Detecting the polarization in χ_{cJ} \to φφ decays to probe hadronic loop effect
DDetecting the polarization in χ cJ → φφ decays to probe hadronic loop e ff ect Qi Huang , ∗ Jun-Zhang Wang , , † Rong-Gang Ping , , ‡ and Xiang Liu , , § University of Chinese Academy of Sciences (UCAS), Beijing 100049, China School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918(1), Beijing 100049, China Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,and Frontier Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China
In this work, we propose that detecting the polarization information of χ cJ → φφ ( J = , ,
2) can be asgood test of the role of hadronic loop e ff ect on these decays. Our results shows that the obtained ratios ofhelicity amplitudes are quite stable, which are | F (0)1 , / F (0)0 , | ≈ . | F (1)1 , / F (1)0 , | = | F (1)1 , / F (1)0 , | = | F (2)1 , | / | F (2)0 , | = | F (2)0 , | / | F (2)0 , | ≈ . , | F (2)1 , − | / | F (2)0 , | = | F (2) − , | / | F (2)0 , | ≈ .
110 and | F (2) − , − | / | F (2)0 , | = | F (2)1 , | / | F (2)0 , | ≈ . (cid:104) t ij (cid:105) , which are directly related tothe determination of helicity amplitudes. We suggest further experiments like BESIII and Belle II to performan analysis on the polarizations of the χ cJ → φφ process in the future, which is important to understand theunderlying decay mechanism existing in χ cJ decays. I. INTRODUCTION
How to quantitatively depict non-perturbative behavior ofstrong interaction is full of challenge and opportunity. In thepast two decades, more and more novel phenomena involvedin hadron spectroscopy were observed with the accumulationof experimental data, which provide an ideal platform to un-derstand non-perturbative behavior of strong interaction.Among these reported novel phenomena, the anomalousdecay behavior of χ cJ decays into two light vector mesons[1, 2] has attracted theorist’s attention to decode the un-derlying mechanism for governing these decays, where thehadronic loop mechanism [3–24] was introduced and foundto be important when reproducing the measured branching ra-tios [6, 7].In fact, the information of branching ratios is not wholeaspect involved in the χ cJ decays into two light vector mesons.Obviously, finding out other crucial evidence of hadronic loope ff ect on the χ cJ decays into two light vector mesons is aninteresting research issue.In this work, with the χ cJ → φφ ( J = , ,
2) decays asexample, we show that detecting the polarization informationof χ cJ → φφ can be as an e ff ective way to probe hadronicloop mechanism. In the present work, we present a concretepolarization analysis on χ cJ → φφ associated with a calcula-tion of the χ cJ → φφ decays when considering hadronic loopmechanism. We find that the obtained ratios of helicity ampli-tudes are quite stable, by which a further Monte-Carlo (MC)events of moments (cid:104) t i j (cid:105) are also generated. By this investi-gation, we strongly suggest that BESIII and Belle II shouldpay more attentions to the measurement of polarization infor-mation of χ cJ → φφ , which may provide crucial test to thehadronic loop mechanism. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
This paper is organized as follows. After the introduction,we give a polarization analysis of the process χ cJ → φφ → 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 re-veal the χ cJ decay mechanism. In the unpolarized e + e − collider, the production of ψ (2 S ) particle is tensor polar-ized, without longitudinal polarization [25]. The subsequent ψ (2 S ) → γχ cJ decay may transfer some polarization to the χ cJ states, which is manifested in the χ cJ → φφ decay, show-ing up the unflat angular distribution of the decayed φ meson.The spin density matrix (SDM) for the φφ system encodesthe full polarization information, transferred from the χ cJ de-cays. In experiment, the measurement on the φφ SDM playsthe role to study the χ cJ decay mechanism, given that the po-larization patten is predicted based on the decay-dynamicalmodels. We follow the standard way to construct the SDM forthe identical particle φφ system.For a spin- s particle, its spin density matrix is given in termsof multipole parameters, r LM , as [26] ρ = s + I + s s (cid:88) L = L (cid:88) M = − L r LM Q [ s , L , M ] , (1)where I denotes a (2 s + × (2 s +
1) dimensional unit ma-trix. 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 a r X i v : . [ h e p - ph ] F e b !" $ % & % ’ $$ ( & ) * + ’ , ) - ) . * - * . !" * / ) ) % & % ’ % & % ’ % & % ’ ) + CM frame frame )) FIG. 1: Helicity system and angles definition for the ψ (2 S ) → γχ cJ , χ cJ → φφ, φ → K + K − process. as ρ φφ = ρ φ ⊗ ρ φ = C I ⊗ I + (cid:88) i = ( C i , Q i ⊗ I + C i I ⊗ Q i ) + (cid:88) i , j = C i j Q i ⊗ Q j . (2)Here, I denotes a 3 × C i j is determined from the φφ production pro-cess, which carry polarization information for the two φ mesons. C i or C i means that the polarization is de-tected only for one φ meson, while C i j measures the po-larization correlation between two φφ mesons. And then, Q = Q [1 , , − , Q = Q [1 , , , Q = Q [1 , , , Q = Q [1 , , − , Q = Q [1 , , − , Q = Q [1 , , , Q = Q [1 , , , Q = Q [1 , , φφ system is unaccessible in a generalpurpose 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 polarizationby studying the implications of the decayed Kaon angular dis-tribution.We formulate the φφ → K + K − ) decays with helicity am-plitude method, which is defined in the helicity system asshown in Fig. 1. One φ decaying into K + K − pair is describedwith helicity angles ( θ , φ ), where θ is the angle spannedbetween the directions of K + and the φ momenta, which aredefined in the rest frames of their respective mother particles.The azimuthal angle φ is defined as the angle between the φφ production plane and the φ decay plane. The helicity an-gles, ( θ , φ ), describing another φ meson decay, is defined byfollowing the same rule (see Table I). Then the joint angulardistribution for φφ → K + K − ) reads as TABLE I: Definitions of helicity angles and amplitudes in the χ cJ → φφ , and φφ → K + K − ) decays.Decay Angles Amplitude χ cJ → φ ( λ ) φ ( λ ) ( θ , φ ) F ( J ) λ ,λ φ ( λ ) → K + K − ( θ , φ ) f φ ( λ ) → K + K − ( θ , φ ) f |M| ∝ Tr[ ρ φφ · M a ⊗ M † b ] = t C + (cid:88) i = ( t i C i + t i C i ) + (cid:88) i , j = t i j C i j (3)with ( M a ) λ ,λ (cid:48) = D ∗ λ , ( φ , θ , D λ (cid:48) , ( φ , θ , f , (4)( M b ) λ ,λ (cid:48) = D ∗ λ , ( φ , θ , D λ (cid:48) , ( φ , θ , f . (5)For simplicity, we take f =
1. The joint angular distributioncan be further decomposed into the φφ polarization in terms ofthe real multipole parameters C i j . The t i j factors play the roleof the spin observables corresponding to the parameters C i j .The term t is the unpolarization cross section, while t L ( t L )corresponds to the observable for detecting one φ polarizationwith rank L , and leaving another φ polarization being unde-tected. The term t i j denotes the spin correlation between thetwo φ ’s. Expressions of t i j factors are given in terms of angles θ i and φ i ( i = ,
3) as shown in Appendix A.The multipole parameters, C i j , in the ρ φφ SDM contain thedynamical information of the χ cJ → φφ decays, which can berelated to the decay helicity amplitudes F ( J ) λ ,λ . Thus, any the-oretical prediction on their values can be tested by measuringtheir spin observables in experiment.We relate the parameter C i j to the helicity amplitude F ( J ) λ ,λ by calculating the spin density matrix ρ φφ of the decay χ cJ → φφ , which reads as ρ φφ = N · ρ J · N † , (6)where ρ J is a spin density matrix for χ cJ with J = , , χ c , χ c and χ c , respectively. N denotes decay matrix, whichcan be written as( N ) λ λ λ (cid:48) λ (cid:48) , M = D J ∗ M ,λ − λ ( φ , θ ,
0) (7) × D JM ,λ (cid:48) − λ (cid:48) ( φ , θ , F ( J ) ∗ λ ,λ F ( J ) λ (cid:48) ,λ (cid:48) , where ( θ , φ ) are the helicity angles describing the φ mesonflying direction as shown in Fig. 1. Azimuthal φ is defined asthe angle between the φ production and decay planes, while θ is the angle spanned between the φ and χ cJ momenta. F ( J ) λ ,λ denotes the helicity amplitude in terms of two φ helicity values λ and λ .A special decay is χ c → φφ , where the χ c spin densityis reduced to Kronecker delta function, i.e. , ρ = δ λ ,λ δ λ (cid:48) ,λ (cid:48) .Then the multipole parametes C i j are calculated to be C = (cid:12)(cid:12)(cid:12) F (0)0 , (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F (0)1 , (cid:12)(cid:12)(cid:12) , (8) C = − (cid:12)(cid:12)(cid:12) F (0)1 , (cid:12)(cid:12)(cid:12) , (9) C = − C = − (cid:16) F (0) ∗ , F (0)0 , + F (0) ∗ , F (0)1 , (cid:17) , (10) C = C = (cid:12)(cid:12)(cid:12) F (0)1 , (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) F (0)0 , (cid:12)(cid:12)(cid:12) , (11) C = (cid:12)(cid:12)(cid:12) F (0)0 , (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F (0)1 , (cid:12)(cid:12)(cid:12) , (12) C = (cid:12)(cid:12)(cid:12) F (0)1 , (cid:12)(cid:12)(cid:12) , (13)while other C i , j parameters are vanishing due to the spin-parity conservation in the χ c → φφ decays.Then, with the helicity amplitude F (0) λ ,λ , the φ angular dis-tribution from the χ c → φφ decay can be expressed as W ∝ cos ( φ ) (cid:104) ( θ ) sin ( θ ) cos ( φ ) (cid:12)(cid:12)(cid:12) F (0)1 , (cid:12)(cid:12)(cid:12) + sin (2 θ ) sin (2 θ ) 2Re(F (0) ∗ , F (0)0 , ) (cid:105) (14) + ( θ ) cos ( θ ) (cid:12)(cid:12)(cid:12) F (0)0 , (cid:12)(cid:12)(cid:12) , where φ = φ + φ .One can see that the φ angular distribution for the χ c → φφ decay is reduced to a uniform distribution either on thecos θ (cos θ ) or φ ( φ ) observables alone. Spin correlationfor φφ system can only be observed by measuring a momentformed by the angles θ i and φ i ( i = ,
3) simultaneously.The strong decay χ c → φφ conserves the parity. Thus,the helicity amplitudes satisfy the relation F (1) − λ , − λ = − F (1) λ ,λ ,especially F (1)0 , =
0. And then, the amplitudes are reduced tothree independent components, i.e., F (1)1 , , F (1)1 , and F (1)0 , . The matrix of amplitudes is taken as (cid:16) F (1) λ ,λ (cid:17) = F (1)1 , F (1)1 , − F (1)1 , − F (1)0 , − F (1)1 , − F (1)1 , . . (15)As for the χ c production from the decay ψ (2 S ) → γχ c , itsSDM is well defined and taken as ρ = diag { , , } [25] inits rest frame. Here, the nonvanishing parameters C i , C i and C i j are calculated and given in Appendix B.The φ meson has nonzero decay width, the masses of two φφ may have di ff erent values from the χ c decay in a givenevent. However, its narrow decay width allows us to treatthe φφ as an identical particle system statistically. Then, Ex-changing two φ mesons yields asymmetry relation F (1)1 , = − F (1)0 , , and F (1)1 , =
0, where the joint angular distribution isindependent on the amplitude, and it reads W ∝ (2 + sin θ )[cos θ sin θ + sin θ cos θ ] . (16)Similarly, we perform the same analysis of the χ c → φφ decay, and we take the χ c SDM as ρ = diag { , , / , , } [25]. Take into consideration of parity conservation in this de-cay, one has the relation F (2) − λ , − λ = F (2) λ ,λ , then the amplitudematrix is reduced to be (cid:16) F (2) λ ,λ (cid:17) = F (2)1 , F (2)1 , F (2)1 , − F (2)0 , F (2)0 , F (2)0 , F (2)1 , − F (2)1 , F (2)1 , . (17)With these considerations, the multipole parameters are cal-culated and given in Appendix C, and these expressions canbe further simplified using the relation F (2) λ ,λ = F (2) λ ,λ if onetakes 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 decaysoccur via the triangle loops composed of D ( ∗ )( s ) and ¯ D ( ∗ )( s ) , wherethese loops play the role of bridge to connect the initia χ cJ andfinal states. In Figs. 2-4, we present the Feynman diagramsdepicting the χ cJ → φφ transitions.To calculate the decay amplitudes shown in Fig. 2-4,we adopt the e ff ective Lagrangian approach, thus at first weshould introduce the e ff ective Lagrangians relevant to our cal-culation. For the interaction between χ cJ and a pair of heavy-light mesons, the general form of the e ff ective Lagrangian canbe constructed under the chiral and heavy quark limits [27] L p = ig Tr (cid:104) P ( Q ¯ Q ) µ ¯ H ( ¯ Qq ) γ µ ¯ H ( Q ¯ q ) (cid:105) + H . c ., (18) FIG. 2: The Feynman diagrams depicting the χ c → φφ process via D meson loop.FIG. 3: The Feynman diagrams depicting the χ c → φφ process via D meson loop.FIG. 4: The Feynman diagrams depicting the χ c → φφ process via D meson loop. where P ( Q ¯ Q ) and H ( Q ¯ q ) denote the P-wave multiplet of char-monia and ( D , D ∗ ) doublet, respectively. Their detailed ex-pressions, as shown in Ref. [10, 27–29], can be written as P ( Q ¯ Q ) µ = + / v (cid:104) χ µα c γ α + √ ε µαβγ v α γ β χ c γ + √ (cid:0) γ µ − v µ (cid:1) χ c + h µ c γ (cid:105) − / v , (19) H ( Q ¯ q ) = + / v (cid:104) D ∗ µ γ µ − D γ (cid:105) , (20)respectively, with definitions D ( ∗ ) † = ( D ( ∗ ) + , D ( ∗ )0 , D ( ∗ )0 s ) and D ( ∗ ) = ( D ( ∗ ) − , ¯ D ( ∗ )0 , ¯ D ( ∗ )0 s ) T . H ( ¯ Qq ) corresponds to the doubletformed by homologous heavy-light anti-mesons, which canbe obtained by applying the charge conjugation operation to H ( Q ¯ q ) .For the interaction between a light vector meson and twoheavy-light mesons, the general form of the Lagrangian readsas [27, 30–34] L V = i β Tr[ H j v µ ( − ρ µ ) ij ¯ H i ] + i λ Tr[ H j σ µν F µν ( ρ ) ¯ H i ] , (21)where ρ µ = i g V √ V µ , (22) F µν ( ρ ) = ∂ µ ρ ν − ∂ ν ρ µ + [ ρ µ , ρ ν ] , (23)and a vector octet V has the form [7] V = ρ √ + κω p + ζφ p ρ + K ∗ + ρ − − ρ √ + κω p + ζφ p K ∗ K ∗− ¯ K ∗ δω p + σφ p , (24)with κ = cos θ √ , ζ = sin θ √ ,δ = − sin θ, σ = cos θ. (25)By expanding the Lagrangians in Eqs. (18) and (21), wecan get the following explicit forms of Lagrangians L χ cJ D ( ∗ ) D ( ∗ ) = − g χ c DD χ c DD † − g χ c D ∗ D ∗ χ c D ∗ µ D ∗ µ † + ig χ c DD ∗ χ µ c ( D ∗ µ D † − DD ∗† µ ) − g χ c DD χ µν c ∂ µ D ∂ ν D † + g χ c D ∗ D ∗ χ µν c ( D ∗ µ D ∗† ν + D ∗ ν D ∗† µ ) − ig χ c D ∗ D ε µναβ ∂ α χ µρ c ( ∂ ρ D ∗ ν ∂ β D † − ∂ β D ∂ ρ D ∗ ν † ) , (26) L D ( ∗ ) D ( ∗ ) V = − ig DDV D † i ↔ ∂ µ D j ( V µ ) ij − f D ∗ DV ε µναβ ( ∂ µ V ν ) ij ( D † i ↔ ∂ α D ∗ β j − D ∗ β † i ↔ ∂ α D j ) + ig D ∗ D ∗ V D ∗ ν † i ↔ ∂ µ D ∗ j ν ( V µ ) ij + i f D ∗ D ∗ V D ∗† i µ ( ∂ µ V ν − ∂ ν V µ ) ij D ∗ j ν . (27)With these Lagrangians given in Eq. (26) and Eq. (27),the amplitudes of χ cJ → φφ then can be written out. For χ c → φφ transition, with ˜ g µν ( p ) ≡ − g µν + p µ p ν m p the amplitudescorresponding to Fig. 2 are M (0 − = (cid:90) d q (2 π ) k − m D k − m D q − m D F ( q ) × [ − g χ c DD ][ − g DD φ (cid:15) ∗ ζφ ( p )( k ζ + q ζ )] × [ − g DD φ (cid:15) ∗ λφ ( p )( q λ − k λ )] , (28) M (0 − = (cid:90) d q (2 π ) k − m D k − m D ˜ g ξσ ( q ) q − m D ∗ F ( q ) × [ − g χ c DD ][ − f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × [2 f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (29) M (0 − = (cid:90) d q (2 π ) ˜ g µξ ( k ) k − m D ∗ ˜ g µσ ( k ) k − m D ∗ q − m D F ( q ) × [ − g χ c D ∗ D ∗ ][2 f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × [ − f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (30) M (0 − = (cid:90) d q (2 π ) ˜ g µψ ( k ) k − m D ∗ ˜ g ιµ ( k ) k − m D ∗ ˜ g γυ ( q ) q − m D ∗ F ( q ) × [ − g χ c D ∗ D ∗ ][ g D ∗ D ∗ ψ g ηγ g ηψ ( k ζ + q ζ ) − f D ∗ D ∗ φ p η ( g γη g ψζ − g γζ g ψη )] (cid:15) ∗ ζφ ( p ) × [ g D ∗ D ∗ φ g ρι g ρυ ( q λ − k λ ) − f D ∗ D ∗ φ p ρ × ( g ιρ g υλ − g ιλ g υρ )] (cid:15) ∗ λφ ( p ) , (31)In the similar way, the amplitudes of χ c → φφ and χ c → φφ can be written out, which are collected into Appendix Dand Appendix E, respectively.In the amplitudes of χ cJ → φφ transitions, a dipole formfactor F ( q ) = ( m E − Λ ) / ( q − Λ ) is introduced to reflectthe structure e ff ect of interaction vertices and the o ff -shell ef-fect of exchanged mesons [7]. In the expression of F ( q ), m E is the mass of the exchanged D ( ∗ ) and Λ denotes the cuto ff ,which is usually parameterized as Λ = m E + α Λ Λ QCD with Λ QCD = .
22 GeV [3, 10–20].With Eqs. (28-31), considering charge conjugation andisospin symmetries, the polarized amplitudes of χ cJ → φφ read as M J ( i , λ , λ ) = (cid:88) j M q ( J − j ) + (cid:88) j M s ( J − j ) , (32)where i , λ and λ denote the helicities of χ cJ and two φ mesons, respectively, M q ( J − j ) and M s ( J − j ) represent that the tri-angle loops are composed of charmed and charmed-strangemesons, respectively.Thus, the helicity amplitudes can be calculated by the fol-lowing expression | F ( J ) λ ,λ | = (cid:88) i ρ J ( i ) |M J ( i , λ , λ ) | , (33) where ρ J is the SDM given in Sec. II, i.e., ρ = , (34) ρ =
14 diag { , , } , (35) ρ =
320 diag { , , , , } . (36)Finally, the general expression of the decay widths of χ cJ → φφ processes reads as Γ χ cJ → φφ = + δ (cid:80) i ρ J ( i ) 18 π | (cid:126) p φ | m χ cJ (cid:88) i ,λ ,λ |M J ( i , λ , λ ) | , (37)where factor δ should be introduced if the final states are iden-tical particles. Thus, for the discussed χ cJ → φφ transitions,we should take δ = IV. NUMERICAL RESULTSA. Helicity amplitudes
With the formula given in Sec. III, now we can estimate allthe helicity amplitudes F ( J ) λ ,λ . Besides the masses taken fromthe Particle Data Group (PDG) [35], other input parametersinclude the coupling constants, the mixing angle θ between ω p and φ p , and the parameter α Λ that appears in the expressionof form factor F ( q ). For the coupling constants relevant tothe interactions between χ cJ and D ( ∗ )( s ) ¯ D ( ∗ )( s ) , in the heavy quarklimit, they are related to one gauge coupling constant g givenin Eq. (18), i.e., g χ c DD = √ g √ m χ c m D , g χ c D ∗ D ∗ = √ g √ m χ c m D ∗ , g χ c DD ∗ = √ g √ m χ c m D m D ∗ , g χ c DD = g √ m χ c m D , g χ c DD ∗ = g (cid:115) m χ c m D ∗ m D , g χ c D ∗ D ∗ = g √ m χ c m D ∗ , where g = − (cid:113) m χ c f χ c is from Refs. [10, 22] and f χ c = .
51 GeV is the decay constant of χ c [10, 22]. Similarly, thecoupling constants of D ( ∗ )( s ) ¯ D ( ∗ )( s ) φ interactions can be extractedfrom Eq. (21) g D s D s φ = g D ∗ s D ∗ s φ = β g V √ σ, f D s D ∗ s φ = f D ∗ s D ∗ s φ m D ∗ s = λ g V √ σ, g DD φ = g D ∗ D ∗ φ = β g V √ ζ, f DD ∗ φ = f D ∗ D ∗ φ m D ∗ = λ g V √ ζ with β = . λ = .
56 GeV − . Additionally, we have g V = m ρ / f π associated with the pion decay constant f π = θ between ω p and φ p , since the exper-imental measurement of the braching ratio of the double-OZIsuppressed process χ c → ωφ is not zero [1, 2], the mixingof ω p and φ p should not be ideal, i.e., θ (cid:44)
0. Thus, follow-ing the results of Refs. [7, 36–38], in this work we also set θ = (3 . ± . ◦ to calculate the helicity amplitudes F ( J ) λ ,λ .Then, by using the experimental data of the branching ratiosof χ cJ → φφ processes, the value α Λ can be determined. Dur-ing our calculation, we find that to reproduce all the branchingratios B ( χ cJ → φφ ) given by PDG [35] simultaneously, α Λ should be in the interval [1.15,1.35], which obeys the require-ment that the cuto ff Λ should not be too far away from thephysical mass of the exchanged mesons [20] and is consistentwith the value given by [7].With the above preparations, finally it is very exciting forus to find that the ratios between these helicity amplitudesare quiet stable when changing α Λ and θ , which are di ff er-ent from the behavior of individual F ( J ) λ ,λ . When scanning α Λ ∈ [1 . , .
35] and θ = (3 . ± . ◦ ranges, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (0)1 , F (0)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . ± . , (38)for the χ c → φφ decay, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (1)1 , F (1)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , (39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (1)1 , F (1)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , (40)for the χ c → φφ decay, and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (2)1 , F (2)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (2)0 , F (2)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . ± . , (41) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (2)1 , − F (2)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (2) − , F (2)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . ± . , (42) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (2) − , − F (2)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (2)1 , F (2)0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . ± . . (43)for the χ c → φφ case.Thus, these ratios of helicity amplitude receives the longdistance contributions and characterize the loop e ff ects in the χ cJ → φφ decays. We expect that they can be measured in thefuture, and used for testing the hadron loop mechanism. B. Polarization observables
Apart from the directly measurements on the ratios given inSec. IV A, the t i j moments, (cid:104) t i j (cid:105) , can also be selected as thespin observables, since their distributions are directly relatedto the helicity amplitude F ( J ) λ ,λ . The t i j observables are con-structed only with the Kaon angles in φ decays. Thus, the (cid:104) t i j (cid:105) moments should be independent on any parameter from the-oretical investigations. In experiment, the (cid:104) t i j (cid:105) moments aredefined as (cid:104) t i j (cid:105) = I (cid:90) t i j |M| d Ω d Ω , (44)where |M| denotes the joint angular distribution for the χ cJ → φφ → K + K − ) decay and d Ω i = d cos θ i d φ i ( i = , I is the normalization fac-tor.One exception is the χ c → φφ decay, in which the multi-pole parameters C i j are independent on the angles of θ or φ .Thus, the (cid:104) t i j (cid:105) moments are uniformly distributed, and theycan not be used as observable. Instead, we chose an observ-able µ = sin θ sin θ to express two φ spin entanglementsproduced from the χ c decays. With the joint angular distribu-tion W , one has (cid:104) µ (cid:105) ∝ + | F (0)1 , / F (0)0 , | cos ( φ + φ ) . (45)An ensemble of events is generated by using the χ c de-cay amplitude W . And the ratio of amplitude is fixed tothe central value of calculation, namely, | F (0)1 , / F (0)0 , | = . (cid:104) sin θ sin θ (cid:105) moment of these TOY Monte-Carlo (MC)events is shown in Fig. 5. One can see that the MC distribu-tion is consistent with the expectation of 1 + ( φ + φ ). ) φ + φ cos( 〉 θ s i n θ s i n 〈 FIG. 5: Distribution of (cid:104) sin θ sin θ (cid:105) moment versus cos( φ + φ )for the χ c → φφ → K + K − ). Histogram is filled with the MCevents, and the curve shows the distribution of 1 + ( φ + φ ). For the χ c → φφ decay, it conserves parity and the de-cay amplitude respects the identical particle symmetry whenexchanging two φ mesons. Thus, the helicity amplitudes areable to factor out as an overall factor in the angular distribu-tion. The φ angular distribution is independent on the ampli-tudes, and it is reduced to dNd cos θ ∝ −
13 cos θ , (46)which corresponds to the observation of moment (cid:104) t (cid:105) for the χ c → φφ decay.We generate an ensemble of MC events for the χ c decaywith the amplitudes constrained by the requirements of par-ity conservation and the identical particle symmetry, namely, F (1)1 , = , F (1)1 , = − F (1)0 , . Figure 6 shows the angular distribu-tion for the φ meson from the χ c decays. One can see that thedistribution is well consistent with the expected one as givenby Eq. (46). θ cos E v en t s FIG. 6: Angular distribution of φ meson in χ c decays. Histogramis filled with the MC events, and the curve shows the distribution of1 − cos θ . One significant feature of t i j moments for χ c decays is thattheir distributions are well determined only with the funda-mental conservation rule and symmetry relations, being inde-pendent on the helicity amplitudes F (1) λ ,λ . For example, some (cid:104) t i j (cid:105) moments are determined to be (cid:104) t (cid:105) ∝ −
12 cos θ , (47) (cid:104) t (cid:105) , (cid:104) t (cid:105) , (cid:104) t (cid:105) ∝ −
13 cos θ , (48) (cid:104) t (cid:105) , (cid:104) t (cid:105) , (cid:104) t (cid:105) , (cid:104) t (cid:105) ∝ − cos θ . (49)Figure 7 shows the (cid:104) t (cid:105) moment distribution filled with the χ c MC events. The curve shows the expected distribution,and it is well consistent with the MC events. θ cos 〉 t 〈 FIG. 7: Moment distribution of (cid:104) t (cid:105) for χ c decays. Histogram isfilled with the MC events, and the curve shows the distribution of1 − cos θ . To show the (cid:104) t i j (cid:105) moments for the χ c → φφ → K + K − )decay, we generated MC events with the central values ofpredicted amplitude ratios, i.e. | F (2)1 , | / | F (2)0 , | = | F (2)0 , | / | F (2)0 , | = . , | F (2)1 , − | / | F (2)0 , | = | F (2) − , | / | F (2)0 , | = .
11 and | F (2)1 , | / | F (2)0 , | = . (cid:104) t (cid:105) moments corresponds to the φ meson angulardistribution. It reads as dNd cos θ ∝ + α cos θ (50)with the angular distribution parameter α = − (cid:104) | F (2)0 , | + (cid:16) −| F (2)1 , − | + | F (2)1 , | + | F (2)1 , | (cid:17)(cid:105) | F (2)0 , | + | F (2)1 , − | + | F (2)1 , | + | F (2)1 , | . (51)Using the amplitude ratios, one has α = . φ meson filled withthe MC events, and the comparison with the predicted angulardistribution (curve). θ cos E v en t s FIG. 8: Angular distribution of the φ meson for χ c decays. His-togram is filled with the MC events, and the curve shows the distri-bution of 1 + .
736 cos θ . Another moment, (cid:104) t (cid:105) or (cid:104) t (cid:105) , can also be used to revealthe amplitude ratios. It distributes with the form (cid:104) t (cid:105) ∝ + α cos θ with α = − (cid:16) | F (2)0 , | + | F (2)1 , − | + | F (2)1 , | − | F (2)1 , | (cid:17) | F (2)0 , | − | F (2)1 , − | + | F (2)1 , | − | F (2)1 , | . (52)Using the predicted amplitude ratios, we get α = .
24. Fig-ure 9 shows the (cid:104) t (cid:105) distribution, filled with the MC events,which is comparable with the predicted distribution with α = . (cid:104) t i j (cid:105) moments in the χ c decays, distributingindependently on the amplitude ratios. After factoring out theamplitudes, we obtain these moment distributions versus x = cos θ , i.e., (cid:104) t (cid:105) ∝ − x / , (53) (cid:104) t (cid:105) , (cid:104) t (cid:105) ∝ x √ − x , (54) (cid:104) t (cid:105) , (cid:104) t (cid:105) ∝ − x . (55)Figure 10 shows the (cid:104) t (cid:105) distribution, for example, for χ c decays, and the comparison with the predicted one. V. SUMMARY
The anomalous decay behaviors of the χ cJ → VV ( VV = ωω, ωφ and φφ ) transitions [1, 2] indicate that the non-perturbative e ff ect of strong interaction cannot be ignored. θ cos 〉 t 〈 FIG. 9: Angular distribution of the φ meson for χ c decays. His-togram is filled with the MC events, and the curve shows the distri-bution of 1 + .
26 cos θ . θ cos 〉 t 〈 FIG. 10: Distribution of the (cid:104) t (cid:105) moment for χ c decays. Histogramis filled with the MC events, and the curve shows the distribution of x √ − x with x = cos θ . For reflecting non-perturbative e ff ect of strong interaction,hadronic loop mechanism is adopted to study the branchingratios of χ cJ → VV ( VV = ωω, ωφ and φφ ) processes [6, 7].Although the measured branching ratios of the χ c → ωω and χ c → φφ processes can be reproduced well, it is not the endof whole story. In fact, we still want to find more crucial infor-mation to reflect the evidence of the hadronic loop mechanismexisting in the χ cJ → ωφ processes [1, 2].Inspired by Refs. [7, 39, 40], we propose that the polar-ization information of the χ cJ → VV decay can be appliedto probe the hadronic loop mechanism, which becomes maintask of this work. The advantage to choose the χ cJ → φφ de-cay is due to the factor that two φ decays provide a rich spinobservables. Another advantage is that these decays are acces-sible in experiment with high detection e ffi ciency and two φ mesons are cleanly reconstructed with low level backgrounds.A high statistics allow one to perform the angular distribu-tion analyses and get the information on the φ polarization,which can shed light on the underlying decay mechanism forthe χ cJ → VV decays.Under the framework of hadronic loop mechanism, we findthat the ratios of the helicity amplitudes of the χ cJ → φφ pro-cesses 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 pa-rameter of the form factor, respectively. Thus, we suggest that these ratios can be as important observable quantities, whichcan be accessible at future experimental measurement as cru-cial test to hadronic loop mechanism.In addition, by using the predicted amplitude ratios, weshow that the observation of moments (cid:104) t i j (cid:105) can be used to man-ifest the nontrivial polarization behavior. For the χ c decays,the choice of the spin observable is quite limited due to the factthat the total spin of the φφ system is constrained to be zero inthe spin triplet. Thus, the spins of two φ mesons are antipar-allel for the χ c decays. For the χ c → φφ decays, the helicityamplitudes can be well determined by considering parity con-servation and by applying the symmetry relationship to takethe φφ as identical particle system. For the χ c → φφ decays,the abundant information of the φφ spin configurations allowsus to directly detect the helicity amplitudes from the observa-tion of di ff erent (cid:104) t i j (cid:105) moments. The patterns of these momentsare presented based on the predicted amplitude ratios, whichcan be tested by expeirment in the near future.In 2019, BESIII released white paper on its future physicsprogram [41]. With the accumulation of charmonium data, wesuggest that BESIII should pay more attentions to the studyof polarization of the corresponding decays, which may pro-vide extra information to reveal underlying mechanism. Obvi-ously, the present work provides a typical example and a newtask for experiment. Acknowledgments
This work is supported by the China National Funds forDistinguished Young Scientists under Grant No. 11825503,National Key Research and Development Program of Chinaunder Contract No. 2020YFA0406400, the 111 Project underGrant No. B20063, and the National Natural Science Foun-dation of China under Grant No. 12047501, 11875262 and11835012.
Appendix A SPIN OBSERVABLE t ij . The obtained spin observables t i j are t = , t = − sin ( θ ) sin (2 φ )3 √ , t = − sin ( θ ) sin (2 φ )3 √ , t =
13 sin ( θ ) sin ( θ ) sin (2 φ ) sin (2 φ ) , t =
13 sin ( θ ) sin (2 θ ) sin (2 φ ) sin ( φ ) , t = sin ( θ ) (3 cos (2 θ ) +
1) sin (2 φ )6 √ , t =
13 sin ( θ ) sin (2 θ ) sin (2 φ ) cos ( φ ) , t =
13 sin ( θ ) sin ( θ ) sin (2 φ ) cos (2 φ ) , t = − sin (2 θ ) sin ( φ )3 √ , t = − sin (2 θ ) sin ( φ )3 √ , t =
13 sin (2 θ ) sin ( θ ) sin ( φ ) sin (2 φ ) , t =
13 sin (2 θ ) sin (2 θ ) sin ( φ ) sin ( φ ) , t = sin (2 θ ) (3 cos (2 θ ) +
1) sin ( φ )6 √ , t =
13 sin (2 θ ) sin (2 θ ) sin ( φ ) cos ( φ ) , t =
13 sin (2 θ ) sin ( θ ) sin ( φ ) cos (2 φ ) , t =
118 ( − θ ) − , t =
118 ( − θ ) − , t = sin ( θ ) (3 cos (2 θ ) +
1) sin (2 φ )6 √ , t = sin (2 θ ) (3 cos (2 θ ) +
1) sin ( φ )6 √ , t =
136 (3 cos (2 θ ) +
1) (3 cos (2 θ ) + , t = sin (2 θ ) (3 cos (2 θ ) +
1) cos ( φ )6 √ , t = sin ( θ ) (3 cos (2 θ ) +
1) cos (2 φ )6 √ , t = − sin (2 θ ) cos ( φ )3 √ , t = − sin (2 θ ) cos ( φ )3 √ , t =
13 sin (2 θ ) sin ( θ ) sin (2 φ ) cos ( φ ) , t =
13 sin (2 θ ) sin (2 θ ) sin ( φ ) cos ( φ ) , t = sin (2 θ ) (3 cos (2 θ ) +
1) cos ( φ )6 √ , t =
13 sin (2 θ ) sin (2 θ ) cos ( φ ) cos ( φ ) , t =
13 sin (2 θ ) sin ( θ ) cos ( φ ) cos (2 φ ) , t = − sin ( θ ) cos (2 φ )3 √ , t = − sin ( θ ) cos (2 φ )3 √ , t =
13 sin ( θ ) sin ( θ ) sin (2 φ ) cos (2 φ ) , t =
13 sin ( θ ) sin (2 θ ) sin ( φ ) cos (2 φ ) , t = sin ( θ ) (3 cos (2 θ ) +
1) cos (2 φ )6 √ , t =
13 sin ( θ ) sin (2 θ ) cos (2 φ ) cos ( φ ) , t =
13 sin ( θ ) sin ( θ ) cos (2 φ ) cos (2 φ ) . Appendix B MULTIPOLE PARAMETERS FOR χ c → φφ We collected the multipole parameters for χ c → φφ , i.e., C = (cid:16) − (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)0 , (cid:12)(cid:12)(cid:12) − (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) + θ ) + (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) (cid:17) , C =
332 sin (2 θ ) (cid:16) F (1) ∗ , F (1)0 , + F (1) ∗ , F (1)1 , (cid:17) , C =
332 (cos (2 θ ) − (cid:16) F (1) ∗ , F (1)0 , + F (1) ∗ , F (1)1 , (cid:17) , C = (cid:16) θ ) − (cid:12)(cid:12)(cid:12) F (1)0 , (cid:12)(cid:12)(cid:12) − (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) + θ ) + (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) (cid:17) , C = (cid:16) (cid:16) (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) + (cos (2 θ ) + (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) (cid:17) − (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)0 , (cid:12)(cid:12)(cid:12) (cid:17) , C = (cid:16) (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)0 , (cid:12)(cid:12)(cid:12) + (cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) + (cos (2 θ ) + (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) (cid:17) , C = √ θ ) (cid:16) F (1) ∗ , F (1)1 , + F (1) ∗ , F (1)1 , (cid:17) , C = − √ ( θ ) (cid:12)(cid:12)(cid:12) F (1)0 , (cid:12)(cid:12)(cid:12) , C = − √ θ ) (cid:16) F (1) ∗ , F (1)0 , + F (1) ∗ , F (1)1 , (cid:17) , C = √ θ ) (cid:16) F (1) ∗ , F (1)1 , + F (1) ∗ , F (1)1 , (cid:17) , C = − √ θ ) (cid:16) F (1) ∗ , F (1)0 , + F (1) ∗ , F (1)1 , (cid:17) , C = (cid:16) F (1) ∗ , F (1)0 , + F (1) ∗ , F (1)1 , (cid:17) , C = −
332 sin (2 θ ) (cid:16) F (1) ∗ , F (1)0 , + F (1) ∗ , F (1)1 , (cid:17) , C = √ ( θ ) (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) , C = √ ( θ ) (cid:12)(cid:12)(cid:12) F (1)0 , (cid:12)(cid:12)(cid:12) , C = − √ ( θ ) (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) , C =
332 sin (2 θ ) (cid:16) F (1) ∗ , F (1)1 , + F (1) ∗ , F (1)1 , (cid:17) , C = −
316 (cos (2 θ ) + (cid:12)(cid:12)(cid:12) F (1)1 , (cid:12)(cid:12)(cid:12) . Appendix C MULTIPOLE PARAMETERS FOR χ c → φφ The multipole parameters for χ c → φφ are C =
940 [ − θ ) × (cid:16)(cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + F (2) ∗ , F (2)0 , − F (2) ∗− , F (2)1 , − (cid:17) + (7 − θ )) (cid:12)(cid:12)(cid:12) F (2)0 , (cid:12)(cid:12)(cid:12) + θ ) + (cid:12)(cid:12)(cid:12) F (2)1 , − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + F (2) ∗ , F (2)0 , + F (2) ∗− , F (2)1 , − (cid:105) , C =
380 (3 cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) , C = − √ θ ) (cid:16) F (2) ∗ , F (2)1 , + F (2) ∗ , F (2)1 , (cid:17) , C = √ θ ) (cid:16) F (2) ∗ , F (2)1 , + F (2) ∗ , F (2)1 , (cid:17) , C = − (cid:16) θ ) − (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + √ ( θ ) (cid:16) F (2) ∗ , − F (2)0 , + F (2) ∗ , F (2)1 , − (cid:17) − (3 cos (2 θ ) − (cid:16) F (2) ∗ , F (2)0 , + F (2) ∗ , F (2)1 , (cid:17)(cid:17) , (56) C = (cid:16) (3 cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (2)0 , (cid:12)(cid:12)(cid:12) + (7 − θ )) (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + θ ) + (cid:12)(cid:12)(cid:12) F (2)1 , − (cid:12)(cid:12)(cid:12) (cid:17) , C = (cid:16) (3 cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (2)0 , (cid:12)(cid:12)(cid:12) + (7 − θ )) (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + θ ) + (cid:12)(cid:12)(cid:12) F (2)1 , − (cid:12)(cid:12)(cid:12) (cid:17) , (57) C = (cid:16) − θ )) (cid:12)(cid:12)(cid:12) F (2)0 , (cid:12)(cid:12)(cid:12) + (7 − θ )) (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + θ ) − (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + θ ) + (cid:12)(cid:12)(cid:12) F (2)1 , − (cid:12)(cid:12)(cid:12) (cid:17) , C = − θ ) (cid:16)(cid:16) F (2) ∗ , − √ F (2) ∗ , − + F (2) ∗ , (cid:17) F (2)1 , + F (2) ∗ , (cid:16) F (2)0 , − √ F (2)1 , − + F (2)1 , (cid:17)(cid:17) , C =
380 sin ( θ ) (cid:16) √ (cid:16) F (2) ∗ , F (2)1 , − + F (2) ∗ , − F (2)1 , (cid:17) − √ (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) (cid:17) , C = θ ) (cid:16)(cid:16) F (2) ∗ , − √ F (2) ∗ , − + F (2) ∗ , (cid:17) F (2)1 , + F (2) ∗ , (cid:16) F (2)0 , − √ F (2)1 , − + F (2)1 , (cid:17)(cid:17) , C = − (cid:16) (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) + √ ( θ ) (cid:16) F (2) ∗ , − F (2)0 , + F (2) ∗ , F (2)1 , − (cid:17) + (3 cos (2 θ ) − (cid:16) F (2) ∗ , F (2)0 , + F (2) ∗ , F (2)1 , (cid:17)(cid:17) , C = − √ θ ) (cid:16) F (2) ∗ , F (2)1 , + F (2) ∗ , F (2)1 , (cid:17) , C = ( θ ) (cid:16) F (2) ∗ , F (2)1 , − + F (2) ∗ , − F (2)1 , (cid:17) √ , C = ( θ ) (cid:16) F (2) ∗ , F (2)1 , − + F (2) ∗ , − F (2)1 , (cid:17) √ , C =
380 sin ( θ ) (cid:16) √ (cid:16) F (2) ∗ , F (2)1 , − + F (2) ∗ , − F (2)1 , (cid:17) − √ (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) (cid:17) , C = √ θ ) (cid:16) F (2) ∗ , F (2)1 , + F (2) ∗ , F (2)1 , (cid:17) , C = −
380 (3 cos (2 θ ) − (cid:12)(cid:12)(cid:12) F (2)1 , (cid:12)(cid:12)(cid:12) . Appendix D AMPLITUDES OF χ c → φφ TRANSITION
For χ c → φφ transition, the amplitudes corresponding toFig. (3) are M (1 − = (cid:90) d q (2 π ) ˜ g µξ ( k ) k − m D ∗ k − m D q − m D F ( q ) × [ − ig χ c DD ∗ (cid:15) µχ c ( p )][ − g DD φ (cid:15) ∗ λφ ( p )( q λ − k λ )] × [2 f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] , (58) M (1 − = (cid:90) d q (2 π ) k − m D ˜ g σµ ( k ) k − m D ∗ q − m D F ( q ) × [ ig χ c DD ∗ (cid:15) µχ c ( p )][ − g DD φ (cid:15) ∗ ζφ ( p )( k ζ + q ζ )] × [ − f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (59)1 M (1 − = (cid:90) d q (2 π ) ˜ g µψ ( k ) k − m D ∗ k − m D ˜ g γσ ( q ) q − m D ∗ F ( q ) × [ − ig χ c DD ∗ (cid:15) µχ c ( p )][ g D ∗ D ∗ ψ g ηγ g ηψ ( k ζ + q ζ ) − f D ∗ D ∗ φ p η ( g γη g ψζ − g γζ g ψη )] (cid:15) ∗ ζφ ( p ) × [2 f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (60) M (1 − = (cid:90) d q (2 π ) k − m D ˜ g µι ( k ) k − m D ∗ ˜ g υξ ( q ) q − m D ∗ F ( q ) × [ − f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × (cid:15) ∗ λφ ( p )[ g D ∗ D ∗ φ g ρι g ρυ ( q λ − k λ ) − f D ∗ D ∗ φ p ρ ( g ιρ g υλ − g ιλ g υρ )] × [ ig χ c DD ∗ (cid:15) µχ c ( p )] , (61) Appendix E AMPLITUDES OF χ c → φφ TRANSITION
The obtained amplitudes of the χ c → φφ transition for Fig.(4) are M (2 − = (cid:90) d q (2 π ) k − m D k − m D q − m D F ( q ) × [ g χ c DD (cid:15) µνχ c ( p k µ k ν ] × [ g DD φ (cid:15) ∗ ζφ ( p )( k ζ + q ζ )] × [ g DD φ (cid:15) ∗ λφ ( p )( q λ − k λ )] , (62) M (2 − = (cid:90) d q (2 π ) ˜ g τξ ( k ) k − m D ∗ k − m D q − m D F ( q ) × [ − g χ c DD ∗ ε µταβ p α (cid:15) µνχ c ( p ) k β k ν ] × [2 f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × [ − g DD φ (cid:15) ∗ λφ ( p )( q λ − k λ )] , (63) M (2 − = (cid:90) d q (2 π ) k − m D ˜ g τσ ( k ) k − m D ∗ q − m D F ( q ) × [ − g χ c DD ∗ ε µταβ p α (cid:15) µνχ c ( p ) k ν k β ] × [ − g DD φ (cid:15) ∗ ζφ ( p )( k ζ + q ζ )] × [ − f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (64) M (2 − = (cid:90) d q (2 π ) k − m D k − m D ˜ g ξσ ( q ) q − m D ∗ F ( q ) × [ g χ c DD (cid:15) µνχ c ( p k µ k ν ] × [ − f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × [2 f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (65) M (2 − = (cid:90) d q (2 π ) ˜ g ωξ ( k ) k − m D ∗ ˜ g χσ ( k ) k − m D ∗ q − m D F ( q ) × [ g χ c D ∗ D ∗ (cid:15) µνχ c ( p )( g νω g µχ + g µω g νχ )] × [2 f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × [ − f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (66) M (2 − = (cid:90) d q (2 π ) ˜ g τψ ( k ) k − m D ∗ k − m D ˜ g γσ ( q ) q − m D ∗ F ( q ) × [ − g χ c DD ∗ ε µταβ p α (cid:15) µνχ c ( p ) k β k ν ] × (cid:15) ∗ ζφ ( p )[ g D ∗ D ∗ ψ g ηγ g ηψ ( k ζ + q ζ ) − f D ∗ D ∗ φ p η ( g γη g ψζ − g γζ g ψη )] × [2 f DD ∗ φ ε λρδσ (cid:15) ∗ φλ ( p ) p ρ ( q δ − k δ )] , (67) M (2 − = (cid:90) d q (2 π ) k − m D ˜ g τι ( k ) k − m D ∗ ˜ g υξ ( q ) q − m D ∗ F ( q ) × [ − g χ c DD ∗ ε µταβ p α (cid:15) µνχ c ( p ) k ν k β ] × [ − f DD ∗ φ ε ζηκξ (cid:15) ∗ φζ ( p ) p η ( k κ + q κ )] × (cid:15) ∗ λφ ( p )[ g D ∗ D ∗ φ g ρι g ρυ ( q λ − k λ ) − f D ∗ D ∗ φ p ρ ( g ιρ g υλ − g ιλ g υρ )] , (68) M (2 − = (cid:90) d q (2 π ) ˜ g ψω ( k ) k − m D ∗ ˜ g ιχ ( k ) k − m D ∗ ˜ g γυ ( q ) q − m D ∗ F ( q ) × [ g χ c D ∗ D ∗ (cid:15) µνχ c ( p )( g νω g µχ + g µω g νχ )] × (cid:15) ∗ ζφ ( p )[ g D ∗ D ∗ ψ g ηγ g ηψ ( k ζ + q ζ ) − f D ∗ D ∗ φ p η ( g γη g ψζ − g γζ g ψη )] × (cid:15) ∗ λφ ( p )[ g D ∗ D ∗ φ g ρι g ρυ ( q λ − k λ ) − f D ∗ D ∗ φ p ρ ( g ιρ g υλ − g ιλ g υρ )] . 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