Transition form factors and angular distributions of the $\bm{\Lambda_b\to\Lambda(1520)(\to N\bar{K})\ell^+\ell^-}$ decay supported by baryon spectroscopy

We calculate the weak transition form factors of the $\Lambda_b\to\Lambda(1520)$ transition, and further calculate the angular distributions of the rare decays $\Lambda_b\to\Lambda(1520)(\to N\bar{K})\ell^{+}\ell^{-}$ ($N\bar{K}=\{pK^-,n\bar{K}^0\}$) with unpolarized $\Lambda_b$ and massive leptons. The form factors are calculated by the three-body light-front quark model with the support of numerical wave functions of $\Lambda_b$ and $\Lambda(1520)$ from solving the semirelativistic potential model associated with the Gaussian expansion method. By fitting the mass spectrum of the observed single bottom and charmed baryons, the parameters of the potential model are fixed, so this strategy can avoid the uncertainties arising from the choice of a simple harmonic oscillator wave function of the baryons. With more data accumulated in the LHCb experiment, our result can help for exploring the $\Lambda_b\to\Lambda(1520)\ell^+\ell^-$ decay and deepen our understanding on the $b\to s\ell^+\ell^-$ processes.


I. INTRODUCTION
The flavor-changing neutral-current (FCNC) processes, including the high-profile b → s + − process, can play a crucial role in indirect searches for physics beyond the Standard Model (SM).These transitions are forbidden at the tree level and can only operate through loop diagrams in the SM, and are therefore highly sensitive to potential new physics (NP) effects, such as the much-discussed R D ( * ) = B(B → D ( * ) τν τ )/B(B → D ( * ) e(µ)ν e(µ) ) [1][2][3][4].These processes thus provided a unique platform to deepen our understanding of both quantum chromodynamics (QCD) and the dynamics of weak processes, and to help hunt for NP signs.Therefore, the rare decays of b → s have attracted the attention of both theorists and experimentalists [5][6][7][8][9][10][11].
With the previous experiences on the decay to the ground state Λ, it is therefore worth to further testing the b → s + − transition in the baryon sector decaying to the excited hyperon with quantum number being J P = 3/2 − .The form factors of the weak transition were calculated by the quark model [21,22], LQCD [28,29], and the heavy quark expansion [30].The angular analysis was performed in Ref. [31] and Ref. [32] for massless and massive leptons, respectively.The authors of Ref. [33] studied the kinematic endpoint relations for Λ b → Λ(1520) + − decays and provided the corresponding angular distributions.Amhis et al. [34] used the dispersive techniques to provide a model-independent parameterization of the form factors of Λ b → Λ(1520) and further investigated the FCNC decay Λ b → Λ(1520) + − with the LQCD data.In addition, Xing et al. also studied the multibody decay Λ b → Λ * J (→ pK − )J/ψ(→ + − ) [35].In addition, Amhis et al. studied the angular distributions of Λ b → Λ(1520) + − and talked about the potential to identify NP effects [36].Obviously, the Λ b → Λ(1520) is less studied.Following this line, we further study the Λ b → Λ(1520)(→ N K) + − with the N K = {pK − , n K0 } process and investigate the corresponding angular observables.
From a theoretical point of view, apart from the consideration of new operators beyond the SM, the calculation of the weak transition form factors is a key issue.In addition, how to solve the three-body system for the Λ b baryon and Λ * hyperon involved is also a challenge.In previous work on baryon weak decays [37][38][39][40][41], the quark-diquark scheme has been widely adopted as an approximate treatment.Meanwhile, the spatial wave functions of hadrons are often approximated as simple harmonic oscillator (SHO) wave functions [37][38][39][40][41][42][43], which makes the results dependent on the relevant parameters.To avoid the correlative uncertainties of the above approximations, in this work we calculate the Λ b → Λ * form factors by the three-body light-front quark model.Moreover, in the realistic calculation, we take the numerical spatial wave functions as input, where the semirelativistic potential model combined with the Gaussian expansion method (GEM) [44][45][46][47] is adopted.By fitting the mass spectrum of the observed single bottom and charmed baryons, the parameters of the semirelativistic potential model can be fixed.Compared with the SHO wave function approximation, our strategy can avoid the uncertainties arising from the selection of the spatial wave functions of the baryons.
The structure of this paper is as follows.After the Introduction, we derive the helicity amplitudes of Λ b → Λ * (→ N K) + − (N K = {pK − , n K0 }) processes and define some angular observables with unpolarized Λ b baryons and massive leptons in Sec.II.The formulas for the weak transition form factors are derived in the three-body light-front quark model in Sec.III.And then, to obtain the spatial wave functions of the involved baryons, the applied semirelativistic potential model and GEM are briefly introduced in Sec.IV.In Sec.V, we present our numerical results, including both the relevant form factors and the physical observables in Λ b → Λ * (→ N K) + − decays.Finally, this paper ends with a short summary in Sec.VI.

II. THE ANGULAR DISTRIBUTION OF
In this paper, we use a model-independent approach with the effective Hamiltonian [48,49] to study the b → s + − process, where G F = 1.16637 × 10 −5 GeV −2 is the Fermi coupling constant and |V tb V * ts | = 0.04088 [12] is the product of the Cabibbo-Kobayashi-Maskawa matrix elements.Furthermore, the Wilson coefficients C i (µ) describe the short-distance physics, while the four fermion operators O i (µ) describe the long-distance physics, where O 1,2 are the current-current operators, O 3−6 are the QCD penguin operators, O 7,8 denote the electromagnetic and chromomagnetic penguin operators respectively, and O 9,10 stand for the semileptonic operators.
Since the quarks are confined in hadron, the weak transition matrix element cannot be calculated in the framework of perturbative QCD.They are conventionally parametrized in terms of eight (axial-)vector and six (pseudo-)tensor type dimensionless form factors [21,25,31,32,[59][60][61]. In this work, we adopt the helicity-based form as [31,32] (2.10)This form defined above is convenient for calculating the corresponding helicity amplitudes, where q 2 is the transferred momentum square and A. The helicity amplitudes of the To calculate the Λ b → Λ * + − process, we define the corresponding helicity amplitudes of the Λ b (s Λ b ) → Λ * (s Λ * ) transition as where µ (λ W = t, ±, 0) are the polarization vectors of the virtual gauge boson in the Λ b rest frame, s Λ b and s Λ * are the polarizations of Λ b and Λ * , respectively.For the vector current, the complete helicity amplitudes ) Analogous expressions for the helicity amplitudes of the axial-vector, tensor, and pseudotensor currents are written as ) respectively.Using the kinematic conventions presented in Appendix B 1, the nonzero terms for the above helicity amplitudes of the vector, axial-vector, tensor, and pseudotensor currents are [31] H respectively.Similarly, we define the leptonic helicity amplitudes as where ¯ µ (λ W = t, ±, 0) are the polarization vectors of the virtual gauge boson in the dilepton rest frame.Using the kinematic conventions presented in Appendix B 2, the nonzero terms are obtained as [32] L V (±1/2, ±1/2, 0) = ±2m cos θ , L V (±1/2, ∓1/2, 0) = − q 2 sin θ , where B. The helicity amplitudes of the Λ * → N K decay We use the effective Lagrangian approach to describe the strong decay process Λ * → N K.The concerned effective Lagrangian is where g Λ * KN is the coupling constant.So the decay amplitude for the Λ * → N K process can be expressed as where k α 2 is the four momentum of the K meson, u Λ * ,α is the Rarita-Schwinger spinor describing the hyperon Λ * , while u N is the Dirac spinor describing the nucleon.The interference terms between matrix elements with different Λ * polarizations can be written as where r ± = (m Λ * ± m N ) 2 − m 2 K , and then the decay width of Λ * → N K can be obtained by where the factor 4 comes from averaging over the polarization of Λ * .With respect to the forms of Rarita-Schwinger spinors and Dirac spinors presented in Appendix B 3, we obtain [31,32] ) with rows and columns corresponding to the polarizations of s a Λ * , s b Λ * = −3/2, −1/2, 1/2, 3/2 from top to bottom and from left to right.We emphasize that Γ Λ * → N K = B Λ * × Γ Λ * , where B Λ * ≡ B Λ * → N K is the corresponding branching ratio and Γ Λ * is the inclusive decay width of the Λ * hyperon.

C. The total amplitudes of
where 4π , and the factor 1/2 comes from the definition of P L(R) .Besides, the relation  (2.32), and the expression of the differential width by considering the narrow-width approximation shown in Eq. (A.7), the differential decay width can be obtained.
As analyzed in Refs.[31,32], the angular distribution for the four-body decay Λ b → Λ * (→ N K) + − can be reduced as (2.34) The complete expressions for the series angular coefficients L i can be found in Appendix G of Ref. [32].

D. Physical observable in the four-body process
By integrating over the angles in the regions θ ∈ [0, π], θ Λ ∈ [0, π], and φ ∈ [0, 2π], the relevant physical observables are listed as follows: (2.35) • The lepton-side forward-backward symmetry A FB is (2.36) • The hadron-side forward-backward symmetry A Λ * FB and the lepton-hadron-side forward-backward symmetry A Λ * FB will undoubtedly disappear, since the decay Λ * → N K is a strong process [31].This can be tested in future experiments.
• The transverse and longitudinal polarization fractions of the dilepton system are defined as [31] respectively.
• We also define the normalized angular observables as

III. THE LIGHT-FRONT QUARK MODEL FOR CALCULATING WEAK TRANSITION FORM FACTORS
In this section, we will calculate the form factors involved in the three-body light-front quark model.First, the vertex function of a baryon B with spin J and momentum P is [42,43,[62][63][64][65][66][67]] where the C αβγ and F q 1 q 2 q 3 are the color and flavor factors, respectively, and λ i and p i (i=1,2,3) are the helicities and lightfront momenta of the on-mass-shell quarks, respectively, defined as To describe the motion of the constituents, the intrinsic variables (x i , k i ) (i = 1, 2, 3) are as follows ) where x i represents the light-front momentum fractions bounded by 0 < x i < 1.
The vertex function should be normalized by As proposed by Refs.[68][69][70], the spin-spatial wave functions for the Λ-type baryon with J P = 1/2 + and 3/2 − are written as respectively, where are the corresponding normalized factors determined by Eq. (3.4).
is the momentum of the λ mode.
In the context of the three-body light-front quark model, the general expression for the weak transition matrix element is written as where the Γ µ i = γ µ , γ µ γ 5 , iσ µν q ν , iσ µν q ν γ 5 .P = p 1 + p 2 + p 3 and P = p 1 + p 2 + p 3 are the light-front momenta for initial and final baryons, respectively, considering p 1 ≡ p 1 and p 2 ≡ p 2 in the spectator scheme.The following kinematics of the constituent quarks as have been used to simplify the above matrix element.
In addition, the ψ b and ψ s are the spatial wave functions of Λ b and Λ * , respectively.Their forms are written as in this paper, where k i = ( k i⊥ , k iz ) with By the way, φ ρ(λ) is the spatial wave function of ρ(λ) mode.The factor N ψ = (4π 3/2 ) 2 for the ground state Λ b and the factor N ψ = (4π 3/2 ) 2 / √ 3 for the P-wave state Λ(1520) are determined by Eq. (3.5).The additional factor /1 √ 3 for the P-wave state comes from different angular components of the spatial wave functions described by the spherical harmonic functions compared to the ground state.
In previous work [40][41][42], the spatial wave function of the baryon is usually adopted as a SHO form with an oscillator parameter β, which causes the β dependence of the form factors.To avoid this uncertainty, we adopt the numerical spatial wave function obtained by solving the three-body Schrödinger equation with the semirelativistic potential model.The detailed discussions are presented in Sec.IV.
The next content discusses how to extract the form factors in the light-front quark model.Here, we consider the q + = 0 and q ⊥ 0 condition.In order to extract the four form factors in vector current, one can multiply the ū(P, J z )Γ V,µβ i u β (P , J z ) on both sides of Eq. (3.8) with specific setting Γ µ i = γ µ and sum over the polarizations of the initial and final states.And then the left side can be replaced by Eq. (2.7), and the right side can be calculated out by carrying out the traces and then the integration.The Lorentz structures are Γ V,µβ i = g βµ , P β γ µ , P β P µ , P β P µ .The complete expressions for the form factors in vector current are obtained by solving where Analogously, the form factors in the axial-vector, tensor, and pseudotensor currents can be obtained by using the structures ū(P, J z )Γ A,µβ i u β (P , J z ), ū(P, J z )Γ T,µβ i u β (P , J z ) and ū(P, J z )Γ T 5,µβ i u β (P , J z ) with setting Γ µ i = γ µ γ 5 , iσ µν q ν and iσ µν q ν γ 5 in Eq. (3.8), respectively.The Lorentz structures are Γ A,µβ i = g βµ γ 5 , P β γ µ γ 5 , P β P µ γ 5 , P β P µ γ 5 , Γ T,µβ i = g βµ , P β γ µ , P β P µ and Γ T 5,µβ i = g βµ γ 5 , P β γ µ γ 5 , P β P µ γ 5 .The complete expressions of the form factors can be obtained by solving ) Tr[(G Λ * ) βα .[formfactors in Eq. (2.10)].(/P + M 0 ).Γ T 5,β (1,2,3),µ ] = This approach has been used to evaluate the form factors of triple heavy baryon transitions from 3/2 → 1/2 cases [71,72].

IV. THE SEMIRELATIVISTIC POTENTIAL MODEL FOR CALCULATING BARYON WAVE FUNCTION
In this section, we will derive the wave function using the GEM with semirelativistic potential model.In general, to obtain the wave function and mass of a baryon, we need to solve the three-body Schrödinger equation, where H is the Hamiltonian and E is the corresponding eigenvalue.It can be solved by using the Rayleigh-Ritz variational principle.Unlike a meson system, a baryon in the traditional quark model is a typical three-body system.In our calculation, the semirelativistic potentials used in Refs.[73][74][75] are applied.
The Hamiltonian in question [73,74] includes the kinetic energy K, the spin-independent linear confinement piece S , the Coulomb-like potential G, and the higher-order terms containing the scalar-type spin-orbit interaction V so(s) , the vector-type spin-orbit interaction V so(v) , the tensor potential V tens , and the spin-dependent contact potential V con .The concrete expressions are given as [73-75] for the spin-independent terms with and the F i • F j = −2/3 for the quark-quark interaction, and .9) for the spin-dependent terms, where m i is the mass of the ith constituent quark, and S i is the corresponding spin operator.Next, a general transformation based on the center of mass of the interacting quarks and the momentum is set up to compensate for the loss of relativistic effect in the nonrelativistic limit [75,76] where i is the energy of the ith constituent quark, the subscript k is used to distinguish the contact, tensor, vector spin-orbit, and scalar spin-orbit terms, and the k is used to denote the relevant modification parameters, which are collected in Table I.
The total wave function of the baryon is composed of color, spin, spatial and flavor wave functions, i.e., where χ color = (rgb − rbg + gbr − grb + brg − bgr)/ √ 6 is the universal color wave function for the baryon.For the affected Λ b and Λ * , their flavor wave functions are chosen as ψ flavor = (ud − du)Q/ √ 2 where Q = b or s.Also, S is the total spin and L is the total orbital angular momentum.ψ spatial L,M L is the spatial wave function, which is composed of the ρ mode and λ mode where the subscripts l ρ l ρ l ρ and l λ l λ l λ represent the orbital angular momentum quanta for the ρ and λ modes, respectively, and the internal Jacobi coordinates are chosen to be  In this calculation, the Gaussian basis [44][45][46], is used to expand the spatial wave functions φ l ρ l ρ l ρ ,ml ρ ml ρ ml ρ and φ l λ l λ l λ ,ml λ ml λ ml λ , where the freedom parameter n max should be chosen from positive integers, and then the Gaussian size parameter ν n can be settled as [77] ν where In our calculation the values of ρ min and ρ max are chosen to be 0.2 fm and 2.0 fm, respectively, and the parameter n ρ max = 6.For the λ mode, we also use the same Gaussian-sized parameters.
With the above preparations, we can calculate the spatial wave functions of Λ b and Λ(1520).Their masses and radial components of spatial wave functions are shown in Table III.It is obvious that the calculated mass of Λ b is consistent with the Particle Data Group (PDG) [58] averaged value, while that of Λ(1520) is about 40 MeV higher than the PDG value.

A. The weak transition form factors
With the input of the numerical wave functions of Λ b and Λ * , and the complete expressions of the form factors obtained by solving Eq. (3.12) and Eqs.(3.17)-(3.19),we present our numerical results of the form factors of the Λ b → Λ * transition.Since the form factors calculated in the light-front quark model are valid in the spacelike region (q 2 < 0), we have to extrapolate them to the timelike region (q 2 > 0).Before we do the extrapolation, we need to talk about some constraints on the form factors at the q 2 = q 2 max point.To make sure that the helicity amplitudes in Eqs.(2.22)-(2.25)have no singularities and are nonzero values in the q 2 → q 2 max limit, we get the constraints in this limit as The form factors that show less singular behavior in the q 2 → q 2 max limit are also reasonable.This would lead the helicity amplitudes to be zero in q 2 = q 2 max .The above features have been discussed in Ref. [31].However, the above requirement is not strict enough, since it gives a broad limit.This will make nonunique extrapolations of the form factors.
Since the LQCD calculation of Λ b → Λ(1520) form factors has been done in Refs.[28,29], and their results work well in the kinematic region near q 2 max , we will talk about the characters of the form factors in the LQCD.The LQCD calculation has been completed in Refs.[28,29].The authors obtained finite values of the form factors of Λ b → Λ(1520) in their definition (i.e., f 0,+,⊥,⊥ , g 0,+,⊥,⊥ , h +,⊥,⊥ , and h+,⊥,⊥ ) in the q 2 = q 2 max limit.Their definition of the form factors can be converted to ours by [28] This shows that in the q 2 = q 2 max limit, the LQCD results [29] show These characters fulfill the requirements.Also we have , where a g 0 1 is a nonzero value.According to Eq. (5.2), the , and this implies, in the q 2 = q 2 max limit, that f A t = O(1).This also satisfies the requirement.
In order to align with the LQCD results, we take the following strategy for the analytical continuation: 1. To do the extrapolations of the form factors f V t,g , f A t,0,⊥,g , f T g , and f T 5 0,⊥,g , the z-series form [34,[79][80][81] f (q 2 ) = 1 is adopted where a f 0 , a f 1 , and a f 2 are free parameters needed to fit in the spacelike region, and (5.5) The parameter t 0 is set to The m f pole is collected in Table IV.

2.
For the form factors f V 0,⊥ and f T 0,⊥ , we use the form as where TABLE IV: The pole masses of the form factors in Eq. (5.4), where the 0 − , 1 − , and 1 + masses are taken from the PDG [58], while the 0 + mass is taken from the LQCD calculation [82].
To determine the parameters a f 0 , a f 1 , and a f 2 , we numerically calculate 24 points for each form factor by Eqs.(3.12)-(3.19)from q 2 = −q 2 max to q 2 = −0.01GeV 2 in the spacelike region, and then fit them using Eq. ( 5.4) and Eq.(5.7) with the MINUIT program.The extrapolated parameters for the form factors of Λ b → Λ * are collected in Table V.The q 2 dependence of the concerned form factors is shown in Fig. 2.
However, as discussed earlier, the less singular behaviors of f V 0,⊥ and f T 0,⊥ in the small-recoil limit are also not forbidden.Therefore, in this work, we also use the formula in Eq. ( 5.4) to perform the extrapolation of the form factors f V 0,⊥ and f T 0,⊥ again.This extrapolation scheme gives different results at the q 2 = q 2 max point for the four form factors, but has no effect on other form factors compared with the previous scheme.For clarification, we compare our results of the form factors in the q 2 = q 2 max point with the two different extrapolation schemes in Table VI.Finally, it should be emphasized that there is no established procedure for the extrapolation.The experimental measurement of Λ b → Λ(1520) + − by the LHCb Collaboration can test the different extrapolation schemes.
As shown in Eqs.(2.7)-(2.10),we need eight (axial-)vector and six (pseudo-)tensor form factors to describe the matrix elements in question.The number can apparently be reduced in the heavy quark limit m b → ∞.We speak separately of two different kinematic situations, i.e., the outgoing Λ * acts softly (the low-recoil limit) and acts energetically (the largerecoil limit).Accordingly, two effective theories, namely heavy quark effective theory (HQET) and soft-collinear effective theory (SCET), are developed to exploit the behaviors of the form factors.
In the low-recoil limit, where HQET is valid [83][84][85][86], the weak transition matrix element can be re-expressed by two Isgur-Wise functions as [30,32,86] (5.8) Here, Γ is an arbitrary Dirac structure, and , where υ = p/m Λ b and υ = p /m Λ * represent the four velocities of the bottom baryon and hyperon, respectively.The eight form factors are derived as two independent form factors ζ 1 (ω) and ζ 2 (ω).In the low-recoil limit this gives q 2 → q 2 max ≡ (m Λ b − m Λ * ) 2 (or ω → 1).With slightly different definitions of the form factors in Refs.[25,60,87],we have [31] max ) f T g (q 2 max ) f T 5 g (q 2 max ) 0, (5.9) while in the large-recoil limit where SCET is valid, we have [60,87,88] where ζ(ω) is the only remaining form factor.This gives, in the large-recoil limit, and four f g form factors will disappear.From Fig. 2, we can see that apart from the f T (T 5) g (q 2 ), which deviates from the predictions, the remaining calculated form factors are consistent with the requirements of HQET and SCET.
In addition, Bordone has completed the heavy quark expansion (HQE) calculation of the Λ b → Λ * form factors beyond the leading order [30].At the zero-recoil limit, the HQE predicts the ratios of the form factors, which are independent of the Isgur-Wise functions, as [30] (5.12) Note that the form factor base used in Ref. [30] is different from ours.By using the conversions collected in Appendix B of Ref. [30] and Eq. ( 5.2), we can get our results of these FIG. 2: The q 2 dependence of the form factors of the vector, axial-vector, tensor, and pseudotensor type currents of the Λ b → Λ * transition, where the red solid curves are central values, and the light red bands are the corresponding errors.The units of the form factors are neglected here.
ratios as: (5.13) Our results are very different from those of the HQE.
In addition, we also compare our results of the form factors with the NRQM [22] and the LQCD [29] at q 2 = 0 and q 2 = q 2 max endpoints in Table VI.Until now, less work has been done on the Λ b → Λ(1520) transition, so more theoretical work is needed to validate these form factors.

B. The branching ratio and angular observables
With the above preparations, we will present our numerical results.The baryon and lepton masses used in our calculation are taken from the PDG [58], as well as τ Λ b = 1.470 ps.We also use B Λ * ≡ B Λ * → N K = 45% [58].To compare with the experimental data, we examine a number of angular observables, including the CP-averaged normal-ized angular coefficients, the differential branching ratios, the lepton's forward-backward asymmetry (A FB ), and the transverse (F T ) and longitudinal (F L ) polarization fractions of the dilepton system.
First, we examine the CP-averaged normalized angular distributions where the angular distributions L i and the differential decay width dΓ/dq 2 are defined in Eqs.(2.34) and (2.35), respectively.For the CP-conjugated mode, the corresponding expression for the angular decay distribution should be written as ) where Li (q 2 ) can be obtained by doing the full conjugation for all weak phases in L i (q 2 ).We should also do the substitutions as where the minus sign is a result from the operations of θ → θ − π and φ → −φ.The differential decay width of the conjugated mode is (5.17) In Fig. 3, we present our results for the q 2 dependent normalized angular coefficients.Since the e channel shows similar behavior to the µ channel, we only present the results of TABLE VI: Theoretical predictions for the form factors of Λ b → Λ(1520) at the endpoints of q 2 = 0 and q 2 = q 2 max using different approaches.This work NRQM [22] LQCD [29] This work [a]  This work [b] NRQM [22] LQCD [29] f V t (0) 0.051 ± 0.007 0.0029 −0.1523 ± 0.0530 f V t (q 2 max ) 0.244 ± 0.008 0.244 ± 0.008 ∞ 0.1726 ± 0.0138 f V 0 (0) 0.051 ± 0.007 0.0029 0.0714 ± 0.0078 a These results, listed in the fourth column, are obtained by the first extrapolation scheme.Here, the f V t,g , f A t,0,⊥,g , f T g and f T 5 0,⊥,g are extrapolated by Eq. (5.4), while the f V 0,⊥ and f T 0,⊥ are extrapolated by Eq. (5.7).b These results, shown in the fifth column, are from the second extrapolation scheme.Here, all form factors are extrapolated by Eq. (5.4).
the µ and the τ channels here.These angular distributions are important physical observables, and can be checked by future experiments.
We further evaluate the differential branching ratios using Eq.(2.35).The q 2 dependence of the differential branching ratios is shown in Fig. 5, where the orange solid curve, the blue dashed curve, and the purple dot-dashed curve are our results for the e, µ, and τ channels, respectively.The gray zones in the regions of the dilepton mass squared 8.0 < q 2 < 11.0 GeV 2 and 12.5 < q 2 < 15.0 GeV 2 show the contributions from the charmonium resonances J/ψ and ψ(2S ), respectively.
Recently, the LHCb collaboration measured the "nonresonant" contributions, which are different from the "resonant" contribution from Λ This indicates that the contribution from Λ(1520) is significant.We find from the PDG [58] that Λ(1600), Λ(1670), and other hyperons can also decay to the N K final state.Their contributions need to be carefully studied.Further studies with more excited Λ hyperons will make a difference to the Λ 0 b → pK − + − decays.
In addition, the q 2 dependence of the lepton-side forwardbackward asymmetry (A FB ), and the transverse (F T ) and longitudinal (F L ) polarization fractions of the dilepton system are presented in Figs. 6, 7 and 8, respectively, where we also show the contributions from the charmonium resonances J/ψ and ψ(2S ) with gray zones.The averaged values of these angular observables for the e and µ channels defined in Eq. (2.39) in the region of 0.1 < q 2 < 6.0 GeV 2 are presented in Table VII.The angular distributions provide a rich set of physical observables to study the weak interaction and the structure of Λ(1520), and are also important to study the NP effects beyond the SM [24,31,32,36,51], so we call for the ongoing LHCb experiment to measure them.

TABLE VII:
The predictions for the averaged lepton-side forwardbackward asymmetry A FB , the averaged transverse polarization fraction F T , and the averaged longitudinal polarization fraction F L in the region of 0.1 < q 2 < 6.0 GeV 2 .

Channels
FIG. 3: The q 2 dependence of the normalized angular coefficients S 1c , S 1cc , S 1ss , S 2c , S 2cc , S 2ss , S 3ss , S 5s , and S 5sc .Here, the red curve and the blue curve are our results of the µ and τ channels, respectively, and the concomitant shadows are corresponding errors.

VI. SUMMARY
With the accumulation of experimental data in the LHCb Collaboration, the experimental exploration of rare decays b → s + − ( =e, µ, τ) in the baryon sector, especially the P-wave final state Λ b → Λ(1520) + − , will attract more attention.Given this opportunity, in this work we focus on the quasi-four-body decay Λ b → Λ(1520)(→ N K) + − , where the angular coefficients, the differential branching ratio, and several angular observables, including the lepton-side forward-backward asymmetry (A FB ), and the transverse and longitudinal polarization fractions (F T (L) ) are investigated.
To describe the weak process, we have worked in the helicity formula, where the relevant weak transition form factors are obtained through the three-body light-front quark model.Our main advantage is the improved treatment of the spatial wave functions of the involved baryons, where a semirelativistic potential model is applied to solve the numerical spatial wave functions of the baryons assisted by the  GEM.Thus, we emphasize that our study of the rare decay Λ b → Λ(1520)(→ N K) + − is supported by the baryon spectroscopy.Our results of the form factors are comparable with the predictions of HQET and SCET, and also with the calculations by the LQCD approach.These form factors will be useful for the study of the corresponding weak decays.
Overall, we have systematically investigated the Λ b → Λ(1520)(→ N K) + − ( = e, µ, τ) processes in the framework of the three-body light-front quark model based on the Gaussian expansion method.We believe that the present work can serve as an essential step toward strong dynamics on the beauty baryon decays.We expect that under the considerable progress on the experimental side, the above predictions could be tested by future LHCb experiments.defined as the angle made by the − with the +z axis in the ( + − ) cms, and (iii) the angle φ between the two decay planes, respectively.

The decay width of the concerned decay
We also take into account the width of Λ * to modify its propagator, but treat it as narrow (Γ Λ * m Λ * ) state 2 .This 2 Checking the PDG [58], we notice that m Λ(1520) = 1519 MeV and Γ Λ(1520) = 16 MeV, indicating that it is reasonable to take the narrowwidth approximation.
gives [24], with the properties of the Dirac delta function Following the above discussion, we can finally obtain where λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2xz − 2yz is the kinematic triangle Källén function.

Appendix B: The kinematic conventions
In this paper, we assign the particle momenta and spin variables for the hadrons in the Λ b → Λ * (→ N K) + − process according to: as shown in Fig. 9.Here we have some relations like q µ = q µ 1 + q µ 2 , k µ = k µ 1 + k µ 2 , and p µ = k µ + q µ .In the following, we will introduce some kinematic conventions that are useful for the calculation of the involved helicity amplitudes.

TABLE I :
The parameters used in the semirelativistic potential model.The quark masses are also chosen to be m u = 220 MeV, m d = 220 MeV, m s = 419 MeV, m c = 1628 MeV, and m b = 4977 MeV [75, 76].

. 14 )
As shown in Fig.1, the Λ b (Λ * ) is considered as a bound state with the u and d quarks bound to form the ρ mode and then bounded to the b (or s) quark to form the λ mode.

FIG. 1 :
FIG. 1: The definition of the internal Jacobi coordinates ρ and λ, where we use green spheres to represent the u and d quarks and yellow spheres to represent the b (or s) quark.

FIG. 5 :
FIG.5:The q 2 dependence of the differential branching ratios for Λ b → Λ * (→ N K) + − ( = e (left panel), µ (center panel), τ (right panel)), where the red, the blue, and the purple curves are our results from the e, µ, and τ channels, respectively, and the concomitant shadows are the corresponding errors.

FIG. 6 :
FIG.6:The q 2 dependence of the lepton-side forward-backward asymmetry parameter (A FB ) for Λ b → Λ * (→ N K) + − ( = e (left panel), µ (center panel), τ (right panel)), where the red, blue, and purple curves are our results from the e, µ, and τ channels, respectively, and the shadows are the corresponding errors.

FIG. 7 :
FIG.7:The q 2 dependence of the transverse polarization fractions (F T ) for Λ b → Λ * (→ N K) + − ( = e (left panel), µ (center panel), τ (right panel)), where the red, blue, and purple curves are our results of the e, µ, and τ channels, respectively, and the shadows are the corresponding errors.

FIG. 8 :
FIG.8:The q 2 dependence of the longitudinal polarization fractions (F L ) for Λ b → Λ * (→ N K) + − ( = e (left panel), µ (center panel), τ (right panel)), where the red, blue, and purple curves are our results of the e, µ, and τ channels, respectively, and the shadows are the corresponding errors.

TABLE II :
Experimentally observed masses of charmed and bottom baryons used to fit the potential model parameters, where only the central values are given.
[58]E III:The comparison of the masses of Λ b and Λ(1520) from our calculation and the PDG[58]data, and the radial components of the spatial wave functions of the concerned Λ b and Λ(1520) from the semirelativistic potential model and GEM.The Gaussian bases (n ρ , n λ ) listed in the fourth column are arranged as [(1, 1), (1, 2), • • • , (1, n λmax ), (

TABLE V :
The form factors of the Λ b → Λ * transition in the standard light-front quark model.