Helicity amplitudes in $B \to D^{*} \bar{\nu} l$ decay

We use a recent formalism of the weak hadronic reactions that maps the transition matrix elements at the quark level into hadronic matrix elements, evaluated with an elaborate angular momentum algebra that allows finally to write the weak matrix elements in terms of easy analytical formulas. In particular they appear explicitly for the different spin third components of the vector mesons involved. We extend the formalism to a general case, with the operator $\gamma^\mu -\alpha\gamma^\mu \gamma_5$, that can accommodate different models beyond the standard model and study in detail the $B \to D^{*} \bar{\nu} l$ reaction for the different helicities of the $D^*$. The results are shown for each amplitude in terms of the $\alpha$ parameter that is different for each model. We show that $\frac{d \Gamma}{d M_{\rm inv}^{(\nu l)}}$ is very different for the different components $M=\pm 1, 0$ and in particular the magnitude $\frac{d \Gamma}{d M_{\rm inv}^{(\nu l)}}|_{M=-1} -\frac{d \Gamma}{d M_{\rm inv}^{(\nu l)}}|_{M=+1} $ is very sensitive to the $\alpha$ parameter, which suggest to use this magnitude to test different models beyond the standard model. We also compare our results with the standard model and find very similar results, and practically identical at the end point of $M_{\rm inv}^{(\nu l)}= m_B- m_{D^*}$.

In the present work we retake this line of research and study the polarization amplitudes in semileptonicB → Vνl decays, applied in particular to theB → D * ν l reaction. We look at the problem from a different perspective to the conventional works where the formalism is based on a parametrization of the decay amplitudes in terms of certain structures involving Wilson coefficients and form factors. A different approach was followed recently in the study of B or D weak decays into two pseudoscalar mesons, one vector and a pseudoscalar and two vectors [40]. Starting from the operators of the standard model at the quark level, a mapping is done to the hadronic level and the detailed angular momentum algebra of the different processes is carried out leading to very simple analytical formulas for the amplitudes. By means of that, reactions likeB 0 → D − s D + , D * − s D + , D − s D * + , D * − s D * + , and others, can be related up to a global form factor that cancels in ratios by virtue of heavy quark symmetry. The approach proves very successful in the heavy quark sector and, due to the angular momentum formalism used, the amplitudes are generated explicitly for different third components of the spin of the vectors involved. In view of this, the formalism is ideally suited to study polarizations in these type of decays.
Work along the line of [40] is also done in [41] in the study of the semileptonic B, B * , D, D * decays intoνl and a pseudoscalar or vector meson. Once again, we can relate different reactions up to a global form factor. If one wished to relate the amplitudes of different spin third components for the same process, the form factor cancels in the ratio and the formalism makes predictions for the standard model without any free parameters.
In the present work we extend the formalism and allow a (γ µ − αγ µ γ 5 ) structure for the weak hadronic vertex which makes it easy to make predictions for different values of α that could occur in different models BSM (α = 1 here for the SM). We evaluate different ratios for the B → D * ν l reaction. Work on this particular reaction, looking at the helicity amplitudes within the standard model, was done in [42]. A recent work on this issue is presented in [43] where the B → D * ν τ τ is studied separating the longitudinal and transverse polarizations. The same reaction, looking into τ and D * polarization, is studied in [44]. Helicity amplitudes are also discussed in the relatedB * → P lν l reactions in the recent paper [45].
The formalism of Ref. [41] produces directly the amplitudes in terms of the third component of the D * spin along the D * direction. This corresponds to helicity amplitudes of the D * . The formulas are very easy for these amplitudes and allow to understand analytically the results that one obtains from the final computations. Not only that, but they indicate which combinations one should take that make the results most sensitive to the parameter α that will differ from unity for models BSM.
We find some observables which are very sensitive to the value of α, which should stimulate experimental work to investigate possible physics BSM.

II. FORMALISM
We want to study the B → D * ν l decay, which is depicted in Fig. 1 for B − → D * 0ν l l − The Hamiltonian of the weak interaction is given by where the C contains the couplings of the weak interaction. The constant C plays no role in our study because we are only concerned about ratios of rates. The leptonic current is given by and the quark current by In the evaluation of B − → D * 0ν l l − decay we need where t is the transition amplitude, and for simplicity which can be easily obtained with the result [46] L αβ = 2 where we adopt the Mandl and Shaw normalization for fermions [47]. In Ref. In evaluating the quark current, we use the ordinary spinors [48] where χ r are the Pauli bispinors and m, p and E p are the mass, momentum and energy of the quark. As in Ref. [46] we take where m B , p B , and E B are the mass, momentum and energy of the B meson, and the same for the c quark related to the D * meson. Theses ratios are tied to the velocity of the quarks or B mesons and neglect the internal motion of the quarks inside the meson. We evaluate the matrix elements in the frame where theνl system is at rest, where p B = p D * = p, with p given by where M (νl) inv is the invariant mass of the νl pair. By using Eq. (8) we can write and A , B would be defined for the D * meson, simply changing the mass in Eq. (7).
In the present work, we are only interested in the B − → D * 0ν l l − decay, which means J = 0, J = 1 decay.
As in [41] we need to evaluate L αβ Q α Q * β which sums over the polarizations ofν l l, but keeping M fixed. We have where M 0 and N i , written in spherical coordinates, is with C(· · · ) a Clebsch-Gordan coefficient.
In addition to the p dependence (and hence M (νl) inv ) of these amplitudes, in [41] there is an extra form factor coming from the matrix element of radial B and D * quark wave functions.
However, in our approach we normalize the different helicity contributions to the total and the effect of this extra form factor disappears.
The magnitude L ij N i N * j can be written in spherical coordinates as M in Eqs. (15), (16), and (17) stands for the third component of the D * spin in the direction of D * . Hence these are the helicities of the D * . Note that in the boost from the B rest frame to the frame where B and D * have the same momentum andνl are at rest, the direction of D * does not change and the helicities are the same. We can see that the sum of these expressions for the three helicities gives the same result as the sum obtained in Ref. [41] using properties of Clebsch-Gordan and Racah coefficients.

III. RESULTS
The differential width is given for where p D * is the D * momentum in the B rest frame and p ν theν momentum in the νl rest frame, The factor m ν m l in the numerator of Eq. (19) is due to the normalization used in [47] and cancels exactly the same factor appearing in the denominator of Eqs. (15), (16) and (17). inv goes to its maximum, then p → 0 and Bp, B p go to zero.
Taking into account the behaviour of Bp and B p depicted in Fig. 2, we can see that when inv goes to its maximum, |t| 2 goes to the same value 2(AA ) 2 It is also interesting to see that This means that the differential width dΓ/dM (νl) inv for this difference goes as (1−BB p 2 )(B p− Bp) and the difference of these two distributions goes to zero, both as M going to its maximum. We show also these results in Figs. 3 and 4. The total differential width is given by , all divided by the total differential width R of Eq. (22).
We also appreciate in Fig. 4 that the ratio of dΓ inv . In Fig. 4 we also see a smooth transition from 1 to 1 3 for the M = 0 case. The rapid transition to zero of some of the amplitudes discussed and the wide change of values for the (a), (b), (c) and (d) cases in the figure make these magnitudes specially suited to look for extra contribution beyond the SM.
To give a further insight into this issue we stress that the reason for the zero strength at inv → 0. However, for M = ±1, M 0 = 0 and N µ goes to zero in that limit. This said, the models beyond the SM which could provide finite contribution for M = ±1, or a sizeably bigger one, are those that go beyond the γ µ − γ µ γ 5 structure in the quark current, like leptoquarks or right-handed quark currents of the type γ µ + γ µ γ 5 [49][50][51]. We discuss this case below.
Some models BSM have quark currents that contain the combination γ µ + γ µ γ 5 . The models mentioned above could be accommodated with an operator We shall call a−b a+b = α and study the distributions for different M as a function of α. We have thus the operator γ µ − αγ µ γ 5 .
Using the same formalism of [41] it is easy to see the results as a function of α. We obtain the following results: and N i written in spherical coordinates is Then, the different helicity contributions are given by 2) M = 1 3) M = −1 Since 1 − BB p 2 and B p − Bp go individually to zero for M   We also see that now Then, it is also interesting to see what happens for the ratio 1 We show these results in Fig. 6. We can see that this magnitude keeps rising up as α goes This means that there is only one independent amplitude for all these processes. This is reminiscent of the heavy quark symmetry [64,65] where all form factors can be cast in terms of only one in the limit of infinite masses of the mesons. In view of this, let us face this issue here to see the heavy quark symmetry implicit in the approach of [41] which we follow here.
The key point in our approach, which allows us to express the quark matrix elements in terms of the meson variables, is Eq. (8). Let is take the first relation p b m b = p B m B . In the B meson at rest there is a distribution of quark momenta due to the internal motion of the quarks, p in . If we make a boost to have the B with a velocity of v, we will have where we have split p in into a longitudinal and transverse part along the direction of v. We can write now The relative correction factor is but since p in,L has positive and negative components the correction is of order Let us remark that around M and their effect is further negligible. One can repeat the argumentation for the second relation of Eq. (8). This indicates that in theνl rest frame, where we evaluate the matrix elements, Eq. (8) is very accurate. However, it is only exact in the strict limit that m B , m D * go to infinite. Hence, it should not be surprising that our method implements automatically the symmetries of heavy quark physics.
In order to test this hypothesis let us first study theB → Dνl transition. We have [66] < D, (M D → M D * for theB → D * ν l transition). Similarly (using 0123 = 1) we have [8,66] <D * ,λ,P |Jµ(0)|B,P > √ m B m D * In the heavy quark limit, with the quark masses going to infinite, one finds [8,66] h with ξ(w) the Isgur Weise function, and with a certain normalization of J µ , ξ(w) at the end This condition appears naturally in the quark model since for w = 1 the momentum transfer is zero and the wave functions with very large quark masses are also equal. Hence the quark transition form factor is unity.
We take the D * polarization vectors consistent with our convention in [41] for the angular momentum states By using these polarization factors we compare the J 0 , J i (Jμ in spherical basis) matrix elements with M 0 and Nμ of the expressions found in [41].
We find Because of our normalization for J µ , all these functions are normalized to the value the symmetry of heavy quark physics, and provides an w dependence for these functions.
It is interesting to compare our results with those of [9]. There a quark model calculation is done. and the quark matrix elements are evaluated, including the transition form factor from B to D * which we do not evaluate with the claim that it cancels in ratios of amplitudes for different M . We see that h + in [9] is qualitatively similar to ours, although it falls faster with w. The difference with us are of the order of 15% at the maximum value of w, indicating in any case a soft transition matrix element.
Next, in order to connect with the standard model we follow the formalism of [18,67] where q µ = P Bµ − P D * µ . Once again, comparing this expression with our results for µ = 0, µ = 1, 2, 3 with M = 0, +1, −1, we obtain the following results: from where we find As in [18] (Eq.(B.5)), we define here h A 1 (w) as In [18,67] the A i , V form factors are parameterized as Our expressions in Eqs. (38), (39), (40), (41), (42) and (43) fulfill these conditions in the strict heavy quark limit with R 0 (w) = 1, R 2 (w) = 1, R 1 (w) = 1, such that R D * A i and R D * V are exactly equal to h A 1 . This is seen in Fig. 8. Diversions from the strict heavy quark limit of the standard model are incorporated in this formalism parameterizing h A 1 (w), R 0 (w),  (38), (39), (42), and (43) as a function of w normalized to 1 at w = 1. The results for h A 1 R D * V , R D * A 0 , R D * A 2 are shown in Fig. 9. Comparison of Fig. 8 with Fig. 9 shows the difference of our approach with the standard model. We can appreciate a bigger slope as a function of w for the standard model (as already seen comparing with Ref. [9]) and also a different normalization at w = 1. Yet, the claim from our approach is that differences become much smaller when we use our approach to calculate ratios of amplitudes.
To see the accuracy of our model to provide ratios, we evaluate again the contribution of M = 0, ±1, divided by the sum of the three contributions, for different values of α, with the form factor of the standard model and compare the results with those obtained in Fig.   5. To evaluate those contributions in the standard model we look at the formulas of Eqs. (26), (27), (28), and looking at the expressions of Eqs. (38), (39), (40), (41), (42) and (43) we substitute, The results are shown in Fig. 10. One can appreciate some differences from to M inv | max the behaviour in both cases is so close can also be traced to the fact that for a certain range of p momentum the A 1 term is still largely dominant. Yet, this could be seen as a manifestation of a general behaviour of the helicity amplitudes close to the end point discussed in Ref. [68].

VI. CONCLUSIONS
We have taken advantage of a recent reformulation of the weak decay of hadrons, where, instead of parameterizing the amplitudes in terms of particular structures with their corresponding form factors, the weak transition matrix elements at the quark level are mapped into hadronic matrix elements and an elaborate angular momentum algebra is performed that allows one to correlate the decay amplitudes for a wide range of reactions. The formal-ism allows one to obtain easy analytical formulas for each reaction in terms of the angular momentum components of the hadrons. One global form factor also appears in the approach related to the radial wave functions of the hadrons involved, but since this form factor is common to many reactions and in particular is exactly the same for the different spin components of the hadrons within the same reaction, it cancels in ratios of amplitudes or differential mass distributions.
In the present paper we have taken this formalism and extended it to the case of hadron matrix elements with an operator γ µ −αγ µ γ 5 , which can accommodate many models beyond the standard model by changing α. We have applied the formalism to study the B → D * ν l reaction and the amplitudes for different helicities of the D * are evaluated. We see that dΓ dM (νl) inv depends strongly on the helicity amplitude and also on the α parameter. In particular the difference dΓ dM (νl) inv | M =+1 is shown to be very sensitive to the α parameter and changes sign when we go from α to −α. Such a magnitude, with its strong sensitivity to this parameter, should be an ideal test to investigate models beyond the standard model and we encourage its measurement in this and analogous reactions, as well as the theoretical calculations for different models.
We have taken advantage to relate our approach to the standard model by calculating the form factors V (q 2 ), A 0 (q 2 ), A 1 (q 2 ), A 2 (q 2 ) in our approach and comparing them to the parameterization of the standard model. The form factors are qualitatively similar but one can observe differences. Yet, when one uses them to evaluate ratios of amplitudes, or partial differential mass distributions, the differences are very small, and near the end point w = 1 the distributions are practically identical.