Physics Letters B 766 (2017) 117–124 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb How to understand the underlying structures of X (4140), X (4274), X (4500) and X (4700) Xiao-Hai Liu Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan a r t i c l e i n f o Article history: Received 20 July 2016 Received in revised form 22 December 2016 Accepted January 2017 Available online 11 January 2017 Editor: J.-P Blaizot Keywords: Exotic state Tetraquark Rescattering effect Triangle singularity a b s t r a c t We investigate the possible rescattering effects which may contribute to the process B + → J /ψφ K + It is shown that the ψ φ rescattering via the ψ K loop can simulate the structure of X (4700) The cusp − effect due to the D ∗+ s D s rescattering may possibly simulate the X (4140) structure, but it depends on the cusp model parameters If the quantum numbers of X (4274) (X (4500)) are 1++ (0++ ), it is hard to ascribe the observation of X (4274) and X (4500) to the P -wave threshold rescattering effects, which implies that X (4274) and X (4500) could be genuine resonances We also suggest that X (4274) may be the conventional orbitally excited state χc1 (3P ) © 2017 The Author Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 Introduction In the Standard Model, quarks and gluons are related to colorsinglet mesons and baryons by the long-distance regime of QCD, which remains the least understood aspect of the theory In experiments, there has been a renewal of hadron spectroscopy in the last decade, initiated by the findings of numerous X Y Z states Most of these states not fit into the predictions of the conventional quark model, which has been proved to be very successful in describing the heavy quarkonia below the open-flavor thresholds Some of the X Y Z states, such as charged charmonium-like Z c states, charged bottomonium-like Z b states, and heavy pentaquark candidates P c (4450) and P c (4380), definitely cannot be the conventional qq¯ -mesons or qqq-baryons These new observations not only enrich our knowledge about the hadron spectroscopy but also bring new challenges We refer to Refs [1–5] for both theoretical and experimental reviews about the recent study on exotic hadrons Very recently, the LHCb collaboration reported the observation of several resonance-like structures in J /ψφ invariant mass distributions in B + → J /ψφ K + decays [6,7] Their masses, widths and favorable quantum numbers are M X (4140) = 4146.5 ± 4.5+ −2.8 MeV, X (4140) 21 PC = 83 ± 21+ = 1++ , −14 MeV, J E-mail address: liuxh@th.phys.titech.ac.jp 17.2 M X (4274) = 4273.3 ± 8.3+ −3.6 MeV, X (4274) PC = 56 ± 11+ = 1++ , −11 MeV, J 12 M X (4500) = 4506 ± 11+ −15 MeV, X (4500) 21 PC = 92 ± 21+ = 0++ , −20 MeV, J 14 M X (4700) = 4704 ± 10+ −24 MeV, X (4700) 42 PC = 120 ± 31+ = 0++ , −33 MeV, J (1) among which the higher states X (4500) and X (4700) are firstly reported by the LHCb collaboration X (4140) and X (4274) were firstly observed by the CDF collaboration in the J /ψφ invariant mass distribution from B → K J /ψφ decays [8,9] The presence of X (4140) in B decays was later confirmed by the CMS and D0 collaborations [10–12] Another state X (4350) was reported by the Belle collaboration from the two photon process γ γ → J /ψφ [13] X (4140) and X (4274) were also expected to be produced in the two photon fusion reaction, but neither of them was observed [13] If the quantum numbers of X (4140) and X (4274) are 1++ , as reported by the LHCb collaboration, their non-observation in the two photo fusion reactions can be understood according to the Landau– Yang theorem, which forbids the transitions between a massive spin-1 particle and two real photons [14,15] But for two virtual photon fusion or one real and one virtual photon fusion, the production of massive spin-1 particles is not forbidden http://dx.doi.org/10.1016/j.physletb.2017.01.008 0370-2693/© 2017 The Author Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 118 X.-H Liu / Physics Letters B 766 (2017) 117–124 Table (∗)+ (∗)− Thresholds for the D s J D s Threshold [MeV] D+ s D ∗+ s D+ s0 (2317) D+ s1 (2460) D+ s1 (2536) D− 3936.6 4080.4 4080.4 4224.2 4286.0 4429.8 4427.8 4571.6 4503.4 4647.2 s D ∗− s M D = 2579.5 ± 3.4 ± 5.5 MeV, Fig Rescattering processes via the open-charm meson loops These resonance-like peaks in the J /ψφ invariant mass spectrum are very intriguing, because they may contain both a c c¯ pair and an s s¯ pair, which implies that these states may be exotic Taking into account their masses and decay modes, some researchers suggest that X (4140) and X (4274) could be the hadronic bound ∗− and D + D − , respectively [16–26] By means of states of D ∗+ s Ds s0 s quark model or QCD sum rules, these “ X ” state observed in J /ψφ distributions are also suggested to be some csc¯ s¯ tetraquark states with proper quantum numbers [27–32] Concerning those exotic states, apart from the genuine resonances explanations, such as molecular states, tetraquark states, or hybrid, some non-resonance explanations were also proposed in literatures There have been many theoretical attempts to try to connect the singularities of the rescattering amplitudes with the resonance-like peaks in experiments, such as the cusp effect [33–38], or the triangle singularity (TS) mechanism The TS mechanism was first notice in 1960s [39–49] Unfortunately, most of those proposed processes were lack of experimental support at that time It was rediscovered by people in recent years and used to interpret some exotic phenomena, such as the largely isospin violation in η(1405) → 3π , the production of a1 (1420), the production of some X Y Z particles and so on [50–64] It is shown that sometimes it is not necessary to introduce a genuine resonance to describe a resonance-like peak, because some kinematic singularities of the rescattering amplitudes could behave themselves as bumps in the corresponding invariant mass distributions, which may bring ambiguities to our understanding about the nature of exotic states Before claiming that one resonance-like peak corresponds to one genuine particle, it is also necessary to exclude or confirm these possibilities In this work, we investigate some possible rescattering effects in the process B + → J /ψφ K + The open-charm mesons rescattering and ψ φ rescattering are studied in Section The conclusions and some discussions are given in Section Rescattering effects 2.1 Open-charm meson rescattering effect In experiments, the rates of B decays into a charmed meson and a charmed-strange meson are found to be quite large among the hadronic decay modes Since the velocities of the open-charm mesons will not be large, they have adequate time to get involved in the final state interactions It has been suggested that the rescattering effect may play an important role in the hadronic decays of bottom meson [65–69] For higher excited charmed mesons, they may further decay into a kaon and a charmed-strange meson, and then we may expect that the rescattering processes illustrated in Fig would contribute to the decay channel B + → K + J /ψφ The (∗) lowest charmed mesons which can decay into D s K could be the ∗ first radially excited states D and D The experimentally observed resonances corresponding to D and D ∗ are D (2550) and D (2600) respectively [70], which is widely accepted According to the latest results of LHCb [71], their masses and widths are D = 177.5 ± 17.8 ± 46.0 MeV, M D ∗ = 2649.2 ± 3.5 ± 3.5 MeV, D∗ = 140.2 ± 17.1 ± 18.6 MeV (2) Since the mass of D is somewhat lower than the threshold of D ∗s K (∼ 2606 MeV), its contribution to the rescattering amplitude is supposed to be smaller compared with D ∗ There exist theoretical and experimental indications that the rates of B decays into excited charmed mesons would be sizable, although such decays are supposed to be suppressed by the heavy quark symmetry (HQS) at the leading order [72–76] This implies that the rescattering effects induced by those excited states may be important Another interesting property of Fig is that the thresholds of (∗)+ (∗)− some D s J D s combinations are rather close to the “ X ” states observed in J /ψφ distributions For convenience, we use D s0 , D s1 and D s1 to represent the P -wave charmed-strange mesons D s0 (2317), D s1 (2460) and D s1 (2536), respectively From Table 1, one can see that the thresholds of D ∗s D s , D ∗s D ∗s (D s0 D s ), and D s1 D s (D s1 D s , D s1 D ∗s , D s0 D ∗s ) are close to the masses of X (4140), X (4274) and X (4500), respectively Due to the singularities around these thresholds may be present in the rescattering amplitudes One may wonder whether there are some connections between the singularities and the “ X ” states There are two intriguing singularities which may appear in the triangle rescattering diagrams When two of the three intermediate states are on-shell, the singularity at threshold is a finite square-root branch point, which corresponds to the cusp effect In some special kinematic configurations, all of the three intermediate states can be on-shell simultaneously, which corresponds to the leading Landau singularity of the triangle diagram This leading Landau singularity is usually called the TS, which may result in narrow peaks in the corresponding spectrum Moreover, when the TS occurs, the triangle rescattering diagram can be interpreted as a classical process in space–time, and the TS will just be located on the physical boundary of the scattering amplitude [45] In Ref [49], it was argued that for the single channel rescattering process, when the corresponding resonance-production tree diagram is added coherently to the triangle rescattering diagram, the effect of the triangle diagram is nothing more than a multiplication of the singularity from the tree diagram by a phase factor Therefore the singularities of triangle diagram cannot produce obvious peaks in the Dalitz plot projections This is the so-called Schmid theorem But for the coupled-channel cases, the situation will be quite different from the single channel case discussed in Ref [49] For the rescattering diagrams studied in this paper, the intermediate and final states are different, therefore the singularities induced by the rescattering processes are still expected to be visible in the Dalitz plot projections The reader is referred to Refs [43,77] for some comments about the Schmid theorem, and Refs [78,79] for more discussions about the coupledchannel case 2.1.1 The model In the factorization approach, if the contributions from penguin (∗) ¯ (∗) operators are neglected, the decays B → D s J D receive contributions only from the external W -emission diagram The weak X.-H Liu / Physics Letters B 766 (2017) 117–124 ¯ (∗) | H W | B can then be factorized into the prodamplitude D s J D (∗) uct of two matrix elements, i.e., (∗) ¯ (∗) Ds J D GF ¯ (∗) ∗ | H W | B = √ V cb V cs a1 D A( D¯ ∗ → D s K ) = |( V − A )μ | B (∗) × D s J |( V − A )μ |0 , A( D¯ ∗ → D ∗s K ) = i (3) with the Wilson-coefficient combination a1 = c + c / N c ¯ (∗) |( V − A )μ | B can be parametrized by The matrix element D a series of hadronic form factors: ¯ |V μ|B = D M B M D h+ (ω)( v + v )μ + h− (ω)( v − v )μ , ¯ | A μ | B = 0, D ¯ ∗ |V μ|B = D M B M D∗ ih V (ω)ε μνα β ν∗ v β v α , ¯ ∗ | Aμ|B = D M B M D∗ h A (ω)(ω + 1) ∗μ − (h A (ω) v μ + h A (ω) v μ )( ∗ · v) , fK g fK M D M D ∗s p · ∗ ( D ∗s ), ¯ ∗ ), M D ∗ M D s p1 · ( D ¯ ∗ ) β∗ ( D ∗s ), M D ∗ M D ∗s ε μνα β p 1μ v ν α ( D (6) where the relative coupling strength between different channels is (∗)+ (∗)− determined by the HQS For the S-wave scattering D s Ds → J /ψφ , a contact interaction which respects the HQS is constructed in Ref [58], and the relevant scattering amplitude takes the form − αβ γ δ v A( D ∗+ α s D s → J /ψφ) = i β S ε ∗+ ∗ ∗ β ( D s ) γ (φ) δ ( J /ψ), αβ γ δ αδ β γ αγ βδ g ∗+ × α (D s ) (4) (∗) the decay constant of the corresponding D s J , which is defined as D s | A μ |0 = f D s M D s v μ , ∗ D ∗s | V μ |0 = f D ∗s M D ∗s μ , D s0 | V μ |0 = f D s0 M D s0 v μ , s1 g fK g ∗− A( D ∗+ s D s → J /ψφ) = β S (− g ¯ (∗) ), ω is the product of velocwhere v (v ) is the velocity of B ( D ¯ ∗ For the decay is the polarization vector of D ities v · v , and (∗) ¯ (∗) (∗) ¯ (∗) process B → D s J D , in the rest frame of B, both D s J and D nearly stay at rest, which is very close to the zero recoil limit (ω = 1) As an approximation, we will set v = v = (1, 0, 0, 0) in the following sections, and calculate the numerical results at the zero recoil limit It should be mentioned that for some of the sub¯ (∗) meson may not be very diagrams in Fig 1, the velocity of the D + ¯ ∗ D ∗+ small For instance, in the process B → D the velocity of s ¯D ∗ is around 0.4 at the rest frame of B + , which is actually not very small This may bring some theoretical uncertainties to the strength of rescattering amplitudes But the positions of the cusp structures (or the TS), which mainly depend on the kinematic configurations of rescattering processes, will not be affected by the ap¯ ∗ D ( )+ , proximation of zero recoil limit For the processes B + → D s1 ∗ ¯ the velocity of D is smaller, and the approximation would be better (∗) In Eq (3), the matrix element D s J |( V − A )μ |0 is related with () ∗ D s1 | A μ |0 = f D ( ) M D ( ) μ A( D¯ → D ∗s K ) = 119 (5) s1 At zero recoil, it can be noticed that in Eq (4) only the form factors h+ and h A have the non-vanishing contributions Correspondingly, for the rescattering processes B + → J /ψφ K + via (∗) ¯ (∗) the D s J D -loops, only the diagrams illustrated in Fig can survive Explicitly, the sub-diagrams involved in the calculations + ¯ ∗− ∗+ ¯ ∗ [ D − ], D ∗+ D ¯ ∗− ¯ ∗ [ D ∗− are D + s D [ D s ], D s0 D [ D s ], D s D s s s ], + ¯∗ + ¯∗ +¯∗ +¯∗ − ∗− − D s1 D [ D s ], D s1 D [ D s ], D s1 D [ D s ], and D s1 D [ D ∗− s ] loops, of which the particles in the brackets represent the ex(∗) ¯ (∗) changed mesons between D s J and D In order to estimate the rescattering amplitudes, we also need to know the relevant strong couplings in Fig To proceed, the momentum for external and internal particles are denoted as (∗)+ ¯ (∗) (q2 )[ D (∗)− B + ( P ) → K + ( p ) J /ψ( p )φ( p ) and D s J (q3 ) D (q1 )], s respectively In the framework of heavy hadron chiral perturba¯ (∗) → D (∗) tion theory [66,80–84], the decay amplitudes for D s K are given by +g ∗− β (D s +g g ∗ ) γ (φ) g ) ∗ δ ( J /ψ), (7) where β S is the coupling constant for the contact interaction It is worth noticing that according to Eqs (4), (6) and (7), the ¯ ∗ [ D ∗− rescattering amplitude corresponding to the D ∗+ s D s ]-loop actually vanishes Because of parity and angular momentum conservations, if D ∗+ and D ∗− scatter in relative S-wave, the quans s ∗+ tum numbers of D s D ∗− system can only be 0++ or 2++ , which s means the quantum numbers of the produced J /ψφ can only be 0++ or 2++ With an anti-symmetric tensor appearing in the rescattering amplitude, this sub-diagram finally gives a vanishing contribution We can understand this conclusion by firstly considering the J /ψφ system as a single scalar particle The + vertex in the D ∗+ D ¯ ∗ D ∗− ¯ ∗ [ D ∗− D s K s s ]-loop is a Vector–Vector– Pseudoscalar type coupling, where an anti-symmetric tensor is required in the corresponding covariant amplitude The loop integrals in the rescattering amplitude with a tensor structure can be reduced to linear combinations of Lorentz-covariant tensors constructed from the metric tensor and a linearly independent set of the external momentum, according to the Passarino–Veltman reduced method [85] These tensor loop integrals will be contracted with the metric tensors, external momentum, polarizations vectors and anti-symmetric tensors to obtain the covariant rescattering ¯ ∗ D ∗s amplitude There is no anti-symmetric tensor from the B D vertex due to the zero recoil limit approximation in Eq (4) For B decaying into K and a scalar particle ( J /ψφ system) via the ¯ ∗ [ D ∗− D ∗+ s D s ]-loop, there are only two linearly independent momentum that will be contracted with the anti-symmetric tensor + vertex, which gives a vanishing rescatter¯ ∗ D ∗− from the D s K ing amplitude The Lorentz indices of the polarization tensors for a tensor particle are symmetric According to the similar analysis, if the quantum numbers of J /ψφ system are set to be 2++ , one will also obtain a vanishing rescattering amplitude corresponding ¯ ∗ [ D ∗− to the D ∗+ s D s ]-loop If the quantum numbers of J /ψφ system are J P C = 0++ or J P C = 1++ , the P -wave and S-wave charmed-strange mesons can scatter into J /ψφ via relative P -wave To simplify the discussion, the quantum numbers of J /ψφ are firstly set to be 0++ , then the P -wave scattering amplitudes which respect the HQS take the form ∗− A( D + s0 D s → J /ψφ) βP q D s0 · ( D ∗s ) = ∗ (φ) · ∗ ( J /ψ), ∗ (φ) · ∗ ( J /ψ), M D s0 M D ∗s A( D s1 D − s → J /ψφ) βP () q D s · ( D s1 ) = ( )+ M D( ) M Ds s1 120 X.-H Liu / Physics Letters B 766 (2017) 117–124 A( D s1 D ∗− s → J /ψφ) ( )+ = iβ P M D ( ) M D ∗s εμνα β qαD ∗s qβ ( ) D s1 s1 μ ( D ∗ ) ν ( D ( ) ) ∗ (φ) · ∗ ( J /ψ), s s1 (8) where β P is the dimensionless coupling constant for the P -wave scattering Other assignments of the quantum numbers of J /ψφ system will be discussed later By means of the above scattering amplitudes, the rescattering amplitude of B + → K + J /ψφ via the open-charm meson loops in Fig is given by (∗) [ D s J D¯ (∗) ] T B + → K + J /ψφ = d4 q i (2π )4 × (∗) (∗) (∗) A( B → D s J D¯ (∗ ) )A( D¯ (∗) → D s K )A( D s J D ∗s → J /ψφ) (q21 − M 2D ∗ )(q22 − M 2¯ (∗) +iM D¯ (∗) s D ¯ (∗) D )(q23 − M (∗) ) D , sJ (9) where the sum over polarizations of intermediate states is implicit The TS corresponds to a logarithmic singularity of the loop integral As long as the kinematic conditions of the TS are fulfilled, it implies that one of the intermediate state in the triangle diagram must be unstable, and we have to take into account the width effect In our model, we use a Breit–Wigner type propagator to replace the normal propagator in the loop integral, as did in Eq (9) The complex mass will remove the singularity from the physical boundary by a small distance, and makes the physical scattering amplitude finite However, this method is just an approximation of some more complete theories In Ref [44], it is argued that this approximation is appropriate for calculating enhancement effects due to the TS Fig The solid line represents the invariant mass distributions of J /ψφ via the rescattering processes of Fig The dotted line represents the distributions expected ¯ (∗) is taken to be that of (a) the first from the phase space The mass (width) of D ¯ (∗) , respectively The vertical dashed and (b) the second radially excited state of D lines indicate the positions of X (4140), X (4274) and X (4500) respectively The experimental data points with error bars are from Ref [6] 2.1.2 Numerical results Ignoring the common coupling constants and form factors, the relative strength between different rescattering amplitudes mainly (∗) depend on the decay constants of D s J and the form factors h+ and h A For the decay constant of D s , we adopt the experimental value, i.e., f D s = 257.5 MeV [70], and make f D ∗s = f D s In the heavy quark limit, we have the following relations [86–89] and the result is displayed in Fig 2(b) There is no experimental measurement concerning the second radially excited charmed mesons D (∗) (3S ), and the following results calculated in the quark model are adopted in calculations [92]: f D s0 = f D s1 , f D In Figs 2(a) and (b), one may notice that there are several cusps which stay around the thresholds of D s1 D s , D s1 D s , D s1 D ∗s and D s1 D ∗s , respectively As discussed previously, the sub-diagram ¯ ∗ [ D ∗− D ∗+ s D s ]-loop gives vanishing contribution, therefore there ∗− threshold Correspondis no cusp appearing around the D ∗+ s Ds ingly, for the rescattering processes studied in this paper, there is no cusp that can simulate the structure of X (4274) Because − the threshold of J /ψφ is larger than that of D ∗+ s D s , there is no ∗+ − cusp corresponding to the D s D s threshold either In the theoretical analysis of the LHCb collaboration [6,7], a cusp model proposed by Swanson was introduced to fit the structures of X (4140) [37] In this cusp model, the introduction of an exponential form factor, with a momentum scale (β0 ) characterizing the hadron size, makes the cusp slightly above the sum of masses of the rescattering mesons In Refs [6,7], the value of β0 obtained by the fit to the data is 297 ± 20 MeV The fitting result indicates that the below- J /ψφ -threshold cusp can simulate the X (4140) structure [6,7] Our model in a sense is similar with the cusp model proposed by Swanson In the rescattering ampli¯ ∗− tudes corresponding to the sub-diagrams D + s D [ D s ]-loop and ∗ [ D − ]-loop of Fig 1, there is also a D ± D ∗∓ -cut corre¯ D ∗+ D s s s s ∗∓ sponding to the D ± s D s -threshold cusp One of the differences between our model and the cusp model is that the exponen- s1 = (10) But these relations are not consistent with the experimental observations very well In the numerical calculation, we adopt the values calculated in the covariant light-front model [90,91], which gives f D s0 = 71 MeV, f D s1 = 121 MeV, f D s1 = 38 MeV (11) For the form factors h+ and h A , we adopt the values calculated in the framework of relativistic quark model [72], which gives h+ (1) 0.012, h A (1) 0.098 (12) Since h A (1) is much larger than h+ (1), correspondingly one can expect that the contribution of Fig 1(b) would be much larger than that of Fig 1(a) The numerical results of J /ψφ invariant mass distributions via the rescattering processes are displayed in Fig The experimental data from Ref [6] is also displayed in Fig to be compared with the numerical result The result in Fig 2(a) is obtained by setting the values of M D¯ (∗) and D¯ (∗) as those in Eq (2) To investigate the dependence on the mass of intermediate states, we also calculate the rescattering amplitudes by setting the values of M D¯ (∗) and D¯ (∗) as those of the second radially excited states, M D (3S ) = 3068 MeV, M D ∗ (3S ) = 3110 MeV, D (3S ) = 106 MeV, D ∗ (3S ) = 103 MeV (13) X.-H Liu / Physics Letters B 766 (2017) 117–124 tial form factor is not introduced in our model As a result, in our model the cusp of the rescattering amplitude just stays at ∗∓ ± ∗∓ the D ± s D s -threshold Since the D s D s -threshold is below the ∗∓ J /ψφ -threshold, one cannot expect that the D ± s D s -cusp appears in the J /ψφ invariant mass distributions To show the influence ∗∓ of the D ± s D s -cut of the rescattering amplitude, the distribution curves expected from the phase space are displayed in Fig It can be seen that in Figs (a) and (b) threshold enhancements appear in the J /ψφ distribution, but there is no narrow peak structure appearing around 4140 MeV In Ref [56], the authors argued that the kinematic threshold cusp cannot produce a narrow peak in the invariant mass distribution in the elastic channel in contrast with a genuine S-matrix pole We agree with this argument It also should be mentioned that for the cusp model introduced in Ref [37], the lineshape of the cusp structure is very sensitive to the parameter β0 in the exponential form factor, which implies that the conclusion drawn from this model is quite model-dependent We refer to Ref [56] for more discussions about the cusp model and its limitations However, taking into account that the X (4140) structure is actually not a narrow peak but rather a broad threshold enhancement in the experimental observations, the rescattering effect (or cusp effect) can not be simply ignored in understanding the nature of X (4140) All of the cusps around X (4500) are too broad and too small to simulate the structure of X (4500) There are two main reasons for this result One is that the kinematic conditions for the presence of TS are not satisfied in those rescattering processes of Fig [57] Therefore the rescattering amplitude cannot get enhancement induced by the TS Another reason is that, if the J /ψ and φ stay in relative S-wave, to preserve the parity, only via P -wave ( ) (∗) (∗) can D s1 D s (or D s0 D s ) scatter into J /ψφ Usually the nearthreshold P -wave scatterings will be highly suppressed due to the small momentum of scattering particles Because of the unsatisfied kinematic conditions of TS and the characteristic of near-threshold P -wave scatterings, no matter what the quantum numbers of the J /ψφ system are (0++ , 1++ , or 2++ ), there will be no obvious narrow peaks appeared in the invariant mass distributions around 4500 MeV It should be mentioned that in Figs 2(a) and (b), the distribution curves are rescaled to be approximately matched with the data points Because some of the couplings in the rescattering processes are not well determined, we only focus on the lineshape behavior, but not aim to fit the data According to the above analysis, it seems hard to ascribe the observation of X (4274) and X (4500) in B decays to the rescattering effects This implies that X (4274) and X (4500) may correspond to some genuine resonances, such as tetraquark states or some higher excited charmonium states The cusp effect due to the − D ∗+ s D s rescattering may possibly simulate the X (4140) structure, but it may depends on the parameters of the cusp model 2.2 ψ φ rescattering effect Fig Rescattering process via the ψ K loop the ratio of the amplitude |A( B → J /ψ K ∗∗ )| to the amplitude |A( B → ψ K ∗∗ )| will be close to f J /ψ / f ψ For J /ψ and ψ , the discrepancy between the decay constants f J /ψ (416 ± MeV) and f ψ (294 ± MeV) is not very large For the decay B + → K + J /ψφ , from LHCb results it is shown that there is a rich spectrum of excited kaon resonances, which has significant contributions via the sequential decays B + → J /ψ K ∗∗+ → J /ψφ K + [6,7] Although the reflection of excited kaons cannot result in obvious resonance-like structures in J /ψφ distributions, they may contribute to the production of those “ X ” states in another way As discussed above, it can be expected that the decay rate of B → ψ K ∗∗ would be at the same order of magnitude with that of B → J /ψ K ∗∗ For some higher excited kaons, their on-shell production may be prohibited due to the phase space, but taking into account their broad decay widths, they can still contribute to the process B + → ψ φ K + Interestingly, it is found that the threshold of ψ φ (∼ 4706 MeV) is very close to the mass of X (4700) (∼ 4704 MeV) Therefore it is possible that the rescattering process illustrated in Fig may result in some resonance-like peaks in J /ψφ distributions around the ψ φ threshold, which may simulate the X (4700) signal Among the excited kaons, the dominant contributions come from the axial vector states, as shown in Refs [6,7] In the following analysis, we will focus on the rescattering effects induced by K , of which the quantum numbers are J P = 1+ The general invariant amplitude for B → ψ K can be written as: A( B → ψ K ) = a + + ∗ (ψ ) · GF ∗ M cc¯ K (∗∗)+ | H W | B + = √ V cb V cs a2 K (∗∗)+ |( V − A )μ | B + (14) with a2 = c + c / N c The matrix element M cc¯ |( V − A )μ |0 is proportional to the decay constant of M cc¯ It can be expected that ∗ (K1) b ( M B + M K )2 c ( M B + M K )2 p B · ∗ (ψ ) p B · ∗ ( K ) μ i εμνα β p B p νK ∗α (ψ ) ∗β ( K ) (15) For B decaying into ψ and the higher excited state K , both ψ and K will nearly stay at rest in the rest frame of B Therefore in the above equation, only the first term on the right hand side will contribute significantly As an approximation, we only keep the fist term in the calculation, and set the form factor a as a constant The axial-vector meson K can decay into φ K in S-wave, and the amplitude takes the form A( K → φ K ) = g K ( K ) · 2.2.1 The model In the naive factorization approach, the amplitude for the color suppressed decays B → M cc¯ K (∗∗) , with M cc¯ and K (∗∗) representing a charmonia and a kaon (excited kaon) meson respectively, is given by × M cc¯ |( V − A )μ |0 , 121 ∗ (16) (φ), where g K is the coupling constant ψ φ can scatter into J /ψφ via exchanging soft gluons To simplify the model, we only construct a contact interaction for this scattering, and the amplitude reads A(ψ φ → J /ψφ) = g C T (ψ ) · (φ) ∗ ( J /ψ) · ∗ (φ), (17) where the quantum numbers of J /ψφ system are 0++ For the rescattering process ψ K → K J /ψφ via strong interactions, the total angular momentum (or spin) of ψ K system should be conserved If we only take into account the first term of Eq (15) in the calculation, it implies that the angular momentum of ψ K system is Furthermore, since K decays into φ K in S-wave, it means the 122 X.-H Liu / Physics Letters B 766 (2017) 117–124 Fig Invariant mass distributions of J /ψφ via the rescattering processes of Fig 3, where the mass/width of the intermediate state K is taken to be 1650/150 MeV (solid line), 1793/365 MeV (dotted line), and 1968/396 MeV (dashed line) separately The vertical dashed line indicates the position of X (4700) The experimental data points with error bars are from Ref [6] angular momentum of the intermediate ψ φ system is 0, which further means the angular momentum of the final J /ψφ system is also As a result, for the S-wave scattering ψ φ → J /ψφ , although the quantum numbers of J /ψφ (ψ φ ) system can be 0++ , 1++ or 2++ , only the 0++ part can survive in the rescattering amplitude This is consistent with the experimental fitting results that the quantum numbers of X (4700) are 0++ The rescattering amplitude of B + → K + J /ψφ via the ψ K loop in Fig is given by [ψ K ] T B + →1K + J /ψφ = d4 q (2π )4 A( B → ψ K )A( K → φ K )A(ψ φ → J /ψφ) , (q21 − M φ2 )(q22 − M 2K + iM K K )(q23 − M ψ ) i (18) where the sum over polarizations of intermediate states is implicit 2.2.2 Numerical results For the moment we just focus on the lineshape behavior of the J /ψφ distributions via the rescattering processes, but ignore the explicit values of the relevant coupling constants and form factors The numerical results are displayed in Fig According to the fitting results of LHCb [6,7], the distributions in Fig are obtained by setting the mass (width) of K to be 1650 MeV (150 MeV), 1793 MeV (365 MeV) and 1968 MeV (396 MeV) separately From Fig 4, one can see that there is a clear peak appearing in the vicinity of 4.7 GeV, which corresponds to the ψ φ threshold The S-wave near threshold scattering makes this peak very obvious Furthermore, although the kinematic configuration of the rescattering process in Fig does not meet the conditions of TS very well, it is already very close to the kinematic region where the triangle singularity can be present [57] Therefore the physical rescattering amplitude will be influenced by the triangle singularity to some extent From Fig 4, it can be seen that the peak induced by the rescattering effect is similar with the structure of X (4700) The Argand plot corresponds to the rescattering amplitude in Eq (18) is displayed in Fig It can be seen that the phase of the amplitude shows a behavior of rapid counter-clockwise change, which is similar with a genuine resonance Conclusion and discussions In conclusion, it is possible that the ψ φ rescattering via the ψ K loop could simulate the X (4700) structure in J /ψφ distribu- Fig Real and imaginary parts of the rescattering amplitude in Eq (18) Motion with the increasing invariant mass M J /ψφ is counter-clockwise − tions The below- J /ψφ -threshold cusp due to the D ∗+ s D s rescattering may possibly simulate the X (4140) structure, but it depends on the cusp model parameters However, if the quantum numbers of X (4274) ( X (4500)) are 1++ (0++ ), due to the conservations of parity and angular momentum, and the suppression of the near threshold P -wave scattering, it is hard to describe the structures of X (4274) and X (4500) with rescattering effects, which implies that X (4274) and X (4500) could be genuine resonances Concerning X (4274), although it is observed in the J /ψφ invariant mass distributions, it does not necessarily mean that it contains s s¯ as the valence quarks We suggest that it may be the conventional orbitally excited state χc1 (3P ) This suggestion is based on the following arguments: • First, the predicted mass of χc1 (3P ) in the framework of quark models is about 4271 MeV or 4317 MeV [93,94], which is very close to the mass of X (4274) The predicted width (∼ 39 MeV) is also close to the observed width of X (4274) (∼ 56 MeV) Although for the higher charmonium states, the prediction of conventional quark models is not very reliable, it can still be taken as a guidance • Second, if the X (4274) is χc1 (3P ), apart from J /ψφ , one may expect that it can also easily decay into J /ψ ω From the Fig in Ref [95], it can be noticed that apart from X (3872), there are also some bumps around 4.3 GeV appearing in the J /ψ ω distributions Although the current statics for B → J /ψ ω K may not be large enough to confirm this point, it can still be taken as an evidence • Third, the rescattering effects can not describe the observation of X (4274), as discussed in this paper and Refs [6,7], which to some extent excludes the possibility of non-resonance explanations • Fourth, under the factorization ansatz, since χc0 , χc2 and their radially excited states can not be produced via the V − A current, it can be expected that the decay rate of B → K χc1 (3P ) will be larger than that of B → K χc0 (3P ) or B → K χc2 (3P ) This may lead to that in the J /ψφ distributions, only the χc1 (3P ) signal is significant If the X (4274) is really χc1 (3P ), one may search for it in the radiative decays of ψ(4415) (ψ(4S )), taking into account that this E1 X.-H Liu / Physics Letters B 766 (2017) 117–124 transition is predicted to have a relatively larger branching ratio in the framework of quark model [93,94] Acknowledgements Helpful discussions with Feng-Kun Guo, Gang Li and Qiang Zhao are acknowledged This work is supported in part by the Japan Society for the Promotion of Science under Contract No.P14324, and the JSPS KAKENHI (Grant No 25247036) References [1] A Esposito, A.L Guerrieri, F Piccinini, A Pilloni, A.D Polosa, Int J Mod Phys A 30 (2015) 1530002, http://dx.doi.org/10.1142/S0217751X15300021, arXiv:1411.5997 [hep-ph] [2] N Brambilla, et al., Eur Phys J C 71 (2011) 1534, http://dx.doi.org/10.1140/ epjc/s10052-010-1534-9, arXiv:1010.5827 [hep-ph] [3] S.L Olsen, Front Phys (Beijing) 10 (2) 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