Electromagnetic form factors of $\Lambda$ hyperon in the vector meson dominance model and a possible explanation of the near-threshold enhancement of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction

Electromagnetic form factors (EMFFs) is an important tool for studying the electromagnetic structure of hadrons Pacetti:2014jai . The measurements of space-like region EMFFs of proton can be done in elastic as well as inelastic $ep$ scattering JeffersonLabHallA:2001qqe . While for $\Lambda$ hyperon, its EMFFs at the space-like region are very hardly been experimentally measured. Instead, the electron-positron annihilation process, $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ allows to study the $\Lambda$ hyperon EMFFS at the time-like region BaBar:2007fsu ; BESIII:2017hyw ; BESIII:2019nep ; Haidenbauer:1991kt ; Baldini:2007qg ; Faldt:2016qee ; Haidenbauer:2016won ; Faldt:2017kgy ; Yang:2017hao . In addition, the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction can be used to study vector mesons with light quark flavors and mass above 2 GeV Bystritskiy:2021frx ; Wang:2021gle , especially for the $\phi$ excitations Xiao:2019qhl ; Cao:2018kos .

Recently, the BESIII collaboration measured the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction with much improved precision. The Born cross-section at the center of mass energy $\sqrt{s}$=2.2324 GeV is determined to be $305\pm 45^{+66}_{-36}$ pb BESIII:2017hyw . This indicates there is an evident threshold enhancement for the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction. The observed value is larger than the previous theoretical predictions, which predicted that the total cross section of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction should be close to zero near the reaction threshold. In fact, before the new measurements of Ref. BESIII:2017hyw , there exist several theoretical studies of this reaction, which proposed the final state interactions Baldini:2007qg ; Haidenbauer:2016won to explain the unexpected features of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ cross sections near threshold.

After the observations of Ref. BESIII:2017hyw by the BESIII collaboration, the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction was investigated in Ref. Cao:2018kos , where it was found that the $\phi(2170)$ is responsible for the threshold enhancement. In Ref. Yang:2019mzq , by using a modified vector meson dominance (VMD) model, an analysis on the EMFFs of $\Lambda$ hyperon and also the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction was performed, where those contributions from $\phi$, $\omega$, $\omega(1420)$, $\omega(1650)$, $\phi(1680)$, and $\phi(2170)$ were taken into account. In Refs. Haidenbauer:2016won ; Haidenbauer:2020wyp , with the role played by the final state interactions of the baryon-anti-baryon pairs, the EMFFs of hyperons ($\Lambda$, $\Sigma$, and $\Xi$) were studied in the timelike region. The threshold enhancement of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction was also investigated in Refs. Haidenbauer:2016won ; Yang:2019mzq ; Haidenbauer:2020wyp from the theoretical side. However, a large finite experimental value on the total cross section of $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction at $\sqrt{s}$=2.2324 GeV cannot be well reproduced Haidenbauer:2016won ; Yang:2019mzq ; Haidenbauer:2020wyp . Further investigations about the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction are mostly welcome.

On the other hand the tails of vectors below threshold have to be detected as large effects in the time-like form factors of $\Lambda$ hyperon and possibly as small structures in $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction near threshold. Indeed, as discussed in Ref. Cao:2018kos , the $\phi(2170)$ plays an important role to reproduce the threshold enhancement. However, the width of $\phi(2170)$, $\Gamma_{\phi(2170)}=165\pm 65$ MeV ParticleDataGroup:2020ssz , is too wide, thus it will affect a large energy region. Besides, with the Godfrey-Isgur model, a narrow $\Lambda\bar{\Lambda}$ bound state with quantum numbers $J^{PC}=1^{--}$ and mass around $2232$ MeV was predicted Xiao:2019qhl , and it has significant couplings to both the $\Lambda\bar{\Lambda}$ and $e^{+}e^{-}$ channels. While in Refs. Zhao:2013ffn ; Zhu:2019ibc , within the one-boson-exchange potential model, a $\Lambda\bar{\Lambda}$ bound state can be also obtained. This narrow $\Lambda\bar{\Lambda}$ bound state, if really existed, will contribute to the threshold enhancement of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction and also the EMFFs of the $\Lambda$ hyperon in the time like region.

In this work we take the achievement of the vector meson dominance model and predictions of the narrow $\Lambda\bar{\Lambda}$ bound state as motivation to explore the electromagnetic form factors of the $\Lambda$ hyperon in the time like region. The EMFFs of baryons have been studied with the VMD model for the proton Iachello:1972nu ; Iachello:2004aq ; Bijker:2004yu ; Bijker:2006id , $\Lambda$ hyperon Yang:2019mzq , $\Sigma$ hyperon Li:2020lsb , and charmed $\Lambda^{+}_{c}$ baryon Wan:2021ncg . Following Ref. Yang:2019mzq , we revisit the EMFFs of $\Lambda$ hyperon and the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction near threshold by using the modified VMD model. In addition to the contributions from the ground $\omega$ and $\phi$ mesons, we consider also a new narrow vector meson with mass around $2232$ MeV, as predicted in Ref. Xiao:2019qhl . Yet, the $\phi(2170)$ resonance is not taken into account in the present work, ²²2The contributions from $\omega(1420)$, $\omega(1650)$, and $\phi(1680)$ resonances are not considered either, since their masses are far from the reaction threshold of $e^{+}e^{-}\to\Lambda\bar{\Lambda}$, and their contributions could be absorbed into the ground $\omega$ and $\phi$ mesons. Besides, we refrain from including such contributions in this work because the model already contains a large number of free parameters. since the experimental information of it is still diverse, and the measured mass and width of $\phi(2170)$ resonance are controversial BESIII:2018ldc ; BESIII:2019ebn ; BESIII:2020gnc ; BESIII:2020kpr ; BESIII:2020vtu ; BESIII:2020xmw ; Huang:2020ocn ; BESIII:2021bjn ; BESIII:2021yam . Indeed, there have also been different theoretical explanations for $\phi(2170)$ resonance Page:1998gz ; Barnes:2002mu ; Ding:2006ya ; Wang:2006ri ; Ding:2007pc ; Chen:2008ej ; MartinezTorres:2008gy ; Coito:2009na ; Ali:2011qi ; Dong:2017rmg ; Ke:2018evd ; Agaev:2019coa ; Li:2020xzs ; Malabarba:2020grf ; Zhao:2019syt .

This article is organized as follows: the theoretical formalism of the $\Lambda$ hyperon in the VMD model are shown in the following section. In Sec. III, we present our numerical results and discussions of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction. A short summary is given in the last section.

In this section, we will briefly review the vector meson dominance model to study the electromagnetic form factors of baryons with spin-$1/2$, and the total cross sections of $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction and the effective form factor of $\Lambda$.

In this work, following Ref. Iachello:2004aq , we will consider only the $\beta_{x}$ term in the Dirac form factor $F_{1}$, and $\alpha_{x}$ is taken as zero. ⁴⁴4In fact, we found that only the sum of the two parameters $\beta_{x}$ and $\alpha_{x}$ can be determined by the $\Lambda$ effective form factor in the energy region of the mass threshold of $\Lambda\bar{\Lambda}$, thus we keep only the $\beta_{x}$ term, that is $\alpha_{x}=0$, since $\alpha_{x}$ is associated to the tensor coupling in Pauli form factor $F_{2}$. Then, we perform seven-parameter ($\gamma$, $\beta_{\omega}$, $\beta_{\phi}$, $\beta_{x}$, $\alpha_{\phi}$, $m_{x}$ and $\Gamma_{x}$) $\chi^{2}$ fits to the experimental data on the effective form factor $G_{\rm eff}$ of $\Lambda$ hyperon and the form factor ratio $R=|G_{E}/G_{M}|$. There are a total of 18 data points. These data correspond to the center of mass energy $\sqrt{s}$ ranging from 3.08 down to 2.2324 GeV. The fitted parameters are compiled in Table 1, with a reasonably small $\chi^{2}/{\rm dof}=0.9$.

In the present work, since both the $\omega$ and $\phi$ are far from the mass threshold of $\Lambda\bar{\Lambda}$, the behaviour of the contributions from them are similar, we have performed a new fit, which only the $\omega$ term was considered. Thus, we have assumed that $\beta_{\phi}=\alpha_{\phi}=0$ and $\beta_{\omega\phi}=\beta_{\omega}$, $m_{\omega}\to(m_{\omega}+m_{\phi})/2$ and $\Gamma_{\omega}\to(\Gamma_{\omega}+\Gamma_{\phi})/2$. The fitted parameters are shown in brackets in Table 1. In the case, we get errors for the fitted model parameters, however, the errors obtained from the fit are large.

In Fig. 1 we depict effective form factor $G_{\rm eff}$ of the $\Lambda$ hyperon obtained with the fitted parameters given in Table 1 of the seven-parameter fit, as a function of $\sqrt{s}$. The experimental data points are taken from Refs. BaBar:2007fsu ; BESIII:2017hyw ; BESIII:2019nep . The red curve is the total contribution, while the blue dashed curve is the contributions from only $\omega$ and $\phi$, with $\beta_{x}=0$. One can see that the experimental data can be well described with the contribution from the new narrow state, especially for the first four data points close to reaction threshold.

Note that the only fifteen available data points about the effective form factor and three data points about the ratio of $R=|G_{E}/G_{M}|$ do not allow to obtain unique values for the model parameters, which we introduced for the Dirac and Pauli form factors. Above $\sqrt{s}=2.3$ GeV, the line shape of $G_{\rm eff}$ is trivial and there are many solutions to describe it. Thus, it is very difficult to get the parameter errors in the fit. In fact, one can also get a good fit to the effective form factor data except the first one with the following parametrized of $G_{\rm eff}$ Bianconi:2015owa ; BESIII:2021dfy :

The fitted parameters are $\gamma=0.33\pm 0.03$ ${\rm GeV}^{-2}$ and $C_{0}=0.10\pm 0.03$. The fitted $G_{\rm eff}$ are shown in Fig. 1 with the green-dash-dotted curve with central values of $\gamma$ and $C_{0}$. One can see that the experimental data can be well reproduced except the first point.

To get more precise information of $m_{x}$ and $\Gamma_{x}$ obtained from the seven-parameter fit, by fixing other parameters with their values as shown in Table 1, within the range of $m_{x}(1\pm 10\%)$ and $(0,2\Gamma_{x})$, we generate random sets of the fitted parameters $(m_{x},\Gamma_{x})$ with a Gaussian distribution. For each set of $(m_{x},\Gamma_{x})$, we perform a $\chi^{2}$ fit to the first four data points of the effective form factor. We collect these sets of the fitted parameters, such that the corresponding $\chi^{2}$ are below $\chi^{2}_{\rm min}+1$, where $\chi^{2}_{\rm min}$ is obtained with these parameters shown in Table 1. With these collected best fitted parameters, we obtain the errors of parameters $m_{x}$ and $\Gamma_{x}$, which are: $m_{x}=2230.9^{+3.4}_{-3.5}$ MeV, and $\Gamma_{x}=4.7^{+2.2}_{-4.7}$ MeV.

One might think that a Flatt${\rm\acute{e}}$ type Flatte:1976xu for $\Gamma_{x}$ might improve the fitting situation, since the mass of the new state is very close to the $\Lambda\bar{\Lambda}$ threshold, and it may also couples strongly the $\Lambda\bar{\Lambda}$ channel. The Flatt${\rm\acute{e}}$ form is useful for coupled-channel analysis, however, we currently have experimental information about only the $\Lambda\bar{\Lambda}$ channel. Yet, we have explored such a possibility. We take

where $\Gamma_{0}$ is a constant and it includes the contributions from the other channels, while $\Gamma_{\Lambda\bar{\Lambda}}(s)$ is the contribution from the $\Lambda\bar{\Lambda}$ channel. For example, with $s$-wave coupling Zou:2002yy for the new state with $J^{PC}=1^{--}$ to the $\Lambda\bar{\Lambda}$ channel, one can get ⁵⁵5In general, there should be also contributions from $d$-wave.:

where $g_{\Lambda\bar{\Lambda}}$ is the unknown $s$-wave coupling constant.

Then we have performed six-parameter ($\gamma$, $\beta_{\omega\phi}$, $\beta_{x}$, $m_{x}$, $\Gamma_{0}$ and $g_{\Lambda\bar{\Lambda}}$) $\chi^{2}$ fits. Indeed, we can also obtain a good fit. The fitted parameters are: $\gamma=0.57\pm 0.21$ ${\rm GeV}^{-2}$, $\beta_{\omega\phi}=-0.30\pm 0.31$, $\beta_{x}=-0.03\pm 0.09$, $m_{x}=2237.7\pm 50.2$ MeV, $\Gamma_{0}=8.8^{+75.9}_{-8.8}$ MeV, and $g_{\Lambda\bar{\Lambda}}=3.0\pm 1.9$. One can find that the fitted errors for the model parameters are very large. On the other hand, one can also look for poles for the Breit-Wigner function, $1/(s-m^{2}_{x}+im_{x}\Gamma_{x})$, which is parametrized by the Flatt${\rm\acute{e}}$ form on the complex plane of $\sqrt{s}$. With the above fitted central values of $m_{x}$, $\Gamma_{0}$ and $g_{\Lambda\bar{\Lambda}}$, we get a pole at $\sqrt{s}=Z_{R}=M_{R}-i\Gamma_{R}/2=(2096.2,-9.9)$ MeV.

As discussed before, only a few experimental data which are very close to and above the reaction threshold need the contribution of the new state. In fact, the Flatt${\rm\acute{e}}$ formulae would push down the Breit-Wigner mass, and there will be a clear drop at the $\Lambda\bar{\Lambda}$ threshold if the value of $g_{\Lambda\bar{\Lambda}}$ is large Xie:2013wfa . However, we donot have any information below the $\Lambda\bar{\Lambda}$ mass threshold, which means that the mass and width of the state still cannot be well determined if we choose the Flatt${\rm\acute{e}}$ formulae. Indeed, we can get good fits by including $\omega(1420)$, $\omega(1650)$, $\phi(1680)$, or the $\phi(2170)$ with a Flatt${\rm\acute{e}}$ form and taking their mass $m_{x}$ and width $\Gamma_{0}$ as quoted in PDG. On the other hand, it is worth to mention that, for very wide regions for these values of $m_{x}$ and $\Gamma_{0}$ one can always get a good fit by adjusting the value of $g_{\Lambda\bar{\Lambda}}$, this is because we donot have information below the $\Lambda\bar{\Lambda}$ mass threshold. In this work since the $\Lambda\bar{\Lambda}$ channel is opened in the considering energy region, we just use a constant total decay width for this new state, in such a way we can also reduce the number of free parameters. This work constitutes a first step in this direction.

Next, we pay attention to the total cross sections of $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction. In Fig. 2, the theoretical fitted results of the total cross sections of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction in the energy range from the reaction threshold to $\sqrt{s}=3.1$ GeV are shown and compared to experimental data taken from Refs. BESIII:2017hyw ; BESIII:2019nep . In this figure, the red solid line displays the theoretical fitted result with total contributions from $\omega$, $\phi$, and the new state $X(2231)$, while the blue dotted cure represents the results without the contribution from $X(2231)$ state. Again, one can see that the near threshold enhancement structure is well reproduced thanks to a significant contribution from a very narrow state $X(2231)$ with mass about 2231 MeV. The narrow peak of this state is clearly seen.

Finally, in Fig. 3 we show the form factor ratio $|G_{E}/G_{M}|$ obtained with the fitted parameters given in Table 1, as a function of $\sqrt{s}$. The experimental data points are taken from Refs. BESIII:2017hyw ; BESIII:2019nep . This ratio is determined to be one at the threshold due to the kinematical restriction, which can be easily obtained from Eqs. (2) and (3).

We find that a narrow vector meson, $X(2231)$, whose mass is close to the mass threshold of $\Lambda\bar{\Lambda}$, is needed to describe the threshold enhancement of $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction. However, its width cannot be well determined through the VMD model by fitting the current experimental data. This state could be a quasi-bound-state of $\Lambda\bar{\Lambda}$, which has significant couplings to the $\Lambda\bar{\Lambda}$ and $e^{+}e^{-}$ channels. In fact, from the analysis of the $p\bar{p}\to\Lambda\bar{\Lambda}$ reaction near threshold, the authors of Ref. Carbonell:1993dt predicted also a narrow $\Lambda\bar{\Lambda}$ subthreshold state with quantum numbers $J^{PC}=1^{--}$ and has a width of a few MeV. However, a later high-statistics measurement of the $p\bar{p}\to\Lambda\bar{\Lambda}$ reaction Barnes:2000be ruled out the existence of such a $\Lambda\bar{\Lambda}$ resonance as predicted in Ref. Carbonell:1993dt , since there is no structure in these new measurements of the $p\bar{p}\to\Lambda\bar{\Lambda}$ reaction near threshold, and the total cross section is observed to grow smoothly from threshold with a mix of $S$- and $P$-wave production.

The near threshold region of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction was also investigated with specific emphasis on the important role played by the $\Lambda\bar{\Lambda}$ final state interaction in Ref. Haidenbauer:2016won , where the $\Lambda\bar{\Lambda}$ potentials were constructed for the analysis of the $p\bar{p}\to\Lambda\bar{\Lambda}$ reaction. The total cross sections reported by the BaBar collaboration BaBar:2007fsu can be well reproduced, but, the new results from the BESIII collaboration BESIII:2017hyw is very difficult to be obtained by the theoretical calculations of Ref. Haidenbauer:2016won . As discussed above, there is no structure in the $p\bar{p}\to\Lambda\bar{\Lambda}$ reaction, and the BESIII results indicate that we do need include such a narrow state which has significant coupling to the $\Lambda\bar{\Lambda}$ channel. ⁶⁶6To explain the new BESIII results, a very narrow resonance with mass around the $\Lambda\bar{\Lambda}$ was also discussed in Ref. Haidenbauer:2016won . Yet, such a narrow state may couple weakly to the $p\bar{p}$ channel, thus it was not appear in the $p\bar{p}\to\Lambda\bar{\Lambda}$ reaction Barnes:2000be .

Moreover, it was found that in the processes of $e^{+}e^{-}\to K^{+}K^{-}K^{+}K^{-}$ and $e^{+}e^{-}\to\phi K^{+}K^{-}$, the cross sections are unusually large at $\rm\sqrt{s}=2.2324$ GeV, which indicates that there should be contributions from a narrow state with mass about $2232.4$ MeV BESIII:2019ebn . This state could be the vector meson $X(2231)$ that we proposed here. On the other hand, in the charmed sector, the $Y(4630)$ and $Y(4660)$ have been studied in the $e^{+}e^{-}\to\Lambda_{c}\bar{\Lambda}_{c}$ reaction by taking into account also the $\Lambda_{c}\bar{\Lambda}_{c}$ final state interaction Guo:2010tk ; Lee:2011rka ; Cao:2019wwt ; Dong:2021juy .

Finally, one knows that $G_{E}$ and $G_{M}$ are complex in the time like region, and there is a relative phase angle $\Delta\Phi$ between these two electromagnetic form factors. In addition to the ratio of $|G_{E}/G_{M}|$, a rather large phase $\Delta\Phi=37^{\circ}\pm 12^{\circ}\pm 6^{\circ}$ was also obtained at $\sqrt{s}=2.396$ GeV by the BESIII collaboration BESIII:2019nep . Because the fitted width of $X(2231)$ is so narrow, we cannot reproduce this large phase at $\sqrt{s}=2.396$ GeV. The large phase will be described by considering these vector mesons with higher masses around $2.3\sim 2.4$ GeV and wide widths as predicted in Refs. Wang:2021gle ; Cao:2018kos . Indeed, a broad vector meson with mass of around $2.34$ GeV was introduced to explain the energy dependent behavior of the cross sections of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction above threshold. Clearly, a further improved investigations needs to consider the contributions from these higher mass resonances. But, including such contributions, the electromagnetic form factors of $\Lambda$ hyperon would become more complex due to additional parameters from the vector meson dominance model, and we cannot determine or constrain these parameters. In the present work, we focus on the near threshold enhancement of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction. Thus, we will leave these contributions to future studies when more precise experimental data become available.

In this work, we have studied the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction near threshold and the electromagnetic form factors of the $\Lambda$ hyperon within the modified vector meson dominance model. In addition to these contributions from ground $\omega$ and $\phi$ meson, we introduce also a new narrow vector meson $X(2231)$ with mass around the mass threshold of $\Lambda\bar{\Lambda}$, and its width is about few MeV. It is found that we can describe the effective form factor $\rm G_{eff}$ and the electromagnetic form factor ratio $|G_{E}/G_{M}|$ of $\Lambda$ hyperon quite well. Especially, the threshold enhancement of the total cross sections of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction at $\sqrt{s}$=2.2324 GeV can be well reproduced. This narrow state could be a $\Lambda\bar{\Lambda}$ quasi-bound-state with quantum numbers $J^{PC}=1^{--}$. Further data in the very close to threshold region with better mass resolution would be very useful to confirm this narrow resonance.

On the other hand, if one take a Flatt${\rm\acute{e}}$ form for the total decay width of $\omega(1420)$, $\omega(1650)$, $\phi(1680)$, and $\phi(2170)$, the experimental data can be also well reproduced with a strong coupling of these resonance to the $\Lambda\bar{\Lambda}$ channel.

The proposed formalism and conclusion here would give insight into the electromagnetic form factors of the $\Lambda$ hyperon and the near threshold enhancement of the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ reaction. The proposed formalism attribute the $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ non-vanishing cross sections near threshold to the contribution of a new narrow vector meson $X(2231)$, which could be the peak structure seen in the $e^{+}e^{-}\to K^{+}K^{-}K^{+}K^{-}$ and $e^{+}e^{-}\to\phi K^{+}K^{-}$ reactions at $\sqrt{s}=2.2324$ GeV. It is expected that this conclusion can be distinguished and may be tested by the future experiments with improved precision at BESIII or the planned Super tau-charm Facility at China Shi:2020nrf ; Sang:2020ksa ; Fan:2021mwp .