Production of $P_c $ states from $\Lambda_b$ decay

In the present work, we investigate $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ production from $\Lambda_b$ decay in a molecular scenario by using an effective Lagrangian approach. We predict the ratio of the branching fraction of $\Lambda_b \to P_c K$, which is weakly dependent on our model parameter. We also find the ratios of the productions of the branching fractions of $\Lambda_b \to P_c K$ and $P_c \to J/\psi p$ can be well interpreted in the molecular scenario. Moreover, the estimated branching fractions of $\Lambda_b \to P_c K$ are of order $10^{-6}$, which could be tested by further measurements in LHCb Collaboration.


I. INTRODUCTION
Searching for hadrons beyond 3-quark baryons and quarkantiquark mesons is one of intriguing frontier of hadron physics, even since the initial period of the quark model. Tremendous process has achieved in the recent decade. A growing number of tetraquark and pentaquark candidates have been observed experimentally (more details can be found in the recent review [1][2][3]). In 2015, the LHCb Collaboration reported two pentaquark candidates, P c (4380) and P c (4450), in theJ/ψp invariant mass spectroscopy of Λ b → K J/ψp process [4]. The two-body mass spectroscopy and angular distributions of three-body final states had been analyzed and the J P quantum numbers of these two tetraquark candidates are preferred to be of opposite parity has J = 3/2 for one state and J = 5/2 for the other one.
Besides, the resonance parameters of the P c states, the production ratio, R ≡ B(Λ c → P c K) × B(P c → J/ψp)/B(Λ c → J/ψpK), were also measured, which are also listed in Table I. The new analysis indicates the production ratios are of order of one percent. The newly measured product ratios are much smaller than those for P + c (4380) and P + c (4450) from their previous analysis, which are (8.4 ± 0.7 ± 4.2)% and (4.1 ± 0.5 ± 1.1)% for P + c (4380) and P + c (4450), respectively. With the PDG average of the branching ratio B(Λ b → J/ψpK) = (3.2 +0.6 −0.5 ) × 10 −4 , the production of the branching ratios for Λ b → P c K and P c → J/ψp are estimated to be, Besides the mass spectra of the P c states, how to under-stand the measured production ratios is an intriguing problem, which could help us to reveal the inner structures of the pentaquark states. In Ref. [58], the partial widths of P c → J/ψp were estimated in a molecular scenario, thus, study the production process Λ b → P c K in the same molecular scenario and compared with the the measured production ratios listed in Eq. (1) can further test the molecular interpretations of P c states, which is the main task of the present work. The present work is organized as follows. After introduction, the formula of the productions of Λ b → P c K are present, including the related effective Lagrangians and production amplitudes. In section III, we present our numerical results and some discussions of the present results. A short summary is presented in Section IV.

II. THE PRODUCTIONS OF Λ b → P c K
We can first analyze the production process of P c states from the quark level. One should notice that P c states are produced accompany with a K − meson. In Fig. 1-(a), the kaon is produced directly from W − meson. Since the P c states have a cc components, thus, the b quark should transits to u quark via W − emission, and the cc components are created from the vacuum. This kind of digram will be suppressed in the P c production, since V ub is about one order of magnitude smaller than V cb . In the second kind of mechanism as shown in Fig.  1-(b), the subprocess of the weak decay is b → ccs. Thec quark and the d quark in the initial Λ b form a anti-charmed meson, such asD ( * ) . The cs quarks and the u quark in the initial Λ b become a baryon, like Ξ ( * ) c . Then the Ξ ( * ) c state emits a kaon and transits into Σ c and the recoiled Σ c andD ( * ) form a P c state. In Fig. 1-(c), the subprocess are the same as the one in Fig. 1-(b), but thecs quark form aD ( * ) s and cdu form a Σ c . By emitting a kaon,D ( * )− s meson transits intoD ( * ) and the recoiledD ( * ) meson and Σ c form a P c state. Comparing to Fig.  1-(c), the mechanism in Fig. 1-(b) is suppressed due to color suppression in the hadronization process, thus in Λ b → P c K process, the mechanism in Fig. 1-(c) is supposed to be dominant. In the present work, we estimate the process Λ b → KP c in the hadronic level and the related diagrams are listed in Fig.  2.

A. Effective Lagrangians
We employ an effective Lagrangian approach to estimate the diagrams in Fig. 2. As for the Λ b →D ( * ) s Σ c , the interac- tions vertexes are the same as the those of Λ b →D ( * ) s Λ c and in the form [59,60] where A, B, A 1 , A 2 , B 1 and B 2 are the recombinations of the form factors, which are, 3) are the transition form factors of Λ b → Σ b , which will be discussed in the next section.

B. Decay amplitudes
With the effective Lagrangians listed above, we can obtain the amplitudes involve in the present work. The decay ampli- Fig. 2-(a) is The decay amplitude of Fig. 2-(b) and (c) are The decay amplitudes of Λ b (p) → D ( * ) s (p 1 )Σ c (p 2 )[D * (q)] → K(p 3 )P c3 (p 4 ) corresponding to Fig. 2-(d) and (e) are In the present work, a monopole form factor is introduced to depict the off-shell effect of the exchanged mesons, which is, where Λ = m + αΛ QCD , Λ QCD = 220 MeV and α is a model parameter, which is of order of unit [63][64][65][66].
With above amplitudes, one can estimated the partial width of Λ b → P c K by where the factor 1/2 results from the average of Λ b spin and p is the momentum of P c or K in the rest frame of Λ b . The overline indicates the sum over the spins of final states.

III. NUMERICAL RESULTS AND DISCUSSIONS
Before we estimate the partial width of Λ b → P c K in the present scenario, we first discuss the transition form factors of Λ b → Σ c . Unfortunately, there are no direct estimation of these transition form factors. One should be notice, the constitute quarks and the spatial part of the Σ c and Λ c are the same, thus the transition form factors of Λ b → Σ c should be the same as those of Λ b → Λ c , but smaller in magnitude due to light quark spin flipping in the transition Λ b → Σ c . Here we defined the suppress ratio R as, In Fig 3, we plot the α-dependence of B(Λ b → P c K)/(R 2 g 2 P c ), which are of order 10 −4 for P + c (4440) and P + c (4457) and 10 −5 for P + c (4312), respectively. As for the coupling constants g P c , they could be estimated by the compositeness condition with the assumption that all three observed P c states are molecular states. In Ref. [58], the coupling constants are estimated depending on a model parameter Λ, which is of order one GeV. When one take Λ = 1 GeV, the coupling constants are estimated to be, g P c1 = 2.25, g P c2 = 1.72 and g P c3 = 1.77, respectively, which are very similar to those in Ref. [47]. With above coupling constants, the ratio of the branching fractions Λ b → P c K are estimated to be, which are independent on R. The center values correspond to α = 1.0 and the uncertainties are resulted from the variation of model parameter α from 0.8 to 1.2. Our estimation indicates the production ratio are very weakly dependent on the model parameter. Our estimation indicates that product ratios are very weakly dependent on the model parameter α. In Ref. [58], the partial widths of P c (4320) → J/ψp, P c (4440) → J/ψp and P c (4450) → J/ψp are estimated to be 5.6, 9.3 and 2.6 MeV , respectively, when we take Λ = 1 GeV. With these estimated partial widths and the measured total widths of P c states, one can get the branching fraction of P c → J/ψp, which are, and then the decay ratio are, With the production and decay ratios estimated in the molecular scenario, we can get the product of the product and decay ratios, i.e., and compare these ratios with the experimental measurement as listed in Eq. (1). We present a comparison of the measured R i j from LHCb Collaboration [33] and the estimation in the present work in Fig. 4. One can find our estimations are consistent with the experimental data from LHCb Collaboration within error, which indicates all three P c states could be interpreted as molecular states. Moreover, taking the branching ratios estimated in the molecular scenario as listed in Eq. (13) back to Eq. (1) one can get the branching ratios of P c production, which are, In the present scenario, the ratio B(Λ b → P c K)/(R 2 g 2 P c ) are estimated to be of 10 −5 for P + c (4312) and 10 −4 for P + c (4440) and P + c (4457), respectively. Considering the light quark spin flip suppression in the Λ b → Σ c D ( * ) process, one can suppose R much smaller than one. In the present work, the estimated product ratio could be consistent with the experimental measurements within error when one take R = 0.09, which are, .02) × 10 −6 , B(Λ b → P + c (4457) + K) = (6.78 ∼ 9.88) × 10 −6 , (17) respectively.
It is interesting to note that the Λ b → Σ c transition requires the spin flip of the involved light quark system. This has been widely expected to be power suppressed in 1/m b . A relevant counterpart is the transition of bottom meson into a charmed scalar meson [68]. An analysis in the QCD sum rules indicates that the helicity-flipped transition form factor is smaller than the ordinary heavy-to-light transition form factor by a factor 3 to 5. A similar power suppression might also happen in Λ b → Σ c compared to Λ b → Λ c . If this were true, it indicates the use of R ∼ 0.1 is reasonable.

IV. SUMMARY
In the present work, we estimated the P c production from Λ b decay in P c molecular scenario, where P + c (4312) is considered as Σ cD molecular with J P = 1 2 − , while P + c (4440) and P + c (4457) are interpreted as Σ cD * molecule with J P = 1 2 − and 3 2 − , respectively. By analyzing the production process in quark level, we find the production process occur via the following process, Λ b could couple with Σ cD ( * ) s and theD ( * ) s transits intoD ( * ) via kaon emission and the recoilD ( * ) and Σ c couple to P c state. The P c production process are investigated in hadronic level with an effective Lagrangian approach. Unfortunately, the transition form factors related to Λ b → Σ c are unknown. In the present work, we borrow the form factors of Λ b → Λ c since the flavor and spatial parts of Λ c and Σ c are the same. However, the transition form factors of Λ b → Σ c should be smaller than those of Λ b → Λ c since the suppression caused by the light quark spin flip. Here, we define a suppression factor R. Our estimation indicates the R independent ratios of B(Λ b → P c K) × B(P c → J/ψp) in the molecular scenario are consistent with the experimental measurements from LHCb Collaboration [33]. Moreover, we find the magnitude of the production of the branching ratios for Λ b → P c K and P c → J/ψp could be reproduced when one take the suppression factor R = 0.09.
In the molecular scenario, the branching ratio of Λ b → P c K are estimated. Together with the branching ratio of P c → J/ψp estimated in Ref. [58], we find we can interpret the decay and production properties of P c states simultaneously in the molecular scenario, which indicates that P c states could be good candidates of Σ cD ( * ) . Furthermore, in the present work and in Ref. [58], we present our estimation of the production and decay branching ratios, which could be tested by further analysis in the LHCb Collaboration.
The transition form factor pf Λ b → Λ c could be parameterized in the form [67], where ζ = Q 2 /m 2 . In Table. II, we collect the parameters related to the transition form factors of Λ b → Λ c [67].
where F V i and F A i (i=1,2,3) are the form factors of Λ b → Λ c .
In the present estimation, we further parameterize the form factors in the form,