Revisiting 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} and Σb→Σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{b}\rightarrow \Sigma _{c}$$\end{document} weak decays in the light-front quark model

In this work, we study 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} and Σb→Σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{b}\rightarrow \Sigma _{c}$$\end{document} weak decays in the light-front quark model. As is well known, the key point for such calculations is properly evaluating the hadronic transition matrix elements which are dominated by the non-perturbative QCD effect. In our calculation, we employ the light-front quark model and rather than the traditional diquark picture, we account the two spectator light quarks as individual ones. Namely during the transition, they retain their color indices, momenta and spin polarizations unchanged. Definitely, the subsystem composed of the two light quarks is still in a color-anti-triplet and possesses a definite spin, but we do not priori assume the two light quarks to be in a bound system–diquark. Our purpose is probing the diquark picture, via comparing the results with the available data, we test the validity and applicability of the diquark structure which turns a three-body problem into a two-body one, so greatly simplifies the calculation. It is indicated that the two approaches (diquark and a subsystem within which the two light quarks are free) lead to similar numerical results even though the model parameters in the two schemes might deviate slightly. Thus, the diquark approach seems sufficiently reasonable.


Introduction
The study on baryon physics is much behind that on mesons so far, because the structure of baryon is remarkably more complicated even though only one more quark gets involved. The case of b , c , b and c which contain one heavy quark is simpler than that with all light quarks, so that corresponding research attracts attention of both experimentalists and theorists of high energy physics. Thanks to the successful operation of LHC, plenty of data on baryons, especially those a e-mail: khw020056@hotmail.com b e-mail: lixq@nankai.edu.cn on heavy baryons have been collected. Thus researchers have a great opportunity to study heavy baryons via their production and decays to gain information of their structure and how the fundamental dynamics works for the baryon case. b is the ground state of b baryons so it can only decay via weak interactions. Indeed besides study on the baryon structure, its weak decays may be valuable for determining the CKM parameter V cb as a compensation to the measurements on mesons and furthermore one can investigate the non-perturbative QCD effects in the heavy baryon system because of the existence of the heavy quark. b is heavier than b which would dominantly decay via the portal b +π , therefore a sizable branching ratio of its weak decays may imply a possible involvement of new physics, so can serve as an ideal laboratory for searching new physics beyond the standard model.
The weak decays of heavy baryons including b and b have been studied. For example: Korner and Kroll [1] explored the weak decays of baryons under the heavy quark limit [2,3] where the quark-diquark picture was employed; in Ref. [4] Ebert et al. used their relativistic quark model to calculate the decay rates of several weak decay modes where baryons consist of a heavy quark and a light diquark; Singleton examined the semileptonic decays of spin- 1 2 baryons in the spectator-quark model [5]; in Ref. [6] Ivanov et al. employed their relativistic three-quark model to study the weak decays of several baryons under the heavy quark limit; lately, Ivanov et al. [7] also studied heavy baryon decays in the Bethe-Salpeter approach under the heavy quark limit. By those works the properties of the weak decays of b and b have been investigated and the non-perturbative QCD effects for baryons (at least for heavy baryons) are partly understood or at least can be approximately handled.
There exists an acute dispute about the diquark structure of baryons yet. As is well known, the fundamental structure of baryons is determined by the Faddeev equations. However, that is an equation group for the three-body system whose solutions are difficult to gain. The diquark picture has been raised for a long while, even at the birth time of quark model [32]. In that picture two quarks are loosely bound into a subsystem which can be a vector, axial vector, scalar or pseudoscalar in color-anti-triplet. This approach definitely is an approximation which turns the three-body problem into a two-body problem, so greatly simplifies the calculation. In the earlier works as listed above where the diquark picture was employed, the diquark was treated as a point-like boson with a definite mass, spin and isospin. When it is involved in the concerned reaction [33], a form factor composed of a few free parameters which are fixed phenomenologically, is introduced. The picture is somehow in analog to the case of elastic electron-proton scattering where the inner structure of proton is manifested in the electric and magnetic form factors.
The assertion needs to be verified in some ways. To test the validity and applicability of the diquark approach, in this work, we treat the three quarks (one heavy and two light) as individual ones and possess their own color indices, spin polarizations(or helicities) and momenta, namely they share the total momentum of the baryon. During the transition, the two light quarks are spectators, i.e. maintain their all quantum numbers (spin, color) and momenta unchanged. In one word, we make a three-body calculation rather than a two-body one. Comparing the upcoming results with that obtained in terms of diquark, one can make a judgement whether the diquark picture indeed works well in the concerned processes where the light-quark subsystem is a spectator. Our results show that when the light quarks can be treated as spectators during the hadronic transitions, the point-like diquark picture is a good approximation, at least at the leading order.
In order to calculate the hadronic transition matrix element one needs to know the effective vertex functions. We construct them at first. Since the isospins of b and c are 0, the light ud subsystem must be an isospin-0 and color3 state. To guarantee the total spin of b(c) to be 1/2 and the spin of the ud system should be zero, i.e. the wavefunction of the ud subsystem is totally antisymmetric for spin×color×isospin. Instead, the isospin of b or c is 1, according to the same principle the spin of the light subsystem (ud) is determined to be 1. Thus the spin polarizations of the two quarks are not free, but correlated.
With the spin arrangements [34] we obtain the vertex function of b , c , b and c (denoting as B Q( ) ). Then following the common approach [8][9][10] we write down the transition matrix element and extract the form factors f 1 , f 2 , g 1 and g 2 which are defined for the transition(see the text for details). We compute these form factors f 1 , f 2 , g 1 and g 2 numerically.
Since the leptons do not participate in the strong interaction, the semileptonic decay is less contaminated by the non-perturbative QCD effect, therefore study on semileptonic decay might more help to test the employed model and/or constrain the model parameters. With the form factors we calculate the widths of the concerned semileptonic decays. Comparing the numerical result with the data the model parameters in the wave function are fixed. Moreover, moving one more step, using those parameters we write out the amplitude of the non-leptonic decay b → c + M. In the case, we suppose that one can factorize out the meson current. Definitely, the factorization does not rigorously holds, but can be a good approximation at the leading order which was thoroughly investigated for the meson decays.
This paper is organized as follows: after the introduction, in Sect. 2 we construct the vertex functions of heavy baryons, then write down the transition amplitude for b → c and b → c in the light-front quark model and deduce the form factors, then we present our numerical results for b → c and b → c along with all necessary input parameters in Sect. 3. Section 4 is devoted to our conclusion and discussions. In our previous work [8][9][10], we employed the quark-diquark picture to study the transitions. Instead, in this work we will estimate the transition rate by treating the three quarks as individual ones. It is noted, the transition occurs between heavy b and c quarks. Even though the other two light quarks are not bound together, the subsystem where they reside in, is still of definite spin, color and isospin and as the two quarks are spectators, all the quantum numbers of the subsystem keep unchanged. In analog to the references [35,36] the vertex functions of a baryon B Q (Q = b, c) with total spin S = 1/2 and momentum P is where C αβγ and F Qdu are the color and flavor factors, λ i (i = 1, 2, 3) and p i (i = 1, 2, 3) are helicities and lightfront momenta of the on-mass-shell quarks defined as In order to describe the motions of the constituents, one needs to introduce intrinsic variables ( where x i are the light-front momentum fractions constrained are independent of the total momentum of the hadron and thus are Lorentz-invariant. The invariant mass square M 2 0 is defined as The invariant mass M 0 is in general different from the hadron mass M which obeys the physical mass-shell condition M 2 = P 2 . This is due to the fact that the baryon, heavy quark and the two-light-quark subsystem cannot be on their mass shells simultaneously. We define the internal momenta as It is easy to obtain where e i denotes the energy of the i-th constituent. The momenta k i⊥ and k iz constitute a momentum vector k i = (k i⊥ , k iz ) and correspond to the components in the transverse and z directions, respectively. Being enlightened by [34] the spin and spatial wave function for Q is written as and for Q SS z where , p 1 is the momentum of the heavy quark Q, p 2 , p 3 are the momenta of the two light quarks,P = p 1 + p 2 + p 3 , γ ⊥α = γ α − v /v α , and λ 1 , λ 2 , λ 3 are the helicities of the constituents.
With the normalization of the state |B Q one can obtain , where p i ·P = e i M 0 (i = 1, 2, 3) is used. The spatial wave function is [35,36] ϕ(

The form factors of b → c in LFQM
The form factors for the weak transition b → c are defined in the standard way as Since S = S = 1/2, we write | b (P, S, S z ) and | c (P , S , S z ) as | b (P, S z ) and | c (P , S z ) respectively.
The lowest order Feynman diagram responsible for the b → c weak decay is shown in Fig. 1. Following the approach given in Refs. [8,10,35,36] the transition matrix element can be calculated with the vertex functions of | b (P, S z ) and | c (P , , S z ) supposing u and d quarks are spectators, where and Q (Q ) represents the heavy quark b (c), p 1 ( p 1 ) denotes the four-momentum of the heavy quark b (c), P (P ) stands as the four- In terms of the approach given in Ref. [35] we extract the form factors defined in Eq. (13) from the Eq. (14) Expanding the traces in above formulas is straightforward, but the expressions are rather tedious, thus we collect he explicit expressions in Appendix A. It is noted that the form factors f 3 (q 2 ) and g 3 (q 2 ) cannot be extracted in terms of the above method for we have imposed the condition q + = 0. However, they do not contribute to the semi-leptonic decays b → c lν l if the masses of electron and neutrino are ignored.

The form factors of b → c in LFQM
Similarly, the hadronic matrix element for transition b → c can also be obtained with the vertex functions of | b (P, S z ) and | c (P , S z ) , For the transition some form factors can also be defined as in Eq. (13), while f v i and g v i replace f s i and g s i (i = 1, 2, 3) in Eq. (13).
The expressions of the form factors are The interested readers can refer to appendix A to simplify the form factors before numerically evaluating.

The b → c form factors and some decay modes
In order to evaluate these form factors numerically one needs the parameters of the concerned model. Here we employ the masses of quarks presented in Ref. [16] and list them in Table 1. Indeed, we know very little about the parameters β 1 and β 23 in the wave function of the initial baryon and β 1 and β 23 in that of the final baryon. Generally the reciprocal of β is related to the electrical radium of the baryon. Since the strong interaction between q and q is half of that between qq if it is a Coulomb-like potential one can expect the electrical radium of qq to be 1/ √ 2 times that of qq i.e. β qq ≈ √ 2β qq . In Ref. [37] in terms of the binding energy the authors also obtained the same results, so in our work we use these β's values which were obtained in the mesons case [38]. With these parameters we calculate the form factors and make theoretical predictions on the transition rates. We set β 1 ≈ √ 2β bs , β 1 ≈ √ 2β cs and suppose β 2 = β 2 ≈ √ 2β ud . It is found that the predicted width of the semilepton decay is larger than the data. Then we readjust the parameter β 2 to reduce the theoretical prediction to be closer to the data and thus we could fix the parameter. At last we obtain β 1 = 0.851 GeV, β 1 = 0.760 GeV and β 2 = β 2 = 0.911 GeV. Here β 2 and β 2 are close to 2.9β ud , which means the distance between u and d quarks is smaller than the normal situation where u and d are evenly distributed in the inner space of the baryon. Indeed, it implies the diquark structure.
It is noted, we derive the form factors in terms of the LFQM in the space-like region, thus applying them to evaluate the transition rates, one needs to extend them to the physical region (time-like). Following the standard scheme, we derive the form factors for the baryonic transitions. The procedure for the derivation was depicted in literature and our previous work in all details, so that we do not keep it in the context, but for readers' convenience, relevant stuff is retained in the attached appendix. Then, we explicitly list the corresponding parameters which are needed for numerical computations in Table 2 (Fig. 2).
Using the form factors, we evaluate the rates of b → c lν l . The decay rates are listed in Table 3 where we update the results which were obtained in our earlier paper [8] with the parameters fixed in [38] as inputs. Comparing them with that obtained by other authors in Refs. [4,[6][7][8], one can notice that the differences among all the results are rather small. We plot the differential decay rates d /dω in Fig. 3a which also is consistent with those given in [4].
From the theoretical aspects, calculating the corresponding quantities for the non-leptonic decays seems to be much more complicated than for the semi-leptonic ones. The theoretical framework adopted in this work is based on the factorization assumption, namely the hadronic transition matrix element is factorized into a product of two independent matrix elements of currents, where the term M |q γ μ (1−γ 5 )q | 0 is determined by the decay constant and the transition b → c is calculated in a Since there exist a mistake in the expressions of P 1 and D (Eq (60)) in [8] and we correct them in Eq. (C5) in this appendix. In addition a factor 2 was missing in the formula for the transition 1 2 → 1 2 + V given in [39] and we have discussed this issue with the authors of Ref. [39], and then they have carefully checked this formula and agreed with us. Therefore the and α of 0 b → + c V in [8] are changed slightly here the previous sections. Since the decays 0 b → c +M − is the so-called color-favored transition, the factorization should be a good approximation. The study on these non-leptonic decays can check how close to reality the obtained form factors for the heavy bottomed baryons would be. In Table 4 we present the results of this work and previous papers together. One can notice that they are very close to each other, and it means that the heavy-quark-light-diquark picture is indeed a good approximation.

b → c form factors and some decay modes
Now we calculate the form factors for the transition b → c . Using the values set for b and c , we determine the parameters β 1 and β 1 . Generally the parameters β 23 and β 23 would be different from those for b and c because the total spin of the ud subsystem in b(c) is 1 but that is 0 in b(c) . The situation is similar to the spin configurations of a pseudoscalar vs a vector. Even though there are no data available, we boldly use the same parameters gained from b → c to make our predictions and the validity or approximation would be tested in the future measurements.
The parameters of the form factor are listed in Table 5. As discussed in the footnote of Table 5 there should be an additional factor m b /(m b + m c ) for the F 0 of f 2 and g 2 [10]. Taking into account of this factor the results are close to those given in [10]. The differential decay rates d /dω are depicted in Fig. 3b whose line-shape is fully consistent with those for b → c in Fig. 8 of Ref. [4]. It is understandable because the decay b → c and b → c are similar under the flavor SU(3) symmetry. Since b does not decay via strong interaction, the weak decays are dominant. The results of semi-leptonic decay of b → c can be found in Table 6. The total width and the longitudinal asymmetry Table 7 The widths (in unit 10 10 s −1 ) and up-down asymmetries of non-leptonic decays b → c M

This work
In [10] Table 7). The non-leptonic decays are presented in Table 7.
The interpretation of the difference is not trivial. When one constructs the vertex function, the first principle is to retain the momentum conservation, then there are two schemes to be selected, i.e. whether let the diquark polarization depend on the total momentum of the baryon P or the momentum of the diquark p 2 . In our earlier paper [40], we discussed the two schemes for constructing the vertex function of b(c) . Comparing the numerical results of the transition rates of b → * c , we found that the values calculated with the two schemes only slightly deviated from each other. Then the conclusion might be that the two schemes are actually equivalent. In our study on the transition [10], we only adopted the scheme where the diquark polarization only depends on momentum p 2 , but not P. One conjecture is that the momentum dependence may lead to the small deviations of the transverse asymmetry a T as is shown in this work. The difference, even though is not large, may imply a distinction of the momentum dependence, which needs to be further investigated and we will do it in our following works.

Conclusions and discussions
In terms of the extended light front quark model we explore the leptonic and non-leptonic weak decays of b → c and b → c with the three-quark picture of baryon. Based on our earlier works for studying hyperon and meson decays in terms of the LFQM, this work has two purposes. The first one is to make a further confirmation of two possible ways of determining the momentum dependence of the diquark polarization during constructing the vertex functions of baryons. In our earlier work [10] , we suggested a possible momentum dependence of the diquark polarization. Namely, we conjectured that the polarization of the diquark should depend on its momentum p 2 . Later [40], we employed another possible scheme where the diquark polarization depends on the total momentum of the baryon P. All the two schemes respect the principle of momentum conservation. The numerical results on the transition rates of b → * c obtained in the two schemes are very close to each other. In this work, we use neither of the two schemes to evaluate the transition rates of b → c in LFQM, because we treat the two light quarks as free individuals rather than demanding them to reside in a "diquark". Then we compare the results with that obtained in terms of the diquark pictures. We notice that the rates obtained in the two approaches are very close, but for longitudinal and transverse asymmetry a T for the semi-leptonic decay of b → c which are more sensitive to the approaches, an obvious deviation appears, i.e the values of a T are different for the two cases. Therefore, there may exist a more profound physics which leads to the difference. In our coming work we will further study those schemes, namely investigate their validity. Beside a theoretical analysis, the experimental data would compose an acceptable probe-stone. Therefore, we suggest the experimentalists to carry out high accuracy measurements on a L and a T of b → c transitions which would a great help.
The second purpose of this work is to test the reasonability and application of the diquark picture which is definitely a good approximation at least at the leading order according to the acute discussions on this topic. Therefore, in this work, we use the three-body picture of baryons, i.e. treat the two spectator light quarks as individual on-mass-shell constituents with definite masses and momenta. Indeed, the two light quarks compose a color-anti-triplet and spin-0 (for b(c) or spin-1 (for b(c) ) subsystem. In this picture, we do not priori assume they are loosely bound into a physical composite: diquark. Comparing the numerical results of the decay rates of b → c and b → c with that obtained in terms of the diquark which is supposed to be a point-like boson with a form factor, we find them to be very close. The consistency implies that the diquark picture indeed is valid, namely, in a heavy baryon ( b(c) , or b(c) etc.) two light quarks may be bound into a physical system and can behave as a point-like boson, especially when it serves as a spectator during the transition between two heavy baryons.