Semileptonic decays of $B^{(*)}$, $D^{(*)}$ into ${\nu} l$ and pseudoscalar or vector mesons

We perform a study of the $B^{(*)}$, $D^{(*)}$ semileptonic decays, using a different method than in conventional approaches, where the matrix elements of the weak operators are evaluated and a detailed spin-angular momentum algebra is performed to obtain very simple expressions at the end for the different decay modes. Using only one experimental decay rate in the $B$ or $D$ sectors, the rates for the rest of decay modes are predicted and they are in good agreement with experiment. Some discrepancies are observed in the $\tau$ decay mode for which we find an explanation. We perform evaluations for $B^*$ and $D^*$ decay rates that can be used in future measurements, now possible in the LHCb collaboration.


I. INTRODUCTION
Semileptonic decays of mesons have been thoroughly studied, and are a source of information on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1][2][3][4][5], chiral dynamics [6], heavy quark symmetry [7]. The process is relatively well understood, to the point that some discrepancies seen in ratios of rates are proposed as signals of new physics [8][9][10]. Concerning the decays of mesons with heavy flavors, the decay ofB → Dνl − and D →Kνl + and the related reactions with B * or D * offer a good ground to study heavy flavor symmetry.
In the conventional approaches the amplitudes of the processes are conveniently parameterized in terms of certain structures and their associated form factors, and some information is taken from experiment. Quark models can provide information on these form factors and structures and have been often used [2, 3,11].
The purpose of this paper is to see how far one can go, assuming basic facts of heavy quark symmetry, with some caution that will be discussed later, which allows us to conclude that the relevant form factors would be the same for D or D * and B or B * . Yet, the structures can be very different due to the angular momentum combinations that the quarks undergo to produce the pseudoscalar or vector meson states. This is what is accomplished in the present work, where a detailed study is done of the amplitudes for each of the four B ( * ) → D ( * )ν l − cases, and the corresponding ones with D ( * ) , evaluating explicitly the weak matrix elements in the rest frame of the νl and performing the angular momentum algebra which relates all the processes. We then fit the results to one experimental branching ratio for the B and D sectors and then the rest of the results are predictions.
The derivation requires some patience, but we succeed using Racah algebra to write the final amplitudes and the sums over polarizations of their modulus square in terms of very simple analytical expressions, which allow us to explain easily some of the features of the reactions, as the relative rates in the different sectors and peculiarities of the differential width distribution in the invariant mass of the νl system.
One of the outputs of the work is the prediction of rates for B * and D * decays, which have received attention recently [12,13] in view of the possibility that such decay rates are observed by the LHCb collaboration. We argue that the method proposed is highly accurate to make predictions for these decay widths.
As to the predictions for the observed rates, the method is rather accurate for the case of production of light leptons, and has some discrepancy for the production of τ lepton, for which a justification is given, but even then ratios of different decay rates with τ leptons in the final state are also well reproduced.

II. FORMALISM
We shall study reactions of the typeB 0 →ν l l − D + , or D + → ν l l +K 0 and the corresponding ones with vector mesons, with the aim of relating them assuming that the form factors do not change practically when changing B → B * or D → D * , which is the essence of heavy quark symmetry [14,15]. Work on these reactions assuming this symmetry is done in Ref. [11]. The process is depicted in Fig. 1  The weak interaction is given by the Hamiltonian where in C one has the couplings of the weak interaction, but, since we are only concerned about ratios of rates, it plays no role in our study. The leptonic current is given by and Q α , the quark current, by In order to obtain theB 0 decay width we need lep pol where L αβ stands for pol L α L β * and is easily evaluated with the result [16] where we adopt the Mandl and Shaw normalization for fermions [17]. The mass of the neutrino and the lepton get cancelled in the final formula of the width.
In Ref. [16] a similar sum and an average and sum over the quark spin third components was done. Here we pay a special attention to the vector or pseudoscalar components, and the coupling of spins to given quantum numbers has to be done prior to the sums over the third components in the final Q α Q * β term. For this purpose we must evaluate explicitly the quark current Q α . We use the ordinary spinors [18] where χ r are the Pauli bispinors and m, p and E p are the mass, momentum and energy of the quark. Next we use [16] and the same for the c quark. Theses ratios are related 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 and both have a sizeable velocity. We have in general where m in , m f in are the masses of the initial, final mesons in the decay, and M (νl) inv is the invariant mass of the νl pair. Using Eq. (7) we can now write In order to work out the angular momentum algebra it is more convenient to evaluate the spherical component of σ i → σ µ , µ = 1, 0, −1 in the last equation, and we define, M 0 for the γ 0 − γ 0 γ 5 matrix element of Eq. (15) and N µ for the matrix element of Eq. (16) substituting The explicit evaluation is done in the Appendix for the M 0 and N µ matrix elements and we show here the results.
2) J = 0, J = 1 3) J = 1, J = 0 2) J = 0, J = 1 3) J = 1, J = 0 C) Next we must evaluate L αβ Q α Q * β and sum and average over polarizations. In terms of the M 0 and N i terms defined before we have the combination The explicit evaluation is done in the Appendix and we show here the final results.

III. RESULTS
The invariant mass distribution dΓ/dM (νl) inv is given for B →νlD by where p D is the D momentum in the B rest frame and p ν theν momentum in the νl rest frame, By integrating dΓ/dM inv we obtain the width that we show in the tables.

A. B and B * decays
We study only the most Cabibbo-favored processes, b → c and c → s. We show in Table I theB,B s andB c semileptonic decays. Since we can only provide ratios, we fix one decay rate to the experiment and the rest are predictions, In this case we fix our rate to B − → D 0 e −ν e . We then observe that the predictions done for six decays are all in agreement with experiment, except for theB 0 → D + τ −ν τ that we will discuss later. The e −ν e and µ −ν µ decay rates are very similar, since the masses of e − and µ − are very small compared to the meson masses. The term proportional to m 2 l in Eq. (B3) is negligible for e − and µ − , but not for τ − . This term is responsible for a bigger rate than expected from phase space for the τ −ν τ decays. In Table I we also show predictions forB 0 s and B − c decays, for which there are not yet experimental data in the PDG [19]. For theB 0 → D + τ −ν τ we get a rate about a factor of two smaller than experiment. This has to be seen from the perspective that we are implicitly using the same form factors as forB 0 → D + e −ν e . However, because of the larger τ − mass, the momentum transfers are smaller in this latter case and by taking the same form factors as inB 0 → D + e −ν e we are reducing theB 0 → D + τ −ν τ rate more than one should. This is telling us implicitly the strength of the form factors in the present reactions. For B − → D 0 τ −ν τ the experimental error is relatively large, such that the rate is compatible with the theoretical one, but also with double its value.
We would like to call the attention to the rates for B − → D 0 e −ν e (B − → D 0 µ −ν µ ). These rates are identical experimentally within experimental errors, and also theoretically (up to small difference due to the different masses of the mesons). This should be the case, since from one reaction to the other the only change has been to substitute thed spectator quark in Fig. 1 by aū.
In Table II we show results forB decays into D * and related reactions. This corresponds to the case J = 0, J = 1 studied in the former sections. We do not fit now one rate, because the idea is to make a prediction for these decays based on the B → D reactions. We can see that the predicted rates for the case of light leptons are compatible with experiment, This should be seen as an accomplishment of the present framework, which shows that assuming the same form factors for D or D * decay, as we would induce from heavy quark symmetry [11], the rates for these two decays are a consequence of the angular momentum structure with the dynamics of the weak interaction.
We can also observe that the branching ratio forB 0 → D * + τ −ν τ is about a factor 0.61 the experimental one, in line with what was observed in Table I. The same happens for where the reduction factor is about 0.55. Yet, if we evaluate the ratio of the rates ofB 0 → D * + τ −ν τ toB 0 → D + τ −ν τ we find a factor of 2.07 against 1.62 ± 0.37 experimentally, or 2.07 for the ratio of rates of 2.44 ± 0.83 experimentally. One expects these two ratios to be the same, and so they are experimentally within errors, and also compatible with the theory.
Once again we make predictions for six more decay modes. It is interesting to observe that the rates for B − →D * ν l are bigger experimentally than those of B − →Dν l , something that is also obtained theoretically.
Recently much work has been devoted to the ratios from where one expects to observe new physics [9,10]. The values in Eq. (34) are taken from the HFLAV collaboration average [20]. The most precise single measurement performed so far is the recent one of the LHCb Collaboration [21] R D * = 0.291 ± 0.019 ± 0.026 ± 0.013.
Our result for these ratios are R D = 0.23, R D * = 0.211. As we can see, both R D and R D * are smaller than experiment for the reasons discussed above. To put our results in perspective we can compare these result with Lattice results [22,23] which give R D = 0.299 ± 0.011, and calculations based on the standard model obtaining ratios of form factors from experiment [9] which give R D * = 0.252 ± 0.03. This latter case is increased to 0.27 in [24] by taking into account in the theoretical evaluation that D * → Dπ, the mode where the D * is observed experimentally.
Another ratio of interest is reported by LHCb [25]. We obtain 0.20 for this ratio, short of the experimental one even considering errors, and one must find the reason in the discussion in former points since momentum transfer for B + c → J/ψ τ + ν τ is smaller than in B + c → J/ψ µ + ν µ . It is also useful to compare our results with other theoretical works based on the standard model which provide value around 0.25 ∼ 0.28 [26][27][28][29] for this ratio.
There is another remark we can do in view of our easy expressions for |t| 2 . If one looks at Eq. (B3) for J = 0, J = 0, one can see that the term independent of p is proportional to m 2 l , which is very small for light leptons. The important term is this case is the second term of that equation, proportional to p 2 . This tells us that nonrelativistic calculations, which would neglect this momentum, or the strict use of heavy quark symmetry, neglecting terms of O( p m Q ) (see B factor in Eq. (9)), would provide very bad results for the rate ofB → D and light leptons. In Table III we also show the rate for the semileptonic decays ofB * → D. These decay rates have not been observed and one reason maybe the fact that B * decays electromagnetically in γB. In Ref. [12] these rates were evaluated using the Bethe-Salpeter approach with the instantaneous approximation. We compare our predictions with those in Ref. [12].
These decay widths are also evaluated in Ref. [13] in the Baner-Stech-Wirbel model and in Table III we also show these results. Our results are qualitatively similar to those of [13], about 25% smaller and also smaller than those of [12] by a factor of about 0.6. One should stress the simplicity of our approach with respect to [12] where 14 form factors are evaluated and the theory relies on several parameters [30], partly constrained from the masses of the mesons. We should stress that the rates for B * → Dνl in our approach involve the same matrix element as for B → D * ν l (J = 1, J = 0, versus J = 0, J = 1 in Eqs. (29), (30)).
Given the accuracy by which we predicted the rate for B → D * ν l in Table II, our predicted rates for B * → Dνl should be equally accurate. Yet, seen from the perspective that these are the first theoretical predictions for B * decay rates, the main message is that the three calculations reported provide very similar numbers and this should be sufficient for planning possible experimental searches.
We complete this part by making predictions for B * → D * νl in Table IV, in which we also compare our results with those obtained in Ref. [12]. We observe now that in both cases the widths are much bigger than for B * → Dνl, and our predictions are again a factor of about 0.55 those of Ref. [12], with a bit bigger discrepancies for decays in the τ mode, as we would expect.
It is also interesting to look into the invariant mass distribution dΓ/dM  (29). As we mentioned before, the B − → D 0 e −ν e reaction gets most of its strength from the p 2 term of Eq. (B3) since the m 2 l term is extremely small. One gets a bigger contribution the larger p 2 , but this means a smaller M   In Table V we show results for D + →K 0 e + ν e and related reactions. We fix our results to the rate for D 0 → K − e + ν e . Our results are in fair agreement for other reactions.
In Table VI we show the results for D + →K * 0 e + ν e reaction and related ones. The agreement with experiment is fair. In particular if we look at ratios for different final vectors we find when experimentally is 0.9 ± 0.11, which is compatible with Eq. (36).
Finally, in Table VII and VIII we give for completeness the rate for D * + →K 0 e + ν e and D * 0 → K − e + ν e and related reactions. Yet, the fact that the D * decays strongly into Dπ,   To finish the section we show in Fig. 4 the mass distribution for D 0 → K − e + ν e , D 0 → K * − e + ν e and D * 0 → K − e + ν e , D * 0 → K * − e + ν e . We can see that the arguments discussed TABLE V. Branching ratios for (P P ) semileptonic decay of D meson. We have taken into account that τ D + τ D 0 = 2.54 and τ D + s τ D 0 = 1.23. For η and η production we consider that the weights for the ss components are 1 3 and 2 3 , respectively.
Decay process BR (Theo.) BR (Exp.) [19] D + →K 0 e + ν e 8.94 × 10 −2 (8.82 ± 0.13) × 10 −2     One of the output of the study is the prediction of decay rates for B * and D * , which have been the object of discussion recently since they could be observed in future measurements of the LHCb collaboration. We justify that our predictions for these decay widths should be very accurate, which can be used in planning experiments to observe them in the future.
For the third term, we have a combination of three Clebsch-Gordan coefficients (CGC).
We follow the angular momentum algebra of Rose [31] and write where W(· · · ) is a Racah coefficient. Incorporating an extra phase in the sum is not trivial. To finally be able to reduce the sum over s to Racah coefficients we write: (−1) 1 2 −s = s|σ z |s = √ 3 C (  1  2  1  1  2 ; s, 0, s) .
A similar approach can be used for the term in all the other cases and one can show that it always vanishes.
c) For the term from L ij N i N * j , this evaluation is made easy recalling that we take p in the z direction and we found in Eq. (A22) that N µ is proportional to δ µ0