Natural Explanation of Recent Results on

We show that the recent experimental data on the cross section of the process near the threshold can be perfectly explained by the final-state interaction of and . The enhancement of the cross section is related to the existence of low-energy real or virtual state in the corresponding potential. We present a simple analytical formula that fits the experimental data very well

Recently, new experimental data have appeared on the cross section of e + e − → Λ Λ annihilation near the threshold [1].These data are consistent with the results of previous works [2][3][4], but have much higher accuracy.All these results demonstrate a strong energy dependence of the cross section near the threshold.A similar phenomenon has been observed in such processes as e + e − → pp [5][6][7][8][9][10][11][12], e + e − → nn [13][14][15], e + e − → Λ c Λc [16,17], e + e − → B B [18], and others.In all these cases the shapes of near-threshold resonances differ significantly from the standard Breit-Wigner parameterization.The origin of the phenomenon is naturally explained by the strong interaction of produced particles near the threshold (the so-called final-state interaction).Since a typical value of the corresponding potential is rather large (hundreds of MeV), existence of either low-energy bound state or virtual state is possible.In the latter case a small deepening of the potential well leads to appearance of a real low-energy bound state.In both cases, the value of the wave function (or its derivative) inside the potential well significantly exceeds the value of the wave function without the final-state interaction.As a result, the energy dependence of the wave function inside the potential well is very strong.Since quarks in e + e − annihilation are produced at small distances of the order of 1/ √ s, a strong energy dependence of the cross section is determined solely by the energy dependence of the wave function of produced pair of hadrons at small distances.Such a natural approach made it possible to describe well the energy dependence of almost all known near-threshold resonances (see Refs. [19][20][21][22][23][24][25] and references therein).
The annihilation e + e − → Λ Λ near the threshold is the most simple for investigation.This is due to the fact that the Λ Λ system has a fixed isotopic spin I = 0, and the pair is produced mainly in the state with an angular momentum l = 0 (the contribution of state with l = 2 can be neglected).Moreover, there is no Coulomb interaction between Λ and Λ.Our analysis shows that the imaginary part of the optical potential of Λ Λ interaction, which takes into account the possibility of annihilation of Λ Λ pair into mesons, has only little effect on the cross section.Therefore, we neglect the imaginary part of the potential.Finally we describe the cross section of the process e + e − → Λ Λ by a simple analytical formula (see Ref. [25] and references therein for more details): where is the dipole form factor, and Λ is some parameter close to 1 GeV.The factor g is related to the probability of pair production at small distance ∼ 1/ √ s and can be considered as a constant independent of energy.In Eq. (1), ψ(0) is the wave function of Λ Λ pair at r = 0.
The cross section (1) is enhanced by the factor |ψ(0)| 2 1 if there is a loosely bound state or a virtual state of Λ Λ pair.In both cases the modulus of scattering length a of Λ and Λ is large compared to the characteristic radius R of Λ Λ interaction potential, |a| R. For a loosely bound state a is positive and the binding energy is ε = −1/M Λ a 2 .For a virtual state a is negative and the energy of virtual state is defined as ε = 1/M Λ a 2 .In both cases |ε| is much smaller than the characteristic depth of the potential well.The energy dependence of |ψ(0)| 2 for near-threshold resonances is more or less universal and is determined by the scattering length a and the effective radius of interaction [26].Therefore, one can use any convenient form of potential U (r) for description of near-threshold resonances.
In the present paper we parametrize the potential as U (r) = −U 0 •θ(R−r).For this potential, the energy dependence of |ψ(0)| 2 is well-known (see, e.g., Ref. [26]): Near the threshold k q and the cross section ( 1) is enhanced if where q 0 = √ M Λ U 0 , and n is an integer.For |δ| 1, the scattering length is a = 1/q 0 δ, where δ > 0 for the bound state, and δ < 0 for the virtual state.
By means of Eq. (3) the expression (2) can be simplified: The corresponding energy dependence of the cross section (1) is equivalent to the Flatté formula [27], which is expressed in terms of the scattering length and the effective radius r 0 of interaction.Note that for the rectangular potential well r 0 = R.One can easily verify that the precise and approximate formulas for the cross section are in good agreement with each other for |δ| 1 and E ε 0 U 0 .Note that |ε 0 | |ε| for both bound and virtual states, namely ε 0 ≈ 2 |ε| a/R.However, the position of peak in the cross section, which is proportional to , is located at energy E ≈ |ε| for both bound and virtual states.
In Fig. 1, we show our predictions for the cross section compared to experimental data [1], as well as the enhancement factor |ψ(0)| 2 .The parameters of the model are U 0 = 584 MeV, R = 0.45 fm, and g = 0.2.In the energy region under consideration the dependence of our predictions on the parameter Λ is very weak.To be specific, we set Λ = 1 GeV.Our model, giving χ 2 /N df = 9.8/13, provides a good description of experimental data [1].Note that account for the enhancement factor |ψ(0)| 2 is of great importance for correct description of experimental data.
In conclusion, the assumption of existence of a low-energy real or virtual state has allowed us to describe perfectly recent and previous experimental data for the cross section of e + e − → Λ Λ annihilation near the threshold.Our model indicates possible existence of a bound Λ Λ state with energy E ≈ −30 MeV.

Figure 1 .
Figure 1.The cross section of e + e − → Λ Λ annihilation (left) and the enhancement factor |ψ(0)| 2 (right) as the of energy E. The parameters of the potential are U0 = 584 MeV and R = 0.45 fm.Experimental data are taken from Ref. [1].