A model to explain angular distributions of $J/\psi$ and $\psi(2S)$ decays into $\Lambda\overline{\Lambda}$ and $\Sigma^0\overline{\Sigma}^0$

BESIII data show a particular angular distribution for the decay of the $J/\psi$ and $\psi(2S)$ mesons into the hyperons $\Lambda\overline{\Lambda}$ and $\Sigma^0\overline{\Sigma}^0$. More in details the angular distribution of the decay $\psi(2S) \to \Sigma^0\overline{\Sigma}^0$ exhibits an opposite trend with respect to that of the other three channels: $J/\psi \to \Lambda\overline{\Lambda}$, $J/\psi \to \Sigma^0\overline{\Sigma}^0$ and $\psi(2S) \to \Lambda\overline{\Lambda}$. We define a model to explain the origin of this phenomenon.


I. INTRODUCTION
Since their discovery, charmonia, i.e., cc mesons, have been representing unique tools to deepen and expand our understanding of the strong interaction dynamics at low and medium energy ranges. Especially in case of the lightest charmonia, decay mechanisms can be studied only by means of effective models, since, due to their low-energy regime, these processes do escape the perturbative description of the quantum chromodynamics. We study the decays of the J/ψ and ψ(2S) mesons into baryon-antibaryon pairs BB = ΛΛ, Σ 0 Σ 0 . The differential cross section of the process e + e − → ψ → BB has the well known parabolic expression in cos θ [1] dN d cos θ where α B is the so-called polarization parameter and θ is the baryon scattering angle, i.e., the angle between the outgoing baryon and the beam direction in the e + e − center of mass frame. As already pointed out in Ref. [2], only the decay J/ψ → Σ 0 Σ 0 has a negative polarization parameter α B . Figure 1 and 2 show BESIII data [3] on the angular distribution of the four decays: J/ψ → ΛΛ, J/ψ → Σ 0 Σ 0 , and ψ(2S) → ΛΛ, ψ(2S) → Σ 0 Σ 0 respectively.

II. AMPLITUDES AND BRANCHING RATIOS
The Feynman amplitude for the decay ψ → BB can be written in terms of the strong magnetic and Dirac FFs

III. EFFECTIVE MODEL
The SU(3) baryon octet states can be described by a matrix notation as follows [4] where the first matrix is for baryons and the latter for antibaryons. We can consider the J/ψ and ψ(2S) mesons as SU (3) where g m is an effective coupling constant. This matrix describes the mass breaking effect due to the mass difference between s and, u and d quarks, where the SU(2) isospin symmetry is assumed, so that m u = m d . This SU(3) breaking is proportional to the 8 th Gell-Mann matrix λ 8 . The EM breaking effect is related to the fact that the photon coupling to quarks, described by the four-current is proportional to the electric charge. This effect can be parametrized using the following spurion matrix where g e is an EM effective coupling constant. The most general SU(3) invariant effective Lagrangian density is given by [5] where g, d, f , d and f are coupling constants. We can extract the Lagrangians describing the J/ψ and ψ(2S) decays into ΛΛ and Σ 0 Σ where G 0 and G 1 are combinations of coupling constants, i.e., By using the same structure of Eq. (2), the BRs can be expressed in terms of electric and magnetic amplitudes as Moreover, as obtained in Eq. (3), such amplitudes can be further decomposed as combinations of leading, E 0 and M 0 , and sub-leading terms, E 1 and M 1 , with opposite relative signs, i.e., where ρ E and ρ M are the phases of the ratios E 0 /E 1 and M 0 /M 1 . Moduli of E 0,1 and M 0,1 amplitudes   [3]. In particular the value of αB for the decay J/ψ → ΛΛ is from Ref. [6].

IV. RESULTS
In this work we have used data from precise measurements [3,6] of the BRs and polarization parameters, reported in table I, based on events collected with the BE-SIII detector at the BEPCII collider. These data are in agreement with the results of other experiments [7][8][9][10][11]. Since for each charmonium state we have six free parameters (four moduli and two relative phases) and only four constrains (two BRs and two polarization parameters), we have to fix the relative phases ρ E and ρ M . The values ρ E = 0 and ρ M = π appear as phenomenologically favored by the data themselves. Indeed, (largely) different phases would give negative, and hence unphysical, val-ues for the moduli |E 0 |, |E 1 |, |M 0 | and |M 1 |. Moreover, as shown in Fig. 5, where the four moduli for J/ψ and ψ(2S) are represented as functions of the phases with ρ E ∈ [−π/2, π/2] and ρ M ∈ [π/2, 3π/2], the obtained results are quite stable, and the central values ρ E = 0, ρ M = π maximise the hierarchy between the moduli of leading, E 0 and M 0 , and sub-leading amplitudes, E 1 and M 1 . Such values for |E 0 |, |E 1 |, |M 0 | and |M 1 | are reported in table II and shown in Fig. 3. The corresponding values of |g E |, |g M | are reported in table III and shown in Fig. 4. The large sub-leading J/ψ amplitudes |E 1 |, |M 1 | (see table II and Fig. 3) are responsible for the inversion of the |g B E |, |g B M | hierarchy (see Fig. 4 and table III).

V. CONCLUSIONS
Different Λ and Σ 0 angular distributions can be explained using an effective model with the SU(3)-driven Lagrangian The interplay between leading G 0 and sub-leading G 1 contributions to the decay amplitudes determines signs and values of polarization parameters α B .
In particular the different behavior of the J/ψ → Σ 0 Σ 0 angular distribution is due to the large values of the subleading amplitudes |E 1 | and |M 1 |. It implies that the SU(3) mass breaking and EM effects, which are responsible for these amplitudes, play a different role in the dynamics of the J/ψ and ψ(2S) decays. It is interesting to notice that angular distributions of Σ 0 (1385) and Σ ± (1385), measured by BESIII [12,13], show the same Σ 0 behavior. The process e + e − → J/ψ → Σ + Σ − is currently under investigation [14], the behavior of its angular distribution would add important pieces of information to the knowledge of the J/ψ decay mechanism.
Appendix A: Production cross section We consider the decay of a charmonium state, a cc vector meson ψ, produced via e + e − annihilation, into a pair baryon-antibaryon BB, i.e., the process where in parentheses are shown the 4-momenta. The Feynman diagram is shown in Fig. 6 and the corresponding amplitude is where J µ B = u(p 1 )Γ µ v(p 2 ) is the baryonic four-current, D ψ q 2 is the ψ propagator, which includes the γ-ψ electromagnetic (EM) coupling, and v(k 2 )γ µ u(k 1 ) is the leptonic four-current, the four-momenta follow the labelling of Eq. (A1). The Γ µ matrix can be written as [15] where M B is the baryon mass and, f B 1 and f B 2 are constant form factors (FFs) that we call "strong" Dirac and Pauli couplings, they weight the vector and tensor part of the ψBB vertex 1 . We can introduce the strong electric and magnetic Sachs couplings [16] g that have the same structure of the EM Sachs FFs [17], M ψ is the mass of the charmonium state. The four quantities f B 1 , f B 2 , g B E and g B M are in general complex numbers. The differential cross section of the process e + e − → ψ → BB, in the e + e − center of mass frame, in terms of the two Sachs couplings, reads is the velocity of the out-going baryon at the ψ mass, θ is the scattering angle, and the polarization parameter α B is given by