Non-leptonic two-body decays of $\Lambda_b^0$ in light-front quark model

We study the non-leptonic two-body weak decays of $\Lambda_b^0 \to p M$ with $ M=(\pi^-,K^-)$ and $(\rho^-,K^{*-})$ in the light-front quark model under the generalized factorization ansatz. By considering the Fermi statistic between quarks and determining spin-flavor structures in baryons, we calculate the branching ratios (${\cal B}$s) and CP-violating rate asymmetries ($\mathcal{A}_{CP}$s) in the decays. Explicitly, we find that ${\cal B}( \Lambda_b^0 \to p \pi^- ,pK^-)=(4.18\pm0.15\pm0.30, 5.76\pm0.88\pm0.23)\times10^{-6}$ and ${\mathcal{A}_{CP}}( \Lambda_b^0 \to p \pi^- ,\,pK^-)=(-3.60\pm0.14\pm0.14, 6.36\pm0.21\pm0.18)\%$ in comparison with the data of ${\cal B}( \Lambda_b^0 \to p \pi^- ,pK^-)=(4.5\pm0.8, 5.4\pm1.0)\times10^{-6}$ and ${\mathcal{A}_{CP}}( \Lambda_b^0 \to p \pi^- ,pK^-)=(-2.5\pm 2.9, -2.5\pm2.2)\%$ given by the Particle Data Group, respectively. We also predict that ${\cal B}( \Lambda_b^0 \to p \rho^-,pK^{*-} )=(12.13\pm3.27\pm0.91, 2.58\pm0.87\pm0.13)\times 10^{-6}$ and ${\mathcal{A}_{CP}}( \Lambda_b^0 \to p \rho^-,pK^{*-} )=(-3.32\pm0.00\pm0.14,19.25\pm0.00\pm0.80)\%$, which could be observed by the experiments at LHCb.


I. INTRODUCTION
It is known that b-physics has been providing us with a nature platform to observe CPviolating phenomena, test the heavy quark effective theory, and explore physics beyond the Standard Model (SM). Among the various processes, the weak decays of Λ 0 b as a complementary of the B-meson ones give us an opportunity to verify the QCD factorization hypothesis.
In the recent years, several interesting Λ 0 b decay processes have been measured by the LHCb Collaboration, such as the two-body non-leptonic mode of Λ 0 b → Λφ [1] and the radiative decay of Λ 0 b → Λγ [2]. In addition, the direct CP violating rate asymmetries in Λ 0 b → pπ − and Λ 0 b → pK − have also been searched with the most recent data of (−2.5 ± 2.9)% and (−2.5 ± 2.2)% [3,4], respectively. One expects that LHCb will accumulate more and more high quality data after the upgrade and lead b-physics into a precision era. There is no doubt that comprehensive studies in the Λ 0 b processes are necessary. Particularly, only theoretical estimations could not be able to understand the future LHCb experimental measurements.
In this work, we concentrate on the non-leptonic two body decays of Λ 0 b → pM with M representing a pseudoscalar (P) or vector (V) meson in the final states. We use the effective Hamiltonian including the QCD-penguin and electroweak-penguin operators and the effective Wilson coefficients are evaluated at the renormalization scale µ = 2.5 GeV in the NLL precision [12,13]. To obtain the decay amplitudes, we follow GFA to split the matrix elements into two pieces, resulting in that the only relevant quantities for the decay amplitudes are meson decay constants and baryon transition form factors. The hardest part to get the decay amplitude is to calculate the baryonic transition form factors in Λ 0 b → p because of the complicated baryon structures and the non-perturbative nature of QCD at the low energy scale. To extract the form factors, we use LFQM, which has been widely used in the B-meson [14][15][16][17] and heavy to heavy baryonic transition systems [18,19]. The greatest advantage of LFQM is that we can deal the baryon states consistently with different momenta because of the boost invariance property in the light front dynamics. As a trade off, we are only allowed to evaluate the form factors in the space-like region to avoid the zero-modes or so called Z-graphs, which are hard to be calculated in LFQM. We analyze the branching ratios of the pseudoscalar modes for Λ 0 b → (pπ − , pK − ) and their corresponding CP-violating asymmetries, and compare them with the current experimental data. We also predict the vector decay modes of Λ 0 b → (pρ − , pK * − ). In particular, we would like to check if the sizable CP-violating rate asymmetry in Λ 0 b → pK * − , predicted to be as large as 20% in GFA [5,6], can be confirmed in LFQM. This paper is organized as follows. In Sec. II, we present our formalism, which contains the defective Hamiltonians, decay widths and asymmetries, vertex functions of the baryons and baryonic transition form factors in LFQM. We show our numerical results of the form factors, branching ratios and CP asymmetries and compare our results with those in the literature in Sec. III. In Sec. IV, we give our conclusions.

A. Effective Hamiltonians
To study the exclusive two-body non-leptonic processes of Λ 0 b → pM with M being the pesudoscalar P = (π − , K − ) and vector V = (ρ − , K * − ) mesons, we start with the effective Hamiltonians of b → quū (q = d, s) at quark level, given by where G F is the Fermi constant, C i stand for the Wilson coefficients evaluated at the renormalization scale µ, V q 1 q 2 represent the CKM quark mixing matrix elements, and O 1−10 are the operators, given as Here, O 1,2 , O 3−6 and O 7−10 correspond to the tree, QCD and electroweak-penguin loop operators, respectively, while Q = u, d, s, c, b for µ = O(m b ). To calculate the decays of Λ 0 b → pM, we need to find the matrix elements for the operators, given by where the explicit expressions for the effective Wilson coefficient c ef f i (µ) can be found in Refs. [12,13]. Note that the physical matrix elements on the left-handed side of Eq. (3) should be independent of the renormalization scheme and scale.
Based on GFA, each element can be written as where the explicit Dirac and color indices are suppressed. As a result, the decay amplitudes are govern by the mesonic and baryonic transitions separately. For the former, we can use the definitions, given by where f M (M = P, V ) are the meson decay constants, m M correspond to the masses of M, and p µ (ǫ µ ) is the momentum (polarization) vector for P (V ). The latter can be related to the baryonic transition form factors, defined by where k µ = p µ i − p µ f and k 2 = m 2 P (V ) . The matrix elements for O 5−8 , which have the (V − A)(V + A) structure, can be calculated by the employment of the Dirac equation after applying the Fierz transformation and factorization, given by with where the quark masses are the current quark ones. Consequently, the amplitude for Λ 0 b → pP is written as where parameterize the non-factorizable QCD effects of the octet-octet operators in Eq. (4). Here, f 2 and g 2 have no contributions to the amplitude due to the anti-symmetric structure of σ µν , while the terms associated with f 3 and g 3 are suppressed by the factor of m 2 P /m 2 Λ b . Similarly, the amplitude for Λ 0 b → pV is given by

B. Decay widths and CP asymmetries
The decay widths of Λ 0 b → pM (M = P, V ) can be found from Eqs. (9) and (10), read as where p c is the momentum in the center mass frame and E p is the energy of the proton.
The direct CP-violating rate asymmetry is defined by where Γ(Λ 0 b → pM) and Γ(Λ 0 b →pM ) are the decay widths of the particle and antiparticle, respectively.
In principle, one could solve the momentum wave function by introducing the QCDinspired effective potential like the one-gluon exchange one. However, the equation will become too complicated to be solved in the three-body case. Therefore, we use the phenomenological Gaussian type wave function with some suitable shape parameters to describe the momentum distributions of the constituent quarks. It is also possible to compare the LFQM wave function with the light-cone distribution amplitude to get the QCD renormalization improvement at different energy scales [29]. The baryon spin-flavor-momentum wave function F abc Ψ SSz (p 1 ,p 2 ,p 3 , λ 1 , λ 2 , λ 3 ) should be totally symmetric under any permutations of the quarks to keep the Fermi statistics. The spin-flavor-momentum wave functions of Λ 0 b and p are given by where the explicit forms for φ 1,2 can be found in Ref. [21], and with N = 2(2π) 3 (β q β Q π) −3/2 and β q,Q being the normalized constant and shape parameters, respectively.
Here, the baryon state is normalized as resulting in the normalization of the momentum wave function, given by We emphasize that the momentum wave functions of φ i are associated with the different shape parameters of β q and β Q in Λ 0 b . For the proton, the momentum distribution functions are the same, i.e. φ = φ 3 (β q = β Q ), for any spin-flavor state because of the isospin symmetry.

D. Baryonic Transition form factors
The baryonic transition form factors of the V − A weak current are given by Eq. (6) with P ′ − P = k. We choose the frame such that P + is conserved (k + = 0, k 2 = −k 2 ⊥ ) to calculate the form factors in order to avoid other diagrams involving particle productions, also known as the Z-graphs in the light-front formalism [20,24]. The matrix elements of the vector and axial-vector currents at quark level correspond to three different effective diagrams as shown in Fig. 1. Since the spin-flavor-momentum wave functions of the baryons are totally symmetric under the permutation of the quarks indices, we have that (a) + (b) + (c) = [20]. We only present the calculation for the diagram (c) in Fig. 1, which contains simpler and cleaner forms with our notation (q ⊥ , Q ⊥ , ξ, η). We can extract the form factors from the matrix elements through the relations,

3(a) = 3(b) = 3(c)
Note that f 3 and g 3 are unobtainable when k + = 0, but they are irrelevant in this work because of the kinematic suppression of m 2 P /m 2 Λ b associated with them. The full expressions of the form factors in Eq. (28) are give by

III. NUMERICAL RESULTS
We choose the Wolfenstein parameterization for the CKM matrix with the corresponding parameters, taken to be [4] Table. I [12,13]. For the quark masses in Eq. (8), we start from the values at µ = 2.0 GeV given by PDG [4], and use the renormalization group equation [13] to determine them at µ = 2.5 GeV, given by    In Table. II, we show our parameter sets of the hadron masses and meson decay constants.
In our numerical calculations, we also set the Λ 0 b life time to be τ Λ 0 b = 146.4 fs [4]. In this work, the effective color number N ef f c is fixed to be 3.
Since the baryonic transition form factors in LFQM can be only evaluated in the spacelike region (k 2 = −k 2 ⊥ ) due to the condition k + = 0, we take some analytic functions to fit f 1(2) (k 2 ) and g 1(2) (k 2 ) in the space-like region and perform their analytical continuations to the physical time-like region (k 2 > 0) as in Refs. [21][22][23][30][31][32]. We employ the numerical values of the constituent quark masses and shape parameters in Table. III. By using Eqs. (29)-(32), we compute totally 32 points for all form factors from k 2 = 0 to k 2 = −9.7 GeV 2 . To fit the k 2 dependences of the form factors in the space-like region, we use the form, given by where q 1,2 are the fitting parameters. Our results for the form factors are presented in Table. IV and compared with the results from light-cone sum rule (LCSR) [11] where the form factors are parameterized by , The definition of these parameters can be found in Ref. [11]. From the table, one has that f 1 ≃ g 1 , which agrees with the relation of f 1 = g 1 in the heavy quark limit. One can also see that f 2 (k 2 = 0) > f 1 (k 2 = 0), which is similar to the cases in the Λ + c decays [21], but The authors in Ref. [11] use the opposite sign convention for f 2 (g 2 ).
different from those in LCSR [11] for the Λ 0 b transitions. For LFQM and LCSR, the results of f 1 (g 1 ) are the same, whereas those of f 2 (g 2 ) are different, in the low-momentum transfer region. The predictions of the branching ratios and CP-asymmetries in these two models are similar because the contributions of f 1 and g 1 are dominated in Λ 0 b → pP (V ). The discrepancies between LFQM and LCSR appear in the Λ 0 b semi-leptonic decays. The decay rates in these semi-leptonic decays for LFQM are much smaller than the LCSR ones as well as the data due to the lack of the information from the B * (1 −(+) ) poles in LFQM, which are important in the calculations of Λ 0 b → pℓ −ν ℓ via the analytical continuation method. Our predictions of the decay branching ratios and direct CP-violating rate asymmetries are summarized in Table V, where the first and second errors contain the uncertainties from the form factors and Wolfenstein parameters in Table IV, respectively. It is interesting to see that our results of A CP (Λ 0 b → pV ) are free of the baryonic uncertainties under the generalized factorization assumption. The reason is that all form-factor dependencies in Γ(Λ 0 b → pV ) can be totally factorized out within the generalized factorization framework so that they get canceled for the CP-violating rate asymmetries of A CP (Λ 0 b → pV ) [5,6]. As a result, the numerical values of A CP (Λ 0 b → pV ) are QCD model-independent. In the table, we also show the recent experimental data [4] as well as other theoretical calculations in the literature [6][7][8][9], where pQCD a and pQCD b represent the conventional and hybrid pQCD approaches [7], while LFQM a [8] and LFQM b [9] refer to those in LFQM with and without the considerations of the QCD and electroweak-penguin loop contributions, respectively. We find that our predictions of B(Λ 0 b → pπ − , pK − ) are consistent with the experimental data and those in GFA and LCSR. In addition, we have that B(Λ 0 b → pK − ) > B(Λ 0 b → pπ − ) as indicated by the data, which is different from those predicted by pQCD [7], LFQM [8,9] and MBM [10]. Our results for B(Λ 0 b → (pρ − , pK * − )) also agree with those in GFA [6]. Note that LFQM b [9] does not consider the loop contributions, so that its results are incompatible with the data as well as all other theoretical ones. For the CP-violating rate asymmetries, we find that A CP (Λ 0 b → pπ − ) = (−3.60 ± 0.14 ± 0.14)%, consistent with the data and those from GFA [6], LFQM a [8] and MBM [10], but different from pQCD [7]. On the other hand, we obtain a positive CP-violating rate asymmetry for Λ 0 b → pK − , which is the same as GFA, LFQM a and MBM, whereas the data of (−2.5 ± 2.2)% [4] along with those from pQCD is negative. It is interesting to note that, similar to GFA and LFQM a , our result in LFQM also predicts a sizable asymmetry of ∼ 20% in Λ 0 b → pK * − .

IV. CONCLUSIONS
We have studied the non-leptonic two body decays of Λ 0 b → pM with LFQM based on the generalized factorization ansatz. By considering the Fermi statistic between quarks and determining spin-flavor structures in baryons, we have evaluated the baryonic form factors in the LFQM. In particular, we have found that f 1 ≃ g 1 , which agrees with the requirement in the heavy quark limit, whereas f 2 (k 2 = 0) > f 1 (k 2 = 0), different from those in the literature for the Λ 0 b transitions. It is possible to include the QCD renormalization effect by comparing the LFQM wave function and light-cone distribution amplitude to understand the uncertainty at different energy scales as the study in Ref. [29], which could be done in our future work. We have compared our form factors with those in LCSR, and shown that they are almost the same in the low-momentum transfer region. However, the LCSR ones TABLE V. Our results in comparison with the experimental data and those in various theoretical calculations in the literature, where Bs and As are in units of 10 −6 and %, respectively. In our results, the first errors come from hadronic uncertainties and the second errors come from the uncertainties of CKM matrix.