Understanding the newly observed heavy pentaquark candidates

We find that several thresholds can contribute to the enhancements of the newly observed heavy pentaquark candidates $P_c^+(4380)$ and $P_c^+(4450)$ via the anomalous triangle singularity (ATS) transitions in the specific kinematics of $\Lambda_b\to J/\psi K^- p$. Apart from the observed two peaks we find that another peaks around 4.5 GeV can also be produced by the ATS. We also show that the $\Sigma_c^{(*)}$ can be produced at leading order in $\Lambda_b$ decay. This process is different from the triangle diagram and its threshold enhancement only appears as CUSP effects if there is no pole structure or the ATS involved. The threshold interaction associated with the presence of the ATS turns out to be a general phenomenon and plays a crucial role in the understanding of candidates for exotic states.


I. INTRODUCTION
The two states P + c (4380) and P + c (4450) observed in the invariant mass spectrum of J/ψp in Λ b → J/ψK − p by the LHCb Collaboration [1] has immediately attracted a lot of attention from the whole community since they could be the long-searching-for pentaquark states in the heavy flavor sector. Recent theoretical studies can be found in Refs. [2][3][4][5][6]. Their masses are 4380 ± 8 ± 29 MeV and 4449.8 ± 1.7 ± 2.5 MeV, respectively, and their widths are 205 ± 18 ± 86 MeV and 39 ± 5 ± 19 MeV, respectively. Their preferred J P are either 3/2 − and 5/2 + , or 3/2 + and 5/2 − , respectively, based on the detailed experimental analysis. The possible existence of multiquark states has always been regarded as a natural consequence of QCD. Although the conventional quark model has made tremendous successes in the description of the hadron spectroscopy, it also raised questions on why and how those multiquark states kept out of our sight for such a long time. The LHCb results and the recent results from Belle [7] and BESIII [8] certainly make a big step forward to our understanding the multiquark system. But at the same time, they also give us chances to ask more questions.
In this Letter we will examine the role of the non-perturbative anomalous triangle singularity (ATS) in the decay of Λ b → J/ψK − p. As recently pointed out in Ref. [9], the pronounced narrow threshold states in the elastic channels should indicate non-perturbative rescatterings which will eventually generate pole structures after sum over all the loops to infinity. It was also stressed that if the ATS is present the threshold peak will be enhanced and mix with the dynamic pole structure in the inelastic channel. Therefore, in order to understand the nature of the threshold enhancements a careful analysis of the triangle process should be necessary.
There are several interesting features arising from the decay channel as the data have shown. For instance, there are clear structures for Λ * resonances in the lower end of the K − p invariant mass spectrum while the higher mass region appear to smooth out. In contrast, the observed P + c (4380) and P + c (4450), in particular, the P + c (4450), are clear structures above the phase space. Note that some interferences occur over a rather broad mass range above the P + c (4450) in the J/ψp invariant mass spectrum (see e.g. Fig. 2 of Ref. [1]). This feature suggests that there are intermediate processes in Λ b → J/ψK − p which will give rise to final state interactions. Our motivation is to investigate whether the ATS will accumulate the events at the two peak positions.
As follows, in the next Section we will first analyze the transition mechanisms for Λ b → J/ψK − p and then investigate the loop diagrams where the ATS can be present. A summary will be given in the last Section.
If the production of the pentaquark candidates P + c (4380) and P + c (4450) is indeed via the Σ ( * ) cD ( * ) (Table IV) or Table II ) interactions, then the relevant thresholds will play non-trivial roles in the coupled-channel decays into e.g. J/ψp. Considering the major rescattering processes arising from Fig. 1, we can reexpress the transitions as Fig. 2 where three type of rescatterings will contribute. Figure 2  interactions.
The loop diagrams as a consequence of Fig. 1 where the ATS and kinematic CUSP can be recognized.
The interesting property of Fig. 2 (a) and (b) is that given the masses of the involved states located within certain ranges it will allow the internal states to be on-shell simultaneously. This is different from the kinematic CUSP effects which only recognizes the on-shell condition for two internal particles and contributions from such a branch point is subleading compared to the ATS [23,24]. Therefore, we do not expect that such CUSP effects produce narrow and strongly enhanced structures in the invariant mass spectrum. In contrast, when the ATS condition is satisfied, the singularity behavior of the integral will produce strong enhancements at the singular points of which the effects can be measured in the experiment. In particular, the singular points will mostly locate in the vicinity of the two-body thresholds but not necessarily to be exactly at the thresholds. It should be realized that the positions of the singularity will not change even when higher partial waves contribute at the interaction vertices. 1 The reason is because the singular term will always be kept in the decomposition of the integrand in the Feynman parametrization. In other words, even though the contribution from the singular term relative to other contributions might be small, its enhancement at the singular point may not be negligible 2 . Nevertheless, in the case of Λ b → J/ψK − p there are several thresholds close to each other. Even a small singularity enhancement can build up and produce measurable effects.
Since quite a lot of thresholds can appear in the decays of Fig. 2 and we are still lack of information about the vertex couplings, we only consider low partial waves and thresholds which are close to the masses of interest and we discuss separately the properties of those three types of loops in Fig. 2. Figure 2 (a) is a consequence of Fig. 1 (a) where the rescattering between χ cJ and an exchanged proton from the decay Λ * → pK − is considered. Note that the mass thresholds for p + χ cJ (J = 0, 1, 2) are close to the peak masses for P + c (4380) and P + c (4450) as listed in Table I. Also, the S-wave scatterings of pχ c2 → J/ψp 3 can access the quantum numbers of 3/2 + and 5/2 + for the threshold enhancement. The χ c1 and p scattering can access the quantum numbers of 1/2 + and 3/2 + . The χ c0 p can reach 1/2 − and 3/2 − via a P wave interaction. It is interesting to notice that the significant enhancement to the χ cJ p (with J = 0, 1, 2) via the ATS would prefer that the mass of Λ * within the mass regions 1.92 ∼ 2.20 GeV, 1.89 ∼ 2.11 GeV, 1.83 ∼ 2.06 GeV, respectively. From Fig. 2 (a) of Ref. [1], it shows that the cross section for K − p is smooth but non-zero. Note that as long as the kinematics approaching the ATS condition, all the cross sections will contribute to the threshold singularity.
In Fig. 3 we show the structures in the invariant mass of J/ψp via the triangle diagram of Fig. 2 (a). As discussed before, since χ c1 p and χ c2 p can access the possible quantum numbers via the S wave, we only consider loops of χ c1 and χ c2 at this moment. On the other hand, since the branching ratio of B + → K + χ c2 is one order of magnitude  Fig. 2 (a). smaller than that of B + → K + χ c1 , we expect that it is also the case in the Λ b decay. Then, the upper limit of the contributions from χ c2 can be estimated by requiring Fig. 3, the ATS can produce significant threshold enhancement of χ c1 p while the effect from the χ c2 p loop is strongly suppressed. Note that the singularity from the χ c1 p loop exactly locates at the mass position of the observed P c (4450). The pole trajectories of the χ c1 p loop and the properties of the poles on different Riemann Sheet were first considered in Ref. [22].
It is possible that the intermediatecs in Fig. 1 sJ states which can decay intoD ( * ) and K − . The intermediateD ( * ) meson will then scatter the Λ c into J/ψp. This process is illustrated in Fig. 2 (b). The accessible thresholds are listed in Table II. Although one can see from Table II that in an S wave none of the thresholds matches the experimental measured masses and favored quantum numbers simultaneously. However, among these 'Λ c 's, the Λ c (2595)+D sJ (2860) loop withD sJ (2860) decay toDK can produce the singularity at the P + c (4450) mass position as shown by the black point in Fig. 4 (b). As a comparison we also show the pole trajectory withD sJ (2860) decay toD * K in Fig. 4 (a). Note that the ATS only works in a very limited kinematic region. It implies that if the kinematics deviate from the ATS condition one should not expect any significant enhancement at the corresponding threshold mass region. This will provide a possibility for experimentalists to pin down the nature of some threshold states. Namely, if they are not caused by kinematic effects the enhancements will still appear in other processes where the ATS condition does not hold.
Although Λ c (2595)D sJ (2860) can access the expected quantum number in a P wave which will be suppressed by the centrifugal barrier, we will show later that the singular points near threshold can still match the observed peak positions. Also, as mentioned earlier that the ATS can still possibly produce observable effects when higher partial waves are present at the interaction vertices, we then investigate the possible partial waves for Fig. 2 (b) and see how the ATS would manifest in the invariant mass spectrum.
In Table III Fig. 2 (b), the charmed baryon and anticharmed meson are chosen as Λ c (2595)/Λ c (2650) andD/D * , respectively, which means that we take into account the contributions from four diagrams. Furthermore, to match the qualitative feature of the experimental observations, we estimate the rescattering amplitudes by assuming relatively smaller branching fractions of Λ b →D   Fig. 2 (a) has been investigated in Ref. [22].  For Fig. 2 (c), the kinematic effects will be just CUSP structures in the J/ψp invariant mass spectrum. We do not discuss the dynamic consequences if the intermediate Σ  (Table IV) may have strong couplings, then they may generate dynamic poles near threshold after proper summation over the bubble loops. Instead, we only show the kinematic CUSP for which has turned out not to lead to pronounced structures as recently studied in Ref. [9]. In another word, the observed pronounced peaks may either be produced by possible pole structures or the ATS mechanisms.
In order to try to distinguish the behavior from a pole structure and the ATS, we generate the invariant mass spectra of J/ψp in different K − p invariant mass regions the same as Fig. 8 of Ref. [1] (see Fig. 6). Since the ATS contributions will vary in terms of different kinematics, such a quantity will be able to distinguish the behavior of the pole and the ATS mechanism. The results for those three loop processes of Fig. 2 (a), (b) and (c) are illustrated by the solid, dashed and dotted lines which are different from the symmetric Breit-Wigner lineshape. In particular, the structures created by the CUSP effects appear to be negligible. Note that the kinematics of higher invariant mass of K − p is favored by Fig. 2 (a). It means even small couplings for χ cJ p scattering are introduced, the ATS enhancement  can still be observable.
Since we introduce several mechanisms to generate the kinematic singularities, it is also necessary to discuss their similar and different characteristics. Both χ c1 p and Λ c (2595)D thresholds are very close to the mass of P + c (4450). Note that χ c1 p can scatter into J/ψp via multi-gluon exchange process, while Λ * cD can scatter into J/ψp via quark interchange process which is a rearrangement of the quark flavors. The P -wave scattering between χ c1 and proton will imply that the quantum numbers of the J/ψp system can be J P = (1/2 − , 3/2 − , 5/2 − ), but the P -wave scattering between Λ c (2595) andD will imply that the quantum numbers can be J P = (1/2 − , 3/2 − ). Some of these quantum numbers are compatible with the experimental fitting results. If one hopes the rescattering mechanism can favor the J P = 5/2 + assignment of P + c (4450), the D-wave rescattering of χ c1 p or Λ c (2595)D will be required. Usually the higher partial wave rescattering amplitudes will be suppressed to some extent. However, the analytic properties of the kinematic singularities will mainly depend on the kinematics of the loop integrals, which will not be affected too much by the coupling forms of the vertices. To clarify whether the χ cJ p rescatterings or the Λ * cD ( * ) rescatterings would be dominant in producing the resonance-like peaks, we suggest that Λ 0 b → K − h c p would be a promising channel. Namely, the transition χ cJ p → h c p would require the heavy quark spin flip, thus, will be suppressed. In contrast, in Λ * cD ( * ) →h c p such a spin flip does not necessarily occur. Furthermore, since the h c p threshold is much larger than the lower thresholds, some higher resonance-like peaks induced by the rescatterings, similar to P + c (4450), can be expected in the h c p invariant mass distributions.

III. SUMMARY
In this work we analyze the role played by anomalous triangle singularity in Λ b → J/ψK − p as a possible contribution to the observed hidden-charm pentaquark candidates P + c (4380) and P + c (4450). We first show that the Σ ( * ) cD ( * ) can be produced at leading order which seems to be overlooked in the literature. We then demonstrate that the ATS can generate threshold enhancements which can mimic the experimental observations. We have to admit that more detailed model construction is needed in order to determine better the ATS behavior and strengths. But we emphasize that the kinematics of Λ b → J/ψK − p are in favor of the ATS mechanism when intermediate states are involved. Since all the triangle amplitudes can build up the contributions from the ATS near the mass region that we are interested in, they seem to be able to produce structures similar to what observed in experiment. Meanwhile, we find that the Λ c (2625)D(1865) threshold can give rise to a third peak around 4.5 GeV. We also discuss the kinematic feature of Σ ( * ) cD ( * ) which can produce kinematic CUSP effects. In this sense, the observed pronounced structures can be either produced by pole structures or the ATS mechanism. The kinematic dependence of the ATS is investigated by looking at the ATS cross sections in different K − p invariant mass regions. We also show that the ATS mechanism should have rather strong dependence of the kinematics by looking at the J/ψp invariant mass distributions at different energy cuts for K − p.
In brief, since the ATS could play a role in the production of threshold states, it may mix with the pole if the singular threshold is close to the pole position. Such a mixing and interference might generate complicated structures such as to distort the lineshape or produce narrower enhancements that behaves very differently from expected pole contributions. In order to better understand the nature of these newly observed pentaquark candidates, a combined study of the ATS and dynamically generated pole structures should be necessary.