Decay widths of $$^3 P_J$$ 3 P J
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Regular Article - Theoretical Physics
Decay widths of 3 P J charmonium to D D, D D∗ , D∗ D∗ and corresponding mass shifts of 3 P J charmonium Hui-Yun Cao, Hai-Qing Zhoua School of Physics, Southeast University, Nanjing 211189, China
Received: 29 August 2020 / Accepted: 13 October 2020 © The Author(s) 2020
Abstract In this work, we calculate the amplitudes of the ¯ 3 PJ ) in the processes cc( ¯ 3 PJ ) → D D, D D ∗ , D ∗ D ∗ → cc( leading order of the nonrelativistic expansion. The imaginary parts of the amplitudes are corresponding to the branch decay widths of the charmonium cc( ¯ 3 PJ ) → D D, D D ∗ , D ∗ D ∗ and the real parts are corresponding to the mass shifts of the charmonium cc( ¯ 3 PJ ) due to these decay channels. After absorbing the polynomial contributions which are pure real, the ratios between the branch decay widths and the corresponding mass shifts are only dependent on the center-ofmass energy. We find the decay widths and the mass shifts of the 3 P2 states are exact zero in the leading order. The ratios between the branch decay widths and the mass shifts for the 3 P0 , 3 P1 states are larger than 5 when the center-ofmass energy is above the D D, D D ∗ , D ∗ D ∗ threshold. The dependence of the mass shifts on the center-of-mass energy is nontrivial especially when the center-of-mass energy is below the threshold. The analytic results can be extended to the b quark sector directly.
1 Introduction The energy spectrum of hadrons is a basic topic in the strong interaction. Up to now, it is still an unsolved problem due to the complex nonperturbative property of QCD. In literatures, many phenomenological models have been developed to study this problem in the quark level, such as the quark model [1–3], QCD sum rules [4–8], Bethe–Salpeter equation [9–22], and etc. In these methods, usually the annihilation effects are neglected since they are much smaller than the non-perturbative potential. Physically, if the annihilation effects can be taken as small comparing with the interaction which binds the quarks, then the imaginary part of the annihilation amplitude is corresponding to the branch decay width and the real part is corresponding to the perturbative mass a e-mail:
shift. Theoretically such annihilation effects should be considered and estimated carefully when aiming to understand the energy spectrum precisely. Experimentally, since 2003 many new charmonium-like states are reported by the collaborations of Belle [23–29], CDF [30,31], D0 [32], BABAR [33,34], Cleo-C [35], LHCb [36], BES [37], and CMS [38,39]. These charmonium-like states cannot be well understood in the traditional quark model and their masses usually lie above the open charm threshold where some new decay modes are opened. In the previous study [40,41], we studied the mass shifts of 1 S and 3 P heavy quarkonia due to the transition q q¯ → 0 J 2g → q q. ¯ Physically, when the masses of the states lie above the threshold of D or D ∗ pairs, the transitions cc¯ to these mesons’ pairs are opened. It is natural that these opened chann
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