Nonleptonic and semileptonic Λb→Λc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda _b} \rightarrow {\Lambda _c}$$\end{document} transitions in a potential quark model

Nonleptonic and semileptonic decay widths of Λb→Λc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda _b} \rightarrow {\Lambda _c}$$\end{document} are analyzed within heavy quark limit and Isgur-Wise formalism. A modified QCD Cornell interaction with the additional logarithmic term in the hyperspherical coordinates is considered and the masses of heavy flavour baryons are calculated. The obtained masses are consequently employed to study the rates of Λb→Λc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda _b} \rightarrow {\Lambda _c}$$\end{document}. The achieved results are motivating.


Introduction
In recent years a huge number of both experimental and theoretical studies on properties of heavy baryons and the masses of ground states of charmed and bottom baryons containing a single heavy quark b or c have been published. Many decay channels of the bottom baryons have been observed due to the operation of LHC and plenty of data on heavy baryons have been collected. Weak decays of singly bottom baryons to final states involving singly charmed baryons are very interesting since they can help to understand the dynamic structure of heavy hadrons based on heavy quark effective theory (HQET) [1,2]. HQET is an effective tool for studying weak decays of heavy baryons. b can only decay via weak interactions. It is interesting and worthy to study b because, besides study on the heavy baryon structure, it may help determine the CKM matrix element V cb and also baryons containing heavy flavours b or c can play a significant role in the understanding of QCD. In the heavy quark limit, the hadronic matrix element of transition b → c can be expressed by a single Isgur-Wise function which can be defined as [3][4][5] a e-mail: s.rahmani120@gmail.com (corresponding author) where is an arbitrary gamma matrix and ω denotes the dot product of four-velocity of initial and final baryons. There have been useful attempts to analyze the nonleptonic decays of b → c using the soft-collinear effective theory [6], in the light-front quark model [7,8], based on HQET [9][10][11] and in the relativistic three-quark model [12]. Zhao obtained the transition form factors of the singly heavy baryons using the light-front approach under the diquark picture [13]. Faustov and Galkin studied semileptonic b decays in a relativistic quark model. They obtained wave functions of the ground, excited baryon states and the ratio of the Cabbibo-Kobayashi-Maskawa matrix elements [14]. Barik et al calculated static properties of baryons based on the Dirac equation with a logarithmic confining potential [15]. The hypercentral constituent quark model with color Coulomb plus power potential has been used to calculate the mass spectra of heavy flavour baryons [16][17][18].
In the next section, we present the Hamiltonian of a three body system including a modified Cornell interaction as effective interaction between quarks and calculate the masses of baryons. In Sect. 3 we study the two body nonleptonic decays b → + c P − and b → + c V − where P and V stand for pseudoscalar and vector mesons respectively. In Sect. 4 semileptonic decay of b → + c is discussed and finally, we present a conclusion.

General framework
To describe the baryon as a bound state of three quarks, the configuration of three particles are defined by two Jacobi vectors,ρ and λ. These Jacobi vectors can be written as One can employ the hyperspherical coordinates which are given by the angles ρ = (θ ρ , φ ρ ) and λ = (θ λ , φ λ ) with the hyperradius, x, and hyperangle ζ given by Therefore, the Hamiltonian of the three-body system which contains three body effects can be expressed as [16][17][18] such that where μ is the reduced mass and the kinetic energy operator takes the form using hyperspherical coordinates. We can write the relation of the eigenvalue of L 2 as [19] Here, the grand-angular momentum γ = 2n + l ρ + l λ with n = 0, 1, and l ρ , l λ are the angular momenta associated with the ρ and λ Jacobi vectors. We have chosen the following logarithmic-type potential [20] which is the effective potential in which a quark is influenced where c F is the color factor which is 2 3 for the baryon, n f is the number of flavors and γ is the Euler constant. Equation (7) has Coulomb potential and also confinement concept. We have plotted the behavior of the considered potential, Eq. (7), and Cornell potential V (x) = − β x + αx by Refs. [18,21] in Fig. 1. In fact Eq. (7) is a Cornell potential modified by a logarithmic function. The use of logarithmic potential was reported by several references [15,[22][23][24] to study hadron properties. The logarithmic term can guarantee quark confinement and also can generate charmonium and bottomonium bound-state masses in reasonable agreement with experiments [22]. The strong running coupling constant is [16] The behavior of the potential Eq. (7), in terms of α s (μ 0 = 1 GeV) = 0.6 . Using virial theorem, the wave function is taken as the hyper Coulomb radial wave function given by [16] ψ n,γ (g, where N is the normalization constant and L 2γ +4 n (2gx) represents the Laguerre polynomial. By using we obtain the variation parameters, g, for baryons as shown in the second column of Table 1. The variational method is one of the ways of solving the radial Schrödinger equation and finding eigenenergies. In this method, one can choose a trial wave function which depends on one or more adjustable parameters known as variation parameter. In our work g is this parameter which can be obtained by minimizing the expectation value of the Hamiltonian and we find an approximated value for the energy. In the variational theorem because the trial energy (E trial = ψ n,γ (g, x)|H |ψ n,γ (g, x) ) is always larger than or equal to the actual energy, we can minimize the trial energy by taking the derivative with respect to g, setting it equal to zero and solving for g as in Eq. (10). From Eqs. (9) and (10), we can see that g depends on quantum numbers and input quark masses . Thus, fixing the quantum numbers as n = 1 and γ = 0, g has different values for

Nonleptonic decays of b → c
Using HQET with factorization method [11], one can obtain the decay widths for the decay processes b → + c P − and b → + c V − given as [11] ( and where G F is the Fermi coupling constant and | p| is the center of mass momentum of the emitted particles that can be obtained from . We assume the IWF as [26] The product is determined by considering the momentum conservation of two body decays [11] v Different parameterizations have been proposed for the baryonic IWF η(v.v ) such as linear, pole-type and exponentialtype [27][28][29][30][31][32][33]. To parametrize the baryonic IW function, Ref.
[3] also considered an exponential form as where η(1) = 1 due to the normalization of IWF at zero recoil point and ρ 2 = 1.35 [3]. We report our values in Table 2 for nonleptonic decays of b → + c .

Semileptonic decays of b → c
The differential semileptonic decay rate of b → c ν is given as [6] d (17) in the physical and kinematical ω range (about 1- . With the calculated masses of b and c by Eq. (11) and IWF as Eq. (14), the decay rate then can be obtained to be ( b → c ν) = 2.88 × 10 −14 GeV . By considering IWF in MIT bag model as [30] η(ω) = 2 ω + 1 3.5+ 1.2 ω (18) we find the semileptonic decay width of b → c as 2.70 × 10 −14 GeV and branching ratio of this decay as 6.03 %. We have also reported the partial semileptonic decay rates of b → c ν in Table 4. It involves three cases for IWF, Eqs. (14), (16) and (18).

Results and discussion
In conclusion, we have presented a model based on Cornell potential plus logarithmic term and calculated semileptonic and nonleptonic decay properties of b → c using HQET.
112 GeV and M π ± = 0.139 GeV. Taking cut off parameter B = 2.12 GeV in IWF leads to successful results. In particular, a good description of the experimental data for b → + c π − and b → + c D − are obtained. We have also computed and plotted the root mean square deviations of the predicted branching ratios of b → c nonleptonic decays of Table 2 considering experimental data. As we have shown in Fig. 2. the standard deviation drops to σ = 0.138 at B = 2.12 GeV . This parameter, B , has been taken different values such as 2.408 GeV in Ref. [36], 2.5 GeV in Ref. [4], 3.037 and 2.408 GeV [12] and in the case of double heavy baryons 2.5 and 3.5 GeV [26].
We have shown the ground states of mass spectrum for some heavy baryons in Table 1 and compared our results with Refs. [16,32]. In the second column of Table 2 we have shown our calculated branching ratios of nonleptonic decays of b → c and compared them with Refs. [9][10][11]35,37]. The second and the third columns of Table 3 show our results for b semileptonic decay parameters regarding to the exponential type of IWF and IWF in MIT bag model respectively which are compared with Refs. [3,7,14,31,37]. In Table 4, the values of partial decay width of b → c are well compatible with Refs. [38,39]. Ke 2 ) =124 MeV stated by Garcilazo et al [41]. We have reported this spacing as 246.31 MeV.
In the scheme adopted in this work, we evaluate the ratio B R( 0 b → + c l −ν l ) B R( 0 b → + reading of our manuscript and for constructive suggestion. Hassan Hassanabadi and Jan Kříž are grateful for the institutional support of the Faculty of Science, University of Hradec Králové, research team "Mathematical physics and differential geometry".

Data Availability Statement
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