Covariant Tensor Formalism For Partial Wave Analyses ψ→Δ(?)

Since the quark model was proposed,the baryon qqq model has provided a powerful explanation for the hadron structure.The quark model predicts that there are 35 Δ*particles,but we know very little about them,and some of them have not been detected experimentally.At present,we use the data from πN and γN to search for Δ*and measure Δ*physical information by multi-orbit wave splitting analysis.It is almost impossible to understand the production and decay mechanism of Δ*by studying charm decay.The charmoquarin decay is an ideal place to study the excited states of baryons.ψ(2S)→Δ(?) is the first decay mode to a baryon pair discovered in particle physics experiments.The study of it can make us understand more about the decay mechanism of charm and may even help us to find the remaining Δ*particles.Since the width of Δ*is larger than 117MeV,the central masses of the two adjacent Δ*particles do not differ much,which results in the overlap of the mass spectra of Δ*.When we study at the decay of ψ→Δ(1232)++Δ(1232)--,Δ*background in data interferes with signal,the interference influence cannot be ignored.It is worth mentioning that the method of partial-wave analysis can give the influence of interference effect in detail.The formulas used in πN or γN scattering experiments are not applicable to the study of the cascaded decay ψ(2S)→Δ(1232)++Δ(1232)--.PWA formulas for ψ→ωpp and ψ→NN*M have also been given in the papers,but neither of these formulas are suitable for the wave splitting formulas of ψ(2S)→Δ(1232)++Δ(1232)--we need now.The partial wave equation of ψ(2S)→Δ(?) is given in this paper by two equivalent ways.The two modes are the covariant tensor form and the helicity amplitude form.The form of covariant tensor has Lorentz invariance and does not need complex coordinate transformation.The angular distribution can be seen intuitively in the form of helicity amplitude.In this paper,the data of BESⅢ is also used for the preliminary data analysis of ψ(2S)→Δ(1232)++Δ(1232)--,Δ(1232)++→pπ+,Δ(1232)--→pπ.