σ-electric form factor in figure 5,our results are in disagreement with the ChPT and LCSR calculations.However,our results for the Σ+are closer to the lattice QCD results described in [34] and the ChPT results in the very low Q2region.It is expected that these theoretical calculations can be tested by future experiments on the EMFFs of the Σ+and Σ-hyperons and will thus provide new insights into the complex internal structure of the baryons.

4.Summary

In this work,we have investigated the electromagnetic form factors of the hyperons Σ+and Σ-within the vector-meson dominance model.The contributions of the ρ,ω and φ mesons were taken into account.The model parameters,γ1and γ2,were determined using the BESIII experimental data for the timelike effective form factors|Geff|of Σ+and Σ-.It was found that the experimental data were able to be reproduced well using only one model parameter.We analytically continued the electromagnetic form factors into the spacelike region and evaluated the spacelike form factors of Σ+and Σ-.The electromagnetic form factors obtained for the Σ+and Σ-and their ratio were qualitatively comparable with those of other model calculations,but slightly different quantitatively.Finally,we would like to stress that the estimations of the Σ form factors in this work,the Λ form factors in[1]and the proton form factors in[37-39]indicate that the vector-meson dominance model is valid for the study of the baryonic electromagnetic form factors.Accurate data for the e+e-→baryon + anti-baryon reaction can be used to improve our knowledge of baryon form factors,which are,at present,poorly known.

Acknowledgments

We thank the anonymous referee for critical comments and suggestions,which were valuable in improving the presentation of the present work.This work is partly supported by the National Natural Science Foundation of China under Grant Nos.12 075 288,11 735 003,and 11 961 141 012.