Disentangling the role of the Y(4260) in e^+e^-\to D^*\bar{D}^* and D_s^*\bar{D}_s^* via line shape studies
aa r X i v : . [ h e p - ph ] F e b Disentangling the role of the Y (4260) in e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s via line shapestudies Si-Run Xue , , a , Hao-Jie Jing , , b , Feng-Kun Guo , , c , and Qiang Zhao , , , d Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, Beijing 100049, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China and Synergetic Innovation Center for Quantum Effects and Applications (SICQEA),Hunan Normal University, Changsha 410081, China (Dated: September 10, 2018)Whether the Y (4260) can couple to open charm channels has been a crucial issue forunderstanding its nature. The available experimental data suggest that the cross sectionline shapes of exclusive processes in e + e − annihilations have nontrivial structures aroundthe mass region of the Y (4260). As part of a series of studies of the Y (4260) as mainly a¯ DD (2420) + c.c. molecular state, we show that the partial widths of the Y (4260) to thetwo-body open charm channels of e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s are much smaller than that to¯ DD ∗ π + c.c. . The line shapes measured by the Belle Collaboration for these two channels canbe well described by the vector charmonium states ψ (4040), ψ (4160) and ψ (4415) togetherwith the Y (4260). It turns out that the interference of the Y (4260) with the other charmoniaproduces a dip around 4.22 GeV in the e + e − → D ∗ ¯ D ∗ cross section line shape. The dataalso show an evidence for the strong coupling of the Y (4260) to the D ¯ D (2420), in line withthe expectation in the hadronic molecular scenario for the Y (4260). PACS numbers: 14.40.Rt, 14.40.Pq a E-mail address: [email protected] b E-mail address: [email protected] c E-mail address: [email protected] d E-mail address: [email protected]
I. INTRODUCTION
The mysterious state Y (4260) has attracted a lot of attention since its observation in 2005 bythe BaBar Collaboration [1]. Although many different models were proposed as solutions in theliterature, it is unfortunate that not all of these scenarios have been systematically studied andcompared with the existing experimental data (see e.g. several recent reviews [2–5] for summaries ofsome theoretical interpretations proposed in the literature). Following a series of recent studies bytreating the Y (4260) as mainly a ¯ DD (2420) + c.c. hadronic molecule, we are motivated to examineas many as possible exclusive processes where the Y (4260) can contribute. Such systematic studieswith more experimental constraints would either support or invalidate the picture of the Y (4260)being a hadronic molecule of ¯ DD (2420) + c.c. and should provide more insights into its intrinsicstructure. Therefore, we investigate the cross section line shapes of the e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s processes which cover the mass region of the Y (4260) and contain several established conventionalcharmonium states, i.e., ψ (4040), ψ (4160) and ψ (4415).So far, it has been demonstrated that most of the puzzling observations in the mass region of Y (4260) in e + e − annihilations can be accounted for in the same framework self-consistently. Forthe strong S -wave interactions between ¯ D and D (2420) (the charge conjugation, D ¯ D (2420), isalways implicated in the calculations), the dynamically generated Y (4260) should contain a largemolecular component of ¯ DD (2420) + c.c. as the long-distance component of its wave function,while a small short-distance component is always allowed. The consequence is that the Y (4260)will dominantly decay into ¯ DD ∗ π + c.c. via the decays of its constituent hadrons [6–8]. Moreover,due to the strong S -wave coupling to the nearby ¯ DD (2420) channel, the cross section line shape forthe e + e − → ¯ DD ∗ π + c.c. process should not be described by a Breit–Wigner parametrization. Thisis generally true for any states that strongly couple to nearby thresholds via an S -wave interaction.Namely, it is natural to expect a nontrivial cross section line shape for e + e − → ¯ DD ∗ π + c.c. aroundthe mass of the Y (4260). This phenomenon has been investigated in detail in Refs. [7, 8] whichare closely correlated with the study of the nature of the charged charmonium states Z c (3900).The experimental data, i.e., the cross sections for e + e − → J/ψππ , h c ππ and the invariant massspectra as well as angular distributions of Y (4260) → ¯ DD ∗ π + c.c. , which were also motivated bythe search for Z c (3900) and Z c (4020) at BESIII [9–14], have provided important constraints on themolecular component of the Y (4260).One interesting question arising from the above mentioned analysis is whether the Y (4260)should have significantly large decay widths into other open charm channels apart from ¯ DD ∗ π + c.c. .Given that the total width of Y (4260) is dominated by the ¯ DD ∗ π + c.c. channel [15], which hasa partial width of about 65 MeV in Ref. [8], while its decays into the hidden charm channels, i.e. J/ψππ , h c ππ , and χ c ω , turn out to be relatively small, the Y (4260) decays into other open charmchannels should also have small widths in order to match the total width extracted in the combinedanalysis of e + e − → J/ψππ , h c ππ , and ¯ DD ∗ π + c.c. In this sense, to accommodate the experimentaldata for e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s in the same framework is a challenge for the molecular picture,and should provide more information about its structure.In this work, we analyze the cross section line shapes of the e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s processesfrom threshold to about 4.6 GeV. These two processes have been measured by the Belle Collab-oration using the initial state radiation (ISR) in e + e − annihilations [16, 17]. One can see thatthe cross sections for e + e − → D ∗ ¯ D ∗ have been measured with a high precision [16], but there arestill large uncertainties in the data for e + e − → D ∗ s ¯ D ∗ s [17]. The former process has been studiedin [18] which considers the P -wave coupled-channel effects due to a pair of ground state charmedmesons and the ψ (4040) but not the Y (4260). In our analysis, in addition to the Y (4260) which isincluded as a ¯ DD (2420) hadronic molecule, we also include several conventional vector charmo-nium states established in this mass region including the ψ (4040), the ψ (4160) and the ψ (4415).We try to understand the behavior of the molecular state Y (4260) in this energy region and itsinterference with other charmonium states in the description of the cross section line shapes. Wenote in advance that our focus is mainly in the vicinity of the Y (4260), i.e. around the threshold of¯ DD (2420). Although there are additional exotic candidates above the ¯ DD (2420) threshold, suchas the Y (4360), to be neglected in this analysis, we find that we can still draw a clear conclusionon the Y (4260) contribution due to the relatively isolated ¯ DD (2420) threshold.In this paper, we first estimate the partial decay width of Y (4260) → D ∗ ¯ D ∗ in the molecularpicture in Sec. II, and then we study the cross section line shapes of e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s considering the Y (4260) and three charmonium states mentioned above in Sec. III. A brief summarywill be given in Sec. IV. II. THE PARTIAL DECAY WIDTH OF Y (4260) → D ∗ ¯ D ∗ In our scenario, the Y (4260) is treated as mainly an S -wave molecule of ¯ DD (2420) + c.c. witha small mixture of a compact c ¯ c core [8]. This treatment recognizes the HQSS breaking in theproduction of Y (4260) via e + e − annihilations. Namely, its production in e + e − annihilations ismainly via the direct coupling to its compact c ¯ c core which contains the S ( c ¯ c ) configuration.Then, the HQSS breaking allows the mixture of the S ( c ¯ c ) core with the long-distance componentof ¯ DD (2420) + c.c. which can couple to D ( c ¯ c ) via an S -wave interaction. The wave functionrenormalization will dress the nonvanishing γ ∗ – S ( c ¯ c ) coupling and the coupling of Y (4260) to¯ DD (2420) + c.c. as investigated in Ref. [8]. As a result of this scenario, it allows for the decayof Y (4260) → D ∗ ¯ D ∗ to occur not only via the dominant ¯ DD (2420) + c.c. component but alsothrough the direct coupling of the c ¯ c core to D ∗ ¯ D ∗ as illustrated in Fig. 1.In the framework of non-relativistic effective field theory (NREFT) the Lagrangians for thecoupling vertices in Fig. 1 can be written as [7, 8, 19, 20] L Y D ¯ D = i y eff √ D † a Y i D i † a − ¯ D i † a Y i D † a ) + H.c., (1) L D D ∗ π = i h ′ f π h D i a ( ∂ i ∂ j φ ab ) D ∗† jb − D i a ( ∂ j ∂ j φ ab ) D ∗† ib + 3 ¯ D i a ( ∂ i ∂ j φ ba ) ¯ D ∗† jb − ¯ D i a ( ∂ j ∂ j φ ba ) ¯ D ∗† ib i + H.c., (2) L D ∗ Dπ = g π (cid:0) D a ∂ i φ ab D ∗ i † b + ¯ D a ∂ i φ ba ¯ D ∗ i † b (cid:1) + H.c., (3)where f π = 132 MeV and the effective coupling for Y (4260) and ¯ DD (2420) is y eff = (3 . ± . − / which has been determined by the combined analysis of e + e − → J/ψππ , h c ππ and¯ DD ∗ π + c.c. [7, 8]; the effective coupling constants h ′ and g π can be determined by the processesof D → D ∗ + π − and D ∗− → D π − , respectively. The direct coupling for Y (4260) → D ∗ ( s ) ¯ D ∗ ( s ) takes the same form as the vector charmonium couplings to D ∗ ( s ) ¯ D ∗ ( s ) and will be given in the nextsection. Y (4260) D ∗ ¯ D ∗ ( a ) ( b ) ¯ D ∗ D ∗ πD ¯ DY (4260) ( c ) D ∗ ¯ D ∗ π ¯ D DY (4260) FIG. 1. Feynman diagrams for the two-body decay Y (4260) → D ∗ ¯ D ∗ in our scenario. The decay amplitude for the loop diagrams in Fig. 1 (b) and (c) can then be expressed as M Loop Y (4260) → D ∗ ¯ D ∗ = 3 y eff h ′ g π √ f π ǫ iY ǫ j ∗ D ∗ ǫ k ∗ ¯ D ∗ Z d l (2 π ) × ( l i l j l k − δ ij l k ~l [( p + l ) − m D + i + ][( p − l ) − m D + i + ][ l − m π + i + ] − l i l j l k − δ ik l j ~l [( p + l ) − m D + i + ][( p − l ) − m D + i + ][ l − m π + i + ] ) ≡ y eff h ′ g π √ f π ǫ iY ǫ j ∗ D ∗ ǫ k ∗ ¯ D ∗ h I ijk − C ijk − I ijk ( p ↔ p ) + C ikj ( p ↔ p ) i = 3 y eff h ′ g π √ f π ǫ iY ǫ j ∗ D ∗ ǫ k ∗ ¯ D ∗ (cid:16) I ijk − C ijk − C ikj (cid:17) , (4)where p , p and l are the four momenta of the D ∗ , ¯ D ∗ and π , respectively. In the last step, wehave used p = p = m D ∗ and p i = − p i in the center-of-mass (c.m.) frame. The factor of 3 / C ijk and I ijk are defined as follows: C ijk ≡ X m =1 δ ij I kmm , (5) I ijk ≡ Z d l (2 π ) l i l j l k [( p + l ) − m D + i + ][( p − l ) − m D + i + ][ l − m π + i + ] . (6)It is interesting to compare the transition of Fig. 1 with the hidden charm decay channels suchas Y (4260) → Z c (3900) π [8, 19] and χ c ω [21]. Following the NREFT power counting scheme ofRefs. [5, 22, 23], it can be seen that the loop amplitude for the Y (4260) → Z c (3900) π is ultraviolet(UV) convergent and scales as 1 /v with v the typical non-relativistic velocity of the intermediatecharmed mesons. With v ≪ Y (4260) → χ c ω is similar. Such a power counting is because of the S -wavecouplings of both the initial and final heavy particles to the intermediate charmed mesons.For the Y (4260) → D ∗ ¯ D ∗ , the velocity scaling is different. Near the mass threshold of¯ DD (2420) the internal charmed mesons carry the typical velocity v ∼ ( | m Y − m D − m D | / ˜ m ) / ≃ .
1, where ˜ m ≡ ( m D + m D ) /
2, and the velocity of the final D ∗ is v f ≃ .
35 in the Y (4260) restframe. For the velocity scaling, we may count v f ∼ v . Then the loop integral measure scales as v ,and all propagators scale as v − . As a result, the triangle loop amplitude scales as v v − v = v ,where the factor of v comes from the vertices, which is significantly suppressed in respect of thecontact interaction. Because of the D - and P -wave pionic couplings given by Eqs. (2) and (3) ,respectively, the loop decay amplitude can be split into P -wave and F -wave parts as I ijk = ~p (cid:16) p i δ jk + p j δ ik + p k δ ij (cid:17) I P + (cid:20) p i p j p k − ~p (cid:16) p i δ jk + p j δ ik + p k δ ij (cid:17)(cid:21) I F . (7)The first term contributes to the decay into the D ∗ ¯ D ∗ in a P -wave, while the second contributesto that in a F -wave. While the F -wave part is UV convergent, the P -wave part diverges andneeds to be regularized and renormalized. The UV divergence can be absorbed by introducing acounterterm. However, the tree-level term of Fig. 1 (a) cannot serve as the counterterm for the loopamplitude of Fig. 1 (b) and (c) since diagram (a) is introduced to incorporate the S ( c ¯ c ) couplingto D ∗ ¯ D ∗ while in diagrams (b) and (c) the S -wave ¯ DD (2420), which leads to the transitions to D ∗ ¯ D ∗ , couples to the D ( c ¯ c ) in the heavy quark limit [24]. This means that the UV divergencehere needs to be absorbed into a different counterterm. Here we will regularize the UV divergencepractically using a form factor with a cutoff, see below. The cutoff will be treated as a freeparameter, which effectively takes the place of the counterterm at a given scale.In order to regularize the UV contributions in the loop integral, we introduce a monopole formfactor for each propagator to take into account the off-shell effects in the loop integral: F (Λ i , m i , l i ) = Λ i − m i Λ i − l i , (8)where Λ ≡ m D + α Λ QCD and Λ ≡ m D + α Λ QCD , with Λ
QCD = 220 MeV and α a parameterof order unity, are defined for the heavy charmed mesons. For the light pion exchange the cut-offΛ π is within a range of 0 . ∼ I ijk can beexpressed as: I ijk = Z d D l (2 π ) D l i l j l k F (Λ , m D , ( p + l ) ) F (Λ , m D , ( p − l ) ) F (Λ π , m π , l )[( p + l ) − m D + i + ][( p − l ) − m D + i + ][ l − m π + i + ]= i π [ p i p j p k A ′ + ( − p i δ jk − p j δ ik − p k δ ij ) B ′ ] , (9)with A ′ ≡ A ( P , p , p , m D , m D , m π ) − A ( P , p , p , Λ , m D , m π ) − A ( P , p , p , m D , Λ , m π ) − A ( P , p , p , m D , m D , Λ π ) + A ( P , p , p , Λ , Λ , m π ) + A ( P , p , p , Λ , m D , Λ π )+ A ( P , p , p , m D , Λ , Λ π ) − A ( P , p , p , Λ , Λ , Λ π ) , (10) B ′ ≡ B ( P , p , p , m D , m D , m π ) − B ( P , p , p , Λ , m D , m π ) − B ( P , p , p , m D , Λ , m π ) − B ( P , p , p , m D , m D , Λ π ) + B ( P , p , p , Λ , Λ , m π ) + B ( P , p , p , Λ , m D , Λ π )+ B ( P , p , p , m D , Λ , Λ π ) − B ( P , p , p , Λ , Λ , Λ π ) . (11)Here the functions A and B are defined as A ( P , p , p , m D , m D , m π ) = Z dx Z y ∆ dy, (12) B ( P , p , p , m D , m D , m π ) = Z dx Z y ln ∆ dy, (13) (cid:1) (cid:2) = (cid:1) (cid:2) = - - - - (cid:3) (cid:4) Y ( ) (cid:5) D * + D * - Loop ( G e V ) FIG. 2. The cutoff-dependence of the partial decay width of Y (4260) → D ∗ + D ∗− from the one-pion exchangediagrams in the molecular scenario. Here the results with two typical Λ π values are shown. where P ≡ p + p is the initial momentum, x and y are the Feynman parameters, and ∆ = y p x + (1 − y ) m π − y ∆ m x with p = xp − (1 − x ) p and ∆ m x = x ( p − m D ) + (1 − x )( p − m D ).In the numerical calculation, we replace m D by m D − i Γ D / D the constant width of the D (2420).Due to the UV divergence in the P -wave part of the loop amplitude, it is impossible to makea definite prediction on the two-body decay partial width of the Y (4260) into a pair of vectorcharm mesons by simply calculating the loop diagrams. The best we can do is to estimate thevalues by varying the cutoffs in the form factors within natural ranges. Thus, in Fig. 2 we showthe dependence of Γ Loop Y (4260) → D ∗ + D ∗− on α with two typical values for Λ π . The result ranges from1 MeV to about 20 MeV in the figure, and for α = 1 it takes a value of 2.4 MeV and 12.3 MeVfor Λ π = 0 . Y (4260) → ¯ DD ∗ π + c.c. One intriguing feature of the Y (4260) is that it does not show up as a peak in the exclusivetwo-body open-charm cross sections. In order to clarify the role played by the Y (4260) in e + e − → D ∗ + D ∗− , we will investigate the cross section line shape of this process in the next section takinginto account contributions from the nearby charmonium states. The idea is to investigate whetherthe cross section line shapes could provide more stringent constraint on Y (4260) or not. A combinedinvestigation of the cross section line shape of e + e − → D ∗ + s D ∗− s will also be presented. ( a ) e + ψ i e − D ∗ ( s ) ¯ D ∗ ( s ) Y (4260) Y (4260) e − e + ( b ) D ¯ D π ( K )¯ D ∗ ( s ) D ∗ ( s ) FIG. 3. Feynman diagrams for e + e − → D ∗ ( s ) ¯ D ∗ ( s ) via (a) intermediate charmonium states ψ i , and (b) Y (4260) as a ¯ DD (2420) + c.c. hadronic molecule state. III. THE LINE SHAPES OF THE CROSS SECTIONS OF e + e − → D ∗ ( s ) ¯ D ∗ ( s ) In order to investigate the line shape of e + e − → D ∗ ( s ) ¯ D ∗ ( s ) in the vicinity of Y (4260), the con-tributions from the nearby charmonium states, ψ (4040), ψ (4160) and ψ (4415), which are normallyconsidered as the 3 S , 2 D and 4 S charmonium states, respectively, should be included. For conve-nience, we use ψ , ψ and ψ to denote ψ (4040), ψ (4160) and ψ (4415), respectively. The processesof e + e − → D ∗ ( s ) ¯ D ∗ ( s ) are depicted in Fig. 3 where the tree-level diagram represents the charmoniumtransitions and the loop diagram illustrates the Y (4260) contribution via its molecular component.As mentioned earlier, the tree diagram also contains the contribution from the short-distance coreof the Y (4260).The effective Lagrangian for the vector charmonium couplings to the virtual photon is describedby the vector meson dominance (VMD) model: L V γ = em V f V V µ A µ , (14)while the strong couplings for ψ i ( i = 1 , ,
3) to the D ∗ ( s ) ¯ D ∗ ( s ) meson pairs are as follows [20, 25]: L ψ S D ∗ ( s ) ¯ D ∗ ( s ) = ig ψ S D ∗ ( s ) ¯ D ∗ ( s ) ψ kS (cid:2) ( δ km δ ln − δ kn δ lm − δ kl δ mn )( ∂ m D ∗† l ( s ) ¯ D ∗† n ( s ) − D ∗† l ( s ) ∂ m ¯ D ∗† n ( s ) ) (cid:3) + H.c., L ψ D D ∗ ( s ) ¯ D ∗ ( s ) = ig ψ D D ∗ ( s ) ¯ D ∗ ( s ) ψ kD (cid:2) (4 δ km δ ln − δ kn δ lm − δ kl δ mn )( ∂ m D ∗† l ( s ) ¯ D ∗† n ( s ) − D ∗† l ( s ) ∂ m ¯ D ∗† n ( s ) ) (cid:3) + H.c., (15)where the coupling constants g ψ i D ∗ ( s ) ¯ D ∗ ( s ) will be determined by fitting the cross section line shapes.Note that ψ S and ψ D denote the S and D wave c ¯ c states of J P C = 1 −− of which the couplings to D ∗ ( s ) ¯ D ∗ ( s ) are different, and the above forms are obtained assuming heavy quark spin symmetry.For the e + e − → D ∗ ¯ D ∗ , we consider all of the three conventional charmonium states mentionedabove and the Y (4260), while for the e + e − → D ∗ s ¯ D ∗ s we only include ψ ( ψ (4160)) and ψ ( ψ (4415))as the contributing charmonium states since ψ ( ψ (4040)) is far below the threshold of D ∗ s ¯ D ∗ s . Thetransition amplitudes for e + e − → D ∗ ( s ) ¯ D ∗ ( s ) can then be expressed as M e + e − → D ∗ ¯ D ∗ = ¯ v ( q ) γ i u ( q ) ( (2 δ ij p k + 2 δ ik p j − δ jk p i ) " − e g ψ D ∗ ¯ D ∗ m ψ exp( − | p f − p | /β + iθ ) f ψ E ( E − m ψ + im ψ Γ ψ )+ − e g ψ D ∗ ¯ D ∗ m ψ exp( − | p f − p | /β + iθ ) f ψ E ( E − m ψ + im ψ Γ ψ ) + − e g eff Y D ∗ ¯ D ∗ m Y exp( − | p f − p Y | /β ) f eff Y E D Y ( E cm ) +(2 δ ij p k + 2 δ ik p j − δ jk p i ) − e g ψ D ∗ ¯ D ∗ m ψ exp( − | p f − p | /β + iθ ) f ψ E ( E − m ψ + im ψ Γ ψ )+ − e y eff h ′ g π m Y √ f π f eff Y E D Y ( E cm ) h I ijk ( α, Λ π ) − C ijk ( α, Λ π ) − C ikj ( α, Λ π ) i ) ǫ j ∗ D ∗ ǫ k ∗ ¯ D ∗ , (16) M e + e − → D ∗ s ¯ D ∗ s = ¯ v ( q ) γ i u ( q ) ( (2 δ ij p k + 2 δ ik p j − δ jk p i ) − e g ψ D ∗ s ¯ D ∗ s m ψ exp( − | p f − p | /β + iθ ) f ψ E ( E − m ψ + im ψ Γ ψ )+(2 δ ij p k + 2 δ ik p j − δ jk p i ) " − e g ψ D ∗ s ¯ D ∗ s m ψ exp( − | p f − p | /β + iθ ) f ψ E ( E − m ψ + im ψ Γ ψ )+ − e g eff Y D ∗ s ¯ D ∗ s m Y exp( − | p f − p Y | /β ) f eff Y E D Y ( E cm ) + − e y eff h ′ g π m Y √ f π f eff Y E D Y ( E cm ) h I ijk ( α, Λ K ) − C ijk ( α, Λ K ) − C ikj ( α, Λ K ) i ) ǫ j ∗ D ∗ s ǫ k ∗ ¯ D ∗ s , (17)where q and q are the incoming four-momenta of the electron and positron, respectively, ~p is theoutgoing momentum of D ∗ ( s ) in the c.m. frame, p f = | ~p | = q E − m D ∗ s ) / p i = q m ψ i − m D ∗ s ) / β controls the suppression. As a reasonableassumption to reduce the number of parameters, we assume that these two processes share thesame value for β which means that the strong couplings for ψ i to D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s have the samesuppression behavior when the resonances become off-shell. The ψ i states and the Y (4260) caninterfere through many possible intermediate hadron loops which can introduce energy-dependentcomplex phases. In order to parameterize such effects, we also introduce a few constant phases,0 TABLE I. The masses, total widths and leptonic partial widths adopted for the charmonium states fromPDG [28]. ψ (4040) ψ (4160) ψ (4415) m ψ (MeV) 4039 ± ± ± ψ (MeV) 80 ±
10 70 ±
10 62 ± e + e − (keV) 0 . ± .
07 0 . ± .
22 0 . ± . denoted by θ i . The constant phase assumption is reasonable as long as the thresholds of theintermediate hadrons are far away. Based on the above argument and taking into account theSU(3) flavor symmetry, we let θ = θ and θ = θ to reduce two more parameters.In Eq. (16) y eff / [ f eff Y D Y ( E cm )] is the product of the bare coupling y/f Y and the Y (4260) prop-agator defined in the molecular picture [8] which has the following expression: y eff f eff Y D Y ( E cm ) ≡ Zy f Y [ E cm − m Y − Z e Σ ( E cm ) + i Γ non − ¯ DD / , (18)where the subtracted self-energy e Σ ( E cm ) = Σ ( E cm ) − ReΣ ( m Y ) − ( E − m Y ) ∂ Σ ( m Y ) /∂E cm [26]with Σ ( E cm ) the Y (4260) self-energy due to the ¯ DD (2420) loop. In the MS subtraction scheme,the self-energy is given by Σ ( E cm ) = µ/ (8 π ) p µ ( E cm − m D − m D ) + iµ Γ D [8]. We use m Y =(4 . ± . non − ¯ DD = (0 . ± . e + e − → J/ψππ and h c ππ [7], and the wave function renormalization constant Z ≃ .
13 isdetermined in Ref. [8]. The values of m ψ i , Γ ψ i → e + e − and Γ ψ i ( i = 1 , ,
3) are taken from thosegiven by the Particle Data Group (PDG) [28], which are listed in Table I. The leptonic decaycoupling constants of the charmonium states defined by the VMD model in Eq. (14) can thus bedetermined.To further reduce the number of parameters we assume the SU(3) flavor symmetry for the strongcouplings of the same charmonium states to D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s so that they take the same value,i.e. g ψ D ∗ ¯ D ∗ = g ψ D ∗ s ¯ D ∗ s and g ψ D ∗ ¯ D ∗ = g ψ D ∗ s ¯ D ∗ s . In total there are 11 parameters to be fitted fromthe cross section data: four cutoff parameters ( α , Λ π , Λ K and β ), four coupling constants ( g ψ i D ∗ ¯ D ∗ and g eff Y D ∗ ¯ D ∗ which is the coupling of the short-distance core of the Y (4260) to the D ∗ ¯ D ∗ ), andthree phases. We note in advance that due to the lack of precise experimental measurements forthe e + e − → D ∗ s ¯ D ∗ s some of the parameters cannot be well constrained in the numerical fitting, andwe anticipate that the main contributions to the χ value will be from the D ∗ ¯ D ∗ channel.In Table II the values of the fitted parameters are listed. The value of β , which bears a largeuncertainty, is consistent with the reasonable order of 1 GeV. The cutoff parameter α is consistent1 TABLE II. Parameters determined by fitting to the Belle experimental data [16, 17].Parameters Fitted values α (1 . ± . β (3 . ± .
34) GeVΛ π (409 . ± .
9) MeVΛ K (544 . ± .
9) MeV θ . ◦ ± . ◦ θ , . ◦ ± . ◦ θ , . ◦ ± . ◦ g ψ D ∗ ¯ D ∗ (1 . ± .
12) GeV − / g ψ D ∗ ( s ) ¯ D ∗ ( s ) (0 . ± .
13) GeV − / g ψ D ∗ ( s ) ¯ D ∗ ( s ) (0 . ± .
06) GeV − / g eff Y D ∗ ( s ) ¯ D ∗ ( s ) (0 . ± .
12) GeV − / χ / d.o.f 1.53TABLE III. The partial decay widths of Y (4260) and ψ i to D ∗ ( s ) ¯ D ∗ ( s ) extracted from this analysis.Widths Y (4260) ψ (4040) ψ (4160) ψ (4415)Γ Tree
Y D ∗ ¯ D ∗ (MeV) 9 . ± .
18 - - -Γ
Loop
Y D ∗ ¯ D ∗ (MeV) 1 . ± .
68 - - -Γ D ∗ ¯ D ∗ (MeV) 10 . ± .
86 10 . ± .
47 4 . ± .
73 8 . ± . D ∗ s ¯ D ∗ s (MeV) - - - 3 . ± . with O (1). The cutoff Λ π can be better constrained in e + e − → D ∗ ¯ D ∗ than Λ K in e + e − → D ∗ s ¯ D ∗ s ,again due to the poor data quality of the latter. With the fitted α and Λ π values, the contributionfrom the ¯ DD (2420) intermediate states to the partial decay width for Y (4260) → D ∗ ¯ D ∗ is givenby Γ Loop Y (4260) → D ∗ + D ∗− = (1 . ± .
68) MeV, while the contribution from the short-distance S c ¯ c core is much larger as listed in Table III. With the fitted couplings for the charmonium states to D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s , we can also obtain their corresponding partial decay widths which are also listedin Table III.With the fitted parameters in Table II, we find that the cross section of e + e − → D ∗ ¯ D ∗ can bewell described. The line shape from the best fit is plotted in Fig. 4 and compared with the Belledata [16]. An apparent feature is that the threshold enhancement in the measured e + e − → D ∗ ¯ D ∗ line shape can be largely accounted for by the ψ (4040) while the contributions from Y (4260) and2other charmonium states are rather small below 4.2 GeV. The bump between 4.1 and 4.2 GeV canbe described well by the contributions from ψ (4040) and ψ (4160) and their interference. Notice thatthe relative phase between these two states leads to destructive interference in the energy regionsof below ψ (4040) or above ψ (4160) while in the region between their masses the interference isconstructive. As a result, the rise of the cross section in the near threshold region is enhanced,although the ψ (4040) coupling to D ∗ ¯ D ∗ is in a P wave.Comparing the dotted curve (denoted by “ ψ i ” in the figure), which is the sum of the contribu-tions from all considered conventional charmonium states, with the solid curve, which is the bestfit result, or with the experimental data, one sees that the dip around 4.22 GeV in the data comesfrom a destructive interference between the charmonia and the Y (4260). The good descriptionof the special shape around 4.3 GeV originates from the strong coupling of the Y (4260) to the¯ DD (2420) + c.c. (see also the curve denoted by “ Y –loop”), which is an essential feature of theconsidered hadronic molecular picture. Although the cross section line shape plotted in Fig. 4 isnot perfectly fitted, it can still clarify the role played by Y (4260). The dominance of the ψ (4040)and ψ (4160) near threshold actually leaves a very limited space for the Y (4260) which is consistentwith the expectation based on the hadronic molecule scenario for the Y (4260). In other words, the Y (4260) does not have a large partial decay width in the D ∗ ¯ D ∗ channel.We also tried a fit without including the Y (4260), and found that the cross section at energiesabove the dip, which correspond to the region around the ¯ DD (2420) + c.c. threshold, cannot bewell described. In fact, a negligibly small contribution from the Y (4260) is consistent with themolecular picture.From Fig. 4 we also see that the cross sections in the region of 4 . ∼ . ψ (4415) and the Y (4260). Despite this, we need tomention that the S -wave open thresholds of D ∗ ¯ D (2420) + c.c. and D ∗ ¯ D (2460) + c.c. have notbeen taken into account, and they could play a role in the region between 4.4 and 4.5 GeV. Weleave their contributions to be investigated more elaborately in future studies when more data areavailable.The transition of Fig. 3 (b) has also access to the kinematics of the so-called “triangle sin-gularity” (TS), which has been broadly investigated recently in the literature [29–43] (see e.g.Refs. [5, 44] for a recent review). For an appropriate input energy of the initial e + e − annihilation,the TS condition corresponds to that the internal particles can approach their on-shell kinematicssimultaneously and the interactions at all vertices can happen as classical processes in space-time.With all the intermediate mesons fixed as D ¯ Dπ as in the figure and final states being D ∗ + D ∗− ,3 – – – – – – – – – –– – – – – – – – – – – – – – – – – – – – – – – – – – – – –– – – – – – – – – – – – – – – – – – –Total (cid:1) i Y (cid:1) (cid:1) (cid:1) Y - TreeY - Loop E cm ( GeV ) (cid:2) e + e - (cid:3) D * + D * - ( nb ) FIG. 4. The fitting results for the cross section of e + e − → D ∗ + D ∗− . The overall cross section is denoted bythe solid line. The exclusive contributions from single states are also presented, i.e. ψ (4040) (long-dashed), ψ (4160) (dot-dashed), ψ (4415) (dot-dashed-dashed), and Y (4260) (dashed). The sum of the contributionsfrom all ψ i states is denoted by the dotted line. The data are from Ref. [16]. Total (cid:1) i Y (cid:1) (cid:1) Y - TreeY - Loop - E cm ( GeV ) (cid:2) e + e - (cid:3) D s * + D s * - ( nb ) FIG. 5. The fitting results for the cross section of e + e − → D ∗ + s D ∗− s . The overall cross section is denoted bythe solid line. The exclusive contributions from single states are also presented, i.e. ψ (4160) (dot-dashed)and ψ (4415) (dot-dashed-dashed), and Y (4260) (dashed). The inclusive contributions from ψ i is denotedby the dotted line. The data are from Ref. [17]. the e + e − c.m. energy for producing a TS is at about 5.35 GeV which is far beyond the region ofFig. 4. This situation is very different from the cases of e + e − → ¯ DD ∗ π [7, 8] and J/ψππ [19, 45].As already mentioned the present experimental data for e + e − → D ∗ s ¯ D ∗ s [17] do not allow areliable determination of the parameters in this channel. As shown in Fig. 5, with the presentfitted parameters only the ψ (4415) can produce a resonance structure in the cross section lineshape. The exclusive contributions from these three states are also presented. The theoretical4curve shows a flattened line shape near threshold which is different from that in e + e − → D ∗ ¯ D ∗ .Although the D ∗ + s D ∗− s threshold, 4.22 GeV, is very close to the mass of the Y (4260), we do not seea near-threshold enhancement due to the Y (4260). We expect that the contribution of the Y (4260)in this process should be smaller than that in the e + e − → D ∗ + D ∗− since the intermediate kaonin Fig. 3 (b) cannot go on shell, contrary to the case of the pion. However, one notices that thepoor data quality do not allow a more quantitative restriction on the Y (4260) contribution to thisprocess. This situation is also reflected by the poor determination of the cutoff energy Λ K shownin Table II. A more precise measurement of the e + e − → D ∗ s ¯ D ∗ s cross section line shape is highlyrecommended. IV. SUMMARY
In this work we have studied the cross section line shapes of the e + e − → D ∗ ¯ D ∗ and D ∗ s ¯ D ∗ s processes from thresholds to about 4.6 GeV. This is the energy region that contributions from the Y (4260) are of great interest since information in addition to those in other processes about thestructure of this mysterious state can be extracted. Our study shows that the cross sections ofthese two processes in this energy region are dominated by the established charmonium states, i.e. ψ (4040), ψ (4160) and ψ (4415), while the contributions from the Y (4260) as a D ¯ D (2420) + c.c. molecule state turns out to be rather small in most of the energy region. This result is consistentwith the observation that the main open charm decay channel of the Y (4260) is D ¯ D ∗ π + c.c. which accounts for most of its decay width. The partial decay width of the Y (4260) → D ∗ ¯ D ∗ isobtained to be (11 ±
8) MeV. The dip around 4.22 GeV in the e + e − → D ∗ ¯ D ∗ cross section is dueto the interference between the Y (4260) and the conventional charmonium states. The hadronicmolecular feature of the Y (4260) in our model shows up as a non-trivial structure at the D ¯ D (2420)threshold. The current data present a clear evidence for such a structure. Yet, more precise dataare necessary to make the conclusion more solid. For the e + e − → D ∗ s ¯ D ∗ s channel, the presentexperimental data from Belle [17] have poor quality. However, although the data do not allow anyconclusion on the role played by charmonium states, we do not expect sizeable contributions fromthe Y (4260). The future precise data from BESIII for these two channels will be able to clarifythe role played by the Y (4260) and provide valuable insights into its internal structure.5 ACKNOWLEDGMENTS
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