Nonleptonic two-body weak decays of $\Lambda_b$ in modified MIT bag model

We study the nonleptonic two-body weak decays of $\Lambda_b$ by modifying the MIT bag model without introducing new parameters to construct the momentum eigenstates of the baryons. We find that the branching ratios of $\Lambda_b^0 \to \Lambda_c^+ \pi^-$, $\Lambda_c^+ K^-$, $p \pi^-$ and $pK^-$ are $4.5\times 10^{-3}$, $3.4\times 10^{-4}$\, $5.0\times 10^{-6}$ and $6.0\times 10^{-6}$, which are all well consistent with the current experimental data, respectively. We also explore P and CP asymmetries for the decays of $\Lambda_b^0 \to p (\pi^-,K^-)$. In particular, we obtain that the direct CP-violating rate asymmetries in $\Lambda_b^0 \to p \pi^-$ and $\Lambda_b^0 \to p K^-$ are $-4.4\%$ and $6.7\%$, in comparison with $(-2.5\pm2.9)\%$ and $(-2.5\pm2.2)\%$ from the Particle Data Group in 2020, respectively.


I. INTRODUCTION
There have been many interesting measurements in the decays of b-baryons by the LHCb Collaboration, including those for the charmful modes of Λ 0 b → Λ + c M [1] and charmless ones of Λ 0 b → pM [2] (M = π − , K − ) as well as the discoveries of the hidden-charm pentaquarks in Λ 0 b → J/ψpM [3,4] and double-charm baryon state of Ξ ++ cc via Ξ ++ cc → Λ + c K − π + π + [5]. In particular, based on the new experimental data from LHCb [6], the Particle Data Group (PDG) of 2020 [7] has updated the average values for the direct CPviolating rate asymmetries (A CP s) in Λ 0 b → pπ − and Λ 0 b → pK − to be respectively. In the standard model, these two decay amplitudes contain V ub and loopinduced penguin operators [8], which could provide weak and strong phases, respectively, resulting in non-vanishing direct CP-violating rate asymmetries (A CP s). Previous theoretical studies of A CP s in Eq. (1) have been performed in various QCD models, such as the perturbative QCD (PQCD) method [9], generalized factorization approach [10] and lightfront quark model (LFQM) [11]. These calculations in the literature are important to check if the results in the standard model are consistent with the experimental measurements. In this work, we would like also to explore the CP-violating asymmetries with the MIT bag model.
It is known that to calculate the decay processes of the baryons, we need to know the details of the baryon wave functions. In the MIT bag model, the quarks in the baryon are confined in a static bag. The model enjoys various successes in its simple structure to explain the mass spectra and magnetic dipole moments of the baryons [12][13][14][15]. However, the construction of the baryon wave functions in the MIT bag model is localized in a fixed space. As a result, the center motions of the baryons are not moved. In other words, the baryon wave functions are not in the momentum eigenstates. Such a defect makes the model questionable when one discusses the decay processes, involving essentially the momentum eigenstates. Nonetheless, many calculations in the bag model have been done by ignoring this problem [16][17][18][19][20][21][22]. In this work, we will modify the MIT bag model to construct the momentum eigenstates for the baryons to study the decays without introducing extra parameters.
On the other hand, it is known that one can examine the heavy quark symmetry in the charmful decays of Λ 0 b → Λ + c M. Particularly, the baryonic matrix element is related to the Isgur Wise function of ξ(ω) [23], given by where ω = v 1 · v 2 with v 1,2 the velocities of the two baryons and u B is the 4 components Dirac spinor for the corresponding baryon of B. In the heavy quark limit, in which the relative velocity between the two baryons vanishes, we have that ξ(ω) We will check if the heavy quark symmetry is valid in our modified MIT bag model This paper is organized as follows. In Sec. II, we introduce the decay amplitudes and parameters. In Sec. III, we constitute the baryon wave functions in the modified MIT bag model, where we sum over the localized baryon wave functions with different centers. In Sec. IV, we compute the form factors accordingly. In Sec. V, we present our numerical results and compare them with the experimental data as well as the theoretical evaluations in the literature. We conclude our study in Sec. VI.

II. DECAY AMPLITUDES AND PARAMETERS
We start with the two-body decays of B i → B q M, where B i(q) is the initial (final) baryon with spin 1/2 and M represents the pseudoscalar meson. In this study, we concentrate on the corresponding decays with B i = Λ 0 b , B q = (Λ + c , p), and M = (π − , K − ). The spin-dependent amplitude for B i → B q M can be written as where A and B are the s-wave and p-wave amplitudes, corresponding to the parity violating and conserving ones, and u B i,q are the baryon Dirac spinors, respectively. In general, A and B are not relatively real, resulting in CP violating effects. The decay branching ratio and forward-backward asymmetry for the initial baryon in the rest frame are given as [24] Γ(B i → B q M) = | p q | 8π where m M is the meson mass and p q represents the 3 momentum of the daughter baryon B q , while α P describes the decay asymmetry between the helicity states of B q , defined by withp q (ŝ q ) the unit vector in the p q ( s q ) direction of B q , provided with the initial baryon B i unpolarized. We can also define the decay asymmetries for the antiparticles, given by The nonzero mean value between α P and α P is a consequence of CP violation, which is defined by [25,26] On the other hand, the direct CP-violating rate asymmetries of the decays are given by In the present work, to relate α P , A CP and A CP directly to the weak interactions, we ignored the final states interactions in our calculations. As a result, the complex phases of A and B solely come from the Wilson coefficients and the CKM elements. We note that possible sizable CP-violating effects in Λ b → p(π − , K − ) could be induced due to the weak phase from the CKM matrix element of V ub .

III. BARYON WAVE FUNCTIONS
In the MIT bag model, the quarks in the baryon are constrained in a certain bag with radius R. Inside the bag, each quark obeys the free Dirac equation, given by due to the mean field approximation of the gluon field, where m q is the current quark mass.
The boundary condition on the surface of the bag is given as where n i is the unit vector toward the surface.
In this work, we only consider the ground states of the baryons with their angular momenta to be J = 1/2, where the spatial parts of the quark wave functions satisfy the spherical symmetry. Consequently, by inserting the boundary condition of Eq. (10) into Eq. (9), we where p q is the magnitude of the 3-momentum for the quark and E q = p 2 q + m 2 q . The lowest momentum given in Eq. (11) corresponds to the ground state. The quark wave function centered in x = 0 is written as where ω ± ≡ 1 ± m q /E q , χ is the two-component spinor of the quark, describing the orientation of the angular momentum, j i represents the i-th spherical Bessel function of the first kind, and N stands for the normalized factor, given by Since the baryon is made of the product of the three quarks, we write its wave function in term of the quark wave functions ψ q i , given by where q i (i = 1, 2, 3) denote the ith quarks and E q i are the corresponding energies. Here, the spin, flavor and color configurations have not been written down explicitly.
However, the above construction of the baryon wave function is valid only around x = 0, which is mainly used to discuss the mass spectrum and magnetic dipole moment of the baryon. To calculate the dynamical processes, we need the baryon state to be in the particular 4-momentum eigenstate of the spacetime translation.
On the other hand, in Eq. (14) one assumes that the baryons are at rest, and the centers of the initial and final baryons locate in the same spot at a particular time, t = t ′ . However, if the baryons are not at relatively rest, the centers will be not coincident at t = t ′ and the results of the dynamical factors will no longer be the same. Clearly, the baryon wave function in Eq. (14) is not the eigenstate of the spacetime translation.
To construct the baryon wave function to be invariant under the space translation, we have to modify the MIT bag model. To do that, we first integrate Eq. (14) with different center locations, leading to where N is the normalization factor. We then consider the space translation where d has only the spatial component. By changing the integration variable x to x − d, we see that the wave function is indeed invariant under the space translation.
The energy of the baryon at rest essentially corresponds to the mass of the baryon, which can be read off from Eq. (15) to be M = E q 1 + E q 2 + E q 3 . We note that the bag energy has not been included in our study. In principle, one can calculate the energy contribution from the complicated gluon and gluon-quark interactions. However, we will not include such effects. Instead, we will simply use M = E q 1 + E q 2 + E q 3 in the integral associated with the energy momentum conservation as an approximation.
The wave function for the baryon at rest is given in Eq. (15). To obtain the function in a certain 4-momentum, we boost it in the z-direction, given by is the pure Lorentz boost matrix for the baryon (quark) 4-spinor in the z-direction with S q v given by where I is the 2 × 2 unity matrix, a ± = 1 2 (γ ± 1) and γ = 1/(1 − v 2 ). In addition, v in the superscript of S q v indicates the Lorentz transformation of the coordinate, given explicitly as where (x q i ) 0 is the time component of the coordinate.
To obtain the proper normalization factor, we calculate the overlap between the two baryon wave functions with different speeds at time t, given by where S q † v = S q v for the pure Lorentz boost and M = E q 1 + E q 2 + E q 3 for the mass of the baryon. To simplify the integral, we adopt the following variables: Now, the overlap integral is read as By normalizing the baryon wave function as p ′ |p = γ(2π) 3 δ 3 ( p − p ′ ), we find that which is clearly independent of the velocity by the construction of the baryon wave functions.
It is worthwhile to take a look at the physical interpretation of the baryon wave function in Eq. (15). In contrast to the original wave function in Eq. (14) with the quarks confined in a static bag located at x = 0, those quarks for Eq. (15) distribute all over the space to fulfill the translation-invariant requirement. Note that the distance between two arbitrary quarks is limited within 2R in the bag model. For the case in Eq. (15), it is due to that if | x q 1 − x q 2 | > 2R, then either | x q 1 − x| or | x q 2 − x| will exceed R, resulting in the vanishing integral. So the quarks are entangling to each other in the spatial part of the wave function, which is not the case in Eq. (14).
The average distance between the quarks is given by In the original MIT bag model, as the quark positions are independent to each other, we have that x q 1 · x q 2 = x q 1 · x q 2 = 0. However, they do not vanish in our modified MIT bag model. For the proton with 3 massless light quarks, the average distance in our modified model is roughly 20% shorter than the original one with the same bag radius.

IV. BARYON FORM FACTORS
To calculate the Λ 0 b decays, we use the factorization approach. In this approach, the amplitudes for Λ 0 b → Λ + c M can be written as where G F is the Fermi constant, a 1 = c 1 + c 2 /3 = 1.02 [8] with c 1,2 the Wilson coefficients, V ij represent the CKM elements with q = d(s) corresponding to M = π − (K − ), f M is the meson decay constant, and the quarks operators are evaluated at x = 0. For the decay of Λ 0 b → pM , the amplitude is given by [10] A where α M (β M ) and α V in Eq. (29) are defined by  [27].

Now, we are left with the matrix elements of the scalar and pseudoscalar operators in
Eqs. (25) and (26), which can be parametrized as where B q represents Λ + c (p) with q being c(u). We evaluate the form factors of f for the scalar (pseudoscalar) matrix element with p M being the 4-momentum of the M meson.
Here, we have used that the initial and final baryons are in the momentum eigenstates to reduce the integral with the Dirac δ function of (2π) 4 δ(p i − p q − p M ). Clearly, one can evaluate the form factors with either the quark operators located at x = 0 in Eq. (28) or the x-dependent ones in the left hand side of Eq. (29). We start with the baryon wave functions in Eq. (17), given by with S q † v γ 0 = γ 0 S q −v for the Lorentz boost and (q 1 , q 2 ) = (u, d). Similar to the case in Eq. (20), we adopt the variable transformations with some modifications, given by where i = 1, 2, 3 and j = 1, 2. Subsequently, we have that Alternatively, one can evaluate the integral with b and q quarks located at x = 0 in Eq. (28) , given as After changing the integral variables by where k, j = 1, 2 , one obtains the identical equation as the one in Eq. (32) with M = Similarly, the pseudoscalar part can be given as With the normalization in Eq. (23), we derive that where η is the overlap factor of the spin-flavor configuration. For Λ b → Λ + c (p), we have η = 1 ( 3/2) [17].

V. NUMERICAL RESULTS AND DISCUSSIONS
We use the bag radius of the Λ + c baryon as R(Λ + c ) = 4.8 GeV −1 from Ref. [14]. In the limit of the heavy quark symmetry, the baryon wave functions for Λ 0 b and Λ + c can be taken to be the same. As a result, the bag radius for the Λ 0 b should be also around 4.8 GeV. For the proton, it is usually chosen to be 5 GeV −1 . However, to simplify our numerical calculations, we choose the same bag radiuses for Λ 0 b , Λ + c and p, i.e. R = R(Λ 0 b ) = R(Λ + c ) = R(p) = 4.8 GeV −1 . In addition, we take that m u = m d = 5 MeV. Note that the variation of the light quark masses from 0 to 10 MeV makes no much difference for the numerical values of the form factors. To determine the heavy quark masses, we assume that the baryon mass differences are related to the corresponding quark energies, i.e.
By using the equations of motion and comparing Eqs. (28) and (38), we obtain the form factors associated with the vector and axial currents from the scalar and pseudoscalar ones, given by  Table I.    factorization approach [10], considering the LFQM for the baryon wave functions [11,32], and using the perturbative QCD method with the hybird scheme [9]. As shown in Table III, the decay branching ratios for Λ 0 b → Λ + c (π − , K − ) from the modified bag model are close to those in LFQM [32] as well as the experimental data. We also find that our predicted branching ratio for Λ 0 b → pK − is about 1.2 times larger than that for Λ 0 b → pπ − , which agrees with the data and that in the generalized factorization approach [10], but different from the results of pQCD [9] and LFQM [11]. On the other hand, our results for the direct CP-violating rate asymmetries of Λ 0 b → p(π − , K − ) are sizable, which are consistent with all other theoretical predictions, and the experimental data except A CP (Λ 0 b → pK − ) PDG . As the experimental value of A CP (Λ 0 b → pK − ) PDG in Eq. (1) is consistent with zero with a negative central value, whereas our prediction of +6.7% along with the others in Refs. [10,11] is positive, it is very interesting to see if such CP asymmetry can be measured precisely by the ongoing experiment at LHCb. In addition, we see that  In Table IV, we illustrate the ratio of in various approaches. In the table, the result of the U-spin symmetry is based on the SU(2) symmetry between d and s quarks, which leads to the naive relation for R, given by In the factorization approach, R receives an extra factor due to the meson decay constants, read as which is consistent with our result and that in LFQM [32] as well as the data [7,33]. Clearly, it shows the evidence that the decays of Λ 0 b → Λ + c (π − , K − ) are factorizable.

VI. CONCLUSIONS
We have studied the decays of Λ 0 b → Λ + c (π − , K − ) and Λ 0 b → p(π − , K − ) in the modified MIT bag model. We have provided a new way to construct the baryon momentum eigenstates in the bag model without introducing new parameters. In particular, we have summed over the localized baryon wave function in Eq. (14) with different centers to fulfill the requirement of the invariant for the space translation.
For Λ 0 b → Λ + c (π − , K − ), we have found that the decay branching ratios are 4.5 × 10 −3 and 3.4 × 10 −4 , which agree well with the experimental data of (4.9 ± 0.4) × 10 −3 and (3.6 ± 0.3) × 10 −4 , respectively. We have also shown that our results of the first derivatives for the form factors f 1 (ω) and g 1 (ω) in Eq. (41) match with the data as well as those in the literature, indicating the validation of the heavy quark symmetry in the decay processes.
In addition, we have explored the CP-violating asymmetries for the decays. Particularly, we have obtained that A CP (Λ 0 b → pπ − ) and A CP (Λ 0 b → pK − ) are −4.4% and 6.7%, in comparison with (−2.5 ± 2.9)% and (−2.5 ± 2.2)% from the Particle Data Group in 2020, respectively. It is also interesting to note that A CP (Λ 0 b → pK − ) is predicted to be −19.6%, which is very large. It is clear that more precise future experimental measurements on these CP violating asymmetries are needed.