Searching for charmoniumlike states with hidden s s ¯
aa r X i v : . [ h e p - ph ] D ec Searching for charmoniumlike states with hidden s ¯ s Xiao-Hai Liu ∗ and Makoto Oka , † Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan and Advanced Science Research Center,JAEA, Tokai, Ibaraki 319-1195, Japan (Dated: November 15, 2018)
Abstract
We investigate the processes e + e − → γJ/ψφ , γJ/ψω and π J/ψη to search for the charmnium-likestates with hidden s ¯ s , such as Y (4140) , Y (4274) , X (4350) and X (3915) . These processes will receivecontributions from the charmed-strange meson rescatterings. When the center-of-mass energies of the e + e − scatterings are taken around the D s (2317) D ∗ s , D s (2460) D s or D s (2460) D ∗ s threshold, the anomaloustriangle singularities can be present in the rescattering amplitudes, which implies a non-resonance explana-tion about the resonance-like structures. The positions of the anomalous triangle singularities are sensitiveto the kinematics, which offers us a criterion to distinguish the kinematic singularities from genuine parti-cles. PACS numbers: 14.40.Rt, 13.25.Gv, 12.39.Mk ∗ [email protected] † [email protected] . INTRODUCTION With the number of charmoniumlike and bottomoniumlike
XY Z states observed in experi-ments increasing, the study on the exotic hadron spectroscopy is experiencing a renaissance inrecent years. In the aspect of theory, most of these
XY Z states do not fit into the conventionalquark model very well, which has been proved to be very successful in describing the heavyquarkonia below the open flavor thresholds. Various theoretical interpretations are then proposedto try to understand the underlying structures of these
XY Z states, such as hadronic molecule,tetraquark, hybrid, hadro-quarkonium, rescattering effect and so on. We refer to Refs. [1–3] forboth experimental and theoretical reviews about the
XY Z states.In this work, we will focus on the exotic candidates which may contain the s ¯ s components, i.e. Y (4140) , Y (4274) , X (4350) and X (3915) . Y (4140) and Y (4274) were firstly observed by theCDF Collaboration in the J/ψφ invariant mass distribution from B → KJ/ψφ decays [4, 5]. Thepresence of Y (4140) in B decays was later confirmed by the CMS and D0 Collaborations [6–8]. X (4350) was observed by the Belle Collaboration from the two photon process γγ → J/ψφ [9]. Y (4140) and Y (4274) were also expected to be produced in the two photon fusion reaction, butneither of them was observed [9]. These resonance-like structures observed in the J/ψφ massspectrum are very intriguing, since they may contain both a c ¯ c pair and and an s ¯ s pair. Althoughtheir masses are well beyond the open charm thresholds, their widths are very narrow, for instance, Γ Y (4140) =15 . +10 . − . MeV, Γ Y (4274) =32 . +23 . − . MeV, and Γ Y (4350) =13 +18 − MeV. The above propertiesimply that these three states may be exotic. Taking into account their masses and decay modes,some people think Y (4140) , Y (4274) and X (4350) are probably the hadronic bound states of D ∗ + s D ∗− s , D + s D − s and D + s D ∗− s respectively [10–20]. The tetraquark state c ¯ cs ¯ s is also a possibleexplanation about them [21, 22]. However, because of the low statistics, the masses and widths ofthese states still have larger uncertainties, even their existence are not well confirmed by differentexperiments [3, 23]. Concerning X (3915) , it is observed in the J/ψω invariant mass distributionfrom both the B decays B → KJ/ψω and the two photon fusion reaction γγ → J/ψω . Although X (3915) is currently taken as the conventional charmonium χ c (2 P ) by PDG [24], there are stillsome serious problems about this assignment. For instance, X (3915) has not been observed inthe D ¯ D invariant mass distribution, but the D ¯ D channel is expected to be the most importantdecay mode of χ c (2 P ) . Furthermore, if X (3915) is χ c (2 P ) , the mass splitting between the wellestablished χ c (2 P ) , of which the mass is about 3927 MeV, and χ c (2 P ) is too small, which isin conflict with the theoretical predictions [1, 25–27]. The width of X (3915) is also very narrow,which is about 20 MeV. We notice that the mass threshold of D + s D − s is about 3937 MeV, whichis less than J/ψφ threshold but close to X (3915) . Since there is a small s ¯ s component in thephysical ω meson, we may wonder whether there are some connections between D + s D − s systemand X (3915) . In Ref. [28], the authors suggest that X (3915) may be the bound state of D + s D − s .Before we claim these XY Z states are genuine particles, such as molecule, tetraquark or hy-brid, it is necessary to study some other possibilities. Some non-resonance explanations are alsoproposed to connect the ”resonance-like” peaks, i.e.
XY Z states, with the kinematic singulari-ties induced by the rescattering effects [29–39]. It is shown that sometimes it is not necessaryto introduce a genuine resonance to describe a resonance-like structure, because some kinematicsingularities of the rescattering amplitudes will behave themselves as bumps in the invariant massdistributions. The similar mechanism actually has been studied many years ago, such as the Peierlsmechanism proposed in 1960s [40–43]. In this paper, we are going to investigate the correlationsbetween the kinematic singularities and some exotic charmonium-like states with hidden s ¯ s .2 I. KINEMATIC SINGULARITY AND ITS OBSERVABLE PHENOMENAA. Radiative transitions ψ (4415) D + s D − s D + s γφJ/ψe + e − (b) ψ (4415) D + s D ∗− s D + s γφJ/ψe + e − (d) γe + φJ/ψψ (4415) D + s D − s D ∗ + s e − (c) φψ (4415) D + s D ∗− s D ∗ + s γJ/ψe + e − (e) φψ (4415) D + s D ∗− s D ∗ + s γJ/ψe + e − (a) FIG. 1: e + e − scattering into γJ/ψφ via ψ (4415) and the charmed-strange meson rescattering loops. Since the charmonium-like states with hidden s ¯ s may decay into J/ψφ , besides the B de-cays and the two photon fusion reactions, we also hope to search for these states in the process e + e − → γJ/ψφ , taking into account the high statics of the modern experimental facilities, such asBESIII and Belle. The process e + e − → γJ/ψφ will receive contributions from the rescattering dia-grams as displayed in Fig. 1. There are several reasons why we expect the rescatterings induced bythese charmed-strange meson loops will be important. Firstly, ψ (4415) is widely accepted as the S -wave charmonium ψ (4 S ) , and it can couple to D s (2317) D ∗ s ( D s (2460) D s , D s (2460) D ∗ s ) inrelative S -wave. This S -wave coupling will respect the heavy quark spin symmetry (HQSS). Thequark model calculation also indicates that this coupling will be strong. Secondly, since M ψ (4415) is very close to the mass thresholds of D s (2317) D ∗ s and D s (2460) D s , if we collect the data sam-ples at the center-of-mass (CM) energies near ψ (4415) , an intriguing kinematic singularity, i.e.,the anomalous triangle singularity (ATS), may emerge in the rescattering amplitude. Another im-portant reason is all of the internal charmed-strange mesons appeared in the rescattering diagramsare very narrow [24], which implies that the effect of the occurrence of the ATS will be obvious[44].The ATS corresponds to a pinch singularity of the loop integral. In Ref. [44], we have discussedthe kinematic conditions under which the ATS can be present. Taking into account the Feynmandiagram displayed in Fig. 2, according to the single dispersion representation of the triangle dia-3 q q q p a p b p c FIG. 2: Triangle diagram under discussion. The internal mass which corresponds to the internal momentum q i is m i ( i =
1, 2, 3). For the external momenta, we define P = s , ( p b + p c ) = s and p a = s . We willuse the same momentum and mass conventions in Figs. 1 and 4. gram, the locations of the ATS for s and s can be determined as s − = ( m + m ) + 12 m [( m + m − s )( s − m − m ) − m m m − λ / ( s , m , m ) λ / ( s , m , m )] , (1)and s − = ( m + m ) + 12 m [( m + m − s )( s − m − m ) − m m m − λ / ( s , m , m ) λ / ( s , m , m )] , (2)respectively, where λ ( x, y, z ) ≡ ( x − y − z ) − yz . s − and s − are the so-called anomalousthresholds. For convenient, we define the normal threshold s N ( s N ) and the critical value s C ( s C ) for s ( s ) as follows, s N = ( m + m ) , s C = ( m + m ) + m m [( m − m ) − s ] , (3) s N = ( m + m ) , s C = ( m + m ) + m m [( m − m ) − s ] . (4)If we fix s and the internal masses m , , , when s increases from s N to s C , the anomalousthreshold s − will move from s C to s N . Likewise, when s increases from s N to s C , s − willmove from s C to s N . This is the kinematic region where the ATS can be present. The discrepan-cies between the normal and anomalous thresholds are defined as follows, ∆ s = q s − − √ s N , ∆ s = q s − − √ s N . (5)Apparently, when s = s N ( s = s N ), we will obtain the maximum value of ∆ s ( ∆ s ), i.e., ∆ max s = √ s C − √ s N , (6) ∆ max s = √ s C − √ s N . (7)Larger ∆ max s and ∆ max s indicate larger kinematic regions where the ATS can emerge, which alsoimplies that it will be easier to detect the ATS in experiments. Notice that as long as s and the4 ABLE I: ∆ max s and ∆ max s for the corresponding triangle diagrams in Fig. 1.[MeV] Fig. 1(a) Fig. 1(b) Fig. 1(c) Fig. 1(d) Fig. 1(e) ∆ max s ∆ max s internal masses m , , are fixed, ∆ max s and ∆ max s are determined. The corresponding ∆ max s and ∆ max s of the diagrams in Fig. 1 are listed in Table I. From Table I, we can see that although ∆ max s and ∆ max s are not very large, they are still sizable. This is because the phase spaces for D ± s (2317) → γD ∗± s and D ± s (2460) → γD ( ∗ ) ± s are relatively larger, as discussed in Ref. [44].The above kinematic analysis indicates that the ATS induced by the charmed-strange mesonloops may emerge in a relatively larger kinematic region. To quantitatively estimate how importantthese rescattering amplitudes are, we will build our model in the framework of heavy hadron chiralperturbation theory (HHChPT) [45–51]. In HHChPT, to encode the HQSS, the charmed mesondoublets with light degrees of freedom J P = 1 / − and / + are collected into the followingsuperfields H a = 1 + v / D ∗ aµ γ µ − D a γ ] , (8) H a = [ ¯ D ∗ aµ γ µ + ¯ D a γ ] 1 − v / , (9) S a = 1 + v / (cid:2) D ′ µ a γ µ γ − D a (cid:3) , (10) S a = (cid:2) ¯ D a − ¯ D ′ µ a γ µ γ (cid:3) − v / , (11) ¯ H a, a = γ H † a, a γ , ¯ S a, a = γ S † a, a γ , (12)where H a ( S a ) is the charge conjugate field of H a ( S a ), and a is the light flavor index. Weidentify the physical states D ± s (2317) and D ± s (2460) as the doublet collected in the superfield S a, a , which is widely accepted. The pertinent effective Lagrangian which respects the HQSSand chiral symmetry takes the form L eff = g S < J ¯ S a ¯ H a + J ¯ H a ¯ S a > + C P < J ¯ H b γ µ γ ¯ H a A µba > + C V < J ¯ H b γ µ ¯ H a ρ µba > + ih < ¯ H a S b γ µ γ A µba > + e ˜ β < ¯ H a S b σ µν F µν Q ba > , (13)where < · · · > means the trace over Dirac matrices, J indicates the S -wave charmonia J = 1 + v / ψ ( nS ) µ γ µ − η c ( nS ) γ ] 1 − v / , (14) A µ is the chiral axial vector containing the Goldstone bosons A µ = 12 (cid:0) ξ † ∂ µ ξ − ξ∂ µ ξ † (cid:1) , (15)5ith ξ = e i M /f π , M = √ π + √ η π + K + π − − √ π + √ η K K − ¯ K − q η , (16) ρ µ is a × matrix for the nonet vector mesons ρ = √ ρ + √ ω ρ + K ∗ + ρ − − √ ρ + √ ω K ∗ K ∗− ¯ K ∗ φ , (17) F µν is the electromagnetic field tensor F µν = ∂ µ A ν − ∂ ν A µ , (18)and Q = diag (2 / , − / , − / . (19)The coupling constants h and ˜ β in Eq.(13), which are in relevant with the strong and radiativedecay rates of the charmed mesons respectively, can be extracted according to the experimentaldata. We will take the averaged values of h and ˜ β estimated in Ref. [51] in our following numericalcalculations, which are h =0.44 and | ˜ β | =0.42 GeV − respectively. For the coupling constant g S , bymatching the decay amplitude of ψ (4 S ) → D + s D ∗− s calculated according to Eq. (13) with that cal-culated in the quark pair creation model [26], we obtain g S ≈ − / . Similarly, by match-ing the scattering amplitudes of D ∗ + s D − s → J/ψη and D ∗ + s D ∗− s → J/ψφ calculated according toEq. (13) with those calculated in the quark-interchange model, we obtain C P ≈ − / and C V ≈
46 GeV − / respectively. We give a brief introduction about the quark-interchange model inAppendix A. Of course the estimation of the coupling constants using quark model will be model-dependent, and may have relatively larger uncertainties, but we expect that the order of magnitudeof this estimation is still reasonable to some extent. Notice that in HHChPT, every heavy filed H will contain a factor √ M H for normalization.According to the effective Lagrangian in Eq. 13, the transition amplitude of ψ (4 S ) → γJ/ψφ corresponding to the rescattering diagram Fig. 1(a) reads T Aψ (4 S ) → γJ/ψφ = 23 g S e ˜ βC V Z d q (2 π ) q − M D ∗ + s )( q − M D + s )( q − M D ∗− s ) × (cid:0) ǫ ψ (4 S ) · ǫ ∗ J/ψ v · ǫ ∗ γ p a · ǫ ∗ φ + ǫ ψ (4 S ) · ǫ ∗ φ v · ǫ ∗ γ p a · ǫ ∗ J/ψ + ǫ ψ (4 S ) · ǫ ∗ γ ǫ ∗ J/ψ · ǫ ∗ φ v · p a − ǫ ψ (4 S ) · ǫ ∗ J/ψ ǫ ∗ γ · ǫ ∗ φ v · p a − ǫ ψ (4 S ) · ǫ ∗ φ ǫ ∗ γ · ǫ ∗ J/ψ v · p a − p a · ǫ ψ (4 S ) v · ǫ ∗ γ ǫ ∗ J/ψ · ǫ ∗ φ (cid:1) , (20)where ǫ γ , ǫ J/ψ , ǫ φ , ǫ ψ (4 S ) are the polarization vectors of the corresponding particles, and the veloc-ity v can be taken as (1 , , , in the static limit. The other transition amplitudes share the similarformula with Eq. (20), which are omitted for brevity. We will introduce a Breit-Wigner type prop-agator of ψ (4415) when calculating the scattering amplitude of e + e − → γJ/ψφ via ψ (4415) andthe charmed-strange meson loops. The propagator takes the form BW [ ψ (4415)] = ( s − M ψ (4415) + iM ψ (4415) Γ ψ (4415) ) − . (21)6he coupling between the virtual photon and ψ (4415) will be determined by means of the vectormeson dominance model [52–54].The numerical results for the invariant mass distribution of J/ψφ in the process e + e − → γJ/ψφ via the charmed-strange meson loops are displayed in Fig. 3(a). We calculate the differential crosssections at several CM energies, i.e. 4.415 GeV and three thresholds. 4.415 GeV is actually outof the kinematic region where the ATS can be present, therefore when √ s =4.415 GeV, there isonly a small cusp appeared in the normal D ∗ + s D ∗− s threshold. Since the J/ψφ threshold is onlybelow the D ∗ + s D ∗− s threshold, but above the D + s D − s and D ∗ + s D − s thresholds, according to Table I,among the five rescattering diagrams of Fig. 1, only in the rescattering amplitudes corresponding toFigs. 1(a) and (e), the ATS can appear in the physical kinematic region. When the CM energy √ s is taken at the D + s D − s threshold, since the ATS can not be present in the rescattering amplitudescorresponding to Figs. 1(b) and (c), there is only a small cusp stay at D ∗ + s D ∗− s threshold (dashedline in Fig. 3(a)). When √ s = M D s + M D ∗ s , the ATS will be present in the rescattering amplitudecorresponding to Fig. 1(a), which lies about 4.6 MeV above the D ∗ + s D ∗− s threshold, as illustratedin Fig. 3(a) (dotted line). When √ s = M D s + M D ∗ s , the ATS will be present in the rescatteringamplitudes corresponding to Fig. 1(e), which lies about 12 MeV above D ∗ + s D ∗− s threshold (dot-dashed line in Fig. 3(a)). However, the CM energy √ s = M D s + M D ∗ s is far away from the peakposition of the resonance ψ (4415) , in which case the contribution of the diagram Fig.1(e) will besuppressed to some extent.Notice that the resonance-like peaks appeared in Fig. 3 are not induced by any genuine reso-nances, and the peak positions and shapes are very sensitive to the kinematics. As we point out inRef. [44], the difference between the genuine particles and the kinematic singularities is that theresonance-like peaks induced by the kinematic singularities will depend on the kinematic config-urations, which means that the peak positons of the resonance-like structures will depend on theproduction modes.The estimated cross section of the process is of the order of magnitude of 1 pico barn. Withthe huge statics of the modern experimental facilities, the effects induced by the ATS may bedetectable at BESIII, Belle or the forthcoming Belle II.As mentioned above, the higher J/ψφ threshold leads to that only the ATS which is in relevantwith the D ∗ + s D ∗− s threshold can emerge in the process e + e − → γJ/ψφ . On the other hand, the J/ψω threshold is even below the D + s D − s threshold, and D ( ∗ )+ s D ( ∗ ) − s can also scatter into J/ψω ,which imply that there will be three ATSs in relevant with three thresholds can be present in therescattering amplitudes of e + e − → γJ/ψω . However, because there is only a small s ¯ s componentin ω , the scattering amplitudes of D ( ∗ )+ s D ( ∗ ) − s → J/ψω will be suppressed.We will estimate the amplitudes of e + e − → γJ/ψω via charmed-strange meson loops by takinginto account the φ - ω mixing. When we introduce the vector nonet matrix ρ µ in Eqs. (13) and (17),we have assumed an ideal mixing between the flavor singlet and octet. The physical states φ and ω are actually not pure s ¯ s and ( u ¯ u + d ¯ d ) / √ , respectively. We rewrite their wave functions asfollows: φ = sin θ φω ( u ¯ u + d ¯ d ) / √ − cos θ φω s ¯ s, (22) ω = cos θ φω ( u ¯ u + d ¯ d ) / √ sin θ φω s ¯ s, (23)where the mixing angle θ φω is approximately equal to 0.065, by means of the quadratic Gell-Mann-Okubo mass formula [55–58]. The numerical results of J/ψω invariant mass distributionsare displayed in Fig. 3(b). Being different from Fig. 3(a), for some CM energies, there are threepeaks staying in the vicinities of D + s D − s , D ∗ + s D − s and D ∗ + s D ∗− s thresholds respectively. Further-more, according to Table I, it seems that in a relatively larger kinematic region these resonance-like7 ABLE II: ∆ max s and ∆ max s for the corresponding triangle diagrams in Fig. 4.[MeV] Fig. 4(a) Fig. 4(b) Fig. 4(c) ∆ max s
13 11 11 ∆ max s
12 10 10 peaks can be observed. However, compared with the process e + e − → γJ/ψφ , the cross sectionof e + e − → γJ/ψω via charmed-strange meson loops is nearly suppressed by two orders of mag-nitude, which will make the observation of the peaks induced by the ATS become difficult. Theprocess e + e − → γJ/ψω will also receive contributions from other rescattering diagrams, such asthe D ¯ D ∗ D and D ′ ¯ DD ∗ loops. But because D and D ′ are too broad, the rescattering amplitudeswill be highly suppressed and can only be taken as the backgrounds, as discussed in Refs. [36, 37].The BESIII Collaboration has ever searched for the charmonium-like state Y (4140) in theprocess e + e − → γJ/ψφ , but no significant signal is observed [59]. This result can be understoodin our scenario. Firstly, if Y (4140) is not a genuine particle but the kinematic threshold effect, itis not strange that people observe it in B decays rather than in e + e − scatterings, because of thedifferent kinematic configurations in these two reactions. Secondly, the BESIII Collaboration usedthe data samples collected at the CM energies 4.23, 4.26 and 4.36 GeV, but unfortunately none ofthese CM energies falls into the kinematic regions where the ATS can be present according toTable I. When the CM energies are taken in the range 4.430 ∼ ∼ J/ψφ invariantmass distributions, which are induced by the ATS. However, in our numerical results Fig 3(a),there are only peaks staying close to the D ∗ + s D ∗− s threshold, which are somewhat far away fromthe peak position of Y (4140) . Since the kinematics and rescattering processes in B decays will beanother story, here we can only point out the possibility but can not verify that the production of Y (4140) is induced by the kinematic threshold effect. B. Isospin-symmetry breaking process
The C -parity of the J/ψφ or J/ψω combination must be positive, but for the D ( ∗ )+ s D ( ∗ ) − s combination, the C -parity can either be positive or negative. In our scenario, we suppose that theresonance-like peaks observed in J/ψφ ( J/ψω ) can be related with the rescattering loops whichcontain the vertices of D ( ∗ )+ s D ( ∗ ) − s scattering into J/ψφ ( J/ψω ). Likewise, we can also expect thesimilar peaks in other final states with the negative C -parity. For instance, since D ( ∗ )+ s D ( ∗ ) − s can also scatter into J/ψη , of which the C -parity is negative, we can then study the process e + e − → π J/ψη via the charmed-strange meson loops. In another word, we hope to searchfor the negative C -parity charmonium-like structures with hidden s ¯ s in e + e − → π J/ψη .The diagrams for e + e − → π J/ψη via the charmed-strange meson loops are displayed inFig. 4. Notice that in these diagrams, there are vertices for D ± s coupling to D ± s π and D ± s cou-pling to D ∗± s π . Although the isospin symmetry is not conserved in these couplings, the processes D ± s → D ± s π and D ± s → D ∗± s π are acturally the most important decay modes for D ± s and D ± s re-spectively. This is because the isospin conserved DK and D ∗ K channels are not open for thesetwo P -wave charmed strange mesons. We therefore expect the rescattering amplitudes corre-sponding to Fig. 4 will be important for e + e − → π J/ψη . To estimate the amplitudes, we will use8he effective Lagrangian in Eq. (13). The decays D ± s → D ± s π and D ± s → D ∗± s π can proceed viathe η - π mixing, and the mixing angle θ ηπ = √ / m d − m u ) / (2 m s − m d − m u ) ≃ D ± s → D ± s π and D ± s → D ∗± s π are larger, the kinematic regions for the occurrence of theATS are also relatively larger. The numerical results of the J/ψη invariant mass distributions aredisplayed in Fig. 3(c). Due to the rescattering diagrams in Fig. 4, when the CM energy √ s issmaller than the D s D ∗ s threshold, only the peaks close to the D ∗ + s D − s threshold can appear. Thereis an intriguing property for the lineshapes of the invariant mass distributions. When the CM en-ergy √ s is taken to be the D s D ∗ s threshold (dotted line in Fig. 3(c)), taking into account Table. II,the kinematic regions of the ATS corresponding to Figs. 4(a) and (b) will overlap, but the loca-tions of the ATSs for √ s ( J/ψη invariant mass) are different. Therefore when √ s = M D s + M D ∗ s ,there will be two peaks appeared in the invariant mass distribution, and both of them stay close tothe D ∗ + s D − s threshold. Notice that there will be no peaks staying close to the D + s D − s threshold,because there is no vertex for D + s D − s → J/ψη included in the diagrams of Fig. 4.The cross sections of this isospin-symmetry breaking process are estimated at the order ofmagnitude of 1 pico barn. In Ref. [60], the BESIII Collaboration reports some results about thecross sections σ ( e + e − → π J/ψη ), of which the upper limits are also at the order of magnitude of1 pico barn. However, for the CM energies where the data are collected in Ref. [60], none of themfalls into the kinematic regions where the ATS can be present according to Table II. To observe theresonance-like peaks induced by the ATS, maybe one should collect the data at other CM energies,of which the range is 4.428 ∼ ∼ e + e − → π J/ψη will also receive contributions from other charmed meson rescat-tering diagrams, such as the D ¯ DD ∗ loop, which has been estimated in Ref. [61]. However, takinginto account that D is much broader than D s and D s , and the scattering e + e − → D ¯ D will besuppressed by the HQSS [37, 62, 63], we suppose that the contribution for e + e − → π J/ψη fromthe charmed meson loops will be smaller than that from the charmed-strange meson loops. Thekinematic regions of the ATS are also different for charmed and charmed-strange meson loops.
III. SUMMARY
In this work, to hunt for the charmonium-like states with hidden s ¯ s , we investigate the ra-diative transition processes e + e − → γJ/ψφ , e + e − → γJ/ψω and the isospin violation process e + e − → π J/ψη . These processes will receive contributions from the rescattering processes viathe charmed-strange meson loops, of which the corresponding amplitudes are demonstrated to bevery important. Especially, when the kinematics of these processes meets some special conditions,the ATSs can be present in the rescattering amplitudes, which will behave themselves as narrowpeaks in the corresponding invariant mass distributions. This implies that the non-resonance ex-planation about the resonance-like structures is possible. The genuine particles, such as tetraquarkstates, molecular states and hybrids, may not be necessary to be introduced when describing theobservations of some
XY Z particles. The ATS is just the kinematic singularity of the S -matrixelements, and the locations of the resonance-like peaks induced by the ATS will mainly depend onthe specific kinematic configurations. In our discussion, usually they will stay close to the D + s D − s , D ∗ + s D − s and D ∗ + s D ∗− s thresholds, which we call normal thresholds. Sometimes the discrepancybetween the normal and anomalous thresholds can be larger. Taking into account the locations ofthe ATSs can move, this offers us a criterion to distinguish kinematic singularities from genuineresonances, because the peak positions of the genuine resonances are usually thought to be rela-9ively stable. However, although the kinematic regions of the ATS for the charmed-strange mesonloops are sizable, they are not too large. To observe the movement of the ATS, the higher energyresolution of the experiments is necessary. Acknowledgments
We thank the helpful discussions with Q. Zhao and G. Li. This work was supported by theJapan Society for the Promotion of Science under Contract No. P14324, and the JSPS KAKENHI(Grant No. 25247036).
Appendix A: Quark-interchange model
In the reactions D ( ∗ )+ s D ( ∗ ) − s → J/ψφ , J/ψω and
J/ψη , c and ¯ c are recombined into a charmo-nium state, which is governed by the short range interaction. To describe these meson-mesonscatterings at the quark level, we will employ the Barnes-Swanson quark-interchange model toestimate the transition amplitudes [64–70]. In this approach, the non-relativistic quark potentialmodel is used, and the hadron-hadron scattering amplitudes are evaluated at Born order with theinterquark Hamiltonian. In the case of the anticharmed meson-charmed meson scatterings, theamplitudes arise from the sum of the four quark exchange diagrams as illustrated in Fig. 5. Theinteraction H ij between constituents i and j is represented by the curly line in Fig.5, and is takento be H ij ≡ λ ( i )2 · λ ( j )2 V ij ( r ij ) = λ ( i )2 · λ ( j )2 ( V conf + V hyp + V constant )= λ ( i )2 · λ ( j )2 (cid:26) α s r ij − b r ij − πα s m i m j S i · S j (cid:18) σ π / (cid:19) e − σ r ij + V constant (cid:27) . (A1)This Hamiltonian contains a Coulomb-plus-linear confining potential V conf and a short range spin-spin hyperfine term V hyp , which is motivated by one gluon exchange.The Born-order T -matrix element T fi can be expressed as the product of three factors for eachof the diagrams in Fig.5, T fi = (2 π ) I flavor I color I spin − space . (A2)Since there is no orbitally excited state involved in our discussion, the factor I spin − space can befurther factored into I spin − space = I spin × I space . (A3)The space factors are evaluated as the overlap integrals of the quark model wave functions. Itis convenient to write these overlap integrals in the momentum-space. For the four diagramsof Fig. 5, in the reaction AB → CD , where AB and CD are the initial and final meson pairs10espectively, the space factors read I C space = Z Z d k d q Φ A (2 k ) Φ B (2 k − P C ) Φ C (2 q − P C ) Φ D (2 k − P C ) V ( k − q ) ,I C space = Z Z d k d q Φ A ( − k ) Φ B ( − k − P C ) Φ C ( − k − P C ) Φ D ( − q − P C ) V ( k − q ) ,I T space = Z Z d k d q Φ A (2 k ) Φ B (2 q − P C ) Φ C (2 q − P C ) Φ D (2 k − P C ) V ( k − q ) ,I T space = Z Z d k d q Φ A ( − k ) Φ B ( − q − P C ) Φ C ( − k − P C ) Φ D ( − q − P C ) V ( k − q ) , (A4)where P C is the center-of mass momentum of meson C , and the potential V ( p ) is obtained viathe Fourier transformation of V ( r ) . [1] S. L. Olsen, Front. Phys. China. , 121 (2015) [arXiv:1411.7738 [hep-ex]].[2] A. Esposito, A. L. Guerrieri, F. Piccinini, A. Pilloni and A. D. Polosa, Int. J. Mod. Phys. A , no.04n05, 1530002 (2014) [arXiv:1411.5997 [hep-ph]].[3] K. YI, Int. J. Mod. Phys. A , 1330020 (2013) [arXiv:1308.0772 [hep-ex]].[4] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. , 242002 (2009) [arXiv:0903.2229 [hep-ex]].[5] T. Aaltonen et al. [CDF Collaboration], arXiv:1101.6058 [hep-ex].[6] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 261 (2014)doi:10.1016/j.physletb.2014.05.055 [arXiv:1309.6920 [hep-ex]].[7] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. D , no. 1, 012004 (2014)doi:10.1103/PhysRevD.89.012004 [arXiv:1309.6580 [hep-ex]].[8] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. , no. 23, 232001 (2015)doi:10.1103/PhysRevLett.115.232001 [arXiv:1508.07846 [hep-ex]].[9] C. P. Shen et al. [Belle Collaboration], Phys. Rev. Lett. , 112004 (2010) [arXiv:0912.2383 [hep-ex]].[10] X. Liu and S. L. Zhu, Phys. Rev. D , 017502 (2009) [Phys. Rev. D , 019902 (2012)][arXiv:0903.2529 [hep-ph]].[11] L. L. Shen, X. L. Chen, Z. G. Luo, P. Z. Huang, S. L. Zhu, P. F. Yu and X. Liu, Eur. Phys. J. C , 183(2010) [arXiv:1005.0994 [hep-ph]].[12] X. Liu, Z. G. Luo and S. L. Zhu, Phys. Lett. B , 341 (2011) [Phys. Lett. B , 577 (2012)][arXiv:1011.1045 [hep-ph]].[13] J. He and X. Liu, Eur. Phys. J. C , 1986 (2012) [arXiv:1102.1127 [hep-ph]].[14] S. I. Finazzo, M. Nielsen and X. Liu, Phys. Lett. B , 101 (2011) [arXiv:1102.2347 [hep-ph]].[15] Z. G. Wang, Int. J. Mod. Phys. A , 4929 (2011) [arXiv:1102.5483 [hep-ph]].[16] Z. G. Wang, Phys. Lett. B , 403 (2010) [arXiv:0912.4626 [hep-ph]].[17] Z. G. Wang, Eur. Phys. J. C , no. 7, 2963 (2014) [arXiv:1403.0810 [hep-ph]].[18] R. M. Albuquerque, M. E. Bracco and M. Nielsen, Phys. Lett. B , 186 (2009) [arXiv:0903.5540[hep-ph]].[19] L. Ma, Z. F. Sun, X. H. Liu, W. Z. Deng, X. Liu and S. L. Zhu, Phys. Rev. D , no. 3, 034020 (2014)[arXiv:1403.7907 [hep-ph]].
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J/ψφ , (b)
J/ψω , and (c)
J/ψη at four CM energy points.The vertical dotted, dashed, dot-dashed grid lines indicate the D + s D − s , D ∗ + s D − s , and D ∗ + s D ∗− s thresholds,respectively. ψ (4415) D + s D ∗− s D + s π J/ψe + e − (a) ηψ (4415) D + s D − s D ∗ + s π J/ψe + e − (b) ηψ (4415) D + s D ∗− s D ∗ + s π J/ψe + e − (c) FIG. 4: e + e − scattering into π J/ψη via ψ (4415) and the charmed-strange meson rescattering loops. q ¯ cc ¯ q q ¯ qc ¯ cq ¯ cc ¯ q q ¯ qc ¯ c q ¯ cc ¯ q q ¯ qc ¯ cq ¯ cc ¯ q q ¯ qc ¯ c C1 C2T1 T2