Zcs states from the Ds∗ D ¯ ∗ and J /ψK∗ coupled channels: Signal in B+ →j /ψφK+ decay ZCS STATES from the DS∗ D ¯ ∗ ... IKENO NATSUMI, MOLINA RAQUEL, and OSET EULOGIO

Zcs states from the D s¯ Dand J=ψKcoupled channels: Signal in B+→J=ψϕK+decay Natsumi Ikeno ,1,* Raquel Molina,2,†and Eulogio Oset2,‡ 1Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori 680-8551, Japan 2Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia, Spain (Received 22 November 2021; accepted 23 December 2021; published 10 January 2022) We study the D s¯ Dsystem in connection with the J=ψKin coupled channels and observe that, within reasonable values of the cutoff used to regularize the loops, the system does not develop a bound state. However, the JP¼2þchannel has enough attraction to create a strong cusp structure that shows up in the J=ψKþinvariant mass distribution in the Bþ→J=ψϕKþdecay at the D s¯ Dthreshold. Such structure is visible in the experimental Bþ→J=ψϕKþand ¯ B0→J=ψKþK−decays, with small statistics, and our results should stimulate further measurements around this region, given the fact that cusp effects provide as valuable information on hadron dynamics as resonances themselves. DOI: 10.1103/PhysRevD.105.014012 I. INTRODUCTION The discovery of the Zcsð3985Þstate by the BESIII collaboration [1] in the mass distribution of ¯ D sDand ¯ DsDadded a new type of exotic meson, with ¯ csc¯ uto the increasing long list of exotic hadronic states (see reviews [2–8]). The reaction of the theoretical community has been fast, and many papers have been devoted to understanding thenatureofthisstate.Asusual,threelineshavebeen followed to describe the state, assuming it to be a tetraquak state, a meson-meson molecular state, or using QCD sum rules. Independent on the picture used, it is an extended idea that the Zcsð3985Þstate is an SU(3) partner of the Zcð3900Þ state, replacing a qquark by a strange squark. In Refs. [9–17] the QCD sum rules method is used to generate the Zcs and the state is obtained as a ¯ D sDconfiguration, as a strangeness analog of the ¯ DDconfiguration of the Zc, with the usual large uncertainties about the mass of the states. Tetraquark calculations, both using diquark-antidiquark configurations ½sc½¯ q¯ c[18–20] or diquark-antidiquark and meson-meson configurations [21,22] have been carried out. In Ref. [21] the molecular component is shown not to bind and the diquark picture is favored. In Ref. [22] the chiral constituent quark model is used and the molecular component, including coupled channels, is shown to lead to a virtual state. A different analysis along these lines is carried out in Ref. [23] where the Zcsð3985Þof Ref. [1] and the Zcsð4000Þobserved by the LHCb collaboration in Ref. [24] are supposed to be two different states and follow a mixing like the one that mixes the K1ð1270Þand K1ð1440Þstates.1 Much work is done considering molecules using directly dynamics for meson interaction [27–41]. By analogy to the Zcð3900Þ, the SU(3) partner in the strange sector Zcsð3985Þ is favored as the D− sD0þD− sD0combination. One exception is Ref. [37], where this combination is preferred for the Zcsð4000Þstate, while the D− sD0−D− sD0combination is proposed for the Zcsð3985Þ, all that assuming that the Zcsð3985Þ,Zcsð4000Þare different states, something not supported in Ref. [22] where the two states are claimed to be the same one. Even admitting the same molecular picture, different works use different dynamics. In Ref. [30] πand ηexchange are considered and the interaction is found too weak to bind. In Ref. [33] the local hidden gauge approach is used and a pole is found in the third Riemann sheet, rather than the ordinary second sheet, indicating not much binding. A clarification of the issue is provided in Refs. [35,42],where heavy quark symmetry is assumed and the source of the interaction is the exchange of vector mesons. The works follow the basic line of Ref. [43] where the heavy quark spin *[email protected] †[email protected].es ‡[email protected].es Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 1This picture would actually be more complicated if one considers the existence of two K1ð1270Þstates coupling differently to Kπand ρK, and appearing at different energies, as found in the chiral unitary approach in Refs. [25,26]. PHYSICAL REVIEW D 105, 014012 (2022) 2470-0010=2022=105(1)=014012(13) 014012-1 Published by the American Physical Society symmetry is assumed, which implies relationships between the different transition potentials and the dynamics is taken from the local hidden gauge approach [44–47].The exchange of light vectors is shown to respect heavy quark spin symmetry because the heavy quarks in the mesons act as spectators [48]. Both in the case of the Zcð3900Þ[49] and in the case of the Zcsð3985Þ, the diagonal interaction with the exchange of light vectors is zero. In Ref. [42] only single channels are considered, and lacking the dominant terms from the exchange of light vectors, only a virtual state has some room around threshold. The presence of a threshold together with some attractive interaction, even if weak, can lead to some structure around threshold as discussed in Ref. [50]. The interaction including coupled channels leads to a stronger attraction than in single channels, and as shown in Ref. [35] is strong enough in the D− sD0þD− sD0 combination to produce a mass distribution for D− sD0 and D− sD0compatible with experiment, while the D− sD0− D− sD0combination is unable to reproduce the experimental shape. Due to the weak primary interaction, in Ref. [36] the exchange of the a1ð1260Þis evaluated providing a small contribution that helps in the binding. A different kind of approach is used in Ref. [51] using an effective range expansion to justify that the Zcsð3985Þis the SU(3) partner of the Zcð3900Þ.AlsoinRef.[52] a discussion is conducted suggesting that the signal of the Zcsð3985Þcould be due to threshold cusps enhanced by a possible triangle singularity. In the present work we extend the molecular picture to the D− sD0,D− sD0and coupled channels sector as shown in detail in Ref. [40]. Within the heavy quark spin symmetry assumptions there is a trivial mapping of the interaction in the two sectors. However, in the charm sector nonleading terms or the interaction in the large Nccounting are not negligible, and as we shall see, in the absence of leading diagonal terms coming from the exchange of light vectors, the contact terms and the exchange of heavy D svectors provide a sizable interaction, which is strong enough to produce some bound states, or threshold structures, also removing the degeneracy of the 0þ;1þ;2þstates. We construct the J=ψKmass distributions in the Bþ→ J=ψϕKþdecay and show that a possible narrow peak observed in Ref. [24] (see the region around 4120 MeV between the Zcsð4000Þand Zcsð4220Þpeaks in the J=ψKþ mass distribution of Fig. 3) can be naturally associated to a near bound (virtual) D s¯ Dstate with JPC ¼2þþ. II. FORMALISM FOR THE INTERACTION For the D s¯ Dwe consider as coupled channels, 1, 2, Dþ s¯ D0ð1Þ;J=ψKþ ð2Þ;ð1Þ and take the interaction from the extension of the local hidden gauge approach [44–47] to the charm sector. The contact term is given by LðcÞ¼g2 2hVμVνVμVν−VνVμVμVνi;ð2Þ with g¼MV=2f(MV¼800 MeV, f¼93 MeV) with hi indicating the trace of the matrices and Vμ¼ 0 B B B B B B @ ρ0 ffiffi2 pþω ffiffi2 pρþKþ ¯ D0 ρ− − ρ0 ffiffi2 pþω ffiffi2 pK0D− K−¯ K0ϕD− s D0Dþ Dþ sJ=ψ 1 C C C C C C Aμ :ð3Þ There is another source of the interaction given by the exchange of vectors based on the three vector vertex LVVV ¼ighðVμ∂νVμ−∂νVμVμÞVνÞi:ð4Þ Working close to threshold allows one to take the approximation of neglecting the three momenta of the external vectors, which implies ϵ0¼0for the external vectors. This implies that Vνin Eq. (4) cannot be external since ν¼1, 2, 3 will involve three vectors through ∂ν. Thus, the field Vνcorresponds to the exchanged vector and the interaction vertex is formally like the VPP (Pfor pseudoscalars) with the additional factor ð−Þϵμϵ0μ¼ϵ·ϵ0 for the external vectors. We have to evaluate the contact term of the diagrams. We work with the interaction in s-wave, but due to the spins of the vertex we can have now spins J¼0, 1, 2 and we must separate the interaction for each of this total spin sectors. This is done with the projectors Pð0Þ,Pð1Þ,Pð2Þof Refs. [53,54], Pð0Þ¼1 3ϵμϵμϵνϵν Pð1Þ¼1 2ðϵμϵνϵμϵν−ϵμϵνϵνϵμÞ Pð2Þ¼1 2ðϵμϵνϵμϵνþϵμϵνϵνϵμÞ− 1 3ϵμϵμϵνϵν:ð5Þ Evaluating the contact term for the interaction of the coupled channels of Eq. (1), we obtain VðcÞ¼g22−4 −40 ;J¼0;ð6Þ VðcÞ¼g230 00 ;J¼1;ð7Þ VðcÞ¼g2−12 20 ;J¼2:ð8Þ IKENO, MOLINA, and OSET PHYS. REV. D 105, 014012 (2022) 014012-2 We can see that the VðcÞinteraction is different for the different spin channels. The diagonal terms are repulsive or zero for J¼0, 1, but attractive for J¼2. This gives us chances that one can find some binding in J¼2, which is usually the most bound channel in all cases of the VV interaction [53–55]. From the exchange of vectors in Fig. 1we obtain the result VðexÞ¼g20 @ −ðp1þp3Þ·ðp2þp4Þ M2 J=ψ − 3s−PM02 i D − 3s−ΣM02 i D01 A;J¼0;2ð9Þ VðexÞ¼g2 −ðp1þp3Þ·ðp2þp4Þ M2 J=ψ 0 00 !;J¼1;ð10Þ where ΣM02 i¼M2 D sþM2 ¯ D0þM2 J=ψþM2 K; D¼M2 J=Ψ −2¯ MDEJ=ψð11Þ with ¯ MD¼1 2ðMD sþM¯ DÞ;E J=ψ¼sþM2 J=ψ −M2 K 2ffiffiffis p;ð12Þ where pirefer to the momenta of the particles as shown in Fig. 1and to simplify the formulas we have taken an average M¯ Dbetween D sand D0in the denominator D. The product ðp1þp3Þ·ðp2þp4Þmust be projected in s-wave, with the results [25] ðp1þp3Þ·ðp2þp4Þ → 1 23s−X i M2 i− 1 sðM2 1 −M2 2ÞðM2 3 −M2 4Þ;ð13Þ where now Mi(i¼1;2;3;4) refer to the particles in the diagrams of Fig. 1with the order expressed there. III. DECAY CHANNELS We should note that the J=ψKstate is 129 MeV below the D s¯ D0threshold, hence any state that we find around the D s¯ D0threshold will decay to J=ψK, except for J¼1 where the transition potential is zero. The case is irrelevant since the interaction is repulsive in that channel. Let us study the decay in other channels. We only study the decay channels of the D s¯ D0component, which is the relevant one in the states that we obtain. We can look at the decay channels of Fig. 2which are not of VV type. It is clear that given the higher thresholds of the diagrams (a), (b) of Fig. 2and the ratio of propagator (a) (b) (c) (d) (e) FIG. 1. Contact term (a) and vector exchange terms (b, c, d, e) involved in the interaction of the coupled channels. (a) (b) (c) (d) FIG. 2. Diagrams containing the decay channels in the intermediate states. The threshold of the intermediate states are (a) 3976 MeV; (b) 3832 MeV; (c) 3593 MeV; (d) 3477 MeV. ZCS STATES FROM THE D S¯ D…PHYS. REV. D 105, 014012 (2022) 014012-3 ðMDs=MηcÞ2¼0.19, the relevant decay channels correspond to diagrams (c), (d), which are the only ones that we consider. Taking into account angular momentum and parity conservation we have the results of Table I. We can see that for JP¼2þboth J=ψKand ηcKdecay channels are possible. For 1þonly the decay to J=ψKis possible and for 0þonly the decay to ηcKis possible. A. ηcKintermediate state We shall evaluate the contribution of the diagram of Fig. 3the Dþ s¯ D0→Dþ s¯ D0potential. The relevant vertices are VPP given by the Lagrangian LVPP ¼−igh½P; ∂μPVμið14Þ with Pgiven now by the matrix [35], P¼ 0 B B B B B B @ η ffiffi3 pþη0 ffiffi6 pþπ0 ffiffi2 pπþKþ¯ D0 π−η ffiffi3 pþη0 ffiffi6 p−π0 ffiffi2 pK0D− K−¯ K0− η ffiffi3 pþffiffi2 3 qη0D− s D0DþDþ sηc 1 C C C C C C A :ð15Þ By neglecting the ϵ0components of the vector, as done before, we obtain for the two diagrams of Fig. 3. −itðaÞ¼Zd4q ð2πÞ4 i q2−M2 Dsþiϵ4g2ϵiðDþ sÞϵjð¯ D0Þqiqj·i ðp2þq−p4Þ2−M2 Dsþiϵ4g2ϵ0 lðDþ sÞϵ0 mð¯ D0Þqlqm ·i ðp1−qÞ2−M2 ηcþiϵ i ðp2þqÞ2−M2 Kþiϵ:ð16Þ From experience [53,55,56], the real part of these box diagrams is small compared to the more important terms stemming from vector exchange, and hence we keep only the imaginary part of the diagrams that contains the new decay channels. Due to this, in the decomposition of the propagator into positive and negative energy parts (ωðqÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2þ q2 p) 1 q2−m2≡1 2ωðqÞ1 q0−ωðqÞþiϵ − 1 q0þωðqÞ−iϵ;ð17Þ we take only the positive energy part for the intermediate ηcand Kstates, and because the Ds,Dstates are massive we can equally neglect the negative energy part. This simplifies the expression of Eq. (16) and we find −itðaÞ¼iZd4q ð2πÞ4ð4g2Þ2ϵiðDþ sÞϵjð¯ D0Þϵ0 lðDþ sÞϵ0 mð¯ D0Þ ·qiqjqlqm 1 2ωDsðqÞ 1 q0−ωDsþiϵ 1 2ωDsðqÞ 1 p0 2þq0−p0 4 −ωDsþiϵ ·1 2ωηcðqÞ 1 p0 1 −q0−ωηcþiϵ 1 2ωKðqÞ 1 p0 2þq0−ωKþiϵ;ð18Þ with ωi¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 iþ q2 p, where the q0integration is immediately performed using Cauchy’s residues, and one gets tðaÞ¼Zd3q ð2πÞ3 1 2ωDsðqÞ 1 2ωDsðqÞ 1 2ωηcðqÞ 1 2ωKðqÞ·ð4g2Þ2ϵiðDþ sÞϵjð¯ D0Þϵ0 lðDþ sÞϵ0 mð¯ D0Þqiqjqlqm ·1 p0 1 −ωηcðqÞ−ωDsðqÞþiϵ21 ffiffiffis p−ωηcðqÞ−ωKðqÞþiϵ:ð19Þ TABLE I. Possible values of Lfor J=ψKand ηcKwhich make the transition of Dþ s¯ D0to J=ψKor ηcKpossible. The × symbol indicates that the transition is forbidden. JPJ=ψKηcK 2þ1−;0−;L¼20 −;0−;L¼2 1þ1−;0−;L¼00 −;0−;L¼1× 0þ1−;0−;L¼1×0−;0−;L¼0 IKENO, MOLINA, and OSET PHYS. REV. D 105, 014012 (2022) 014012-4 A further simplification can be done using Zd3q ð2πÞ3fð q2Þqiqjqlqm¼Zd3q ð2πÞ3fð q2Þ q41 15 ðδijδlmþδilδjm þδimδjlÞ;ð20Þ which leads to the combination of polarization vectors in order 1,2,3,4 of the ordering of Fig. 3 ϵiϵiϵjϵjþϵiϵjϵiϵjþϵiϵjϵjϵi;ð21Þ which by virtue of the spin projectors of Eq. (5) gives 3Pð0ÞþPð0ÞþPð1ÞþPð2ÞþPð2Þ−Pð1ÞþPð0Þ¼5Pð0Þþ2Pð2Þ;ð22Þ and Pð1Þdoes not appear as anticipated in Table I. Finally we obtain an easy formula for tðaÞ tðaÞ¼Ws 1 15 ð4g2Þ2Zd3q ð2πÞ31 2ωDsðqÞ21 2ωηcðqÞ 1 2ωKðqÞ  q4·1 p0 1 −ωηcðqÞ−ωDsðqÞþiϵ21 ffiffiffis p−ωηcðqÞ−ωKðqÞþiϵ; ð23Þ where Ws¼8 > > < > > : 5 0 2 9 > > = > > ; J¼0 J¼1 J¼2ð24Þ The imaginary part of Eq. (23) is readily obtained and we find: ImVðaÞ¼−Ws 1 15 ð4g2Þ21 8π 1 ffiffiffis pq51 2ωDsðqÞ 1 p0 1 −ωηcðqÞ−ωDsðqÞ2 FðqÞ4FHQ;ð25Þ where q¼λ1=2ðs; M2 ηc;M2 KÞ 2ffiffiffis p;p0 1¼sþM2 D s −M2 ¯ D0 2ffiffiffis p;ð26Þ and we have added a form factor for each vertex as in Ref. [57], FðqÞ¼eq2=Λ2¼eðq02−  q2Þ=Λ2;q 0¼p0 1 −ωηcðqÞ;ð27Þ with Λ¼1200 MeV, and the factor FHQ required by the normalization we use with the heavy quarks and corrections needed in the space components of the VPP vertices that we use [48], FHQ ¼MD MK4 :ð28Þ (a) (b) FIG. 3. Diagram for Dþ s¯ D0decay to ηcKþ. ZCS STATES FROM THE D S¯ D…PHYS. REV. D 105, 014012 (2022) 014012-5 Following the same steps we find ImVðbÞ¼−Ws 1 15 ð4g2Þ21 8π 1 ffiffiffis pq051 2ωD0ðq0Þ 1 p0 1 −ωKðq0Þ−ωD0ðq0Þ2 Fðq0Þ4FHQ;ð29Þ where Fðq0Þ¼eq02=Λ2¼eðq002−  q02Þ=Λ2;q 00¼p0 1 −ωKðq0Þ;ð30Þ q0¼λ1=2ðs; M2 ηc;M 2 KÞ 2ffiffiffis p¼q: ð31Þ The two expressions for ImVðaÞ,ImVðbÞare now added to the diagonal Dþ s¯ D0,Dþ s¯ D0as VDþ s¯ D0;Dþ s¯ D0→VDþ s¯ D0;Dþ s¯ D0þiImVðaÞþiImVðbÞð32Þ B. J=ψKintermediate state The diagrams accounting for this intermediate state are shown in Fig. 4. The vertices now involve the anomalous VVP couplings. The Lagrangian is now given by Refs. [58,59]. L¼G0 ffiffiffi 2 pϵμναβh∂μVν∂αVβPi;ð33Þ with G0¼3g0 4π2f;g0¼− GVmρ ffiffi2 pf2,GV¼55 MeV, f¼93 MeV. Following the same steps as in the former subsection we obtain now: ImV0ðaÞ¼−W0 s 1 8π 1 ffiffiffis pG0gMD s ffiffiffi 2 p24 15 q51 2ωDsðqÞ 1 p0 1 −ωJ=ψðqÞ−ωDsðqÞ2 FðqÞ4F0 HQ:ð34Þ Now F0 HQ is different F0 HQ ¼MD MK2 ;ð35Þ because we have two anomalous couplings which are proportional to the external vector masses and do not require correction [48]. In Eq. (34),W0 sis given by W0 s¼8 < : 0 5 39 = ; J¼0 J¼1 J¼2;ð36Þ q¼λ1=2ðs; M2 J=ψ;M2 KÞ 2ffiffiffis p;q0¼p0 1 −ωJ=ψðqÞð37Þ (a) (b) FIG. 4. Diagram for Dþ s¯ D0decay to J=ψK. IKENO, MOLINA, and OSET PHYS. REV. D 105, 014012 (2022) 014012-6 and ImV0ðbÞ¼−W0 s 1 8π 1 ffiffiffis pG0gMD s ffiffiffi 2 p24 15 q5 ×1 2ωD0ðqÞ 1 p0 1 −ωKðqÞ−ωD0ðqÞ2 FðqÞ4F0 HQ; ð38Þ and finally we include all these decay channels taking for the Dþ s¯ D0→Dþ s¯ D0transition, VDþ s¯ D0→Dþ s¯ D0þiImVðaÞþiImVðbÞþiImV0ðaÞþiImV0ðbÞ: ð39Þ Note that W0 s¼0for J¼0in agreement with the findings of Table I. C. J=ψKdistribution in B+→J=ψϕK+decay In Ref. [24], the Bþ→J=ψϕKþdecay is studied and several mass distributions are shown. Two clear peaks are seen which are associated to the states Zcsð4000Þand Zcsð4220Þ. The Zcsð4000Þstate could correspond to the BESIII Zcsð3985Þ, but the Zcsð4220Þis definitely a new structure. We would like to call attention to the fact that in the vicinity of the D s¯ Dthreshold there are two points striking out of the LHCb fit, which could be indication of a dynamical structure, which we discuss here. These points can be observed in the J=ψKþdistribution of Fig. 3 of Ref. [24] between the two peaks associated to the Zcsð4000Þand Zcsð4220Þstates. These two latter states are assumed to have JP¼1þand 1−respectively, so the structure that we obtain with JP¼2þcannot be associated to any of these states and is a genuine new structure. Let us look at how the Bþ→J=ψϕKþdecay proceeds at the microscopical level. We look at the charge conjugate reaction to work with bquarks. In Fig. 5, we have a mechanism for this decay at the quark level. However, we have investigated the D s¯ Dstructures and their decay to J=ψK. Thus, we can also have the mechanism of Fig. 6which involves external emission and is in principle favored. The mechanism of Fig. 6produces ϕD− sD0, but upon rescattering of D− sD0→J=ψK−, as shown in Fig. 6(b), we can have J=ψϕK−at the end. This mechanism would reveal any structure tied to a possible D− s¯ D0molecular state. Indeed, the B−decay amplitude will have two structures, which ignoring the spin dependence will read as, tð1Þ¼A0;ð40Þ tð2Þ¼B0GD− sD0tD− sD0;J=ψK−;ð41Þ and we will assume that they do not interfere. Actually for the only relevant case of J¼2for D sD0, the structure of tð2Þis quite different form the s-wave ϵðJ=ψÞϵðϕÞof tð1Þ. Then we will have a structure at the end for jtj2summed over spin polarization of the vectors as, jtj2¼jAj2þjBj2jGD− sD0j2jtD− sD0;J=ψK−j2;ð42Þ where tD− sD0;J=ψKis a transition matrix from the state D− sD0to J=ψK−. We can obtain this transition matrix from the previous evaluation of ImV0ðaÞof Eq. (34). This latter magnitude was evaluated from the diagram of Fig. 4 and was included as a source of potential, see Eq. (39),in FIG. 5. A mechanism for B− →J=ψϕK−decay based on internal emission. (a) (b) FIG. 6. (a) Mechanism for B− →ϕD− sD0; (b) Rescattering mechanism leading to ϕJ=ψK. ZCS STATES FROM THE D S¯ D…PHYS. REV. D 105, 014012 (2022) 014012-7 the evaluation of the Bethe-Sapeter equation. This equation will generate the diagrams of Fig. 7which can be summed up as2 ð1þGD− sD0tD− sD0;D− sD0ÞVD− sD0;J=ψK−GJ=ψK−VD− sD0;J=ψK−ð1þGD− sD0tD− sD0;D− sD0Þð43Þ which upon the use of the Bethe-Salpeter equation, tD− sD0;D− sD0¼VD− sD0;D− sD0þVD− sD0;D− sD0GD− sD0tD− sD0;D− sD0ð44Þ can be rewritten as tD− sD0;D− sD0 VD− sD0;D− sD0 VD− sD0;J=ψK−GJ=ψK−VD− sD0;D− sD0 tD− sD0;D− sD0 VD− sD0;D− sD0 :ð45Þ If we evaluate the imaginary part of this magnitude corresponding to placing J=ψK−on shell we get  tD− sD0;D− sD0 VD− sD0;D− sD0 2 ðVD− sD0;J=ψK−Þ2ImGJ=ψK−ð46Þ which corresponds to jtD− sD0;J=ψK−j2ImGJ=ψK−;ð47Þ with ImGJ=ψK−¼− 1 8π 1 MinvðJ=ψKÞq; q ¼λ1=2ðM2 invðJ=ψKÞ;M 2 J=ψ;M 2 KÞ 2MinvðJ=ψKÞð48Þ Since ðVD− sD0;J=ψK−Þ2ImGJ=ψK−¼ImV0ðaÞþImV0ðbÞð49Þ Eq. (42) will become jtj2¼jAj2þjBj2jGD− sD0j2 tD− sD0;D− sD0 VD− sD0;D− sD0 2 ·ð−Þ8πMinvðJ=ψKÞ qðImV0ðaÞþImV0ðbÞÞð50Þ FIG. 7. Diagrams for the transition D− sD0→J=ψK− →D− sD0. Intermediate K−J=ψstates are implicitly assumed. 2We simplify the formalism and ignore the J=ψKchannel. The purpose is to show that tD− sD0;J=ψK−is proportional to tD− sD0;D− sD0. IKENO, MOLINA, and OSET PHYS. REV. D 105, 014012 (2022) 014012-8 The mass distribution for B− →J=ψϕK−is then given by dΓ dMinvðJ=ψKÞ¼1 ð2πÞ3 1 4M2 B pϕ ˜ pKjtj2ð51Þ with pϕ¼λ1=2ðM2 B;M 2 ϕ;M 2 invðJ=ψKÞÞ 2MB ; ˜ pK¼λ1=2ðM2 invðJ=ψKÞ;M 2 J=ψ;M 2 KÞ 2MinvðJ=ψKÞð52Þ IV. RESULTS With the potential obtained in the former section, we solve now the Bethe-Salpeter equation in coupled channels, T¼½1−VG−1V; ð53Þ where Gis the diagonal meson baryon loop function G¼GD sD00 0GJ=ψK;ð54Þ for which we take the formula with cutoff method, Gl¼Zd3q ð2πÞ3 ω1þω2 2ω1ω2 1 ðP0Þ2−ðω1þω2Þ2þiϵ;ð55Þ with ω1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1þ q2 p,ω2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 2þ q2 p(m1and m2are the vector masses of the lchannel). As for the cutoff parameters of the Gfunction in Eq. (55), we use different values of qmax and q0 max for the different channels. For qmax we use values between 450 MeV and 650 MeV. Values around qmax ¼700–850 MeV were used in Ref. [35] to get an enhancement in the ¯ DsDþ¯ D sDmass distribution close to threshold which was proposed as an explanation of the Zcsð3985Þ. On the other hand, values around 420–450MeVareusedinRef.[60] to explain the Tcc state as a molecule of DD. For the J=ψKchannel we take a larger value of q0 max to avoid having the on shell J=ψK momentum at energies close to the D s¯ Dthreshold bigger than the cutoff. The results are shown for different values of q0 max ranging from 700–900 MeV. As shown in Eqs. (6) to (10), the potential for different spins, J¼0,1,2aredifferent.WeseethatJ¼1is the most unlikely case to develop a bound state because the contact term is repulsive. The attraction from VðexÞ, coming from J=ψexchange is very small and there is no connection between D s¯ Dand J=ψK.Thecaseof J¼0is more favorable because now there is a coupled channel effect from a nonvanishing D s¯ D→J=ψK transition. Yet, the diagonal interaction is very weak and the contact term is still repulsive. On the other hand, the J¼2case is the most favorable, since the contact term is attractive and one also has the D s¯ D→J=ψKtransition. We should note that all the interaction terms are subleading in the heavy quark counting and do not follow the heavy quark spin symmetry rules. Indeed, both J=ψor D s,Dexchange are subleading with respect to light vector exchange. In this latter case the heavy quarks are spectators in the vector exchange process and hence the matrix elements are independent on the heavy quarks. Thus, the light vector exchange terms automatically fulfill heavy quark spin symmetry, but not the other terms where one exchanges some heavy quarks. The contact term is also subleading since it is not proportional to the external energies of the mesons, unlike the vector exchange. Because all the terms of the interaction are small, it is unlikely that one can form a bound state of the system, but like the case of the Zcsð3985Þwe could also have cusp effects around the D s¯ Dthreshold. We can see the repercussions of the former discussion in the values of jTj2whichweshowinFig.8. Indeed, we can see that the strength of jTj2for J¼1in the diagonal D s¯ D→D s¯ Dtransition is very weak and only a tiny cusp is seen in the D s¯ Dthreshold. On the other hand, a cusp like structure is seen for jT11j2for J¼0both at the J=ψKand D s¯ Dthresholds, particularly in the second channel. The same occurs for J¼2, but here the strength of jT11j2is a factor of 15 times larger than for J¼0. Because of that, we should associate any structure observed at the D s¯ Dthreshold to J¼2.InFig.9we show again jT11j2for J¼0and J¼2for qmax ¼450 MeV instead of 650 MeV used in Fig. 8. We observe that the cusp structure around the two thresholds is very similar and the strength of the magnitude has not changed much. Particularly visible is the cusp structure around the D s¯ Dthreshold (4119 MeV). This structure is typical of a barely “missed”bound state, or virtual state. The reason why in the case of J¼1there is only one curve independent on q0 max is that, as one can see in Eqs. (7) and (10), the matrix elements including the J=ψKþ state are all zero and hence the loops including q0 max do not appear in the scheme. Finally we would like to show the results for the J=ψK− distribution in the B− →J=ψϕK−decay. In Fig. 10,we show the mass distribution for the case of J¼2based on Eq. (51). We see a pronounced sharp peak around the D s¯ D threshold on top of a background created by the tree level [jAj2term of Eq. (42)] which should be visible in an experiment. In Fig. 10(a) the results are shown for qmax ¼ 650 MeV and different values of q0 max, while in Fig. 10(b) we show results for qmax ¼450 MeV. The features are qualitatively similar. In Fig. 11, we show the same mass spectrum for J¼0and J¼1. As we can see, there is no signal for J¼0, since W0 s¼0for J¼0[see Eq. (36)] and the cusp effect at the D s¯ Dthreshold for J¼1 ZCS STATES FROM THE D S¯ D…PHYS. REV. D 105, 014012 (2022) 014012-9