$X$ structures in $B^+\to J/\psi\phi K^+$ as one-loop and double-triangle threshold cusps

The LHCb data on $B^+\to J/\psi\phi K^+$ show four peaks and three dips in the $J/\psi\phi$ invariant mass distribution, and the peaks are interpreted as $X(4140)$, $X(4274)$, $X(4500)$ and $X(4685)/X(4700)$ resonance contributions. Interestingly, all the peaks and dips are located at (or close to) $D^*_{s}\bar{D}^{(*)}_{s}$, $D_{s0}^*(2317)\bar{D}^{(*)}_{s}$, $D_{s1}(2536)\bar{D}^{(*)}_{s}$, and $\psi'\phi$ thresholds. These coincidences suggest a close connection between the structures and the thresholds, which however has not been seriously considered in previous theoretical studies on the $X$ structures. In fact, if we utilize this connection and interpret the $X$ structures as common $s$-wave threshold cusps, we face a difficulty: $X(4274)$ and $X(4500)$ have spin-parity that conflict with the experimentally determined ones. In this work, we introduce double triangle mechanisms that cause threshold cusps significantly sharper than the ordinary one-loop ones of the same spin-parity. We demonstrate that all the $X$ and dip structures are well described by a combination of one-loop and double-triangle threshold cusps, thereby proposing a novel interpretation of the $X$ and dip structures.

It is recognized [53,54] that the X(4274) and X(4500) peak positions are virtually at the D * s0 (2317)D s and D s1 (2536)D s thresholds, respectively, and X(4700) and X(4685) are at the ψ ′ φ threshold; see channels through kinematical effects such as threshold cusps and triangle singularities [55].
Indeed, the LHCb confirmed that the X(4140) structure can be described with a D * sDs threshold cusp, albeit using a rather small cutoff in form factors [11,56] 3 . Similarly, Liu studied triangle diagrams that cause D * sDs and ψ ′ φ threshold cusps, and found X(4140)-and X(4700)like enhancements, respectively [53]. Dong et al. also suggested that X(4140) could be caused by a D * sD s virtual state and the associated threshold cusp [54]. X(4140) as the kinematical effect may be supported by a lattice QCD that found no J P C = 1 ++ cscs state below 4.2 GeV [57]. On the other hand, X(4274) [X(4500)] as an s-wave D * s0 (2317)D s [D s1 (2536)D s ] threshold cusp has J P that conflicts with the experimentally determined ones [11,12] 4 . Non s-wave threshold cusps from oneloop diagrams are unlikely either, since they should be suppressed [53]. Thus, until the present work, there exists no explanation of X(4274) and X(4500) based on kinematical effects. Now let us assume negligibly small D * s0 (2317)D D s1 (2536)D ( * ) s → J/ψφ transition strengths caused by short-range (e.g., quark-exchange) interactions. This assumption may seem reasonable for the D * s0 (2317) cases, because previous theoretical studies [58][59][60][61]  s . The DT mechanisms are worthwhile studying to understand the X and dip structures and their locations. The DT mechanisms cause threshold cusps that are significantly sharper than ordinary oneloop ones with the same J P . This is because the DT is close to causing the leading kinematical singularity. Thus, the DT can generate X-like and dip structures at the D ( * ) sJD ( * ) s thresholds. In this paper, we develop a B + → J/ψφK + decay model. The model includes the DT mechanisms that cause enhanced threshold cusps at the D and ψ ′ φ threshold cusps are also generated by one-loop mechanisms. We first examine singular behaviors of the DT amplitudes. We then analyze the M J/ψφ distribution from the LHCb. Since onedimensional analysis would not reliably extract partial wave amplitudes or determine spin-parity of resonances, this is not our intention. The present one-dimensional analysis keeps J P of X from the LHCb analysis. Under this constraint, we demonstrate that all the X and dip structures can be well described with the threshold cusps. The purpose of this work is to propose a novel interpretation of the X and dip structures in this way.

II. MODEL
In describing B + → J/ψφK + , we explicitly consider mechanisms that generate the structures in the M J/ψφ distribution through kinematical effects or resonance excitations; others are subsumed in contact mechanisms. Thus we consider diagrams shown in Fig. 1. To derive the corresponding amplitudes, we write down effective Lagrangians of relevant hadrons and their matrix elements, and combine them following the time-ordered perturbation theory. We consider the DT diagrams [ Fig. 1(a)] that include p-wave pairs of (1) where J P of a pair is indicated in the parenthesis. In principle, more quantum numbers are possible such as J P from s-wave pairs and J P = 2 + from p-wave pairs which the LHCb did not find relevant to the X structures. While the kinematical effects can generate structures in lineshapes almost model-independently, it is the dynamics that determines the strength of the kinematical effects. Since the relevant dynamical information is scarce, we need to rely on the LHCb analysis to select the quantum numbers to take into account in the model. Most phenomenological models share this limitation of predicting quantum numbers relevant to the process. We assume that contributions from the other quantum numbers are relatively minor and can be absorbed by mechanisms included in the model. We also do not consider D s1 (2460)D ( * ) s pairs since their threshold cusps are either not clear in the data or replaceable by a D * s0 (2317)D * s threshold cusp. The one-loop diagram [ Fig. 1 is forbidden by the Cparity conservation. We denote the DT and one-loop amplitudes by A DT We consider Z cs excitations [ Fig. 1(c)] since the data [12] shows their effects on the M J/ψφ distribution. In particular, Z cs (4000) seems to enhance the X(4274) peak through an interference. The LHCb presented the Z cs (4000) and Z cs (4220) properties. Meanwhile, coupled-channel analyses [62][63][64] found virtual states below the D Z cs , we use a p-wave B + → Z cs φ decay vertex which contributes to the 1 + J/ψφ final state. Our fits visibly favored the threshold-cusp-based Z cs ; we thus use them hereafter.
All the other mechanisms such as non-resonant and K ( * ) J -excitations are simulated by two independent direct decay mechanisms [ Fig. 1(d)] creating J/ψφ(0 + , 1 + ). We consider J/ψφ(0 + , 1 + ) partial waves. Although the LHCb amplitude analysis found resonances in 1 − and 2 − partial waves, their contributions are rather small in the M J/ψφ spectrum. We confirmed that the 1 − and 2 − resonance contributions only marginally improved our fits; we thus do not consider them.
The DT and one-loop diagrams are respectively initiated by B + → D s K + generally have independent decay strengths, the corresponding DT and one-loop amplitudes have the same singular behaviors as the original ones. Thus we do not explicitly consider the charge analogous processes, but their effects and projections onto positive C-parity states are understood to be taken into account in coupling strengths of the considered processes.
We present amplitude formulas for representative cases; see Appendix A for complete formulas. We use the particle mass and width values from Ref. [1] unless otherwise specified, and denote the energy, momentum, and polarization vector of a particle x by E x , p x , and ǫ x , respectively. A DT diagram [ Fig. 1(a)] that includes D * s0 (2317)D s (1 + ) consists of four vertices such as B + → D * s0Ds K + , D * s0 → DK, DD s → J/ψK, and KK → φ given as respectively; p ab ≡ p a − p b . We have introduced dipole form factors F LL ′ ijk,l , f L ij , and f L ij,k including a cutoff Λ. We use a common cutoff value Λ = 1 GeV in all form factors unless otherwise stated. We used a p-wave DD s → J/ψK interaction; s-wave is forbidden by the spin-parity conservation 5 . The coupling c KK,φ can be determined by the φ → KK decay width. Experimental information for the other couplings (c D * s0D s (1 + ) , c DK,D * s0 , c 1 − ψK,DDs ) are unavailable. Thus we determine their product, which is generally a complex value, by fitting the data. The DT amplitude from the above ingredients is where the summation over D + K 0K 0 and D 0 K +K − intermediates states with the charge dependent particle masses is implicit; K + in the final state is denoted by K f , and W is related to the total energy E by s0 → D s π decay is isospin-violating. Experimentally, only an upper limit has been set: Γ D * s0 < 3.8 MeV [65]. Theoretically, Γ D * s0 ∼ 0.1 MeV (0.01 MeV) has been given by a hadron molecule model [58] (cs models [66,67]). We use Γ D * s0 = 0.1 MeV; our results do not significantly change for Γ D * s0 < 1 MeV. Similarly, we consider other p-wave D ( * ) sJD ( * ) s pairs of Eq. (1) in DT diagrams. The D s1Ds cusp needs to be 0 + to be consistent with the LHCb result for X(4500). Interestingly, the DT amplitudes of the s-wave D s1Ds (1 − ) and p-wave D s1Ds (1 + ) share the same D * D s → J/ψK interaction of Eq. (A16), while the p-wave D s1Ds (0 + ) DT includes Eq. (A15). Thus the dominance of 0 + and hindered 1 ± might hint that the D * D s → J/ψK interaction of Eq. (A16) is weaker than that of Eq. (A15). Cusps from the D * s0D * s and D s1D * s pairs occur at the dips in the M J/ψφ distribution, and the LHCb did not assign any spin-parity to these structures; we can thus choose their spin-parity to obtain a good fit.
The D * sD s (1 + ) one-loop amplitude includes B + → D * sDs K + and D * sDs → J/ψφ vertices given by respectively, from which the one-loop amplitude is The D * s width is expected to be tiny (∼ 0.1 keV [68,69]) and thus neglected.
For the D *   threshold. Similar virtual poles are also obtained in Ref. [54] where a contact interaction saturated by a φ-exchange mechanism is used. An attractive D * sD ( * ) s interaction makes the threshold cusp significantly sharper [71]. Yet, the fit quality does not largely change even when a = 0 after adjusting other coupling parameters. 6 The scattering length (a) is related to the phase shift (δ) by  Fig. 2(a,b) and Fig. 2(c,d), respectively.

A. Singular behaviors of double triangle amplitudes
A DT amplitude (A DT ) such as Fig. 1(a) can cause a kinematical singularity (anomalous threshold) [72]. The DT singularity can generate a resonancelike structure in a decay spectrum, as first demonstrated in Refs. [70,73]. According to the Coleman-Norton theorem [74], A DT has the leading singularity if the whole DT process is kinematically allowed at the classical level: the energy and momentum are always conserved; in Fig. 1(a), all internal momenta are collinear in the D ( * ) sJD ( * ) s center-ofmass frame; D ( * ) andD ( * ) s (K and K) are moving to the same direction and the former is faster than the latter. Whether a given diagram has a singularity is solely determined by the participating particles' masses.
The DT amplitudes included in our B + → J/ψφK + model do not cause the leading singularity. Yet, the leading singularity is close to being caused 7 , and its effect is expected to be visible as an enhancement of the threshold cusp. We thus study the singular behavior of A DT numerically 8 . In Fig. 2(a,b), we show A DT of Eq. (6) by the red solid curve, and a one-loop amplitude A 1L by the green dotted curve. The p-wave pair of D * s0 (2317)D s (1 + ) is included in A DT and A 1L . While both amplitudes have threshold cusps at the D * s0 (2317)D s threshold, A DT is sharper. This can be seen more clearly by taking a ratio A DT /A 1L as shown in Fig. 3(a). The ratio is still singu-lar; the derivative of the imaginary part with respect to M J/ψφ seems divergent at the threshold. The ratio may also serve to isolate from A DT the kinematical singularity effect other than the ordinary threshold cusp. Similarly, A DT and A 1L including D s1 (2536)D s (0 + ) p-wave pairs and their ratio are shown in Fig. 2(c,d) and Fig. 3(b), respectively. At the threshold, A DT is even sharper and the imaginary part of the ratio is singular. The quantitative difference in the singular behavior between A DT

B. Analysis of the LHCb data
To analyze the B + → J/ψφK + data, we have seven DT diagrams, four one-loop diagrams, two Z cs -excitation diagrams, and two direct decay diagrams. Each of the diagrams has a complex overall factor that comes from the product of unknown coupling constants. The overall normalization and phases of the 0 + and 1 + full amplitudes are arbitrary. We totally have 20 fitting parameters from the coupling constants after removing relatively unimportant parameters; see Tables II and III in Appendix A for coupling parameters and fit fractions. Also, we use cutoffs different from the common value for the direct decay diagrams so that their M J/ψφ distributions are similar to the phase-space shape. We note that no parameter can adjust the DT and one-loop threshold cusp positions where the experimental peaks are located. Fitting the M J/ψφ -distribution lineshape requires the adjustable parameters. In contrast, the quark and hadron-molecule models need adjustable parameters to get pole positions at the experimental peak positions; parameters for fitting the lineshape are needed additionally. It is therefore likely that our model can fit the M J/ψφ -distribution with fewer parameters.
We compare in Fig. 4 our calculation with the M J/ψφ distribution data. Theoretical curves are smeared with the experimental bin width. The data are well fitted by the full model (red solid curve). In particular, the resonancelike four peaks and three dips are well described by threshold cusps from the DT and one-loop amplitudes. We used common cutoff values over Λ = 0.8 − 1.5 GeV and confirmed the stability of the fit quality. This is understandable since the structures in the spectrum are generated by the threshold cusps that are insensitive to a particular choice of the form factors.
In the same figure, we plot the 1 + partial wave contribution without Z cs [blue dashed curve]. There are two clear resonancelike cusps from A 1L interferes with Z cs (4000) accompanied by p-wave φ to create the prominent X(4274) structure as seen in the brown dash-two-dotted curve.
Similarly, threshold cusps play a major role to form resonancelike and dip structures in the 0 + contribution [green dotted curve]. The cusp from A DT Ds1Ds(0 + ) develops the X(4500) structure. This structure is made even sharper by the neighboring two dips at M J/ψφ ∼ 4.46 and 4.66 GeV due to the A DT cusps, respectively. There is another peak at M J/ψφ ∼ 4.75 GeV, near the X(4700) peak, that is caused by the dip at M J/ψφ ∼ 4.66 GeV and the rapidly shrinking phase-space near the kinematical endpoint. Another dip is created at M J/ψφ ∼ 4.23 GeV by A 1L D * sD * s (0 + ) . The contributions from the lighter and heavier Z cs are shown by the magenta dash-dotted and gray two-dash-two-dotted curves, respectively. These Z cs contributions without interference are similar to those of the LHCb analysis [12].
The above partial wave decomposition might change by including more data and more partial waves as done in the LHCb amplitude analysis [12]; this will be a future work (see also footnote 4). The objective here is to demonstrate that the X and dip structures can be well described with the kinematical effects. We also add that the present analysis by no means excludes other interpretations for the X states based on the quark and hadron The red solid curve additionally includes diagrams of Fig. 1(a) with the KK → φ vertex removed. Data are from Ref. [11]. The small window shows the enlarged φ-tail region.
molecule models. We need more experimental and lattice QCD inputs to judge the different interpretations. The LHCb presented M K + K − distribution for B + → J/ψK + K − K + [11]. In Fig. 5, the data show the φ peak and a small backgroundlike contribution. The data actually put a constraint on the contribution from the DT diagrams of Fig. 1(a). This is because when the DT diagrams followed by φ → K + K − contribute to B + → J/ψK + K − K + , there must be a contribution from the diagrams of Fig. 1(a) with the last KK → φ vertex removed. This single triangle contribution has to be smaller than the backgroundlike data. Thus, in Fig. 5, we plot the M K + K − distribution from our model with (red solid curve) and without (blue dashed curve) the single triangle contribution. The single triangle contribution does not significantly change the φ peak and slightly enhances the φ-tail region well within the experimental constraint. The unexplained part of the backgroundlike data should be from non-φ mechanisms not considered here.
The key assumption in our model is that shortrange D The assumption naturally leads to the DT mechanisms that cause enhanced threshold cusps consistent with the LHCb data. If the assumption is wrong, the initial B + → D  Fig. 1(b), and ordinary threshold cusps are expected. However, s-wave cusps are disfavored by the LHCb data, and p-wave cusps are too suppressed to fit the data as shown in Ref. [53]. Thus the LHCb's result seems to be in favor of the assumption.
For D ( * ) sJ = D * s0 (2317), the assumption is also partly supported by previous theoretical works that analyzed lattice QCD energy spectrum and found D * s0 (2317) to be mainly a DK molecule [58][59][60][61]. On the other hand, D s1 (2536) have been mostly considered to be a p-wave cs [61,76] and, thus, the assumption is not intuitively understandable. Yet, D s1 (2536) is known to have a strong coupling to D * K, which has been utilized in our model.
Here and in what follows, the initial vertices contributing to the final 0 + and 1 + J/ψφ partial waves are parity-conserving and parity-violating, respectively. We have used dipole form factors F LL ′ ijk,l defined by where q ij (p k ) is the momentum of i (k) in the ij (total) center-of-mass frame. The second vertices D * s0 → DK and D s1 → D * K are given by with form factors f L ij,k ≡ f L ij / √ E k and The third vertices D ( * )D ( * ) s → J/ψK are p-wave interactions between the pairs with the same spin-parity j p = 0 − or 1 − , and are given with coupling constants respectively, where a notation of p ab ≡ p a − p b has been used. The fourth vertex KK → φ is common for all the DT diagram, and is given as We denote the DT amplitudes including D . The DT amplitudes are constructed with the above ingredients, and are given by where, in each amplitude, the summation over D ( * )+ K 0K 0 and D ( * )0 K +K − intermediates states with the charge dependent particle masses is implicit; K + in the final state is denoted by K f , and W ≡ E − E K f .
For a given DT amplitude, we can analytically integrate the angular part of the loop-integrals by ignoring smaller angle dependences from denominators and form factors. If the DT amplitude is (non-)vanishing after this angular integral, the DT integrand has a suppressed (favored) tensor structure. Next we present formulas for one-loop amplitudes of Fig. 1 s K + and B + → ψ ′ φK + vertices given as → J/ψK + . The s-and p-wave decays are parity-conserving and parity-violating, respectively, and contribute to the 0 + and 1 + J/ψφ final states, respectively. The corresponding amplitudes A J P Z (′) cs are given by where the Z cs and Z ′ cs masses are 3975 MeV and 4119 MeV from the D + sD * 0 and D * + sD * 0 thresholds, respectively; their widths are set to be 100 MeV (constants).
The direct decay amplitudes [ Fig. 1(d)] can be projected onto the J/ψφ(J P ) partial waves. Thus we employ TABLE II. Fit fractions and parameter values. The common cutoff value Λ = 1 GeV is used. The first column lists each mechanism considered in our model, and the second column is its fit fraction (%) defined in Eq. (A43). The third column lists the product of coupling constants to fit the data, and its value and unit are given in the fourth and fifth columns, respectively. Amplitude formulas are given in the equations in the last column.
a form as follows: where c J P dir is a coupling constant for the J/ψφ(J P ) partial wave amplitude.
We basically use a common cutoff value (1 GeV unless otherwise stated) in the form factors for all the interaction vertices discussed above. One exception applies to Eqs. (A41) and (A42) where we adjust Λ ′ of Eq. (A8) so that the M J/ψφ distribution from the direct decay amplitude is similar to the phase-space shape.
In numerical calculations, for convenience, the above amplitudes are evaluated in the J/ψφ center-of-mass frame. An exception is the Z (′) cs amplitudes that are evaluated in the total center-of-mass frame. With the relevant kinematical factors multiplied to the amplitudes, the invariant amplitudes are obtained and plugged into the Dalitz plot distribution formula; see Appendix B of Ref. [78] for details.
Parameter values obtained from and not from the fit are listed in Tables II and III, respectively. In Table II, we also list each mechanism's fit fraction defined by where Γ full and Γ Ax are B + → J/ψφK + decay rates calculated with the full model and with an amplitude A x only, respectively. In Table II, A 1L ψ ′ φ(1 + ) seems to have a rather large fit fraction of ∼ 36%. This mechanism causes a threshold cusp at M J/ψπ ∼ 4.7 GeV, and its height (without interference) is about 80% of the data. The mechanism also has a long tail toward the lower M J/ψπ region, which makes its fit fraction rather large. The amplitudes A 0 + dir and A 1 + dir also have large fit fractions of ∼ 67% and ∼ 45%, respectively. This may be because K ( * ) J -excitation mechanisms have been subsumed in this mechanism. The LHCb analysis [12] found large fit frac- The Coleman-Norton theorem [74] states that a DT amplitude like Fig. 1(a) has the leading singularity if the whole DT process is kinematically allowed at the classical level: the energy and momentum are always conserved; in Fig. 1(a), all internal momenta are collinear in the D ( * ) sJD ( * ) s center-of-mass frame; D ( * ) andD ( * ) s (K and K) are moving to the same direction and the former is faster than the latter.
The DT amplitudes presented in Eqs. (A20)-(A26) do not exactly satisfy the above kinematical condition, and thus do not have the leading singularity. Yet, their threshold cusps are significantly enhanced compared with an ordinary one-loop threshold cusp. This is because the DT amplitudes are fairly close to satisfying the kinematical condition of the leading singularity, and here we examine how close.
Let us study A DT Ds1Ds(0 + ) of Eq. (A23) that generates an X(4500)-like threshold cusp. Apart from the coupling constants and the dependence on the external K + , we can express Eq. (A23) as with G(p K ) = dΩ pK p 2 K pK K · ǫ φ pK ψ · ǫ ψ f 1 KK,φ f 1 where G(p K ) and H(p K ) have been implicitly projected onto 0 + of the J/ψφ pair. Here, we suppose that G(p K ) and H(p K ) include only K = K + ,K = K − , D * = D * 0 in the two-loop, although their isospin partners are also included in Eq. (A23). Now we plot in Fig. 6 G(p K ) and H(p K ) for W = m Ds1 + mD s + 1.4 MeV ∼ 4505 MeV where the DT amplitude is close to causing the leading singularity. Both G(p K ) and H(p K ) show singular behaviors. The real part of H(p K ) [red solid curve] shows a peak at p K ∼ 170 MeV (peak A). The peak A is due to a triangle singularity from the D + s1 D − s D * 0 triangle loop; see Fig. 1(a). Meanwhile, the angular integral of the KKJ/ψ energy denominator in G(p K ) causes a logarithmic end-point singularity. Thus the real part of G(p K ) [blue dashed curve] shows two peaks, and the one at p K ∼ 240 MeV (peak B) is relevant to the leading singularity. The other peak does not create a singular behavior in the amplitude of Eq. (B1). If the peaks A and B occurred at the same p K , the DT leading singularity (pinch singularity) would have occurred. Yet, Fig. 6 indicates a substantial overlap between the peaks A and B, and this is the cause of the enhancement of the DT threshold cusps. The proximity of A DT Ds1Ds(0 + ) to the leading singularity condition is due to the fact that: (i) each vertex is kinematically allowed to occur at on-shell; (ii) the mass deficit in D * D − s →KJ/ψ enables the relatively lightK to chase K with a velocity faster than K. In fact, if the exchanged K − mass were in the range of 445 < ∼ m K − < ∼ 455 MeV, A DT Ds1Ds(0 + ) would have hit the leading singularity.