# A model for NL and SL decays by $\overline{B}^{0}\to D^{**}$ transitions
with $BR(j=1/2)\ll BR(j=3/2)$ using the $LLSW$ scheme

Alain Le Yaouanc  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,
France  Jean-Pierre Leroy  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  Patrick Roudeau  Université Paris-Saclay, CNRS/IN2P3, IJCLab,
91405 Orsay, France

###### Abstract

We present a model for the vector and axial form factors of the transitions
$\overline{B}^{0}\to D^{**}$ in good agreement with the presently available
data and based on the present theoretical knowledge, combining a) the safe
lattice QCD predictions at $m_{Q}=\infty$ and $w=1$ ; b) the predictions at
general $w$ of a relativistic, covariant quark model at $m_{Q}=\infty$,
including the well tested Godfrey and Isgur spectroscopic model and which
agrees with lattice QCD at $w=1$ ; c) the constraint of Bjorken and Neubert
relating Semi-leptonic (SL) and Class I Non-leptonic (NL) decays, which shows
that $\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-}$ strongly constrains
$\tau_{1/2}(w)$ to be much smaller than $\tau_{3/2}(w)$, in agreement with the
theoretical expectation; d) the general HQET expansion which constrains the
$1/m_{Q}$ corrections (cf LLSW , denoted hereafter as $LLSW$).

An important element in the understanding of data is the large contribution of
virtual $D^{(*)}_{V}$ to the broad structures seen in SL decays at low
$D^{(*)}\pi$ masses - which makes it difficult to isolate the broad resonances
denoted as $D_{1/2}$ in the following.

##  I Introduction

There is insufficient knowledge of both non-leptonic (NL) and semi-leptonic
(SL) transitions to charmed orbital excitations, generically termed as
$D^{**}$. The latter (SL) are especially interesting

1) in themselves,

for a) striking theoretically expected features which contradict the naive
idea of strong similarity between SL transitions to $j=1/2$ and $j=3/2$
$D^{**}$,

and b) for the apparent contradiction between this expectation and the
presently widespread interpretation of experimental data on broad states, a
contradiction which we claim to have resolved by the interplay of the possibly
large $D^{*}_{V}$ background roudeau1 .

2) for some applications, like the background to $\overline{B}\to
D^{(*)}\ell^{-}(\tau^{-})\overline{\nu}_{\ell(\tau)}$. Ultimately, one would
like to have a description of the full system of semi-leptonic
$\overline{B}\to D^{**}$ form factors, but this appears a hard task.

Indeed, there are no full calculations of the latter from first principles, as
are provided in simpler cases by lattice QCD. Presently, the latter gives
results only at $m_{Q}=\infty$ and $w=1$. The reasons are given below, Section
II.

On the other hand, the experimental data are scarce : total branching ratios,
some points in the $d\Gamma/dw$ of certain transitions… Then, as explained in
more details below, Sections II and VIII, a more complicated path is to be
followed, combining theoretical inputs and experience, the former being used
in addition in several different ways.

We want indeed to use the very useful general ideas of the extensive analyses
of ref:leibo ; ref:ligeti1 ; ref:ligeti2  111From now on we shall refer to
this set of analyses as $LLSWB$. When comparing such type of analyses with our
model we will use the acronym $LLSWBi$ where the “i” means “inspire”, implying
slight modifications to $LLSWB$ required for a fair comparison with our own
model, and which are explained in Appendix D relying on their HQET analysis,
but to take into account
1) quantitative dynamical results at $m_{Q}=\infty$ of lattice QCD (completed
by quark models in the Bakamjian-Thomas (BT) approach) as well as
2) the very important experimental measurements of
$\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-}$ which constrain strongly the
transition to $j=1/2$ to be small,  both of them having been disregarded in
the $LLSWB$ analyses.

Finally we underline the very important role of the $D^{*}_{V}$ background in
the SL decays, which we have already emphasized in roudeau1 , and which, we
think, has not been fully taken into account up to now.

All these elements, taken together, change drastically the conclusions with
respect to the ones of the above mentioned analyses, especially the claim that
the Isgur-Wise (IW) functions are roughly equal at $w=1$, as we explain now.

The $LLSW$ approach ref:leibo , provides a framework to parameterize $1/m_{Q}$
corrections in corresponding hadronic form factors, and applies factorization
to relate semi-leptonic and Class I non-leptonic decays. Such analyses
ref:ligeti1 ; ref:ligeti2  which take, as input, quoted values ref:hfag2016 ;
ref:pdg2019  for $\overline{B}^{0}\to D^{**,+}\ell^{-}\overline{\nu}_{\ell}$
conclude that production of narrow and broad $D^{**}$ states are similar.
Meanwhile, to reach this conclusion, they have to discard the measurement of
$\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-}$ which implies, as a consequence
of factorization, a very small value for the production of the ${D_{0}(2300)}$
meson in semi-leptonic decays, when compared with the rates measured for
narrow states. This could seem to be justified because such a low value,
evaluated for
$\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\overline{\nu}_{\ell}$, appears
to be in contradiction with the rates measured in experiments for this
channel. However, on the opposite, we think that the non-leptonic data for the
$D_{1/2}$ are much more trustable than the semi-leptonic ones. Indeed, the
identification of the ${D_{0}(2300)}$ in the non-leptonic channel is supported
by 1) the extraction of the $D_{V}^{*}$ and 2) the measure of the phase shift,
while no such work has been done in the semi-leptonic case (except, for the
$D_{V}^{*}$, the work reported in ref:belle_dsstarl ). Moreover, from a
theoretical point of view, the smallness of the transitions to the $D_{1/2}$
states with respect to the $D_{3/2}$ ones has been anticipated using,
initially, quark models and, later on, by direct LQCD evaluations.

We have considered again this problem and propose a model, based also on the
$LLSW$ parameterization and which uses all the measured non-leptonic Class I
decays in the framework of factorization. This model agrees with LQCD and
relativistic quark models (RQM) expectations. Because, in this model,
production of broad $D^{**}$ mesons is expected to be much smaller than the
production of narrow states, it is necessary to add another broad component to
be able to explain the broad mass distributions measured in
$\overline{B}^{0}\to D^{(*)}\pi\ell^{-}\overline{\nu}_{\ell}$ decays. We find
that $D^{(*)}_{V}$ decays can fill such a gap. In the following we detail our
model and provide comparisons with the $LLSWBi$ model in which the broad
$D^{(*)}\pi$ mass distributions are explained by the contributions from
$D^{**}$ decays alone, as done in previous analyses and where the measurement
from $\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-}$ is not used.

Finally one is led to a solution with a $j=1/2$ rate much smaller than the
$j=3/2$ one.

##  II Theoretical inputs

We speak of "inputs" because one lacks a systematical theoretical treatment :
rather, one is led to use a mixture of procedures, including the very
experimental data which are to be explained, and fitting.

In addition, several ingredients a)b)c)d), as enumerated in the abstract, are
available.

We can classify them also into two categories :

###  II.1 Dynamical results at $m_{Q}=\infty$

In the infinite quark mass limit, hadronic form factors that describe
$\overline{B}\to D^{**}$, are determined by the two Isgur-Wise functions:
$\tau_{3/2}(w)$ and $\tau_{1/2}(w)$. Their values at $w=1$ and their $w$
dependence are constrained by theory:

a) lattice QCD for $w=1$ at $N_{F}=0,2$ ref:lqcd-tau ; at $N_{F}=2$ :

$\tau_{3/2}=0.526\pm 0.023,\,\tau_{1/2}=0.296\pm 0.026$ (1)

showing a striking difference between them, in contrast with a naive non-
relativistic (NR) expectation according to which these two quantities are
equal.

These are trustable results which cannot be disregarded. In the most recent
simulation the lattice spacing is reasonably small, $a=0.085fm$, and the
volume reasonably large: $24^{3}\times 48$, although certain systematic errors
are not estimated. Of course, they should be improved. One notes that, at
$N_{F}=2$, the inequality between $j=1/2$ and $j=3/2$ values is appreciably
reinforced with respect to the older $N_{F}=0$ ones.

b) quark models, which, although purely phenomenological, have the advantage
of providing results for $w\neq 1$ as well.

Of course, there is a very large variety of quark models. We consider a class
of models for current matrix elements where the calculation is decomposed into
two steps : 1) the determination of the wave functions at rest and, 2) a
procedure to derive from it the state in motion. Then there is first a variety
of spectroscopic models, describing the spectrum and the wave functions at
rest, i.e internal quark motion ; secondly, another variety comes from the way
one describes the hadron motion.

As to spectroscopic models Among many, there is an outstanding one by Godfrey
and Isgur (GI) godfreyj , which we therefore decide to use. This model is
unique in its covering of a very large number of hadronic states, both with
light quarks and with heavy ones. One must underline that most predictions
have been confirmed by later experiments. Although it is a complicated model
with many parameters, these can be determined thanks to this covering of a
very extended spectrum. This model has a relativistic kinetic energy and its
success confirms the necessity of a relativistic treatment of quark internal
motion inside hadrons, which is already implied by the fact that excitation
energies are of the order of the reduced mass.

As to the description of hadron motion Here we shall use a specific framework,
the one of Bakamjian and Thomas, to describe states in relativistic motion at
$m_{Q}=\infty$. It has several advantages:

  * $\alpha$)

it uses the standard three-dimensional wave functions at rest provided by
spectroscopic models ;

  * $\beta$)

it is relativistic as to hadronic motion and even covariant;

  * $\gamma$)

it satisfies the standard set of HQET sum rules, like Bjorken Taron  and
curvature curvature  sum rules, Uraltsev sum rule Uraltsev …

The last two points are important advantages as compared to non relativistic
treatment of quark motion, even if we were adopting the GI spectroscopic
model. Note that a NR treatment of hadron motion is not satisfactory for the
full range of $w$ since the 3-velocity at $w_{max}$ is large even in the
"Equal Velocity Frame" which minimizes velocities : $v=\sqrt{(w-1)/2}\simeq
0.4$.

Moreover these differences lead to quite different quantitative results. In
the NR quark models with non-relativistic treatment of both internal quark
velocities and hadron motion, a general statement 222 In Ref.IsgurWise , the
calculation is done for harmonic oscillator wave functions, but this does not
alter the generality of the conclusion. Note also that a “relativisation”
factor $\kappa$ has to be disregarded in their Eq. (44). Possible additional
factors expressed as powers of $(w+1)/2$ encountered in the literature must
also be disregarded anyway since such factors give a 1/2 contribution to the
slope while the dominant NR approximation gives a slope that goes $\to\infty$
as $m^{2}R^{2}$ (see Eq. (44) of IsgurWise ) would be :

$\displaystyle\tau_{1/2}(w)=\tau_{3/2}(w)$ (2)

in clear contradiction with the above lattice QCD results. The equality (2)
assumes that there is no sizable spin-orbit force surviving at $m_{Q}=\infty$,
which is not a theorem but seems reasonable from the spectroscopic models
studies, whence identical rest frame wave functions.

In the BT framework, using the well known spectroscopic model of Godfrey and
Isgur godfreyj , one finds very different values for $j=1/2$ and $j=3/2$
ref:morenas :

$\displaystyle\tau_{3/2}(1)\simeq 0.5,\tau_{1/2}(1)\simeq 0.25$ (3)

in full agreement with QCD. This strikingly large difference is a general
feature of the BT approach, which provides an intuitive explanation: it is due
to the typically relativistic effect of Wigner rotations of spin, included in
the BT approach, which acts differently on $j=3/2$ and $j=1/2$ states in
motion and which is completely independent of the presence of possible spin-
dependent forces in the potential. Note that, more generally, the inequality
of the $\tau(1)$’s is suggested by the Uraltsev sum rule :

$\displaystyle\Sigma_{n}\left(|\tau^{(n)}_{3/2}(1)|^{2}-|\tau^{(n)}_{1/2}(1)|^{2}\right)=1/4$
(4)

which contradicts the equality (2) under the assumption of dominance of $n=0$.

In summary, one should consider the marked difference between $\tau_{3/2}(1)$
and $\tau_{1/2}(1)$ as a solidly settled result (see the extensive discussions
by Bigi et al.bigi_et_al , see also DBecirevic  and the work on zero-recoil
sum rules of BigShif  and Gambino . In the latter reference the point of view
of the authors on this matter is expressed at the end of subsection 6.3.2).
Combined with the appropriate kinematical factors, and converted into
branching ratios, this leads to the still more striking inequality :

$\displaystyle{\cal B}(\overline{B}^{0}\to D_{1/2})\ll{\cal
B}(\overline{B}^{0}\to D_{3/2})$ (5)

by around one order of magnitude, and we claim to provide a model which both
satisfies this strong inequality and fits well the data.

The quantitative agreement of the BT prediction with the lattice QCD one gives
encouragement to trust the predicted shapes at $w\neq 1$, which are another
crucial theoretical input.

The full shape is well approximated by a relativistic quark-model inspired
description ref:bt_model :

$\displaystyle\tau_{3/2\,(1/2)}(w)=\tau_{3/2\,(1/2)}(1)\times\left({{2}\over{w+1}}\right)^{2\,\sigma^{2}_{3/2\,(1/2)}}$
(6)

Numerically :

$\displaystyle\sigma^{2}_{3/2}\simeq 1.5,\quad\quad\sigma^{2}_{1/2}\simeq 0.8$
(7)

These values are in contrast with what would be given in a fully NR treatment
namely a common and much lower value, of order 0.4 (obtained from IsgurWise
and skipping the relativisation factor $\kappa$.)

Other analyses use instead a linear approximation for the $\tau$-functions :
$\tau_{i}(w)=\tau_{i}(1)\times\left(1-\sigma_{i,lin.}^{2}\,(w-1)\right)$. It
can be checked that $\sigma_{i,lin.}^{2}=\sigma_{i}^{2}$ when $w=1$. In
practice the two descriptions are rather similar because the $w$ variation
range, between 1 and
$w_{max}=(m_{B}^{2}+m_{D^{**}}^{2})/(2\,m_{B}\,m_{D^{**}})$ is limited anyway.
Meanwhile, significant differences are expected, from the two
parameterizations, when comparing semi-leptonic decays with a light or an
heavy lepton, as evaluated in Table 13.

Why start from $m_{Q}=\infty$?

The visible difficulty is that one has presently trustable quantitative
statements only for $m_{Q}=\infty$. This derives from different reasons :

1) as to lattice QCD, it is the fact that it is very difficult to treat
directly the finite masses $m_{b},m_{c},$ all the more for transitions to
excitations. Thus one has been led presently to use a $m_{Q}=\infty$
framework. Then, to treat finite $w\neq 1$, one would require infinite
momenta, whence one is restricted to $w=1$.

2) quark models have in general no such limitations. However the BT framework
which we want to use is satisfactory only at $m_{Q}=\infty$. Indeed, it has
been shown that the finite mass corrections cannot be trusted : they break
covariance as well as certain important sum rules, in contrast with the
$m_{Q}=\infty$ limit. One has not been able to cure this defect.

The consequence is that these dynamical results must be complemented by other
statements, of a more general nature, which provide constraints on the
physical transitions (of course at finite mass) or, equivalently, on finite
mass corrections to the above dynamical results. Those questions are
considered in points c) and d) below.

###  II.2 Use of general statements or relations

c) First of all, we rely on the validity of the factorization, as does in
principle $LLSWB$. This phenomenon has been firmly established from the
theoretical point of view (specially in the thorough demonstration and
discussions by BBNS BBNS ). Moreover the order of magnitude of the departure
from asymptotic BBNS factorization ($a_{1}=1+{\cal O}(10\%))$ as observed from
the phenomenological side, is such that it does not seem possible to doubt
about this validity for the present decays. Of course, BBNS gives only an
asymptotic statement, and a departure from this limit is quite expected
(presence of various $1/m_{Q}$ corrections, $m_{c}$ not very large, etc…).
Factorization implies the well known Bjorken-Neubert relation ref:bjorken
between the NL and SL decays at some $q^{2}$ (for example $m_{\pi}^{2}$ for
decay by pion emission), which is a very strong and useful constraint for the
otherwise ill known SL differential $q^{2}$ distributions.

Practically, since we have no theoretical quantitative estimates of the
departure of $a_{1}$ from the asymptotic BBNS result, we will apply
factorisation in the following way. We fit $a_{1,\,eff.}$ in $\overline{B}\to
D_{3/2}$ decays and find a value compatible with those obtained by analysing
$\overline{B}\to D^{(*)}$ transitions (see Sec. VI). Then, in our final
analysis, we impose a constraint in the fits : $a_{1,\,eff.}=0.93\pm 0.07$.
This $a_{1,\,eff.}$ will also be used for cases where experimental data are
lacking or not precise enough. This procedure is, in principle, valid for
Class I decays with emission of a light meson. But we shall extend it to
predict $NL$ processes where a charmed strange meson is emitted like
$\overline{B}\to D^{**}D_{s}^{(*)}$; although in such a case the asymptotic
theorem of BBNS does not apply, it is a fact that, once more, one obtains a
value of $a_{1,\,eff.}$ close to 1 from the measured processes.

d) HQET and $1/m_{Q}$ expansion,

Let us comment about d). The constraints from HQET expansion as performed in
ref:leibo  and further work, including order ${\cal O}(1/m_{Q})$ corrections,
are crucial too as a necessary complement to the $m_{Q}=\infty$ predictions.

On the determination of the ${\cal O}(1/m_{Q})$ subdominant functions.

One must stress that the constraints d) do not lead to quantitative statements
on the finite mass corrections; they rather yield a parameterization of these
finite corrections. They leave us with a large number of unknown parameters
and functions of $w$. Some of those parameters, like $\bar{\Lambda}$, (quoted
as Set 1 in the following) can be estimated otherwise, as explained in Section
II.3. One could think of fixing the remaining unknown ones (Set 2) by fits to
the experimental data. But there remains at least a host of unknown functions
of $w$. For instance, for $j=3/2$, one needs in principle the ten functions :
$\tau_{1}(w),\tau_{2}(w),\eta^{b,c}_{ke}(w),\eta^{b,c}_{1,2,3}(w)$ in $LLSW$
notation333beware not to confuse the subindices to $\tau$’s (1,2) with the
previous subindices $j=1/2,3/2$ concerning the dominant Isgur-Wise functions.,
which is of course too much to be fitted at present. This number is reduced
thanks to the fact that the $\eta_{i}^{b}$’s and $\eta^{b}_{ke}$ are present
only in one combination,
$\eta^{b}=\eta^{b}_{ke}+6\eta_{1}^{b}-2\eta_{2}^{b}(w-1)+\eta_{3}^{b}$ but the
problem remains.

Many further assumptions must then be added :

e) an additional but reasonable assumption of rough proportionality of the
latter functions to the dominant Isgur-Wise functions reduces them to
numerical parameters, denoted with "hats", but one remains however with a host
of unknown numbers. For $D_{3/2}$ mesons, present data are sensitive to the
values of $\hat{\eta}_{1,\,3}$ and $\hat{\tau}_{1}$, whose values we have
determined, while some guesses have to be used to define a possible range of
variation for the other two quantities: $\hat{\eta}_{2}$ and $\hat{\tau}_{2}$.
For $D_{1/2}$ mesons, data are much less accurate and fits have the same
sensitivity to $\hat{\chi}_{1,\,2}$ or $\hat{\zeta}_{1}$ and one can obtain
the value for only one of these quantities. We choose, arbitrarily, to fit
$\hat{\chi}_{1}$.

f) of course, one would like, however, to find some reasons for that
selection, or to find consistency checks for the physical soundness of the
values of the parameters found in this way.

-$LLSW$ LLSW  proposed an additional expansion in $w-1$ (the latter too being small, of magnitude at most $\simeq 0.3\simeq\epsilon_{c}=1/2m_{c}$) in the narrow width approximation ($w_{max}\simeq 1.3,m_{c}\simeq 1.5$ $GeV$). Were this valid, it would allow to skip $\tau_{1,\,2}$ terms as being of higher order and to be left with the $\eta$’s. However, of course, this supposes that the $\tau_{1,\,2}$ are not large, a point for which there is no guarantee (in fact one often retains the reverse, i.e. dropping the $\eta$’s in favor of the $\tau_{1,\,2}$). In the present analysis we have not used any expansion in $w-1$ and we take the full $1/m_{Q}$ expansions of the Lorentz invariant form factors given in ref:leibo . The validity of this framework requires that unknown quantities that enter in the expansion are of order $\overline{\Lambda}$. This is what we do verify for fitted quantities, therefore we have assumed that quantities, not determined by the fit, are also of that order to evaluate their contribution to systematic uncertainties.

-inspiration from quark models can help to check the soundness of fit results, at least qualitatively. As explained above, one cannot trust the BT approach for finite mass corrections, and this is the main obstacle to get physical results in this quark model approach. Then one may return to the NR model of center of mass motion, but only to get a qualitative understanding. An example is the $\eta$ functions, which correspond in naive terms to the modification of the wave functions induced by the change of $m_{Q}$ from $\infty$ to its real value in the Schrödinger equation.

Knowing theoretically the "true" (infinite mass) $\tau_{3/2}(w=1)$, the
magnitude of the corrections at $w=1$ which appear in the combination
$\frac{\eta^{b}}{2m_{b}}+\frac{\eta^{c}_{ke}}{2m_{c}}$ (corresponding to our
$\hat{\epsilon}_{3/2}$ below ), can be rather well determined with a stable
value in the various fits to the data, and it is found to be neatly negative.

A fully NR calculation of the effect of ${\cal O}_{ke}(v=(1,0,0,0))$ (naively
interpreted as the effect of the change of the kinetic energy on the wave
functions in the rest frame) suggests a negative $\eta^{c}_{ke}(w=1)$
corresponding to the effect on the final state. $\eta^{b}_{ke}$ on the
contrary should be positive, but the combination $\eta^{b}$ corresponds to the
total effect on the state vectors, including ${\cal O}_{mag}(v=(1,0,0,0))$,
and, naively interpreted, includes the large spin-spin force present at finite
mass. In the GI model, it is found to be neatly negative by numerical
calculation.

Then it is encouraging to find consistency between the theoretically expected
sign and the finding of the fits. One could hope to estimate similarly the
signs and order of magnitude of the remaining $\eta^{c}_{i}$’s, but this
requires interpreting them separately in the quark model (see Appendix B).

g) on the whole, at present, there is no theoretical estimate for the values
of the different parameters that enter in the $LLSW$ expansion, except,
perhaps, a qualitative one of the $\eta$’s if one follows the arguments above
(see also Appendix B).

###  II.3 Evaluation of HQET parameters

In the framework of HQET, masses of charm and beauty mesons are used to
evaluate values of the parameters (named Set 1 in the following) that enter in
some of the $1/m_{Q}$ corrections ref:leibo , the relation being:

$m_{H_{\pm}}\,=\,m_{Q}\,+\overline{\Lambda}^{H}\,-\frac{\lambda_{1}^{H}}{2\,m_{Q}}\,\pm\frac{n_{\mp}\lambda_{2}^{H}}{2\,m_{Q}}\,+\,...$
(8)

The total spin ($J_{\pm}$) of the resonance ($H_{\pm}$) is expressed in terms
of the total spin ($s_{l}$) of the light hadronic system as: $J_{\pm}=s_{l}\pm
1/2$. while $n_{\pm}=2\,J_{\pm}+1$ is the number of spin states. Values of
these different quantities and those of meson masses, adopted in our analysis,
are indicated in Table 1. We have not considered I-spin averaged masses and
used only charged $(c\bar{d})$ states for charm and neutral states for beauty
$(b\bar{d})$.

meson | $J^{P}$ | $s_{l}^{\pi}$ | $n_{\pm}$ | charm mass | beauty mass
---|---|---|---|---|---
$D^{+}/\overline{B}^{0}$ | $0^{-}$ | $1/2^{-}$ | $n_{-}=1$ | $1869.65\pm 0.05$ | $5279.65\pm 0.12$
$D^{*\,+}/\bar{B}^{*\,0}$ | $1^{-}$ | $1/2^{-}$ | $n+=3$ | $2010.26\pm 0.05$ | $5324.70\pm 0.21$
$D_{1}^{+}/B_{1}^{0}$ | $1^{+}$ | $3/2^{+}$ | $n_{-}=3$ | $2423.8\pm 1.1$ | $5726.1\pm 1.4$
$D_{2}^{*\,+}/B_{2}^{*\,0}$ | $1^{+}$ | $3/2^{+}$ | $n_{+}=5$ | $2465.4\pm 1.3$ | $5739.5\pm 0.7$
${D_{0}(2300)^{+}}$ | $0^{+}$ | $1/2^{+}$ | $n_{-}=1$ | $2330\pm 20$ | $(?)$
${D_{1}(2430)^{+}}$ | $1^{+}$ | $1/2^{+}$ | $n_{+}=3$ | $2452\pm(?)$ | $(?)$
Table 1: The three doublets of heavy mesons having, respectively, a total spin
and parity of their light component, $s_{l}^{\pi},$ equal to $1/2^{-}$,
$3/2^{+}$ and $1/2^{+}$. The masses are given in $MeV$.

We use, for $\overline{\Lambda}^{H}$, the notations $\overline{\Lambda}$,
$\overline{\Lambda}_{3/2}$ and $\overline{\Lambda}_{1/2}$ for the
$s_{l}^{\pi}$ doublets $1/2^{-}$, $3/2^{+}$ and $1/2^{+}$ respectively.

Considering only the first order expansion in $1/m_{Q}$ of Eq. (8), the ratio
($r_{Q}$) between charm and beauty quark masses is equal to:

$r_{Q}=\frac{m_{c}}{m_{b}}=\frac{m_{B^{*}}-m_{B}}{m_{D^{*}}-m_{D}}=0.3215\pm
0.0015.$ (9)

The difference between the $\overline{\Lambda}^{H}$ values for the three
doublets can be expressed in terms of the ratio between heavy quark masses and
of the values of spin averaged masses ($\overline{m}$) ref:leibo . We have
obtained:

$\overline{\Lambda}_{3/2}-\overline{\Lambda}=(396.0\pm 1.5)\,MeV.$ (10)

For the $1/2^{+}$ doublet the corresponding estimate is more uncertain because
the associated $B$-meson states are still un-measured. From the masses of
charm states we estimate: $\overline{\Lambda}_{1/2}-\overline{\Lambda}\sim
366\,MeV$.

These values for $\overline{\Lambda}_{3/2}-\overline{\Lambda}$ and
$\overline{\Lambda}_{1/2}-\overline{\Lambda}$ agree with previous
determinations.

To evaluate absolute values for heavy quark masses and $\overline{\Lambda}$
one needs a value for $\lambda_{1}$. In previous analyses, the value
$\lambda_{1}=-0.2\,GeV^{2}$ was used; in a recent report from the HFLAV
collaboration ref:hfag2016 , the value $\lambda_{1}=(-0.362\pm
0.067)\,GeV^{2}$ was obtained from an analysis of $B$-meson semi-leptonic
decays. In Table 2 a summary is given for the values of the different
parameters entering in the analysis (first line, with
$\lambda_{1}=-0.362\,(GeV)^{2}$).

$\lambda_{1}$ | $\overline{\Lambda}$ | $\overline{\Lambda}_{3/2}$ | $\overline{\Lambda}_{1/2}$ | $m_{c}$ | $m_{b}$
---|---|---|---|---|---
$-0.362$ | $0.245$ | $0.641$ | $0.611$ | $1.618$ | $5.032$
$-0.2$ | $0.4$ | $0.8$ | $0.76$ | $1.4$ | $4.8$
Table 2: Values of HQET parameters used in the evaluation of $1/m_{Q}$
corrections (first line). The last line gives the values used in previous
analyses. The various quantities are expressed in $GeV$, except for
$\lambda_{1}\,(GeV^{2})$

###  II.4 Evaluation of parameters entering in the $LLSW$ parameterization

In the infinite quark mass limit, hadronic form factors that describe
$\overline{B}\to D^{**}$, are determined by the two Isgur-Wise functions:
$\tau_{3/2}(w)$ and $\tau_{1/2}(w)$ respectively, which have been introduced
in Eq. (6). It has been noted ref:leibo  that one can define useful effective
functions including finite mass corrections to replace the IW functions:
namely, particularising at $w=1$, one writes:

$\displaystyle\tau_{3/2}^{eff.}(1)$ $\displaystyle=$
$\displaystyle\tau_{3/2}(1)+\frac{\eta_{ke}(1)}{2\,m_{c}}+\frac{\eta_{b}(1)}{2\,m_{b}}=\tau_{3/2}(1)\times|1+\hat{\epsilon}_{3/2}|$
(11) $\displaystyle\tau_{1/2}^{eff.}(1)$ $\displaystyle=$
$\displaystyle\tau_{1/2}(1)+\frac{\chi_{ke}(1)}{2\,m_{c}}+\frac{\chi_{b}(1)}{2\,m_{b}}=\tau_{1/2}(1)\times|1+\hat{\epsilon}_{1/2}|$

In Table 3 are reminded the parameters that enter in the expressions of
Lorentz-invariant form factors for the $LLSW$ parameterization of
$\overline{B}\to D^{**}$ transitions.

| parameters | evaluation | constraints
---|---|---|---
| list |  | from theory
Set 1 | $m_{b,\,c},\,\overline{\Lambda},$ | using HQET | see Section II.3
| $\overline{\Lambda}_{3/2},\,\overline{\Lambda}_{1/2}$ |  |
| $\tau_{3/2}(1)$ | fitted | $0.53\pm 0.03$
Set 2 for | $\hat{\epsilon}_{3/2}$ | fitted |
$D_{3/2}$ | $\sigma_{3/2}^{2}$ | fitted | $1.5\pm 0.5$
mesons | $\hat{\eta}_{1,\,3},\,\hat{\tau}_{1}$ | fitted |
| $\hat{\eta}_{2},\,\hat{\tau}_{2}$ | set to zero |  $\pm 0.5\,GeV$ for syst.
| $\tau_{1/2}^{eff.}(1)$ | fitted | $0.20\pm 0.06$
Set 2 for | $\sigma_{1/2}^{2}$ | fitted | $\sigma_{3/2}^{2}-\sigma_{1/2}^{2}=0.7\pm 0.5$
$D_{1/2}$ | $\hat{\chi}_{1}$ | fitted |
mesons | $\hat{\chi}_{2},\,\hat{\zeta}_{1}$ | set to zero |  $\pm 0.5\,GeV$ for syst.
Table 3: List of the parameters that enter in $LLSW$ formalism. The two last
columns indicate how values of these quantities are obtained in the present
analysis. When no information is available on a parameter, it is set to zero
and a range of $\pm 0.5\,GeV\sim\pm\overline{\Lambda}$ is used to evaluate a
plausible contribution to systematic uncertainties.

##  III $\overline{B}^{0}\to D^{**}$ experimental results used as constraints

Input data, in this analysis, are obtained by averaging branching fraction
measurements of non-leptonic Class I, $\overline{B}^{0}\to
D^{**,\,+}\pi^{-}(K^{-})$, and semi-leptonic decay $\overline{B}\to
D^{**}\ell^{-}\overline{\nu}_{\ell}$ channels. These last values, obtained
separately for charged and neutral $B$-mesons, are combined, assuming the
equality of corresponding partial decay widths for charged and neutral
$B$-mesons, and averaged values are expressed in terms of the
$\overline{B}^{0}$. We have taken into account possible correlations between
the different uncertainties, corrected intermediate branching fractions
ref:pdg2019 , and used the hypotheses explained in Section III.1 to evaluate
$D^{**}$ absolute decay branching fractions. Values obtained in this way are
compared with those used in a previous analysis ref:ligeti1  in Table 4. The
first lines are relative to the production of narrow ($D_{3/2}$) states, then
they correspond to broad ($D_{1/2}$) states and the last line reports a ratio
between the production of narrow and broad states. Therefore, apart for the
last constraint, it is possible to investigate the production of narrow and
broad states, independently.

decay channel | this analysis | analysis ref:ligeti1  | PDG(2020)
---|---|---|---
|  | (2016) | or HFAG $(^{3})$
${\cal B}(\overline{B}^{0}\to D_{2}^{*+}\pi^{-})\times 10^{4}$ | $5.85\pm 0.42$ | $5.9\pm 1.3$ | $5.85\pm 0.43$
${\cal B}(\overline{B}^{0}\to D_{2}^{*+}K^{-})\times 10^{5}$ | $4.7\pm 0.8$ | not used | $5.0\pm 0.9$
${\cal B}(\overline{B}^{0}\to D_{2}^{*+}\ell^{-}\bar{\nu}_{\ell})\times 10^{3}$ | $3.09\pm 0.32$ | $2.8\pm 0.4$ | $3.18\pm 0.26$
${\cal B}(\overline{B}^{0}\to D_{1}^{+}\pi^{-})\times 10^{4}$ | $7.12\pm 1.13$ | $7.5\pm 1.6$ | $6.6\pm 2.0$
${\cal B}(\overline{B}^{0}\to D_{1}^{+}\ell^{-}\bar{\nu}_{\ell})\times 10^{3}$ | $6.40\pm 0.44$ | $6.2\pm 0.5$ | $6.24\pm 0.54$
${\cal B}(\overline{B}^{0}\to D_{0}(2300)^{+}\pi^{-})\times 10^{4}$ | $1.19\pm 0.12$ | not used | $1.14\pm 0.12$
${\cal B}(\overline{B}^{0}\to D_{0}(2300)^{+}\ell^{-}\bar{\nu}_{\ell})\times 10^{3}$ | $2.2\pm 1.2$ | $4.1\pm 0.7$ | $3.9\pm 0.7\,(^{4})$
${\cal B}(\overline{B}^{0}\to D_{1}(2430)^{+}\pi^{-})\times 10^{4}$ | $0.21\pm 0.27\,(^{1})$ | not used | not quoted
${\cal B}(\overline{B}^{0}\to D_{1}(2430)^{+}\ell^{-}\bar{\nu}_{\ell})\times 10^{3}$ | $1.4\pm 1.3$ | $1.9\pm 0.5$ | $1.8\pm 1.5\,(^{4})$
${\cal B}(\overline{B}^{0}\to D_{0}(2300)^{+}K^{-})/{\cal B}(\overline{B}^{0}\to D_{2}^{*+}K^{-})$$\,(^{2})$ | $0.84\pm 0.36$ | not used |
Table 4: Measured branching fractions used as constraints in the present
analysis. In the two last columns are indicated the values used in a previous
analysis and those quoted in PDG ref:pdg2019  or obtained by HFAG ref:hfag2016
. As compared with our numbers, these values differ mainly in the estimate of
the production of broad $D^{**}$ states in $B$-meson semi-leptonic decays.
Analysis ref:ligeti1  differs also by the fact that non-leptonic measurements
of $D_{1/2}$ states are not included. $(^{1})$: we have not used the measured
value for ${\cal B}(\overline{B}^{0}\to D_{1}(2430)^{+}\pi^{-})$
ref:belle_d12430pi  as constraint in our nominal fit because this is still a
preliminary result and we have instead compared this value to our expectation.
$(^{2})$: $D^{**,\,+}$ mesons are reconstructed in the $D^{0}\pi^{+}$ final
state. $(^{3})$: values from PDG and HFAG have been modified using results
quoted in Table 6. $(^{4})$: values from HFAG ref:hfag2016 , the uncertainty
on ${\cal B}(\overline{B}^{0}\to D_{1}(2430)^{+}\ell^{-}\bar{\nu}_{\ell})$ is
multiplied by a factor three to account for the poor compatibility between the
three measurements used to obtain the average value.

Production of $D_{1/2}$ mesons in semi-leptonic decays, reported in the two
last columns of Table 4, are derived from HFAG ref:hfag2016  whereas those
used in our analysis are obtained using a different approach, as explained in
Section III.2. As illustrated from the values given in Table 4 and from the
results of the fits quoted in Tables 13 and 18, estimates for ${D_{1}(2430)}$
are not accurate, with about 100$\%$ uncertainty, once taking into account the
fact that existing measurements are rather incompatible, with
$\chi^{2}/NDF=18/2$, and if we scale the uncertainty, obtained on the average
value, using the "PDG recipe" that corresponds to a factor three. Such a
scaling factor is not included in the ref:ligeti1  analysis and the present
value from PDG ($\sim(4\pm 1)\times 10^{-3}$), not quoted in Table 4, is based
on the measurement from BaBar alone, not including the other two results that
enter in the HFAG average, and which are rather different, in particular the
one from Belle which does not see a signal and quotes a stringent limit.

In addition, the Belle collaboration ref:belle_dsstarl  has measured the semi-
leptonic decay branching fraction $\overline{B}^{0}\to
D_{2}^{*+}\ell^{-}\bar{\nu}_{\ell}$ in four bins of the variable $w=v_{D}\cdot
v_{B}=(m_{B}^{2}+m^{2}_{D_{2}^{*}}-q^{2})/2\,m_{B}\,m_{D_{2}^{*}};\,q^{2}=m^{2}(\ell^{-}\bar{\nu}_{\ell})$.
The sum of these fractions is normalized to unity therefore, in the following,
because we do not have the full error matrix on these measurements available,
we use the first three results and assume that they are independent.

$w$ bin | fraction $(\%)$
---|---
$[1.00,\,1.08]$ | $6.0\pm 2.3$
$[1.08,\,1.16]$ | $30.0\pm 5.4$
$[1.16,\,1.24]$ | $37.5\pm 6.2$
$[1.24,\,1.32]$ | $26.5\pm 6.2$
Table 5: Measured fractions of the $\overline{B}^{0}\to
D_{2}^{*+}\ell^{-}\bar{\nu}_{\ell}$ decay width in several $w$ bins.

Values quoted in the third column of Table 4 are used in Appendix C to
demonstrate that our code is able to reproduce the values obtained in
ref:ligeti1  when using similar input values and hypotheses. Those given in
the last column allow to perform a comparison between experimental values
retained in our analysis and those quoted in "official" compilations.

###  III.1 Evaluation of absolute $D^{**,\,+}$ decay branching fractions

At present, absolute $D^{**}$ decay branching fractions are obtained using
hypotheses on the contribution of the various decay channels. For the $D_{1}$,
it is assumed that it decays only into $D^{*}\pi$ and $D\pi\pi$ (through
${D_{0}(2300)}\pi$). The ${D_{2}^{*}}$ is expected to decay only into $D\pi$
and $D^{*}\pi$. For broad states, we consider that the ${D_{0}(2300)}$ and the
${D_{1}(2430)}$ decay only into $D\pi$ and $D^{*}\pi$, respectively. Expected
branching fractions are given in Table 6; they are similar to those used in
ref:ligeti1 .

${\cal B}({D_{2}^{*}}\to D\pi)$ | $0.61\pm 0.02$
---|---
${\cal B}({D_{1}^{+}}\to D^{*0}\pi^{+})$ | $0.45\pm 0.03$
${\cal B}({D_{1}^{+}}\to D^{+}\pi^{+}\pi^{-})$ | $0.15\pm 0.02$
${\cal B}({D_{1}(2430)}\to D^{*}\pi)$ | $1$
${\cal B}({D_{0}(2300)}\to D\pi)$ | $1$
Table 6: Expected absolute $D^{**}$ mesons decay branching fractions.

###  III.2 Estimates for ${\cal B}(\overline{B}^{0}\to
D_{1/2}^{+}\ell^{-}\bar{\nu}_{\ell})$

Values for ${D_{0}(2300)}$ and ${D_{1}(2430)}$ production in $\overline{B}$
hadron semi-leptonic decays, quoted in PDG or in HFAG, are obtained by fitting
expected $D^{(*)}\pi$ mass distributions on measured $\overline{B}\to
D^{(*)}\pi\ell^{-}\bar{\nu}_{\ell}$ events. This approach is reliable for
$D_{3/2}$ mesons which appear as relatively narrow mass peaks. The
determination of broad $D_{1/2}$ production is more difficult because one
expects also contributions from $D_{V}^{(*)}\to D^{(*)}\pi$ and it is not
clear how these components have been included in the analyses. For these
reasons we have adopted another approach.

To evaluate ${\cal B}(\overline{B}^{0}\to
D_{1/2}^{+}\ell^{-}\bar{\nu}_{\ell})$ we use ${\cal B}(\bar{B}\to
D^{(*)}\pi\ell^{-}\bar{\nu}_{\ell})$ exclusive measurements from BaBar
ref:babardpi  and Belle ref:belledpi , from which we subtract the expected
contributions from $D_{3/2}$ and $D^{(*)}_{V}$ decays.

It has to be noted that, at present, in all these evaluations, contributions
from higher mass resonances, which can decay into $D^{(*)}\pi$, are not
specifically evaluated and therefore are fully or partially included in the
rates estimated for $D_{1/2}$ mesons, depending on the approach.

$D^{(*)}_{V}$ components are normalized in an absolute way using values for
${\cal B}(\overline{B}^{0}\to D^{(*)+}\ell^{-}\bar{\nu}_{\ell})$ in which
$D^{(*)}$ are on-mass shell, albeit with rather large uncertainties related to
their mass dependence roudeau1 .

In the $D\pi$ channel, only the $D^{*}_{V}$ component contributes. It is
expected to decrease with the $D\pi$ mass value. For the $D^{*0}_{V}$ channel,
there is a natural threshold in the decay to $D^{+}\pi^{-}$ because
$m_{D^{*0}}<m_{D^{+}}+m_{\pi^{-}}$. For other charge combinations, the
separation between so called $D^{*}$ and $D^{*}_{V}$ components is arbitrary,
and therefore the absolute value of the $D^{*}_{V}$ component depends on the
considered threshold. In Belle, they use the following mass range
$m_{D^{(*)}\pi}\in[2.05,\,3]\,GeV/c^{2}$ whereas, in BaBar they require
$m_{D^{0}\pi^{+}}-m_{D^{0}}>0.18\,GeV/c^{2}$. These cuts are rather similar,
being typically $40\,MeV/c^{2}$ above the nominal $D^{*}$mass.

To evaluate the uncertainty on these estimates we assume that the
$D^{(*)}_{V}$ follows a relativistic Breit-Wigner mass distribution, modified
by a Blatt-Weisskopf damping term with parameter $r_{BW}=1.85\,(GeV/c)^{-1}$
which is varied in the range $[1.0,\,3.0]\,(GeV/c)^{-1}$.

####  III.2.1 ${\cal
B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\bar{\nu}_{\ell})$

Averaging experimental measurements one obtains: ${\cal B}(\overline{B}^{0}\to
D\pi\ell^{-}\bar{\nu}_{\ell})=(6.14\pm 0.53)\times 10^{-3}$. We estimate the
$D^{*}_{V}\to D\pi$ contribution, with $m_{D\pi}>2.05\,GeV$, to be equal to:
$(2.0\pm 0.6)\times 10^{-3}$, in which the uncertainty corresponds to the
variation range used for $r_{BW}$. The ${D_{2}^{*}}$ contribution is obtained
from Tables 4 and 6; it amounts to $(1.90\pm 0.18)\times 10^{-3}$.

This gives:

${\cal
B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\bar{\nu}_{\ell})=(2.2\pm
1.2)\times 10^{-3}$ (12)

as reported, after rounding, in Table 4. The uncertainty on the $D^{*}_{V}$
estimate is added linearly to the total uncertainty evaluated for the other
sources.

####  III.2.2 ${\cal
B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\ell^{-}\bar{\nu}_{\ell})$

We use the same approach for the $D^{*}\pi$ hadronic final state. Averaging
experimental measurements, one obtains: ${\cal B}(\overline{B}^{0}\to
D^{*}\pi\ell^{-}\bar{\nu}_{\ell})=(8.39\pm 0.54)\times 10^{-3}$. We estimate
the $D^{*+}_{V}$ contribution, using the coupling
$g_{D^{*}D^{*}\pi}=\sqrt{2\frac{m_{D^{*}}}{m_{D}}}g_{D^{*}D\pi}$ which
corresponds to a fictitious $\Gamma(D^{*}\to D^{*}\pi)=2\,\Gamma({D}^{\ast}\to
D\pi)$, and we have added the expected (small) contribution from $D_{V}\to
D^{*}\pi$. This gives: $(1.4\pm 0.6)\times 10^{-3}$. $D_{V}$ and $D^{*}_{V}$
contributions are added incoherently because of helicity conservation and of
the vanishing of $A-V$ interferences for the zero-helicity after the angular
integration.

Subtracting the ${D_{1}^{+}}$ ($(4.32\pm 0.41)\times 10^{-3}$)and
${D_{2}^{*+}}$ ($(1.21\pm 0.14)\times 10^{-3}$) contributions, obtained from
Tables 4 and 6, this gives:

${\cal
B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\ell^{-}\bar{\nu}_{\ell})=(1.4\pm
1.3)\times 10^{-3}$ (13)

as reported in Table 4. The uncertainty on the $D^{(*)}_{V}$ estimate is added
linearly.

##  IV Parameterization of semi-leptonic and non-leptonic decay widths

We indicate here the expressions we use to compute semi-leptonic and non-
leptonic Class I decay branching fractions and explain how we have taken into
account effects from contributions of virtual states.

###  IV.1 Semi-leptonic transitions to real, discrete charmed states

Differential semi-leptonic $\overline{B}\to D_{X}\ell^{-}\bar{\nu}_{\ell}$
decay widths, with $D_{X}=D^{(*,\,**)}$, can be written as ref:ligeti1 :

$\displaystyle\frac{d\Gamma}{dq^{2}}$ $\displaystyle=$ $\displaystyle
C|\vec{p}|q^{2}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}$ (14)
$\displaystyle\times$
$\displaystyle\left[(H_{+}^{2}+H_{-}^{2}+H_{0}^{2})\left(1+\frac{m_{\ell}^{2}}{q^{2}}\right)+\frac{3m_{\ell}^{2}}{2q^{2}}H_{t}^{2}\right]$

with $C=G_{F}^{2}\left|{\rm
V}_{cb}\right|^{2}\eta_{EW}^{2}/(96\,\pi^{3}\,m_{B}^{2})$. $\vec{p}$ is the
three-momentum of the $D_{X}$ in the $B$ rest frame.

$|\vec{p}|=m_{D_{X}}\,\sqrt{w_{D_{X}}^{2}-1}$ (15)

in which $w_{D_{X}}=v_{B}\cdot v_{D_{X}}$ is the product of the 4-velocities
of the two mesons.

$H_{\pm,0,t}$ are helicity form factors which are expressed in terms of,
$q^{2}$ dependent, Lorentz invariant form factors ($FF(q^{2})$) and depend on
the considered meson $D_{X}$. Accurate parameterizations of $FF(q^{2})$ are
obtained for $D$ (we use ref:ff_dlnu ) and $D^{*}$ mesons (we use ref:cln ;
ref:fajfer ). For $D^{**}$ mesons, expressions are taken from ref:leibo . They
correspond to expansions at first order in $1/m_{c,b}$ and $\alpha_{s}.$ For
$D_{3/2}$ mesons, expressions that contain $1/m_{c,b}\times\alpha_{s}$
corrections are also available and have been included.

###  IV.2 Semi-leptonic transitions to virtual charmed states

Let us now consider the physical case where the charmed state terminates on a
two-body continuum like $D^{(*)}\pi$. As a useful intermediate step, one now
considers fictitious weak transitions to intermediate virtual $D^{**}$ or
$D^{(*)}$, which represent the weak vertex part of the overall process, with
the charmed leg bearing a momentum squared
$p^{2}=m^{2}\left(=m^{2}_{D^{(*)}\pi}\right)$, different from the nominal
squared mass of the state $m^{2}_{D_{X}}$ (of course, in the overall process,
leading for instance to a $D\pi$ final state, one has to introduce a $D^{*}$
propagator relating the weak vertex to the strong vertex which couples the
virtual state to $D\pi$). If one considers production of virtual $D^{**}$
mesons (that will get a Breit-Wigner distribution for $m$) or of virtual
$D^{(*)},$ noted usually $D^{(*)}_{V},$ mesons, with $m$ higher than their
nominal mass value $m_{D^{(*)}},$ the invariant form factors at the weak
vertex are expected to be also dependent on $m\neq m_{D_{X}}$.

However, in the absence of a theoretical knowledge of this dependence we
assume simply:

$FF(m,q^{2})=FF(m_{D_{X}},q^{2})$ (16)

in the expressions of helicity form factors, while keeping $m\neq m_{D_{X}}$
dependence of the kinetic quantities, such as momenta, in the additional
factors entering in these expressions. This procedure is illustrated in the
following example of the production of a virtual $D_{V}$ meson of mass $m\neq
m_{D}$. There are two invariant form factors noted : $f_{+,0}(m,q^{2})$.
Helicity form factors are expressed as:

$\displaystyle q\,H_{0}$ $\displaystyle=$ $\displaystyle
2\,m_{B}\,|\vec{p}|\,f_{+}(m_{D},q^{2})$ $\displaystyle q\,H_{t}$
$\displaystyle=$ $\displaystyle(m_{B}^{2}-m^{2})\,f_{0}(m_{D},q^{2}).$ (17)

The two other form factors, $H_{\pm}$, vanish because of helicity
conservation. In these expressions and in Eq. (14), the decay momentum is
evaluated at the virtual mass $m$: $|\vec{p}|=m\,\sqrt{w^{2}-1}$ with
$w=(m_{B}^{2}+m^{2}-q^{2})/(2\,m\,m_{B})$. Expressions relating helicity and
invariant form factors when a $D^{*}$ meson is emitted can be found in
ref:fajfer  and in ref:ligeti1  in case of $D^{**}$ mesons. Since these
formula are devised for real, discrete charmed states, one must modify the
factors affecting the form factors when considering virtual charmed states.

###  IV.3 Non-leptonic decays

Thanks to factorization, non-leptonic Class I decay widths
$\Gamma(\overline{B}^{0}\to D_{X}^{+}P^{-})$ are related to corresponding
differential semi-leptonic decay widths, through the $H_{t}^{D_{X}}$ helicity
form factor:

$\begin{split}\Gamma&(\overline{B}^{0}\to D_{X}^{+}P^{-})\\\
&=\frac{\left|a_{1,eff.}^{D_{X}\pi}\right|^{2}f_{P}^{2}\,G_{F}^{2}\,|V_{ij}|^{2}|V_{cb}|^{2}|\vec{p}|}{16\pi\,m_{B}^{2}}\left.q^{2}(H_{t}^{D_{X}})^{2}\right|_{w_{P}}.\end{split}$
(18)

In this expression, $q^{2}=m_{P}^{2}$, $f_{P}$ is the leptonic decay constant
of the emitted charged meson $P$, and $V_{ij}$ is the corresponding CKM matrix
element.

$w_{P}=\frac{m_{B}^{2}+m_{D_{X}}^{2}-m_{P}^{2}}{2\,m_{B}\,m_{D_{X}}}.$ (19)

If the virtual “mass” $m$ of the $D_{X}$ meson doesn’t have the nominal value
$m_{D_{X}}$, we will still use values of invariant form factors evaluated at
$q^{2}=m_{P}^{2}$ as above while taking the running mass to compute the other
terms that enter in $H_{t}^{D_{X}}$ and in $\vec{p}$.

We term the factor $a_{1,\,eff.}^{D_{X}\pi}$ "effective" because the non-
leptonic decay width, that enters in Eq. (18), corresponds to the sum of the
Class I diagram amplitude and of subdominant terms which correspond to
exchange or penguin mechanisms, while the remaining factors in the right hand
are those provided by the analytic expression for a pure Class I process.

Expressions for $\overline{B}^{0}\to D^{**,\,+}P^{-}$ partial decay widths,
with $P\,=\,\pi^{-}$ or ${D}_{s}^{-}$, and for $\overline{B}^{0}\to
D^{**,\,+}D_{s}^{*-}$ are given in Appendix A. Corresponding quantities,
obtained in the infinite quark mass limit, for non-leptonic -
$\Gamma(\overline{B}^{0}\to D^{**}\pi^{-})$ \- and for the differential semi-
leptonic - $d\Gamma(\overline{B}^{0}\to
D^{**,\,+}\ell^{-}\bar{\nu}_{\ell})/dw$ \- can be found, for example, in
ref:jugeau .

###  IV.4 Finite width effects

To include effects from the mass distribution of resonances we express the
differential decay width for a process in which a $D^{**}$ is reconstructed in
a given final state ($i$) as roudeau1 :

$\frac{d\Gamma_{i}}{ds}=\frac{1}{\pi}\frac{\Gamma_{0}(s)\,\sqrt{s}\,\Gamma_{i,\,D^{**}}(s)}{(s-m_{D^{**}}^{2})^{2}+s\,\Gamma^{2}_{D^{**}}(s)}.$
(20)

$\Gamma_{0}(s)$ is the decay width for the process computed in the hypothesis
of a virtual $D^{**}$ of mass equal to $\sqrt{s}$. $\Gamma_{i,\,D^{**}}(s)$ is
the partial decay width for the $D^{**}$, of mass $\sqrt{s}$, reconstructed in
the $i$ observed channel. $\Gamma_{D^{**}}(s)$ is the total $D^{**}$ decay
width at the mass $\sqrt{s}$.

When the current mass ($\sqrt{s}$) is higher than the threshold for the $i$
decay channel:

$\Gamma_{i,\,D^{**}}(s)=\Gamma_{i,\,D^{**}}(m_{D^{**}}^{2})\left(\frac{p_{i}}{p_{i}^{0}}\right)^{2\,L+1}\left(\frac{m_{D^{**}}}{\sqrt{s}}\right)^{2}\left(\frac{F_{i,\,L}(p_{i})}{F_{i,\,L}(p_{i}^{0})}\right)^{2}.$
(21)

$p_{i}$ and $p_{i}^{0}$ are the break-up momenta for the $D^{**}$ decaying
into the $i$-channel at masses equal to $\sqrt{s}$ and $m_{D^{**}}$,
respectively. $F_{i,\,L}(p_{i})$ is a Blatt-Weisskopf damping factor. For
decay channels with a threshold above the resonance mass, $p_{i}^{0}$ becomes
imaginary and $F_{i}(p_{i}^{0})$ cannot be evaluated. In these conditions we
take:

$\Gamma_{i,\,D^{**}}(s)=\frac{g_{i}^{2}}{24\,\pi}\frac{p_{i}^{2L+1}}{s}F_{i}^{2}(p_{i})$
(22)

in which $g_{i}$ is the coupling of the $D^{**}$ to the $i$-channel. This
expression is used, in the following, to evaluate contributions from virtual
$D$ or $D^{*}$ mesons to $D^{(*)}\pi$ final states.

$\Gamma_{D^{**}}(s)$ is the sum of all partial decay widths
$\Gamma_{i,\,D^{**}}(s)$, opened at the mass $\sqrt{s}$.

In non-leptonic decays, the expression in Eq. (20) is multiplied by another
damping factor (noted usually
$(F_{B,\,i}(p^{\prime}_{i})/F_{B,\,i}(p^{\prime,\,0}_{i}))^{2}$) to account
for strong interaction effects due to the hadron emitted with the $D^{**}$.
$p^{\prime}_{i}$ is the momentum of the $B$ decay products, evaluated in the
$B$ rest-frame 444In some analyses, $p^{\prime}$ is evaluated in the resonance
rest-frame. We consider that our choice is more physical. Effects of changing
the convention to compute $p^{\prime}$ are given in Table 13..

##  V Constraints used in our analysis

The measurements used in our analysis are listed in Table 4, second column.

As to the theoretical constraints, there are three of them (see Table 3):

1) factorization. The validity of the factorization is checked, using
$D_{3/2}$ events (see Section VI), and then it is used as a constraint in the
final fits with $a_{1,\,eff.}^{D^{**}\pi}=0.93\pm 0.07$ for all $D^{**}$
states.

2) at $w=1$ we use the constraint : $\tau_{3/2}(1)=0.53\pm 0.03$ as expected
from LQCD ref:lqcd-tau . Because $\tau_{3/2}$ is defined in the infinite quark
mass limit it is necessary to introduce mass corrections and one parameter
characterizing them $\hat{\epsilon}_{3/2}$, (see Eq. (II.4)) which is a useful
combination of the basic ones introduced by $LLSW$. $\hat{\epsilon}_{3/2}$ is
fitted, as well as a part of the other parameters. The data on $D_{1/2}$
mesons are less accurate and it is not possible to fit $\hat{\epsilon}_{1/2}$.

3) at $w\neq 1$, LQCD does not provide any information on the variation with
respect to $w$ of the two IW functions for which we use quark models, namely
the BT calculations explained in the second section. Unfortunately, quark
models cannot provide errors. Therefore, to use them in a fit, we have
considered a rather large range of values of the slopes around the predicted
one.

###  V.1 Comparison with the analysis of ref:ligeti2 .

To validate our code, we check that, using the same input data (given in the
third column of Table 4) and the same hypotheses, we reproduce the results
published in ref:ligeti2 (see Appendix C).

##  VI Production of $D_{3/2}$ mesons : a check of factorization

In a first step, the analysis is restricted to the production of $D_{3/2}$
mesons to quantify the importance of $1/m_{Q}$ corrections and to check the
applicability of the factorization property. We have required that
$\tau_{3/2}(1)=0.53\pm 0.03$ and fitted the $\hat{\epsilon}_{3/2}$ correction.
This is essentially equivalent to fitting directly $\tau^{eff.}_{3/2}$. No
constraint is used on $a_{1,\,eff.}^{D_{3/2}\pi}$ and $\sigma_{3/2}^{2}$.

Data are first analyzed without any $1/m_{Q}$ correction. The fit probability
is below $10^{-13}$. This is mainly due to the fact that, in the infinite
quark mass limit, theory predicts a production higher for the $D_{2}^{*}$ as
compared to the $D_{1}$ whereas it is measured two times lower.

Adding the Set 1 of $1/m_{Q}$ corrections improves the situation but the fit
probability is still below $10^{-4}$.

Therefore it is necessary to fit additional parameters which control the other
$1/m_{Q}$ corrections. Fitting successively one among all other parameters, we
end up with the results given in Table 7. All fit probabilities are higher
than $10\%$.

X param. | $a_{1,\,eff.}^{D_{3/2}\pi}$ | $\sigma_{3/2}^{2}$ | $\hat{\epsilon}_{3/2}$ | X $(GeV)$ | $\chi^{2}/NDF$
---|---|---|---|---|---
$\hat{\eta}_{1}$ | $0.90\pm 0.05$ | $-0.4\pm 0.7$ | $-0.50\pm 0.09$ | $-0.40\pm 0.11$ | $3.4/4$
$\hat{\eta}_{3}$ | $0.82\pm 0.05$ | $-2.1\pm 0.7$ | $-0.77\pm 0.07$ | $3.2\pm 1.1$ | $7.0/4$
$\hat{\tau}_{1}$ | $0.81\pm 0.07$ | $1.0\pm 0.7$ | $-0.34\pm 0.12$ | $0.75\pm 0.24$ | $6.4/4$
$\hat{\tau}_{2}$ | $0.89\pm 0.06$ | $2.1\pm 0.5$ | $-0.19\pm 0.11$ | $2.9\pm 0.8$ | $2.5/4$
$\hat{\eta}_{2}$ | $0.86\pm 0.06$ | $1.2\pm 0.6$ | $-0.30\pm 0.11$ | $-1.63\pm 0.44$ | $3.8/4$
Table 7: Results obtained using the constraint from theory on $\tau_{3/2}(1)$.
Set 1 $1/m_{Q}$ corrections are used and one additional parameter from Set 2
is fitted, in addition to $\hat{\epsilon}_{3/2}$.

It can be noted that, depending on the choice of the additional fitted
parameter, values of the slope ($\sigma_{3/2}^{2}$) and of the correction
($\hat{\epsilon}_{3/2}$ ) to the normalization of the IW function fluctuate.
This comes from the fact that fitted individual parameters are also changing
the $w$ dependence of form factors and there are not enough measurements of
the differential decay branching fractions versus $w$ to constrain these
variations. It results that fitted values for the IW slope can be highly
correlated with the value fitted for some of the additional parameters. In the
following we therefore use the constraint expected from QM :
$\sigma_{3/2}^{2}=1.5\pm 0.5$ so that the $w$-dependence of the IW function
verifies expectations from theory while we have no constraints on the precise
values of all parameters entering in $1/m_{Q}$ corrections. One has to check,
a posteriori, that such quantities are not too large so that the model we are
using remains valid.

Results obtained for all possible fitted pairs of parameters are given in
Table 8. The fitted correction $\hat{\epsilon}_{3/2}$ is now rather
independent on the choice of the fitted pair.

X-Y | $a_{1,\,eff.}^{D_{3/2}\pi}$ | $\sigma_{3/2}^{2}$ | $\hat{\epsilon}_{3/2}$ | X | Y | $\chi^{2}/NDF$
---|---|---|---|---|---|---
param. |  |  |  | $(GeV)$ | $(GeV)$ |
$\hat{\eta}_{1}-\hat{\eta}_{3}$ | $1.06\pm 0.08$ | $1.45\pm 0.45$ | $-0.19\pm 0.11$ | $-0.45\pm 0.08$ | $-0.78\pm 0.25$ | $1.4/4$
$\hat{\eta}_{1}-\hat{\tau}_{1}$ | $0.83\pm 0.11$ | $1.34\pm 0.57$ | $-0.28\pm 0.10$ | $0.05\pm 0.26$ | $0.68\pm 0.73$ | $6.7/4$
$\hat{\eta}_{1}-\hat{\tau}_{2}$ | $0.88\pm 0.06$ | $1.55\pm 0.50$ | $-0.27\pm 0.09$ | $-0.07\pm 0.10$ | $2.34\pm 1.24$ | $2.7/4$
$\hat{\eta}_{1}-\hat{\eta}_{2}$ | $0.86\pm 0.06$ | $1.55\pm 0.50$ | $-0.24\pm 0.10$ | $0.09\pm 0.17$ | $-2.1\pm 1.1$ | $3.7/4$
$\hat{\eta}_{3}-\hat{\tau}_{1}$ | $0.82\pm 0.05$ | $1.75\pm 0.50$ | $-0.18\pm 0.12$ | $-0.52\pm 0.33$ | $1.04\pm 0.31$ | $4.6/4$
$\hat{\eta}_{3}-\hat{\tau}_{2}$ | $0.86\pm 0.05$ | $1.66\pm 0.50$ | $-0.27\pm 0.11$ | $0.10\pm 0.23$ | $2.9\pm 0.9$ | $3.0/4$
$\hat{\eta}_{3}-\hat{\eta}_{2}$ | $0.89\pm 0.05$ | $1.67\pm 0.48$ | $-0.19\pm 0.11$ | $-0.30\pm 0.26$ | $-2.0\pm 0.5$ | $2.8/4$
$\hat{\tau}_{1}-\hat{\tau}_{2}$ | $0.86\pm 0.06$ | $1.75\pm 0.46$ | $-0.25\pm 0.09$ | $0.07\pm 0.33$ | $2.8\pm 1.5$ | $3.2/4$
$\hat{\tau}_{1}-\hat{\eta}_{2}$ | $0.98\pm 0.10$ | $1.60\pm 0.41$ | $-0.20\pm 0.10$ | $-1.1\pm 0.7$ | $-3.9\pm 1.6$ | $1.9/4$
$\hat{\tau}_{2}-\hat{\eta}_{2}$ | $0.87\pm 0.05$ | $1.65\pm 0.48$ | $-0.25\pm 0.09$ | $2.1\pm 2.2$ | $-0.5\pm 1.1$ | $3.0/4$
Table 8: Results obtained using the two constraints from theory on
$\tau_{3/2}(1)$ and $\sigma_{3/2}^{2}$. Set 1 $1/m_{Q}$ corrections are used
and two additional parameters from set 2 are fitted, in addition to
$\hat{\epsilon}_{3/2}$.

The value of $a_{1,\,eff.}^{D_{3/2}\pi}$, fitted in each model, (see Tables 7
and 8), varies between $0.81$ and $1.06$ with uncertainties between $0.05$ and
$0.11$. This is compatible with estimates of $a_{1,\,eff.}^{D\pi}=0.880\pm
0.024$ and $a_{1,\,eff.}^{D^{*}\pi}=0.981\pm 0.025$ we obtain by analysing
corresponding decay channels. These values differ somewhat from the one given
by BBNS, but this is not unexpected since we are far from the asymptotic
situation considered by those authors.

Therefore, in our final results we add, as a constraint, that
$a_{1,\,eff.}^{D^{**}\pi}=0.93\pm 0.07$ obtained from the measurements when
either a $D$ or a $D^{*}$ is produced. This constraint is important, when
evaluating systematic uncertainties, to avoid that effects of a variation of a
given parameter induce a large variation on $a_{1}$ and therefore correspond
to effects that are outside the fact that the present analysis is done in the
framework of factorization. This constraint is softer than the one used for
this same property in previous analyses of these channels.

###  VI.1 Fitted model parameters for $D_{3/2}$ production

As observed in Tables 7 and 8, the $\chi^{2}$ obtained when fitting present
data is mainly sensitive to the parameters $\hat{\eta}_{1}$, $\hat{\eta}_{3}$,
and $\hat{\tau}_{1}$. There are enough measurements and constraints to
determine these quantities with some accuracy. Values of the two other
parameters, $\hat{\tau}_{2}$ and $\hat{\eta}_{2}$ cannot be determined from
present data. We have evaluated effects from this indetermination by changing
the values of these two quantities by $\pm 0.5\,GeV$ and by redoing the fit of
the other parameters. The variation range we consider is obtained by noting
that these quantities can have, at most, values similar to
$\overline{\Lambda}$ so that the model remains valid.

###  VI.2 Summary on $D_{3/2}$ production

In summary, production of $D_{3/2}$ in non-leptonic Class I $B$-meson decays
is compatible with factorization.

The analysis can be done using the value expected from LQCD for
$\tau_{3/2}(1)=0.53\pm 0.03$ but this differs from the $\tau_{3/2}^{eff.}(1)$
introduced in equation (II.4) by the quantity $\hat{\epsilon}_{3/2}=-0.2\pm
0.1$, whose value has been fitted. We have verified, in Appendix B, that the
minus sign of this correction agrees with theory. Combining these values we
obtain:

$\tau_{3/2}^{eff.}(1)=0.42\pm 0.06,$ (23)

in agreement with previous analyses.

To have reasonable agreement between data and expectations we find that it is
necessary to fit at least one, among the five parameters that control
$1/m_{Q}$ corrections. In this case there remain four parameters that are
unknown and it is difficult to obtain model uncertainties. Hopefully, present
measurements and constraints from theory allow us to evaluate values of the
three most important parameters that control the model. In this way we
estimate to have a better control of systematic uncertainties that come from
estimates of the values of quantities that are not fitted on data. We obtain:

$\displaystyle\hat{\eta}_{1}\,(GeV)$ $\displaystyle=$ $\displaystyle-0.32\pm
0.13\pm 0.03$ $\displaystyle\hat{\eta}_{3}\,(GeV)$ $\displaystyle=$
$\displaystyle-0.77\pm 0.28\pm 0.21$ $\displaystyle\hat{\tau}_{1}\,(GeV)$
$\displaystyle=$ $\displaystyle 0.36\pm 0.35\pm 0.35.$ (24)

in which the second uncertainty corresponds to the largest variations induced
by changing the values of $\hat{\tau}_{2}$ and $\hat{\eta}_{2}$ by $\pm
0.5\,GeV$. We consider that this model uncertainty has to be added linearly to
the one coming from the fit because we cannot favour any value for
$\hat{\tau}_{2}$ and $\hat{\eta}_{2}$, within their considered variation
range.

One cannot directly compare values obtained for $\hat{\eta}_{1,\,3}$ and
$\hat{\tau}_{1}$ with previous determinations in which one or at most two of
these parameters have been fitted on data. It can be noted that fitted values
do not preclude the assumed validity of the $1/m_{Q}$ expansion because these
quantities are of order $\overline{\Lambda}$. In Table 9 we compare expected
values for the ratios ${\cal R}_{D_{3/2}}={\cal B}(\overline{B}^{0}\to
D_{3/2}^{+}\tau^{-}\bar{\nu}_{\tau})/{\cal B}(\overline{B}^{0}\to
D_{3/2}^{+}\ell^{-}\bar{\nu}_{\ell})$ with previous determinations.

${\cal R}_{D_{3/2}}\,(\%)$ | our analysis |  ref:ligeti1 (2016) |  ref:ligeti2 (2017)
---|---|---|---
${\cal R}_{D_{2}^{*}}$ | $6.1\pm 0.5\pm 0.2$ | $7\pm 1$ | $7\pm 1$
${\cal R}_{D_{1}}$ | $9.9\pm 0.7\pm 0.1$ | $10\pm 1$ | $10\pm 2$
Table 9: Comparison between expected values for ${\cal R}_{D_{3/2}}$.

##  VII Production of $D_{1/2}$ mesons:
$\tau_{1/2}^{eff.}(1)\ll\tau_{3/2}^{eff.}(1)$

Data on $B$-meson semi-leptonic decays into $D_{1/2}$ states are rather
uncertain. In agreement with previous analyses (see Table 17 in Appendix C),
we find that present data do not allow to determine the slope
$\sigma_{1/2}^{2}$ of the corresponding IW function. In addition, the effet of
$1/m_{Q}$ parameters is to modify the observed $w$ dependence of form factors,
therefore it is important, as explained in the previous section, to ensure
that the variation of the IW function remains physical. Following relativistic
quark model (RQM) expectations, we use $\sigma_{1/2}^{2}=0.8$ with a
conventional error $\pm 0.5$.

For the same reasons it is not possible to fit the parameter
$\hat{\epsilon}_{1/2}$.

Using measurements, relative to $D_{1/2}$ production in Table 4, apart for
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\pi^{-})$ which is not
published, and factorization with $a_{1}=1$, we have fitted
$\tau_{1/2}^{eff.}(1)$ (see Table 10).

Without fitting any additional $1/m_{Q}$ parameter, we obtain:

$\tau_{1/2}^{eff.}(1)=0.147\pm 0.025,$ (25)

with a $24\,\%$ fit probability. This value is much smaller than
$\tau_{3/2}^{eff.}(1)$ given in Eq. (23), in agreement with the theoretical
expectations and in contradiction with $LLSWB$.

Let us now take into account the $1/m_{Q}$ corrections. The
$\hat{\chi}_{1,\,2}$ and $\hat{\zeta}_{1}$ correction parameters provide
enough flexibility in decay rate expressions to accommodate essentially any
measured values, with an acceptable $\chi^{2}$ probability. It is therefore
needed to use additional constraints from theory.

LQCD expectation gives $\tau_{1/2}(1)=0.296\pm 0.026$. We have no predicted
value for the quantity $\hat{\epsilon}_{1/2}$ which corresponds to $1/m_{Q}$
corrections on $\tau_{1/2}(1)$. If we assume that
$\hat{\epsilon}_{1/2}=\hat{\epsilon}_{3/2}$, we expect
$\tau_{1/2}^{eff.}(1)=0.24\pm 0.02$, which is higher than the measurement in
Eq. (25). This can be due to either the fact that $\hat{\epsilon}_{1/2}$
differs from $\hat{\epsilon}_{3/2}$ or that other $1/m_{Q}$ corrections, not
considered for the evaluation in Eq. (25) can have some effect. For these
reasons, in the following, we use, as a constraint, the value:
$\tau_{1/2}^{eff.}(1)=0.20\pm 0.06$ where the uncertainty is large enough to
cover the two previous estimates.

Using these two constraints (on $\tau_{1/2}^{eff.}(1)$ and $\sigma_{1/2}^{2}$
), and fitting one additional parameter, we obtain the values given in Table
10.

X param. | $\tau_{1/2}^{eff.}(1)$ | $\sigma_{1/2}^{2}$ | X | $\chi^{2}/NDF$
---|---|---|---|---
no X | $0.155\pm 0.025$ | $1.0\pm 0.5$ | no value | $3.5/3$
$\chi_{1}$ | $0.21\pm 0.06$ | $0.80\pm 0.50$ | $-0.27\pm 0.22$ | $2.6/2$
$\chi_{2}$ | $0.21\pm 0.06$ | $0.78\pm 0.50$ | $0.37\pm 0.27$ | $2.5/2$
$\zeta_{1}$ | $0.22\pm 0.06$ | $0.76\pm 0.50$ | $0.75\pm 0.44$ | $2.3/2$
Table 10: Fitted values of one of the parameters entering in $1/m_{Q}$
corrections. $\tau_{1/2}^{eff.}(1)$ and $\sigma_{1/2}^{2}$ are constrained as
explained in the text.

Fit probabilities are close to $30\,\%$ and values for the different
parameters are reasonable, of the order of $\overline{\Lambda}$, but comparing
with $D_{3/2}$ production one cannot identify a parameter to which the
analysis is most sensitive and we are not able to fit more than one of these
quantities, with reasonable accuracy, using present data. In the following we
adjust $\hat{\chi}_{1}$ and evaluate model systematic uncertainties changing
$\hat{\chi}_{2}$ and $\hat{\zeta}_{1}$ by $\pm 0.5\,GeV$.

##  VIII Production of $D_{3/2}$ and $D_{1/2}$ mesons: combined analysis and
systematic uncertainties

We include in the analysis the data given in Table 4 excluding the unpublished
measurement of ${\cal B}(\overline{B}^{0}\to D_{1}(2430)^{+}\pi^{-})$.
Constraints from theory on the parameters of the IW functions and the list of
fitted quantities are given in Table 3, Section II.4.

The analysis is done taking into account the validity of factorization. for
all $D^{**}$-meson production and using: $a_{1,\,eff.}^{D^{**}\pi}=0.93\pm
0.07$, see Section VI.

The ratio $\chi^{2}/NDF=6.3/7$ corresponds to a fit probability of $51\,\%$.

Values of fitted parameters, given in Table 11, are almost identical with
those obtained when considering separate productions of $D_{3/2}$ and
$D_{1/2}$ mesons (see Sections VI.2 and VII).

$a_{1,\,eff.}^{D^{**}\pi}$ | $\tau_{3/2}(1)$ | $\sigma_{3/2}^{2}$ | $\tau_{1/2}^{eff.}(1)$ | $\sigma_{1/2}^{2}$
---|---|---|---|---
$0.944\pm 0.062$ | $0.53\pm 0.03$ | $1.50\pm 0.50$ | $0.21\pm 0.06$ | $0.80\pm 0.70$
$\hat{\epsilon}_{3/2}$ | $\hat{\eta}_{1}\,(GeV)$ | $\hat{\eta}_{3}\,(GeV)$ | $\hat{\tau}_{1}\,(GeV)$ | $\hat{\chi}_{1}\,(GeV)$
$-0.18\pm 0.11$ | $-0.32\pm 0.13$ | $-0.77\pm 0.28$ | $0.36\pm 0.35$ | $-0.24\pm 0.26$
Table 11: Fitted values of the reference model parameters, for $B\to D^{**}$
Lorentz invariant form factors. The first line contains parameters that are
constrained by theory or from external measurements
($a_{1,\,eff.}^{D^{**}\pi}$). Quoted uncertainties are obtained from the fit.

Fitted quantities allow to obtain values for different branching fractions of
a $\overline{B}^{0}$ meson decaying into $D^{**}\ell^{-}\bar{\nu}_{\ell}$,
with a light or the $\tau$ lepton, as well as for non-leptonic decays. We have
also considered $\overline{B}^{0}\to D^{**,\,+}D_{s}^{(*),-}$ decays, in the
framework of factorization.

To evaluate systematic uncertainties, values of not-fitted parameters are
changed to $\pm 0.5\,GeV$ and the largest variation on a fitted or derived
quantity, that depends on fitted values, is used as systematic model
uncertainty. For some of these quantities, mainly related to the production of
$D_{1/2}$ mesons, these variations are asymmetric, when compared to the value
obtained with the reference model (in which the non-fitted parameters are set
to zero). In this case we have symmetrized systematic uncertainties and
corrected central values accordingly. Model uncertainties are added linearly
to those from the fit because there is no reason to prefer a value for
unfitted quantities, within their variation range.

Other sources of systematic uncertainties are considered as:

  * •

effects from uncertainties on HQET parameters (Set 1). They are illustrated by
using the values adopted in previous analyses (see Table 2);

  * •

effects of using a linear parameterization for the IW functions, versus $w$,
in place of the dipole distribution (Eq. (6));

  * •

the effect of changing the parameterization of Blatt-Weisskopf terms in non-
leptonic decays when the momentum of the emitted hadron is computed in the
frame of the resonance in place of the $B$-meson rest frame (the former has
been used by some experimental collaborations).

These observed variations are only indicative and cannot be considered as
really representative of corresponding systematic uncertainties values. In
most cases, these sources are less important than uncertainties from the fit
or from the model.

When evaluating a ratio between two derived quantities, correlations between
the different uncertainties are taken into account.

In the following we detail our expectations and give comparisons with those
obtained in another analysis, quoted as $LLSWBi$, which is close to those done
in previous publications ref:ligeti1 ; ref:ligeti2 . Differences relative to
our approach are listed in Appendix D. Numerical values, obtained in this way,
are quoted also in appendices whereas corresponding expected distributions are
compared with our results on the different Figures that follow.

###  VIII.1 Comparison between our model and the $LLSWBi$ analysis for
$D_{1/2}$ production in non-leptonic Class I decays

In Table 12 we illustrate the differences between our model and the $LLSWBi$
analysis for Class I non-leptonic decays.

channel | measured | our model |  $LLSWBi$ model
---|---|---|---
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-})\times 10^{4}$ | $1.19\pm 0.12$ | $1.21\pm 0.12$ | $10.0\pm 2.5$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\pi^{-})\times 10^{4}$ | $0.21\pm 0.27$ | $0.7\pm 0.7$ | $3.2\pm 2.8$
${\cal R_{K}}({D_{0}(2300)},{D_{2}^{*}})$ | $0.84\pm 0.36$ | $0.35\pm 0.04$ | $2.8\pm 0.7$
Table 12: Comparison between measured and expected values for non-leptonic
$\overline{B}^{0}\to D_{1/2}$ transitions. ${\cal
R_{K}}({D_{0}(2300)},{D_{2}^{*}})$ is the ratio between the branching
fractions ${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}K^{-})$ and ${\cal
B}(\overline{B}^{0}\to{D_{2}^{*+}}K^{-})$, with the two $D^{**}$ mesons
decaying into $D^{0}\pi^{+}$.

The measured values for ${\cal
B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-})$ and ${\cal
R}({D_{0}(2300)},{D_{2}^{*}})$ enter our model through the use of
factorization. Therefore it is a check of consistency that the corresponding
fitted values are in agreement with the data, as well as with theory, which
predicts indeed that $j=1/2$ transitions should be much smaller than $j=3/2$
ones.

On the other hand, one can see that predictions of $LLSWBi$ are in
disagreement with the data by more that four standard deviations and too large
by around one order of magnitude, which leads to discard the model. Keeping
this in mind, it may nevertheless be useful to apply the same model to semi-
leptonic decays for the sake of comparison with our own results, especially
since $LLSW$ is used by experimentalists in the analyses of the background to
decays such as $\overline{{B}}\to D^{*}\ell\overline{\nu_{\ell}}$.

##  IX $D^{**}$ meson production in $\overline{B}^{0}$ semi-leptonic decays

Our results on decay branching fractions of $\overline{B}^{0}$ mesons into the
four $D^{**,\,+}$ mesons are explained. In Sections IX.2 and IX.3, expected
hadronic mass and $q^{2}$ distributions are obtained for the $D^{*}\pi$ and
$D\pi$ hadronic final states, respectively. Spectra are compared with the
$LLSWBi$ analysis and, to ease the comparison, total decay rates, expected in
the two cases, have been scaled (only for plotting purposes) to the central
values measured for the considered final states. Corresponding uncertainties
are scaled also to agree with those obtained on measurements for ${\cal
B}(\overline{B}^{0}\to D^{(*),+}\pi^{-}\ell^{-}\bar{\nu}_{\ell})$, with light
leptons. Additional uncertainties, from the fit and the model, which appear
when considering decays with a $\tau$ lepton or a $D_{s}^{-}$ meson, are
included.

###  IX.1 Expected values for ${\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}\ell^{-}\bar{\nu}_{\ell})$ and corresponding $q^{2}$
distributions.

Expected $q^{2}$ distributions in $\overline{B}^{0}$ semi-leptonic decays with
a light or a $\tau$ lepton are given in Figures 1, 2, 3 , and 4 for
${D_{2}^{*+}}$, ${D_{1}^{+}}$, ${D_{0}(2300)^{+}}$, and ${D_{1}(2430)^{+}}$,
respectively. Hatched areas correspond to uncertainties from the fit. Curves
indicated with dots are the expected systematic uncertainties from the un-
measured $\hat{\eta}_{2}$ and $\hat{\tau}_{2}$ parameters. Those indicated
with triangles are from the un-measured $\hat{\chi}_{2}$ and $\hat{\zeta}_{1}$
parameters. In each figure the left plot is for light leptons and the right
one for the $\tau$.

Figure 1:  Expected $q^{2}$ distributions for
$\overline{B}^{0}\to{D_{2}^{*+}}$ in semi-leptonic decays. Hatched areas
correspond to uncertainties from the fit, curves with dots indicate the model
uncertainty. Other systematic uncertainties, indicated in Table 13, are not
displayed.

Figure 2:  Expected $q^{2}$ distributions for $\overline{B}^{0}\to{D_{1}^{+}}$
in semi-leptonic decays. Same conventions are used as in Fig.1.

Figure 3:  Expected $q^{2}$ distributions for
$\overline{B}^{0}\to{D_{0}(2300)^{+}}$ in semi-leptonic decays. Hatched areas
correspond to uncertainties from the fit, curves with dots indicate the model
uncertainty expected from $\hat{\tau}_{2}$ and $\hat{\eta}_{2}$ parameters
whereas, those with triangles, correspond to the uncertainty from
$\hat{\chi}_{2}$ and $\hat{\zeta}_{1}$. Other systematic uncertainties,
invoked in Table 13, are not displayed.

Figure 4:  Expected $q^{2}$ distributions for
$\overline{B}^{0}\to{D_{1}(2430)^{+}}$ in semi-leptonic decays. Same
conventions are used as in Fig.3

Expected values for semi-leptonic branching fractions, with a light and the
$\tau$ lepton, are given in Table 13. Those obtained in the $LLSWBi$ analysis
are given in Appendix D.1. To evaluate uncertainties on quoted values we have
added linearly uncertainties from the fit and from the model. Values for other
possible sources of uncertainties, quoted in Table 13 are simply indicative
and usually are not dominant when compared with the others. It can be noted
that ratios of branching fractions with a $\tau$ or a light lepton have a
better accuracy because of correlations between the different uncertainties.

channel | value $\pm$ fit | model | HQET | IW linear | Blatt-W.
---|---|---|---|---|---
${\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}\ell^{-}\bar{\nu}_{\ell})\times 10^{3}$ | $3.15\pm 0.30$ | $0.$ | $0.$ | $0.01$ | $0.02$
${\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}\tau^{-}\bar{\nu}_{\tau})\times 10^{4}$ | $1.90\pm 0.29$ | $0.05$ | $0.$ | $0.09$ | $0.03$
${\cal R}_{{D_{2}^{*}}}\times 10^{2}$ | $6.03\pm 0.52$ | $0.15$ | $0.02$ | $0.24$ | $0.06$
${\cal B}(\overline{B}^{0}\to{D_{1}^{+}}\ell^{-}\bar{\nu}_{\ell})\times 10^{3}$ | $6.40\pm 0.44$ | $0.$ | $0.$ | $0.$ | $0.$
${\cal B}(\overline{B}^{0}\to{D_{1}^{+}}\tau^{-}\bar{\nu}_{\tau})\times 10^{4}$ | $6.30\pm 0.59$ | $0.10$ | $0.30$ | $0.70$ | $0.07$
${\cal R}_{D_{1}}\times 10^{2}$ | $9.84\pm 0.68$ | $0.15$ | $0.47$ | $1.10$ | $0.11$
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\bar{\nu}_{\ell})\times 10^{4}$ | $5.1\pm 1.2$ | $1.2$ | $-0.2$ | $0.4$ | $-0.1$
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\tau^{-}\bar{\nu}_{\tau})\times 10^{5}$ | $5.0\pm 1.3$ | $1.7$ | $-0.1$ | $0.6$ | $0.1$
${\cal R}_{{D_{0}(2300)}}\times 10^{2}$ | $9.9\pm 1.5$ | $1.0$ | $0.4$ | $0.4$ | $0.1$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\ell^{-}\bar{\nu}_{\ell})\times 10^{4}$ | $4.6\pm 3.7$ | $0.9$ | $0.3$ | $-0.5$ | $0.$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\tau^{-}\bar{\nu}_{\tau})\times 10^{5}$ | $3.4\pm 2.7$ | $0.6$ | $0.3$ | $-0.3$ | $0.$
${\cal R}_{{D_{1}(2430)}}\times 10^{2}$ | $7.4\pm 1.2$ | $1.6$ | $0.1$ | $0.4$ | $0.1$
Table 13: Our expectations for semi-leptonic branching fractions with a light
or a $\tau$ lepton and their ratio for individual $D^{**}$-meson states.

####  IX.1.1 $D_{3/2}$ production

Values for semi-leptonic branching fractions with a light lepton and a
$D_{3/2}$ meson are essentially identical with input measurements. This is
because one has basically no measurement of the $q^{2}$ dependence of the
different decay rates and because the normalisation is fitted through the
$\tau_{3/2}$ and $\hat{\epsilon}_{3/2}$ parameters. Expected uncertainties on
the production of $D_{1}$ and $D_{2}^{*}$, with a $\tau$ lepton are of about
$20\,\%$. It can be noted that, in $D_{1}$ production, uncertainties on Set 1
(HQET) parameters and on the $w$ expected dependence of $\tau_{3/2}$ may be
not negligible. This is expected because HQET Set 1 corrections change the
computed branching fraction with a $D_{1}$ by a large factor, as compared with
the infinite quark mass limit prediction.

####  IX.1.2 $D_{1/2}$ production

The value expected for ${\cal
B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\bar{\nu}_{\ell})=(5.1\pm
2.4)\times 10^{-4}$ is much smaller than the one usually anticipated from
$LLSWBi$: $(39.1\pm 7.2)\times 10^{-4}$, as given in Appendix D.1. This result
is a direct consequence of the use of factorization. Values expected for
${D_{1}(2430)}$ are similar but being affected by larger uncertainties one
cannot draw any conclusion. Production of $D_{1/2}$ mesons in $B$-meson semi-
leptonic decays is expected to be an order of magnitude smaller than the one
of narrow states. In the production of $D_{1/2}$ mesons, model uncertainties
are dominant when compared with the other considered sources of systematic
uncertainties.

Expected branching fractions for ${D_{0}(2300)}$ and ${D_{1}(2430)}$
production have about $50\,\%$ and $100\,\%$ uncertainty, respectively. These
relative uncertainties are even larger when a $\tau$ lepton is emitted.

####  IX.1.3 Conclusions

In our analysis, the expected low production rates of broad, relative to
narrow, $D^{**}$ mesons comes from theoretical arguments and is in agreement
with the factorization property. This low value implies that it has to be
complemented by another source of events, to explain the measured broad mass
distributions in $D^{(*)}\pi$ hadronic final states. We examine, in the
following sections, if the contributions expected from $D^{(*)}_{V}$
components can fill these gaps. Such components have been ignored in previous
analyses which consider that $D_{1/2}$ decays, alone, are enough to explain
measurements.

###  IX.2 Analysis and predictions for the $\overline{B}^{0}\to
D^{*}\pi\ell^{-}\bar{\nu}_{\ell}$ final state

We examine if measured $\overline{B}^{0}\to D^{*}\pi\ell^{-}\bar{\nu}_{\ell}$
decays can be explained using a sum of $D^{**}$ and $D^{(*)}_{V}$ components.
All quoted numbers are relative to the sum of $D^{*0}\pi^{+}$ and
$D^{*+}\pi^{0}$ final states and we refer to Section III.2.2 for input
measurements.

The measured branching fraction into broad components amounts to:

${{\cal
B}(\overline{B}^{0}\to\left.D^{*}\pi\right|_{broad}\ell^{-}\bar{\nu}_{\ell})=(2.86\pm
0.69)\times 10^{-3}.}$ (26)

In the hypothesis that contributions from higher mass resonances can be
neglected, the expected contribution from ${D_{1}(2430)^{+}}$ decays, equal to
$(0.46\pm 0.46)\times 10^{-3}$, has to be complemented by a $D^{(*)}_{V}$
component equal to: $(2.4\pm 0.8)\times 10^{-3}$. For $r_{BW}$ values varying
between $1$ and $3\,GeV^{-1}$ our estimates for this component are in the
range $[2.0,\,0.9]\times 10^{-3}$ (see Section III.2.2). These values are
compatible with the needed contribution.

In Figure 5, expected $D^{*}\pi$ mass distributions from our model (full line)
and from the $LLSWBi$ analysis (dashed line), are compared. To ease the
comparison, central values expected from the two models are scaled to agree
with the measured one. To do so, in our model, the $D^{(*)}_{V}$ component
only is scaled while, for other components, expected values are used. In
$LLSWBi$, the ${D_{1}(2430)^{+}}$ component, only, is scaled. Scaling factors
are obtained for decays into light leptons and their values are used for the
other analyzed transitions which involve a $\tau$ lepton or a $D_{s}$ meson.

Spectra are dominated by the contributions from $D_{3/2}$ mesons.

Figure 5:  Expected $D^{*}\pi$ mass distributions for $\overline{B}^{0}\to
D^{*}\pi$ in semi-leptonic decays. Full lines correspond to our model and the
dashed line is for $LLSWBi$. The line with stars gives the $D^{**}$
contribution and the one with squares is for the $D^{(*)}_{V}$ component. The
red line corresponds to the sum of these two contributions. Only central
values of the distributions are displayed.

####  IX.2.1 Expected differences between our model and $LLSWBi$, for broad
mass components

In the following we compare expected mass and $q^{2}$ distributions, for the
$D^{*}\pi$ broad mass component, in our model and in $LLSWBi$, and after
having normalized expectations to agree with the measured central value, for
light leptons. Hatched areas correspond, only, to the uncertainty quoted in
Eq. (26). Therefore, they are mainly indicative and do not illustrate the
uncertainty on the shape of the distributions which comes from the model
dependence of the two analyses.

Figure 6:  Expected broad $D^{*}\pi$ mass distributions for
$\overline{B}^{0}\to D^{*}\pi$ in semi-leptonic decays. The same conventions
as in Figure 5 are used. Hatched areas include only the experimental
uncertainty given in Eq. (26).

For $\tau$ events one expects 1.3 times more events in our analysis whereas
estimates were equal, by convention, for light leptons. This comes from the
different dependence of $D^{(*)}_{V}$ and ${D_{1}(2430)}$ components versus
$q^{2}$. The $D^{(*)}_{V}$ component corresponds to a $D^{*}\pi$ mass
distribution which can mimic a broad resonance in the absence of an analysis
of the angular distribution.

Figure 7:  Expected $q^{2}$ distributions for $\overline{B}^{0}\to D^{*}\pi$,
broad mass component, in semi-leptonic decays. The same conventions as in
Figure 5 are used.

###  IX.3 Analysis and predictions for the $\overline{B}^{0}\to
D\pi\ell^{-}\bar{\nu}_{\ell}$ final state

All quoted numbers are relative to the sum of $D^{0}\pi^{+}$ and
$D^{+}\pi^{0}$ final states. Input measurements are given in Section III.2.1.

The measured branching fraction into broad components amounts to:

${\cal
B}(\overline{B}^{0}\to\left.D\pi\right|_{broad}\ell^{-}\bar{\nu}_{\ell})=(4.24\pm
0.56)\times 10^{-3}.$ (27)

In the hypothesis that contributions from higher mass resonances can be
neglected, the expected contribution from $D_{1/2}$ decays is equal to
$(0.51\pm 0.24)\times 10^{-3}$. To account for the observed broad mass $D\pi$
distribution, the ${D_{0}(2300)^{+}}$ contribution has to be complemented by a
$D^{*}_{V}$ component equal to: $(3.7\pm 0.6)\times 10^{-3}$. For $r_{BW}$
values varying between $1$ and $3\,GeV^{-1}$ our estimates are in the range
$[2.6,\,1.5]\times 10^{-3}$. Therefore, taking into account present
uncertainties, this scenario is compatible (marginally) with present
measurements.

In Figure 8, we compare expected $D\pi$ mass distributions from our model
(full line) and from the $LLSWBi$ analysis (dashed line). There is a narrow
peak from the ${D_{2}^{*+}}$ meson located on top of a broad mass distribution
which is very different in the two scenarios. As in the $D^{*}\pi$ channel
analysis, broad mass distributions have been scaled so that total decay rates
agree with measurements in the case of light leptons.

Figure 8:  Expected $D\pi$ mass distributions for $\overline{B}^{0}\to D\pi$
in semi-leptonic decays. Only central values are displayed.

####  IX.3.1 Expected differences between our model and $LLSWBi$, for broad
mass components

In the following we compare expected mass and $q^{2}$ distributions, for the
$D\pi$ broad mass component, in our model and $LLSWBi$, and after having
normalized expectations to agree with the measured central value, for light
leptons.

Figure 9:  Expected broad $D\pi$ mass distributions for $\overline{B}^{0}\to
D\pi$ in semi-leptonic decays.

Our model and $LLSWBi$ show marked differences. In particular, for $\tau$
events one expects two times more candidates in our model and very different
mass and $q^{2}$ distributions. Dashed areas correspond to measured
uncertainties of the broad mass component, used for the normalization, as
given in Eq. (27) and do not include model uncertainties which can affect also
the shape of the distributions.

Figure 10:  Expected $q^{2}$ distributions for $\overline{B}^{0}\to D\pi$,
broad mass component, in semi-leptonic decays.

##  X Predictions for $\overline{B}^{0}\to D^{**,\,+}{D}_{s}^{-}$ decays

After having described our predictions for decay branching fractions of
$\overline{B}^{0}$ mesons into the four $D^{**,\,+}$ accompanied by a
${D}_{s}^{-}$, we give expectations for the hadronic $D^{*}\pi$ and $D\pi$
final states.

Because the $D_{s}$ meson and the $\tau$ lepton have a similar mass, we have
also evaluated ref:GW  the ratios:

${\cal R}^{\tau,\,D_{s}}_{D^{**}_{i}}=\frac{{\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}\tau^{-}\bar{\nu}_{\tau})}{{\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}{D}_{s}^{-})}$ (28)

expecting that, because of correlations between the different sources of
uncertainties, they are more accurate than individual measurements of the
corresponding decay rates, as observed already for semi-leptonic channels (see
Table 13).

Results can be used also for corresponding charged $B$-meson decays, $B^{-}\to
D^{**,\,0}{D}_{s}^{-}$, because they are of Class I (once penguin terms are
neglected). Branching fractions have simply to be corrected by the ratio
between charged and neutral $B$-meson lifetimes and ratios ${\cal
R}^{\tau,\,D_{s}}_{D^{**}_{i}}$ are the same.

The validity of factorization in $B$ meson decays, with emission of a heavy
meson has been questionned, see the comment in Sec. II.2 supra, and the
previous experimental tests had large uncertainties. Using ${\cal
B}(\bar{B}\to D^{(*)}{D}_{s}^{-})$ and ${\cal B}(\bar{B}\to
D^{(*)}\ell^{-}\bar{\nu}_{\ell})$ measurements, we have checked that this
property is verified, once the evaluated contribution from penguin terms is
taken into account. Therefore we use the same constraint,
$a_{1,\,eff.}=0.93\pm 0.07$ and, in the absence of theoretical evaluations for
penguin amplitudes in $\bar{B}\to D^{**}{D}_{s}^{-}$ decays, we assume that
they can be neglected.

###  X.1 Predicted values for ${\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}{D}_{s}^{-})$

Predicted values for $\overline{B}^{0}\to D_{i}^{**,\,+}{D}_{s}^{-}$ decay
branching fractions, in our model, are given in Table 14.

channel | value $\pm$ fit | model | HQET | IW linear | Blatt-W.
---|---|---|---|---|---
${\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}{D}_{s}^{-})\times 10^{4}$ | $5.8\pm 0.8$ | $0.6$ | $0.1$ | $1.7$ | $0.1$
${\cal R}_{{D_{2}^{*}}}^{\tau,\,D_{s}}$ | $0.33\pm 0.06$ | $0.03$ | $0.$ | $-0.07$ | $0.02$
${\cal B}(\overline{B}^{0}\to{D_{1}^{+}}{D}_{s}^{-})\times 10^{4}$ | $13.1\pm 3.5$ | $0.9$ | $-0.1$ | $6.4$ | $0.9$
${\cal R}_{D_{1}}^{\tau,\,D_{s}}$ | $0.48\pm 0.12$ | $0.03$ | $0.03$ | $-0.15$ | $-0.03$
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}{D}_{s}^{-})\times 10^{4}$ | $2.3\pm 0.4$ | $0.2$ | $0.1$ | $0.2$ | $0.0$
${\cal R}_{{D_{0}(2300)}}^{\tau,\,D_{s}}$ | $0.22\pm 0.04$ | $0.05$ | $-0.01$ | $0.08$ | $-0.03$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}{D}_{s}^{-})\times 10^{4}$ | $1.4\pm 1.1$ | $0.9$ | $0.1$ | $-0.1$ | $0.$
${\cal R}_{{D_{1}(2430)}}^{\tau,\,D_{s}}$ | $0.25\pm 0.05$ | ${}^{+0.45}_{-0.10}$ | $0.01$ | $0.04$ | $-0.03$
Table 14: Our expectations for ${\cal B}(\overline{B}^{0}\to
D^{**,\,+}_{i}{D}_{s}^{-})$ branching fractions, and their ratio, to
corresponding semi-leptonic decays with a $\tau$ lepton. The first quoted
uncertainty corresponds to the error from the fit. The model uncertainty is
evaluated by changing the values of parameters, that are fixed to zero, by
$\pm 0.5\,GeV$. Then, as explained in the text, are indicated variations of
fitted parameters which correspond to different hypotheses that enter in the
parameterization of fitted expressions.

Values obtained in the $LLSWBi$ analysis are given in Table 19 in Appendix
D.2.

Values for ${D_{1}(2430)^{+}}$ are very uncertain because of the lack of
control of two parameters that enter in $1/m_{Q}$ corrections
($\hat{\chi}_{2}$ and $\hat{\zeta}_{1}$ in the present analysis). These model
uncertainties affect also the ratio ${\cal R}^{\tau,\,D_{s}}_{{D_{1}(2430)}}$.
Measurements of non-leptonic Class I $B$ decays with ${D_{1}(2430)}$ emission
are therefore desirable to improve the present situation.

To evaluate branching fractions when the hadronic final state, accompanying
the ${D}_{s}^{-}$ meson, is $D^{*}\pi$ or $D\pi$, we have to multiply values
given in Table 14 for the production of the different $D^{**}_{i}$ meson by
corresponding branching fractions listed in Table 6. In our model we have also
evaluated the $\overline{B}^{0}\to
D_{V}^{(*),\,+}{D}_{s}^{-},\,D_{V}^{(*),\,+}\to D^{(*)}\pi$ expected
contributions.

In practice it is not possible to compute really the expected mass
distributions because of strong phase-shifts between the different hadronic
amplitudes, that are unknown. One can evaluate only absolute values of these
amplitudes. Therefore mass distributions, displayed in the following, are
obtained by adding incoherently contributing individual channels.

###  X.2 Analysis of the $\overline{B}^{0}\to D^{*}\pi{D}_{s}^{-}$ final state

All quoted numbers are relative to the sum of $D^{*0}\pi^{+}$ and
$D^{*+}\pi^{0}$ final states. Expected branching fractions for $D^{**}$
channels, multiplied by ${\cal B}(D^{**}\to D^{*}\pi)$, are the following:

$\displaystyle{\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}{D}_{s}^{-})$
$\displaystyle=$ $\displaystyle(2.3\pm 0.3\pm 0.2)\times 10^{-4}$
$\displaystyle{\cal B}(\overline{B}^{0}\to{D_{1}^{+}}{D}_{s}^{-})$
$\displaystyle=$ $\displaystyle(8.8\pm 2.3\pm 0.6)\times 10^{-4}$
$\displaystyle{\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}{D}_{s}^{-})$
$\displaystyle=$ $\displaystyle(1.4\pm 1.1\pm 0.9)\times 10^{-4}$ (29)

To evaluate the $D^{*}\pi$ mass distribution, $D^{**}$ channels are
complemented by the $D_{V}^{(*),\,+}$ contribution of $(3.0\pm 1.0)\times
10^{-4}$. This gives a total broad $D^{*}\pi$ component of $(4.4\pm 3.0)\times
10^{-4}$ which can be compared with the estimate from $LLSWBi$: $(8.8\pm
7.4)\times 10^{-4}$.

It can be noted that the $D^{(*)}_{V}$ contribution can mimic a broad
resonance and an analysis of the alignment distribution is necessary to
separate the two possibilities.

Figure 11:  Expected $D^{*}\pi$ mass distributions from our analysis (full red
curve) and from $LLSWBi$ (dashed blue curve). Left plots correspond to all
expected contributing components whereas the plots on the right are for broad
mass components only.

###  X.3 Analysis of the $\overline{B}^{0}\to D\pi{D}_{s}^{-}$ final state

All quoted numbers are relative to the sum of $D^{0}\pi^{+}$ and
$D^{+}\pi^{0}/\gamma$ final states. Expected branching fractions for channels
with a $D^{**}$ meson, multiplied by ${\cal B}(D^{**}\to D\pi)$, are the
following:

$\displaystyle{\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}{D}_{s}^{-})$
$\displaystyle=$ $\displaystyle(3.48\pm 0.46\pm 0.33)\times 10^{-4}$
$\displaystyle{\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}{D}_{s}^{-})$
$\displaystyle=$ $\displaystyle(2.30\pm 0.43\pm 0.22)\times 10^{-4}$ (30)

To evaluate the $D\pi$ mass distribution, the $D^{**}$ component is
complemented by a $D_{V}^{*}$ contribution evaluated to be $(5.5\pm 0.9)\times
10^{-4}$. This gives a total broad $D\pi$ component of $(7.8\pm 1.5)\times
10^{-4}$ which can be compared with the estimate from the $LLSWBi$ analysis:
$(17\pm 8)\times 10^{-4}$.

Figure 12:  Expected $D\pi$ mass distributions obtained in our model (red full
line) and in $LLSWBi$ (blue dashed line). Left plots correspond to all
expected contributing components whereas, plots in the right are for broad
mass components only.

##  XI Predictions for $\overline{B}^{0}\to D^{**,\,+}{D}^{\ast-}_{s}$ decays

Expected results on decay branching fractions of $\overline{B}^{0}$ mesons
into the four $D^{**,\,+}_{i}$ accompanied by a ${D}^{\ast-}_{s}$ are given in
Table 15, including different ratios that compare these branching fractions
with those expected for $\overline{B}^{0}\to D^{**,\,+}_{i}{D}_{s}^{-}$ and
$\overline{B}^{0}\to D_{i}^{**,\,+}\tau^{-}\bar{\nu}_{\tau}$ decays.
Expressions for ${\cal B}(\overline{B}^{0}\to D_{i}^{**,\,+}{D}^{\ast-}_{s})$
are given in Appendix A. They are obtained using factorization and considering
only Class I decay amplitudes. These expressions are valid for charged or
neutral $B$-meson decays.

${\cal R}^{\tau,\,D_{s}^{*}}_{D^{**}_{i}}=\frac{{\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}\tau^{-}\bar{\nu}_{\tau})}{{\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}{D}^{\ast-}_{s})}$ (31)

and

${\cal R}^{D_{s},\,D_{s}^{*}}_{D^{**}_{i}}=\frac{{\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}{D}_{s}^{-})}{{\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}{D}^{\ast-}_{s})}$ (32)

channel | value $\pm$ fit | model
---|---|---
${\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}{D}^{\ast-}_{s})\times 10^{3}$ | $1.9\pm 0.4$ | $0.$
${\cal R}_{{D_{2}^{*}}}^{\tau,\,D_{s}^{*}}$ | $0.100\pm 0.014$ | $0.$
${\cal R}_{{D_{2}^{*}}}^{D_{s},\,D_{s}^{*}}$ | $0.30\pm 0.08$ | $0.03$
${\cal B}(\overline{B}^{0}\to{D_{1}^{+}}{D}^{\ast-}_{s})\times 10^{3}$ | $4.7\pm 0.9$ | $0.1$
${\cal R}_{D_{1}}^{\tau,\,D_{s}^{*}}$ | $0.134\pm 0.024$ | $0.$
${\cal R}_{D_{1}}^{D_{s},\,D_{s}^{*}}$ | $0.28\pm 0.10$ | $0.02$
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}{D}^{\ast-}_{s})\times 10^{4}$ | $2.4\pm 0.7$ | $1.0$
${\cal R}_{{D_{0}(2300)}}^{\tau,\,D_{s}^{*}}$ | $0.20\pm 0.04$ | $0.02$
${\cal R}_{{D_{0}(2300)}}^{D_{s},\,D_{s}^{*}}$ | $0.96\pm 0.26$ | ${}^{+0.44}_{-0.24}$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}{D}^{\ast-}_{s})\times 10^{4}$ | $2.2\pm 2.0$ | ${}^{+0.6}_{-0.1}$
${\cal R}_{{D_{1}(2430)}}^{\tau,\,D_{s}^{*}}$ | $0.14\pm 0.02$ | $0.02$
${\cal R}_{{D_{1}(2430)}}^{D_{s},\,D_{s}^{*}}$ | $0.52\pm 0.02$ | ${}^{+0.52}_{-0.36}$
Table 15: Our expectations for ${\cal B}(\overline{B}^{0}\to
D^{**,\,+}_{i}{D}^{\ast-}_{s})$ branching fractions, and their ratio, to
corresponding semi-leptonic decays with a $\tau$ lepton and non-leptonic
${D}_{s}^{-}$ production. The first quoted uncertainty corresponds to the
error from the fit. The model uncertainty is evaluated by changing the values
of parameters, that are fixed to zero, by $\pm 0.5\,GeV$.

##  XII Conclusions

We have analyzed $\bar{B}\to D^{**}$ decays in semi-leptonic and non-leptonic
Class I processes that can be related using factorization.

We have verified that factorization is well satisfied in $\bar{B}\to D_{3/2}$
decays and also in Class I non-leptonic transitions with $D_{s}^{-}$ emission.

Assuming naturally that this factorization is valid also in $\bar{B}\to
D_{1/2}$ decays, one expects a quite small contribution from $D_{1/2}$
relative to $D_{3/2}$ mesons in semi-leptonic decays as is the case in non-
leptonic Class I processes. This is in contrast with the results of $LLSWBi$
and in agreement with LQCD computations of the corresponding IW functions at
maximum transfer ref:lqcd-tau  and with relativistic QM calculations
ref:morenas .

To evaluate branching fractions for the different decay channels in which a
$D^{**}$ meson is produced, we use the expressions derived in ref:leibo . They
depend on several parameters that control $1/m_{Q}$ corrections. Using present
experimental measurements and constraints from theory we have determined the
most important of these parameters for $D_{3/2}$ emission. In particular we
find that the $1/m_{Q}$ correction $\hat{\epsilon}_{3/2}$ included in the
auxiliary $\tau^{eff}_{3/2}$, is of the order of $-0.2\pm 0.1$, the minus sign
confirming the agreement with a quark model calculation. The three other
quantities we obtain are of the order of $\bar{\Lambda}$, as expected in a
$1/m_{Q}$ expansion, which is encouraging in view of the roughness of the
method. For $D_{1/2}$ mesons, estimates of the parameters are more uncertain
but this does not change our conclusion on the smallness of the contribution.

To explain measurements of exclusive $D^{(*)}\pi$ broad hadronic final states
in $B$ meson semi-leptonic decays we have evaluated the contribution from
$D_{V}^{(*)}$ components, in addition to $D_{1/2}$ decays. These components
can be normalized by using $\bar{B}\to D^{(*)}\ell^{-}\bar{\nu}_{\ell}$
measurements but the mass dependence of the $D^{(*)}\pi$ mass distribution
remains highly undetermined.

We propose a model which accounts for $\bar{B}\to
D^{(*)}\pi\ell^{-}\bar{\nu}_{\ell}$ measurements by adding $D_{V}^{(*)}$ and
$D^{**}$ contributions. We have, at present, not considered contributions from
higher mass hadronic states. Results have been compared with the $LLSWBi$
model in which $D_{1/2}$ mesons alone explain the broad mass spectra.

These two models give very different expectations for the broad $D^{(*)}\pi$
mass and $q^{2}$ distributions regarding light leptons. The $D\pi$ final
state, in particular can provide clear informations on the relative importance
of the $D^{*}_{V}$ and ${D_{0}(2300)}$ components. The two models have also
very different expectations for semi-leptonic decays with a $\tau$ lepton and
in Class I non-leptonic transitions with $D_{s}^{(*)}$ emission.

##  Appendix A Expressions for Class I non-leptonic decays

Expressions for Lorentz invariant form factors are those of $LLSW$ ref:leibo ,
therefore, for the IW functions, one has to use the correspondance
$\tau(w)=\sqrt{3}\,\tau_{3/2}(w)$ and $\zeta(w)=2\,\tau_{1/2}(w)$. In
following formulas, Lorentz invariant form-factors are evaluated at
$w_{D}=(m_{B}^{2}-m_{P(V)}^{2}+m_{D^{**}}^{2})/(2m_{B}m_{D^{**}})$ while the
quantity $w$, which appears in the rest of the expressions, is computed at the
running mass value of the $D^{**}$ resonance. Expressions for the traditional
Blatt-Weisskopf damping factors in $\overline{B}\to D^{**}X$ decays in which
$X$ is a stable particle and which occur in an angular momentum
$L=1,\,2,\,\rm{and}\,3$ are the following:

$\displaystyle F_{B,1}(p^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+z}}$ (33) $\displaystyle F_{B,2}(p^{\prime})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{9+3\,z+z^{2}}}$ $\displaystyle
F_{B,3}(p^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{225+45\,z+6\,z^{2}+z^{3}}}$

with $z=(r_{BW}\,p^{\prime})^{2}$, in which $p^{\prime}$ is the decay momentum
evaluated in the $\overline{B}$ rest-frame. For the damping term, we use
$r_{BW}=3\,(GeV)^{-1}$. We define the quantities:

$B_{B,L}(w)=\left(\frac{F_{B,L}(p^{\prime})}{F_{B,L}(p_{0}^{\prime})}\right)^{2}$
(34)

that are the ratios of the previous functions evaluated at the running mass of
the $D^{**}$ resonance and at its nominal mass, $m_{D^{**}}$.

###  A.1 $\overline{B}^{0}\to D^{**,+}P^{-}$ decays

We write the following expressions for a generic pseudo-scalar meson denoted
as "$P$". $a_{1,eff}^{{D_{X}},P}$ is an effective parameter describing the
deviation from strict factorization which includes also possible contributions
from exchange, annihilation or penguin amplitudes, $V_{{q_{1}}{q_{2}}}$ is the
relevant light-quarks $CKM$ matrix element and $f_{P}$ the annihilation
constant. Those parameters are to be adapted to the case under consideration.

The damping factor $B_{B,2}(w)$ is used for $\overline{B}\to D_{3/2}$ decays,
that occur in a D-wave and, as usual, no damping is considered for the decays
into $D_{1/2}$ states, that are S-wave. Each expression is followed by its
$m_{Q}=\infty$ limit (between parentheses). This is valid also in the case of
a final vector meson.

$\Gamma_{\overline{B}^{0}\to{D_{2}^{*+}}P^{-}}=|a_{1,eff}^{{{D^{*}_{2}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{24\pi}f_{P}^{2}m_{B}m_{D^{*}_{2}}^{2}|k_{A_{1}}+k_{A_{2}}(1-rw)+k_{A_{3}}(w-r)|^{2}(w^{2}-1)^{5/2}{B_{B,2}(w)}$
($|a_{1,eff}^{{{D^{*}_{2}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{24\pi}f_{P}^{2}m_{B}m_{D^{*}_{2}}^{2}\tau(w)^{2}(1-r)^{2}(1+w)^{2}(w^{2}-1)^{3/2}$
in the $m_{Q}\to\infty$ limit)

$\Gamma_{\overline{B}^{0}\to{D_{1}^{+}}P^{-}}=|a_{1,eff}^{{{D_{1}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{P}^{2}m_{B}m_{D_{1}}^{2}|f_{V_{1}}+f_{V_{2}}(1-rw)+f_{V_{3}}(w-r)|^{2}(w^{2}-1)^{3/2}{B_{B,2}(w)}$
($|a_{1,eff}^{{{D_{1}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{24\pi}f_{P}^{2}m_{B}m_{D_{1}}^{2}\tau^{2}(1-r)^{2}(1+w)^{2}(w^{2}-1)^{3/2}$)

$\Gamma_{\overline{B}^{0}\to{D_{0}(2300)^{+}}P^{-}}=|a_{1,eff}^{{{D^{*}_{0}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{P}^{2}m_{B}m_{D^{*}_{0}}^{2}|g_{+}(1-r)(1+w)+g_{-}(1+r)(1-w)|^{2}(w^{2}-1)^{1/2}$
($|a_{1,eff}^{{{D^{*}_{0}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{P}^{2}m_{B}m_{D^{*}_{0}}^{2}\zeta^{2}(1+r)^{2}(1-w)^{2}(w^{2}-1)^{1/2}$)

$\Gamma_{\overline{B}^{0}\to
D_{1}^{\ast+}P^{-}}=|a_{1,eff}^{{{D^{*}_{1}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{P}^{2}m_{B}m_{D^{*}_{1}}^{2}(w^{2}-1)^{3/2}|g_{V1}+g_{V2}(1-rw)+g_{V3}(w-r)|^{2}$
(
$|a_{1,eff}^{{{D^{*}_{1}}},P}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{P}^{2}m_{B}m_{D^{*}_{1}}^{2}(1-r)^{2}\zeta^{2}(w^{2}-1)^{3/2})$

###  A.2 $\overline{B}^{0}\to D^{**,+}V^{-}$ decays

These expressions are obtained by a direct calculation. Several partial waves
are contributing in each decay channel and the appropriate Blatt and Weisskopf
damping factor has to be used for each contribution. It can be identified from
the power of the momentum dependence.
$\Gamma_{\overline{B}^{0}\to{D_{2}^{*+}}V^{-}}=|a_{1,eff}^{{{D_{2}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{48\pi}f_{V}^{2}\frac{m_{V}^{2}m_{D^{*}_{2}}^{2}}{m_{B}}(w^{2}-1)^{3/2}\left[3|k_{V}|^{2}(w^{2}-1){B_{B,2}(w)}\right.$
$\left.+|k_{A_{1}}|^{2}\left(5{\,B_{B,1}(w)}+2\frac{m_{B}^{2}}{m_{V}^{2}}(w^{2}-1){B_{B,2}(w)}\right)+2|k_{A_{2}}+\frac{1}{r}k_{A_{3}}|^{2}(w^{2}-1)^{2}\frac{m_{D^{*}_{2}}^{2}}{m_{V}^{2}}{B_{B,3}(w)}\right.$
$\left.+4Re(k_{A_{1}}^{*}(k_{A_{2}}+\frac{1}{r}k_{A_{3}}))(w^{2}-1)(w-r)\frac{m_{B}m_{D^{*}_{2}}}{m_{V}^{2}}{\sqrt{B_{B,1}(w)}}{\sqrt{B_{B,3}(w)}}\right]$
($|a_{1,eff}^{{{D_{2}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{48\pi}f_{V}^{2}\frac{m_{V}^{2}m_{D^{*}_{2}}^{2}}{m_{B}}\tau^{2}(5(1+w)^{2}+(w^{2}-1)(3+4\frac{m_{B}m_{D}}{m_{V}^{2}}(1+w)))(w^{2}-1)^{3/2}$)

$\Gamma_{\overline{B}^{0}\to{D_{1}^{+}}V^{-}}=|a_{1,eff}^{{{D_{1}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{V}^{2}m_{V}^{2}m_{D_{1}}^{2}/m_{B}(w^{2}-1)^{1/2}\left[|f_{V_{1}}|^{2}(3+\frac{m_{B}^{2}}{m_{V}^{2}}(w^{2}-1){B_{B,1}(w)})\right.$
$\left.+2|f_{A}|^{2}(w^{2}-1){B_{B,1}(w)}+|f_{V_{2}}+\frac{1}{r}f_{V_{3}}|^{2}(w^{2}-1)^{2}\frac{m_{D_{1}}^{2}}{m_{V}^{2}}{B_{B,2}(w)}\right.$
$\left.+2Re(f_{V_{1}}^{*}(f_{V_{2}}+\frac{1}{r}f_{V_{3}}))(w^{2}-1)(w-r)\frac{m_{B}m_{D_{1}}}{m_{V}^{2}}{\sqrt{B_{B,1}(w)}}{\sqrt{B_{B,2}(w)}}\right]$
($|a_{1,eff}^{{{D_{1}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{96\pi}f_{V}^{2}\frac{m_{V}^{2}m_{D_{1}}^{2}}{m_{B}}\tau^{2}\left[2(1+w)^{2}+(w^{2}-1)(3+\frac{m_{B}^{2}}{m_{V}^{2}}(2r(1+w)+3(1+r)^{2}))\right]\\\
(w^{2}-1)^{3/2}$)

$\Gamma_{\overline{B}^{0}\to{D_{0}(2300)^{+}}V^{-}}=|a_{1,eff}^{{{D_{0}^{*}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{V}^{2}\frac{m_{D_{0}^{*}}^{4}}{m_{B}}(w^{2}-1)^{3/2}|g_{+}(1+\frac{1}{r})+g_{-}(1-\frac{1}{r})|^{2}{B_{B,1}(w)}$
($|a_{1,eff}^{{{D_{0}^{*}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{V}^{2}m_{D_{0}^{*}}^{4}/m_{B}\zeta^{2}(1-1/r)^{2}(w^{2}-1)^{3/2}$)

$\Gamma_{\overline{B}^{0}\to
D_{1}^{\ast+}V^{-}}=|a_{1,eff}^{{{D_{1}^{*}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{V}^{2}\frac{m_{V}^{2}m_{D_{1}^{*}}^{2}}{m_{B}}(w^{2}-1)^{1/2}\left[|g_{V_{1}}|^{2}(3+\frac{m_{B}^{2}}{m_{V}^{2}}(w^{2}-1){B_{B,1}(w)})+\right.$
$\left.2|g_{A}|^{2}(w^{2}-1){B_{B,1}(w)}+|g_{V_{2}}+\frac{1}{r}g_{V_{3}})|^{2}(w^{2}-1)^{2}\frac{m_{D_{1}^{*}}^{2}}{m_{V}^{2}}{B_{B,2}(w)}\right.$
$\left.+2Re(g_{V_{1}}^{*}(g_{V_{2}}+\frac{1}{r}g_{V_{3}}))(w^{2}-1)(w-r)\frac{m_{B}m_{D_{1}^{*}}}{m_{V}^{2}}{\sqrt{B_{B,1}(w)}}{\sqrt{B_{B,2}(w)}}\right]$
($|a_{1,eff}^{{{D_{1}^{*}}},V}|^{2}|V_{cb}V_{{q_{1}}{q_{2}}}^{*}|^{2}\frac{G_{F}^{2}}{16\pi}f_{V}^{2}m_{B}m_{D_{1}^{*}}^{2}\zeta^{2}(3(m_{V}/m_{B})^{2}(w-1)/(w+1)+2((1-r)^{2}+r(1-w)))(w^{2}-1)^{3/2}$)

##  Appendix B Calculations of $\eta^{b(c)}_{ke}$ and $\eta^{b}$ with NR
treatment of the c.o.m motion

1) according to the standard analysis, the ${\cal O}(1/m_{Q})$ corrections to
form factors fall into two main categories, due to modifications of :

  * •

the state vectors;

  * •

the current operators.

Here we are concerned with the first category, and the $\eta$’s or $\chi$’s
parameters are precisely characterizing the corrections to the form factors
due to this modification of state vectors, when the current operator is kept
to the infinite mass limit.

More precisely, we want to consider $\eta^{b(c)}_{ke}$, which parameterize the
effect of the kinetic operator ${\cal O}_{ke}$ on respectively the initial
ground state and on the $j=3/2$ final state, with specification to $w=1$.

2) in terms of the quark model, the modification of vector states at $w=1$ is
identified with the one of the rest frame wave functions of the mesons, and
the aim is then to evaluate the effect of this modification on form factors,
i.e. on matrix elements of currents when the current operator is kept to its
infinite mass limit.

As concerns $\eta^{b(c)}_{ke}$, one can give an intuitive interpretation by
assuming that they are generated by the addition of the heavy quark kinetic
energy in the Schrödinger equation :

$p^{2}/2m_{q}\to p^{2}/2m_{q}+p^{2}/2m_{Q}$

$m_{q},m_{Q}$ being respectively the light and heavy quark masses. But this
amounts simply to replace $m_{q}$ by the reduced mass
$\mu=\frac{m_{q}m_{Q}}{m_{q}+m_{Q}}$ :

$p^{2}/2m_{q}\to p^{2}/2\mu$

This fact implies that the wave equation is the same as in the heavy quark
limit, except for the change of mass. In the case of power-like potential
$r^{\alpha}$, this allows to use the scale invariance to deduce the wave
functions from the heavy mass limit ones $\phi^{\infty}$. Let the equation be
:

$(p^{2}/2m_{q}+br^{\alpha})\phi^{\infty}(\vec{r})=E_{n}\phi^{\infty}(\vec{r})$

Performing the substitution $\vec{r}\to\lambda\vec{r}$, one has :

$(\frac{p^{2}}{2m_{q}\lambda^{(2+\alpha)}}+br^{\alpha})\phi^{\infty}(\lambda\vec{r})=\frac{E_{n}}{\lambda^{\alpha}}\phi^{\infty}(\lambda\vec{r})$

Therefore if one chooses
$\lambda=(\mu/m_{q})^{(1/(2+\alpha))}=(1+(m_{q}/m_{Q}))^{(-1/(2+\alpha))}$,
one sees that $\phi^{\infty}(\lambda\vec{r})$ is the solution of the finite
mass equation. The normalised wave function is :

$\lambda^{3/2}\phi^{\infty}(\lambda\vec{r}).$

Now, the non relativistic (NR) (in fact the familiar dipolar expression) for
$\tau$ is :

$\tau(w=1)=m(\phi^{\infty}_{f}|z|\phi^{\infty}_{i})$

Note that j=1/2 and j=3/2 are degenerate at $m_{Q}=\infty$ in the NR approach
(no Wigner rotation) if one disregards any spin-orbit force. In fact, all the
spin dependent forces are relativistic effects in the sense that they are
${\cal O}(v^{2}/c^{2})$ with respect to internal velocities. Therefore, even
at finite $m_{Q}$, if one considers the fully NR approach, there are no spin
independent forces. This is what is done in the following paragraphs 1),2),3)
below where we mean only to evaluate kinetic energy corrections to $B$ and
$D^{(**)}$.

On the other hand, for $\eta^{b}$, which combines all the type of $\eta$’s
relative to the $B$, one has to consider the spin-dependent forces, since this
contains the magnetic contributions generating the $\eta$’s (1,2,3). This is
done in paragraph 4), where we use the GI model which contains all the
relevant forces. An important contribution is obviously the one of spin-spin
force, although it is $1/m_{Q}$. It lifts the degeneracy with the $B^{*}$.

We now define :

$\tau^{b,c}(w=1)=m(\phi^{c}_{f}|z|\phi^{b}_{i}),$

with the substitution by the finite mass wave functions, whence the
corrections are represented by 555One must note the difference of definition
of the Isgur-Wise functions in $LLSW$. This difference disappears in
corrections with a hat since they represent the quotient by the Isgur-Wise
functions themselves:

$\displaystyle\tau^{b,c}(w=1)-\tau(w=1)$ (35)

or, assuming the proportionality to the $\tau$ and defining reduced
corrections as $\hat{\eta}$,

$\displaystyle\tau^{b,c}(w=1)/\tau(w=1)=1+\frac{\hat{\eta}^{b}_{ke}}{2m_{b}}+\frac{\hat{\eta}^{c}_{ke}}{2m_{c}}+...$
$\displaystyle=\lambda^{3/2}_{b}\lambda^{3/2}_{c}\frac{\int{d_{3}\vec{r}\leavevmode\nobreak\
z\leavevmode\nobreak\
\phi^{\infty}_{1}(\lambda_{c}\vec{r})\phi^{\infty}_{0}(\lambda_{b}\vec{r})}}{\int{d_{3}\vec{r}\leavevmode\nobreak\
z\leavevmode\nobreak\ \phi_{1}^{\infty}(\vec{r})\phi^{\infty}_{0}(\vec{r})}}$
(36)

where $0,1$ denote the orbital angular momentum $L$. If one passes to radial
wave functions by means of an angular integration :

$\displaystyle\tau^{b,c}(w=1)/\tau(w=1)=\lambda^{3/2}_{b}\lambda^{3/2}_{c}\frac{\int{r^{3}dr\leavevmode\nobreak\
\phi^{\infty}_{1}(\lambda_{c}r)\phi^{\infty}_{0}(\lambda_{b}r)}}{\int{r^{3}dr\leavevmode\nobreak\
\phi_{1}^{\infty}(r)\phi^{\infty}_{0}(r)}}$

If one makes $m_{b}=m_{c}=m_{Q}$, one finds :

$\displaystyle\tau^{b,c}(w=1)/\tau(w=1)$
$\displaystyle=1+\frac{\hat{\eta}^{b}_{ke}}{2m_{Q}}+\frac{\hat{\eta}^{c}_{ke}}{2m_{Q}}+...$
(37) $\displaystyle=1/\lambda_{Q}\ $
$\displaystyle=1+\frac{m_{q}/m_{Q}}{2+\alpha}+...$ (38)
$\hat{\eta}^{b}_{ke}+\hat{\eta}^{c}_{ke}=\frac{2m_{q}}{2+\alpha}$

To go further and separate $\hat{\eta}^{b}_{ke}$ and $\hat{\eta}^{c}_{ke}$,
one needs explicit wave functions, which is possible for $\alpha=2$ (harmonic
oscillator) or $-1$ (Coulomb).

Harmonic oscillator ($\alpha=2$): $\phi^{\infty}_{0}(r)\propto
e^{(-r^{2}/(2R^{2}))}$ , $\phi^{\infty}_{1}(r)\propto r\leavevmode\nobreak\
e^{-r^{2}/(2R^{2})}$

$\displaystyle\tau^{\infty,c}(w=1)/\tau(w=1)=\left(\frac{2\lambda_{c}}{\lambda_{c}^{2}+1}\right)^{5/2}$
(39)

whence $\hat{\eta}^{c}_{ke}=0$, and
$\hat{\eta}^{b}_{ke}=(2m_{q})/(2+2)=m_{q}/2$

Coulomb ($\alpha=-1$) :$\,\phi^{\infty}_{0}(r)\propto e^{(-r/r_{0})}$ ,
$\phi^{\infty}_{1}(r)\propto r\leavevmode\nobreak\ e^{(-r/2r_{0})}$

$\displaystyle\tau^{\infty,c}(w=1)/\tau(w=1)=\left(\frac{3\lambda^{1/2}_{c}}{\lambda_{c}^{2}+2}\right)^{5}$
(40)

whence $\hat{\eta}^{c}_{ke}=-10/6\,m_{q}$, and
$\hat{\eta}^{b}_{ke}=(2\,m_{q})/(2-1)+10/6\,m_{q}=11/3\,m_{q}$.

3) numerical calculation for a linear+Coulomb potential

Of course, the physical potential is rather of linear+Coulomb type. The result
may be expected to lay between the HO and the Coulomb one. But one has no
analytical solution. Therefore we perform a numerical calculation, using the
particular wave functions of reference BC  with a potential close to
linear+Coulomb and find, with $m_{Q}$ respectively infinite or equal to
$5\,GeV$ masses (light mass : $m_{q}=0.45\,GeV$)666The numerical calculations
can be performed for fictitious heavy quark masses because we need only the
coefficient of the dependence.:

$\displaystyle\tau^{m_{b}=5,m_{c}=\infty}(w=1)/\tau(w=1)=1+$
$\displaystyle\frac{\hat{\eta}^{b}_{ke}}{2m_{b}}=1.053\,$ (41)
$\displaystyle\tau^{m_{b}=\infty,m_{c}=5}(w=1)/\tau(w=1)=1+$
$\displaystyle\frac{\hat{\eta}^{c}_{ke}}{2m_{c}}=0.9898\,$ (42)
$\displaystyle\tau^{m_{b}=5,m_{c}=5}(w=1)/\tau(w=1)=1+$
$\displaystyle\frac{\hat{\eta}^{b}_{ke}+\hat{\eta}^{c}_{ke}}{2m_{Q}}=1.045\,$
(43)

whence approximately: $\hat{\eta}^{b}_{ke}\simeq 0.5\,GeV$,
$\hat{\eta}^{c}_{ke}\simeq-0.1\,GeV$, which is indeed intermediate between the
results from HO and Coulomb potentials. In fact
$\hat{\eta}^{b}_{ke}+\hat{\eta}^{c}_{ke}\simeq 0.5\,GeV$ corresponds roughly
to what is expected from $\alpha=0$ (Eq. above), i.e. $m_{q}$, and it is
indeed well known that such a power potential $\alpha\simeq 0$ or a log one
approximate roughly the linear+Coulomb one (e.g. Martin potentials).

The conclusion up to now is that $\hat{\eta}^{c}_{ke}<0$. In addition,
$\hat{\eta}^{b}_{ke}>0$ but what must be estimated is $\hat{\eta}^{b}$, a
common combination appearing in all the form factors, and which can therefore
be interpreted as the effect of the full Lagrangian contribution, i.e.,
intuitively, one needs to include spin dependent forces, which are not present
in the potentials which have been considered up to now.

4) numerical calculation of $\hat{\eta}^{b}$ for the GI model.

We consider then the Godfrey and Isgur spectroscopic model with all relevant
forces, and moreover a relativistic kinetic energy. To obtain
$\hat{\eta}^{b}$, we calculate the variation of $\tau$ with the initial $B$
respectively at infinite and finite mass. We find finally
$\hat{\eta}^{b}\simeq-0.26\,GeV$ : this indicates that the effect of spin-spin
force is large, dominating the kinetic energy effect.

##  Appendix C  Repeating the $LLSWB$ analysis of ref:ligeti1 ; ref:ligeti2

This comparison is intended to show that, using the same input data and
constraints we find, using our own code, the same values for fitted parameters
as in ref:ligeti1 ; ref:ligeti2 , with the $LLSWB$ approach.

For this purpose measurements of $B$-meson semi-leptonic decays and
$\overline{B}^{0}\to D_{3/2}^{+}\pi^{-}$ non-leptonic decays, reported in the
third column of Table 4, are used to constrain parameterizations of hadronic
form factors. In addition, measurements from Belle ref:belle_dsstarl  which
provide, respectively, 4 and 5 values for the production fractions of
${D_{2}^{*}}$ and ${D_{0}(2300)}$ mesons, in different bins of the $w$
variable are used. The $w$ dependence of the two IW functions is assumed to be
linear. The validity of factorization , with $a_{1}=1$, is assumed, to relate
semi and non-leptonic decays in which $D_{3/2}$ mesons, only, are emitted.

Therefore $\overline{B}^{0}\to D_{1/2}^{+}\pi^{-}$ non-leptonic decays are not
included in the analysis. $D^{**}$ mesons are assumed to be stable (no mass
distribution is considered).

We have modified accordingly our analysis but some differences remain:

  * •

for the fractions measured in different $w$ bins, we have not used one of the
measurements in each of the two samples because these quantities are not
independent (their sum is equal to one);

  * •

the parameterization of the different form factors is derived from the
original article of ref:leibo , without using different approximations;

  * •

in addition to $\alpha_{s}$ corrections, we have also included, for $D_{3/2}$
mesons, those at order $1/m_{Q}\times\alpha_{s}$, provided in ref:leibo ;

###  C.1 Numerical aspects

Production of $D_{3/2}$ and $D_{1/2}$ mesons are evaluated separately. In
addition to the normalization and slope of the IW functions, the same
parameters which determine $1/m_{Q}$ corrections, as in ref:ligeti2 , are
fitted.

Considering $D_{3/2}$ mesons only, values of fitted parameters are compared in
Table 16.

We obtain very similar results. The numbers of degree of freedom differ by one
unit, in the two analyses, because we have not used one of the measurements
for the $w$ dependence of ${D_{2}^{*}}$ production in semi-leptonic decays. A
similar comparison is done, see Table 17, fitting only data relative to
$D_{1/2}$ mesons, measured in semi-leptonic decays (in ref:ligeti2
measurements of non-leptonic transitions are not used). We have also modified
our code to use the zero width formulation of our expressions as done in
ref:ligeti2 .

analysis | $\tau_{3/2}^{eff.}$ | $\sigma_{3/2}^{2}$ | $\hat{\tau}_{1}(CeV)$ | $\hat{\tau}_{2}(GeV)$ | $\chi^{2}/NDF$
---|---|---|---|---|---
ref:ligeti2  | $0.40\pm 0.04$ | $1.6\pm 0.2$ | $-0.5\pm 0.3\,$ | $2.9\pm 1.4\,$ | $2.4/4$
our code | $0.41\pm 0.06$ | $1.60\pm 0.25$ | $-0.66\pm 0.42\,$ | $5.\pm 2.\,$ | $1.8/3$
Table 16: Comparison between the values of fitted parameters obtained in ref:ligeti2  and with our own code, modified to be similar to the previous analysis and using the same input measurements. analysis | $\tau_{1/2}^{eff.}$ | $\sigma_{1/2}^{2}$ | $\hat{\zeta}_{1}\small$ | $\chi^{2}/NDF$
---|---|---|---|---
ref:ligeti2  | $0.35\pm 0.11$ | $0.2\pm 1.4$ | $0.6\pm 0.3$ | $9.1/4$
our code ($\Gamma(D_{1/2})=0$) | $0.37\pm 0.11$ | $0.26\pm 1.23$ | $0.23\pm 0.31$ | $7.0/3$
our code ($\Gamma(D_{1/2})\neq 0$) | $0.30\pm 0.18$ | $-1.6\pm 3.2$ | $0.45\pm 0.28$ | $6.0/3$
Table 17: Comparison between the values of fitted parameters obtained in
ref:ligeti2  and with our own code, modified to be similar to the previous
analysis. In the third line, values are obtained using the physical widths for
the broad $D_{1/2}$ resonances.

Very similar results are obtained in the two analyses when considering zero
width resonances. In the third line, obtained using the physical resonance
widths, the central value of the IW function slope comes out negative, which
is unexpected and inconclusive because the corresponding uncertainty is large.
From all fits, with or without a finite resonance width, it can be concluded
that data are not able to measure really this slope.

###  C.2 The main difficulty of the $LLSWB$ analysis

Values obtained in this way, for $\tau_{1/2}^{eff.}(1)$ and
$\tau_{3/2}^{eff.}(1)$ are compatible. This comes simply from the fact that
the measurement ${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\pi^{-})$ is not
included in ref:ligeti1 ; ref:ligeti2  analyses. Meanwhile, expectations from
relativistic quark models and LQCD ref:lqcd-tau , obtained in the
$m_{b,\,c}\to\infty$ limit are very different with
$\tau_{1/2}(1)\ll\tau_{3/2}(1)$.

If one assumes factorization ($a_{1}=1$), the branching fractions of the NL
$\bar{B}\to D_{1/2}$ Class I decays predicted in $LLSWBi$ are, at present,
higher than the measured values by at least four standard deviations (when
including the two channels)

This is illustrated in Section VIII.1, Table 12, where we compare expectations
from our model and from the $LLSWBi$ analysis, for non-leptonic Class I
$\overline{B}\to D_{1/2}$ decays.

Then we show that it is possible to fit data in a way which satisfies
factorization for narrow and broad states and which is compatible with present
theoretical expectations.

##  Appendix D A summary of how the $LLSWBi$ analysis differs from the $LLSWB$
one and from our model

The $LLSWBi$ analysis uses the same hypotheses as $LLSWB$ does but, to make it
directly comparable with our model we use the same input measurements and the
same constraints from theory, when possible.

$LLSWBi$ differs from our model by the following points:

  * •

constraints from factorization are ignored in the production of $D_{1/2}$
mesons;

  * •

possible contributions from $D^{(*)}_{V}$ decays are ignored;

  * •

semi-leptonic branching fractions, ${\cal B}(\overline{B}^{0}\to
D_{1/2}^{+}\ell^{-}\bar{\nu}_{\ell})$, are taken from the last column of Table
4. It can be noted that the uncertainty taken for ${\cal
B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\ell^{-}\bar{\nu}_{\ell})$ is three
times larger than the one assumed in ref:ligeti1 ; ref:ligeti2  to account for
the fact that the corresponding central value is obtained from an average of
not compatible experimental results (the factor three is evaluated using the
usual PDG recipe to scale uncertainties in this situation);

  * •

no constraint is used on $\tau^{eff.}_{1/2}$;

  * •

the measured fractions, in several $w$ bins, attributed by Belle
ref:belle_dsstarl , to the
$\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\bar{\nu}_{\ell}$ decay
distribution are used;

Because $LLSWBi$ violates factorization in ${D_{0}(2300)}$ production and
theoretical expectations for $\tau_{1/2}(1)$ it cannot be considered as a
possible alternative to our model. We simply mean to illustrate the large
expected differences between our model and previous analyses that can be
confronted with data, when available.

###  D.1 Expected values for ${\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}\ell^{-}\bar{\nu}_{\ell})$

Expected values for semi-leptonic branching fractions, with a light or the
$\tau$ lepton, in $LLSWBi$, are given in Table 18. As expected, they differ
mainly from those quoted in Table 13 on the production of $D_{1/2}$ mesons and
are now similar to those obtained for $D_{3/2}$ mesons. Values obtained in
this approach, are essentially identical with the input values given in Table
4. Therefore estimates for the ${D_{1}(2430)}$ are quite inaccurate.

channel |  $e$ or $\mu$ | $\tau$ | ${\cal R}_{D^{**}}$
---|---|---|---
| $\times 10^{3}$ | $\times 10^{4}$ | $(\%)$
${\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}\ell^{-}\bar{\nu}_{\ell})$ | $3.16\pm 0.30$ | $1.90\pm 0.27\pm.07$ | $6.01\pm 0.49\pm 0.19$
${\cal B}(\overline{B}^{0}\to{D_{1}^{+}}\ell^{-}\bar{\nu}_{\ell})$ | $6.40\pm 0.44$ | $6.19\pm 0.56\pm 0.15$ | $9.67\pm 0.62\pm 0.24$
| $\times 10^{4}$ | $\times 10^{5}$ | $(\%)$
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}\ell^{-}\bar{\nu}_{\ell})$ | $39.1\pm 7.0\pm 0.2$ | $31.9\pm 7.9\pm 2.0$ | $8.2\pm 1.5\pm 0.5$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}\ell^{-}\bar{\nu}_{\ell})$ | $17.\pm 15.\pm 2.$ | $12.8\pm 11.5\pm 5.0$ | $7.6\pm 1.0\pm 2.0$
Table 18: $LLSWBi$ model: expected semi-leptonic branching fractions with a
light or a $\tau$ lepton, and their ratio, for the individual $D^{**}$ mesons.
Only model systematic uncertainties are quoted.

###  D.2 Expected values for ${\cal B}(\overline{B}^{0}\to
D_{i}^{**,\,+}{D}_{s}^{-})$

Values for $\overline{B}^{0}\to D_{3/2}{D}_{s}^{-}$ branching fractions are
essentially identical with those obtained in our analysis. On the contrary,
for $D_{1/2}$ mesons, they differ by about an order of magnitude, as expected.

channel | value $\pm$ fit | model
---|---|---
${\cal B}(\overline{B}^{0}\to{D_{2}^{*+}}{D}_{s}^{-})\times 10^{4}$ | $5.7\pm 0.7$ | $0.7$
${\cal R}_{{D_{2}^{*}}}^{\tau,\,D_{s}}$ | $0.34\pm 0.06$ | $0.03$
${\cal B}(\overline{B}^{0}\to{D_{1}^{+}}{D}_{s}^{-})\times 10^{4}$ | $12.3\pm 3.2$ | $1.3$
${\cal R}_{D_{1}}^{\tau,\,D_{s}}$ | $0.50\pm 0.12$ | $0.01$
${\cal B}(\overline{B}^{0}\to{D_{0}(2300)^{+}}{D}_{s}^{-})\times 10^{4}$ | $16.0\pm 4.1$ | $3.0$
${\cal R}_{{D_{0}(2300)}}^{\tau,\,D_{s}}$ | $0.20\pm 0.03$ | $0.04$
${\cal B}(\overline{B}^{0}\to{D_{1}(2430)^{+}}{D}_{s}^{-})\times 10^{4}$ | $5.3\pm 4.5$ | $3.2$
${\cal R}_{{D_{1}(2430)}}^{\tau,\,D_{s}}$ | $0.24\pm 0.04$ | ${}^{+0.54}_{-0.12}$
Table 19: $LLSWBi$ model : ${\cal B}(\overline{B}^{0}\to
D^{**,\,+}_{i}{D}_{s}^{-})$ branching fractions, and their ratio, to
corresponding semi-leptonic decays with a $\tau$ lepton. Only model systematic
uncertainties are quoted.

##  Appendix E Acknowledgements

We would like to thank colleagues who have provided interest and help on
various aspects of this analysis, namely, D. Becirevic, I. Bigi, B. Blossier,
S. Descotes-Genon and S. Fajfer. This study has been triggered by G. Wormser
who asked us some advice about the interest of measuring $\bar{B}\to
D^{**}D_{s}^{-}$ decays to improve our knowledge on $D^{**}$ production in
$\bar{B}$ decays.

## References

  * (1) A. K. Leibovich, Z. Ligeti, I. W. Stewart and M. B. Wise, Phys.Rev.Lett. 78 (1997) 3995, (hep-ph/970321)
  * (2) A. Le Yaouanc, J.-P. Leroy and P. Roudeau,  Phys. Rev. D 99 (2019) 073010, (hep-ph/1806.09853)
  * (3) A.K. Leibovich, Z. Ligeti, I.W. Stewart and M.B. Wise, Phys. Rev. D 57 (1998) 308, (hep-ph/9705467).
  * (4) F.U. Bernlochner and Z. Ligeti, Phys. Rev. D 95 (2017) 014022, (arXiv:1606.09300).
  * (5) F.U. Bernlochner, Z. Ligeti, and D.J. Robinson, Phys. Rev. D 97 (2018) 075011, (arXiv:1711.03110).
  * (6) Y. Amhis et al. [HFLAV], Eur. Phys. J. C 77 (2017) 895, (arXiv:1612.07233]).
  * (7) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98 (2018) 030001.
  * (8) D. Liventsev et al. [Belle Collaboration], Phys. Rev. D 77 (2008) 091503.
  * (9) B. Blossier et al. [European Twisted Mass Collaboration], JHEP 0906 (2009) 022, (arXiv:0903.2298).
  * (10) S. Godfrey and N. Isgur,  Phys. Rev. D 32 (1985) 189
  * (11) J.D. Bjorken, I. Dunietz and J. Taron, Nucl. Phys. 371 (1992) 111 (SLAC-PUB-5586).
  * (12) A. Le Yaouanc, L. Oliver and J.-C. Raynal, Phys. Rev. D 69 (2004) 094022 (hep-ph/0307197).
  * (13) N. Uraltsev, Phys. Lett. B 501 (2001) 86 (hep-ph/0011124).
  * (14) A. Le Yaouanc, V. Morénas, L. Oliver, O. Pène and J.-C. Raynal, Phys. Rev. D 56 (1997) 5668 (hep-ph/9706265).
  * (15) I.I. Bigi, B. Blossier, A. Le Yaouanc, L. Oliver and O. Pene, Eur.Phys.J.C 52 (2007) 975 (arXiv:0708.1627).
  * (16) D. Becirevic, A. Le Yaouanc, L. Oliver, J.-C. Raynal, P. Roudeau and J. Serrano, Phys.Rev.D 87 (2013) 054007 (arXiv:1206.5869).
  * (17) I. Bigi, M. Shifman, N.G. Uraltsev and A. Vainshtein, Phys. Rev. D 52 (1995) 196 (hep-ph/9405410)
  * (18) P. Gambino, T. Mannel and N.G. Uraltsev, JHEP 10 (2012) 169, (arXiv:1206.2296)
  * (19) N. Isgur and M.B. Wise, Phys. Rev. D43 (1991) 819.
  * (20) M. Neubert, Phys. Rept. 245 (1994) 259 (hep-ph/9306320)
  * (21) M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Nucl.Phys.B 591 (2000) 313 (hep-ph/000612)
  * (22) J.D. Bjorken, Nucl. Phys. B (Proc. Suppl.) 11, 325 (1989).
  * (23) K. Abe et al. [Belle Collaboration], ICHEP04 11-0710, (hep-ex/0412072).
  * (24) B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 100 (2008) 151802 (arXiv:0712.3503).
  * (25) A. Vossen et al. [Belle Collaboration], Phys. Rev. D 98 (2018) 012005 (arXiv:1803.06444).
  * (26) J.A. Bailey et al. [MILC Collaboration], Phys. Rev. D 92 (2015) 034506 (arXiv:1503.07237).
  * (27) I. Caprini, L. Lellouch, and M. Neubert, Nucl. Phys. B 530 (1998) 153 (hep-ph/9712417).
  * (28) S. Fajfer, J.F. Kamenik, and I. Nisandzic, Phys. Rev. D 85 (2012) 094025 (arXiv:1203.2654).
  * (29) F. Jugeau, A. Le Yaouanc, L. Oliver, and J.-C. Raynal, Phys. Rev. D 72 (2005) 094010 (hep-ph/0504206).
  * (30) This study was initiated following a proposal from G. Wormser.
  * (31) C. S. Kim, Y. Kwon, J. Lee and W. Namgung, Phys. Rev. D 63 (2001) 094506 (hep-ph/0010157).
  * (32) B. Chen, K.-W. Wei, X. Liu and T. Matsuki, Eur. Phys. J. C 77 (2017) 154 (arXiv:1609.07967)