# Helicity amplitudes in \(B \rightarrow D^{*} \bar{\nu } l\) decay

## Abstract

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 \(B \rightarrow D^{*} \bar{\nu } l\) 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 \(\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}\) is very different for the different components \(M=\pm \,1, 0\) and in particular the magnitude \(\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}|_{M=-1} -\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}|_{M=+1} \) is very sensitive to the \(\alpha \) parameter, which suggest to use this magnitude to test different models beyond the standard model. We show that our formalism implies the heavy quark limit and compare our results with calculations that include higher order corrections in heavy quark effective theory. We find very similar results for both approaches in normalized distributions, which are practically identical at the end point of \( M_\mathrm{inv}^{(\nu l)}= m_B- m_{D^*}\).

## 1 Introduction

Semileptonic decays of hadrons have been thoroughly studied and have brought much information on the nature of weak interactions and some aspects of QCD [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The relative good control of the reactions within the Standard Model (SM) has led to new work searching for evidence of new physics beyond the standard model (BSM) [17, 18, 19].

One of the magnitudes that has captured attention as a source of information of new physics BSM is the polarization of vector mesons in *B* decays. One intriguing feature was observed in the \(B \rightarrow \phi K^*\) decays, where naively it was expected that the transverse amplitudes would be highly suppressed while the experiment showed equal strength for longitudinal and transverse polarizations [20, 21]. Theoretical papers have followed [22, 23], as well as new experimental measurements on related reactions, like \(B^0_s \rightarrow \phi \phi \) [24], \(B^+ \rightarrow \rho ^0 K^{*+}\) [25], \(B^0_s \rightarrow K^{*0} \bar{K}^{*0} \) [26], which had been addressed in papers dealing with \(B \rightarrow VV\) decays [27, 28], \(B \rightarrow V T\) decays [29], and some particular reactions as the \(B_{(s)} \rightarrow D^{(*)}_{(s)} \bar{D}_s^*\) [30]. More recently the topic has caught up in studies of weak decays into a vector and two leptons as the experiments on \(B \rightarrow K^* l^+ l^-\) [31], \(B \rightarrow K^* l^+ l^-\) [32], \(B^0 \rightarrow K^{*0} \mu ^+ \mu ^-\) [33, 34, 35], and theoretical works on \(B \rightarrow K^* \nu \bar{\nu }\) [36, 37], \(B \rightarrow K^* l^+ l^-\) [37], \(B \rightarrow K^{*0} l^+ l^-\) [38], \(B \rightarrow K_J^* l^+ l^-\) [39] and \(B \rightarrow K_2^* \mu ^+ \mu ^-\) [40].

In the present work we retake this line of research and study the polarization amplitudes in semileptonic \(\bar{B} \rightarrow V \bar{\nu } l\) decays, applied in particular to the \(\bar{B} \rightarrow D^* \bar{\nu } l\) 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 [41]. 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 like \(\bar{B}^0 \rightarrow D^{-}_s D^+, D^{*-}_s D^+, D^{-}_s D^{*+}, D^{*-}_s D^{*+}\), 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 [41] is also done in [42] in the study of the semileptonic \(B, B^*,D, D^*\) decays into \(\bar{\nu } l\) 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 \((\gamma ^\mu - \alpha \gamma ^\mu \gamma _5)\) structure for the weak hadronic vertex which makes it easy to make predictions for different values of \(\alpha \) that could occur in different models BSM (\(\alpha =1\) here for the SM). We evaluate different ratios for the \(B \rightarrow D^* \bar{\nu } l\) reaction. Work on this particular reaction, looking at the helicity amplitudes within the Standard Model, was done in [43]. A recent work on this issue is presented in [44] where the \(B \rightarrow D^* \bar{\nu }_\tau \tau \) is studied separating the longitudinal and transverse polarizations. The same reaction, looking into \(\tau \) and \(D^*\) polarization, is studied in [45]. Helicity amplitudes are also discussed in the related \({\bar{B^*}} \rightarrow P l \bar{\nu }_l\) reactions in the recent paper [46].

The formalism of Ref. [42] 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 \(\alpha \) that will differ from unity for models BSM.

We find some observables which are very sensitive to the value of \(\alpha \), which should stimulate experimental work to investigate possible physics BSM.

## 2 Formalism

*t*is the transition amplitude, and for simplicity

*M*and \(M'\) components of Eq. (4) was done in Ref. [42]. Here we shall keep track of the individual

*M*and \(M'\) contributions.

*m*,

*p*and \(E_p\) are the mass, momentum and energy of the quark. As in Ref. [47] we take

*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 the \({\bar{\nu }} l\) system is at rest, where \({{\varvec{p}}}_B={{\varvec{p}}}_{D^*}={{\varvec{p}}}\), with

*p*given by

In the present work, we are only interested in the \(B^- \rightarrow D^{*0} \bar{\nu }_{l} l^-\) decay, which means \(J=0,J'=1\) decay.

In addition to the *p* dependence (and hence \(M_\mathrm{inv}^{(\nu l)}\)) of these amplitudes, in [42] 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.

- 1)\(M'=0\)$$\begin{aligned} \sum |t|^2= & {} \frac{m^2_l}{m_{\nu } m_l} \frac{M_\mathrm{inv}^{2(\nu l)}-m^2_l}{M_\mathrm{inv}^{2(\nu l)}} \big \{A A' (B+B')p \big \}^2 \nonumber \\&+ \frac{2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \big \{A A' (1+B B' p^2) \big \}^2 \, . \nonumber \\ \end{aligned}$$(15)
- 2)\(M'=1\)$$\begin{aligned} \sum |t|^2= & {} \frac{2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \nonumber \\&\times \big \{A A' [(1-B B'p^2) +(B p-B' p)]\big \}^2 \, . \nonumber \\ \end{aligned}$$(16)
- 3)\(M'=-1\)$$\begin{aligned} \sum |t|^2= & {} \frac{2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \nonumber \\&\times \big \{A A' [(1-B B'p^2) -(B p-B' p)]\big \}^2 \, . \nonumber \\ \end{aligned}$$(17)

*B*rest frame to the frame where

*B*and \(D^*\) have the same momentum and \(\bar{\nu } l\) 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. [42] using properties of Clebsch-Gordan and Racah coefficients.

## 3 Results

*B*rest frame and \(\widetilde{p}_\nu \) the \(\bar{\nu }\) momentum in the \(\nu l\) rest frame,

It is interestig to look individually at the distribution of the three third components of the \(D^*\) spin. For this we plot *Bp*, \(B' p\) as a function of \(M_\mathrm{inv}^{(\nu l)}\) in Fig. 2. We can see that \(B'p\) is always bigger than *Bp* and that both *Bp* and \(B' p\) go to unity as \(M_\mathrm{inv}^{(\nu l)} \rightarrow 0\) (for \(m_l=m_\nu =0\)). Moreover, when \(M_\mathrm{inv}^{(\nu l)}\) goes to its maximum, then \(p \rightarrow 0\) 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 \(M_\mathrm{inv}^{(\nu l)} \rightarrow 0\) then \(\sum |t|^2 \) goes to \(\frac{2 (A A')^2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) F\), with \(F \rightarrow 4, 0, 0\) for \(M'=0,1,-1\) respectively, with \((A A')^2\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \) going to a constant. Conversely, when \(M_\mathrm{inv}^{(\nu l)} \) goes to its maximum, \(\sum |t|^2 \) goes to the same value \(\frac{2 (A A')^2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \) for \(M'=0,1,-1\) cases.

In Fig. 3 we show the individual contribution of each \(M'\) and the total. In Fig. 4 we show the contribution of each \(M'\) and the difference of \(M'=-1\) and \(M'=+1\), divided by the total differential width *R*. In this latter figure we can see how fast the individual \(M'=-1\) and \(M'=+1\) components go to zero.

We also appreciate in Fig. 4 that the ratio of \( \frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}|_{M'=-1}-\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}|_{M'=+1}\) divided by the total differential width goes fast to zero for \(M_\mathrm{inv}^{(\nu l)} \rightarrow 0\) or maximum, while individually each of the contributions goes to \(\frac{1}{3}\) at the maximum of \(M_\mathrm{inv}^{(\nu l)}\). In Fig. 4 we also see a smooth transition from 1 to \(\frac{1}{3}\) 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 \(M_\mathrm{inv}^{(\nu l)} \rightarrow 0 \) in the case of \(M'=\pm 1\), is tied not to the lepton current, since we always get \((A A')^2\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \), which goes to a constant for \(M_\mathrm{inv}^{(\nu l)} \rightarrow 0\), but to the quark current. Indeed, if we look at Eqs. (12), (13), we can see that both \(\frac{M_0}{AA'}\) and \(\frac{N_\mu }{AA'} (\mu =0)\) are different from zero for \(M'=0\) in the limit of \(M_\mathrm{inv}^{(\nu l)} \rightarrow 0\). However, for \(M'=\pm 1\), \(M_0=0\) and \(N_\mu \) goes to zero in that limit. This said, the models beyond the SM which could provide finite contribution for \(M'=\pm 1\), or a sizeably bigger one, are those that go beyond the \(\gamma ^\mu - \gamma ^\mu \gamma _5\) structure in the quark current, like leptoquarks or right-handed quark currents of the type \(\gamma ^\mu + \gamma ^\mu \gamma _5\) [50, 51, 52]. We discuss this case below.

## 4 Consideration of right-handed quark currents

The literature about models BSM is large and this is not the place to discuss it. Yet, we would like to mention more recent papers on models which could be easily tested within the present approach, minimal gauge extensions of the SM [53, 54], leptoquarks [55], scalar leptoquarks [56, 57], vector leptoquarks [58, 59, 60], Pati-Salam gauge models [61, 62, 63] and right-handed models [19, 64].

- 1)\(M'=0\)$$\begin{aligned} \sum |t|^2= & {} \frac{m^2_l}{m_{\nu } m_l} \frac{M_\mathrm{inv}^{2(\nu l)}-m^2_l}{M_\mathrm{inv}^{2(\nu l)}} \big \{A A' (B+B')p \big \}^2 \alpha ^2 \nonumber \\ \,&+ \frac{2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \nonumber \\&\times \big \{A A' (1+B B' p^2) \big \}^2 \alpha ^2 \, . \nonumber \\ \end{aligned}$$(26)
- 2)\(M'=1\)$$\begin{aligned} \sum |t|^2= & {} \frac{2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \nonumber \\ \,&\times \big \{A A' [(1-B B'p^2)\alpha +(B p-B' p)]\big \}^2 \, . \nonumber \\ \end{aligned}$$(27)
- 3)\(M'=-1\)$$\begin{aligned} \sum |t|^2= & {} \frac{2}{m_{\nu } m_l} \,\left( \widetilde{E}_\nu \widetilde{E}_l +\frac{1}{3} \widetilde{p}_\nu ^2 \right) \nonumber \\ \,&\times \big \{A A' [(1-B B'p^2)\alpha -(B p-B' p)]\big \}^2 \, . \nonumber \\ \end{aligned}$$(28)

## 5 Connection with the conventional formalism and discussion on heavy quark symmetry

In Ref. [42] one relates the weak amplitudes for the \({B} \rightarrow D \bar{\nu } l, D^* \bar{\nu } l\), \({B}^* \rightarrow D \bar{\nu } l, D^* \bar{\nu } l\). This means that there is only one independent amplitude for all these processes. This is reminiscent of the heavy quark symmetry [65, 66] 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 [42] which we follow here.

*B*meson at rest there is a distribution of quark momenta due to the internal motion of the quarks, \({{{\varvec{p}}}_{in}}\). If we make a boost to have the

*B*with a velocity of \({\varvec{v}}\), we will have

*p*from Eq. (9) becomes infinite and thus \(v=1\). Hence, the correction terms are of the order of \(\frac{1}{3} \frac{{p}^2_{in}}{m_B^2 v^2}\). With typical values of \(|{\varvec{p}}_{in}|\simeq 300\) MeV, this is a correction of one permil. For the \(D^*\) meson is a correction of less than \(1\%\). We can make

*v*smaller as \(M_\mathrm{inv}^{(\nu l)} \) grows and still keep these numbers very small. Certainly, when we go to the end point, for \(M_\mathrm{inv}^{(\nu l)}/\mathrm{max}\), when both

*B*and \(D^*\) are at rest, the argument would fail since \(v=0\). However, in this case the approximation is equally good since the

*Bp*term is zero and only \(A,A'\) matter and \(E_b=m_b\) at the level of \(\frac{p^2_{in}}{2m^2_B}\) and \(E_B=M_B\), hence \(\widetilde{A}\) in Eq. (7) and

*A*in Eq. (10) are again remarkably close. Incidentally, the transverse components in the boosted frame lead to a correction of

- 1)\(J=0,J'=0\) (\(\bar{B} \rightarrow D \bar{\nu } l\) )$$\begin{aligned} M_0= & {} A A' (1+B B'P^2) \, \delta _{M 0} \, \delta _{M' 0} \,, \nonumber \\ N_{\tilde{\mu }}= & {} -A A' (B+B') \, p \,\delta _{M0} \,\delta _{M' 0} \, \delta _{\mu 0} \,. \end{aligned}$$(37)
- 2)
\(J=0,J'=1\) (\(\bar{B} \rightarrow D^* \bar{\nu } l\))

\(M_0\) and \(N_{\tilde{\mu }}\) are given by Eqs. (12) and (13).

*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 since it cancels in ratios of amplitudes for different \(M'\). This transition form factor, evaluated in the quark model, is close to 1 at \(w=1\), where there is no momentum transfer, and decreases as the momentum transfer increases. This allows us to compare our calculated \(h_{+}\) factor normalized to 1 at \(w=1\) with \(h_{+}\) of Ref. [9]. 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.

*w*for the improved model (as already seen comparing with Ref. [9]) and also a different normalization at \(w=1\). The differences are not small, since the values of \(R_i (1)\) in Eq. (48) compared with those in the heavy quark limit implicit in our approach where \(R_i=1\), contain deviations from \(14\%\) to \(40\%\). Yet, the claim we make here is that the 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 \(M'=0,\pm 1\), divided by the sum of the three contributions, for different values of \(\alpha \), with the form factor of the improved model and compare the results with those obtained in Fig. 5. To evaluate those contributions in the improved model we look at the formulas of Eqs. (26), (27), (28), and looking at the expressions of Eqs. (40), (41), (42), (43), (44) and (45) we substitute,

*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. [70].

The comparison of these two approaches is useful. Once again, we can see some differences in the distributions of Figs. 5 and 10 for low and intermediate values of \(M_\mathrm{inv}^{(\nu l)}\). Should one see in an experiment some discrepancy with respect to our predictions of Fig. 5 this should not be taken as evidence of physics BSM. Indeed, there are corrections in these distributions in Fig. 10 when one includes higher order corrections in HQEF. The differences with respect to Fig. 10 would be more significant. Yet, the most significant thing is the large sensitivity of the difference of the \(M=-1\) and \(M=1\) contributions to small changes of \(\alpha \), which is shared in both approaches. The other point worth stressing is that since close to the end point our approach and the improved one are practically indistinguishable, the predictions in that region are rather model independent, and, yet, different for different values of \(\alpha \). Differences found in experiment with respect to these predictions for \(\alpha =1\) in that region would clearly manifest some physics beyond the Standard Model.

## 6 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 formalism 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 \(\gamma ^\mu -\alpha \gamma ^\mu \gamma _5\), which can accommodate many models beyond the standard model by changing \(\alpha \). We have applied the formalism to study the \(B \rightarrow D^{*} \bar{\nu } l\) reaction and the amplitudes for different helicities of the \(D^*\) are evaluated. We see that \(\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}\) depends strongly on the helicity amplitude and also on the \(\alpha \) parameter. In particular the difference \(\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}|_{M=-1} -\frac{d \varGamma }{d M_\mathrm{inv}^{(\nu l)}}|_{M=+1} \) is shown to be very sensitive to the \(\alpha \) parameter and changes sign when we go from \(\alpha \) to \(-\alpha \). 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, which implies the heavy quark limit, to the conventional one using explicit polarization vectors, 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 improved conventional model which incorporates higher order corrections in HQEF and phenomenological information. The form factors are qualitatively similar but one can observe clear differences. Yet, when one uses them to evaluate ratios of amplitudes, or partial differential mass distributions, the differences are small, and near the end point \(w=1\) the distributions are practically identical.

## Notes

### Acknowledgements

We wish to express our thanks to Jose Valle, Martin Hirsch, Avelino Vicente and Xiao-Gang He for useful discussions. Also discussions with Juan Nieves and Eliecer Hernandez are much appreciated. LRD acknowledges the support from the National Natural Science Foundation of China (Grant No. 11575076) and the State Scholarship Fund of China (No. 201708210057). This work is partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B and No. FIS2017-84038-C2-2-P B, and the Generalitat Valenciana in the program Prometeo II-2014/068, and the project Severo Ochoa of IFIC, SEV-2014-0398 (EO).

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