A possible explanation of the threshold enhancement in the process $e^+e^-\rightarrow \Lambda\bar{\Lambda}$

Inspired by the recent measurement of the process $e^+e^-\rightarrow \Lambda\bar{\Lambda}$, we calculate the mass spectrum of the $\phi$ meson with the GI model. For the excited vector strangeonium states $\phi(3S,~4S,~5S,~6S)$ and $\phi(2D,~3D,~4D,~5D)$, we further investigate the electronic decay width with the Van Royen-Weisskopf formula, and the partial widths of the $\Lambda\bar{\Lambda}$, $\Xi^{-(*)}\bar{\Xi}^+$, and $\Sigma^{+(*)}\bar{\Sigma}^{-(*)}$ decay modes with the extended quark pair creation model. We find that the electronic decay width of the $D$-wave vector strangeonium is about $3\sim8$ times larger than that of the $S$-wave vector strangeonium. Around 2232 MeV the partial decay width of the $\Lambda\bar{\Lambda}$ mode can reach up to several MeV for $\phi(3^3S_1)$, while the partial $\Lambda\bar{\Lambda}$ decay width of $\phi(2^3D_1)$ is $\mathcal{O}(10^{-3})$ keV. If the threshold enhancement reported by the BESIII Collaboration arises from the strangeonium meson, this state is very likely to be the $\phi(3^3S_1)$ state. We also note that the $\Lambda\bar{\Lambda}$ and $\Sigma^{+}\bar{\Sigma}^{-}$ partial decay widths of the states $\phi(3^3D_1)$ and $\phi(4^3S_1)$ are about several MeV, respectively, which are enough to be observed in future experiments.

Very recently, the BESIII Collaboration studied the process e + e − → ΛΛ with improved precision [22]. The Born cross section at √ s = 2.2324 GeV, which is 1.0 MeV above the ΛΛ mass threshold, is measured to be 305 ± 45 +66 −36 pb. Is the unexpected feature in the near-threshold region due to an unobserved strangeonium meson resonance? In the present work, we will try to answer this question.
We will calculate the spectrum of the ss system in the framework of the Godfrey-Isgur (GI) model [23], which has achieved a good description of the known mesons and baryons [23][24][25]. After we obtain the masses of the higher excited strangeonium states, we further estimate the electronic decay width of the J PC = 1 −− states φ(2D, 3D, 4D, 5D) and φ(3S , 4S , 5S , 6S ) with the Van Royen-Weisskopf formula [26]. Meanwhile, we use the extended quark pair creation model [27,28] to calculate the partial ΛΛ, Ξ −( * )Ξ+ , and Σ +( * )Σ−( * ) decay widths of those vector states with the obtained spatial wave functions. Considering there existing many theoretical calculations of the two-body strong decays of the ss system with various models in the literature [29][30][31][32][33][34], in the present work we will emphasise on the baryon-antibaryon decay mode and electronic decay properties.
According to the theoretical predictions from various models, the masses of φ(3 3 S 1 ) and φ(2 3 D 1 ) mesons are about 2.2 GeV (see Table I). Therefore, we calculate the e + e − and ΛΛ partial decay widths of the excited vector states φ(3 3 S 1 ) and φ(2 3 D 1 ). We find that the electronic decay width of φ(3 3 S 1 ) is about 1/3 times smaller than that of φ(2 3 D 1 ). However around 2232 MeV the partial decay width of the ΛΛ mode can reach up to several MeV for φ(3 3 S 1 ), while the partial ΛΛ decay width of the states φ(2 3 D 1 ) is a very small value O(10 −3 ) keV. The threshold enhancement in the process e + e − → ΛΛ observed by the BESIII Collaboration [22] may be caused by φ(3 3 S 1 ). We also notice that the ΛΛ and Σ +Σ− partial decay widths of the states φ(3 3 D 1 ) and φ(4 3 S 1 ) are about several MeV, respectively. These two states have a good potential to be observed in future experiments via their corresponding main baryon-antibaryon decay channel. This paper is organized as follows. In Sec. II we give a brief introduction of the GI model and calculate the spectrum of the ss system. Then we present the Van Royen-Weisskopf formula and give the electronic decay properties in Sec. III. In Sec. IV we discuss the extended quark pair creation model and baryon-antibaryon decay results. We give a short summary in Sec. V.
In the nonrelativistic limit, it can reduce to the familiar Breit-

Fermi interaction
Here, H conf 12 is the spin-independent linear confinement and Coulomb-type interaction; H hyp 12 is the color-hyperfine interaction and H so 12 is the spin-orbit interaction. To incorporate the relativistic effects, Godfrey and Isgur further built a semiquantitative model [23]. By introducing the smearing function for a meson q iq j the OGE potential G(r) = −4α(r)/3r and confining potential S (r) = br + c are smeared toG(r) andS (r) viã Through the introduction of the momentum-dependent factors, the Coulomb term is modified according tõ and the contact, tensor, vector spin-orbit, and scalar spin-orbit potentials were modified according tõ where ǫ i corresponds to the contact (c), tensor (t), vector spinorbit [so(v)], and scalar spin-orbit [so(s)].
With the notation we have wherẽ The spin-orbit termH so 12 can be decomposed into a symmetric partH so (12) and an antisymmetric partH so [12] , while theH so [12] vanishes when m 1 = m 2 .
We adopt the free parameters in the original work of the GI model [23], and diagonalize the Hamiltonian in the simple harmonic oscillator bases |n 2S +1 L J . The resulting mass spectrum of the strangeonium are shown in Fig. 1. Meanwhile, we compare our predicted mass of the higher vector φ mesons with various models predictions, as listed in Table I.

III. THE ELECTRONIC DECAYS
With the Van Royen-Weisskopf formula [26,38], the electronic decay width of the excited vector strangeonium states is given by Here, α = 1 137 denotes the fine structure constant. m s = 450 MeV and e s = − 1 3 are the strange quark constituent mass and charge in unit of electron charge, respectively. M nS (M nD ) is the mass for φ(nS )(φ(nD)). R nS (0) represents the radial S wave function at the origin, and R ′′ nD (0) represents the second derivative of the radial D wave function at the origin.
In the present calculation, we adopt the simple harmonic oscillator (SHO) wave functions for the space-wave functions of the initial meson. According to the wave functions obtained in mass spectrum calculations, we get the root mean square radius of the vector states. Then, we determine the value of harmonic oscillator strength β th between the two strange quarks for the initial mesons (as listed in Table II).
According to PDG [37], the electronic decay branching ra- Combining this ratio with its total decay widths(Γ = 4.249 ± 0.013 MeV), the central value of the electronic decay width is Γ[φ(1S ) → e + e − ] = 1.26 keV. Then, from the formulas (12)-(13), we can obtain electronic decay width ratios of between the higher excited vector strangeonium states and the state φ(1S ). Thus, we can get those states electronic decay widths, as shown in Table II.  From the table, the ratio R is smaller than one. The electronic decay widths of the excited vector strangeonium states φ(3S , 4S , 5S , 6S ) and φ(2D, 3D, 4D, 5D) are smaller than that of the state φ(1S ). Meanwhile, the electronic decay width of the D-wave vector strangeonium is about 3 ∼ 8 times larger than that of the S -wave vector strangeonium. For the S -wave states, our predictions are in accordance with ref. [39], while for the D-wave states, our predictions are about 3 times larger than those of ref. [39].
Considering the uncertainties of the predicted mass and harmonic oscillator strength β th , we plot the variation of the electronic decay width ratio R as a function of the mass with different values of β=β th + 20 MeV, β th , and β th − 20 MeV, re-spectively, in Fig. 2. It is obvious that the ratio R decreases with the mass with the same β values.

IV. DOUBLE BARYON DECAY MODE
A. The 3 P 0 model The quark pair creation ( 3 P 0 ) model was first proposed by Micu [40], Carlitz and Kislinger [41], and further developed by the Orsay group [42][43][44], which has been widely used to study the OZI-allowed two-body strong decays of hadrons. Very recently, the 3 P 0 model was extended to study some OZIallowed three-body strong decays [28] as well. In the framework of this model, the interaction Hamiltonian for one quark pair creation was described as [45][46][47] Here, γ is a dimensionless parameter and usually determined by fitting the experimental data. m f denotes the constituent quark mass of flavor f and ψ f stands for a Dirac quark field. In our previous work [27], we extended the 3 P 0 model to study the partial decay width of the Λ cΛc mode for the charmonium system. In this work, we further use this model to study the process φ * (A) → B(B) +B(C), where φ * denotes the excited strangeonium states. As pointed out in Ref. [27], two light quark pairs should be created for this type of reaction (as shown in Fig. 3

), and the helicity amplitude M M J A M J B M J B reads
Here, p I (I = A, B, C) denotes the momentum of the hadron. E A(k) represents the energy of the initial(intermediate) state A(k). Considering the quark-hadron duality [48], we simplify the calculations via taking E k − E A as a constant, namely E k − E A ≈ 2m q . Here, m q is the constituent quark mass of the created quark. We adopt this crude approximation because the intermediate state differs from the initial state by two created additional quarks at the quark level [27,28]. Thus, we can rewrite the Eq. (16) as Then, the transition operator for the two quark pairs creation in the nonrelativistic limit reads where p i (i=3, 4, 5, 6) stands for the three-vector momentum of the ith quark. ϕ 0 = (uū + dd + ss)/ √ 3 corresponds to the flavor function and ω 0 = δ i j represents the color singlet of the quark pairs created from vacuum. χ 1,−m(m ′ ) are the spin triplet states for the created quark pairs. The solid harmonic polynomial Y m(m ′ ) 1 (p) ≡ |p|Y m(m ′ ) 1 (θ p , φ p ) denotes the P-wave quark pairs. a † i b † j is the creation operator representing the quark pair creation in the vacuum.
Finally, the hadronic decay width Γ in the relativistic phase space reads Here, p represents the momentum of the daughter baryon. M A and J A are the mass and total angular quantum number of the parent baryon A, respectively. In the center of mass frame of the parent baryon A, p reads  [37], as listed in Table. III. We adopt the simple harmonic oscillator (SHO) wave functions for the space-wave functions of the hadrons. The harmonic oscillator strength β th between the two strange quarks for the initial mesons is determined by the spatial wave functions obtained in mass spectrum calculations (as listed in Table I). The harmonic oscillator strength between the two light quarks for final baryons is taken as α = 400 MeV. As to the strength of the quark pair creation from the vacuum, we adopt the same value as in Ref. [49], γ = 6.95. The uncertainty of the strength γ is about 30% [47,[50][51][52], and the partial decay widths are proportional to γ 4 . Thus our predictions may bare a quite large uncertainty.

States around the ΛΛ threshold
In 2007, the BABAR Collaboration measured the cross section for e + e − → ΛΛ from threshold up to 3 GeV [9] and observed a possible nonvanishing cross section at threshold. Recently, the BESIII Collaboration published a measurement of the process e + e − → ΛΛ [22] with improved precision. The Born cross section at √ s = 2232.4 MeV, which is 1.0 MeV above the ΛΛ mass threshold, is measured to be 305 ± 45 +66 −36 pb, which indicates an obvious threshold enhancement. According to various model predictions (see Table I), there are two strangeonium meson resonances φ(3 3 S 1 ) and φ(2 3 D 1 ) with both masses around 2.2 GeV and J P = 1 −− . As a possible source of the observed threshold enhancement, it is crucial to study the decay properties of the states φ(3 3 S 1 ) and φ(2 3 D 1 ).
We first explore the ΛΛ partial decay width of the state with a mass of M = 2232 MeV (see Table IV). This partial decay width is large enough to be observed in experiments, and indicates that the observed threshold enhancement may arise from this state. Although the phase space is suppressed seriously around threshold, the transition amplitude for this decay mode is quite large. Hence, the partial decay width of the ΛΛ mode for the state φ(3 3 S 1 ) reaches several MeV. Considering the uncertainties of the predicted mass, we study the variation of the ΛΛ decay width as a function of the mass of the state φ(3 3 S 1 ). The decay width increases rapidly with the mass in the range of (2233-2300) MeV. Then, we investigate the decay properties of the state φ(2 3 D 1 ). Fixing the mass at M = 2232 MeV, we get This width seems too small to be observed in experiments.
Combining the predicted partial decay width of φ(3 3 S 1 ), we further obtain The decay ratio of φ(3 3 S 1 ) into the ΛΛ channel is about O(10 6 ) larger than that of φ(2 3 D 1 ) into the ΛΛ channel. Combining their electronic decay width we calculated in Sec. III, we obtain that if the threshold enhancement reported by the BESIII Collaboration in the process e + e − → ΛΛ were related to an unobserved strangeonium meson resonance, this state should most likely be φ(3 3 S 1 ). Besides the uncertainties coming from the predicted mass and harmonic oscillator strength β th , the results of φ(3 3 S 1 ) and φ(2 3 D 1 ) may have large uncertainties due to their lower masses. At the hadron level, the energy of the intermediate states with the spin parity J PC = 1 −− , such as molecular states KK 1 (1270), K * (892)K * 0 (700), K * (892)K 1 (1270), and φ(1020)a 0 (980) and so on, is about 1.7 Gev∼2.1 GeV. Thus the E k − E A are small and sensitive to the masses of the intermediates state. In this case, taking E k − E A =2m q as a constant will introduce a large uncertainty in this calculation.

higher states
Besides φ(3 3 S 1 ) and φ(2 3 D 1 ), we also analyze the decay properties of the S-wave states φ(4 3 S 1 , 5 3 S 1 , 6 3 S 1 ) and the  Table V. From the table, we get that the ΛΛ partial decay width of φ(3 3 D 1 ) can reach up to Γ ∼ 3.5 MeV, which is the largest compared to five other vector states we considered in this work. The sizeable width indicates that this state has a good potential to be observed in the ΛΛ decay channel. Similarly, taking the uncertainties of the theoretical masses and harmonic oscillator strength β th into account, we plot the ΛΛ partial decay widths of those states as functions of the masses in Fig. 4 with different values of β=β th +20 MeV, β th , and β th -20 MeV, respectively. According to Fig. 4, for the state φ(3 3 D 1 ), the variation curve likes a bowel structure when the mass varies from 2550 MeV to 2850 MeV, and the partial width can reach up to Γ ∼ 3.7 MeV with β = β th . The ΛΛ partial decay width for φ(5 3 D 1 ) is the smallest. The decay width is less than Γ < 0.4 MeV with the mass in the range of M = (3150 − 3450) MeV. As to φ(4 3 S 1 ), its ΛΛ decay width is very sensitive to the mass (see Fig. 4). When β = β th , the width varies in the range of Γ ∼ (0.0−4.8) MeV with the mass in the range of M = (2450−2750) MeV. If the mass of φ(4 3 S 1 ) lies in (2496-2590) MeV, the decay width of the ΛΛ mode is less than one MeV. Most of the ΛΛ partial decay widths for the other three states, φ(4 3 D 1 ), φ(5 3 S 1 ) and φ(6 3 S 1 ), are less than one MeV (see Fig. 4). These partial widths seem to be sizeable as well.
From the Table, we notice that the Σ +Σ− partial decay width of φ(4 3 S 1 ) and φ(3 3 D 1 ) can reach up to Γ ∼ 2.9 MeV and Γ ∼ 1.5 MeV, respectively, which are large enough to be observed in future experiments. Meanwhile, the Ξ −Ξ+ and Σ + * Σ− * partial decay widths of the state φ(5 3 S 1 ) are both larger than one MeV.
In addition, we also plot the decay properties of the states φ(4S , 5S , 6S ) and φ(3D, 4D, 5D) as a function of the mass   in Fig. 5.
To investigate the uncertainties of the parameter β th , we further consider the partial decay properties with different β th values. The theoretical numerical results are not shown in the present work. According to our calculations, our main predictions hold in a reasonable range of the parameter β th .
For the electronic decay widths, we obtain that the electronic decay widths of the excited vector strangeonium states φ(3S , 4S , 5S , 6S ) and φ(2D, 3D, 4D, 5D) are smaller than that of the state φ(1S ). Meanwhile, the electronic decay width of the D-wave vector strangeonium is about 3 ∼ 8 times larger than that of the S -wave vector strangeonium.
For the double baryons decay widths, the partial decay width of the ΛΛ mode can reach up to ∼ 5.84 MeV for φ(3S ), while the partial ΛΛ decay width of the states φ(2D) is about O(10 −3 ) keV. Thus, the ΛΛ decay width ratio between the states φ(3 3 S 1 ) and φ(2 3 D 1 ) is O(10 6 ). If the threshold enhancement reported by the BESIII Collaboration in process e + e − → ΛΛ does arise from an unobserved strangeonium meson, the resonance is most likely to be the strangeonium state φ(3S ). We also notice that the ΛΛ and Σ +Σ− partial decay widths of the states φ(3 3 D 1 ) and φ(4 3 S 1 ) are about several MeV, respectively, which are enough to be observed in future experiments. The double baryons decay modes provide a unique probe of the excited vector strangeonium resonances, which may be produced and investigated at BESIII and BelleII.