Heavy baryon wave functions , Bakamjian-Thomas approach to form factors , and observables in Λ b → Λ c ( 1 2 ± ) ` ν transitions

Motivated by the calculation of observables in the decays Λb → Λc ( 1 2 ± ) `ν, as possible tests of Lepton Flavor Universality (LFU), we present a calculation of the necessary form factors in the quark model. Our scheme combines a spectroscopic model, providing the internal wave functions, and the Bakamjian-Thomas (BT) relativistic formalism to deduce the wave functions in motion, and then the current matrix elements, that amount in the heavy quark limit to the Isgur-Wise (IW) function. This limit is covariant and satisfies a large set of sum rules. This framework has been successfully applied to mesons. On the other hand, for baryons, we meet difficulties using standard spectroscopic models. This leads us to propose a provisory spectroscopic phenomenological model : a Q-pointlikediquark model, non-relativistic, with harmonic oscillator forces, flexible enough to give both a reasonable low-lying spectrum and the expected slope of the IW function through the BT formalism. To begin, we extract this slope from Lattice QCD data and find it to be around ρΛ ∼ 2, which we use as a guideline. Then we find and try to explain why we are not able to reproduce the right ρΛ when using certain typical standard linear + Coulomb potential models, both with three quarks Qqq or in a Q-pointlike-diquark picture, since we get too large or too small ρΛ. These difficulties do not question the BT formalism itself, but seem to derive from the high sensitivity of ρΛ to the structure of the light quark subsystem in a relativistic scheme, in contrast with a non-relativistic treatment. Finally we present the interim model, and after fixing its parameters to yield the correct spectrum Unité Mixte de Recherche UMR 8627 CNRS Unité Mixte de Recherche UMR 6533 CNRS 1 ar X iv :2 00 6. 07 13 0v 1 [ he pph ] 1 2 Ju n 20 20 and ρΛ ∼ 2, we apply it to the calculation of observables. By studying Bjorken sum rule we show that the inelastic IW function has to be large, and therefore the transitions Λb → Λc ( 1 2 − , 32 − ) `ν could be studied at LHCb. Interestingly, some observables in the τ case present zeroes for specific values of q2 that could be tests of the Standard Model. For example, the forward-backward asymmetry for both Λb → Λc ( 1 2 ± ) τν presents a zero for q2 ' mτ √ mb −mc .


Introduction
Possible physics beyond the Standard Model (SM), suggesting Lepton Flavor Universality Violation (LFUV), has been pointed out by data of different experiments on B → D ( * ) ν [1,2,3], summarized in [4]. These experimental results have attracted much attention in terms of analyses within the Standard Model (SM) and also using models for New Physics (NP) [5] - [12]. In particular, following the lattice calculations of form factors in the SM [13], ref. [14] examines Λ b → Λ c τ ν τ with NP operators.
With the intention of providing predictions for observables in Λ b → Λ ( * ) c ν, we have considered the quark model, which can describe simultaneously the ground state and the excitations, not calculated in the present works on Lattice QCD. Moreover, we use the BT relativistic framework, that yields a Lorentz invariant Isgur-Wise (IW) function in terms of internal hadron wave functions. The latter are deduced from a quark model spectroscopic Hamiltonian describing the states at rest, and fitting the observed spectrum. The resulting IW function gives the leading order of the form factors in a heavy quark expansion.
We have used this guideline in the meson case B → D ( * ) ν, for ground state [15] and orbitally excited D mesons [16]. In the meson case we did use, as spectroscopic Hamitonian, the one of Godfrey and Isgur (GI), that describes a wealth of meson data for the qq and Qq systems [17].
In this way, it was obtained a reasonable and theoretically founded description of IW functions, both elastic and inelastic [18].

Relevance of the BT scheme for hadron form factors
One must underline in detail the relevance of the BT scheme for the calculation of form factors of heavy hadron transitions by heavy currents. BT is an approach to hadron motion which can be combined with any internal (rest frame) wave function. In quark model calculations, like [19,20], the spectroscopic model providing these wave functions could be either non-relativistic, as in [19], or possibly relativistic, as in Pervin et al. [20], that consider both cases. But whatever the type of spectroscopic equation, both groups apply the usual non-relativistic treatment for the hadron motion. In the calculation of [19], although a very careful calculation of the spectrum and the wave functions is done, a very small IW slope is found ρ 2 Λ 0.6 − 1., instead of ρ 2 Λ 2 for the Λ b , as indicated by our fit below to lattice QCD [13] and the LHCb data [21].
As has been shown previously in detail in the meson case [15,18,22], the BT calculation gives a large enhancement for the IW slope with respect to the non-relativistic calculation with the same internal wave functions. This is due to the Lorentz transformation of the spatial arguments (i.e. quark momenta) of the wave function for hadrons in motion. This effect gives, with respect to the non-relativistic slope (ρ 2 N R = 1 2 m 2 R 2 for a Gaussian), an additional contribution that is (i) independent of the wave function shape and parameters, and (ii) very large since it is roughly δρ 2 0.75 for a model with a scalar light quark, and δρ 2 1 for a meson (see for instance our discussion in [22]). The Bonn group [23] seems to find such an enhancement in a Bethe-Salpeter approach by applying also the full Lorentz transformation.
We can write an illustrating explicit simple formula for the slope in the BT scheme if we consider a Gaussian wave function exp (−R 2 p 2 /2) for a Q − q bound state, neglecting the light quark spin and the Jacobian factor. The product of the initial and final wave functions with Lorentz transformation along Oz gives Integrating over p, and expanding around w = 1, one finds the slope with neglect of the Jacobian factor. The neglect of the latter factor allows to get a completely analytic result : This +1 is the enhancement with respect to the non relativistic result, which is the first term.
The Jacobian can be expanded in terms of the internal velocity, and the lowest term gives −0.25, whence the final enhancement 0.75 + O(v 2 /c 2 ) for a j P = 0 + light cloud (diquark model of baryons, see Section 4). On the other hand, for a meson, another contribution + 1 4 must be added, corresponding to the heavy quark current. Finally, the slope for a meson is around 1, as observed.
For baryons, the more complex structure of the three-quark wave functions and of the BT expression for the IW function makes this enhancement effect more difficult to evaluate, and strongly dependent on this structure. However, the general expectation is that the enhancement of ρ 2 should be larger than for mesons.
In the simple case of wave functions factorised in | r 2 |, | r 3 | one would have an enhancement for two light quarks twice the one for one light quark, δρ 2 2 × 0.75 = 1.5, pointing naturally towards ρ 2 Λ 2 or more. But as we show below in Section 4, it could be much larger (and too large) for a wave function of the type of ref. [20], which causes new problems. As a general fact, one observes a very strong dependence of ρ 2 Λ on the structure of the wave function, for instance for a gaussian in the relative λ, ρ coordinates, it depends strongly on the ratio R ρ /R λ , and may acquire much too large values. See the analysis of Subsection 3.2.
It should be noted that the Lorentz transformation also implies Wigner rotations of spins, but their effect is found to be small for the ground state IW function.
Another important feature of the BT approach is that it implements automatically the HQET sum rules like Bjorken's or the curvature sum rules, which help to constrain rather efficiently the contributions of higher states.

Failures in the attempt to calculate the baryon IW function from standard spectroscopic models
In trying to apply this scheme to heavy baryons, we have found a number of problems. There are several quark model approaches which could provide the required internal wave functions.
Among the most standard ones (i.e. with linear+Coulomb potential), we quote first the work parallel to GI for mesons, the relativistic Hamiltonian of Capstick and Isgur for the Qqq system [24]. Unfortunately, this is a rather complicated model, which reproduces a very large spectrum of states, but for which it is not easy to obtain the corresponding wave functions. Second, the work of Albertus et al. [19], using a non relativistic kinetic energy, with a very complete study of the states, and which writes explicitly the wave functions, but these are not easy to use in our calculation. Third, there is the quark model study of Pervin, Roberts and Capstick for Λ Q baryons [20], more manageable than the former two models, and to which we refer now.
In the present paper we have computed the IW function Λ b → Λ c in terms of a generic internal Qqq wave function. Then, we have used one of the internal wave functions given by ref. [20] in an harmonic oscillator basis in order to compute numerically the IW function and the corresponding slope. As pointed out in detail below, using the parameters of Pervin et al. [20], we have found a slope ρ 2 Λ 4. This is much larger than the estimate by LHCb, ρ 2 Λ 1.8 [21], and the value that follows from Lattice QCD calculations. Indeed, we describe below a fit to the Lattice data of Detmold et al. [13], that gives ρ 2 Λ 2.
Note that LHCb does not perform properly a determination or measurement of the true ρ 2 Λ that we need, since it would require an extraction of the 1/m Q corrections for each form factor, which do not separate. It is, as qualified by the authors, a "measurement of the shape of the differential decay rate".
We identify the mathematical origin of the large value of the slope obtained from the spectrum and the BT scheme, and we comment on the related work by Cardarelli and Simula in the Light Front formulation of the BT approach [25].
Then, we turn to the simpler scheme of a quark-diquark model, a bound state of a heavy quark and a color triplet pointlike diquark. This model has been widely used in the literature to compute the heavy baryon spectrum and heavy baryon form factors appearing in different processes [28]. Concerning the spectrum, there is the interesting paper by Bing Chen et al. [29], a non-relativistic model with QCD-inspired potential, that, as we will show, presents also problems for the description of the IW slope, that turns out to be too small.

A simple provisory model for calculation of observables
On the other hand, within the quark-diquark scheme, but renouncing to QCD-inspired potentials, a simple non relativistic harmonic oscillator model can be adjusted to give reasonable level spacings, and one can get also the IW slope in the BT scheme ρ 2 Λ 2. In this paper we will adopt, for the moment, this simple model for the internal wave functions in view of the When this paper was in progress, the Mainz group has issued a paper [30] on some observables that could be useful to test LFUV in Λ b → Λ c 1 2 ± , 3 2 − ν transitions, one of the objects of the present paper. However, as our approach is different, we still present our results, and compare with their work and other related literature.

Plan of the paper
The paper is organized as follows. In Section 2 we present a fit to the Lattice QCD data, that yields a slope of the IW function ρ 2 Λ 2, and we quote the value of ρ 2 Λ given by LHCb. In Section 3 we expose the numerical problem that we find on computing the IW slope for Qqq baryons with the wave functions of ref. [20], and we trace back the mathematical origin of this difficulty using a generic gaussian wave function in the spirit of Cardarelli and Simula [25].
As an alternative model, we turn to the quark-diquark model in Section 4, we compute the IW functions for the elastic case and for the L = 0 → L = 1 transitions in the BT scheme from the wave functions of the Bing Chen et al. Hamiltonian [29], and we find a much too low value compared to the lattice result. In Section 5, in front of these difficulties, to compute the observables [31] that could be sensitive to LFUV for 1 2 + → 1 2 ± transitions, we renounce to models with QCD-inspired potentials, and use quark-diquark wave functions deduced from a non-relativistic harmonic oscillator quark model, adjusted to give the desired ρ 2 Λ 2. In Appendix A we define the baryon form factors using different needed conventions, in B we give some details of the involved calculation of the Qqq elastic IW function in the BT scheme, in C we compute the elastic and inelastic quark-diquark IW functions in the BT scheme, in D we make explicit the quark-diquark wave functions within the Bing Chen et al. scheme, and finally in E we give the expressions for the helicity amplitudes and the observables as formulated by the Mainz group, that we have used for our applications in Section 5.
2 LHCb measurement of the dΓ/dq 2 shape, Lattice form factors and slope of the IW function 2.1 LHCb measurement of the dΓ/dq 2 shape The differential rate of the decay Λ b → Λ c + − + ν writes, for m = 0 and heavy quark limit form factors, Of course, here ξ(w) is not the real IW function, but a rough approximation to it, with unspecified 1/m Q errors. In next subsection, we try to extract the IW function from the form factors calculated in Lattice QCD by taking into account the 1/m Q corrections. This is the way one must actually define the IW function ξ Λ (w) and its slope, in accordance with the 1/m Q expansion, and it is this ρ 2 Λ which we use in the discussions which follows. With the "dipole" ansatz LHCb finds the value of his ρ 2 dip parameter [21] ρ 2 dip = −ξ (1) = 1.82 ± 0.03 (4) and the curvature Of course, the very small errors in eqns. (4,5) are not to be taken as the actual errors on the real IW slope and curvature.
The main reason for adopting the shape (3) is that a number of theorems have been obtained that constrain the successive zero recoil derivatives of the baryon IW function [32,33], in particular the bound on the curvature It has been established [33] that the "dipole" form (3), that depends on a single parameter, satisfies these constraints provided that ρ 2 Λ ≥ 1 4 .

Fits to lattice data on form factors
Early studies of the Λ b → Λ c ν lattice form factors in the quenched approximation were provided in refs. [34] and [35]. In the former study, a value was given for the slope of the IW function, ρ 2 Λ 2.4 with a 15% error, showing no dependence on the heavy quark masses and a value was also obtained for the HQET parameter Λ 0.75 with a 20% error.
A great wealth of new precise data in Lattice QCD has been obtained recently by W.
Detmold, C. Lehner and S. Meinel [13], that have given results for all the form factors entering in the process Λ b → Λ c ν within the Standard Model.
Our aim is now to try to extract information on the slope of the IW function ξ Λ (w) and other parameters, Λ and the heavy quark masses m Q (Q = b, c), from these lattice calculations, that are summarized in Fig. 12 of ref. [13].
We adopt a simple HQET model, keeping the form factors up to first order in 1/m Q included, as given in the formulas of Appendix A. Unlike ref. [26] we do not take into account explicitly the QCD perturbative corrections to HQET, and therefore it must be understood that our slope ρ 2 Λ accounts by itself roughly for such effects. Inspection of the formulas of Appendix A shows that, at this first order, besides the dependence of the form factors on the IW function ξ Λ (w), on the heavy quark masses m Q (Q = b, c) and on the HQET parameter Λ, there is another subleading function A(w) that, due to Luke's theorem [27] must vanish at w = 1. For this function, since the domain in w is not large, we adopt the parametrization Moreover, we will adopt the explicit "dipole" form (3) for the leading IW function.

The IW function slope from lattice form factors
In the approximation that we adopt, HQET up to first order in 1/m Q included, there are two quantities that isolate the IW function, where all dependence on Λ and the parameter A (1) defined by (7) cancels. These quantities are differences of ratios that, up to O(1/m 2 Q ) corrections, are identical to the IW function ξ Λ (w), Inspection of the formulas of Appendix A shows indeed that these ratios do not depend on Λ and on the parameter A (1), that cancel in these quantities, To have information on the IW function, we will use the z-expansion parametrization in [13], that we will fit with our HQET model of form factors, that includes up to O(1/m Q ) corrections, made explicit in Appendix A. The lattice data is parametrized by the z-expansion [36] for each form factor where t 0 = (m Λ b − m Λc ) 2 , m f pole and t f + are given in Table VII, a f 0 , a f 1 up to O(z) in Table VIII, and a f 0 , a f 1 , a f 2 up to O(z 2 ) in Table X of [13]. We do not pretend to make a fit on the two ratios (8,9) with their errors. We just take the z-expansion central values at face value, to see if for the two ratios we find reasonable consistent values for the IW function slope ρ 2 Λ , using both expansions up to O(z) and up to O(z 2 ). For the IW function we adopt the "dipole" parametrization (3), that satisfies the rigurous results that constrain the successive zero recoil derivatives of the baryon IW function [32,33].
To perform the fits we select a number of points of the z-expansions for the lattice values for these ratios, up to first order and up to second order in z, and we use the Mathematica package FindFit. For the IW function ξ(w) we consider the domain 1 ≤ w ≤ 1.2 where there are data points measured on the lattice. For the individual form factors we will consider below the z-expansions and the fits for the whole phase space. We can safely conclude that the slope is consistent with the following ranges obtained from the fit. For the first order z expansion we get the domain ρ 2 Λ 2.20 − 2.24 (12) while for the z 2 order we obtain the range ρ 2 Λ 2.03 − 2.25 (13) Although our fits are somewhat naive, from (12,13) we can safely conclude that the data on Λ b → Λ c form factors [13] can be described in HQET up to O(1/m Q ) included, with the slope of the "dipole" form for the IW function (3),

Fits to the different form factors
We do not want to make an overall fit on the whole set of form factors with their errors (errors that we do not master), but just to study individually each form factor of Fig. 12 of [13] by making a fit to the central values of these domains, given by the z-expansion up to order z.
We take these central values at face value, to see if for each form factor we find reasonable results for the set of parameters, and how these parameters compare between the different form factors, i.e. how dispersed they are.
Since we have an independent estimate of the slope of the IW function (12,13), we now fix ρ 2 Λ 2.15 (16) and we use FindFit to perform constrained fits for m c and m b and Λ, assuming the following domains, from different analyses within HQET [37,38], in GeV units : (17) and the slope of the 1/m Q form factor A (1) as a free parameter.
We take now the data for each form factor, the central values of Fig. 12 of [13], that are  Table 1.
We summarize the values for the parameters obtained from the fits for the different form factors in Table 1. Let us comment on Fig. 3 and Table 1. First, the values obtained for the parameters m c , m b and Λ are of course within the imposed limits (17). The fits are quite good for all form factors, except for g + (Λ b → Λ c ) and g ⊥ (Λ b → Λ c ) at large q 2 or near zero recoil w = 1. In the lattice data one sees that g ⊥ (q 2 max ), g + (q 2 max ) < 1, while the calculation of the model gives g ⊥ (q 2 max ) = g + (q 2 max ) = 1. The discrepancy is due to the same 1/m 2 Q correction at w = 1, since one has Indeed, it is well-known that at zero recoil w = 1 there is a negative 1/m 2 Q correction for g ⊥ (q 2 max ), and this explains the discrepancy between the lattice data and the model. The 1/m 2 Q correction satisfies a sum rule that gives its sign, see for example the discussion of the meson form factor F D * at zero recoil in the review paper [39].
As a numerical example, from the range (14), we adopt for ρ 2 Λ 2.15 we find in Table 1 the following ranges for the quark masses and subleading parameters, It is worth to emphasize that the parameter A (1) turns out to be negative and sizeable. This is a new result from the present analysis of form factors. Interestingly, the sign and magnitude is in qualitative agreement with the expression obtained in the non-relativistic quark model,

Correlation between ρ 2 Λ and the slope parameter A (1)
There is a correlation between ρ 2 Λ and the slope parameter A (1). Indeed, taking the heavy quark limit in the expression of the form factors except for A (1), one finds, for all 6 form factors, for small w − 1, that yields an effective slope Of course, one must take into account that the terms O Λ m Q contribute to the coefficient of ρ 2 Λ , so that the relation is somewhat different according to the form factors (Appendix A). For instance and different expressions depending on Λ 2m Q ρ 2 Λ for the derivatives of the other form factors. These differences allow us to determine separately ρ 2 Λ and A (1), but the tendency of the correlation remains the same. This correlation indicates that with some increase of ρ 2 Λ , as it is possible from the above discussion, A (1), which is found negative for ρ 2 Λ 2.15, should decrease in magnitude, or even change its sign.
Note that in the paper [26], the parameter A (1) has been absorbed into the slope, introducing , dependent on the quark masses, and common to all form factors. Their ζ (1) therefore differs from our −ρ 2 Λ . But also, as we have shown above, A (1) can be estimated separately from the lattice data, although with a rather large error.
The values of Table 1 correspond to the choice ρ 2 Λ = 2.15, central value of the domain (14). Had we adopted a higher value for the slope, then A (1) would be negative but with a smaller absolute magnitude, and for ρ 2 Λ 2.5 − 2.6 there is a change of sign for A (1), although this depends on the particular form factor.
On the other hand, in ref. [26] the curvature appears to be rather small in comparison with our fits. Indeed, with a ρ 2 Λ around 2.15 with our dipole fit, which satisfies the well established lower bound (6) on the curvature, we find that σ 2 Λ 5.7, i.e. a term +2.8(w − 1) 2 in the expansion. However, this is partly compensated by our next negative term −3.0(w − 1) 3 , which is still not negligible.
3 Elastic IW function for the Qqq system in the Bakamjian-

Thomas quark model
For the ground state we have the total wave function The flavor wave function writes ϕ , the spin wave function χ µ 231 is antisymmetric in the 2,3 quarks and the full antisymmetry of the baryon wave function follows from the antisymmetry of the color singlet wave function.
For the simple case of the non-relativistic harmonic oscillator, the ground state internal wave function (see for example Appendix A of ref. [40]) writes : where the relative momentum variables p ρ and p λ for m 2 = m 3 = m are defined by The wave function (24) is normalized according to Some words of caution concerning the wave function (24). First, this expression is valid in the limiting case of equal masses for the two light quarks [40] but, in general, crossed terms of the form p ρ .p λ could appear. Here we restrict ourselves to the simplest case of the non-relativistic harmonic oscillator with two light quarks of equal mass.
Assuming that the harmonic oscillator spring constant is flavor-independent, the reduced radii R ρ and R λ are given, in terms of the equal mass baryon radius R, by the expressions : In the center-of-mass p 1 + p 2 + p 3 = 0, relations (25) give In the heavy quark limit m << m 1 , the reduced radii (28) become Finally we can obtain the explicit form of the baryon IW function with harmonic oscillator wave functions by replacing the expression (24) for the initial and final states in the general formula (146) of Appendix B.
We find where the factor (p 2 .v)(p 3 .v)(p 2 .v )(p 3 .v ) in the first line is due to the Jacobian of the change of variables in the BT scheme, and the complicated factor in the last line comes from the Wigner rotations [15], computed in the baryon case in Appendix B.

An attempt to a concrete calculation of the IW function
In Pervin et al. [20], the spectrum of heavy baryonic states has been studied with a linear + Coulomb Hamiltonian, with diagonalisation in harmonic oscillator (HO) or pseudocoulombic (PC) bases and with kinetic energy either non-relativistic or relativistic. In practice, the ground state wave function seems to be well represented by one gaussian or one PC wave function.
Choosing the Hamiltonian with relativistic energy, and the HO basis, we read from their tables in the HOSR entry, the light quark mass and the ground state internal wave function necessary for the BT calculation, the latter being well approximated by one gaussian, eqn. (24). They are given as follows : We have computed the slope of the IW function (31) and have found This value is much larger than the ranges (12,13) determined in Section 2.
Notice that the the last factor in (31), due to the Wigner rotations, gives a very small numerical contribution to the slope.
It must be emphasized that this value is different and larger from the one given by the authors, the reason being that they use a non-relativistic treatment to calculate the form factors, where in principle ρ 2 Λ = 3m 2 R 2 λ . This shows the tendency of the relativistic BT treatment to enhance the slope, which is what one would like. But of course the enhancement is too large, and it is worse with the PC basis.
On trying to understand this disapointing result one notices that, as found by the authors of [20], there could be artefacts due to the smallness of the HO or PC expansion bases. On the other hand, in BT there is for baryons, in contrast with mesons, a particular sensitivity of the value of ρ 2 Λ to the detailed structure of the wave function, as we argue below. This emphasizes the need for larger bases.

General discussion
In the meantime, in view of this conclusion concerning the above gaussian wave function, we proceed as follows. We pursue the investigation with the gaussian shape (24) now considered as a model, with the objective of investigating the dependence of ρ 2 Λ on the shape of the generic internal wave functions in the BT scheme, and in particular to understand the high value obtained above, ρ 2 Λ 4. In fact, a somewhat similar discussion has been done numerically by Cardarelli and Simula [25] in the null plane formalism, which is known to be equivalent to the BT formalism in the heavy quark limit. However, one must avoid to give a physical interpretation to the gaussian wave function, as we will see below. Here we will rather develop a mathematical analysis to understand the variations of the slope ρ 2 Λ . The formula (31) and its expansion at small velocity to extract ρ 2 Λ , keeping for simplicity the terms coming from the gaussians, and disregarding the contributions from the Jacobian and from the Wigner rotations, gives : with 2, 3 labelling the two light quarks, and ... denoting averages on the wave functions. The first term is the non-relativistic contribution, the two others are relativistic corrections. The third term corresponds to crossed terms that, of course, are absent in mesons.
This formula shows that ρ 2 Λ depends on two parameters, instead of one in the non-relativistic limit (the term ρ 2 Λ = 3m 2 R 2 λ ), and one can get very high values because for R 2 ρ < 3R 2 λ the last two terms in the expression are positive, and when R 2 ρ becomes small, the ... averages become large. Indeed, i.e. the momenta are equal, large and antiparallel in average (and of course p 0 2 p 0 3 becomes also large). Then, the magnitude of ρ 2 Λ is controlled by the ratio R 2 λ /R 2 ρ , and ρ 2 Λ diverges when R 2 ρ → 0 at fixed R λ , in agreement with the numerical findings of Cardarelli and Simula in the null plane formalism [25].
Though, the interpretation of this limit as corresponding to the point-like diquark model given in this reference is at odds with our analysis of the quark-diquark model, which gives a small ρ 2 Λ , analogous to mesons (Section 4 of the present paper). This is understandable, since in this limit R 2 ρ → 0 the gaussian is not the physical solution calculated from a QCD inspired Hamiltonian.
In fact, the last two terms in eqn. (35) diverge for R ρ /R λ → 0 or ∞, but in one case they have the same sign, and whence ρ 2 Λ diverges, while in the other case the divergences cancel when R λ is help fixed, and ρ 2 Λ tends to a finite positive value. It can be seen that these large values are related to the crossed term in the arguments of the two gaussians with the coefficient −2( p 2 . p 3 ) 3R 2 λ − R 2 ρ give a large positive contribution balancing the factorisable one −( p 2 2 + p 3 2 ) 3R 2 λ + R 2 ρ when R 2 ρ approaches 0. On the other hand, imposing which corresponds to cancelling the "crossed" terms, the expression (35) simplifies very much and one finds : corresponding to the factorization of the wave function in p 2 , p 3 . The value (39) is not at odds with the slope determined from the lattice data in Section 2, ρ 2 Λ 2.
To repeat, the BT result is quite unlike the non-relativistic treatment, which gives always it depends now strongly on R ρ /R λ . One sees that in the relativistic treatment ρ 2 Λ can get arbitrary large values, while none of the two radii is supposed to be large.
Of course, let us recall that there is no claim to a dynamical calculation in all this discussion, but only an analysis of the relation between a generic gaussian internal wave function and ρ 2 Λ , specific to the relativistic BT formalism.
However, it is interesting to note that the condition (39), which corresponds to a reasonable value of ρ 2 Λ , corresponds also to a situation where the distance between the two light quarks would be larger than the distance between each light quark and the heavy quark. This seems consistent with the intuition that the Compton wave length of each light quark is large, and this is in fact the situation which seems to be found in dynamical calculations, like the one of Hernandez et al. in the non-relativistic quark model [41], as well as in lattice studies [42] [43].
Indeed, it is very important to recall that also in lattice QCD calculations one finds a qq system with a large separation. Let us emphasize that in conclusion of these calculations, the term diquark must be taken with care since it is often meant on the contrary as a pointlike diquark, especially when speaking of diquark models. And, of course, these calculations question the very idea of a point-diquark model, at least when claiming to QCD inspired models, as we discuss in the next section.
Let us recall now another important conclusion coming from the above discussion. In the BT scheme, the value of the IW function slope for the Qqq system depends strongly on the spatial configuration of the light diquark. This illustrates strikingly the contrast between the BT scheme and the non-relativistic treatment of the center-of-mass motion of hadrons, for which there is no dependence of the slope on R ρ , but only on R λ . Therefore, in this relativistic scheme there is a need to have a very good calculation of the wave function.
Interestingly, in ref. [19]) the wave function has been calculated very carefully, although in a spectroscopic model with non-relativistic kinetic energy, which may be less worrying for a baryon. As to the authors themselves, they propose a rather low value ρ 2 Λ 0.6 − 1., too low of course. But this value derives from the non-relativistic treatment of the center-of-mass motion of the baryons.
It would be worth applying the BT formalism to the wave function of [19]) to see whether it yields a correct slope. Indeed, the relativistic BT treatment could enlarge the value appreciably, as explained above and in Subsection 1.1.

The Q-pointlike-diquark models
As a possible alternative, the models with a point-like diquark instead of two light quarks would be attractive because of their simplicity. One must note that the diquark may be also considered as extended, like in the works of Ebert et al. [28], but this is a different idea, outside of the present discussion (see also ref. [44]). The quark-diquark model has been widely used to compute properties of the baryon spectrum, and also relevant form factors in heavy baryon transitions [28].
Nevertheless, considering the several findings that have been recalled in the previous section, showing definitely that the qq light quark subsystem has a large size, comparable with the one of the whole baryon, it is paradoxical to appeal to a point-like diquark model. And indeed, our conclusion below in subsection 4.2 is that such a model is not valid in the context of the QCD-inspired potentials, since it leads to a too low value ρ 2 Λ 1 as for mesons, which is quite logical since they are both two-body bound states with one heavy quark, and the potential is quite similar to the one for mesons.
On the other hand, this negative argument does not apply if we renounce to a QCD-inspired potential and introduce a non standard harmonic oscillator potential, whose strength can be freely adjusted. And indeed, we develop such a model as a provisory solution in the next section.
In subsection 4.1 we first develop the general BT framework for models with scalar point-like diquark model, which will serve for both sections and then apply it to the model of Bing Chen et al., with a standard QCD-inspired potential, in subsection 4.2.

Isgur-Wise functions in the BT scheme
Let us indeed present the general calculation of the IW functions for a scalar 0 + and 3 under color, point-like particle, in the field of a heavy quark. As we will see now, there are no Wigner rotations in this case, and the BT results for IW functions simplify enormously.

Elastic IW function
One finds the simple expression (149) of Appendix C, where m D denotes is the scalar diquark mass, of the order of twice the light quark mass, m D 2m.

IW function for L = 1 excited states
In this case one finds expression (154) of Appendix C, where one can see that the 1/(w − 1) singularity in the overall factor cancels with the numerator , that vanishes when w → 1.

An improved bound on the slope
However, in this Q-diquark model, one can demonstrate a better lower bound due to the absence of the Wigner rotations, just by using the careful analysis of the different contributions to the meson IW slope given in ref. [22].
One finds that the expression for the slope writes where and one has demonstrated that the lower bound of expression (48) is given by i.e., an improved bound relatively to the general bound (47).
Moreover, this bound was obtained on general grounds for a heavy baryon with light cloud j = 0 [33] for the shape (3) of the IW function.

Elastic and inelastic IW functions from wave functions in a QCDinspired potential model
The heavy baryon spectrum has been studied by Bing Chen et al. within the Q-diquark description with a QCD-inspired Hamiltonian [29], where µ is the reduced mass, and p is the relative momentum of the heavy quark Q and light point-like diquark of mass m D .
One notices that the potential in (51) is very similar to the one for a meson. This is easily understood : the interquark potential inside baryons is known to be half the one inside mesons, but on the other hand there are two quarks on a diquark. This leads to a similarity in wave functions and finally for ρ 2 Λ , except that the mass here is heavier than for a light quark. The wave functions corresponding to the spin-independent part of the Hamiltonian (51) are given in Appendix D for the heavy quark limit, and with the free β parameter characterizing the variational basis chosen to β = 0.4. Inserting the heavy quark limit wave function ϕ  one finds the elastic IW function ξ Λ (w) of Fig. 4. An excellent fit is the "dipole" function (3) with the slope and curvature The slope is lower than the value ρ 2 Λ 2 obtained from the data of Lattice QCD described in Section 2. This low value is easily understood because this model amounts to a meson system, except for details of spin and for the light diquark mass which should be larger than for one quark. One must also take into account the difference of definition for baryons against mesons, that have a +1/4 for the slope.
Consequently, the much too low value of ρ 2 Λ (55) compels us to abandon the model, at least for form factors, as this is seen to be an unavoidable consequence of the scheme.  Why one fails in the case of the Bing Chen et al. is clear from the discussion : the pointlike diquark assumption directly contradicts the dynamical calculations of the three quark system, in particular those of lattice QCD, which show definitely that the diquark system has a large extension. In fact the model is close to a heavy meson, with a similar potential, and the BT formalism yields consistently ρ 2 Λ not much above 1. For the Pervin et al. wave function, one has no reason to suspect the linear + Coulomb spectroscopic Hamiltonian, and the reason is less obvious : the calculation of the wave function clearly requires larger bases, since the authors have observed a very large discrepancy between the HO and PC bases -all the more since in a three-quark system, as we have shown in Section 3, the ρ 2 Λ deduced from BT is very sensitive to the details of the wave functions, in contrast with a non-relativistic treatment.
Then, leaving the correct solution of the three-quark case for a further investigation, we turn presently, for the phenomenological purpose of computing the observables, to a very simple model that is able to fit the observed ρ 2 Λ . It is a point-like diquark model, but quite different from the one above with a QCD-inspired potential, with now a harmonic oscillator potential of arbitrary strength, which can be fitted both to the low-lying spectrum and to the desired ρ 2 Λ . Such a model is analogous in spirit with HO models used in the beginning of the quark model, except for the further simplification of using a pointlike diquark picture. The reason to expect sensible results from such a rough model is the fact that, in a first approximation, ρ 2 Λ seems the main parameter controlling the heavy limit of the form factors, because dipole fits describe well the overall shape of ξ Λ (w) both in the model and on the lattice.
Let us assume harmonic-oscillator wave functions for the ground state Λ Q (Q = b, c) and for the tower of the radially excited L = 1 states (n ≥ 0) according to the Hamiltonian where the reduced mass µ is given by (52)  According to (58) the wave functions read that are normalized according to dp (2π) 3 | ϕ(p) | 2 = 1 We will consider also the wave functions for n > 0 and L = 1 in order to verify that Bjorken SR holds.
Let us consider harmonic oscillator parameters that describe qualitatively the spectrum data, GeV. Therefore, the spring tension (59) will be Although the quantum numbers are still not confirmed, we consider now the natural candidate for the radial excitation, as assumed in [29].
However, since the IW function is defined in the heavy quark limit, we should take the reduced mass for m Q → ∞. To summarize, the spring tension K is kept fixed and the reduced mass becomes in the heavy quark limit µ → m D where m D is the diquark mass. One has then, in the heavy quark limit, the radius squared of the wave function, One finds for some illustrative cases, for m c = 1.25 GeV, the radius squared, the elastic slope ρ 2 Λ and, using formula (42), the square of the n = 0 inelastic IW function L = 0 → L = 1 at zero recoil | σ Notice that within HQET one has, for baryons, the parameter Λ, Λ m D (71) and the value Λ = m D 0.8 GeV is precisely the one adopted in the HQET study of Leibovich and Stewart [37].
Here, to compute the interesting observables, we would like to adjust m D in order to obtain the central value for the slope obtained from the lattice data. We get roughly, We plot in Fig. 6 the elastic IW function for the set of parameters (72). A very good fit to the IW function of Fig. 6 is given by the "dipole" form with ρ 2 Λ = 2.15. Comparing the values for ρ 2 Λ and | σ (0) Λ (1) | 2 , we observe that the lowest radial excitation n = 0 largely dominates the r.h.s. of Bjorken SR (46). Therefore, we conclude that the lowest inelastic IW function (L = 0, n = 0) → (L = 1, n = 0) is large, and thus there is a good prospect for this transition to be well observed at LHCb.
We plot in Fig. 7 the inelastic IW function σ Λ (w) with the set of parameters (72).
A reasonable "dipole" fit to Fig. 7 is given by

Observables in Λ
The Mainz group has extensively formulated a number of different observables that could allow to test Lepton Flavor Universality Violation [31]. The expressions for the observables in terms of helicity amplitudes are given below in Appendix E.

τ ) transitions
For the numerical calculations of the form factors and helicity amplitudes we adopt the "dipole" shape expression for the IW function, with the slope (14) determined from the lattice data in Section 2, The ansatz (75) is close to the numerical calculation in the BT model within the Q-diquark scheme with HO NR internal wave function (62) with parameters (72).

For the function A(w) we adopt
where the function f (w) satisfies f (1) = 1 and is introduced to soften the behaviour of A(w) for large w, near w max , because lattice data give only the slope (19). As an example, we could use simply f (w) = ξ Λ (w). So, we take We will below comment further the role of the function f (w) in (76), when discussing the comparison of the spectrum with experiment in section 6.1.1. Moreover, for m c , m b we use the central values (19), and for Λ we adopt the value of our model (72), that agrees within errors with the lattice determination (19), The observables are given in Appendix E in terms of the helicity amplitudes H V /A λ 2 ,λ W , that are given in terms of the form factors by the expressions, In the physical processes the V − A chiral combination (159) appears, and one has the parity relations between the V /A helicity amplitudes The form factors in (79) are described in Appendix A.1.

The normalized theoretical rate compared to LHCb data
Among the observables, only the shape of the LHCb data on the differential rate is known, but not the absolute magnitude [21]. We compare the LHCb rate normalized to one with the predictions of our model.
As a first remark, let us notice that our model of form factors up to O(1/m Q ) included, with essentially a single main parameter ρ 2 Λ can well reproduce the LHCb normalized rate, as shown in Fig. 8. We have used the "dipolar" shape (75) with a slope slightly lower than the domain obtained from the lattice, ρ 2 Λ = 2, the mass parameters (78), and the assumption A(w) = 0. Figure 8: Normalized rate dΓ dq 2 /Γ compared to the LHCb data. We use the "dipolar" shape with ρ 2 Λ = 2, the mass parameters (78), and we assume A(w) = 0.
We consider next the comparison with the parameters obtained from the lattice.  We compare these different choices to the data in Fig. 9. We observe that the set of parameters (1) describes the data very poorly, in particular due to the linear behaviour of A(w) and also due to a slightly too large slope. The set (2) is somewhat better, due to the softening of A(w) at large w. Finally, the parameters (3) describe the data rather well, although not as well as the naive choice of Fig. 8. The main conclusion of this discussion is that the LHCb normalized rate agrees within errors with the fit to the lattice data of form factors performed in Section 2, that are summarized in formulas (75,76,78).

Other observables
We have seen that there are no sizeable differences between the set of parameters (ρ 2 Λ , A (1)) = (2.05, −0.20) (Fig. 9) and the naive ansatz (ρ 2 Λ , A (1)) = (2, 0) (Fig. 8). For the calculation of the rest of the observables we will use for simplicity the latter set. Moreover, all observables that are given by ratios of squared of helicity amplitudes are not sensitive to the small differences between the parameters used in Fig. 8 and the set (3) in Fig. 9.

Comments on other observables Λ
It is interesting to observe the shape of the forward-backward asymmetry in Fig. 11 for the electron mass and for the τ mass, where we observe a zero in the distribution.
From (165), for m 0, the electron case, the FB asymmetry is given by Figure 10: dΓ dq 2 for the electron and tau modes. In the electron case, one has dΓ dq 2 → 0 for q 2 → 0.     : Longitudinal lepton polarization P z (q 2 ) for the electron and tau modes. P e z (q 2 ) is very close to −1, unlike P τ z (q 2 ). Figure 16: Transverse lepton polarization P x (q 2 ) for the electron and tau modes. P e x (q 2 ) is very small, unlike P τ x (q 2 ).
From the V − A structure of the theory, the left-handed final baryon dominates, and therefore we expect to have the inequality and similar inequalities for other helicity amplitudes. From (82) and (81) we expect A e F B (q 2 ) to be positive for all values of q 2 , as we observe in Fig. 11.
Let us see how the inequality (82) holds in our model of the form factors described in Appendix A.1. Just for illustration, keeping only the heavy quark limit terms, one finds indeed In the presence of a non-vanishing lepton mass m , the FB asymmetry (165) presents a zero.
In particular, for the τ case, one has a zero in the F B asymmetry as shown in Fig. 11. It is interesting to have a theoretical idea of the position of this zero, which, keeping only the heavy quark limit terms, is which qualitatively agrees with the one of Fig. 11, computed taking into account 1/m Q subleading terms.
We observe in Fig. 15 that in the τ case, the longitudinal lepton polarization P τ z (q 2 ) has a zero in the neighborhood of q 2 4. GeV 2 . Indeed, performing an expansion in powers of mτ mb and mc mb , one finds the position of this zero that is numerically reasonable.

τ ) transitions
To compute the form factors we refer to the expressions and discussion of Appendix A.2, taken from Leibovich and Stewart [37]. We neglect the subleading Lagrangian perturbations (109), that amounts to take (cf. (111))σ(w) ≡ σ Λ (w) and the central value of (115),σ 1 = 0. We are left with the leading and subleading contributions proportional to the inelastic IW function.  The helicity amplitudes H V /A λ 2 ,λ W are in this case, In the physical processes the V − A chiral combination (159) appears, and the parity relations between helicity amplitudes are now, The interesting observables of Appendix E are given in Figs. 17-23. For the electron case, the FB asymmetry is given by (81) and again, although the final state parity has changed, from the V − A structure of the theory, the left-handed final baryon dominates, and we expect to have the inequality (82) and A e F B (q 2 ) to be positive for all values of q 2 , as we indeed observe in Fig. 18.
It is interesting to see how the inequality (82) holds in our model of the form factors described in Appendix A.2. Similarly to what we have done above for the ground state, keeping only the heavy quark limit terms, one finds indeed In the presence of a non-vanishing lepton mass m , the inequality (91) does not follow for all values of q 2 . Figure 17: dΓ dq 2 for the electron and tau modes. In the electron case, one has dΓ dq 2 → 0 for q 2 → 0.
For the τ case, one has a zero in the F B asymmetry. Keeping only the heavy quark limit terms one finds the same value for the position of this zero as in the elastic case (85), Also, we observe in Fig. 22 that, in the τ case, the longitudinal lepton polarization P τ z (q 2 ) has a zero in the neighborhood of q      : Longitudinal lepton polarization P z (q 2 ) for the electron and tau modes. P e z (q 2 ) is very close to −1, unlike P τ z (q 2 ). Figure 23: Transverse lepton polarization P x (q 2 ) for the electron and tau modes. P e x (q 2 ) is very small, unlike P τ x (q 2 ).

τ / observables sensitive to LFUV
We now compute the relevant ratio of rates to test LFU (Lepton Flavor Universality), and we find, for the ground state, and for the transition to the excited state, An interesting observable is the forward-backward asymmetry A F B , that has a very different behaviour for the light leptons and for the τ . In this latter case A τ F B , unlike A e F B , presents a zero at We would like to compare some aspects of their approach with our own.
First, in the CCQM, baryons are composites of Qqq, with the quantum numbers of the different states given by local interpolating fields with the correct quantum numbers, a compositeness condition, and a simple universal gaussian form for the vertex functions. Moreover, the calculation is done at finite mass [46].
Our approach is a naive quark model in a Q-diquark scheme with a harmonic-oscillator potential, reproducing qualitatively the spectrum and giving within the BT scheme a reasonable  [30], so that they seem to be on a firm ground.

Conclusions
Our objective has been the calculation of the observables in the decays Λ b → Λ ± c ν, that could provide tests of Lepton Flavor Universality Violation.
We have done this in a quark model that, unlike present Lattice QCD calculations, allows not only the computation of transitions within the ground state, but also those to the L = 1 excitations.
The BT method is very suited for such a calculation, the wave functions are three-dimensional but the result is covariant in the heavy quark limit.
The BT approach can explain the value of the slope of the baryon IW function ρ 2 Λ , very different of the non-relativistic value, as it happens for mesons.
The slope ρ 2 Λ is a very important parameter to describe the form factors, and it is at the same time a discriminant of the different models. The adopted dipolar fit for the IW function satisfies a number of theorems that the different derivatives of the IW function must satisfy.
To test the different models, we have analysed the lattice data on the form factors, and we have shown that the slope of the IW function is of the order ρ 2 Λ 2. On the other hand, we have demonstrated that the slope of the subleading form factor A (1) can be determined independently of ρ 2 Λ . We have found a number of difficulties of standard QCD-inspired Hamiltonian approaches, both in the three quark Qqq model and also in quark pointlike-diquark models. We have thoroughly discussed these problems.
In view of these difficulties, we have adopted a preliminary quark pointlike-diquark model that allows a qualitative description of the spectrum, and of the IW slope ρ 2 Λ 2, in agreement with lattice data.
We have computed the different observables proposed by the Mainz group for both transitions We have seen that some observables, the forward-backward asymmetries and the longitudinal lepton polarization, present a zero at some characteristic value of q 2 for the τ case. The positions of these zeros could provide tests of physics beyond the Standard Model. In particular, the forward-backward asymmetry for both Λ b → Λ c 1 2 ± τ ν cases presents a zero for q 2 m τ m 2 b − m 2 c . We do not study for the moment the case of the inelastic transitions Λ b → Λ c 3 2 − ν ( = e, τ ). In the quark model one has to consider a S = 1 diquark coupled to L = 1. We postpone this study since we would like to analyze and compute the different observables that could be interesting for our purpose, besides the one computed in ref. [30].
As a word of caution, we have to say that our results for the observables are preliminary, as we will need in the future to treat systematically the three quark system Qqq to study the spectrum and the IW function, and then turn to phenomenological applications.

Appendix A
A.1 Baryon form factors Λ b → Λ c 1 2 + up to order 1/m Q From Falk and Neubert [47], the 6 conventional form factors f i , g i (i = 1, 2, 3) write The form factors f 3 , g 3 will contribute to Λ b → Λ c τ ν.
The alternative notation, convenient for HQET, is given in terms of the four-velocities The form factors f i , g i write, in terms of the F i , G i : In terms of HQET form factors up to order 1 m Q one has where the 1 m Q corrections read The terms proportional to Λξ Λ (w) correspond to the first order Current perturbation in HQET, while the form factor A(w) corresponds to the Lagrangian insertion perturbation Luke's theorem [27] implies at zero recoil An interesting feature of formulas (98,99) is that the form factors f 2 , f 3 and g 2 , g 3 are of order 1/m 2 Q . This fact can have consequences for the comparison between Λ b → Λ c τ ν and Λ b → Λ c ν.
Let us finally give the notation for the form factors used in the Lattice calculations [13]. In terms of the form factors (96,97) the defintion used in [13] is the following.
For the vector form factors : and for the axial form factors : The matrix elements read, Notice the presence (absence) of γ 5 in the V (A) matrix elements for Λ b → Λ c 1 2 − due to the intrinsic negative parity of the final state.
The alternative notation in terms of the four-velocities is given by [37] < and the relation between both notations is Neglecting for the moment the subleading terms dependent on Lagrangian insertions and keeping only the subleading 1/m Q terms that are proportional to the inelastic IW function σ(w), the form factors (106) are given by the expressions of Leibovich and Stewart [37], The subleading Lagrangian perturbations give the following extra contributions to the preceding form factors [37], According to Leibovich and Stewart, the chromomagnetic functions φ mag are expected to be small because the j P = 1 − doublet mass splittings are small, and they are taken The functions φ (Q) kin can be absorbed by the Isgur-Wise function by replacing σ with Moreover [37] assume as predicted by QCD in the large N c limit, and thereforẽ One is left then with two IW functions,σ(w) and σ 1 (w) and defining the ratiô Leibovich and Stewart assume a constant ratio forσ 1 (w) = constant =σ 1 within the range The Qqq elastic IW function ξ Λ (w) in the BT scheme Let us begin with the general formula for a transition matrix element in the Bakamjian-Thomas relativistic quark model in terms of 2 × 2 matrices [15] : where 1 labels the active quark, the matrix element of the currrent operator O is : and the vectors k i , the 0-components k 0 i and p 0 i , M 0 and the Wigner rotations R i are functions of the p i defined as follows : where B p is the boost ( p 2 , 0) → p and D i (R) is the matrix of rotation R for the spin S i .
The internal wave function of the baryon Λ Q with heavy quarks Q = b or c and polarization µ will write because the spectator quarks 2, 3 are in a state of spin and isospin 0, and the notation χ Considering the polarized states Λ Q , the matrix element (116) writes then < P , µ | O | P, µ > = dp 2 (2π) 3 dp 3 (2π) 3 From the wave function (119) one gets and using the relation one obtains < P , µ | O | P, µ > = 1 2 dp 2 (2π) 3 dp 3 (2π) 3 where we have omitted the index indicating on which quark the Wigner rotation acts, and keep it only on the rotation, because all these matrices act on the spin 1 2 . Equation (121) is the final formula in the 2 × 2 matrix formalism and at finite mass.
We now pass to a 4 × 4 matrix formulation : and will have, for the spinor matrix element : In terms of the boosted spinors the spinor matrix element in (125) writes and using the identities We have now to compute the trace in formula (123) that reads, in the 4 × 4 Dirac matrix formalism and in terms of the boost matrices : u because 1 + γ 0 commutes with the Wigner rotations, the quarks 2, 3 are spectators and then one has p 2 = p 2 and p 3 = p 3 and we have inserted the products B −1 u B u = B −1 u B u = 1 within the trace.
Consider now the product of wave functions The radial wave functions are rotational invariant, so that they can be redefined as follows : and similarly for ϕ(k 2 , k 3 ).