Hadronic transitionχc1→ηcππat the Beijing Spectrometer BES and the Cornell CLEO-c

Abstract

Hadronic transitions of the χcj states have not been studied yet. We calculate the rate of the hadronic transition χc1 → ηcππ in the framework of QCD multipole expansion. We show that this process can be studied experimentally at the upgraded Beijing Spectrometer BES III and the Cornell CLEO-c. PACS number(s): 13.20. Gd, 13.30.Eg, 14.40.Cs It has been shown that the theory of hadronic transitions based on QCD multipole expansion [1,2,3,4] can make quite successful predictions for many hadronic transition rates in the cc and bb systems [5,6]. So far, hadronic transitions of the χ cj states have not been studied yet. The observed χ cj decays are mainly hadronic decays (decaying into light hadrons), and the hadronic widths of the three χ cj states are rather different. From the point of view of perturbative QCD, the hadronic decay rates of χ c0 and χ c2 are proportional to α 2 s , while that of χ c1 is proportional to α 3 s [7]. Hence χ c1 has the smallest hadronic decay rate, and is thus the most interesting one among the three χ cj states for studying hadronic transitions. The main hadronic transition process of χ c1 is χ c1 → η c ππ. In the framework of QCD multipole expansion, this is dominated by the E1-M1 transition, and the rate is significantly smaller than those of E1-E1 transitions such as ψ ′ → J/ψ ππ. In addition, χ c1 cannot be directly produced in e + e − collision. It can only be produced via the radiative transition ψ ′ → γχ c1 . This is why χ c1 → η c ππ is not easy to detect at the original Beijing Spectrometer BES II. The designed luminosity of the upgraded Beijing Electron-Positron Collider BEPC II is about 1 − 2 orders of magnitude higher than that of the original BEPC I, and the ability of the upgraded Beijing Spectrometer BES III will be significantly improved compared with the original BES II, especially its photon detector. So that BES III will be able to detect processes like χ c1 → η c ππ which BES II can hardly do. In this paper, we calculate the transition rate of χ c1 → η c ππ in the framework of QCD multipole expansion, and our result shows that both the upgraded BES III and the Cornell * Mailing address CLEO-c have a good chance to study χ c1 → η c ππ. and the theoretical approaches to hadronic transitions can be further tested.According to the formula for E1-M1 transitions [2], the transition amplitude for χ c1 → η c ππ iswhere E a and B a are color electric and magnetic fields; g E and g M are the corresponding coupling constants for E1 and M1 gluon emissions; x is the relative separation between c andc in the quarkonium; s c (sc) is the spin of the c (c) quark; E I is the energy of the initial state χ c1 ; E KL and |KL are the energy and eigenstate of the intermediate state with the principal quantum number K and orbital angular momentum L, respectively. As described in Ref.[2], we take the quark confining string model [8] to describe the intermediate states.The hadronization factor ππ|E a i B a j |0 is a second rank pseudo tensor at the light hadron scale. In the rest frame of...

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