Detecting the polarization in χcJ → φφ decays to probe hadronic loop effect

Qi Huang1,∗ Jun-Zhang Wang2,3,† Rong-Gang Ping1,4,‡ and Xiang Liu2,3,5§ 1University of Chinese Academy of Sciences (UCAS), Beijing 100049, China 2School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China 3Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China 4Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918(1), Beijing 100049, China 5Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Frontier Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China


I. INTRODUCTION
How to quantitatively depict non-perturbative behavior of strong interaction is full of challenge and opportunity. In the past two decades, more and more novel phenomena involved in hadron spectroscopy were observed with the accumulation of experimental data, which provide an ideal platform to understand non-perturbative behavior of strong interaction.
Among these reported novel phenomena, the anomalous decay behavior of χ cJ decays into two light vector mesons [1,2] has attracted theorist's attention to decode the underlying mechanism for governing these decays, where the hadronic loop mechanism  was introduced and found to be important when reproducing the measured branching ratios [6,7].
In fact, the information of branching ratios is not whole aspect involved in the χ cJ decays into two light vector mesons. Obviously, finding out other crucial evidence of hadronic loop effect on the χ cJ decays into two light vector mesons is an interesting research issue.
In this work, with the χ cJ → φφ (J = 0, 1, 2) decays as example, we show that detecting the polarization information of χ cJ → φφ can be as an effective way to probe hadronic loop mechanism. In the present work, we present a concrete polarization analysis on χ cJ → φφ associated with a calculation of the χ cJ → φφ decays when considering hadronic loop mechanism. We find that the obtained ratios of helicity amplitudes are quite stable, by which a further Monte-Carlo (MC) events of moments t i j are also generated. By this investigation, we strongly suggest that BESIII and Belle II should pay more attentions to the measurement of polarization information of χ cJ → φφ, which may provide crucial test to the hadronic loop mechanism. * Electronic address: huangqi@ucas.ac.cn † Electronic address: wangjzh2012@lzu.edu.cn ‡ Electronic address: pingrg@ihep.ac.cn § Electronic address: xiangliu@lzu.edu.cn This paper is organized as follows. After the introduction, we give a polarization analysis of the process χ cJ → φφ → 2(K + K − ) in Sec. II. And then, the detailed calculation of χ cJ → φφ via charmed meson loop is presented in Sec. III. After that, the numerical results are given in Sec. IV. Finally, this paper ends with a summary.

II. POLARIZATION ANALYSIS
We analyze the φφ polarization with the motivation to reveal the χ cJ decay mechanism. In the unpolarized e + e − collider, the production of ψ(2S ) particle is tensor polarized, without longitudinal polarization [25]. The subsequent ψ(2S ) → γχ cJ decay may transfer some polarization to the χ cJ states, which is manifested in the χ cJ → φφ decay, showing up the unflat angular distribution of the decayed φ meson.
The spin density matrix (SDM) for the φφ system encodes the full polarization information, transferred from the χ cJ decays. In experiment, the measurement on the φφ SDM plays the role to study the χ cJ decay mechanism, given that the polarization patten is predicted based on the decay-dynamical models. We follow the standard way to construct the SDM for the identical particle φφ system.
For a spin-s particle, its spin density matrix is given in terms of multipole parameters, r L M , as [26] where I denotes a (2s + 1) × (2s + 1) dimensional unit matrix. The SDM for φφ system can be constructed from the φ individual ones, and an easy way is to decompose it into Q matrices multiplied by a set of real parameters, which reads arXiv:2102.07104v1 [hep-ph] 14 Feb 2021 FIG. 1: Helicity system and angles definition for the ψ(2S ) → γχ cJ , χ cJ → φφ, φ → K + K − process. as Here, I 3 denotes a 3 × 3 identity matrix. And, the real parameters C i j is determined from the φφ production process, which carry polarization information for the two φ mesons. C i0 or C 0i means that the polarization is detected only for one φ meson, while C i j measures the polarization correlation between two φφ mesons. And then, The polarization of φφ system is unaccessible in a general purpose of electromagnetic spectrometer at the modern e + e − colliders. Nonetheless, the subsequential decay, φ → K + K − , can be used as the polarimeter to measure the φ polarization by studying the implications of the decayed Kaon angular distribution.
We formulate the φφ → 2(K + K − ) decays with helicity amplitude method, which is defined in the helicity system as shown in Fig. 1. One φ decaying into K + K − pair is described with helicity angles (θ 2 , φ 2 ), where θ 2 is the angle spanned between the directions of K + and the φ momenta, which are defined in the rest frames of their respective mother particles. The azimuthal angle φ 2 is defined as the angle between the φφ production plane and the φ decay plane. The helicity angles, (θ 3 , φ 3 ), describing another φ meson decay, is defined by following the same rule (see Table I). Then the joint angular distribution for φφ → 2(K + K − ) reads as

Decay
Angles Amplitude For simplicity, we take f 2 = 1. The joint angular distribution can be further decomposed into the φφ polarization in terms of the real multipole parameters C i j . The t i j factors play the role of the spin observables corresponding to the parameters C i j . The term t 00 is the unpolarization cross section, while t 0L (t L0 ) corresponds to the observable for detecting one φ polarization with rank L, and leaving another φ polarization being undetected. The term t i j denotes the spin correlation between the two φ's. Expressions of t i j factors are given in terms of angles θ i and φ i (i = 2, 3) as shown in Appendix A. The multipole parameters, C i j , in the ρ φφ SDM contain the dynamical information of the χ cJ → φφ decays, which can be related to the decay helicity amplitudes F (J) λ 1 ,λ 2 . Thus, any theoretical prediction on their values can be tested by measuring their spin observables in experiment.
We relate the parameter C i j to the helicity amplitude F (J) λ 1 ,λ 2 by calculating the spin density matrix ρ φφ of the decay χ cJ → φφ, which reads as where ρ J is a spin density matrix for χ cJ with J = 0, 1, 2 for χ c0 , χ c1 and χ c2 , respectively. N denotes decay matrix, which can be written as where (θ 1 , φ 1 ) are the helicity angles describing the φ meson flying direction as shown in Fig. 1. Azimuthal φ 1 is defined as the angle between the φ production and decay planes, while θ 1 is the angle spanned between the φ and χ cJ momenta. F (J) denotes the helicity amplitude in terms of two φ helicity values λ 1 and λ 2 . A special decay is χ c0 → φφ, where the χ c0 spin density is reduced to Kronecker delta function, i.e., ρ 0 = δ λ 1 ,λ 2 δ λ 1 ,λ 2 . Then the multipole parametes C i j are calculated to be while other C i, j parameters are vanishing due to the spinparity conservation in the χ c0 → φφ decays. Then, with the helicity amplitude F (0) λ 1 ,λ 2 , the φ angular distribution from the χ c0 → φφ decay can be expressed as W 0 ∝ cos (φ 23 ) 4 sin 2 (θ 2 ) sin 2 (θ 3 ) cos (φ 23 One can see that the φ angular distribution for the χ c0 → φφ decay is reduced to a uniform distribution either on the cos θ 2 (cos θ 3 ) or φ 2 (φ 3 ) observables alone. Spin correlation for φφ system can only be observed by measuring a moment formed by the angles θ i and φ i (i = 2, 3) simultaneously.
The φ meson has nonzero decay width, the masses of two φφ may have different values from the χ c1 decay in a given event. However, its narrow decay width allows us to treat the φφ as an identical particle system statistically. Then, Exchanging two φ mesons yields asymmetry relation 0,1 , and F (1) 1,1 = 0, where the joint angular distribution is independent on the amplitude, and it reads Similarly, we perform the same analysis of the χ c2 → φφ decay, and we take the χ c2 SDM as ρ 2 = 3 20 diag{2, 1, 2/3, 1, 2} [25]. Take into consideration of parity conservation in this decay, one has the relation With these considerations, the multipole parameters are calculated and given in Appendix C, and these expressions can be further simplified using the relation F (2) λ 1 ,λ 2 = F (2) λ 2 ,λ 1 if one takes the φφ as an identical particle system.

III. MESON LOOP EFFECTS IN χ cJ → φφ DECAY
Under the hadronic loop mechanism, the χ cJ → φφ decays occur via the triangle loops composed of D ( * ) (s) andD ( * ) (s) , where these loops play the role of bridge to connect the initia χ cJ and final states. In Figs. 2-4, we present the Feynman diagrams depicting the χ cJ → φφ transitions.
To calculate the decay amplitudes shown in Fig. 2-4, we adopt the effective Lagrangian approach, thus at first we should introduce the effective Lagrangians relevant to our calculation. For the interaction between χ cJ and a pair of heavylight mesons, the general form of the effective Lagrangian can be constructed under the chiral and heavy quark limits [27]  where P (QQ) and H (Qq) denote the P-wave multiplet of charmonia and (D, D * ) doublet, respectively. Their detailed expressions, as shown in Ref. [10,[27][28][29], can be written as For the interaction between a light vector meson and two heavy-light mesons, the general form of the Lagrangian reads as [27,[30][31][32][33][34] and a vector octet V has the form By expanding the Lagrangians in Eqs. (18) and (21), we can get the following explicit forms of Lagrangians With these Lagrangians given in Eq. (26) and Eq. (27), the amplitudes of χ cJ → φφ then can be written out. For χ c0 → φφ transition, withg µν (p) ≡ −g µν + p µ p ν m 2 p the amplitudes corresponding to Fig. 2 are In the similar way, the amplitudes of χ c1 → φφ and χ c2 → φφ can be written out, which are collected into Appendix D and Appendix E, respectively.
With Eqs. (28)(29)(30)(31), considering charge conjugation and isospin symmetries, the polarized amplitudes of χ cJ → φφ read as where i, λ 1 and λ 2 denote the helicities of χ cJ and two φ mesons, respectively, M q (J− j) and M s (J− j) represent that the triangle loops are composed of charmed and charmed-strange mesons, respectively.
Thus, the helicity amplitudes can be calculated by the following expression where ρ J is the SDM given in Sec. II, i.e., Finally, the general expression of the decay widths of χ cJ → φφ processes reads as where factor δ should be introduced if the final states are identical particles. Thus, for the discussed χ cJ → φφ transitions, we should take δ = 1.

A. Helicity amplitudes
With the formula given in Sec. III, now we can estimate all the helicity amplitudes F (J) λ 1 ,λ 2 . Besides the masses taken from the Particle Data Group (PDG) [35], other input parameters include the coupling constants, the mixing angle θ between ω p and φ p , and the parameter α Λ that appears in the expression of form factor F (q 2 ). For the coupling constants relevant to the interactions between χ cJ and D ( * ) (s)D ( * ) (s) , in the heavy quark limit, they are related to one gauge coupling constant g 1 given in Eq. (18), i.e., is from Refs. [10,22] and f χ c0 = 0.51 GeV is the decay constant of χ c0 [10,22]. Similarly, the coupling constants of D ( * ) with β = 0.9 and λ = 0.56 GeV −1 . Additionally, we have g V = m ρ / f π associated with the pion decay constant f π = 132 MeV [30][31][32][33].
For the mixing angle θ between ω p and φ p , since the experimental measurement of the braching ratio of the double-OZI suppressed process χ c1 → ωφ is not zero [1,2], the mixing of ω p and φ p should not be ideal, i.e., θ 0. Thus, following the results of Refs. [7,[36][37][38], in this work we also set θ = (3.4 ± 0.2) • to calculate the helicity amplitudes F (J) λ 1 ,λ 2 . Then, by using the experimental data of the branching ratios of χ cJ → φφ processes, the value α Λ can be determined. During our calculation, we find that to reproduce all the branching ratios B(χ cJ → φφ) given by PDG [35] simultaneously, α Λ should be in the interval [1.15,1.35], which obeys the requirement that the cutoff Λ should not be too far away from the physical mass of the exchanged mesons [20] and is consistent with the value given by [7].
With the above preparations, finally it is very exciting for us to find that the ratios between these helicity amplitudes are quiet stable when changing α Λ and θ, which are different from the behavior of individual F (J) λ 1 ,λ 2 . When scanning α Λ ∈ [1.15, 1.35] and θ = (3.4 ± 0.2) • ranges, we get for the χ c0 → φφ decay, for the χ c1 → φφ decay, and for the χ c2 → φφ case. Thus, these ratios of helicity amplitude receives the long distance contributions and characterize the loop effects in the χ cJ → φφ decays. We expect that they can be measured in the future, and used for testing the hadron loop mechanism.

B. Polarization observables
Apart from the directly measurements on the ratios given in Sec. IV A, the t i j moments, t i j , can also be selected as the spin observables, since their distributions are directly related to the helicity amplitude F (J) λ 1 ,λ 2 . The t i j observables are constructed only with the Kaon angles in φ decays. Thus, the t i j moments should be independent on any parameter from theoretical investigations. In experiment, the t i j moments are defined as where |M| 2 denotes the joint angular distribution for the χ cJ → φφ → 2(K + K − ) decay and dΩ i = d cos θ i dφ i (i = 2, 3) is the angles to be integrated out. I 0 is the normalization factor.
One exception is the χ c0 → φφ decay, in which the multipole parameters C i j are independent on the angles of θ 1 or φ 1 . Thus, the t i j moments are uniformly distributed, and they can not be used as observable. Instead, we chose an observable µ = sin 2 θ 2 sin 2 θ 3 to express two φ spin entanglements produced from the χ c0 decays. With the joint angular distribution W 0 , one has An ensemble of events is generated by using the χ c0 decay amplitude W 0 . And the ratio of amplitude is fixed to the central value of calculation, namely, |F (0) 1,1 /F (0) 0,0 | = 0.359. The sin 2 θ 2 sin 2 θ 3 moment of these TOY Monte-Carlo (MC) events is shown in Fig. 5. One can see that the MC distribution is consistent with the expectation of 1 + 2 cos 2 (φ 2 + φ 3 ). For the χ c1 → φφ decay, it conserves parity and the decay amplitude respects the identical particle symmetry when exchanging two φ mesons. Thus, the helicity amplitudes are able to factor out as an overall factor in the angular distribution. The φ angular distribution is independent on the amplitudes, and it is reduced to which corresponds to the observation of moment t 00 for the χ c1 → φφ decay. We generate an ensemble of MC events for the χ c1 decay with the amplitudes constrained by the requirements of parity conservation and the identical particle symmetry, namely, Figure 6 shows the angular distribution for the φ meson from the χ c1 decays. One can see that the distribution is well consistent with the expected one as given by Eq. (46). One significant feature of t i j moments for χ c1 decays is that their distributions are well determined only with the fundamental conservation rule and symmetry relations, being independent on the helicity amplitudes F (1) λ 1 ,λ 2 . For example, some t i j moments are determined to be t 80 , t 08 , t 68 , t 86 ∝ 1 − cos 2 θ 1 .

V. SUMMARY
The anomalous decay behaviors of the χ cJ → VV (VV = ωω, ωφ and φφ) transitions [1,2] indicate that the nonperturbative effect of strong interaction cannot be ignored. For reflecting non-perturbative effect of strong interaction, hadronic loop mechanism is adopted to study the branching ratios of χ cJ → VV (VV = ωω, ωφ and φφ) processes [6,7]. Although the measured branching ratios of the χ c1 → ωω and χ c1 → φφ processes can be reproduced well, it is not the end of whole story. In fact, we still want to find more crucial information to reflect the evidence of the hadronic loop mechanism existing in the χ cJ → ωφ processes [1,2]. Inspired by Refs. [7,39,40], we propose that the polarization information of the χ cJ → VV decay can be applied to probe the hadronic loop mechanism, which becomes main task of this work. The advantage to choose the χ cJ → φφ decay is due to the factor that two φ decays provide a rich spin observables. Another advantage is that these decays are accessible in experiment with high detection efficiency and two φ mesons are cleanly reconstructed with low level backgrounds. A high statistics allow one to perform the angular distribution analyses and get the information on the φ polarization, which can shed light on the underlying decay mechanism for the χ cJ → VV decays.
Under the framework of hadronic loop mechanism, we find that the ratios of the helicity amplitudes of the χ cJ → φφ processes are quiet stable, where we scan the ranges of θ and α Λ , which are the mixing angle between ω p and φ p and the free parameter of the form factor, respectively. Thus, we suggest that these ratios can be as important observable quantities, which can be accessible at future experimental measurement as crucial test to hadronic loop mechanism.
In addition, by using the predicted amplitude ratios, we show that the observation of moments t i j can be used to manifest the nontrivial polarization behavior. For the χ c0 decays, the choice of the spin observable is quite limited due to the fact that the total spin of the φφ system is constrained to be zero in the spin triplet. Thus, the spins of two φ mesons are antiparallel for the χ c0 decays. For the χ c1 → φφ decays, the helicity amplitudes can be well determined by considering parity conservation and by applying the symmetry relationship to take the φφ as identical particle system. For the χ c2 → φφ decays, the abundant information of the φφ spin configurations allows us to directly detect the helicity amplitudes from the observation of different t i j moments. The patterns of these moments are presented based on the predicted amplitude ratios, which can be tested by expeirment in the near future.
In 2019, BESIII released white paper on its future physics program [41]. With the accumulation of charmonium data, we suggest that BESIII should pay more attentions to the study of polarization of the corresponding decays, which may provide extra information to reveal underlying mechanism. Obviously, the present work provides a typical example and a new task for experiment.
The multipole parameters for χ c2 → φφ are