#  The $a_{1}$ factorisation coefficient in $\overline{B}^{0}\to
D^{(*)+}M^{-}$ and $\overline{B}^{0}\to D^{(*)+}D_{s}^{(*)-}$ decays:
measurements versus theory.

###### Abstract

Using recent measurements of exclusive B-meson decays we extract the $a_{1}$
factorization parameter in $\overline{B}^{0}_{d}\to D^{(*)^{+}}K^{-}/\pi^{-}$
decay channels. The values obtained in the four channels are very similar, in
agreement with the theoretical expectations obtained in the $m_{Q}\to\infty$
limit, but the measured values differ definitely from the expected central
values. Such differences have already been observed. Using recent data, we
improve the accuracy by a factor close to two in this comparison. We study
possible interpretations for such a difference and conclude that the claimed
accuracy of corrections due to soft gluon contributions and finite mass
effects has to be revisited before arguing for possible "New-Physics" effects.
We observe that the corrections to the $m_{Q}\to\infty$ limit are larger in
the spectator topology than effects from exchange amplitude contributions. We
discuss also the way expected ratios of $a_{1}$ values, involving hypotheses
from theory, are used to obtain the fraction of ${B^{0}_{s}}$ meson production
in jets at LHC and conclude that the uncertainties attached to this approach
are not well established. Finally, the $a_{1}$ values extracted from
$\overline{B}^{0}_{d}\to D^{(*)+}D_{s}^{-}$ decays are found to be similar,
within uncertainties, to the ones obtained with $K^{-}$ emission, once
expected penguin contributions are corrected. Using the same approach, we note
that $a_{1}$ values measured in channels with a $D_{s}^{*-}$ emission are
about two standard deviations lower than $a_{1}(D^{(*)+}K^{-}$).

Alain Le<EMAIL_ADDRESS>Jean-Pierre Leroy222jean-
<EMAIL_ADDRESS>Patrick
<EMAIL_ADDRESS>
Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

##  1 Introduction

The factorization property of non-leptonic weak decays of heavy mesons has
been, for over thirty years and under various forms, the object of extensive
studies. On the theoretical side, an approximate expression for the amplitude
of decays to heavy-light mesons is through the product of heavy-quark meson
form factors and of the annihilation constant of the light meson. The ratio of
the exact result to this approximation, is expressed by means of the parameter
$a_{1}$, the value of which would be $1$ in the absence of further strong
interactions effects. However a reliable calculation of this parameter lies
presently beyond our theoretical capacities, except for the unphysical
$m_{Q}\to\infty$ limit, and, consequently, one has to rely on phenomenological
and experimental studies to get a precise knowledge of its value.

On the other hand, an accurate value for $a_{1}$ is, in turn, a necessary
ingredient in the determination of $B_{s}$ decays through a quantity
$f_{s}/f_{d}$ relating them to $B_{d}$ decays. It allows for example a precise
determination of $B_{s}$ decay branching fractions at high energy colliders,
and other quantities which are important to test the Standard Model (see below
for more details). It is then essential that, when doing so, uncertainties
(both from theory and experiment) be as small as possible and well under
control. Unfortunately, at present, there has appeared a definite and large
numerical discrepancy between the experimental results and the theoretical
expectations in the asymptotic limit, which are both very accurate on their
own side. It must be stressed that, while experimental studies are progressing
much, theory is not able to provide an accurate evaluation of the $a_{1}$’s.

In principle the deviation from the calculable $m_{Q}\to\infty$ limit should
be attributed to $1/m_{Q}$ corrections but no safe evaluation of these effects
is presently available.

Since both the theoretical expectations and the experimental data have evolved
with time, we shall now briefly review some of the efforts that have been
devoted to these problems.

As to theory:

  1. 1.

in the seminal paper of Beneke, Buchalla, Neubert and Sachrajda (BBNS) [1],
the computations were performed at order NLO and the value of $a_{1}=1.06\pm
0.02$ in the infinite quark mass limit was obtained. The authors have
considered effects from $1/m_{Q}$ corrections and expected them to be small (a
few percent).

Meanwhile, the latter computation of $1/m_{Q}$ corrections is rather crude and
uncertain, and the authors were conscious of that. Indeed, there is here a
question of principle: a quantitative calculation (that is using perturbative
QCD and leaving aside lattice QCD which, here, is not tractable ) is only
possible in the $m_{Q}\to\infty$ limit because, in this case, only hard gluons
contribute to the exchanges between $B\to D$ and the emitted light meson
thanks to the occurrence of an exceptional cancellation of the soft gluons.
However, the latter remain in the $1/m_{Q}$ corrections, a situation which
impedes a safe calculation of the corrections, as underlined briefly after Eq.
(28) of BBNS for the case of spectator quark corrections : the scale of the
strong coupling constant $\alpha_{s}$ in front of the correction is
$k^{2}\simeq\Lambda^{2}_{QCD}$, not a heavy scale, so that, as emphasised, the
calculation cannot be safe.

After all, this should be no surprise, since the long demonstration of BBNS
aims at showing that soft gluons can be eliminated in the asymptotic limit,
opening the way to a systematic calculation, but by construction this holds
only in the limit. It was rightly considered a remarkable success, precisely,
that one could get rid of soft gluons but, obviously, these still keep raising
a barrier away from this limit to a tractable quantitative treatment.

Being quite aware of this barrier, the authors were formulating rather prudent
perspectives, relying mainly on future experimental progress rather than on
theory. This is clearly seen for instance in the following quotation of the
beginning of their section 4.

“Rather, what we may do is ask whether data supports the prediction of a
quasi-universal parameter |$a_{1}$| $\approx 1.05$ in these decays. If this is
indeed the case, it would support the usefulness of the heavy-quark limit in
analyzing non-leptonic decay amplitudes. If, on the other hand, we were to
find large non-universal effects, this would point towards the existence of
sizable power corrections to our predictions. We will see that with present
experimental errors the data are in good agreement with our prediction of a
quasi universal $a_{1}$ parameter. However, a reduction of the experimental
uncertainties to the percent level would be very desirable for obtaining a
more conclusive picture.”

But to understand these conclusions, one must recall that at that time, around
year 2000, the experimental value was $\left|a_{1}^{expt.}\right|=1.08\pm
0.07$, which agreed with the asymptotic estimate but left room, also, for
sizable $1/m_{Q}$ corrections.

  2. 2.

More recently, [2] has extended the BBNS analysis in the infinite mass limit
to a higher order in $\alpha_{s}$. A value slightly higher and more accurate
than the previous one, $a_{1}=1.070\pm 0.011$, was obtained.

At the same time, the experimental values have decreased, becoming
$\left|a_{1}^{expt.}\right|\simeq 0.90\pm 0.05$ for the most accurate ones.
Therefore there is a deviation at the $2-3\sigma$ level between experiment and
the infinite mass limit evaluation [2]. The authors of this latter reference
have considered a possible contribution from $1/m_{Q}$ corrections, using the
previous BBNS estimates. They say:

"One possible interpretation of our findings that, on the one hand, the non-
leptonic to semi-leptonic ratios come out larger in theory compared to
experiment, and on the other hand the non-leptonic ratios in general agree
with experiment, might be non negligible power corrections which could be
negative in sign and 10 to 15 % in size on the amplitude level. They would
render the factorization test via non-leptonic to semi-leptonic ratios better,
and at the same time could cancel out in the nonleptonic ratios, especially if
they were of a certain universality."

Experiments indicate that the corrections are quite large, in contradiction
with the expectations of [3]. However, as to us, we consider that it is
premature to resort to "New Physics", and we rather consider, in view of all
these discussions, that the crucial $1/m_{Q}$ corrections cannot be, at
present, accurately estimated, and not even roughly, for lack of a systematic
method.

Note now that, at variance with $\left|a_{1}\right|$ absolute values,
experimental results for ratios between $\left|a_{1}\right|$ values obtained
in different reactions are close to 1, as already observed, suggesting a
property of universality of $a_{1}$. This is not unexpected, if applied to the
so-called “tree topology” (T-topology). But it should be at least corrected by
the contribution of the subleading “$W$-exchange topologies” to part of the
reactions.

Unfortunately, as long as the knowledge of the corrections remains only
qualitative, it is difficult to attribute an uncertainty on those ratios,
apart for those that are measured. Therefore it is not appropriate to base the
determination of ${B^{0}_{s}}$ production at high energy colliders on
techniques that are using universality of the $a_{1}$ parameter as we explain
in Section 2.3.3

As announced above, in the case of light meson emission, $a_{1}$ universality
(up to some corrections like annihilation contribution) is used to relate
$\overline{B}^{0}_{d}\to D^{(*)+}K^{-}/\pi^{-}$ and $\overline{B}^{0}_{s}\to
D_{s}^{(*)+}\pi^{-}$ decays in view of obtaining, at high energy colliders, a
value for the ratio $f_{s}/f_{d}$ between the production fractions of strange
and non-strange $\overline{{B}}$ mesons [6] (see section 2.3.3 for a more
detailed discussion). This then allows to evaluate absolute values of
$\overline{B}^{0}_{s}$ decay branching fractions by reference to
$\overline{B}^{0}_{d}$ decay branching fractions, measured at B-factories. In
this exercise, we consider that the corresponding uncertainties, expected from
theory, are not really established. In addition we find that the expression
used to correct for the exchange in $\overline{B}^{0}_{d}\to
D^{(\star)+}\pi^{-}$ amplitudes in [7] is not correct.

In the case of charm meson emission, the present study is related to our
previous publication [8] on ${\cal B}(\overline{{B}}\to
D^{**}\ell^{-}\bar{\nu}_{\ell})$ transitions, in which we study, in addition,
$\bar{B}\to D^{**}D_{s}^{(*)-}$ decays.

We consider more specifically Class I decays, i.e. $\overline{B}^{0}_{d}\to
D^{(*)+}M^{-}$ channels, in which the decay mechanism is dominated by the
external emission, from the $b\to c$ quark transition, of a $W$ coupled to the
meson $M$. Quite surprisingly, the hypothesis works as well -or as roughly-
for the emission of a charmed meson, contrarily to the theoretical argument
given in [1].

Our discussion will address first the dominating external emission
contribution. It seems to us that various definitions and statements require
clarification; an attempt to this is proposed in the following section.

###  1.1 Discussion of the factorization hypothesis

The expression itself, “factorisation hypothesis”, needs clarification. For
Class I processes, it may be understood in a loose sense as meaning, for
instance in the case444chosen here because it is the simplest case, being
purely Class I. of $\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$, that the amplitude
for the decay can be written as :

$\displaystyle{\cal
M}=(G_{F}/\sqrt{2})~{}a_{1}\,(m_{B}^{2}-m_{D}^{2})\,f_{K}\,F_{0}(m_{K}^{2})$
(1)

where $f_{K}$ describes the emission of the $K$ meson while $F_{0}(m_{K}^{2})$
is the form factor of the $b\to c$ transition at $q^{2}=m_{K}^{2}$, with a
coefficient $a_{1}$ close to 1, which may depend on the particular transition.
“Factorisation” points to the fact that, approximately, the matrix element is
given by the product of simple hadronic factors. More precisely, $a_{1}$
represents the effects of QCD which are not included in $f_{K}$ and $F_{0}$.
Note that $a_{1}$ can be obtained by comparing measurements with expectations
using Eq. (1). Meanwhile, in this case, the value of $a_{1}$ depends also
directly on those of decay constants, form factors and $V_{cb}$.

However, there may be additional contributions to the same process, the main
one consisting in the exchange of a $W$ between the two initial quarks. In
this situation the initial $\overline{{B}}$ is annihilated, in the sense that
there is no preserved (“spectator”) quark. This is the case, for example, when
considering $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$. Then, traditionally, this
contribution has been termed “annihilation”, but now one speaks mostly of
“exchange” topology and this is the expression we use in this note555 One must
stress that there exist also pure exchange processes, such as
$\overline{B}^{0}_{d}\to D_{s}^{+}{K}^{-}$, which allow to scale this latter
contribution.. Finally there can also be “penguin” and “color-
suppressed”666which however, for the latter, play no role in the present
discussion. contributions. When considering such decays, we use the same
expression for the decay amplitude as given in Eq. (1), but we change the
quantity $a_{1}$ into $a_{1,eff.}$. The subscript “eff.” refers to the above
mentioned extra-contributions. Similar quantities, obtained from theory, will
be indicated by using the superscript “th.”.

Therefore we adopt the following notations for the various situations in which
the $a_{1}$ coefficient is considered:

  * •

$a_{1}^{\infty,\,th.}(\overline{{B}}_{Y},\,D_{X}M)$ is an estimate from
theory, in the $m_{Q}\to\infty$ limit. Consequently, it does not include
$W$-exchange contributions, nor other $1/m_{Q}$ corrections (remaining inside
the tree topology);

  * •

$a_{1,\,eff.}(\overline{{B}}_{Y},\,D_{X}M)$ is the total measurable value in
Class I processes where, possibly, additional decay amplitudes contribute as
those mentioned above (such as in $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$);

  * •

$a_{1}(\overline{{B}},\,D_{X}M)$ is the measured value of the coefficient in
processes described by the tree topology alone (as
$\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$), in which case
$a_{1}(\overline{B}^{0}_{d},\,D^{+}K^{-})\equiv
a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{+}K^{-})$. Also in those cases in which
there are (subleading) contributions of other topologies, it is the result one
obtains by subtracting these contributions (as in
$\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$ by subtracting the W-exchange
contribution). This subtraction can be performed if the contribution can be
evaluated separately as we show below for the exchange one, following the
approach of [7].

The statement that $\left|a_{1}\right|$ is “close to 1” is only qualitative.
It has been the object of long efforts for finding a precise theoretical
justification or even trying to calculate $a_{1}^{th.}$ – this attempt
culminating in BBNS [1]– but one must warn from the beginning that in fact
only a rough approximation is possible and that many illusions and
misinterpretations have circulated. Even the minus sign of the departure from
$1$ has not been predicted, although it is likely that there is a negative
contribution stemming from spectator $1/m_{Q}$ corrections [1]. Empirical
recipes could perhaps be formulated, but most probably they will also remain
very approximate.

From the beginning, the hypothesis was supported by the idea that the soft
gluons linking the $b\to c$ current to the pion or other emitted mesons could
be neglected when the momentum transfer is large, whence all the non-
perturbative physics would be included in the $F_{0}$ form factor and the pion
annihilation constant. But let us stress that, from the very idea, it was
clear that an exact conclusion -if any- could be valid only in an asymptotic
limit of large energy of the pion, and can otherwise hold only as an
approximation, with unknown uncertainty.

As stated previously, the most naive statement would be $a_{1}^{th.}=1$. But,
first, let us remark that hard gluons cannot be neglected anyway. Actually the
basic weak interaction effective Hamiltonian reads :

$\displaystyle{\cal H}_{W}=(G_{F}/\sqrt{2})~{}(C_{1}O_{1}+C_{2}O_{2})$ (2)

where
$O_{1}=\bar{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})u_{\alpha}~{}~{}\bar{c}_{\beta}\gamma^{\mu}(1-\gamma_{5})b_{\beta}$,
$O_{2}=\bar{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})u_{\beta}~{}~{}\bar{c}_{\beta}\gamma^{\mu}(1-\gamma_{5})b_{\alpha}$
(with contracted greek indices $\alpha,\beta$ for colour) and the coefficients
$C_{1}=1+{\cal O}(\alpha_{s})$, $C_{2}={\cal O}(\alpha_{s})$ containing
effects of such hard gluons. Whence
$a_{1}^{th.}=C_{1}+\frac{C_{2}}{N_{c}}=1+{\cal O}(\alpha_{s})$. Before
proceeding further, let us quote numbers including the leading correction at
$\mu\simeq m_{b}$ 777Those values are the ones obtained by Buras [9] in the
t’Hooft-Veltman scheme with $\Lambda_{\overline{MS}}=225\,{\rm MeV}$ and
$\alpha_{s}(M_{Z})=.117$.:

$C_{1}\simeq 1.1,~{}C_{2}\simeq-0.24,~{}a_{1}^{th.}\simeq 1.03.$

But it has been clear for long that this cannot be the full account of hard
gluons. Indeed, the above estimate of $a_{1}^{th.}$ depends on the
regularisation and renormalisation schemes, in particular, it would depend on
the renormalisation point $\mu$. On the other hand, the physical amplitude
${\cal M}$ should of course show no such dependence (and the factors
$f_{K},F_{0}(m_{\pi}^{2})$ do not either).

The correct expression for $a_{1}^{th.}$, in a well defined heavy quark mass
asymptotic limit (made precise below), is systematically calculable in
perturbative QCD, up to distribution functions, which however give very small
additional corrections, and a very accurate estimate has been given by BBNS
[1], including ${\cal O}(\alpha_{s}^{2})$ contributions (further improved by
Huber et al.,[2]). Numerically : $a^{\infty,\,th.}_{1}=1.070\pm 0.011$ (with
essentially no difference between an emitted pion or kaon).

Now, let us make explicit what ${\cal O}(1/m_{Q})$ means. In the presentation
of BBNS, it denotes ${\cal O}(1/m_{b,c})$ with the ratio $m_{c}/m_{b}$ kept
fixed. Another demonstration is given by Bauer, Pirjol and Stewart [10] who
require $m_{b},m_{c}$ and $E_{\pi}$ to be very large. In any case, the limit
is of course very far from the physical situation. In particular, it must be
observed that in both cases, $m_{c}$ is required to be very large, which poses
an obvious problem. Moreover, as it is so often the case in HQET, these
corrections cannot be computed, all the more for orders $(1/m_{Q})^{n},n>1$.
It must be stressed that, at present, the decay $\overline{B}^{0}_{d}\to
D^{+}{K}^{-}$ can most probably not be treated in lattice QCD, in view of the
difficulty of this calculation. Then, because of these ${\cal O}(1/m_{Q})$
terms, the high precision : ${\cal O}(\pm 0.01$), obtained in the asymptotic
result is actually of little help, since these terms cannot be evaluated
systematically and cannot be expected to be very small. This high precision
has even been deceptive since it has suggested that one could attain high
precision in non leptonic decays. Rough estimates of ${\cal O}(1/m_{Q})$
corrections at lowest order, $n=1$, have been given by BBNS [1], but anyway
they meet objections as explained below888A first general problem is that, in
order to estimate the corrections, they do not use the same expansion
conditions as previously, namely they use a fixed, small value of $m_{c}$
while $m_{b}$ goes to $\infty$. This induces a change in the counting rules..

What is really ascertained is that the statement of factorisation hypothesis,
i.e. the proximity of $a_{1,\,eff.}$ to $1,$ has received a clear foundation.
On the other side, it remains a qualitative statement [1]. This is clearly
recognised by BBNS themselves, as we said above; it may be that one is indeed
too far from the asymptotic limit for it to be useful.

###  1.2 Exchange contributions

Among the ${\cal O}((1/m_{Q})^{n})$ corrections, beginning with $n=1$, there
are the “exchange” ones. This is in addition to the “spectator contribution”
which remains within the tree topology. These contributions correspond to
transitions in which there is no “spectator” $\bar{d}$ quark but, instead, the
two initial $b\bar{d}$ quarks annihilate against each other. The general
structure of the matrix element would then be
$<D\pi|J^{\prime}_{\mu}|0><0|J^{\prime}_{\mu}|\bar{B}>$, where the
$J^{\prime}_{\mu}$ are either currents directly present in $O_{1,2}$ or ones
generated by the Fierz transformation.

Formally, the expression may seem similar to the factorisation, with the
product of an annihilation constant and of a form factor. However, here,
contrarily to the standard $m_{Q}=\infty$ factorisation contribution, the
creation of a $\bar{d}d$ final pair to form the $D^{+}\pi^{-}$ requires
essentially a gluon emission by one of the $b,c,d,u$ quarks, whence at least
an $\alpha_{s}$ factor. Such a contribution is termed generically “non-
factorisable”.

Attempts have been made to estimate the exchange contributions, which we shall
discuss in confrontation with experiment, with the help of the pure exchange
process $\overline{B}^{0}_{d}\to D_{s}^{+}{K}^{-}$. Obviously, these estimates
are not at the same level of theory as the asymptotic result described above.

###  1.3 Electro-weak corrections

In the present analysis, our aim is to consider only, for clarity and
simplicity, purely hadronic quantities, i.e. those deriving only from QCD, all
the more since the question of electroweak corrections presents some
complexity. On the other hand, we depend on experimental data, e.g. we use the
measured non leptonic branching ratios, and then, for coherence, we should
have to discard the electroweak effects from these data. But for reasons
similar to the above mentioned ones, we do not attempt to proceed to this
extraction and keep the experimental data as they are presented.

Therefore, in $a_{1,\,eff.}$ evaluations, using nonleptonic and semileptonic
$B$-decays we have not considered effects from radiative corrections that can
be different for the two decay modes. Considering the present experimental
accuracy attained in such branching fraction measurements, it seems of
interest to evaluate, in the future, the real effects of these corrections,
that can be of the percent order.

##  2 Factorization in light meson emission reactions

First we consider the experimental evaluations of
$a_{1,\,(eff.)}(\overline{{B}}_{Y},\,D^{(*)+}M^{-})$ in Class I nonleptonic
decays in which the meson $M^{-}$ is light.

###  2.1 Experimental determination of
$a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{+}K^{-}/\pi^{-})$

In $\overline{B}^{0}_{d}\to D^{+}P^{-}$ channels, in which $P^{-}$ is a
pseudo-scalar meson, the dominant decay mechanism corresponds to the external
emission of a virtual W which is coupled to the emitted meson (tree process
noted T in the following). In the factorization hypothesis the corresponding
partial decay width is written as:

$\Gamma(\overline{B}^{0}_{d}\to{D}^{+}P^{-})=6\pi^{2}C\left|a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{+}P^{-})\right|^{2}\,|\vec{p}_{D}(m_{P}^{2})|(f_{P}\,|V_{uq}|)^{2}\\\
\times(m_{B}^{2}-m_{D}^{2})^{2}F_{0}^{2}(m_{P}^{2})$ (3)

with $C=G_{F}^{2}\left|{\rm V}_{cb}\right|^{2}/(96\,\pi^{3}\,m_{B}^{2})$.
$\vec{p}_{D}(m_{P}^{2})$ is the ${D}^{+}$ three-momentum evaluated in the
$\overline{B}^{0}_{d}$ rest frame, $f_{P}$ the charged meson leptonic decay
constant and $V_{uq}$ the appropriate CKM matrix element999If $B_{d}^{0}$’s
are not separated from $\overline{B}^{0}_{d}$ mesons, contributions from $b\to
u$ transitions are included in some of the measurements that are considered.
In practice, however, this component can be neglected because it is doubly-
Cabibbo suppressed (DCS).

$F_{0}(q^{2})$ – which appears here for $q^{2}=m_{P}^{2}$ – is a hadronic form
factor which enters also in the $\overline{B}^{0}_{d}\to
D^{+}\ell^{-}\bar{\nu}_{\ell}$ semileptonic decay rate through a term
proportional to the charged lepton mass squared. The hadronic current involves
another hadronic form factor, $F_{+}(q^{2})$, which is measured in decays
leading to a light lepton (see Eq. 4):

$\frac{d{\Gamma}}{dq^{2}}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})=4Cm_{B}^{2}|\vec{p}_{D}(q^{2})|^{3}F_{+}^{2}(q^{2}).$
(4)

Kinematics implies that $F_{0}(0)=F_{+}(0)$. For $q^{2}\neq 0,$ theoretical
parameterizations of the $q^{2}$ dependence of hadronic form factors, whose
parameters have been fitted on data, are used [11].

The value of $a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{+}P^{-})$ is obtained
by taking the ratio between Eq. (3) and Eq. (4)[12]:

$\left|a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{+}P^{-})\right|^{2}=\frac{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}P^{-})}{\left.\frac{d{\cal
B}}{dq^{2}}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})\right|_{q^{2}=m_{P}^{2}}}\frac{1}{6\pi^{2}}\\\
\times\frac{4\,m_{B}^{2}\,|\vec{p}_{D}(m_{P}^{2})|^{2}}{(m_{B}^{2}-m_{D}^{2})^{2}(f_{P}\,|V_{uq}|)^{2}}\frac{F_{+}^{2}(m_{P}^{2})}{F_{0}^{2}(m_{P}^{2})}.$
(5)

The result is independent of the value of $V_{cb}$. It does not depend either
on the absolute values and parameterization of hadronic form factors because
$F_{+}(m_{P}^{2})\simeq F_{0}(m_{P}^{2})$ when the light hadron is a $\pi$ or
a $K$ meson. Moreover, the quantities $f_{P}\,|V_{uq}|$ are accurately
measured for these hadrons.

To evaluate $|a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{+}P^{-})|$, in addition
to the values and corresponding uncertainties (reminded below), one needs the
parameters which govern the $q^{2}$ variation of $F_{+}(q^{2})$. We use the
ones obtained with the CLN parameterization101010These results include charged
and neutral semileptonic $\overline{{B}}$ decays measurements, which are
combined by assuming that the corresponding semileptonic exclusive partial
decay widths are equal.[11] :

$\displaystyle\eta_{EW}\,G_{1}(1)\,\left|{\rm V}_{cb}\right|$ $\displaystyle=$
$\displaystyle(41.53\pm 0.98)\times 10^{-3}$ $\displaystyle\rho^{2}$
$\displaystyle=$ $\displaystyle 1.129\pm 0.033$ $\displaystyle
C_{\eta_{EW}\,G_{1}(1)\,\left|{\rm V}_{cb}\right|,\,\rho^{2}}$
$\displaystyle=$ $\displaystyle 0.758$ (6)

with ${\cal G}^{2}(w)=4r/(1+r)^{2}F_{+}^{2}(w)$ and $r=m_{D}/m_{B}$;
$w=(m_{B}^{2}+m_{D}^{2}-q^{2})/2m_{B}m_{D}$. For the form factor ratio, we use
the linear parameterization $F_{0}(q^{2})/F_{+}(q^{2})=1-r_{F}\,q^{2}$, with
$r_{F}=(0.022\pm 0.001)GeV^{-2}$ as obtained in [13]. The input measurements
are listed in Table 1. We have also rescaled the value obtained by integrating
the CLN parameterization ($(2.13\pm 0.07)\%$) such that it agrees with the
value we are using for ${\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})$ given in this
table.

| PDG [14] | our estimate
---|---|---
${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})$ | $(2.25\pm 0.08)\times 10^{-2}$ | $(2.18\pm 0.06)\times 10^{-2}$
${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})$ | $(2.56\pm 0.08)\times 10^{-3}$ | $(2.57\pm 0.08)\times 10^{-3}$
$R_{DK/D\pi}$ | $(8.19\pm 0.20)\times 10^{-2}$ | $(8.19\pm 0.20)\times 10^{-2}$
Table 1:  Measurements used to evaluate
$\left|a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{+}P^{-})\right|$. The value
for ${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})$, in our
estimate, is the average of charged and neutral $B$-meson decays, assuming
that their semileptonic partial decay widths are equal.
$R_{DK/D\pi}\equiv{\cal B}(\overline{B}^{0}_{d}\to{D}^{+}K^{-})/{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})$

In Table 1, the difference between the values quoted for ${\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})$ is explained by
the fact that we are using the evaluation of this quantity obtained by
averaging the measurements for charged and neutral $\overline{{B}}$ mesons,
assuming the equality of their corresponding partial decay widths. For ${\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})$, our result agrees with PDG2022
[14] and is more precise than the value quoted by HFLAV ($(2.56\pm
0.13)\%$)[11] which does not include a recent measurement from Belle [15].

####  2.1.1 $\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$

Because the decay $\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$ is described by the
“tree” topology contribution alone (which does not mean, of course, tree
approximation in the sense of quantum field theory since it includes gluon
dressing), this process makes it possible, using Eq. (5), to measure directly
$\left|a_{1}(\overline{B}^{0}_{d},\,D^{+}K^{-})\right|$:

$\left|a_{1}(\overline{B}^{0}_{d},\,D^{+}K^{-})\right|\equiv\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{+}K^{-})\right|=0.877\pm
0.023.$ (7)

The quoted uncertainty is obtained by summing in quadrature the following
components: $\pm 0.018$ from ${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})$
and $R_{DK/D\pi}$ measurements, $\pm 0.014$ from fitted CLN parameters and
$\pm 0.0015$ from $f_{K}\times V_{us}$.

What is striking in this measurement is the large magnitude of the deviation
from the theoretical $m_{Q}\to\infty$ limit
($\left|a_{1}^{\infty,\,th.}\right|-\left|a_{1}(\overline{B}^{0}_{d},\,D^{+}K^{-})\right|=0.194\pm
0.026$, amounting to more than $7~{}\sigma$, given the high precision on both
sides).

This deviation is quite larger than the one announced by the theoretical
estimates of BBNS [1] for order ${\cal O}(1/m_{Q})_{n=1}$ as given in their
Section $6.5$. In fact for the $DK$ final state the only relevant contribution
is the so-called spectator one. $T_{spec}/T_{lead}$ is given in their Eq.
(243), and amounts numerically to only $-3\%$ (the ratio should be roughly the
same as for the $D\pi$ final state).

However, one must take into account the fact that the uncertainty affecting
this theoretical result is probably itself very large. To support this, we
will follow comments taken from the authors themselves.

In the same section, while commenting on their numerical evaluations of the
${\cal O}(1/m_{Q})_{n=1}$ corrections, they conclude : “the typical size of
power corrections to the heavy-quark limit is at the level of $10\%$ or less”.
This means that they expect a large uncertainty, all the more since, in the
$D^{+}\pi^{-}$ case which they are considering, the spectator correction is
counterbalanced by the exchange one, ending with a very small numerical
estimate of the correction $-3+4=1\%$, far below $10\%$.

A more explicit idea on the cause of such a large uncertainty in the
evaluation of the ${\cal O}(1/m_{Q})_{n=1}$ correction to the spectator
amplitude is given below their Eq. (28), where they discuss their evaluation
of the spectator correction. It is explained that their calculation,
consisting in introducing a soft gluon through a propagator linking the hadron
wave functions, cannot be “justified” since the momentum of the gluon (or
“virtuality”) is small, of order ${\cal O}(\Lambda_{QCD})$. Indeed, it is
clear that a perturbative treatment of soft gluons is not possible. It seems
also reasonable to expect that the real effect of the soft gluon should be
larger than expected from the naive insertion of the perturbative $\alpha_{s}$
at scale $m_{b}$. It is useful to note that the problem does not appear in the
$m_{Q}\to\infty$ limit because, there, the soft gluons cancel against each
other.

The authors of references [2] and [3] have updated the discussion of BBNS on
the $1/m_{Q}$ corrections to the asymptotic limit, first by a more complete
theoretical evaluation of this limit (higher orders in $\alpha_{s}$), secondly
by new experimental estimates close to ours (see Table 9 in Section 4), and
finally, in the second paper, by a more quantitative estimate of the matrix
element in Eq. (75) of BBNS [1], which controls the “spectator” $1/m_{Q}$
correction, and which they find to be very small. Therefore, they conclude to
a large discrepancy between theory at finite mass and experiment, which would
be of course puzzling. We do not try to discuss their calculation, but rather
we just underline that LCSR results in general are very uncertain or affected
with large errors even in the simplest cases (one may take as example the LCSR
calculation of semileptonic $\overline{{B}}\to
D^{*}\ell^{-}\overline{\nu}_{\ell}$ decay at $q^{2}=0$ by the authors of [4]).
It must be much aggravated in the present case where the LCSR method is
iterated, first at the level of a three particle distribution, then at the one
of the calculation of the final matrix element for the $1/m_{Q}$ correction.
This is very clearly illustrated in the recent recalculation of the same
quantity, with a similar method, by Piscopo and Rusov [5] who give much more
details : they conclude that error bands are very large, about $100\%$ of the
central value for the relevant matrix element $C_{2}<O_{2}>$.

####  2.1.2 $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$

The decay $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$ is described by the sum of
external $W$-emission and $W$-exchange graphs; therefore what is measured is
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|$ and, using
Eq. (5) and the values listed in Table 1, one obtains:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|=0.846\pm
0.019.$ (8)

The quoted uncertainty is obtained by summing in quadrature: $\pm 0.014$ from
the $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$ branching fraction measurement and
$\pm 0.013$ from fitted CLN parameters. Other uncertainties can be neglected.

The same observation can be addressed to this result as it has been above for
$\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$ : the deviation is much larger than
expected. In addition, since the expected exchange contribution is positive
($+0.04$, see Eq. (242) of BBNS), it is expected to nearly cancel the
spectator one ($-0.03$, see Eq. (243) of BBNS). At variance to this, the
measurements indicate that the exchange contribution is much less important
than the spectator correction. In Section 2.3 we investigate in more details
the exchange contribution.

###  2.2 Experimental determination of
$a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-}/\pi^{-})$

We have repeated the same analyses as for $\bar{B}\to D$ transitions. The
values of $a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-}/\pi^{-})$ are
obtained using ratios between corresponding nonleptonic and differential
semileptonic branching fraction measurements, the latter being evaluated at
$q^{2}=m_{P}^{2}$.

Expressions relating helicity and Lorentz invariant form factors at $w=1$,
which corresponds to $q^{2}=q_{max}^{2}$, can be found in [16]. We have used
the CLN parameterization of these form factors [17] which depend on a single
form factor, $h_{A_{1}}(w)$, with
$w=(m_{B}^{2}+m_{D^{(*)}}^{2}-m_{P}^{2})/(2\,m_{B}\,m_{D^{(*)}})$, and on form
factor ratios $R_{1,\,2,\,0}(w)$. Their variation versus $w$ is given in [17]
and [16].

The fitted values of the different parameters, obtained by HFLAV [11], at w=1,
are the following:

$\displaystyle\eta_{EW}\,h_{A_{1}}(1)\,\left|{\rm V}_{cb}\right|$
$\displaystyle=$ $\displaystyle(35.00\pm 0.36)\times 10^{-3}$
$\displaystyle\rho^{2}$ $\displaystyle=$ $\displaystyle 1.121\pm 0.024$
$\displaystyle R_{1}(1)$ $\displaystyle=$ $\displaystyle 1.269\pm 0.026$
$\displaystyle R_{2}(1)$ $\displaystyle=$ $\displaystyle 0.853\pm 0.017.$ (9)

The correlations between these quantities are given in [11]. The value for
$R_{0}(1)$ and the corresponding uncertainty are obtained using the constraint
imposed by kinematics at $q^{2}=0$:

$A_{0}(0)=\frac{m_{B}+m_{D^{*}}}{2m_{D^{*}}}A_{1}(0)-\frac{m_{B}-m_{D^{*}}}{2m_{D^{*}}}A_{2}(0).$
(10) $\Gamma(\overline{B}^{0}_{d}\to
D^{\ast+}P^{-})=24\,\pi^{2}C\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}P^{-})\right|^{2}|\vec{p}_{D^{*}}(m_{P}^{2})|^{3}\,\,m_{B}^{2}\,\\\
\times\left(f_{P}\,V_{uq}\,A_{0}(m_{P}^{2})\right)^{2}.$ (11)

To evaluate $|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}P^{-})|$ we use
also, in addition to the values and corresponding uncertainties of the CLN
parameters given in Eq. (9), the measurements given in Table 2. We rescale the
parameter fitted by HFLAV, $\eta_{EW}\,h_{A_{1}}(1)\,\left|{\rm
V}_{cb}\right|$, which fixes the normalisation for the decay rate so that the
integral of the CLN parameterization agrees with the value we have used for
${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\ell^{-}\bar{\nu}_{\ell})$.

| PDG [14] | our estimate
---|---|---
${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\ell^{-}\bar{\nu}_{\ell})$ | $(5.16\pm 0.16)\times 10^{-2}$ | $(5.17\pm 0.13)\times 10^{-2}$
${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-})$ | $(2.74\pm 0.13)\times 10^{-3}$ | $(2.64\pm 0.11)\times 10^{-3}$
$R_{D^{*}K/D^{*}\pi}$ | $(7.74\pm 0.36)\times 10^{-2}$ | $(7.75\pm 0.30)\times 10^{-2}$
Table 2:  Measurements used to evaluate
$\left|a_{1,\,(eff.)}(\overline{B}^{0}_{d},\,D^{\ast+}P^{-})\right|$, in
addition to the parameters quoted in Eq. (9), relative to
$\overline{B}^{0}_{d}\to D^{\ast+}\ell^{-}\overline{\nu}_{\ell}$.
$R_{D^{*}K/D^{*}\pi}\equiv{\cal B}(\overline{B}^{0}_{d}\to
D^{\ast+}K^{-})/{\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-})$.

The difference observed on the values for ${\cal B}(\overline{B}^{0}_{d}\to
D^{\ast+}\pi^{-})$ comes from the fact that PDG [14] has not included a
measurement from Belle [18]; our result agrees with HFLAV [11], $(2.63\pm
0.10)\times 10^{-3}$, which has used this measurement.

####  2.2.1 Results for $\overline{B}^{0}_{d}\to D^{\ast+}{K}^{-}/\pi^{-}$

Using the same procedure as for the $D^{+}P^{-}$ channel and the measurements
listed in Table 2, we obtain:

$\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-})\right|\equiv\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-})\right|=0.870\pm
0.027$ (12)

with a dominant contribution to the uncertainty ($\pm 0.025$) from the
measurements of ${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-})$ and
$R_{D^{*}K/D^{*}\pi}$. Other sources of uncertainties are $\pm 0.009$ from CLN
parameters and $\pm 0.0015$ from $f_{K}\times V_{us}$.

In this channel also, the difference between the experimental result and the
expected value is close to $20\%$ and is larger than $7\sigma$.

For the decay $\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-}$, we obtain:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})\right|=0.854\pm
0.021$ (13)

with a dominant contribution ($\pm 0.018$) to the uncertainty from the
measurement of ${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-})$. The
comparison of the measured values for
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-})\right|$ and
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})\right|$ shows
clearly that the difference with the infinite quark mass prediction from
theory comes mainly from spectator effects and not from the exchange
contribution, as observed already in the corresponding decay channels with a
${D}^{+}$.

###  2.3 Evaluation of $W$-exchange amplitude contributions in
$\overline{B}^{0}_{d}\to D^{(*)+}\pi^{-}$ decays, $SU(3)$ assumptions.

We have shown, in the previous section, that the measured and expected values
for $\left|a_{1}\right|$, in channels with an emitted kaon that are described
by the T-topology alone, differ by large amounts and that this difference most
probably indicates large ${\cal O}(1/m_{Q})$ contributions, even if present
evaluations of such corrections [3] appear to be very small, in contradiction
with previous results [1] that leave room for larger uncertainties on these
evaluations. In this section we compare the measured and expected results for
the ratio between the exchange and the tree amplitudes.

The considered decays, with an emitted pion, are the sum of external
$W$-emission ($T$) and $W$-exchange $(E)$ amplitudes. Therefore, using the
fact that the corresponding decays with an emitted kaon are governed by the
sole $T$-topology, it is possible to evaluate the contribution from the
$W$-exchange amplitude, once terms violating the SU(3) symmetry (such as
$f_{K}$ used in place of $f_{\pi}$) are taken into account. Moreover, one is
helped by the existence of purely W-exchange processes, a circumstance which
allows to fix directly the magnitude of this contribution. In the following we
have adopted the definitions and procedures described in [7]. The results have
been updated by using recent measurements, in particular for those that use
the $\overline{B}^{0}_{d}\to D^{(*)+}K^{-}$ decay.

In addition, we obtain an expression for the correction factor to the pure
T-topology contribution which is the inverse of the one quoted in [7].

We detail the evaluations for decays with a ${D}^{+}$ while only results are
quoted for $D^{\ast+}$ channels since the procedures are similar.

####  2.3.1  $\overline{B}^{0}_{d}\to D^{+}\pi^{-}$

Using the definitions for the $T$ and $E$ amplitudes given in [7], we get:

$\Gamma(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})=6\pi^{2}C|\vec{p}_{D}(m_{\pi}^{2})|\,|V_{ud}|^{2}(m_{B}^{2}-m_{D}^{2})^{2}\\\
\times\left|T_{D\pi}+E_{D\pi}\right|^{2}.$ (14)

The expression for $T_{D\pi}$ is obtained by comparing the previous equation
with Eq. (3):

$T_{D\pi}=a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})f_{\pi}F_{0}(m_{\pi}^{2}).$
(15)

The $a_{1}$ parameter is no longer said to be “effective” because it
corresponds to the $T$-topology alone and, using Eq. (3), this implies that:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|=\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|\times\left|1+r_{D\pi}\right|$
(16)

with $r_{D\pi}=\frac{E_{D\pi}}{T_{D\pi}}.$

The measured ratio $R_{DK/D\pi}$ is used to evaluate the contribution from the
$W$-exchange amplitude in $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$.

$\displaystyle R_{DK/D\pi}$ $\displaystyle\equiv\frac{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}{K}^{-})}{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})}\hphantom{\left(\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{\pi}\,\left|{\rm
V}_{ud}\right|}\frac{F_{0}(m_{K}^{2})}{F_{0}(m_{\pi}^{2})}\frac{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|}\frac{1}{\left|1+r_{D\pi}\right|}\right)}$
$\displaystyle=\frac{|\vec{p}_{D}(m_{K}^{2})|}{|\vec{p}_{D}(m_{\pi}^{2})|}\left(\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{\pi}\,\left|{\rm
V}_{ud}\right|}\frac{F_{0}(m_{K}^{2})}{F_{0}(m_{\pi}^{2})}\frac{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|}\frac{1}{\left|1+r_{D\pi}\right|}\right)^{2}.$
(17)

This gives:

$\left|1+r_{D\pi}\right|=\frac{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|}\times(0.964\pm
0.012).$ (18)

The numerical quantity which appears in Eq. (18) is simply the ratio between
the values of
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|$ and
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|$, given in Eq. (8)
and Eq. (7), respectively. It is completely fixed by experiment. Its accuracy
is better than the one on individual quantities because the uncertainty on the
semileptonic branching fraction does not contribute. Moreover, the
uncertainties stemming from the form factors are also reduced because $F_{0}$
is evaluated at two nearby $q^{2}$ values and experiments have measured
directly the ratio between the two decay modes.

To obtain more information on the ratio $|r_{D\pi}|=|E_{D\pi}|/|T_{D\pi}|$ and
on the relative phase between the two amplitudes we follow the analysis of [7]
and use the $\overline{B}^{0}_{d}\to D_{s}^{+}{K}^{-}$ decay. This channel is
a pure $W$-exchange process but it differs from the contribution entering
$\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$ because an $s\bar{s}$ pair is created
from vacuum in place of a $d\bar{d}$ pair. The corresponding partial decay
width is expressed as [7]:

$\Gamma(\overline{B}^{0}_{d}\to
D_{s}^{+}{K}^{-})=\frac{G_{F}^{2}(m_{B}^{2}-m_{D_{s}}^{2})^{2}}{16\pi
m_{B}^{2}}|\vec{p}_{D_{s}}(m_{K}^{2})|\left|{\rm V}_{ud}\right|^{2}\left|{\rm
V}_{cb}\right|^{2}\left|E_{D_{s}K}\right|^{2}.$ (19)

In [7] the $W$-exchange amplitudes $E_{D_{s}K}$ and $E_{D\pi}$ are linked
through the relation:

$|E_{D_{s}K}|=|E_{D\pi}|\frac{f_{K}f_{D_{s}}}{f_{\pi}f_{D}}$ (20)

which can be derived from the BBNS expression given in Section 6.5, Eq. (241)
of their paper [1]. Again, it relies on a definite way to treat $SU(3)$
breaking, namely by assuming somewhat the validity of the expansion to order
$(1/m_{Q})^{n=1}$.

The ratio between the branching fractions of the $\overline{B}^{0}_{d}$ into
$D_{s}^{+}{K}^{-}$ and ${D}^{+}\pi^{-}$ becomes:

$\displaystyle R_{D_{s}K/D\pi}$ $\displaystyle\equiv$
$\displaystyle\frac{{\cal B}(\overline{B}^{0}_{d}\to D_{s}^{+}{K}^{-})}{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})\hfill}$ $\displaystyle=$
$\displaystyle\frac{(m_{B}^{2}-m_{D_{s}}^{2})^{2}}{(m_{B}^{2}-m_{D}^{2})^{2}}\frac{|\vec{p}_{D_{s}}(m_{K}^{2})|}{|\vec{p}_{D}(m_{\pi}^{2})|}\left(\frac{f_{K}}{f_{\pi}}\frac{f_{D_{s}}}{f_{D}}\frac{\left|E_{D\pi}/T_{D\pi}\right|}{\left|1+E_{D\pi}/T_{D\pi}\right|}\right)^{2}.$
(21)

$B$-factories have measured the branching fraction ${\cal
B}(\overline{B}^{0}_{d}\to D_{s}^{+}{K}^{-})$ whereas LHCb has measured
directly the ratio, relative to the ${D}^{+}\pi^{-}$ decay channel. Because
values obtained for $R_{D_{s}K/D\pi}$ at $B$-factories and LHCb differ by
about three standard deviations the uncertainty quoted for the average of
these two quantities is scaled according to the PDG recipe 111111It consists
in scaling the uncertainty on the average of N measurements by
$\sqrt{\chi^{2}/(N-1)}$. This gives:

$R_{D_{s}K/D\pi}=(1.12\pm 0.22)\times 10^{-2}.$ (22)

To go further it is necessary to use additional hypotheses, from theory,
because there are more unknowns ($r_{D\pi}$ modulus and phase, value of the
ratio
$r_{a_{1}}\equiv\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|/\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|$
and possible corrections to the expression relating $\left|E_{D_{s}K}\right|$
and $\left|E_{D\pi}\right|$ in Eq. (20)) than the two constraints provided by
experiment. In this respect, three situations are envisaged in the following
paragraphs:

##### analysis using the BBNS approach, including recent updates

In the BBNS approach [1] the values for
$\left|a_{1}^{th.}(\overline{B}^{0}_{d},\,D^{(*)+}M^{-})\right|$ are
“universal”. Thus one has:

$\frac{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|}\simeq\frac{\left|a_{1}^{th.}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|}{\left|a_{1}^{th.}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|}=1.00\pm(<0.01).$
(23)

As explained before, Eq. (20) is also valid using this approach.

Therefore, using Eqs. (18), (20), (23) and the measured value for the ratio
$R_{D_{s}K/D\pi}$ we obtain the modulus and the phase of the ratio $r_{D\pi}$:

$\left|r_{D\pi}\right|=(7.3\pm 0.7)\times 10^{-2},\,\phi(r_{D\pi})=(121\pm
11)^{\circ}.$ (24)

Because the analysis is not sensitive to the sign of the imaginary part, we
assume that it is positive.

The phase between the $E_{D\pi}$ and $T_{D\pi}$ amplitudes is higher than
$90^{\circ}$ and close to this value. The value for $\left|r_{D\pi}\right|$ is
compatible with the initial estimates from [1] and the comments formulated in
[2].

The phase, close to $\pi/2$, is more compelling : it completely contradicts
the typical feature of the BBNS approximation at this order, which gives real
numbers. Let us emphasize that now, one goes beyond the known statement
(Fleischer [7]) that the phase does not vanish, as expected, in the
$1/m_{Q}\to\infty$ limit. Apparently, predicting the phase requires going
beyond $1/m_{Q}$.

We have therefore considered whether it is possible to have a phase equal to
zero or $\pi$ between the two amplitudes by relaxing, in turn, each of the
previously mentioned additional hypotheses.

##### analysis without the constraint on the exchange amplitude

Because ${\cal O}(1/m_{Q})$ terms can be quite important, they should also
enter the expression which relates $E_{D_{s}K}$ to $E_{D\pi}$. In this test we
still assume that the value of $a_{1}$, which originates from the T-topology
contribution alone, is universal and that $r_{D\pi}$ is real. However, we
modify Eq. (20) by introducing the quantity $r_{E}$, that will correspond to
such ${\cal O}(1/m_{Q})$ corrections:

$\left|E^{expt.}_{D_{s}K}\right|=r_{E}\,\left|E^{th.}_{D_{s}K}\right|=r_{E}\,\left|E_{D\pi}\right|\frac{f_{K}f_{D_{s}}}{f_{\pi}f_{D}}$
(25)

and fit its value.

We obtain:

$\displaystyle r_{D\pi}=$ $\displaystyle(-3.6\pm 1.2)\times 10^{-2}$ (26)
$\displaystyle\,{\rm and\qquad}\,r_{E}=$ $\displaystyle 2.0\pm 0.7.$ (27)

Thus, if Eq. (20) is modified by including a correction of ${\cal O}(1)$ given
by the value of the parameter $r_{E}$, measurements can accomodate a real
(negative) value for $r_{D\pi}$ at the few percent level.

##### analysis without assuming T-topology $a_{1}$ value universality

We have redone the fit to data for the quantities, supposed to be real,
$r_{D\pi}$ and
$r_{a_{1}}=\left|a_{1}(\overline{B}^{0}_{d},DK)\right|/\left|a_{1}(\overline{B}^{0}_{d},D\pi)\right|$,
and using Eq. (20) in the constraint provided by $R_{D_{s}K/D\pi}$. We assume
$r_{D\pi}$ to be negative; for positive values, the deviation from unity of
$r_{a_{1}}$ would be larger. We obtain:

$\displaystyle r_{D\pi}=$ $\displaystyle(-7.1\pm 0.6)\times 10^{-2}$ (28)
$\displaystyle\,{\rm and\qquad}\,r_{a_{1}}=$ $\displaystyle 1.037\pm 0.015.$
(29)

There is a $2\sigma$ deviation from the universality.

####  2.3.2 $\overline{B}^{0}_{d}\to D^{*+}\pi^{-}$

We proceed in full parallelism with the ${D}^{+}\pi^{-}$ case (see the
discussion in the previous section 2.3.1).

The expression for $T_{D^{*}\pi}$ is:

$T_{D^{*}\pi}=a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})f_{\pi}A_{0}(m_{\pi}^{2})$
(30)

and:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})\right|=\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})\right|\times\left|1+r_{D^{*}\pi}\right|$
(31)

with $r_{D^{*}\pi}=\frac{E_{D^{*}\pi}}{T_{D^{*}\pi}}$.

The ratio $R_{D^{*}K/D^{*}\pi}$ is used to evaluate the contribution from the
$W$-exchange amplitude in $\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-}$.

$\displaystyle R_{D^{*}K/D^{*}\pi}$ $\displaystyle=$ $\displaystyle\frac{{\cal
B}(\overline{B}^{0}_{d}\to D^{\ast+}{K}^{-})}{{\cal B}(\overline{B}^{0}_{d}\to
D^{\ast+}\pi^{-})}$ $\displaystyle=$
$\displaystyle\left(\frac{|\vec{p}_{D^{*}}(m_{K}^{2})|}{|\vec{p}_{D^{*}}(m_{\pi}^{2})|}\right)^{3}\left(\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{\pi}\,\left|{\rm
V}_{ud}\right|}\frac{A_{0}(m_{K}^{2})}{A_{0}(m_{\pi}^{2})}\frac{\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})\right|}\frac{1}{\left|1+r_{D^{*}\pi}\right|}\right)^{2}.$

This gives:

$\left|1+r_{D^{*}\pi}\right|=\frac{\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}\pi^{-})\right|}\times(0.987\pm
0.020).$ (33)

To have more information on the ratio $|E_{D^{*}\pi}|/|T_{D^{*}\pi}|$ and on
the relative phase between the two amplitudes we follow the same approach as
in section 2.3.1, using now the $\overline{B}^{0}_{d}\to D_{s}^{*+}{K}^{-}$
decay.

The $W$-exchange amplitudes, $E_{D_{s}^{*}K}$ and $E_{D^{*}\pi}$, are related
through the new $SU(3)$ relation [7], with the same justification as in the
case of transitions to a pseudoscalar:

$|E_{D_{s}^{*}K}|=|E_{D^{*}\pi}|\frac{f_{K}f_{D_{s}^{*}}}{f_{\pi}f_{D^{*}}}.$
(34)

Therefore, the ratio between the branching fractions of the
$\overline{B}^{0}_{d}$ into $D_{s}^{*+}{K}^{-}$ and $D^{\ast+}\pi^{-}$
becomes:

$\displaystyle R_{D_{s}^{*}K/D^{*}\pi}$ $\displaystyle=$
$\displaystyle\frac{{\cal B}(\overline{B}^{0}_{d}\to D_{s}^{*+}{K}^{-})}{{\cal
B}(\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-})}$ (35) $\displaystyle=$
$\displaystyle\left(\frac{|\vec{p}_{D_{s}^{*}}(m_{K}^{2})|}{|\vec{p}_{D^{*}}(m_{\pi}^{2})|}\right)^{3}\left(\frac{f_{K}}{f_{\pi}}\frac{f_{D_{s}^{*}}}{f_{D^{*}}}\frac{\left|r_{D^{*}\pi}\right|}{\left|1+r_{D^{*}\pi}\right|}\right)^{2}.$

For numerical evaluations, it has been assumed that
$\frac{f_{D_{s}^{*}}}{f_{D^{*}}}=\frac{f_{D_{s}}}{f_{D}}$.

$B$-factories have measured the branching fraction ${\cal
B}(\overline{B}^{0}_{d}\to D_{s}^{*+}{K}^{-})$ and, dividing by the value of
${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-})$ given in Table 2, this
gives:

$R_{D_{s}^{*}K/D^{*}\pi}=(0.83\pm 0.12)\times 10^{-2}.$ (36)

Similarly to what we have done for pseudoscalar mesons, we consider three
possible scenarios; the comments made above apply here too, mutatis mutandis.

##### analysis using the BBNS approach, including recent updates

$\left|r_{D^{*}\pi}\right|=(6.7\pm 0.5)\times
10^{-2},\,\phi(r_{D^{*}\pi})=(107\pm 17)^{\circ}$ (37)

##### analysis without the constraint on the exchange amplitude

$r_{D^{*}\pi}=(-1.8\pm 2.0)\times 10^{-2}\\\ ,{\rm}\,r_{E^{*}}=3.7\pm 4.2$
(38)

##### analysis without assuming T-topology $a_{1}$ value universality

We obtain:

$\displaystyle r_{D^{*}\pi}=$ $\displaystyle(-6.4\pm 0.4)\times 10^{-2}$ (39)
$\displaystyle{\rm and}\qquad r_{a_{1}^{*}}=$ $\displaystyle 1.05\pm 0.02.$
(40)

####  2.3.3 Comparison with a previous similar analysis [7] and comments on
$f_{s}/f_{d}$ measurement at LHCb

The values we obtain for the quantity
$\left|1+E_{D^{(*)}\pi}/T_{D^{(*)}\pi}\right|$ are compared with the results
of [7] in Table 3.

| our analysis | [7]
---|---|---
$R_{DK/D\pi}\,(\%)$ | $8.19\pm 0.20$ | $7.5\pm 2.3$
$\left|1+E_{D\pi}/T_{D\pi}\right|$ | $0.964\pm 0.012$ | $1.01\pm 0.15$
$R_{D^{*}K/D^{*}\pi}\,(\%)$ | $7.75\pm 0.30$ | $7.76\pm 0.45$
$\left|1+E_{D^{*}\pi}/T_{D^{*}\pi}\right|$ | $0.987\pm 0.020$ | $1.017\pm 0.029$
Table 3:  Comparison between the values, evaluated for the contribution of the
exchange amplitude, in our analysis and in [7]. Values for the ratios $R$ are
those from experiments, as available at the time of the two analyses.

Once it has been assumed that all SU(3) corrections consist in changing the
value of the light emitted meson decay constant, the values of $|1+E/T|$
depend only on the measured ratio of branching fractions
$R_{D^{(*)}K/D^{(*)}\pi}$. For decays where a $D^{\ast+}$ is emitted, the
uncertainty has been slightly reduced thanks to the use of new experimental
data. It can be noted that, while values for $R_{D^{*}K/D^{*}\pi}$ are
identical in the two analyses, the contribution from the exchange amplitude
differs in sign. We wonder whether, in [7], what was evaluated would not be
$\left|1+E_{D^{*}\pi}/T_{D^{*}\pi}\right|^{-1}$ rather than the inverse.

For decays with an emitted ${D}^{+}$, using measurements that were not
available at the time of [7] produces a reduction of the uncertainty by an
order of magnitude and the result becomes even more accurate than with an
emitted $D^{\ast+}$. The contribution from the $W$-exchange amplitude is
similar, for ${D}^{+}$ and $D^{\ast+}$ channels, within uncertainties.

The difference between [7] and our analysis implies a significant effect on
the correction factor used, at present, by LHCb to obtain the value for the
ratio $f_{s}/f_{d}$ using the $\overline{B}^{0}_{d}\to D^{+}\pi^{-}$ decay
channel.

Comments on $f_{s}/f_{d}$ measurement at LHCb using the
$\overline{B}^{0}_{d}\to D^{(*)+}\pi^{-}$ decay channel.  To measure the
$\overline{B}^{0}_{s}$ relative to $\overline{B}^{0}_{d}$ meson production
rate at a hadron collider (usually noted as $f_{s}/f_{d}$), it has been
proposed to use, among other methods, the corresponding decay channels
$\overline{B}^{0}_{s}\to D_{s}^{+}\pi^{-}$ and
$\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$ because, using factorisation, one can
predict the ratio between these two branching fractions. Considering that the
$\overline{B}^{0}_{d}$ decay channel is not purely of the T-topology, it is
necessary to correct for the effect of the exchange contribution. For this
purpose the quantities ${\cal N}_{E^{(*)}}$ have been introduced [7]:

${\cal
N}_{E^{(*)}}=\left|\frac{1}{1+E_{D^{(*)}\pi}/T_{D^{(*)}\pi}}\right|^{2},$ (41)

for, respectively, the $\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$ and
$\overline{B}^{0}_{d}\to D^{\ast+}\pi^{-}$ decay channels. The value, obtained
in [7], from the analysis of the channel with an emitted $D^{\ast+}$ (Table 3)
is ${\cal N}_{E^{*}}=0.966\pm 0.056$, from which they evaluate ${\cal
N}_{E}={\cal N}_{E^{*}}\pm 0.05$. The additional uncertainty is supposed to
account for the fact that the correction, evaluated with an emitted
$D^{\ast+}$, is used to correct the measurement of the decay
$\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}$, with an emitted ${D}^{+}$. In present
LHCb analyses [19], this procedure is still adopted, and they use ${\cal
N}_{E}=0.966\pm 0.062$. The central value remains the same as in the previous
evaluation and the total uncertainty too is similar.

For the channel with an emitted $D^{\ast+}$, we obtain ${\cal
N}_{E^{*}}=1.027\pm 0.042$. As already mentioned, the difference between the
two ${\cal N}_{E^{*}}$ central values most probably originates from a
confusion between ${\cal N}_{E^{*}}$ and $1/{\cal N}_{E^{*}}$ in [7].

In addition, it can be noted also that, thanks to progress in the experimental
results, it is no longer needed to use decays with a $D^{\ast+}$ to correct
measurements with an emitted ${D}^{+}$. Using the corresponding values quoted
in Table 3, we obtain directly: ${\cal N}_{E}=1.076\pm 0.027$, which,
according to us, is the value that has to be used in practice (in place of
$0.966\pm 0.062$). It differs by $11\%$ from the one used at present and it is
at least two times more accurate, with the benefit that there is no need to
add an uncertainty (having an essentially arbitrary value) to cope for the
difference between an emitted ${D}^{+}$ or $D^{\ast+}$.

It is difficult to estimate the effect of such a correction on the present
LHCb complex fitting procedure to obtain the value of $f_{s}/f_{d}$ [19]
because there are also $\overline{B}^{0}_{d}\to{D}^{+}K^{-}$ (which has no
annihilation contribution), semileptonic and ${B^{+}}\to J/\Psi K^{+}$ decay
channels that are used. It can be noted that central values for $f_{s}/f_{d}$
are higher by about one standard deviation in fits with theoretical
constraints [19], indicating that such constraints, currently used, have some
effect thereon.

According to the results given in Tables (9) and (10) of the LHCb publication
[19], the uncertainties on $f_{s}/f_{d}$ are similar whether one uses
theoretical constraints or not; therefore it seems more appropriate to use, at
present, the values obtained without such constraints.

The reason for this, in addition to the comments we made before about ${\cal
N}_{E}$, is that in all these evaluations the equality

$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|=\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|=\left|a_{1}(\overline{B}^{0}_{s},\,D_{s}^{+}\pi^{-})\right|$

is assumed, although no well established uncertainty is available as long as
there is no theoretical explanation for the large difference between the
measured and the theoretical expected values for
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-}/\pi^{-})\right|$.

One should note also that, although neither $\overline{B}^{0}_{d}\to
D^{(*)+}{K}^{-}$ nor $\overline{B}^{0}_{s}\to D_{s}^{(*)+}\pi^{-}$ decays
receive a contribution from an exchange term, the spectator quarks are
different for the two channels, being $d$ and $s$ respectively. The comparison
between the values of
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}\pi^{-})\right|$ and
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|$, which is
sensitive to the effect of exchange terms, provides no information on the
effect of spectator quarks with different flavours. Therefore, the assumption
that
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|=\left|a_{1}(\overline{B}^{0}_{s},\,D_{s}^{+}\pi^{-})\right|$
induces an uncertainty which is not controlled.

##### Comments on $f_{s}/f_{d}$ determination at LHCb using inclusive
semileptonic decays.

From the discussion we have presented so far, one cannot claim, from the
theory side, for precision physics in non-leptonic decays, as long as the
origin of the discrepancy between the expected values and the measured ones
for $\left|a_{1}\right|$ is not understood.

In this respect, using inclusive semileptonic decays, for which theory expects
$\Gamma_{sl.}({B^{0}_{s}})/\Gamma_{sl.}({B^{0}_{d}})=(1-0.018\pm 0.008)$ [20],
appears more promising. Meanwhile, in this case, the main difficulty comes
from the experimental side because it is very difficult to measure such
inclusive decay widths. Experiments select samples of events that are
enriched, respectively, in strange and non-strange $\overline{{B}}$-meson
decays and cross-contaminations are corrected. Most of these corrections are
measured using the same apparatus but there are contributions from some
channels that need to be corrected using external measurements or even
guessed. A typical example is $\overline{{B}}\to
D_{s}^{+}{K}^{-}X\ell^{-}\bar{\nu}_{\ell}$ decays which contaminate the
$D_{s}^{+}X\ell^{-}$, $\overline{B}^{0}_{s}$ enriched sample. The
corresponding correction is obtained using ${\cal B}({B^{-}}\to
D_{s}^{(*)+}{K}^{-}\ell^{-}\bar{\nu}_{\ell})=(6.1\pm 1.0)\times 10^{-4}$
measured at B-factories [14]. This is a quite small branching fraction but the
correction to the selected $D_{s}^{+}X\ell^{-}$, amounting to $5-6\%$, is
enhanced because strange $B$-mesons are much less abundant than non-strange
ones. As can be noted, B-factories have measured only one exclusive decay
channel with a ${K}^{-}$ produced in addition to the $D_{s}^{(*)+}$, but there
might be additional pions. $b$-baryons can have similar decays with an hyperon
produced in addition to the strange charm meson. It seems that such channels
have not been considered in the present LHCb analysis. If one assumes, as a
simple exercise, that ${\cal B}(\Lambda_{b}\to
D_{s}^{(*)+}Y\ell^{-}\bar{\nu}_{\ell})={\cal B}({B^{-}}\to
D_{s}^{(*)+}{K}^{-}\ell^{-}\bar{\nu}_{\ell})$ and that ${\cal B}({B^{-}}\to
D_{s}^{(*)+}K\pi\ell^{-}\bar{\nu}_{\ell})=1/5\times{\cal B}({B^{-}}\to
D_{s}^{(*)+}{K}^{-}\ell^{-}\bar{\nu}_{\ell})$, the previous correction
increases to $7-9\%$, depending on the $b$-hadron transverse momentum.

#####  $f_{s}/f_{d}$ determination using exclusive semileptonic decays.

The determination of the ratio $f_{s}/f_{d}$ depends on the knowledge, from
theory, of the ratio between two decay branching fractions. As we have
explained in the previous sections, uncontrolled uncertainties are attached to
the ratios between exclusive nonleptonic channels as ${\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-}/{K}^{-})/{\cal
B}(\overline{B}^{0}_{s}\to D_{s}^{+}\pi^{-})$. The ratio between inclusive
Cabibbo-allowed semileptonic decays of strange and non-strange $B$-mesons
seems to be in better control from theory because the attached uncertainties,
are estimated to be small as compared with the experimental ones.

At present, and in particular because of recent progress in the evaluation of
hadronic form factors in
$\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell}$ and
$\overline{B}^{0}_{s}\to D_{s}^{+}\ell^{-}\bar{\nu}_{\ell}$ exclusive
semileptonic decay channels [3, 21], it seems promising to measure the ratio
$f_{s}/f_{d}$ using these transitions. As already demonstrated by LHCb [22],
it is possible to use the decay channels
${D}^{+}/D_{s}^{+}\to{K}^{+}{K}^{-}\pi^{+}$ to obtain the same particles, in
the final state, for the $\overline{B}^{0}_{d}$ and $\overline{B}^{0}_{s}$
mesons. In this way, a large amount of systematic uncertainties cancel in the
ratio (trigger, particle identification, tracking efficiency, …), when
measuring $f_{s}/f_{d}$. The accuracy of this measurement will depend on the
precision attained on charmed mesons decay branching fractions, which are, at
present [14]: ${\cal B}({D}^{+}\to{K}^{+}{K}^{-}\pi^{+})=(9.68\pm 0.18)\times
10^{-3}$ and ${\cal B}(D_{s}^{+}\to{K}^{+}{K}^{-}\pi^{+})=(5.38\pm 0.10)\times
10^{-2}$. They correspond to a relative uncertainty of 2.6$\%$.

Unfortunately, up to now, LHCb has used their measurements to obtain a value
for the CKM matrix element $V_{cb}$, in $\overline{B}^{0}_{s}\to
D_{s}^{(*)+}\mu^{-}\bar{\nu}_{\mu}$ decays, under the assumption that their
determination of $f_{s}/f_{d}$ is well under control. In our proposal, it
seems more appropriate to invoke the fact that the value of $V_{cb}$ is
universal and does not enter when considering the ratio between
$\overline{B}^{0}_{s}$ and $\overline{B}^{0}_{d}$ semileptonic decays, to
obtain a value for $f_{s}/f_{d}$.

$\frac{f_{s}}{f_{d}}=\frac{N^{rec.}_{B_{s}\to D_{s}\mu\nu}}{N^{rec.}_{B_{d}\to
D\mu\nu}}\frac{\int{d\Gamma(B_{d}\to D\mu\nu)/dw\,\epsilon_{B_{d}\to
D\mu\nu}(w)dw}}{\int{d\Gamma(B_{s}\to D_{s}\mu\nu)/dw\,\epsilon_{B_{s}\to
D_{s}\mu\nu}(w)dw}}\frac{\tau(B_{d})}{\tau(B_{s})}\frac{{\cal B}(D^{+})}{{\cal
B}(D_{s}^{+})}$ (42)

In this expression, the first ratio corresponds to the measured
$\overline{B}^{0}_{s}$ and $\overline{B}^{0}_{d}$ exclusive semileptonic
decays. The second term is the ratio between the integrated semileptonic
rates, taken from theory, mutiplied by the corresponding reconstruction
efficiencies. At the end of this expression, there is the ratio between the
${D}^{+}$ and $D_{s}^{+}$ branching fractions, into the
${K}^{+}{K}^{-}\pi^{+}$ decay channel. It is not possible for us to evaluate
the part, in this expression, that depends on the experiment performances. The
same expression can be used, replacing $D$ by $D^{*}$ mesons and considering
more kinematic variables in the integrals.

##  3 $\left|a_{1}(\overline{B}^{0}_{s},D_{s}^{+}\pi^{-})\right|$ absolute
determination

Accurate evaluations of the $\left|a_{1}\right|$ parameters in several
$\overline{{B}}$ decay channels may help to understand the large differences
with corresponding theoretical determinations.

A possibility to evaluate
$\left|a_{1}(\overline{B}^{0}_{s},D_{s}^{+}\pi^{-})\right|$ is to use the same
approach as in Section 2.

For this purpose, we use recent results from LHCb [19]:

$\displaystyle{\cal B}(\overline{B}^{0}_{s}\to D_{s}^{+}\pi^{-})\times 10^{3}$
$\displaystyle=$ $\displaystyle 3.20\pm 0.10\pm 0.16\,\,\,(3.02\pm 0.10)$ (43)
$\displaystyle{\cal B}(\overline{B}^{0}_{s}\to
D_{s}^{+}\mu^{-}\overline{\nu}_{\mu})\times 10^{2}$ $\displaystyle=$
$\displaystyle 2.40\pm 0.12\pm 0.15\pm 0.06\pm 0.10\,(2.24\pm 0.18).$

These values are corrected to remove the common uncertainty coming from the
evaluation of $f_{s}/f_{d}$. The central values and the corresponding
uncertainties are slightly modified when they depend on the values of external
decay branching fractions, to be in agreement with PDG or with the
corresponding values used in previous sections. The value of ${\cal
B}(\overline{B}^{0}_{s}\to D_{s}^{+}\pi^{-})$ depends on ${\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\pi^{-})$ for which we use the value given in
Table 1. The value of the exclusive semileptonic $\overline{B}^{0}_{s}$ decay
branching fraction depends on the values of ${\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}\mu^{-}\overline{\nu}_{\mu})$ (Table 1) and
of the ${D}^{+}/D_{s}^{+}$ decay branching fractions to
${K}^{+}{K}^{-}\pi^{+}$ [14]. The values obtained in this way are quoted
within parentheses in Eq. (43). To extract from the experimental data the
value of $\left|a_{1}(\overline{B}^{0}_{s},D_{s}^{+}\pi^{-})\right|$, one
needs also the values of the CLN parameters, fitted on the differential
distribution $d{\cal B}(\overline{B}^{0}_{s}\to
D_{s}^{+}\mu^{-}\overline{\nu}_{\mu})/dq^{2}$, which have been published in
[22]. We have integrated the corresponding expression and scaled the result
such that it agrees with the quoted measured branching fraction in Eq. (43).

Using an expression similar to the one quoted in Eq. (5), which was relative
to the decay of the ${B^{0}_{d}}$ meson we obtain:

$\left|a_{1}(\overline{B}^{0}_{s},\,D_{s}^{+}\pi^{-})\right|=0.922\pm 0.026.$
(44)

This result does not depend on the value assumed for the ratio $f_{s}/f_{d}$.
The main contributions to the uncertainty are those which stem from the
semileptonic and nonleptonic branching fraction measurements: $\pm 0.021$ and
$\pm 0.015$, respectively. The value of
$\left|a_{1}(\overline{B}^{0}_{s},\,D_{s}^{+}\pi^{-})\right|$ can be compared
with the one obtained previously for
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|=0.877\pm 0.023$,
which is expected to be the same according to [7]. Present results agree with
this expectation:
$\left|a_{1}(\overline{B}^{0}_{s},\,D_{s}^{+}\pi^{-})\right|-\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|=0.045\pm
0.034$.

##  4 Factorization in heavy meson emission reactions

According to [1], non-leptonic Class I decays of the type $\overline{{B}}\to
D^{(*)}D_{s}^{(*)-}$ in which a charmed meson is emitted in place of a light
one are not expected to satisfy factorization, i.e. one does not expect
$\left|a_{1}(\overline{{B}},D^{(*)}D_{s}^{(*)-})\right|$ to be (even roughly)
close to $1$, because one does not expect
$\left|a^{\infty,th}_{1}(\overline{{B}},D^{(*)}D_{s}^{(*)-})\right|=1$ in the
$m_{Q}\to\infty$ limit. Moreover, in these decays, penguin amplitudes are
contributing and their effect has to be estimated before comparing with
factorization estimates.

Conversely to decays in which a light meson is emitted, the effects of the
non-dominant amplitudes are not expected to give a significant difference
between charged and neutral $B$-mesons decays because:

  * •

the penguin amplitudes, denoted "$P$" in the following, are the same;

  * •

in final states with a $D_{s}^{(*)-}$, the exchange amplitude, which
contributes in charged $B$ decays only, is DCS.

Therefore, the measurements with a charged or a neutral $B$-meson can be
averaged (and, in the following, we express the result in terms of the neutral
state). We have evaluated our own averages using existing measurements,
updating values of intermediate decay branching fractions and taking into
account correlations.

Within the factorization framework, one can use a formulation similar to the
one of Section 2.3 to express the effect of penguin contributions by defining:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|\\\
=\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|\times\left|1+r_{D^{(*)}D_{s}^{(*)-}}\right|$
(45)

with
$r_{D^{(*)}D_{s}^{(*)-}}=\frac{P_{D^{(*)}D_{s}^{(*)-}}}{T_{D^{(*)}D_{s}^{(*)-}}}$.
Values for the induced correction to the $\left|a_{1}\right|$ parameter,
$\left|1+r_{D^{(*)}D_{s}^{(*)-}}\right|$, have been evaluated in [23] and are
reminded of in Table 4:

decay channel | $\left|1+r_{D^{(*)}D_{s}^{(*)-}}\right|$ | $\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|/\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}{K}^{-})\right|$
---|---|---
| [23] | as obtained in our analysis
$\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{-}$ | $0.847$ | $0.901\pm 0.054$
$\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{-}$ | $1.037$ | $1.155\pm 0.075$
$\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{*-}$ | $0.962$ | $0.850\pm 0.068$
$\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{*-}$ | $0.962$ | $0.838\pm 0.042$
Table 4:  Expected corrections to the $\left|a_{1}\right|$ parameter in
channels with an emitted $D_{s}^{(*)-}$ as obtained in [23]. In the last
column, similar quantities are given, under the hypothesis that
$\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|=\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}{K}^{-})\right|$
(see the following sections).

In the following, for each one of the different decay channels that are
considered, we repeatedly use the following procedure:

  * •

we first estimate, from the experimental data, the ratio
$\frac{\left|a_{1,eff.}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}{K}^{-})\right|}$
where the $K^{-}$-channel, in the denominator, has been chosen because it
involves no exchange contribution. The result is then compared with the
estimated effects of the penguin amplitudes as obtained in [23] assuming that
$\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|=\left|a_{1}(\overline{B}^{0}_{d},\,D^{(*)}{K}^{-})\right|$,
which implies that factorization applies also to decays with heavy meson
emission;

  * •

then we provide values for
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{(*)}D_{s}^{(*)-})\right|$ using,
in addition to the nonleptonic decay channels, the corresponding semileptonic
ones, as was already done in Section 2 in the case of light meson emission.

We use the expressions for ${\cal B}(\overline{B}\to D^{(*)}D_{s}^{*-})$
derived in [8].

###  4.1 The $\overline{B}\to DD_{s}^{-}$ decay channel

The branching fractions used in this analysis are given in Table 5.

decay channel |  PDG [14] |  HFLAV [11] | our average
---|---|---|---
${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{-})\times 10^{3}$ | $7.24\pm 0.77$ | $7.82\pm 0.72$ | $7.30\pm 0.72$
${\cal B}({B^{-}}\to{D}^{0}D_{s}^{-})\times 10^{3}$ | $9.01\pm 0.94$ | $9.58\pm 0.98$ | $10.0\pm 1.35$
Table 5:  Branching fractions for $\overline{{B}}$ meson decays into a
$D_{s}^{-}$ and another charm pseudo-scalar meson, obtained in PDG 2022, HFLAV
2022 and in our analysis.

The smaller uncertainty obtained for $\overline{B}^{0}_{d}$ than for ${B^{-}}$
results from the fact that, for $\overline{B}^{0}_{d}$-mesons only, there is
an additional result from Belle [25] which is more accurate than other
measurements. Our uncertainty for the ${B^{-}}$ is also higher because we have
not included the ratio between the results obtained by LHCb for neutral and
charged $B$-meson decays, having considered that it should correspond to the
one expected from the corresponding lifetimes, from which it differs by less
than two sigmas. Therefore this is simply a consistency check of the LHCb
analysis.

From the values given in Table 5 and assuming that their partial decay widths
are equal, we combine results for charged and neutral $B$-mesons to obtain:

${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{-})=(7.76\pm 0.84)\times
10^{-3}$ (46)

in which the quoted uncertainty has been scaled by 1.37, according to the PDG
recipe.

####  4.1.1 Penguin contribution (step 1 of the procedure explained above)

To evaluate the penguin amplitude contribution we compare the decay channel
under consideration with $\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$. In oher
words, we consider the ratio
$|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{-})/a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})|$
obtained by using an expression similar to Eq. (17):

$\displaystyle\frac{\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})\right|}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{-})}{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}K^{-})}\frac{|\vec{p}_{D}(m_{K}^{2})|}{|\vec{p}_{D}(m_{D_{s}}^{2})|}}\,\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{D_{s}}\,|V_{cs}|}\,\frac{F_{0}(m_{K}^{2})}{F_{0}(m_{D_{s}}^{2})}$
(47) $\displaystyle=$ $\displaystyle 0.901\pm 0.054.$

This value agrees within one standard deviation with the expectation from
theory : $0.847$ [23].

####  4.1.2
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}{D}_{s}^{-})\right|$
evaluation (step 2)

We evaluate
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{-})\right|$ using the
semileptonic decay channel as in Eq. (5):

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}{D}_{s}^{-})\right|^{2}=\frac{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}{D}_{s}^{-})}{\left.\frac{d{\cal
B}}{dq^{2}}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})\right|_{q^{2}=m_{D_{s}}^{2}}}\frac{1}{6\pi^{2}}\\\
\times\frac{4\,m_{B}^{2}\,|\vec{p}_{D}(q^{2})|^{2}}{(m_{B}^{2}-m_{D}^{2})^{2}(f_{D_{s}}\,|V_{cs}|)^{2}}\frac{F_{+}^{2}(m_{D_{s}}^{2})}{F_{0}^{2}(m_{D_{s}}^{2})}$
(48)

which gives:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}{D}_{s}^{-})\right|=0.790\pm
0.046$ (49)

in which the dominant contributions to the uncertainty are : $\pm 0.043$ from
the ${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}{D}_{s}^{-})$ measurement, $\pm
0.009$ from the $f_{D_{s}}\,|V_{cs}|$ determination and $\pm 0.013$ from the
CLN parameterization of semileptonic form factors.

After extraction of the penguin contribution, expected from theory:

$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{-})\right|=(0.790\pm
0.046)/0.847=0.933\pm 0.054.$ (50)

The result is compatible, within one standard deviation, with the number
obtained from light meson emission:
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}K^{-})\right|=0.877\pm 0.023$.

###  4.2 The $\overline{B}\to DD_{s}^{*-}$ decay channel

We consider now decay channels with the emission of a vector meson instead of
a pseudo-scalar one and averaging, as for the previous channel, measurements
obtained with charged and neutral $B$-mesons we obtain :

decay channel | PDG [14] | HFLAV [11] | our average
---|---|---|---
${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{*-})\times 10^{3}$ | $7.4\pm 1.6$ | $7.5\pm 1.9$ | $7.8\pm 1.6$
${\cal B}({B^{-}}\to{D}^{0}D_{s}^{*-})\times 10^{3}$ | $7.6\pm 1.6$ | $8.3\pm 2.0$ | $7.8\pm 1.5$
Table 6:  Branching fractions for $\overline{{B}}$ mesons decaying into a $D$
and a strange charm vector meson, obtained in PDG 2022, HFLAV 2022 and in our
analysis.

${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{*-})=(7.45\pm 1.07)\times
10^{-3}.$ (51)

####  4.2.1 Penguin contribution

To evaluate the penguin amplitude contribution we have compared, similarly,
this decay channel with $\overline{B}^{0}_{d}\to{D}^{+}{K}^{-}$. Therefore we
consider the ratio
$|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{*-})/a_{1}(\overline{B}^{0}_{d},\,{D}^{+}{K}^{-})|$
obtained by using Eq. (52) to compute the partial decay width for the decay
$\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{*-}$.

$\Gamma_{Tree}(\overline{B}^{0}_{d}\to{D}^{+}V^{-})=24\pi^{2}C|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}V^{-})|^{2}\,|\vec{p}_{D}(m_{V}^{2})|^{3}\,(f_{V}\,|V_{q_{1}q_{2}}|)^{2}\\\
\times\,m_{B}^{2}\,F_{+}^{2}(m_{V}^{2})$ (52)

where $q_{1}$ and $q_{2}$ are a generic notation for the appropriate quarks
constituting the specific vector meson under consideration. This gives:

$\displaystyle\frac{\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}V^{-})\right|}{\left|a_{1}({\overline{B}^{0}_{d},\,{D}^{+}{K}^{-}}\right|})$
$\displaystyle=\sqrt{\frac{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}V^{-})}{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}K^{-})}\frac{|\vec{p}_{D}(m_{K}^{2})|\,(m_{B}^{2}-m_{D}^{2})^{2}}{4\,|\vec{p}_{D}(m_{V}^{2})|^{3}\,m_{B}^{2}}}\,\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{V}\,|V_{q_{1}q_{2}}|}$ (53)
$\displaystyle\times\frac{F_{0}(m_{K}^{2})}{F_{+}(m_{V}^{2})}$
$\displaystyle={0.850\pm 0.068.}$ (54)

For the $D_{s}^{*}$ decay constant we use $f_{D_{s}^{*}}/f_{D_{s}}=1.26\pm
0.03$ [24].

The dominant contributions to the uncertainty on the obtained ratio are $\pm
0.063$ from the branching fraction measurement, $\pm 0.022$ from the decay
constants and CKM elements determination. The value obtained in Eq. (54) is
about two standard deviations lower than the expectation from theory : $0.962$
[23].

####  4.2.2
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{*-})\right|$
evaluation

We evaluate
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{*-})\right|$ using
the semileptonic decay channel as in Eq. (5):

$\displaystyle\begin{split}\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{*-})\right|^{2}=\frac{{\cal
B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{*-})}{\left.\frac{d{\cal
B}}{dq^{2}}(\overline{B}^{0}_{d}\to{D}^{+}\ell^{-}\bar{\nu}_{\ell})\right|_{q^{2}=m_{D_{s}^{*}}^{2}}}\frac{1}{6\pi^{2}}\frac{1}{(f_{D_{s}^{*}}\,|V_{cs}|)^{2}}\end{split}$
(55)

which gives:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{*-})\right|=0.746\pm
0.058$ (56)

in which the dominant contribution to the uncertainty is $\pm 0.055$ and comes
from the ${\cal B}(\overline{B}^{0}_{d}\to{D}^{+}D_{s}^{*-})$ measurement.

After extraction of the penguin contribution expected from theory one obtains:

$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}D_{s}^{*-})\right|=(0.746\pm
0.058)/0.962=0.775\pm 0.060.$ (57)

The result is about two standard deviations lower than the light meson
emission value:
$\left|a_{1}(\overline{B}^{0}_{d},\,{D}^{+}K^{-})\right|=0.877\pm 0.023$.

###  4.3 The $\overline{{B}}\to D^{*+}{D}_{s}^{-}$ decay channel

We have repeated the same analysis as in Section 4.1 for $\overline{{B}}\to
D^{*}$ transitions. The measured branching fractions are listed in Table 7.

decay channel | PDG [14] | HFLAV [11] | our average
---|---|---|---
${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{-})\times 10^{3}$ | $8.0\pm 1.1$ | $8.1\pm 1.4$ | $9.1\pm 1.2$
${\cal B}({B^{-}}\to{D}^{\ast 0}D_{s}^{-})\times 10^{3}$ | $8.2\pm 1.7$ | $7.9\pm 1.6$ | $8.1\pm 1.7$
Table 7:  Branching fractions for $\overline{{B}}\to D^{*}D_{s}^{-}$ decays
obtained in PDG 2022, HFLAV 2022 and in our analysis.

Our value for the neutral $B$-meson differs from the other averages because of
the way we correct the measurements from BaBar. From the values given in Table
7 and assuming that their partial decay widths are equal, we combine results
for charged and neutral $B$-meson to evaluate:

${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{-})=(8.6\pm 1.0)\times
10^{-3}.$ (58)

####  4.3.1 Penguin contribution

Using Eq.(2.3.2) we evaluate the ratio :

$\displaystyle\begin{split}\frac{\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}{K}^{-})\right|}&=\sqrt{\frac{{\cal
B}(\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{-})}{{\cal
B}(\overline{B}^{0}_{d}\to
D^{\ast+}K^{-})}\frac{|\vec{p}_{D^{*}}(m_{K}^{2})|^{3}}{|\vec{p}_{D^{*}}(m_{D_{s}}^{2})|^{3}}}\,\\\
&\times\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{{D_{s}}}\,|V_{cs}|}\frac{A_{0}(m_{K}^{2})}{A_{0}(m_{D_{s}}^{2})}\\\
&=1.155\pm 0.075.\end{split}$ (59)

The value obtained agrees with the theoretical estimate -$1.037$ [23]\- which
includes contributions from the penguin amplitude in
$a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{-})$.

####  4.3.2
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{D}_{s}^{-})\right|$
evaluation

We evaluate
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{D}_{s}^{-})\right|$ using
the semileptonic decay channel, as in Section 2.2:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}{D}_{s}^{-})\right|=1.000\pm
0.066$ (60)

in which the dominant uncertainties are $\pm 0.059$ from the ${\cal
B}(\overline{B}^{0}_{d}\to D^{\ast+}{D}_{s}^{-})$ measurement, $\pm 0.011$
from the $f_{D_{s}}\,|V_{cs}|$ determination and $\pm 0.027$ from the CLN
parameterization of semileptonic form factors.

After extraction of the penguin contribution one obtains:

$\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{-})\right|=(1.000\pm
0.066)/1.037=0.964\pm 0.064.$ (61)

The result is not significantly different from the light meson emission value
: $0.870\pm 0.027$ for $D^{\ast+}{K}^{-}$.

###  4.4 The $\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{*-}$ decay channel

If, in place of a pseudo-scalar, we consider decay channels with vector meson
emission, the corresponding decay width becomes:

$\displaystyle\begin{split}\Gamma_{Tree}(\overline{B}^{0}_{d}\to
D^{\ast+}V^{-})=&6\pi^{2}C\left|a_{1}(D^{*}V)\right|^{2}\,(f_{V}\,|V_{q_{1}q_{2}}|)^{2}\\\
\times
m_{V}^{2}\,m_{D^{*}}&\,m_{B}|\vec{p}_{D^{*}}(m_{V}^{2})|FF_{D^{*}V}^{2}(m_{V}^{2})\end{split}$
(62)

with:

$\displaystyle\begin{split}FF_{D^{*}V}^{2}(m_{V}^{2})=A_{1}^{2}(m_{V}^{2})&\frac{(1+r)^{2}}{r}\left(3+\frac{m_{B}^{2}|\vec{p}_{D^{*}}(m_{V}^{2})|^{2}}{m^{2}_{V}m^{2}_{D^{*}}}\right)+\\\
\frac{4r}{(1+r)^{2}}\frac{|\vec{p}_{D^{*}}(m_{V}^{2})|^{2}}{m_{D^{*}}^{2}}&\left(V^{2}(m_{V}^{2})+A_{2}^{2}(m_{V}^{2})\frac{m_{B}^{2}|\vec{p}_{D^{*}}(m_{V}^{2})|^{2}}{2m^{2}_{V}m^{2}_{D^{*}}}\right)\\\
-4A_{1}(m_{V}^{2})&\,A_{2}(m_{V}^{2})\frac{m_{B}^{2}|\vec{p}_{D^{*}}(m_{V}^{2})|^{3}}{m^{2}_{V}m^{3}_{D^{*}}}\end{split}$
(63)

in which $r=m_{D^{*}}/m_{B}$.

The measured branching fractions for the corresponding decay channels are
listed in Table 8.

decay channel |  PDG [14] |  HFLAV [11] | our average
---|---|---|---
${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{*-})\times 10^{3}$ | $17.7\pm 1.4$ | $18.7\pm 1.5$ | $17.8\pm 1.3$
${\cal B}({B^{-}}\to{D}^{\ast 0}D_{s}^{*-})\times 10^{3}$ | $17.1\pm 2.4$ | $17.9\pm 2.7$ | $17.6\pm 2.4$
Table 8:  Branching fractions for $\overline{{B}}$ mesons decaying into a
$D^{*}$ and a strange charm vector meson, obtained in PDG 2022, HFLAV 2022 and
in our analysis.

Assuming that their partial decay widths are equal, we combine results for
charged and neutral $B$-mesons to evaluate:

${\cal B}(\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{*-})=(17.5\pm 1.1)\times
10^{-3}.$ (64)

####  4.4.1 Penguin contribution

It is evaluated by considering the ratio of
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{*-})\right|$ and
$\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}K^{-})\right|$:

$\displaystyle\frac{\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}V^{-})\right|}{\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}K^{-})\right|}$
$\displaystyle=$ $\displaystyle 2\,\sqrt{\frac{{\cal
B}(\overline{B}^{0}_{d}\to D^{\ast+}V^{-})}{{\cal B}(\overline{B}^{0}_{d}\to
D^{\ast+}K^{-})}\frac{m_{B}\,|\vec{p}_{D^{*}}(m_{K}^{2})|^{3}}{|\vec{p}_{D^{*}}(m_{V}^{2})|\,m_{V}^{2}\,m_{D^{*}}}}$
(65) $\displaystyle\times$ $\displaystyle\frac{f_{K}\,\left|{\rm
V}_{us}\right|}{f_{V}\,|V_{q_{1}q_{2}}|}\,\frac{A_{0}(m_{K}^{2})}{FF_{D^{*}V}(m_{V}^{2})}$
$\displaystyle=$ $\displaystyle 0.838\pm 0.042.$

####  4.4.2
$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{*-})\right|$
evaluation

We evaluate
$\left|a_{1,\,eff.}((\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{*-}))\right|$
using the semileptonic decay channel, as in Section 2.2:

$\left|a_{1,\,eff.}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{*-})\right|=0.729\pm
0.033$ (66)

in which the dominant sources of uncertainty are : $\pm 0.023$ from the ${\cal
B}(\overline{B}^{0}_{d}\to D^{\ast+}D_{s}^{*-})$ measurement, $\pm 0.015$ from
the CLN parameterization of semileptonic form factors, $\pm 0.008$ from
$f_{D_{s}}\,|V_{cs}|$ determination, and $\pm 0.017$ from the
$f_{D_{s}^{*}}/f_{D_{s}}$ estimate.

After extraction of the penguin contribution, expected from theory, one
obtains:

$\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}D_{s}^{*-})\right|=(0.729\pm
0.033)/0.962=0.758\pm 0.034.$ (67)

This value differs by more than three standard deviations from :
$\left|a_{1}(\overline{B}^{0}_{d},\,D^{\ast+}K^{-})\right|=0.870\pm 0.027$.

##  5 Comparison between measurements and theoretical expectations

| Huber[2] | this work
---|---|---
$|a_{1,eff.}(\overline{B}^{0}_{d},D^{+}\pi^{-})|$ | $0.89\pm 0.05$ | $0.846\pm 0.019$
$|a_{1,eff.}(\overline{B}^{0}_{d},D^{*+}\pi^{-})|$ | $0.96\pm 0.03$ | $0.854\pm 0.021$
$|a_{1,eff}.(\overline{B}^{0}_{d},D^{+}K)|$ | $0.87\pm 0.06$ | $0.877\pm 0.023$
$|a_{1,eff}.(\overline{B}^{0}_{d},D^{*+}K)|$ | $0.97\pm 0.04$ | $0.870\pm 0.027$
Table 9:  Comparison between the different determinations of the $|a_{1}|$
parameter obtained from experimental measurements in [2] (2016) and in our
analysis (2022).

###  5.1 Values for $|a_{1}|$ with an emitted light meson

The statement of factorisation as a qualitative one -i.e. that $a_{1}$ is not
very far from $1$\- is reasonable, but it does not have the status of a
definite quantitative estimation, and theory is unable to provide it.

The values we obtain from experimental data,
$|a_{1,eff.}(\overline{B}^{0}_{d},D^{+}K^{-})|=0.877\pm 0.023$ and
$|a_{1,eff.}(\overline{B}^{0}_{d},D^{*+}K^{-})|=0.870\pm 0.027$ are, on the
other hand, significantly lower than the theoretical expectation in the
infinite $m_{Q}$ limit:
$a_{1}^{\infty,th.}(\overline{B}^{0}_{d},D^{+}K^{-})=1.055\pm 0.018$ at NLO or
$a_{1}^{\infty,th.}(\overline{B}^{0}_{d},D^{*+}K^{-})=1.070\pm 0.011$ at NNLO.
The difference amounts to seven standard deviations and therefore is very well
established.

There is no problem in observing a difference, since of course we are not in
the theoretical limit : there should be $1/m_{Q}$ corrections.

The real problem is twofold :

1) As in most processes of $B$ physics except the simplest ones, there does
not exist presently a tractable, safe and systematic theoretical method to
evaluate exact amplitudes from QCD. The only one in principle is lattice QCD,
but it is far outside our present capacities for these NL decays.

2) Various evaluations of the $1/m_{Q}$ corrections to the known
$m_{Q}\to\infty$ limit by approximate methods have been proposed, but they
suffer from serious doubts. In fact they disagree between themselves and they
are far from the observed values.

The $1/m_{Q}$ corrections have been considered by BBNS in the framework of
semi-perturbative QCD methods involving distribution functions [1] and the
authors believed them to be small. There would remain however to include
${\cal O}(1/m_{Q}^{n})$ corrections (with $n>1$). Moreover, as stressed by the
same authors themselves, all these are controlled by soft gluon effects, which
are beyond a rigourous treatment. The attempt of [3], making iterated use of
LCSR, concludes to a very small $1/m_{Q}$ spectator correction, contradicting
strongly experimental data, but we think that the conclusion cannot be trusted
because of the very large uncertainties which must be expected in this method
(see the indications of the end of Section 2.1.1 for a further discussion, and
in particular the reference to the more recent paper of Piscopo and Rusov
[5]).

Therefore we consider that there is presently no compelling reason to invoke
New Physics effects to explain such differences as is done in [3]; present
uncertainties on ${\cal O}(1/m_{Q}^{n})$ corrections leave room for a standard
explanation.

Such differences and discrepancies with observation must not cast doubt on the
whole theory. In our opinion, it means only that these approaches to $1/m_{Q}$
corrections are doubtful, which is not surprising in view of the many
approximations involved, in contrast with the rigorous treatment of the
$m_{Q}\to\infty$ limit.

In the case of the BBNS analysis, as we have already noticed, it is recognised
by the authors themselves below their Eq. (28), as concerns the spectator
contribution : it is noted there that the $\alpha_{s}(k^{2})$ factor for the
perturbatively treated gluon cannot be trusted since $k$ is small. The
uncertainty cannot be estimated. On the other hand, as to Bordone et al. [3],
LCSR are commonly recognised to be rather uncertain. This is implicitly
acknowledged by the authors as they include a factor $10$ of uncertainty.

###  5.2 Values for $|a_{1}|$ with an emitted heavy meson

We have found that factorisation still applies to a reasonable degree of
approximation when the emitted meson is a charmed one. That is, $a_{1,eff}$ is
found experimentally to be not too far from $1$, all the more when the penguin
contributions are subtracted, as is necessary. We must stress that this is not
what was expected from the theoretical statement in the heavy quark limit,
since it applies only to light meson emission (see, in this respect, section
3.5.1 of BBNS). In the present case, one must emphasize that the main
statement, namely that the factorisable contribution dominates in the heavy
quark limit, is no longer guaranteed because this contribution is now
suppressed by a power of $1/m_{Q}$ with respect to the estimate for light
meson emission. Indeed, in the limit $m_{Q}\to\infty$ with $m_{c}/m_{b}$
fixed, one still gets $F_{0}\to\mathrm{cste}$ at $q^{2}=m^{2}_{D}$, but
$f_{D}\propto(m_{c})^{-1/2}$ (if, on the other hand, we had kept $m_{c}$
fixed, we would have obtained $F_{0}\to(m_{b})^{-3/2}$).

##  6 Conclusions

We have seen that, while the $a_{1}$ evaluations obtained in the
$m_{Q}\to\infty$ limit have reached the $1\%$ accuracy, they differ from
experimental values, obtained with a similar accuracy, by about $20\%.$ In
fact, the progresses made both in the experimental measurements and in the
theoretical evaluation of the asymptotic limit have created a gap between them
which is so large that it implies the necessity of a safe evaluation of the
$1/m_{Q}$ corrections. Let us stress indeed that this gap is observed in all
considered channels, whether the process is purely of the tree topology type
or exchange terms are also present; this fact seemingly shows that the
exchange contributions are small. But now, as we argued in the preceding
section, such a safe evaluation is lacking: the two proposed evaluations, by
BBNS [1] and by Bordone et al. [3] seem quite unsatisfactory. Therefore also,
one should not base “Precision physics” - for instance the precise evaluation
of $f_{s}/f_{d}$ \- on theoretical estimates of $\overline{B}^{0}\to
D^{(*)+}P^{-}$ non leptonic decays.

On the other hand, one can formulate some qualitative, hypothetical,
statements on the basis of observation of the presently available values of
$a_{1}$.

1) The differences are similar whether a $D$ or a $D^{*}$ is produced, as
expected in the $m_{Q}\to\infty$ limit.

2) Similarly, some approximate $SU(3)$ symmetry properties appear. One can
reasonably suspect that $SU(3)$ breaking in the ratios of various $a_{1}$
factors is moderate, and smaller than the departure from $1$ of the $a_{1}$’s
themselves, but it is not a quantitative statement.

It has been advocated that predicted ratios between $a_{1}$ values obtained in
that limit can be more reliable than absolute $a_{1}$ values. As a result,
some sort of universality of these corrections has been invoked to use them to
relate $\overline{B}^{0}_{d}\to D^{(*)+}K^{-}$ and $\overline{B}^{0}_{s}\to
D_{s}^{(*)+}\pi^{-}$ decay channels and to consider that
$a_{1}(\overline{B}^{0}_{d},D^{(*)+}K^{-})=a_{1}(\overline{B}^{0}_{s},D_{s}^{(*)+}\pi^{-})$.
This equality is used in one of the method to obtain $f_{s}/f_{d}$ at LHC.
Meanwhile, at present, there is no theoretical proof, nor experimental
control, that such an equality is valid. We advocate to use the direct
measurement of $a_{1}(\overline{B}^{0}_{s},D_{s}^{(*)+}\pi^{-})$ using
exclusive nonleptonic and semileptonic decays to test this hypothesis.

Finally we want to stress the unexpected conclusion that charmed meson
emission obeys also factorisation with a coefficient quite similar to class I
light meson emission, i.e. not too far from $1$. This is a purely empirical
fact, not expected from BBNS asymptotic theory, which suggests that
factorisation should not extend to heavy meson emission.

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