F eb 2 01 9 The electromagnetic form factors of Λ hyperon in the vector meson dominance model

We perform an analysis on the electromagnetic form factors of theΛ hyperon in the time-like reaction e+e− → ΛΛ̄ by using a modified vector meson dominance model. We consider both the intrinsic structure components and the meson clouds components. For the latter one, we not only include the contributions from the φ and ω mesons, but also take into account the contributions from the resonance states ω(1420), ω(1650), φ(1680) and φ(2170). We extract the model parameters by combined fit to the time-like effective form factor |Geff |, the electromagnetic form factor ratio |GE/GM | and the relative phase ∆Φ of the Λ hyperon from the BaBar and BESIII Collaborations. We find that the vector meson dominance model can simultaneously describe these observables. Particularly, the inclusion of the resonance states in the model is necessary for explaining the ratio |GE/GM | in a wide range of √ s as well as the large phase angle. With the fitted parameters, we predict the single and double polarization observables, which could be measured in polarized annihilation reactions. Moreover, we analytically continue the expression of the form factors to space-like region and estimate the space-like form factors of Λ hyperon.

In recent years, the EMFFs of charmed hyperon Λ c attracts a lot of interest both theoretically [29][30][31][32] and experimentally [33][34][35][36][37], since the Λ c hyperon is the lightest baryon containing the charm quark. Similar to the Λ hyperon, the Λ c target is unfeasible and the EMFFs of Λ c can not be accessed from exclusive experiments in the space-like region [27,38]. On the other hand, the cross section of reaction e + e − → Λ + cΛ − c have been measured by the Belle and the BESIII Collaborations [34,36]. The ratio between the electric form factor G E and the magnetic form factor G M near threshold region is also available [36]. These measurements provide great opportunity to study the dynamics on the production of charmed baryon pairs and the time-like EMFFs of Λ c . Furthermore, by analytic continuation, the knowledge of EMFFs in the time-like region could be extend to study the EMFFs in the space-like region.
The vector meson dominance (VMD) model has been rec-ognized as a reliable theoretical approach in the study of the space-like electromagnetic form factors of hadrons. It can describe the existing data of proton and neutron EMFFs in the space-like region quite well. The approach was also extended to investigate the EMFFs of the Λ hyperon [27] in the timelike region. In the VMD model, the electromagnetic form factors receive contributions from two parts. One is the intrinsic structure defined by the valence quarks, the other is the contribution form the meson clouds in terms of vector mesons. Due to the isoscalar property of Λ, the contribution of ρ meson and its resonances should excluded. In order to introduce a complex structure of EMFFs in time-like region, the decay widths of the vector mesons and their resonance states are taken into account [27,28,39]. Particularly, the contributions from the resonance states below the threshold of ΛΛ pair are involved. The study shows that the inclusion of these resonance states are essential to simultaneously describe the experiment data of the effective form factors, ratio |G E /G M | and relative phase ∆Φ for Λ hyperon in a wide range of √ s. Encouraged by the success on the nucleon and Λ hyperon, in this work, we extend the VMD model to explore the EMFFs of the Λ c hyperon. There is some difference between the VMD model for the Λ hyperon and that for the Λ c hyperon. Firstly, the production of the Λ + cΛ − c pair are related to cc pair, which has the quantum numbers Beside ω, φ and their resonance states, the J/Ψ and their resonance states below the threshold should be included. Thus, we take into account the contributions from all the resonances states below the threshold: ω(782), ω(1420), ω(1650), φ(1020), φ(1680) and φ(2170), as well as ψ(1S ), ψ(2S ), ψ(3770), ψ(4040), ψ(4160) and ψ(4415). Thirdly, the Coulomb final-state interactions should be considered, which is similar to the case of the proton [28,40]. Based on the above consideration, we can obtain the formula of the time-like form factors for Λ c by analytic continuation of the space-like form factors.
The remained content of the paper is organized as follows. In Section II, we present a detailed framework on the form factors of Λ c hyperon in the VMD model. In Section III, we fit theoretical expressions for G E and G M the to experimental data of reaction e + e − → Λ + cΛ − c from the Belle and the BESIII collaborations. We also provide our predictions for the single and double polarization observables, the relative phase angle ∆Φ, as well as space-like form factors of Λ c . We summarize the paper in Section IV.

II. FORM FACTORS OF Λ c HYPERON IN THE VMD MODEL
The process e + e − → Λ + cΛ − c which we study in the framework of the VMD model is shown in Fig. 1. That is, the photons formed in e + e − annihilation are first transformed into neutral vector mesons, the latter ones then decay into Λ + cΛ − c pairs through some specified couplings. Since one-photon exchange dominates the production of spin −1/2 baryons B, the Born cross section of the process e + e − → BB can be parameterized [3] in terms of EMFFs. Generally, the integrated cross section of the Λ c hyperon pairs production can be given in the following way: Here, G E (s) and G M (s) are the electric form factor and the magnetic form factor in the time-like region, respectively, α is the fine-structure constant, s the square of the center of mass (c.m.) energy, τ = s/4M 2 Λ c , and β = √ 1 − 1/τ is the velocity of the Λ c hyperon. The Coulomb factor C Λ c = ε R parameterizes the electromagnetic interaction between the outgoing baryon and antibaryon, with ε = πα/β an enhancement factor resulting in a nonzero cross section at threshold and R = 1/(1 − e −πα/β ) the Sommerfeld resummation factor [36,41].
In the space-like region, the EMFFs G E (Q 2 ) and G M (Q 2 ) of Λ c hyperon can be expressed as where τ = Q 2 /4M 2 Λ c , and the F 1 (Q 2 ) and F 2 (Q 2 ) are the Dirac form factor and Pauli form factor respectively, which can be decomposed into where F S i and F V i denote the isoscalar and isovector components of the form factors, respectively. Since Λ c hyperon is an isospin singlet, the contribution from the isovector part F V i should be excluded, which is similar to the case of the Λ hyperon. We note that the kinematic constraint [42]. Previously, the VMD model have been widely used to study the EMFFs of the nucleon and the Λ hyperon, showing that it has the advantage to well describe the experimental data in the space-like and time-like region [15,27,[43][44][45][46]. Very recently, it has also been applied to investigate the EMFFs for Σ + and Σ − [25]. Encouraged by its success, we extend the model to study the EMFFs of the Λ c hyperon. In the VMD model, two parts contributes to the Dirac form factor, one is the the intrinsic structure, the other is the vector meson clouds; while the Pauli form factor only receives the contribution from the meson cloud [45]. Due to the isoscalar property of the Λ c hyperon, we consider the contributions of vector mesons ω, φ and their resonance states. Moreover, as Λ c is charmed hyperon, so we should also consider the contribution of charmed mesons J/ψ and their resonance states. Thus, in our modified model, the contributing meson resonance states below the threshold of Λ + cΛ − c are ω(782), ω(1420), ω(1650), φ(1020), φ(1680), φ(2170), and ψ(1S ), ψ(2S ), ψ(3770), ψ(4040), ψ(4160), ψ(4415), and we assume that the expression of the form factors from the ω and φ resonance states have the same form as those from the vector mesons ω(782) and φ(1020) [47]. As for J/ψ and its resonance states, we compare them to ω(782) and φ(1020) in light of the magnitude of the mass of the charmed states, and put them into Ω and Φ. At Q 2 = 0, the EMFFs of Λ c hyperon can be normalized as follow, where the magneton µ Λ c = 0.48μ N is predicted in Ref. [29]. One should note that result of the magnetic moment of the Λ c hyperon is given in unit of nucleon magneton. Thus the magnetic moments with units of the Λ c hyperon natural magneton can be expressed as [27]. Taking into account all the above constraints, we can write the parameterized forms of scalar parts of the Dirac and Pauli form factors in VMD model as follows: where N = 6. Ω i (i = 1, 2, 3, 4, 5, 6) denotes the vector meson states ω(782), ω(1420), ω(1650), ψ(1S ), ψ(3770), and ψ(4160), Φ i (i = 1, 2, 3, 4, 5, 6) represents the vector meson states φ(1020), φ(1680), φ(2170) and ψ(2S ), ψ(4040), ψ(4415). The intrinsic structure factor is a characteristic of valence quark structure and is chosen in a dipole form g(Q 2 ) = (1 + γQ 2 ) −2 , which is consistent with pQCD and fits the EMFFs of nucleon well [43,44,47]. In the large Q region, the forms also satisfy the constraints of the asymptotic behavior, F 1 ∼ 1/Q 4 and F 2 ∼ 1/Q 6 . Furthermore, the coefficients β Ω i , β Φ i , α Φ i can be naturally interpreted as the products of a Vγ coupling constant and a V BB coupling constant [47], respectively. The parameter γ in g(Q 2 ) and the coefficients (6) are free parameters the values of which can be obtained be fitting to the data of EMFFs. By proper analytic continuation on the complex plane, we can obtain the form factors in the time-like region on the basis of the form factors in space-like region [11,15]. The analytic continuation in the time-like region is based on the following relation [11]: Therefore, in the time-like region, the intrinsic structure g(q 2 ) has an analytical continuation form: where γ is a parameter larger than zero. Thus, there is a pole in g(q 2 ) in the position q 2 = 1/γ. There are two methods to remove the pole, one is to change the relations in Eq. (7) with Q 2 → q 2 e iθ (θ π) [15,42], the other is to impose the constraint γ > 1/(4m 2 Λ c ) for the Λ c form factors [27]. In this work, we will choose the latter one. For the contribution of the meson cloud to the form factor, we take into account the widths of the vector mesons ω, φ, J/ψ and their resonance states in order to introduce the complex structure of the EMFFs in the time-like region [27,28]. This leads to the following replacement for Eq. (7) β In this way we obtain the modified VMD model in the timelike region for Λ c hyperon. These terms are crucial for constructing the complex structure and reproducing the relative phase angle of the time-like EMFFs of Λ c .

A. Fit the time-like form factors
We fit the expressions of the form factors in Eqs. (5)-(6) and the replacement in Eq. (9) to the experimental data of Born scattering cross section and EMFFs ratio measured by the Belle [34] and BESIII [36] Collaborations. The data are in the range 4.59 GeV < √ s < 5.39 GeV. The masses and widths of the isoscalar vector mesons used in the fit are taken from Table I Table II.
It should be noted that the value of intrinsic parameter γ in our model is fitted to be 0.0601 GeV −2 , corresponding q = 4.049 GeV. Thus the poles of the intrinsic structure are restricted in the unphysical region and satisfied the constraint γ > 1/(4m 2 Λ c ) in this scenario. Since the poles are below the threshold and we focus on the region above the Λ + cΛ − c threshold, we can ignore the effect of the pole in the first place.
In Fig. 2, we show our fit (solid line) to the Born cross sections in reaction e + e − → Λ + cΛ − c measured by the Belle (filled square) and BESIII Collaboration (filled circle). The vertical lines depict the error bars of the data. We also provide the theoretical band corresponds corresponding to the uncertainty of parameters obtained from the errors of the data. The Belle data cover the region 4.58GeV < √ s < 5.4GeV, while the BESIII data concentrate in the range 4.57GeV < √ s < 4.6GeV. The comparison shows that the modified VMD model can describe the Belle and BESIII data after the theoretical error band is included. The enhancement effect near the threshold of the Λ + c Λ − c pair due to the contribution of the resonance states of the vector mesons is also observed.
In Fig. 3, we present the model result of the ratio |G E /G M | (solid line) and compare it with the BESIII data (filled circle) which are near the threshold. Again, the band corresponds to the uncertainty of parameters. It is shown that the VMD FIG. 3: The same as Fig. 2 but for the ratio G E /G M . The experimental data denote by the circles are from BESIII [36] Collaboration. model can qualitatively describe the ratio. It is worth noting that, according to the kinematic constraint, this ratio is equal to 1 at the threshold, which is an important constraint for the form factors in the time-like region. Furthermore, the ratio increases with increasing √ s in the near threshold region and reaches the maximum value 1.6 at around √ s = 4.7 GeV. This trend is also consistent with the BESIII data. In the larger √ s region, the ratio decreases with increasing √ s. These features are similar to those of the Λ hyperon [27,49]. Due the VMD model, the asymptotic behaviors of form factors, the ratio of EMFFs satisfy a fact that the result tends to be constant in the limit of q 2 → ∞ [27].
As G E and G M in the time-like region are complex, there is a relative phase ∆Φ between the two EMFFs. The measurement of this phase at different √ s could provide additional information of EMFFs which can not been revealed by |G E | and |G M |. Using the values of the parameter extracted from the Belle and BESIII data, we predict the relative phase of the EMFFs of the Λ c as function of √ s, as shown in Fig. 4. One should be note that ∆Φ = 0 at the Λ + cΛ − c threshold due to G E = G M at s = 4M 2 Λ c . our numerical result also satisfies the constraint that the phase goes to zero as s → ∞. be expressed in terms of the EMFFs: [51,52] A y = −2M Λ √ s sin(2θ) Im(G M G * E ) D c − D s sin 2 (θ) , , θ is the scattering angle defined in the c.m.frame, and D c = 2s|G M | 2 , D s = s|G M | 2 − 4M 2 |G E | 2 . In Fig. 5, we present our numerical results of the single polarization observable A y vs √ s, which is dependent on the the imaginary of product of G M G * E . As a demonstration, in the calculation we fixed the scattering angle θ = 45 • The prediction shows that the shape of A y is similar to that of the relative phase ∆Φ in Fig. 4, since Im(G M G * E ) is proportional to ∼ sin(∆Φ). This indicates that exact information of ∆Φ could be obtained from the precise measurement of the single spin polarization A y . In addition, we plot the double polarization observables A xz , A xx , A yy and A zz vs √ s in Fig. 6. It is found that in the near threshold region, the polarization observables changes drastically with √ s, while in the large √ s region, the double polarization observables almost remain unchanged. The shape and the √ sdependence of the polarization observables should be sensitive to the model assumptions on EMFFs. Therefore, precise measurements on these observables will be useful for examining the validity of our model.

C. Form factors in space-like region
The EMFFs of Λ c in the space-like region can be directly calculated using Eqs. (5), (6) and the model parameters in Table II. We perform the numerical calculation on the space-like G M and G E of Λ c vs Q 2 and present the results in the left panel of Fig. 7, which shows that the magnitude and the shape of G E are similar to those of G M . A more clear picture about the relative size of G E and G M can be revealed by the ratio µ Λ c G E /G M , as dipicted in the right panel of Fig. 7. It is found that in the region Q 2 < 1 GeV the ratio can be slightly larger than 1. In the region Q 2 > 1 GeV the ratio decreases with increasing Q 2 , which is similar to the case of the proton [11,53]. However, it is larger than the ratio of the proton EMFFs.
Finally, using the space-like G E (Q 2 ) and G M (Q 2 ) of Λ c , we estimate the magnetic and charge radius of the Λ c defined by and obtain the following results: Our result for r 2 E is less than the result r 2 E ∼ 0.7 fm 2 in Ref. [31] and the relativistic quark model result r 2 E = 0.5 fm 2 [54], but is larger than r 2 E = 0.117 fm 2 from the heavy quark effective theory [55].

IV. SUMMARY
In this work, we have investigated both the time-like and space-like EMFFs of Λ c using a modified VMD model. In this model, the EMFFs are contributed by two parts. One is the intrinsic structure part, the other is meson clouds part. Similar to the Λ hyperon, the contributions from the isovector components to the Dirac and Pauli form factors vanish due to the isoscalar property of the Λ c hyperon. We have taken into account all the isospin-singlet vector mesons below the Λ + c to the data from the Belle and BE-SIII experiments. We have also included the ratio |G E /G M | measured by BESIII. We find that the modified VMD model can describe the Born cross section of e + e − → Λ + c Λ − c at Belle and BESIII simultaneously. The enhancement effect near the threshold of the Λ + cΛ − c pair is reproduced due to the contribution of the resonance states of the vector mesons. It is also shown that the VMD model can qualitatively describe the ratio. We have presented the numerical results of the relative phase ∆Φ, and predicted the single and double polarization observables in the process e + e − → Λ + cΛ − c . The measurement of these quantities could be used to verify the validity of the model. Finally, we have extended the time-like EMFFs to space-like region using the parameter values obtained from the fit. The numerical results show that the magnitudes and shapes of the G E and G M is rather similar. Furthermore, we obtain that the electric radius squared r 2 E and magnetic radius squared r 2 M of Λ c hyperon are 0.259 fm 2 and 0.297 fm 2 , respectively.