Study on possible molecular states composed of Λ c D̄ ( Λ b B ) and Σ c D̄ ( Σ b B ) within the Bethe-Salpeter framework

X iv :1 90 9. 12 50 9v 1 [ he pph ] 2 7 Se p 20 19 Study on possible molecular states composed of ΛcD̄ (ΛbB) and ΣcD̄ (ΣbB) within the Bethe-Salpeter framework Hong-Wei Ke ∗, Mei Li, Xiao-Hai Liu †and Xue-Qian Li2‡, 1 School of Science, Tianjin University, Tianjin 300072, China 2 School of Physics, Nankai University, Tianjin 300071, China Abstract Pc(4312) observed by LHCb collaboration is confirmed as a pentaquark and its structure, production and decay behaviors attract great attention of theorists and experimentalists. Since its mass is very close to sum of Σc and D̄ masses, it is naturally tempted to be considered as a molecular state composed of Σc and D̄. Moreover, Pc(4312) is observed in the channel with J/ψp finial state, requiring isospin conservation Pc(4312) should be an isospin-1/2 eigenstate. In literature, several groups used various models to estimate its spectrum. We are going to systematically study the pentaquarks within the framework of the Bethe-Salpetr equation, thus Pc(4312) is an excellent target because of the available data. We re-calculate the spectrum of Pc(4312) in terms of the Bethe-Salpter equations and further study its decay modes. Some predictions on other possible pentaquark states which can be tested in the future experiments, are made.


I. INTRODUCTION
Due to the innovation of experimental techniques and facilities as well as the advances in theory of recent years several exotic states have been experimentally observed and theoretically confirmed. Indeed, more constituents would cause more ambiguities, unlike the simplest qq for mesons and qqq for baryons. The inner structures of the exotic states are still not clear yet, those discoveries stir up large numbers of discussions [1]. Indeed the theoretical exploration is crucial for getting a better understanding on the quark model and obtaining valuable information about non-perturbative physics. Definitely, to complete the theoretical job achieving more accurate data would compose the key.
Some hidden charm or bottom states were measured in two-meson finial states [2][3][4][5][6][7][8][9][10][11]. They are regarded as tetraquark states or meson-meson molecular states. In 2003 a baryon was measured by LEPS [12] which was conjectured as a pentaquark, however later the allegation was negated by further more accurate experiments. Breaking the frustration on existence of pentaquark which was predicted by Gell-Mann in his first paper on quark model, the LHCb collaboration reported two pentaquark states observed in Λ b decays where peaks appear at the J/ψp finial states [13].
Recently another narrow pentaquark state P c (4312) [14] has also been observed in the J/ψp mass spectrum. Its mass and width are 4311.9 ± 0.7 +6. 8 −0.6 MeV and 9.8 ± 2.7 +3.7 −4.5 MeV respectively. Since its mass is very close to the sum of Σ c andD masses, it is natural to regard it as a molecular state of Σ cD [15][16][17][18][19][20][21][22][23]. Furthermore its width is rather wide in accord with the property of molecular states, so the phenomenon further supports the proposal of its molecular structure. Some other theorists conjecture P c (4312) as a compact pentaquark. [23,24] instead. In Ref. [25] the authors think the interaction between Σ c andD is too weak to bind them into a bound state. It is worth of deeper explorations about whether the molecule picture is reasonable. In this work we will calculate the mass spectrum of P c (4312) based on the assumption that it is a stable bound state of Σ c andD. Additionally we also study other possible bound states of Λ cD , Λ b B and Σ b B and see if they can be formed.
We will employ the Bethe-Salpeter (B-S) equation to study the possible bound state which consists of a baryon and a meson. The B-S equation is a relativistic equation to deal with the bound state and established on the basis of quantum field theory [26]. Initially, people use the B-S equation to study the bound state of two fermions [27,28] and the system of onefermion-one-boson [29]. In Ref. [30,31] the authors employed the Bethe-Salpeter equation to study the KK or BK molecular state and their decays. With the approach we studied the molecular state of Bπ [32], D ( * ) D ( * ) and B ( * ) B ( * ) [33]. Recently the approach is extended to explore double charmed baryons [34,35] and pentaquarks which are assumed to be two-body bound systems. In Ref. [36] the authors studied possible bound states of Λ (Σ) andK. In this work we will a similar approach to study the possible bound states of Σ cD , Λ cD , Λ b B and Σ b B.
At present pentaquark states P c (4312), P c (4380), P c (4440) and P c (4457) have been measured in decays of Λ b where the pentaquark states peak up at the invariant mass spectrum of J/ψp, so their isospin is 1 2 because of isospin conservation. Thus we require that the two hadron constituents reside in an isospin eigenstate. Instead, for the Λ cD (as well Λ b B) system its isospin must be 1 2 but the Σ cD ( or Σ b B) system may reside in either isospin 1 2 or 3 2 . Certainly, for a bound system with spin-parity 1 2 − the two constituents are in the S−state.
For carrying on our calculation the interactions between two constituents are needed. According to the quantum field theory two particles interact via exchanging certain mediate particles. Since two constituents in a pentaquark are color-singlet hadrons the exchanged particles are some light hadrons such as ρ or (and) ω etc..
The effective interaction deduced from the chiral lagrangian can be written as L = L MMV + αL BBV + ... where L BBV and L MMV are for baryon and meson parts respectively ( See Appendix A) and the ellipsis represents other possible parts. It is noted that in the effective lagrangian a free phase factor α exists which is not determined by any fundamental theory. Principally, it can be a complex phase, but in this work, we just require it to be either 1 or -1. The kernel between a baryon and a meson is proportional to the hadronic matrix element K =< P |T (L MMV L BBV |P > where |P > is a spin-isospin eigensate corresponding to the required pentaquark which is a molecular state composed by a charmed baryon B and a charmed meson M. Just as Dyson noticed [37], a factor (he was discussing interaction between two electrons in comparison with that between an electron and a positron) is crucial for the physics, since it can reverse a repulsive interaction into an attractive one. Generally, it is a classical phenomenon, but also applies to quantum field theory. This effect results in divergence of perturbation of QED, however, Dyson also pointed that it does not affect the phenomenological application as long as the physical input is set. Similarly, in our case, the free phase factor α can only be determined by experiments. Since its value would determine if the interaction (potential) is attractive or repulsive, so that is playing a key role for a definite spin-isospin composition for the pentaquark state. Namely the factor would determine if the spin-isospin structure is stable. Actually, we do not prior set the value of α, but let the experiment decide.
With the effective interactions we derive the kernel and obtain the corresponding B-S equation. In our calculation we include two different cases corresponding to α = 1 and -1. In the case I α takes 1 i.e. the kernel is calculated using the effective interaction presented in the appendix A directly while in the case II α takes -1 i.e. a minus sign is added into the kernel obtained in the case I.
With a reasonable parameter set, the B-S equation is solved. For a spin-isospin eigenstate, if the equation does not possess a solution, then we conclude that the corresponding bound state should not exist in the nature, by contraries, a solution of the B-S equation implies the bound state being formed. At the same time the B-S wave function is obtained and we are able to use the corresponding formula for calculating the rates of strong decay P c (4312) → proton + V (vector) which can be compared with the data. This paper is organized as follows: after this introduction we will derive the B-S equations related to possible bound states composed of a baryon and a meson and the formula for its strong decay. Then in section III we will solve the B-S equation numerically and present our results. Section IV is devoted to a brief summary.
FIG. 1: the bound states of Λ cD (a) and Σ cD (b)formed by exchanging light vector meson(s) .

II. THE BOUND STATES OF Λ cD AND Σ cD
Since the newly found pentaquarks P c (4312), P c (4380), P c (4440) and P c (4457) are all hadrons containing hidden charms(or hidden bottoms), so we will focus on the molecular structures composed of one charmed (bottomed) baryon and an anti-charmed(anti-bottomed) meson. Concretely, in this paper we will study Λ cD , Σ cD , Λ b B and Σ b B systems whose spinparity is 1 2 − i.e. the spatial wave function is in S−wave. In this section as an example we only formulate the corresponding quantities for Λ cD and Σ cD systems. These formulas can be equally applied to Λ b B and Σ b B systems.
A. The isospin states of Λ cD and Σ cD The isospin structure of the possible bound state of Λ cD is We will use P ′ where B represents Λ c or Σ c and M denotes D. In the effective theory a meson and a baryon can interact by exchanging mesons. The Feynman diagram at the leading order are depicted in Fig. 1. The relative and total momenta of the bound state in the equations are defined as where p and q are the relative momenta at the two sides of the effective vertex, p 1 (q 1 ) and p 2 (q 2 ) are those momenta of the constituents, P is the total momentum of the bound state, k is the momentum of the exchanged meson, η i = m i /(m 1 + m 2 ) and m i (i = 1, 2) is the mass of the i-th constituent meson. The bound state composed of a baryon and a meson can be written as The B-S wave function is a Fourier transformation of that in momentum space In the so-called ladder approximation the corresponding B-S equation was deduced in earlier references as where S B (p 1 ) is the propagator of the baryon (Λ c or Σ c ), S M (p 2 ) is that of the meson (D) and K(P, p, q) is the kernel which can be obtained by calculating the Feynman diagram in Fig.  1. For the later convenience the relative momentum p is decomposed into the longitudinal p l (≡ p · v) and transverse projection p µ t (≡ p µ − p l v µ )=(0, p T ) according to the momentum of the bound state P (v = P M ).
where M is the total energy of the bound state, E i = p l 2 + m 2 i and m 1 (m 2 ) is the mass of baryon (meson).
By the Feynman diagram the kernel K(P, p, q) is written as where m V is the mass of the exchanged meson, g M M V , g BBV and κ BBρ are the concerned coupling constants, C I,Iz is the isospin coefficient which is given in Appendix B. Apparently the contribution of the tensor term is much smaller than that of the first term, thus we can be ignored it in practical computations. Indeed, a numerical estimate verifies this allegation.
Since the constituents of the molecule (meson and baryon) are not point particles, a form factor at each effective vertex should be introduced. The form factor is suggested by many researchers is of the form: where Λ is a cutoff parameter. Since the form factor is not derived from a fundamental principle, the concerned cutoff parameter is neither determined theoretically, thus until now we know little about the cutoff parameter Λ. In some Refs. [38][39][40][41] the form factor is parameterized as λΛ QCD + m V with Λ QCD = 220 MeV and the dimensionless parameter λ being of order of unit. We will employ the expression Λ = λΛ QCD + m V in our calculation. The three-dimension B-S wave function is obtained after integrating over p l For the S−wave system, the spatial wave function can be easily derived [34][35][36] where f 1 (|p t |) and f 2 (|p t |) are the radial wave functions, u(v, s), v and s are the spinor, velocity and total spin of the pentaquark.
Multiplying dp l (2π) on the both sides of Eq. (9), integrating over p l and q l , substituting Eqs. (12), (15) into Eq.(9) and using the so-called covariant instantaneous approximation q l = p l we obtain Now let us fix the expressions of f 1 (|p T |) and f 2 (|p T |). Multiplyingū(v, s) on both sides of Eq.(17), we get an expression which only contains f 1 whereas multiplyingū(v, s)p t / the expression for f 2 is obtained, then taking a trace, the resultant formulaes are To extract f 1 (|p T |) and f 2 (|p T |) from the above equations, instead of the procedure adopted in earlier works, we multiplyū(v) from the right side of the equation and sum over the spin projections of u(v), then taking a trace of the modified equation, the job is done. The advantage of this procedure is to keep the equation of motion v /u(v, s) = u(v, s).

C. The normalization condition for the B-S wave function
The normalization condition for the B-S wave function of a bound state is [30,34] i where P 0 is the energy of the bound state and the spinor relation s u(v, s)ū(v, s) = v /+1 2 is used. I(P, p, q) is the reciprocal of the four-point propagator For the molecular sates composed of two mesons the second term in the normalization condition is several orders smaller than the first term [32,33], thus we have all reasons to believe that the rule also applies to the case where the molecule is composed of a baryon and a meson, consequently the term ∂ ∂P 0 K(P, p, q) can be ignored and then Let us define the transverse projections of the B-S wave function as follows: the normalization condition is Substituting the expressionχ P (p) (Eq. (9)) into the Eqs. (23) under the covariant instantaneous approximation one can obtain the expressions of α P (p) and β P (p), for example and α P (p) and β P (p) can be parameterized into with Substituting Eqs. (10), (11) and equation group (26) into Eq. (24) we obtain After the contour integration on p l and the azimuthal integration the normalization condition can be calculated numerically and the values of f 1 (|p T |) and f 2 (|p T |) are fixed at the same time.
D. the decay of P c → V+proton Now we investigate the strong decays of P c in terms of the framework formulated above.
FIG. 2: the decay of P c by exchanging mesons .
The amplitudes corresponding to the two diagrams in Fig. 2 are, where C I is the isospin coefficient of the transition, k = p − (η 2 q 1 − η 1 q 2 ), B denotes the charmed baryon in the molecular state: Σ c or Λ c , ǫ is the polarization vector of V and B ′ represents proton. We still take the approximation k 0 = 0 to carry out the calculation. The total amplitude is The factors g 1 , g 2 and g 3 can be extracted from the expressions of A 1 and A 2 . Then the partial width is expressed
With these parameters and the corresponding isospin factors a complete B-S equation (the coupled equations (19)) is established. These coupled equations are complicated integral equations, thus to numerically solve them, the standard way is to discretize them, namely we would convert them into algebraic equations. Concretely, we set a reasonable finite range for |p T | and |q T |, and let the variables take n ( n=129 in our calculation) discrete values Q 1 , Q 2 ,...Q n which distribute with equal gap from Q 1 =0.001 GeV to Q n =2 GeV . The gap between two adjacent values is ∆|p T |=(1.999/128) GeV. For clarity, we let n values of f 1 (|p T |) and n values of f 2 (|p T |) constitute a column matrix with 2n rows and the 2n elements f 1 (|q T |), f 1 (|q T |) construct another column matrix residing on the right side of the equation as shown below. The column matrix composed of f 1 (|p T |) and f 2 (|p T |) is associated with the right column matrix of f 1 (|q T |) and f 2 (|q T |) by a 2n × 2n matrix whose elements are the coefficients given in Eq. (19). The standard way to treat the equation is to let |p T | and |q T | take the same sequential values Q 1 , Q 2 ,...Q n for discretizing the integral equation. ... ...
As a matter of fact, it is a homogeneous linear equation group. If it possesses non-trivial solutions, the necessary and sufficient condition is the coefficient determination to be zero. In our case, it is |A(∆E, λ) − I| = 0. By calculating the determinant of |A(∆E, λ) − I| (I is the unit matrix) where A(∆E, λ) is a function of the binding energy ∆E = m 1 +m 2 −M and parameter λ. Our strategy is following: we arbitrarily vary ∆E within a possible range, by requiring |A(∆E, λ) − I| = 0, we obtain a corresponding λ. In Ref. [38] λ was fixed to be 3. In our earlier paper [41] we change the value of λ from 1 to 3 to explore possible dependence of the results on it, it seems that a value of λ within the range of 0 ∼ 4 is reasonable for forming a bound state of two hadrons. Consequently, if the obtained λ is much beyond the range, we would conclude that the resonance cannot exist.
To get the wavefunction T (f 1 (Q 1 ), f 1 (Q − 2)..., f 2 (Q 1 )...f 2 (Q 129 ), we adopt a special method. Namely, we suppose a matrix equation (A(∆E, λ) ij )(f (j)) = β(f (i)), it is an eigenequation, in terms of the standard software, we can find all the possible "eigenvalues" β, among them only β = 1 is the solution we expect, then the corresponding wavefunction is gained which just is the solution of the B-S equation.
For |A(∆E, λ) − I| = 0, inputting some binding energies, we would check whether we can obtain reasonable values for λ. If yes, we substitute the values of λ and the binding energy into the matrix equation to obtain the B-S wavefunctions.
With this strategy, let us investigate the molecular structure of Λ c andD as well as that of Σ c andD.
If the exchanging particles are limited to light vector meson, only ω and ρ can be exchanged between charmed baryons and D. Of course, exchanging two ρ mesons between Λ c (Σ c ) andD can also induce a potential, but it undergoes a loop suppression, therefore, we do not consider that contribution.

case I
As aforementioned, in the chiral lagrangian L = L MMV + αL BBV there is a free phase factor α which could be either +1 or -1. In the case I, we set α = +1.
As the first trial, let us study a simple compound, namely we explore the possible bound states of Λ c andD which is an I = 1 2 state, therefore only ω can be exchanged between Λ c andD. We find that there is no solution for the B-S equation, therefore we would conclude that the interaction induced by the single ω exchange is repulsive.
With the same procedure, we study a molecule composed of Σ c andD whose isospin could be either 1/2 or 3/2 and C 1 2 , 1 2 = 1. Since P C (4312) is observed in the J/ψp portal, it is confirmed to be a state of I = 1/2. In this case both ω and and ρ exchanges between the two ingredients are allowed. The isospin factor for the ρ exchange is −2, namely plays an opposite role to the ω exchange. We try to solve the equation |A(∆E, Λ) − I| = 0 for some chosen ∆E and find a solution for Σ cD with the quantum number I(J) = 1 2 ( 1 2 ) where the factor λ can span a large range.
The result indicates that although, the ω exchange contributes a repulsive interaction, for Σ cD molecule, the total interaction can be attractive due to the ρ exchange. Numerically, the obtained values of λ and corresponding ∆E for Σ cD system are presented in Tab. I. Our numerical computation also confirms that the tensor coupling in the L BBV has little effect on the results. For example setting ∆E = 8 MeV one can fix λ = 3.77 GeV and 3.88 GeV with and without the tensor contribution the obtained wave functions are very close to each other so we can ignore the tensor coupling in the vertex L BBV . Apparently when ∆E is very small the obtained λ is smaller than 4 GeV so Σ c andD should form a weak bound state. At present the pentaquark P c (4312) was experimentally observed in Λ b → J/ψpK portal, which is peaked at the invariant mass spectrum of J/ψp and has the invariant mass of about 4312 MeV. Apparently its isospin is 1 2 , thus the majority of authors [15-19, 22, 23] regarded this pentaquark as a bound state of Σ c andD.
Following our observation given above, for the state with I = 3 2 the isospin factor is 1 for exchanging either ω or ρ, therefore the total interaction is repulsive, it means that Σ c and D cannot form a bound state with I = 3 2 .

case II
Just for our theoretical interest, we take an alternative phase factor rather than that in the case I. Here we adopt the phase factor α to be −1. As the sign of the concerned coupling may change the whole physical picture, we would like to see what consequences can be induced. When one changes the situation from an electron-electron system to an electronpositron system, the interaction converts from repulsive to attractive. As Dyson noted, the well-known classical physical phenomenon can be manifested in quantum electrodynamics. We should consider that similar situation can also show up at QCD and related fields. Now we deliberately change the relative phase factor in the lagrangian, then the behavior of the interaction immediately changes. The bound states predicted in the case I would disappear while some other possible pentaquarks which were sentenced to death in case I would resurvive. Let us analyze the change.
In fact, the relative sign between the coupling L BBV and L PPV is not determined by a fundamental principle, so that we just keep it as a free phase to be fixed by experimental measurements. When the sign of L BBV or that of L PPV is changed, the sign of the transition For Σ c and D system with I = 1 2 the total interaction turns to be repulsive now because the isospin coefficient C 1 2 , 1 2 is −2 for exchanging ρ which means the interaction to be repulsive whereas exchanging ω it is +1 (attractive). Now, we try to solve the equation |A(∆E, λ) − I| = 0 for some ∆E and find that no λ satisfies the equation. For the state with I = 3 2 the isospin coefficient C 3 2 is 1 for exchanging ρ and ω, so the force is attractive. The obtained values of λ and corresponding ∆E for the Σ c and D system are presented in Tab. III. Apparently within a certain range of ∆E the obtained λ locates within a reasonable range, a bound state of Σ c and D system with I = 3 2 is formed. Since the conservation of isospin P c( 3 2 , 1 2 ) and P c( 3 2 , 3 2 ) can decay into ρ + p. Supposing the binding energy is 40 MeV, with the coupling constants g BB ′ D = 3.0, g BB ′ D * = 2.7, g DD * ρ = 2.8 GeV −1 [45] we obtain g 1 = 0.222 GeV, g 2 = 1.043, g 3 = 0.00390 GeV −1 and the decay width Γ[P c( 3 2 , 1 2 ) → ρ 0 p] = 16.1 MeV. The two different cases induce different physical consequences. According to the present knowledge, only experiment can make a judgement. The available data about P c (4312) seems to support the case I. However, if in new experiments a pentaquark is observed from the invariant mass spectrum of ρp the case II would be favored.

B. predictions about pentaquark P b
The isospin of the Λ b B + system is The isospin of the Σ b B system can be also | 1 2 , ± 1 2 | 3 2 , ± 1 2 and | 3 2 , ± 3 2 . Let us work on the isospin states and Since the phase convention in case I of last subsection can lead to results which are consistent with data, we employ that to explore the Λ b B, and Σ b B systems. Using the masses of Λ b , Σ b and B presented in Ref. [42] and other parameter are listed in previous sections, we solve those B-S equations. It is found that only the equation for the Σ b B system with I = 1 2 has a solution. The binding energies and corresponding λs are presented in Tab

IV. CONCLUSION AND DISCUSSION
Within the B-S framework we explore several bound states which are composed of a baryon and a meson. Their total spin and parity is 1 2 − i.e. the the orbital angular momentum L = 0 (S-wave). We try to solve the B-S equation for getting possible spatial wave functions for Λ cD , Σ cD , Λ b B and Σ b B systems. If the B-S equation for a supposed molecular structure has a stable solution, we would conclude that the concerned pentaquark could exist in the nature, oppositely, no-solution means the supposed pentaquark cannot appear as a resonance or the molecular state is not an appropriate structure. The criteria can apply for a check of the pentaquark states which have already been or will be experimentally observed. In this scenario, the two constituents interact by exchanging light vector mesons. For the Λ cD (Λ b B) system only ω is the exchanged mediate meson, while for the Σ cD system (Σ b B) both ω and ρ contribute. The chiral interaction determines if those molecular states can be formed.
For Within the B-S framework, we systematically investigate the molecular structure of pentaquarks. We pay a special attention to P c (4312) because it is experimentally well measured. From that study, we have accumulated valuable knowledge on probable molecular structure of pentaquarks which can be applied to the future research. Definitely, the discovery of pentaquarks opens a window for understanding the quark model established by Gell-Mann and several other predecessors. Deeper study on their structure and concerned effective interaction which binds the ingredients to form a molecule would greatly enrich our theoretical asset. So we will continue to do research along the line. useful discussions.

Appendix B: The isospin factors in the kernel
To gain the characteristic hadronic property of the pentaquark, one needs to project the bound states on the vacuum via the field operators B 1 , B 2 , M 1 and M 2 and where χ I P (x 1 , x 2 ) is the B-S wave function for the bound state with isospin I. The isospin coefficients C 22 2 ) for Λ c D bound state is 1, the isospin coefficients for Σ c D bound states are Then corresponding B-S equation was deduced in Ref. [36] as where K ij,lk (P, p, q) is still the kernel and its superscripts ij and lk denote the initial and finial components.