Vector mesons and electromagnetic form factor of the $\Lambda$ hyperon

The measured electromagnetic form factors of $\Lambda$ hyperon in the time-like region are significantly deviated from pQCD prediction. We attribute the non-vanishing cross section near threshold to be the contribution of below-threshold $\phi$(2170) state, supporting its exotic structure. Above the threshold, we find significant role of a wide vector meson with the mass of around 2.34 GeV, which would be the same state present in $p\bar{p}$ annihilation reactions. As a result, we give a satisfactory description of the behavior of existing data without modifying pQCD expectation.


I. INTRODUCTION
The electromagnetic form factors (EMFFs) are essential probe of electromagnetic structure of bound states and can deepen our understanding of perturbative and non-perturbative quantum chromodynamics (QCD) effects encoded in hadrons. The nucleon EMFFs have been extensively explored from both experimental and theoretical sides for more than sixty years (see reviews [1,2] and references therein). The general behavior of available data [3][4][5][6] in the time-like region tends to be consistent with naive quark counting rules and the pQCD prediction at large-q 2 [7,8]. It also agrees with the simple extension of the space-like dipole model to the time-like region and most of the nucleon models, e.g. the constituent quark model [9,10]. The pQCD predicted that EFFs of other baryons should follow the same energy dependence. It was only recently that the EMFFs of the Λ baryon was measured with good precision [11][12][13][14], so this conclusion could be tested for hyperon for the first time.
However, it seems that the data deviate significantly from the parametrization motivated by pQCD analysis [15]. Besides, a very close-to-threshold enhancement is observed by BESIII collaboration [13]. It was shown that final state interaction [16] or valence quark Coulomb enhancement factor [17] are deficient in accounting for it. These anomalous behaviours are still waiting for a reasonable explanation.
As a matter of fact, new structures beyond the pQCD power-law behavior were seen in the data of e + e − → pp by BaBar collaboration [3]. It was found that these peaks are possibly related of to the resonances [18]. The most distinct peak would be the known φ(2170) with the width of around 80 MeV observed by several experimental groups (see [18][19][20] for a summary of data). The higher structure can be fitted well with a broader vector meson at about 2.43 GeV. Though alternative explanations, e.g. thresholds opening [18] or large imaginary optical potential in the inelastic rescattering [9,10], are not completely excluded, these results give us enlightening insight into the e + e − → ΛΛ. In this paper, we are attempting to investigate the role of vector mesons besides the usual pQCD contribution in this reaction.

II. FORMALISM AND RESULTS
Assuming that one photon exchange dominates the production, the Born cross section of baryon anti-baryon pairs through virtual photon in e + e − annihilation can be written in terms of the effective form factor G eff (s), with τ = s/4m 2 and β = 1 − 1/τ . Here α ≈ 1/137 is the electromagnetic fine-structure constant, m the baryon mass, and s the square of the center of mass (c.m.) energy. The Coulomb factor C is accounting for the electromagnetic interaction of the point-like baryon anti-baryon pairs in the final states [17]. It is relevant to the close-to-threshold enhancement in pp final states but reads as 1 for the neutral ΛΛ interaction here.
The effective form factor can be generally expressed in terms of electric and magnetic form factors G E and G M : which can be extracted from experimental measurements of σ e + e − →γ * →BB (s) by Eq. (1).
Based on the pQCD predictions [7,8], the |G M | and |G E | in the time-like region can be parameterized as follows which are the same with the nucleon EFFs. Here Λ QCD = 0.3 GeV is the QCD scale parameter [9,10,18]. The only free parameter A B can be obtained by fitting the experimental data. The factor ln(s/Λ 2 QCD ) represents the logarithmic corrections from QCD [8], which enables a good fit to the nucleon EFFs.
with tan ∆ n = M n Γ tot n /(s − M 2 n ) for n-th resonance with mass M n and total width Γ tot n . When only one (n = 1) resonance is present, it reduces to be an usual form [22,23] So the interference phase between two amplitudes is also contained in the parameter q n .
Phenomenological model can be constructed to explicitly calculate these amplitudes, as in the case of DD reaction [22,23]. Here the transition into continuum has been calculated by pQCD, but that into resonant states need the knowledge of the inner structure of the corresponding states, which is beyond the scope of present paper. As can be seen from above equation, q n is an energy dependent complex variable in the Fano scheme, but here we parameterize it as real for simplicity: where Γ tot is introduced to make the q n dimensionless and q 0 n is a constant determined by the data. The q n could be regraded as a constant in the limited energy range, e.g. for narrow state ψ(3770) in DD reaction [22,23]. We use the s-wave energy dependent width for below-threshold vector meson decaying to ΛΛ,  with p cm = s/4 − m 2 being the final baryon momentum in the c.m. system. We use Γ ΛΛ (s) = const. for the resonances above threshold.
We find that two resonant mesons are needed to describe the available data of e + e − → ΛΛ.
One of them is the φ(2170), which is required to explain the close-to-threshold enhancement.
In view of the fact that this state is below the ΛΛ threshold, its line shape is not fully present in the cross section of e + e − → ΛΛ. It is impossible to unambiguously determine its properties here. We fix its mass and width to be 2.188 ± 0.010 GeV and 83.0 ± 12.0 MeV quoted from the Particle Data Group (PDG) [20], respectively. Another one is a broad meson with the mass of around 2.340 GeV (hereafter labeled as X(2340) for simplicity), which is demanded by the energy dependent behaviour of the cross section above threshold. We let its mass and width be free parameters. The resulted curves are depicted in Figure 1. It needs to be pointed out that due to the larger uncertainty, the point marked by green open-circle from DM2 measurement [11] is not included into the fit. A very good quality of agreement is achieved with 16 data points and five free parameters, i.e. χ 2 /d.o.f = 7.7/11 = 0.7. The obtained numerical parameters are summarized in Table I. Here the second uncertainties are associated with the mass and width of φ(2170) resonance, which are evaluated by changing the mass and width by one standard deviation from the PDG values.
As seen in Fig. 1, the EFFs from the pQCD calculation contribute as a smooth background. The overall normalization factor A Λ is smaller than the A p ∼ 5.0 for e + e − → pp [5].
The ratio A Λ /A p is roughly consistent with (m u,d /m s ) 2 in the relativized constitute quark model [24].
We show the curves for total contribution of pQCD and one resonance by the big error bars of the data at 2.2324 ± 0.00048 GeV, which is 1.0 MeV above the threshold.
If the error of this energy point can be reduced or there are more precise measurements at different near-threshold energies in the future, we can obtain more information about φ(2170) and study its nature through model calculation of its q 0 1 by Eq. (6). The vector meson with the mass of around 2.340 GeV is broad, with a width of 257 ± 159 MeV. The large uncertainty is originated from the big error bars of the data in this area. It provides the correct energy dependency of the cross section over a wide energy range.
Whether these states in e + e − → ΛΛ are related to the structures in e + e − → pp is not clear at present. The most distinct peak in e + e − → pp is around 2.125 GeV, with a width of around 90 MeV [18], both compatible with the properties of φ(2170) here. The position of the second peak is around 2.43 GeV in e + e − → pp, which is higher than the second resonance here, though their widths are roughly close [18]. BESIII [25] and Belle-II [26] collaborations have the chance to clarify these problems by measuring both e + e − → ΛΛ and e + e − → pp with higher precision at more energy points with more statistics in future.

III. DISCUSSION AND CONCLUSION
In a brief summary, we investigate the role of vector mesons in the e + e − → ΛΛ reaction.
Our results support the argument that the EFFs of Λ hyperon obey the pQCD prediction.
In other words, the deviation of the e + e − → ΛΛ to pQCD calculation is attributed to the contribution of two resonant mesons. We find that the φ(2170) is responsible for the close-to-threshold enhancement shown by the data, and another wide meson state with the mass around 2.34 GeV explains the energy dependency above threshold. If the structures in e + e − → pp are confirmed by more accurate experiment and really related to the vector mesons here, we could further determine some of their properties, for example, separating their coupling to e + e − , ΛΛ and pp by a combined analysis of these reactions. Our formalism and conclusion would give insight into the EMFFs of other baryons, e.g. Σ [12], Λ(1520) [27] and Λ + c [28,29] after they are measured with higher precision. The nature of φ(2170) state is widely studied in the literature [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. The possible explanations include the strangeonium hybrid [30,31], dynamical generated states [32][33][34][35][36][37], tetraquark [38][39][40] or ΛΛ molecular state [41,42], et al. Our finding that φ(2170) state is important in the close-to-threshold ΛΛ production would support its nature of ΛΛ molecule. Actually some evidence of ΛΛ threshold enhancement has been also observed in the J/ψ → γΛΛ decay [45], which is possibly associated with the ΛΛ 1 S 0 bound state [42].
There are several possible candidates for the higher lying state X(2340). We suggest that the ω(2290) with a mass of 2.290 ± 0.020 GeV and a width of 275 ± 35 MeV found in the partial wave analysis of pp → ΛΛ [46] is probably the same state in our fit here.
Their values of mass and width are both highly overlapped with the parameters in Table I within errors. Furthermore, they both couple strongly to the ΛΛ. Among the further light unflavored states listed by PDG [20], another two ω states are also possible candidates with bigger spread of mass and width. They are ω(2330) with a mass of 2.330 ± 0.030 GeV and a width of 435 ± 75 MeV found in multi-pions production in γp reaction [47], and ω(2205) with a mass of 2.205 ± 0.030 GeV and a width of 350 ± 90 MeV seen in pp → ωη and ωππ [20]. We also would like to point out that the recent data of multi-mesons production in e + e − annihilation show a state with the mass of around 2.4 GeV and the width of around 100 MeV, with large uncertainties due to the small statistical significance [40]. It would be the possible partner state of the φ(2170) as suggested in QCD sum rule [40]. Further study of pp annihilation by PANDA collaboration is definitely welcome in order to drive a firm conclusion about the vector mesons in this energy range [48]. Moreover, the goal of BESIII data taking [25] is to accumulate 10 billion J/ψ and 3 billion ψ(3686) events, thus these vector mesons can also be searched and studied in charmonium decays, such as J/ψ, ψ(3686) → ΛΛπ 0 /η, χ cJ → ΛΛγ/φ/ω and so on.