Investigating possible decay modes of Y(4260) under the D 1 (2420) D ¯ +c.c molecular state ansatz
aa r X i v : . [ h e p - ph ] O c t Investigating possible decay modes of Y (4260) under the D (2420) ¯ D + c.c molecularstate ansatz Gang Li ∗ and Xiao-Hai Liu †
1) Department of Physics, Qufu Normal University, Qufu 273165, People’s Republic of China and2) Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, People’s Republic of China (Dated: August 8, 2018)By assuming that Y (4260) is a D ¯ D molecular state, we investigate some hidden-charm andcharmed pair decay channels of Y (4260) via intermediate D ¯ D meson loops with an effective La-grangian approach. Through investigating the α -dependence of branching ratios and ratios betweendifferent decay channels, we show that the intermediate D ¯ D meson loops are crucial for drivingthese transitions of Y (4260) studied here. The coupled channel effects turn out to be more importantin Y (4260) → D ∗ ¯ D ∗ , which can be tested in the future experiments. PACS numbers: 13.25.GV, 13.75.Lb, 14.40.Pq
I. INTRODUCTION
During the past years, the experimental observation of a large number of so-called
XY Z states has initiated tremen-dous efforts to unravel their nature beyond the conventional quark model (for recent reviews, see, e.g. Refs [1–5]). Y (4260) was reported by the BaBar Collaboration in the π + π − J/ψ invariant spectrum in e + e − → γ ISR π + π − J/ψ [6],which has been confirmed both by the CLEO and Belle collaboration [7, 8]. Its mass and total width are well deter-mined as m = 4263 +8 − MeV and Γ Y = 95 ±
14 MeV, respectively [9]. The new datum from BESIII confirms the signalin Y (4260) → J/ψπ + π − with much higher statistics [10]. The mass of Y (4260) does not agree to what is predicted bythe potential quark model. Further more, the most mysterious fact is that as a charmonium state with J P C = 1 −− ,it is only “seen” as a bump in the two pion transitions to J/ψ , but not in any open charm decay channels like D ¯ D , D ∗ ¯ D + c.c. and D ∗ ¯ D ∗ , or other measured channels. The line shapes of the cross section for e + e − annihilations into D ( ∗ ) meson pairs appear to have a dip at its peak mass 4 .
26 GeV instead of a bump.Since the observation of the Y (4260), many theoretical investigations have been carried out (for a review seeRef. [11]). It has variously been identified as a conventional ψ (4 S ) based on a relativistic quark model [12], atetraquark c ¯ cs ¯ s state [13], a charmonium hybrid [14–16], hadronic molecule of D ¯ D [17–19], [1] χ c ω [23], χ c ρ [24], J/ψf [25], a cusp [26, 27] or a non-resonance explanation [28, 29] etc. The dynamical calculation of tetraquark statesindicated that Y (4260) can not be interpreted as P-wave 1 −− state of charm-strange diquark-antidiquark, because thecorresponding mass is found to be 200 MeV heavier [30]. In Ref. [31], the authors also studied the possibility of Y (4260)as P-wave 1 −− state of charm-strange diquark-antidiquark state in the framework of QCD sum rules and arrived thesame conclusion as Ref. [30]. Some lattice calculations give the mass of vector hybrid within this mass region [32],which is very close to the new charmonium-like state Y(4360) [33]. With the D ¯ D molecular ansatz, a consistentdescription of some of the experimental observations can be obtained, such as its non observation in open charmdecays, or the observation of Z c (3900) as mentioned in Ref. [19], the threshold behavior in its main decay channels areinvestigated in Ref. [34] and the production of X (3872) is studied in the radiative decays of Y (4260) [22]. Under such amolecular state assumption, a consistent description of many experimental observations could be obtained. However,as studied in [35], the production of an S-wave D ¯ D pair in e + e − annihilation is forbidden in the limit of exactheavy quark spin symmetry, which substantially weakens the arguments for considering the Y (4260) charmonium-likeresonance as a D ¯ D molecular state.The intermediate meson loop (IML) transition is one of the possible nonperturbative dynamical mechanisms, espe-cially when we investigate the pertinent issues in the energy region of charmonium [36–61]. During the last decade,many interesting observations were announced by Belle, BaBar, CLEO, BESIII, and so on. And in theoretical study,it is widely recognized that the IML may be closely related to a lot of nonperturbative phenomena observed in exper-iments [44–64], e.g. apparent OZI-rule violations, sizeable non- D ¯ D decay branching ratios for ψ (3770) [44–49], theHSR violations in charmonium decays [56–58], the hidden charmonium decays of the newly discovered Z c [62], etc. ∗ [email protected] † [email protected][1] Notice that there are two D states of similar masses, and the one in question should be the narrower one, i.e. the D (2420) (Γ = 27MeV), the D (2430)(Γ ≃
384 MeV) is too broad to form a molecular state [20–22].
In this work, we will investigate the hidden-charm decays of Y (4260) and Y (4260) → D ( ∗ ) ¯ D ( ∗ ) via D ¯ D loop withan effective Lagrangian approach (ELA) under the D ¯ D molecular assumption. The paper is organized as follows.In Sec. II, we will introduce the ELA briefly and give some relevant formulae. In Sec. III, the numerical results arepresented. The summary will be given in Sec. IV. II. THE MODEL Y (4260) ¯ DD ρ ( ω )¯ DD ( a ) Y (4260) ¯ DD ρ ( ω )¯ D ∗ D ( b ) Y (4260) ¯ DD π ¯ D ∗ D ∗ ( c )FIG. 1: The hadron-level diagrams for Y (4260) → D ( ∗ ) ¯ D ( ∗ ) with D ¯ D as the intermediate states. Y (4260) ¯ DD ¯ D ∗ ψP ( a ) Y (4260) ¯ DD ¯ D ∗ h c P ( b )FIG. 2: The hadron-level diagrams for hidden-charm decays of Y (4260) with D ¯ D as intermediate states. P denotes thepseudoscalar meson π or η . In order to calculate the leading contributions from the charmed meson loops, we need the leading order effectiveLagrangian for the couplings. Based on the heavy quark symmetry and chiral symmetry [65, 66], the relevant effectiveLagrangian used in this work read L ψD ( ∗ ) D ( ∗ ) = ig ψDD ψ µ ( ∂ µ D ¯ D − D∂ µ ¯ D ) − g ψD ∗ D ε µναβ ∂ µ ψ ν ( ∂ α D ∗ β ¯ D + D∂ α ¯ D ∗ β ) − ig ψD ∗ D ∗ (cid:8) ψ µ ( ∂ µ D ∗ ν ¯ D ∗ ν − D ∗ ν ∂ µ ¯ D ∗ ν ) + ( ∂ µ ψ ν D ∗ ν − ψ ν ∂ µ D ∗ ν ) ¯ D ∗ µ + D ∗ µ ( ψ ν ∂ µ ¯ D ∗ ν − ∂ µ ψ ν ¯ D ∗ ν ) (cid:9) , (1) L h c D ( ∗ ) D ( ∗ ) = g h c D ∗ D h µc ( D ¯ D ∗ µ + D ∗ µ ¯ D ) + ig h c D ∗ D ∗ ε µναβ ∂ µ h cν D ∗ α ¯ D ∗ β , (2)where D ( ∗ ) = (cid:16) D ( ∗ )+ , D ( ∗ )0 , D ( ∗ )+ s (cid:17) and ¯ D ( ∗ ) T = (cid:16) D ( ∗ ) − , ¯ D ( ∗ )0 , D ( ∗ ) − s (cid:17) correspond to the charmed meson isodoublets.The following couplings are adopted in the numerical calculations, g ψDD = 2 g √ m ψ m D , g ψD ∗ D = g ψDD √ m D m D ∗ , g ψD ∗ D ∗ = g ψD ∗ D r m D ∗ m D m D ∗ . (3)In principle, the parameter g should be computed with nonperturbative methods. It shows that vector mesondominance (VMD) would provide an estimate of these quantities [65]. The coupling g can be related to the J/ψ leptonic constant f ψ which is defined by the matrix element h | ¯ cγ µ c | J/ψ ( p, ǫ ) i = f ψ m ψ ǫ µ , and g = √ m ψ / m D f ψ , where f ψ = 405 ±
14 MeV, and we have applied the relation g ψDD = m ψ /f ψ . The ratio of the coupling constants g ψ ′ DD to g ψDD is fixed as that in Ref. [57], i.e., g ψ ′ DD g ψDD = 0 . . (4)In addition, the coupling constants in Eq. (2) are determined as g h c DD ∗ = − g √ m h c m D m D ∗ , g h c D ∗ D ∗ = 2 g m D ∗ √ m h c , (5)with g = − p m χ c / /f χ c , where m χ c and f χ c are the mass and decay constant of χ c (1 P ), respectively [67]. Wetake f χ c = 510 ±
40 MeV [68].The light vector mesons nonet can be introduced by using the hidden gauge symmetry approach, and the effectiveLagrangian containing these particles are as follows [69, 70], L D ∗ D V = ig D ∗ D V ǫ αβµν (D b ↔ ∂ α D ∗ β † a − D ∗ β † b ↔ ∂ α D ja )( ∂ µ V ν ) ba + ig D ∗ DV ǫ αβµν (D b ↔ ∂ α D ∗ β † a − D ∗ β † b ↔ ∂ α D ja )( ∂ µ V ν ) ab + h.c , L DD V = ig DD V (D b ↔ ∂ µ D † a ) V µba + ig D D V (D b ↔ ∂ µ D † a ) V µab , L DD V = g DD V D µ b V µba D † a + g ′ DD V (D µ b ↔ ∂ ν D † a )( ∂ µ V ν − ∂ ν V µ ) ba + g D D V D † a V µab D µ b + g ′ D D V (D µ b ↔ ∂ ν D † a )( ∂ µ V ν − ∂ ν V µ ) ab + h.c. , L D ∗ D ∗ V = ig D ∗ D ∗ V (D ∗ bν ↔ ∂ µ D ∗ ν † a ) V µba + ig ′ D ∗ D ∗ V (D ∗ µb D ∗ ν † a − D ∗ µ † a D ∗ νb )( ∂ µ V ν − ∂ ν V µ ) ba + ig D ∗ D ∗ V (D ∗ bν ↔ ∂ µ D ∗ ν † a ) V µab + ig ′ D ∗ D ∗ V (D ∗ µb D ∗ ν † a − D ∗ µ † a D ∗ νb )( ∂ µ V ν − ∂ ν V µ ) ab . (6)And the coupling constants read g DD V = − g D D V = 1 √ βg V ,g DD V = − g D D V = − √ ζ g V √ m D m D ,g ′ DD V = − g ′ D D V = 1 √ µ g V ,g D ∗ D ∗ V = − g D ∗ D ∗ V = − √ βg V ,g ′ D ∗ D ∗ V = − g ′ D ∗ D ∗ V = −√ λg V m D ∗ , (7)where f π = 132 MeV is the pion decay constant, and the parameter g V is given by g V = m ρ /f π [66]. We take λ = 0 .
56 GeV − , g = 0 .
59 and β = 0 . L D D ∗ P = g D D ∗ P [3 D µ a ( ∂ µ ∂ ν φ ) ab D ∗† νb − D µ a ( ∂ ν ∂ ν φ ) ab D ∗† bµ ]+ g ¯ D ¯ D ∗ P [3 ¯ D ∗† µa ( ∂ µ ∂ ν φ ) ab ¯ D ν b − ¯ D ∗† µa ( ∂ ν ∂ ν φ ) ab ¯ D bν ] + H.c. , (8) L DD ∗ P = g DD ∗ P D b ( ∂ µ φ ) ba D ∗ µ † a + g DD ∗ P D ∗ µb ( ∂ µ φ ) ba D † a + g D D ∗ P D ∗ µ † a ( ∂ µ φ ) ab D b + g D D ∗ P D † a ( ∂ µ φ ) ab D ∗ µb , (9)with D ( ∗ ) = (cid:16) D ( ∗ )+ , D ( ∗ )0 , D ( ∗ )+ s (cid:17) and ¯ D ( ∗ ) = (cid:16) D ( ∗ ) − , ¯ D ( ∗ )0 , D ( ∗ ) − s (cid:17) . φ is the 3 × g DD ∗ P = − g D D ∗ P = − gf π √ m D m D ∗ , (10) g D ∗ D P = g D ∗ D P = − √ h ′ Λ χ f π √ m D ∗ m D , (11)with the chiral symmetry breaking scale Λ χ ≃ h ′ = 0 .
65 [75].By assuming Y (4260) is a D ¯ D molecular state, the effective Lagrangian is constructed as L Y (4260) D D = i y √ D † a Y µ D µ † a − ¯ D µ † a Y µ D † a ) + H.c., (12)which is an S -wave coupling. Since the mass Y (4260) is slightly below an S-wave D ¯ D threshold, the effective coupling g Y (4260) D D is related to the probability of finding D D component in the physical wave function of the bound state, c , and the binding energy, δE = m D + m D − m Y [22, 76, 77], g ≡ π ( m D + m D ) c s δEµ [1 + O ( p µǫr )] , (13)where µ = m D m D / ( m D + m D ) and r is the reduced mass and the range of the forces. The coupling constants inEq. (12) is given by the first term in the above equation. The coupling constant gets maximized for a pure boundstate, which corresponds to c = 1 by definition. In the following, we present the numerical results with c = 1.With the mass m Y = 4263 +8 − MeV, and the averaged masses of the D and D mesons [9], we obtain the massdifferences between the Y (4260) and their corresponding thresholds, m D + m D − m Y = 27 +9 − MeV , (14)and with c = 1, we obtain | y | = 14 . +1 . − . ± .
20 GeV (15)where the first errors are from the uncertainties of the binding energies, and the second ones are due the the approx-imate nature of the approximate nature of Eq. (13).The loop transition amplitudes for the transitions in Figs. 1 and 2 can be expressed in a general form in the effectiveLagrangian approach as follows, M fi = Z d q (2 π ) X D ∗ pol. T T T a a a F ( m , q ) (16)where T i and a i = q i − m i ( i = 1 , ,
3) are the vertex functions and the denominators of the intermediate mesonpropagators. For example, in Fig. 2 (a), T i ( i = 1 , ,
3) are the vertex functions for the initial Y (4260), finalcharmonium and final light pseudoscalar mesons, respectively. a i ( i = 1 , ,
3) are the denominators for the intermediate¯ D , D ∗ and D mesons, respectively. We introduce a dipole form factor, F ( m , q ) ≡ (cid:18) Λ − m Λ − q (cid:19) , (17)where Λ ≡ m + α Λ QCD and the QCD energy scale Λ
QCD = 220 MeV. This form factor is supposed to kill thedivergence, compensate the off-shell effects arising from the intermediate exchanged particle and the non-local effectsof the vertex functions [36, 78, 79].
III. NUMERICAL RESULTS B R (a) B R (b) FIG. 3: (a). The α -dependence of the branching ratios of Y (4260) → D ¯ D (solid line) and D ∗ ¯ D + c.c. (dashed line). (b). The α -dependence of the branching ratios of Y (4260) → D ∗ ¯ D ∗ . B R (a) B R (b) FIG. 4: (a). The α -dependence of the branching ratios of Y (4260) → J/ψη (solid line) and
J/ψπ (dashed line). (b). The α -dependence of the branching ratios of Y (4260) → ψ ′ η (solid line) and ψ ′ π (dashed line). B R FIG. 5: The α -dependence of the branching ratios of Y (4260) → h c η (solid line) and h c π (dashed line). Since Y (4260) has a large width 95 ±
14 MeV, so one has to take into account the mass distribution of the Y (4260)when calculating its decay widths. Its two-body decay width can then be calculated as follow [80],Γ( Y (4260)) − body = 1 W Z ( m Y +2Γ Y ) ( m Y − Y ) ds (2 π ) √ s Z d Φ |M| π Im( − s − m Y + im Y Γ Y ) (18) R d Φ is the two-body phase space [9]. M are the loop transition amplitudes for the processes in Figs. 1 and 2. Thefactor 1 /W with W = 1 π Z ( m Y +2Γ Y ) ( m Y − Y ) Im( − s − m Y + im Y Γ Y ) ds (19) TABLE I: The predicted branching ratios of Y(4260) decays with different α values. The uncertainties is dominated by the useof Eq. (13). Final states α = 0 . α = 1 . α = 1 . α = 2 . D ¯ D (3 . +3 . − . ) × − (4 . +4 . − . ) × − (1 . +1 . − . ) × − (3 . +4 . − . ) × − D ∗ ¯ D + c.c. (9 . +10 . − . ) × − (1 . +1 . − . ) × − (4 . +5 . − . ) × − (1 . +1 . − . ) × − D ∗ ¯ D ∗ (1 . +1 . − . ) × − (2 . +2 . − . ) × − (16 . +17 . − . )% (52 . +54 . − . )% J/ψη (7 . +7 . − . ) × − (8 . +8 . − . ) × − (2 . +3 . − . ) × − (6 . +7 . − . ) × − J/ψπ (3 . +3 . − . ) × − (3 . +3 . − . ) × − (1 . +1 . − . ) × − (2 . +2 . − . ) × − ψ ′ η (4 . +4 . − . ) × − (2 . +2 . − . ) × − (6 . +6 . − . ) × − (1 . +1 . − . ) × − ψ ′ π (1 . +1 . − . ) × − (9 . +10 . − . ) × − (2 . +2 . − . ) × − (3 . +3 . − . ) × − h c η (3 . +4 . − . ) × − (2 . +3 . − . ) × − (8 . +8 . − . ) × − (15 . +15 . − . )% h c π (1 . +1 . − . ) × − (9 . +9 . − . ) × − (2 . +2 . − . ) × − (4 . +4 . − . ) × − R a t i o (a) R a t i o (b) FIG. 6: (a). The α -dependence of the ratios of R (solid line) and R (dashed line) defined in Eq. (22). (b). The α -dependenceof the ratios of r (solid line), r (dashed line) and r (dotted line) defined in Eq. (23). is considered in order to normalize the spectral function of the Y (4260) state.The numerical results are presented in Figs. 3-5. In Table. I, we list the predicted branching ratios of Y (4260) atdifferent α values and the errors are from the uncertainties of the the coupling constants in Eq. (15). We have checkedthat including the width for the D only causes a minor change of about 1%-3%.In Fig. 3(a), we plot the α -dependence of the branching ratios of Y (4260) → D ¯ D (solid line) and Y (4260) → D ∗ ¯ D + c.c. (dashed line), respectively. The branching ratios of Y (4260) → D ∗ ¯ D ∗ in terms of α are shown in Fig. 3(b).In this figure, no cusp structure appear. This is because that the mass of Y (4260) lies below the intermediate D ¯ D threshold. The α dependence of the branching ratios are not drastically sensitive to some extent, which indicates areasonable cutoff of the ultraviolet contributions by the empirical form factors. As shown in this figure, at the same α ,the intermediate D ¯ D meson loops turns out to be more important in Y (4260) → D ∗ ¯ D ∗ than that in Y (4260) → D ¯ D and D ∗ ¯ D + c.c. . This behavior can also be seen from Table. I. As a result, a smaller value of α is favored in Y (4260) → D ∗ ¯ D ∗ . This phenomenon can be easily explained from Fig. 1. For the decay Y (4260) → D ∗ ¯ D ∗ , theoff-shell effects of intermediate mesons D D ( π ) are not significant, which makes this decay favor a relatively smaller α value. For the decay Y (4260) → D ¯ D and D ∗ ¯ D + c.c. , since the exchanged mesons of the intermediate meson loopsare ρ and ω , which makes their off-effects are relatively significant, which makes this decay favor a relatively larger α value.In a fit to the total hadronic cross sections measured by BES [81], authors set an upper limit on Γ e + e − for Y (4260)to be less than 580 eV at 90% confidence level (C.L.) [82]. This implies that its branching fraction to J/ψπ + π − is greater than 0.6% at 90% C.L. [82]. Recently, BESIII has reported a study of e + e − → h c π + π − , and observesa state with a mass of 4021 . ± . ± . , MeV and a width of 5 . ± . ± . h c π ± mass distribution,called the Z c (4020). The Belle collaboration did a comprehensive search for Y (4260) decays to all possible final statescontaining open charmed mesons pairs and found no sign of a Y (4260) signal in any of them [84–89]. The BaBarCollaboration measured some upper limits of the ratios B ( Y (4260) → D ¯ D ) / B ( Y (4260) → J/ψπ + π − ) < . B ( Y (4260) → D ∗ ¯ D ) / B ( Y (4260) → J/ψπ + π − ) <
34 and B ( Y (4260) → D ∗ ¯ D ∗ ) / B ( Y (4260) → J/ψπ + π − ) <
40 at 90% C.L. [91], respectively. Within the parameter range considered in this work, the results displayed in Table Icould be compatible with these available experimental limits. However, since there are still several uncertaintiescoming from the undetermined coupling constants, and the cutoff energy dependence of the amplitude is not quitestable, the numerical results would be lacking in high accuracy. Especially, since the kinematics, off-shell effectsarising from the exchanged particles and the divergence of the loops in theses open charmed channels studied here aredifferent, the cutoff parameter can also be different in different decay channels. We expect more precise experimentalmeasurements on these open charmed pairs to test this point in the near future.In Ref. [59], a nonrelativistic effective field theory (NREFT) method was introduced to study the meson loop effectsin ψ ′ → J/ψπ transitions. And a power counting scheme was proposed to estimate the contribution of the loopeffects, which is helpful to judge how important the coupled-channel effects are. This power counting scheme wasanalyzed in detail in Ref. [61]. Recently, the authors study that the S-wave threshold plays more important rolethan P-wave, especially for the S-wave molecule with large coupling to its components, such as Y (4260) coupling to D ¯ D in Ref. [22]. Before giving the explicit numerical results, we will follow the similar power counting scheme toqualitatively estimate the contributions of the coupled-channel effects discussed in this work. Corresponding to thediagrams Fig. 2(a) and Fig. 2(b), the amplitudes for Y (4260) → J/ψπ ( J/ψη , ψ ′ π , ψ ′ η ) and Y (4260) → h c π ( h c η )scale as v ( v ) q ∆ v ∼ q ∆ v (20)and v ( v ) q ∆ v ∼ q ∆ v , (21)respectively. There are two scaling parameters v and q appeared in the above two formulae. As illustrated inRef. [92], v is understood as the average velocity of the intermediated charmed meson. q denotes the momentumof the outgoing pseudoscalar meson. And ∆ denotes the charmed meson mass difference, which is introduced toaccount for the isospin or SU(3) symmetry violation. For the π and η production processes, the factors ∆ are about M D + + M D − − M D and M D + + M D − M D s , respectively. According to Eqs. (20) and (21), it can be concludedthat the contributions of the coupled channel effects would be significant here since the amplitudes scale as O (1 /v ).And the branching ratio of Y (4260) → h c π is expected to be larger than that of Y (4260) → J/ψπ , because thecorresponding amplitudes scale as O ( q ) and O ( q ) respectively. However, the momentum q in Y (4260) → J/ψπ islarger than that in Y (4260) → h c π , which may compensate this discrepancy to some extent.For the open charmed decays in Fig. 1, the exchanged intermediate mesons are light vector mesons or light pseu-doscalar mesons which will introduce different scale. Since we cannot separate different scales, so we just give possiblenumerical results in the form factor scheme.For the hidden-charm transitions Y (4260) → J/ψη ( π ), we plot the α -dependence of the branching ratios of Y (4260) → J/ψη ( π ) in Fig. 4(a) as shown by the solid and dashed lines, respectively. The π - η mixing has beentaken into account. (Using Dashen’s theorem [93], one may express the mixing angle in terms of the masses of theGoldstone bosons at leading order in chiral perturbation theory and the value is about 0.01). Some points can belearned from this figure: (1). A predominant feature is that the branching ratios are not drastically sensitive to thecutoff parameter, which indicates a reasonable cutoff of the ultraviolet contributions by the empirical form factors tosome extent. (2). The leading contributions to the Y (4260) → J/ψπ are given by the differences between the neutraland charged charmed meson loops and also from the π - η mixing through the loops contributing to the eta transition.(3). At the same α , the branching ratios for Y (4260) → J/ψη transition are 2-3 orders of magnitude larger than thatof Y (4260) → J/ψπ . It is because that there is no cancelations between the charged and neutral meson loops.The branching ratios of Y (4260) → ψ ′ η (solid line) and Y (4260) → ψ ′ π (dashed line) in terms of α are shown inFig. 4(b). The behavior is similar to that of Fig. 4(a). Since the mass of ψ ′ is closer to the thresholds of ¯ DD ∗ than J/ψ , it should give rise to important threshold effects in Y (4260) → ψ ′ η ( π ) than in Y (4260) → J/ψη ( π ). At thesame α value, the obtained branching ratios of Y (4260) → ψ ′ π is larger than that of Y (4260) → J/ψπ . Since thethree-momentum of final η is only about 167 MeV in Y (4260) → ψ ′ η , which lead to a smaller branching rations in Y (4260) → J/ψη than that in Y (4260) → J/ψη at the same α value.In Fig. 5, we plot the α -dependence of the branching ratios of Y (4260) → h c π (solid line) and Y (4260) → h c η (dashed line), respectively. The branching ratios for Y (4260) → h c π ( η ) are larger than that of Y (4260) → J/ψπ ( η )and ψ ′ π ( η ), which is consistent with the power counting analysis in Eqs. (20) and (21).In order to study the exclusive threshold effects via the intermediate mesons loops, we define the following ratios, R ≡ |M Y (4260) → ψ ′ π | |M Y (4260) → J/ψπ | , R ≡ |M Y (4260) → ψ ′ η | |M Y (4260) → J/ψη | , (22)and r ≡ |M Y (4260) → J/ψπ | |M Y (4260) → J/ψη | , r ≡ |M Y (4260) → ψ ′ π | |M Y (4260) → ψ ′ η | , r ≡ |M Y (4260) → h c π | |M Y (4260) → h c η | . (23)These ratios are plotted in Fig. 6(a) and (b), respectively. The stabilities of the ratios in terms of α indicate areasonably controlled cutoff for each channels by the form factor. Since the coupling vertices are the same for thosedecay channels when taking the ratio, the stability of the ratios suggests that the transitions of Y (4260) → J/ψπ ( η )and ψ ′ π ( η ) are largely driven by the open threshold effects via the intermediate D ¯ D meson loops to some extent.The future experimental measurements of these decays can help us investigate this issue deeply. IV. SUMMARY
In this work, we have investigated the hidden-charm decays of Y (4260) and the decays Y (4260) → D ¯ D , D ¯ D ∗ and D ∗ ¯ D ∗ in ELA. In this calculation, Y (4260) is assumed to be the D ¯ D molecular state. Our results show that the α dependence of the branching ratios are not drastically sensitive, which indicate the dominant mechanism driven bythe intermediate meson loops with a fairly well control of the ultraviolet contributions.For the hidden charmonium decays, we also carried out the power counting analysis and our results for these decaysin ELA are qualitatively consistent with the power counting analysis. For the open charmed decays Y (4260) → D ¯ D , D ¯ D ∗ and D ∗ ¯ D ∗ , the exchanged intermediate mesons are light vector mesons or light pseudoscalar mesons which willintroduce different scale, so we cannot separate different scales and only give possible numerical results in the formfactor scheme. For the decay Y (4260) → D ∗ ¯ D ∗ , the exchanged mesons π is almost on-shell, so the coupled channeleffects are more important than other channels studied here. We expect the experiments to search for the hidden-charm and charmed meson pairs decays of Y (4260), which will help us investigate the nature and decay mechanismsof Y (4260) deeply. Acknowlegements
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