Pentaquarks from intrinsic charms in $\Lambda_b$ decays

We study the three-body $\Lambda_b$ decays of $\Lambda_b\to J/\psi pM$ with $M=K^-$ and $\pi^-$. The two new states ${\cal P}_{c1}\equiv {\cal P}_c(4380)^+$ and ${\cal P}_{c2}\equiv {\cal P}_c(4450)^+$ observed recently as the resonances in the $J/\psi p$ invariant mass spectrum of $\Lambda_b\to J/\psi p K^-$ can be identified to consist of five quarks, $uudc\bar c$, being consistent with the existence of the pentaquark states. We argue that, in the doubly charmful $\Lambda_b$ decays of $\Lambda_b\to J/\psi pK^-$ through $b\to c\bar c s$, apart from those through the non-resonant $\Lambda_b\to pK^-$ and resonant $\Lambda_b\to \Lambda^*\to pK^-$ transitions, the third contribution with the non-factorizable effects is not the dominant part for the resonant $\Lambda_b\to K^-{\cal P}_{c1,c2}, {\cal P}_{c1,c2}\to J/\psi p$ processes, such that we propose that the ${\cal P}_{c1,c2}$ productions are mainly from the charmless $\Lambda_b$ decays through $b\to \bar u u s$, in which the $c\bar c$ content in ${\cal P}_{c1,c2}$ arises from the intrinsic charms within the $\Lambda_b$ baryon. We hence predict the observables related to the branching ratios and the direct CP violating asymmetries to be ${\cal B}(\Lambda_b\to \pi^-({\cal P}_{c1,c2}\to) J/\psi p)/{\cal B}(\Lambda_b\to K^-({\cal P}_{c1,c2}\to) J/\psi p)=0.8\pm 0.1$, ${\cal A}_{CP}(\Lambda_b\to \pi^-({\cal P}_{c1,c2}\to)J/\psi p)=(-3.9\pm 0.2)\%$, and ${\cal A}_{CP}(\Lambda_b\to K^-({\cal P}_{c1,c2}\to)J/\psi p)=(5.8\pm 0.2)\%$, which can alleviate the inconsistency between the theoretical expectations from the three contributions in the doubly charmful $\Lambda_b$ decays and the observed data.


I. INTRODUCTION
According to the recent observations of the three-body b-baryon decays of Λ b → J/ψpM with M = K − and π − [1][2][3], apart from the non-resonant Λ b → J/ψpM and resonant Λ b → J/ψB * , B * → pM (B * =Λ * (N * ) for M = K − (π − )) contributions, depicted in Figs. 1a and 1b, respectively, there can be another resonant process in Λ b → J/ψpM as shown in Fig. 1c. The LHCb collaboration has presented the compelling evidence for the new resonant states, being consistent with the existence of the pentaquark states as the five-quark bound states, while the P c (4380) + and P c (4450) + states are observed as the two resonances in the J/ψp invariant mass spectrum of Λ b → J/ψK − p, with the significance for each state to be more than 9 standard deviations, which can be regarded to be composed of uudcc. We note that, in the same principle, the two new states should also exist in Λ b → J/ψpπ − . The processes of P c → J/ψp in Fig. 1c are theoretically known to be dominated by the nonfactorizable effects, calculated non-perturbatively with the scatterings of the soft hadrons, such that the pentaquarks are considered as the molecular states [4][5][6]. In spite of the non-factorizable diagrams shown in Fig. 1c, which may get enhanced when the strong FSI interactions occur near the threshold to explain the pentaquark productions, we propose another possibility based on the factorizable effects. We note that this type of the processes in Fig. 1c has not been observed in the previous searches of the lighter pentaquarks than P c . For example, the resonant B + →pΘ(1710) ++ , Θ(1710) ++ → pK + decay is measured to be B(B + →pΘ(1710) ++ , Θ(1710) ++ → pK + ) < 9.1 × 10 −8 with the upper bound about 60−70 times smaller than the observed branching ratio of B + → ppK + [7,8] smaller than B(B 0 → ppK 0 s ) [8,9]. Similar to the charmless B → ppK decays, the upper bound on B(B 0 → Θ cp π + , Θ c → D ( * )− p) is expected to be about 30 − 40 times smaller than B(B 0 → ppπ + D ( * )− ) [10]. In contrast, since the resonant Λ b → K − P c , P c → J/ψp decays can contribute to the branching ratio as much as 10%, this leads to the question that if there can be other processes, which are responsible for the resonant P c → J/ψP decays, other than the ones in Fig. 1c.
In Ref. [1], the LHCb collaboration has given where the first and second errors are from the statistical and systematic uncertainties, respectively. The data in Eq. (1) indicate some new Λ b → MP c , P c → J/ψp processes beyond the non-factorizable ones in Fig. 1c with reasons as follows.
Second, ∆A CP ∼ 5.7% in Eq. (1) with the significance of 2.2σ suggests that a new contribution must proceed with V ub to provide the weak CP phase, otherwise ∆A CP = 0 as the case in the doubly charmful Λ b decays in Fig. 1, in which such a phase is vanishingly small.
We hence propose that the resonant Λ b → MP c , P c → J/ψp processes can be the new contributions to the charmless Λ b decays as depicted in Fig. 2, where the cc content comes from the intrinsic charm (IC) in the Λ b baryon. In the followings, we will assume that these new processes in Fig. 2 It is not surprising that the Λ b baryon contains the ICs, which are presented in the Fock state decomposition [14,15] as |Λ b = Ψ bud |bud +Ψ budcc |budcc +···. In fact, the existence of the IC was first suggested in the proton to explain the large D + and Λ + c productions at large energies in the proton-proton scattering [14,15]. In addition, as a possible solution to the socalled ρ-π puzzle [16], the IC in ρ for J/ψ → ρ + π − allows a strong decay not through the J/ψ annihilation suppressed by the OKubo-Zweig-Iizuka (OZI) rule. For a heavier hadron, since the gluon fluctuation, such as gg → cc, can easily occur without costing a large energy [21], it is expected that [22] the IC component in Λ b (m Λ b > m B > m p ) can be larger than the proton and the B mesons, estimated to be 1% and 4%, respectively. Consequently, in the Λ b → p transition, we only consider the ICs in Λ b since the heavier baryon would contribute a larger cc production. Note that, to distinguish the IC in the proton from that in Λ b , the J/ψ photoproduction can be useful [17][18][19], which is in accordance with Ref. [20]. While the study of the ICs in the B decays has been done extensively in the literature [21][22][23][24], it is not well examined in Λ b , which should be a suitable scenario.
In this paper, since we propose that the two new resonant P c states, i.e. the pentaquark states, in the m J/ψp spectrum of the Λ b → J/ψpK − decay can be traced back to the charmless Λ b decays from b → uūs, while the cc content in J/ψ is from the IC in the Λ b baryon, we will study the branching ratios and the direct CP violating asymmetries, and check if our results will be able to understand the inconsistency between the theoretical estimations in the doubly charmful Λ b decays and the observed data in Eq. (1).

II. FORMALISM
In terms of the effective Hamiltonian for b → ccq at the quark level, the amplitude of Λ b → J/ψpM from Figs. 1a and 1b is given by where G F is the Fermi constant, V stands for the CKM mixing matrix, q = s(d) corresponds where δ 1 is the strong phase from the on-shell resonant B * → pM decay and F M is the parameter with F π /F K ≃ (f π /f K ) representing the flavor SU(3) symmetry breaking. As a result, we rewrite the amplitude in Eq. (2) as with ε= ε µ * · γ µ . From Fig. 2, which depicts the charmless Λ b decays of Λ b → P c M, P c → J/ψM decays, with cc in P c coming from the IC in Λ b , the amplitudes of Λ b → MP c , P c → J/ψp can be derived as [25] A where f M is the meson decay constant, defined by M|q 1 γ µ γ 5 q 2 |0 = −if M q µ with the four-momentum q µ . The constant α M (β M ) in Eq. (5) is related to the (pseudo)scalar quark current, given by  [25]. In Eq. (5), the matrix elements for the resonant Λ b → P c , P c → J/ψp transition can be given as where the Breit-Wigner factor R Pc for P c is described as an intermediate state, given by with m Pc and Γ Pc the mass and the decay width for the P c state, respectively. Despite the fact that there is no sufficient information for the detailed parameterization of J/ψp|P c P c |ū(γ 5 )b|Λ b , the matrix elements of J/ψp|ū(γ 5 )b|Λ b in Eq. (7) can still be reduced as This is due to the fact that, after the summations of the intermediate P c spins with spin=3/2 or 5/2, all Lorentz indices are in fact coupled to be a scalar quantity, which can be parameterized as F S and F P . In general, F S,P are momentum dependent, but they can be taken as nearly constants around the threshold area of t ≃ m 2 Pc , at which the threshold effect dominates the decay branching ratio. Besides, we take F S = F P ≡ F Pc as a consequence of the Λ b transition [11]. We hence obtain such that the total amplitude for the two resonant P c states is in the form of with F 2 e iδ 2 = R P c1 F P c1 +R P c2 F P c2 , where δ 2 is the strong phase from the on-shell P c → J/ψp decays, and P c1 and P c2 denote P c (4380) + and P c (4380) + , respectively. Note that P c1,c2 have been observed to have the masses and the decay widths as (m, Γ) = (4380 ± 8 ± 29, 205 ± 18 ± 86) MeV and (4449.8 ± 1.7 ± 2.5, 39 ± 5 ± 19) MeV, respectively, while their quantum numbers for J P can be (3/2 − , 5/2 + ) or (3/2 + , 5/2 − ). However, the information of P c1,c2 can be cast into the to-be-determined parameters F 2 e iδ 2 , without losing generality.

III. NUMERICAL ANALYSIS AND DISCUSSIONS
For the numerical analysis, the theoretical inputs of the meson decay constants and Wolfenstein parameters in the CKM matrix are taken as [26,27] (f J/ψ , f π , f K ) = (418 ± 9, 130.4 ± 0.2, 156.2 ± 0.7) MeV , (λ, A, ρ, η) = (0.225, 0.814, 0.120 ± 0.022, 0.362 ± 0.013) , while the parameters a 1,4,6 can be adopted from Refs. [11,28], along with a 2 = 0.2 [29]. The data in the fitting are given in Table I. As a result, we obtain F K = 2.8 ± 0.2 , F P c1 = 19.6 ± 3.1 , F P c2 = 5.5 ± 1.0 , δ 1 = (54.8 ± 31.9) • , with the fitted numbers in column 2 of Table I to be consistent with the data. First, for the three-body Λ b decays only from the resonant Λ b → MP c , P c → J/ψp contributions in Fig. 2, we obtain where the parameters F 2 e iδ 2 in Eq. (10) have been canceled by the ratio. In the doubly charmful Λ b decays, since the three contributions are all through b → ccq at the quark level (see Fig. 1), the ratio of R πK defined in Eq. (1) should be (V cd /V cs ) 2 ≃ 0.05, which is not approved by the data in Eq. (1). However, by adding the contributions from the charmless , so that the value of R πK is able to increase from 0.05 to a larger one to meet the data in Eq. (1), as the fitted result of (8.38 ± 0.77)% in Table I.
Second, the direct CP violating asymmetries from the resonant Λ b → MP c , P c → J/ψp parts are evaluated to be However, since the measurement by the LHCb in Ref. [2] has suggested that the doubly charmful Λ b → J/ψpK − mode dominates the corresponding decay, it leaves little room for the interference effects with the charmless ones of Λ b → K − (P c1 , P c2 →)J/ψ that provide the weak CP phase, of which A raw CP (Λ b → J/ψpK − ) = (1.1±0.9)% from the LHCb [1] agrees with the fitted result of A CP (Λ b → J/ψpK − ) = (−0.22 ± 0.16)%. Note that ∆A CP = 0 from b → ccq to be different from the data of ∆A CP = 5.7% in Eq. (1) requires the interference between the two compatible Λ b → π − (P c1,c2 →)J/ψp and Λ b → J/ψ(N * (1440), N * (1520) →)pπ − channels. It is found that the contributions from b → ccd with the strong phase δ 1 = 54.8 • and the contributions from b → uūd with the weak phase by V ub gives ∆A CP = (2.9 ±1.4)%, which is in good agreement with the data. Finally, the branching ratio for Λ b → J/ψpπ − is predicted as which includes the compatible contribution from Λ b → π − (P c1,c2 →)J/ψp to agree well with B(Λ b → J/ψpπ − ) = (2.51 ± 0.08 ± 0.13 +0.45 −0.35 ) × 10 −5 measured by the LHCb [3], whereas the contributions only from the doubly charmful Λ b decays give B(Λ b → J/ψpπ − ) = (1.68 ± 0.24) × 10 −5 , which is around 0.05B(Λ b → J/ψpK − ), borne by the relation of In sum, the charmless processes of Λ b → M(P c1 , P c2 →)J/ψ provide us with a possible way to understand the CP asymmetry in Eq. (1) due to the origin of the weak phase from V ub . Furthermore, to realize the ratio of R πK in Eq. (1), which is unable to be explained from b → ccq, the contributions apart from the non-perturbative processes in Fig. 1c have to be included.