Effects of heavy meson loops on higher charmonia radiative transitions

Radiative transitions between charmonium states offer an insight into the internal structure of heavy quark bound states within QCD. In this work, we performed a systematically investigation of the radiative transitions of ψ(nS)→γχcJ(mP)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (nS) \rightarrow \gamma \chi _{cJ}(mP)$$\end{document} and ψ(nD)→γχcJ(mP)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (nD) \rightarrow \gamma \chi _{cJ}(mP)$$\end{document} via intermediate charmed meson loop with a non-relativistic effective field theory. We only focus on the line-shape behavior of the distributions via different intermediate meson loops. Our results show that the coupled-channel effects of these decays are relatively weak, while the coupled channel effects has obvious enhancement when the inial (final) states are closed to the thresholds of charmed-mesons pair.


Introduction
Heavy quarkonia decays have played an important role in the hadronic physics since the first charmonium J/ψ was discovered in November 1974 [1][2][3]. Until now, experimental progresses of the heavy quarkonia [4] have provided great opportunities for examining many interesting properties of Quantum Chromodynamics (QCD). Meanwhile, many theoretical studies, such as potential models, lattice QCD (LQCD), non-relativistic QCD (NRQCD), QCD sum rules, have been performed [5][6][7]. However, there are still many unresolved problems in charmonium physics [5]. More and more XYZ states have been announced by different experiments [4]. Not all of them can be described by conventional qq quark model [7][8][9][10]. Many investigations on the production and decay of these XYZ states have been carried out in order to understand their nature (see Refs. [11][12][13] for a review).
The IML transition is regarded as an important nonperturbative transition mechanisms which has a long history [27][28][29] and recently are widely used to study the production and decays of ordinary and exotic states . Radiative transitions of charmonium states are of interest largely because they provide one of the few pathways between charmonium states with different quantum numbers. In this work, we will investigate the radiative transitions ψ(nS)/ψ(n D) → γ χ cJ (m P) via the intermediate charmed meson loops (IML) in a nonrelativistic effective field theory (NREFT). This paper is organized as follows: In Sect. 2, we will introduce the formulas of the relevant effective lagrangian employed in this work. In Sect. 3, the numerical results are presented. A summary will be given in Sect. 4.

Radiative decay
In principle, one should include all the possible intermediate-meson-exchange loops in the calculation, however, the break-down of the local quark-hadron duality allows us pick up the leading contributions as a reasonable approximation [29,30]. The initial charmonium with J PC = 1 −− can couple to either two S-wave charmed mesons in a P-wave, or one S-wave and one P-wave charmed mesons in an S-or Dwave. As is discussed in Ref. [62], the mechanism with an S-wave coupling to the initial 1 −− charmonium will greatly facilitate these similar processes. Therefore, we investigate the meson loops listed in Table 1 as the major contributions to ψ(nS)/ψ(n D) → γ χ cJ (m P).
In Table 1, we list the possible charmed meson loops contributing to ψ(nS)/ψ(n D) → γ χ cJ (m P). There are three charmed mesons in each loop. To be specific, we denote the one connecting the initial charmonium and the photon as M1, the exchanged meson as M2 and one connecting two charmonia as M3. The meson Mi has mass m i . For example, in Fig. 1, we plot all the dominant meson loops contributing to ψ(nS) → γ χ c0 (m P), M1, M2 and M3 are the D 1 ,D and D, respectively in Fig. 1a.
Based on the heavy quark symmetry, the relevant effective Lagrangians used in this work are as follows [63,64], L S = g S T r R ccS2iH1i + H.c., with where v μ is the heavy quark four-velocity. H A and K A are the effective fields of the various members of the multiplets with total spin J = A. The charmed meson doublets are collected into the following superfields [64]: where ) and i is the light flavor index. Here, we should notice that every heavy field H will contain a factor √ m H for normalization in the heavy hadron chiral perturbation theory. Table 2 The initial and final charmonium mass spectrum (in unit of GeV) used in this work. The measured masses from the PDG [4], the calculated with screened potential (SP) in Ref. [102], linear potential (LP) in Ref. [67], and the results with SP and LP in [74]   In addition, the lagrangian relevant to the charmed mesons and photon are expressed as [63,65], where F μν = ∂μA ν − ∂ν A μ is the electromagnetic field tensor, and Q = diag{2/3, −1/3, −1/3}. The explicit expression of transition for ψ(nS) → γ χ cJ (n P) and ψ(n D) → γ χ cJ (n P) amplitudes are given in Appendix.

Numerical results
The ψ(4040) and ψ(4160) are widely accepted as the ψ(3S) state [67][68][69][70] and ψ(2D) state [67][68][69], respectively. In Ref. [69], the author argue that the ψ(4260) state can be assigned as 4S charmonium state under the Dirac potential model. The ψ(4360) state was explained as a 3D state from a quark potential model calculation [71], while the ψ(4360) was explained to be the 4S state [69]. In Refs. [62,67,68], the ψ(4415) was widely assigning to be the 4S state, while the ψ(4415) is taken to be ψ(5S) state in [73]. The ψ (4415) was also interpreted as 3D or 4D charmonium state depending on different Dirac potential models [69]. In Ref. [71], ψ(4660) was explained as the 5S vector charmonium state. The author proposed that ψ(4660) belongs to the 4D state under potential models [69]. The X (3927) was observed in the γ γ → DD process by Belle [75] and Babar [76] collaborations, and has been a good candidate for χ c2 (2P) state. In the LP model the calculated mass of χ c0 (3P) is about 4310 MeV [74], which is very close to the mass of X (4350). The higher charmonium states with J PC = 1 ++ , such as χ c1 (2P), and χ c1 (3P), are still not established. The X (3872) resonance has the same quantum numbers as χ c1 (2P) but with a much lighter mass than potential quark model predictions. The quantum numbers of the charmoniumlike states X (4140) and X (4274) observed by the LHCb Collaboration [16] are determined to be J PC = 1 ++ . According to the predicted mass from the linear potential model, the X (4274) might be a good candidate of χ c1 (3P). However, within the screened potential model, X (4140) seems to favor the χ c1 (3P) state. The charmonium-like state X * (3860) observed in the process e + e − → J/ψ DD by the Belle collaboration [72] serves as a good candidate for the χ c0 (2P) state. The measured mass and width fit the expectation of the χ c0 (2P) state predicted in the potential models [74]. In Table 2, we list the initial and final charmonium mass spectrum used in this work. Here, we mainly focus on the line-shape behavior of the distributions via different intermediate meson loops.

ψ(nS) → γ χ cJ (m P) radiative decays
In Fig. 2 We notice that there are two anomalous peaks appears in Fig. 4b, whose positions are neither the genuine particles nor the threshold of D s JDs . Therefore it is possible that the threshold effects illustrated in Fig. 4b may result in some new resonance. So, our results shows that the coupled-channels effects may be more important when the initial states are close to the thresholds of charmed-mesons pair. It also shows that the decay widths of the excited state χ c0 (3P) and χ c1 (3P) could be quite small.
The numerical results for χ cJ invariant mass distributions corresponding to these rescattering processes ψ(n D) → γ χ cJ via intermediate meson loops are displayed in Fig. 5. We choose four different masses of ψ(n D) [4,67].  Fig. 5a. Two obvious peaks in Fig. 5b also appeared in the position of D 0D * 0 and D * + s D − s threshold. In Fig. 5c, there are two peaks in the position of D * + D * − and D * + s D * − s thresholds for ψ(n D) → γ χ c2 (m P). Here, one should notice the coupled channels effects and phase space effects will both affect these cusps. For ψ(4D) → γ χ c0,1 (m P) in Fig. 5a, b, the phase space may affect the cusp strength more important with the increasing mass of χ c0,1 (m P). For ψ(4D) → γ χ c2 (m P) in Fig. 5c, the coupled channel effects may dominate the cusp strength.

Summary
In this work, we have systematically investigated these radiative transitions of ψ(nS) → γ χ cJ (m P) and ψ(n D) → γ χ cJ (m P) via intermediate charmed meson loop(IML) with NREFT. We present the line-shape behaviors of the distributions via different intermediate meson loops. We only focus on the line-shape behavior of the distributions via different intermediate meson loops. Our results show that the coupledchannel effects of these decays are relatively weak, while the coupled channel effects has obvious enhancement when the inial (final) states are closed to the thresholds of charmedmesons pair. The coupled-channel effects in the channels of excited states should be testable experimentally in the future. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors' comment: This is a theoretical study and no experimental data has been listed.] Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP 3 .

Appendix: The transitions amplitude
In this appendix, we present the transitions amplitudes for the intermediate meson loops in Table 1. p 1 , p 2 , p 3 are the four-vector momenta for the initial charmonium, final photon and final charmonia, respectively. q 1 , q 2 , and q 3 are the fourvector momenta for the intermediate charmed mesons. 1 , 2 and 3 are the polarization vectors for the initial stat, final photon, and final charmonium, respectively. The velocity v can be taken as (1, 0, 0, 0) in the static limit. αβμν is the antisymmetric Levi-Civita tensor and 0123 = +1.
Here the coupling constants g S , g P , and g D may be determined by the relevant data of Refs. [74,104]. Recently, the ψ(4415) is widely accepted to be ψ(4S) state. With  [74,104], we get g P 1.28 GeV −1/2 . We get g D 0.30GeV −1/2 with M ψ(3D) = 4.486 GeV and ψ(3D)→D D 1 = 9.8 MeV. The coupling constantsβ is relevant to radiative decay rates of the charmed mesons and can be extracted according to the experimental data. We take the values of |β| = 0.42 GeV −1 estimated in Ref. [65]. There is no experimental measurement on the radiative decays of the P-wave charmed mesons. We take theμ 0.47 GeV −1 with the help of the predictions of (D 0 1 → D ( * )0 γ ) [66].