\bar q \, i \, \overleftarrow{D}_{\mu} \, b  & \to & D_0^{(v)}(\mu) \, \overline{m}_b(\mu) \,\,  \bar q \, \gamma_{\mu} \, h_v
+ D_1^{(v)}(\mu) \, \overline{m}_b(\mu) \,\,  \bar q \, v_{\mu} \, h_v
+  \bar q \, i \, \overleftarrow{D}_{\mu} \, h_v +  ... \,,
\end{eqnarray}

we can readily derive an improved Isgur-Wise relation between the semileptonic bottom-meson decay form factors
in the small recoil region [156, 158]

\begin{eqnarray}
\frac{f_{B_{q^{\prime}} M}^{T}(q^2)} {f_{B_{q^{\prime}} M}^{+}(q^2)}
= \frac{m_{B_{q^{\prime}}}  (m_{B_{q^{\prime}}} + m_M) }{q^2}
\left [{\cal C}_{\rm T+}(\mu) + \left ( {2 \over m_{B_q^{\prime}}} \right )
\frac{\delta_{+}(q^2)}{f_{B_{q^{\prime}} M}^{+}(q^2) } \right ]
+ {\cal O} \left ( \left ({\Lambda_{\rm QCD} \over m_b} \right )^2 \right ),
\hspace{0.8 cm}
\label{soft-pion relation of fplus and fT}
\end{eqnarray}

The short-distance matching function ${\cal C}_{\rm T+}$ can be evidently expressed in terms of the Wilson coefficients
of the HQET currents [157]

\begin{eqnarray}
{\cal C}_{\rm T+}(\mu) = \left [ 1 + \frac{2 \, D_0^{(v)}(\mu)}{C_0^{(v)}(\mu)}  \right ] \,
\frac{\overline{m}_b(\mu)} {m_{B_{q^{\prime}}}} \,,
\end{eqnarray}

where the analytical expressions of $C_0^{(v)}$ and $D_0^{(v)}$ at the  one-loop accuracy
can be written as

\begin{eqnarray}
C_0^{(v)}(\mu) &=&  1 - {\alpha_s(\mu) \, C_F \over 4 \, \pi} \, \left ( 3 \, \ln {\mu \over m_b}  + 4 \right )
+ {\cal O} (\alpha_s^2) \,,
\nonumber \\
D_0^{(v)}(\mu) &=& 0 +   {\alpha_s(\mu) \, C_F \over 4 \, \pi} \, \left ( 2 \, \ln {\mu \over m_b}  + 2 \right )
+ {\cal O} (\alpha_s^2) \,.
\end{eqnarray}

Applying further the HQET equation of motion for the effective field $h_v$ enables us to
derive an important constraint between the two subleading form factors [156]

\begin{eqnarray}
(m_{B_{q^{\prime}}} + v \cdot p) \, \delta_{+}(q^2)
+ (m_{B_{q^{\prime}}} - v \cdot p) \, \delta_{-}(q^2)  = 0   \,.
\end{eqnarray}

Evaluating the effective matrix element (<ref>)
with the aid of the heavy-hadron chiral perturbation theory at the lowest order
in the $v \cdot p / \Lambda_{\rm CSB}$ expansion
(with the notation $\Lambda_{\rm CSB}$ characterizing the chiral-symmetry-breaking scale)
leads to the model-independent prediction

\begin{eqnarray}
\delta_{+}(q^2)  - \delta_{-}(q^2) =
\frac{2 \,  \bar \Lambda \, f_{B_{q^{\prime}}} } {3 \, f_M}  \,
\left ( \frac{\hat{g}_{\rm eff}}{ 1 - q^2 /  m_{B_q^{\ast}}^2} \right )  \,,
\end{eqnarray}

where the hadronic parameter $\bar \Lambda$ stands for the “effective mass" of the bottom-meson state
as previously defined in (<ref>).
It is then straightforward to determine the desired soft function $\delta_{+}(q^2)$
dictating the considered form-factor ratio (<ref>)

\begin{eqnarray}
\delta_{+}(q^2)  =
\frac{\bar \Lambda \, f_{B_{q^{\prime}}} } {3 \, f_M}  \,
\left ( { m_{B_q^{\prime}}^2  + q^2 \over 2 \, m_{B_q^{\prime}}^2} \right ) \,
\left ( \frac{\hat{g}_{\rm eff}}{ 1 - q^2 /  m_{B_q^{\ast}}^2} \right )  \,.
\end{eqnarray}

We present the yielding theory predictions for the very form-factor ratio
$\left [  q^2  \, f_{B_{q^{\prime}} M}^{T}(q^2) \right ]
: \left [ m_{B_{q^{\prime}}}  (m_{B_{q^{\prime}}} + m_M) \,  f_{B_{q^{\prime}} M}^{+}(q^2) \right ]$
from fitting the BCL $z$-series expansion against the LCSR and lattice QCD data points
in the lower-recoil region in Figure <ref>,
where the theoretical expectations from  the improved Isgur-Wise relation (<ref>)
in the soft final-state meson approximation  in virtue of the heavy quark spin symmetry
are further  shown for a numerical comparison.

The low-recoil theory predictions for the intriguing form-factor ratio $\left [  q^2  \, f_{B_{q^{\prime}} M}^{T}(q^2) \right ]
: \left [ m_{B_{q^{\prime}}} \,  (m_{B_{q^{\prime}}} + m_M) \,  f_{B_{q^{\prime}} M}^{+}(q^2) \right ]$
for the exclusive semileptonic $B_{d, s}  \to \pi, K$  decay processes  determined
from the combined BCL $z$-series expansion fitting against both the LCSR and lattice QCD data points
(blue bands) and from the theoretical expectations of the improved Isgur-Wise relation (<ref>)
at the NLP accuracy due to the combination of heavy quark and chiral symmetries (grey bands) [156, 157].

Subsequently, we  proceed to carry out the combined BCL $z$-expansion fitting of the semileptonic $B \to K$ form factors
against the newly determined LCSR data points at three representative values of
$q^2= \left \{-4.0, \, 0, \, 4.0  \right \} \, {\rm GeV^2}$,  in combination with  the small-recoil lattice QCD results
achieved with the three-flavor gauge-field ensembles  generated by the MILC Collaboration  [14]
(employing further the Sheikholeslami-Wohlert (SW) action with the Fermilab interpretation for the bottom quark)
as well as obtained for the first time with $N_f=2+1+1$ gluon field ensembles [15]
(in the meanwhile adopting the highly improved staggered quark (HISQ) formalism for all valence and sea quarks
developed by the HPQCD Collaboration).
Since neither of these lattice collaborations provide explicitly the resulting  physical data points
for the three exclusive $B \to K$ form factors in their publications,
we are then required to generate the  correlated synthetic data points  in the kinematic region
$16.8  \, {\rm GeV}^2 \leq q^2 \leq  22.9 \, {\rm GeV}^2$ from their BCL expansion  fit results
with the truncation $N=3$ as documented in [14, 15].
We remark in passing that the HPQCD Collaboration actually adopted the so-called “modified $z$-expansion" strategy
by simultaneously extrapolate the obtained lattice simulation data to the physical light-quark masses
and zero lattice spacing, and to interpolate their lattice results in the momentum transfer
(see [16] for more technical discussions on this alternative approach).
Implementing now the binned $\chi^2$ fit for both the LCSR and lattice simulation results
with the standard BCL parametrization (<ref>) and
(<ref>)  leads to our final numerical predictions
of the eight form-factor parameters (with their correlation matrix)
indispensable for the theory description of the electroweak penguin $B \to K \ell \bar \ell$ decays [108, 109] as summarized in Table <ref>.
We further observe that our BCL expansion fit brings about a minimal $\chi^2=14.12$
for $16$ degrees of freedom in the fitting procedure, thus corresponding to
an encouraging $p$-value of $59 \, \%$ numerically.
Adopting the tabulated $z$-series coefficients $b_k^{+, 0, T}$ allows us to predict
the desired momentum-transfer dependence for the three semileptonic  $B \to K$ form factors
versus  $z$ (left panel) and versus $q^2$ (right panel) in the entire kinematic region
in Figure <ref>,
where we also display the corresponding theory predictions from performing an independent
$z$-series fitting to the “only lattice QCD" data points [14, 15]
exclusively for the convenience.
It turns out that  the rather remarkable precision for the whole lattice data points
at high momentum transfer from both the FNAL/MILC Collaboration  [14]
(with the total uncertainties, including both statistical and systematic errors,
less than $4.0 \, \%$ for all the three $B \to K$ form factors)
and the HPQCD Collaboration [15]
(with the uncertainties below $\{ 2.0 \, \%, 4.0 \, \%,  5.5 \, \% \}$ for
the three form factors $\{f_{B K}^{+}, \,  f_{B K}^{0},  \, f_{B K}^{T} \}$ in consequence)
makes it  challenging to carry out the combined BCL $z$-series fit with high quantity,
by simultaneously  accommodating  the achieved LCSR predictions at low momentum transfer
within the individual $1.0 \, \sigma$ intervals
(albeit with the very sizeable theory uncertainties of the order of $50 \, \%$).
Actually, this intriguing observation  can be attributed to the very fact that
our updated LCSR computations with the bottom-meson distribution amplitudes will bring about
the strongly correlated numerical results for the exclusive $B \to K$ form factors
at the different kinematic points  (despite of the quite uncertain central values as explained above)
in the large hadronic recoil region, which  exhibit the delicacy tension with
the extrapolated lattice QCD predictions with the extraordinary high accuracy in the low recoil region.
In this respect, it would be in high demand to deepen further our understanding,
on the one hand,  towards the momentum dependence of the HQET bottom-meson distribution amplitudes
$\phi_B^{\pm}(\omega, \mu_0)$ at the renormalization scale $\mu_0=1.0 \, {\rm GeV}$
from the first field-theoretical principles on the bottom-meson LCSR aspect,
and on the other hand, towards the unquantified systematic uncertainties of the lattice simulation method
(for instance, several potential concerns with the “modified $z$-expansion" proposal
as previously discussed in [16]).

Furthermore, we confront the combined BCL expansion fit results for the two particular form-factor ratios
(<ref>) and   (<ref>)
at high momentum transfer with the counterpart model-independent  predictions from
the combination of the heavy quark spin symmetry and the current algebra technique

in Figures <ref>
and <ref>.
Generally, the resulting BCL $z$-fit predictions for the considered low-recoil symmetry breaking corrections
appear to be in reasonable agreement with the theoretical expectations from the HQET symmetry relations
at the unphysical kinematic point $q^2= 27.9 \, {\rm GeV^2}$ within the obtained uncertainties.
By contrast, our BCL fit result for the scalar form-factor ratio at the zero-recoil limit
differs from the counterpart prediction with the heavy quark symmetry and the soft-kaon approximation
by an enormous amount of ${\cal O}(50 \, \%)$,
thus confirming the previous lattice simulation results  from the  FNAL/MILC Collaboration [14].
We are then led to conclude immediately that employing  the derived low-recoil symmetry relations
for the exclusive $B \to K \ell \bar \ell$ phenomenological applications
could result in the substantial derivations from the direct QCD predictions
due to the numerically pronounced NLP corrections in the heavy quark expansion.

2|c||$B \to K$   Form Factors      8|c|Correlation Matrix

Parameters     Values     $b_0^+$     $b_1^+$     $b_2^+$     $b_0^0$     $b_1^0$     $b_0^T$     $b_1^T$     $b_2^T$

$b_0^+$     $0.465(8)$     $1$     $-0.035$     $-0.494$     $0.711$     $0.283$     $0.744$     $0.017$     $-0.281$

$b_1^+$     $-0.925(53)$           $1$     $-0.034$     $0.160$     $0.831$     $-0.111$     $0.691$     $-0.056$

$b_2^+$     $-0.355(257)$               $1$     $-0.038$     $0.166$     $-0.299$     $-0.099$     $0.533$

$b_0^0$     $0.290(4)$                    $1$     $0.422$     $0.594$     $0.061$     $-0.015$

$b_1^0$     $0.246(38)$                         $1$     $0.141$     $0.595$     $0.14$

$b_0^T$     $0.479(10)$                              $1$     $-0.023$     $-0.283$

$b_1^T$     $-0.759(74)$                                    $1$     $0.305$

$b_2^T$     $-0.479(324)$                                         $1$

Theory predictions for the correlated $z$-series coefficients
in the vector, scalar and tensor $B \to K$ form factors
determined by fitting the BGL parametrization simultaneously
against our LCSR results including a variety of the subleading-power corrections
and the available  lattice QCD data points from [14, 15]
with the preferred truncation $N=3$.

Theory predictions for the complete set of the semileptonic $B \to K$ form factors
versus  $z$ (left panel) and versus $q^2$ (right panel) in the entire kinematic region
obtained by carrying out the combined BCL $z$-fit of the updated LCSR (from this work)
and lattice simulation (from [14, 15]) data points.
We further display the yielding numerical results of these essential form factors by performing an independent $z$-series fit
against the “only lattice QCD" data points [14, 15]
exclusively for an instructive  comparison.

Along the same vein, we will continue to perform the combined BCL expansion fitting of
the semileptonic $B_s \to K$ form factors against  our improved LCSR predictions
and the available lattice results for the vector and scalar form factors
[4, 8, 11] simultaneously.
On account of the very absence of the lattice simulation results for the tensor form factor,
we prefer to take advantage of the determined  LCSR data points for $f_{B_s K}^{T}(q^2)$
at five representative values of
$q^2 = \left \{-8.0, \, -4.0, \, 0, \, 4.0, \, 8.0   \right \} \, {\rm GeV^2}$,
while adopting  the large-recoil LCSR  predictions for  the vector and scalar form factors $f_{B_s K}^{+, \, 0}(q^2)$
at three distinct kinematic points of $q^2 = \left \{-4.0, \, 0, \, 4.0  \right \} \, {\rm GeV^2}$
as the same as before.
While the RBC/UKQCD Collaboration [4] provides explicitly the synthetic data points of
$f_{B_s K}^{+}(q^2)$ and $f_{B_s K}^{0}(q^2)$
at three representative values of $q^2 = \left \{17.6, \, 20.8, \, 23.4  \right \} \, {\rm GeV^2}$
with the normalized statistical and systematic correlation matrices,
both the HPQCD Collaboration [8] and the FNAL/MILC Collaboration [11]
only provide their BCL $z$-fit results for the form-factor shape parameters with the truncations
$N=3$ and $N=4$,  respectively.
Consequently, we are then required to produce the necessary lattice data points for $f_{B_s K}^{+, \, 0}(q^2)$ in the kinematic region
$17.0  \, {\rm GeV}^2 \leq q^2 \leq  23.7 \, {\rm GeV}^2$
in order to utilize the complete information of the lattice QCD fits from  [8, 11].
Performing now the combined BCL fit against both our LCSR predictions in the large recoil region
and the yielding lattice simulation data points in the low recoil region
gives rise to the final predictions for the $z$-expansion coefficients (with their correlation matrix)
for the three semileptonic $B_s \to K$ form factors as collected in Table <ref>.
In addition,  our numerical fit leads to a slightly larger number  of  $\chi^2=28.58$
for $19$ degrees of freedom in the fitting program.
Unsurprisingly, the obtained BCL fit results for the three coefficients $b_{0, 1, 2}^T$
in Table <ref>
turn out to be more uncertain compared with the corresponding predictions
for the $z$-expansion parameters of the tensor $B \to \pi$ form factor
collected in Table <ref>,
due to the unavailable lattice data points for the form factor $f_{B_s K}^{T}(q^2)$
at large momentum transfer.
Under such circumstance, achieving the lattice simulation determination
for the very tensor form factor $f_{B_s K}^{T}(q^2)$ at high $q^2$ will be evidently crucial
to pin down the current theory uncertainties from the BCL extrapolation of the LCSR results,
thus providing the fundamental ingredient for the model-independent description
of the exclusive electroweak penguin $\bar B_s \to \bar K^0 \ell \bar \ell$  decays.
Employing the tabulated $z$-series coefficients $b_k^{+, 0, T}$ further enables us
to predict the desired momentum-transfer dependence for the exclusive  $B_s \to K$ form factors
versus  $z$ (left panel) and versus $q^2$ (right panel) in the entire kinematic region
in Figure <ref>,
where we also display the alternative BCL fit results for the vector and scalar form factors
with the “only lattice QCD" data points [4, 8, 11]
In the light of the high-precision lattice data points for $f_{B_s K}^{+, \, 0}(q^2)$ at small hadronic recoil
from the RBC/UKQCD Collaboration [4] (with the total uncertainties below
$\left \{6.2 \, \%, \, 7.1 \, \% \right \}$ for the vector and scalar form factors, respectively),
from HPQCD Collaboration [8]
(with the combined uncertainties below $\left \{5.0 \, \%, \, 6.0 \, \% \right \}$
for the two form factors $\{f_{B_s K}^{+}, \,  f_{B_s K}^{0} \}$ in consequence)
and from the FNAL/MILC Collaboration [11]
(with the uncertainties less than $3.0 \, \%$ for both the two form factors),
the resulting theory benefits  from  the combined BCL $z$-expansion fit
to both the LCSR and lattice simulation results  consist in
the rather moderate improvements (at the level of ${\cal O}(20 \, \%)$ numerically)
of the large-recoil form factor predictions  on the counterpart BCL fitting procedure
with the “only lattice QCD" data points.

We proceed to compare the combined BCL expansion fitting predictions for the two form-factor ratios
(<ref>) and   (<ref>)
in the low recoil region  with the corresponding model-independent computations based upon
the combination of heavy quark and chiral symmetries
in Figures <ref>
and <ref>.
We can draw an analogous conclusion (as previously observed in the context of the $B \to K$  form factors)
that the derived  HQET symmetry relations  for the exclusive $B_s \to K$ form factors appear to be well respected
at the unphysical  kinematic point $q^2=28.8 \, {\rm GeV^2}$ within the theory uncertainties.
Apparently, our numerical result for the particular form-factor ratio
$\left [  q^2  \, f_{B_s K}^{T}(q^2) \right ]
: \left [ m_{B_s}  (m_{B_{s}} + m_K) \,  f_{B_s K}^{+}(q^2) \right ]$
suffers from the more pronounced theory uncertainty as displayed
in Figure <ref>,
due to the relatively less precise BCL-fitting  prediction for the tensor form factor $f_{B_s K}^{+}(q^2)$.
Moreover, the very low-recoil symmetry breaking correction to the scalar form-factor ratio
$\left [ \left (1 - q^2 /  m_{B^{\ast}}^2 \right ) \, f_{B_{s} K}^{+}(q^2) \right ]  :
\left [ f_{B_{s} K}^{0}(q^2) \right ]$  at the maximal momentum transfer
turns out to be even greater than the counterpart theory predictions
for both the semileptonic $B \to \pi, K$ decay form-factor ratios
as shown in  Figure <ref>.

2|c||$B_s \to K$   Form Factors      8|c|Correlation Matrix

Parameters     Values     $b_0^+$     $b_1^+$     $b_2^+$     $b_0^0$     $b_1^0$     $b_0^T$     $b_1^T$     $b_2^T$

$b_0^+$     $0.373(10)$     $1$     $0.202$     $-0.294$     $0.653$     $0.135$     $0.422$     $-0.277$     $0.103$

$b_1^+$     $-0.731(41)$           $1$     $0.075$     $0.255$     $0.683$     $0.488$     $-0.301$     $0.092$

$b_2^+$     $-0.473(146)$               $1$     $0.099$     $0.366$     $0.215$     $-0.121$     $0.025$

$b_0^0$     $0.443(10)$                    $1$     $-0.024$     $0.407$     $-0.268$     $0.100$

$b_1^0$     $-1.427(45)$                         $1$     $0.433$     $-0.262$     $0.075$

$b_0^T$     $0.437(46)$                              $1$     $-0.835$     $0.550$

$b_1^T$     $-0.900(167)$                                    $1$     $-0.914$

$b_2^T$     $0.091(172)$                                         $1$

Theory predictions for the correlated $z$-series coefficients
in the vector, scalar and tensor $B_s \to K$ form factors
determined by fitting the BGL parametrization simultaneously
against our LCSR results including a variety of the subleading-power corrections
and the available  lattice QCD data points from
[4, 8, 11]
with the preferred truncation $N=3$.

Theory predictions for the complete set of the semileptonic $B_s \to K$ form factors
versus  $z$ (left panel) and versus $q^2$ (right panel) in the entire kinematic region
obtained by carrying out the combined BCL $z$-fit of the updated LCSR (from this work)
and lattice simulation (merely for the two form factors $f_{B_s K}^{+, \, 0}(q^2)$
from [4, 8, 11]) data points.
We further display the yielding numerical results for the vector and scalar  form factors
by performing an independent $z$-series fit against the
“only lattice QCD" data points [4, 8, 11]
exclusively for the illustration purpose.

§.§ Phenomenological analysis of the   $B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$ observables

Having at our disposal the combined BCL $z$-fit results  for the exclusive $B \to \pi, K$ form factors,
we are now prepared to explore their phenomenological implications on the semileptonic
$B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$ decay observables
constructed from the corresponding full angular distributions,
such as  the differential branching fractions, the normalized forward-backward asymmetries,
the new-physics (NP) sensitive “flat terms" (vanishing in the massless lepton limit in the SM),
and the lepton-flavour universality ratios, the lepton polarization asymmetries.
In particular,  the ever-increasing precision measurements
on the binned $q^2$ distributions  for  the golden exclusive  $B \to \pi \ell \bar \nu_{\ell}$
(with $\ell=e, \, \mu$) decay processes from the BaBar Collaboration [63, 64],
the Belle Collaboration [65, 66]
as well as the  Belle II [67] Collaboration enable us to
further extract the desired CKM matrix element $|V_{ub}|$ straightforwardly
in combination with  our improved determination of the vector form factor $f_{B \pi}^{+}(q^2)$
in the entire kinematic regime.
In order to achieve this goal, we first present the explicit expression
for  the full differential decay distribution of $B_{q^{\prime}} \to M \ell \bar \nu_{\ell}$
with respect to the two kinematic variables $q^2$ and $\cos \theta_{\ell}$
(dropping out the very intricate but numerically subdominant electromagnetic correction)

\begin{eqnarray}
\frac{d^2 \Gamma(B_{q^{\prime}} \to M \ell \bar \nu_{\ell})}
{d q^2 \, d \cos \theta_{\ell} } = a_{\theta_{\ell}}(q^2)
+ b_{\theta_{\ell}}(q^2) \, \cos \theta_{\ell}
+  c_{\theta_{\ell}}(q^2) \, \cos^2 \theta_{\ell}  \,,
\label{angular distribution of B to M l nu}
\end{eqnarray}

where the three $q^2$-dependent angular coefficient functions are given by [159]

\begin{eqnarray}
a_{\theta_{\ell}}(q^2)  &=& {\cal N}_{\rm ew} \,\,  \lambda^{3/2} \,
\left ( 1 - {m_{\ell}^2 \over q^2} \right  )^2 \,
\left [ \left | f_{B_{q^{\prime}} M}^{+}(q^2) \right |^2
+ {1 \over \lambda} \,\, {m_{\ell}^2 \over q^2} \,  \left ( 1 - { m_M^2  \over m_{B_{q^{\prime}} }^2 } \right )^2  \,
\left | f_{B_{q^{\prime}} M}^{0}(q^2) \right |^2 \right ] \,,
\hspace{1.0 cm}
\\
b_{\theta_{\ell}}(q^2)  &=&  2 \, {\cal N}_{\rm ew} \,  \lambda \, \left ( 1 - {m_{\ell}^2 \over q^2} \right  )^2 \,
{m_{\ell}^2 \over q^2} \,  \left ( 1 - { m_M^2  \over m_{B_{q^{\prime}} }^2 } \right )  \,
{\rm Re} \left [ f_{B_{q^{\prime}} M}^{+}(q^2) \, f_{B_{q^{\prime}} M}^{0 \, \ast}(q^2) \right ] \,,
\\
c_{\theta_{\ell}}(q^2)  &=& -  {\cal N}_{\rm ew} \,\,  \lambda^{3/2} \,
\left ( 1 - {m_{\ell}^2 \over q^2} \right  )^3 \, \left | f_{B_{q^{\prime}} M}^{+}(q^2) \right |^2 \,,
\end{eqnarray}

and we have introduced the following shorthand notations for convenience

\begin{eqnarray}
&& {\cal N}_{\rm ew} = \frac{G_F^2 \, |V_{ub}|^2 \,  m_{B_{q^{\prime}}}^3 }{256 \, \pi^3} \,,
\qquad
\lambda \equiv \lambda \left (1, \, {m_M^2 \over m_{B_{q^{\prime}}}^2}, \, {q^2 \over m_{B_{q^{\prime}}}^2} \right )\,,
\nonumber \\
&& \lambda(a, b, c) \equiv  a^2 + b^2 + c^2 - 2 \, (ab + ac +bc) \,.
\end{eqnarray}

In addition,  the helicity angle $\theta_{\ell}$ is defined as the angle between the $\ell^{-}$ direction of flight
and the final-state meson  momentum in the dilepton rest frame.
We can immediately observe two interesting algebra relations for the angular functions
$b_{\theta_{\ell}}(q^2) =0$ and  $a_{\theta_{\ell}}(q^2) + c_{\theta_{\ell}}(q^2) =0$
in the massless lepton limit.

0.80 !

Parameters      Values     6|c|Correlation Matrix

$b_0^+$     $0.409(12)$     $1$     $0.111$     $-0.452$     $0.298$     $0.119$     $-0.87$

$b_1^+$     $-0.507(42)$          $1$     $-0.793$     $-0.088$     $0.203$     $-0.298$

$b_2^+$     $-0.267(152)$               $1$     $0.008$     $-0.096$     $0.405$

$b_0^0$     $0.507(18)$                    $1$     $-0.445$     $-0.266$

$b_1^0$     $-1.446(45)$                        $1$     $-0.201$

$|V_{ub}|$     $3.76(13) \times 10^{-3}$                             $1$

Theory predictions for the  $B \to \pi \ell \bar \nu_{\ell}$
form-factor shape parameters and the CKM matrix element $|V_{ub}|$
(with their correlation matrix) from carrying out the simultaneous fit
against the SCET sum rules, lattice QCD and experimental data points
with the aid of the truncated BCL $z$-parameterizations at $N=3$.

Integrating over the helicity angle $\theta_{\ell}$ allows for spelling out the expression
for the differential decay rate of $B_{q^{\prime}} \to M \ell \bar \nu_{\ell}$
in the bottom-meson rest frame

\begin{eqnarray}
\frac{d \Gamma(B_{q^{\prime}} \to M \ell \bar \nu_{\ell})}
{d q^2 } &=& \int_{-1}^{1} d \cos \theta_{\ell}  \,\,
\frac{d^2 \Gamma(B_{q^{\prime}} \to M \ell \bar \nu_{\ell})}
{d q^2 \, d \cos \theta_{\ell} }
= 2 \, \left [  a_{\theta_{\ell}}(q^2) +  {1 \over 3} \, c_{\theta_{\ell}}(q^2) \right ]
\nonumber \\
&=& \frac{G_F^2 \, |V_{ub}|^2 \,  m_{B_{q^{\prime}}}^3 }{192 \, \pi^3} \, \lambda^{3/2} \,
\left ( 1 - {m_{\ell}^2 \over q^2} \right  )^2 \,
\bigg \{  \left ( 1 + {m_{\ell}^2 \over 2 \, q^2} \right ) \,
\left | f_{B_{q^{\prime}} M}^{+}(q^2) \right |^2
\nonumber \\
&&  + \, {1 \over \lambda} \,    {3 \, m_{\ell}^2 \over 2 \, q^2} \,
\left ( 1 - { m_M^2  \over m_{B_{q^{\prime}} }^2 } \right )^2  \,
\left | f_{B_{q^{\prime}} M}^{0}(q^2) \right |^2  \bigg \}  \,,
\end{eqnarray}

which can be further employed to determine the  measurable $q^2$-binned branching fractions.
Following the strategy presented in [16, 160, 161, 162],
we will turn  to  extract the magnitude of the CKM matrix element $|V_{ub}|$
by carrying out a simultaneous fit to the SCET sum rules, lattice QCD and experimental data points
with the aid of the constrained BCL $z$-series parameterizations,
thus leaving their relative normalization $|V_{ub}|$ as a free parameter.
As emphasized previously in [16],
this attractive  fitting  strategy combines the theoretical and experimental inputs in a more efficient manner,
yielding a somewhat smaller uncertainty on $|V_{ub}|$ numerically.
Taking advantage of the available state-of-the-art experimental data sets
from the three untagged measurements by the BaBar Collaboration [64, 64]
and the Belle Collaboration [65] assuming isospin symmetry,
from the two tagged measurements of $\bar B^0 \to \pi^{+} \ell \bar \nu_{\ell}$
and  $B^{-} \to \pi^{0} \ell \bar \nu_{\ell}$ by the Belle Collaboration [66],
and from the untagged  $B^0 \to \pi^{-} \ell \bar \nu_{\ell}$ measurements
by the Belle II Collaboration [67],
we display the numerical fit results for both the form-factor shape parameters
with the truncation $N=3$ for the BCL $z$-expansion and $|V_{ub}|$
in Table <ref>, including their correlation matrix.
The quality of the binned maximum-likelihood fit can be understood from
the resulting chi-square per degree of freedom
$\chi^2/{\rm dof} = 86.79 / (73-6) \approx 1.30$.
In particular, the newly achieved predictions for the five BCL parameters $b_k^{+, \, 0}$
entering in the vector and scalar $B \to \pi$ form factors
are compatible with the corresponding numerical results presented in
Table <ref>, at the $1.0 \, \sigma$ level,
from fitting against only the LCSR and lattice simulation data points.
Additionally,  the thus-far determined interval for $|V_{ub}|$ from our nominal fit model

\begin{eqnarray}
\left |V_{ub} \right |_{B \to \pi \ell \bar \nu_{\ell}} = (3.76 \pm 0.13) \times \, 10^{-3} \,,
\qquad
({\rm BCL \,\, fit \,\, with}  \,\, N=3)
\label{Vub value: N=3}
\end{eqnarray}

appears to be in excellent agreement with the counterpart numerical result
from the analogous  fitting  strategy  but with the LCSR input data points
generated by the traditional dispersive  technique
with the $\pi$-meson distribution amplitudes [160]
and from the combined BCL fit against the lattice and experimental results [16].

For the sake of  understanding quantitatively  the systematic uncertainties
from the truncations of the BCL series expansions,
we repeat our numerical fit procedure to the simultaneous  determinations of the vector and scalar form-factor shape parameters
as well as the CKM matrix element $|V_{ub}|$ with the different truncation $N=4$,
yielding the correlated numerical predictions shown in Table  <ref>.
Moreover, this particular BCL expansion fit turns out to generate a minimal $\chi^2=85.31$
for $65$ degrees of freedom, thus corresponding to the equally good  fit quantity
when compared with the former case with the truncation $N=3$.
Unsurprisingly, both the yielding  central value and theory uncertainty
for the numerical result  of $|V_{ub}|$

\begin{eqnarray}
\left |V_{ub} \right |_{B \to \pi \ell \bar \nu_{\ell}} = (3.72 \pm 0.14) \times \, 10^{-3} \,,
\qquad
({\rm BCL \,\, fit \,\, with}  \,\, N=4)
\end{eqnarray}

coincide with the previous BCL fitting results with $N=3$ perfectly.
Apparently, the combined BCL fit results for the $z$-series coefficients
of the semileptonic $B \to \pi$ form factors also stabilize at $N=3$
and do not change notably by increasing the expansion order to $N=4$.
We are therefore led to conclude that truncating the $z$-series expansions at the order $N=3$
in the numerical fit procedure  will be indeed  sufficient to provide us
the reliable and satisfactory theory predictions.

Parameters      Values     8|c|Correlation Matrix

$b_0^+$     $0.409(12)$     $1$     $0.095$     $-0.367$     $-0.028$     $0.214$     $0.075$     $0.031$     $-0.840$

$b_1^+$     $-0.477(52)$         $1$     $-0.109$     $-0.649$     $0.07$     $0.089$     $-0.075$     $-0.390$

$b_2^+$     $-0.127(211)$               $1$     $-0.565$     $0.219$     $-0.089$     $-0.202$     $0.153$

$b_3^+$     $-0.732(745)$                     $1$     $-0.295$     $0.062$     $0.295$     $0.275$

$b_0^0$     $0.506(22)$                        $1$     $-0.579$     $-0.777$     $-0.234$

$b_1^0$     $-1.341(172)$                            $1$     $0.715$     $-0.126$

$b_2^0$     $1.913(324)$                              $1$     $-0.027$

$|V_{ub}|$     $3.72(14) \times 10^{-3}$                                     $1$

Theory predictions for the  $B \to \pi \ell \bar \nu_{\ell}$
form-factor shape parameters and the CKM matrix element $|V_{ub}|$
(with their correlation matrix) from carrying out the simultaneous fit
against the SCET sum rules, lattice QCD and experimental data points
with the aid of the truncated BCL $z$-parameterizations at $N=4$.

Theory predictions for the differential $q^2$ distributions
of the exclusive semileptonic $B \to \pi  \ell \bar \nu_{\ell}$  (left panel)
and $B_s \to K \ell \bar \nu_{\ell}$ (right panel) decay  processes
in the entire kinematic region with the distinct BCL $z$-series fits of the form-factor shape parameters.
The available experimental measurements on the binned $q^2$ distributions
of the “golden"  decay process $B \to \pi  \mu \bar \nu_{\mu}$
from the BaBar [63, 64], Belle [65, 66]
and Belle II [67] Collaborations are further displayed for an exploratory comparison.

We further display our final theory predictions for the differential $q^2$ distributions
of the semileptonic $B_{(s)} \to \pi (K) \ell \bar \nu_{\ell}$ (with $\ell = \mu, \, \tau$)
decay processes  in the entire kinematic region in Figure <ref>,
where the experimental measurements of the  $B \to \pi  \ell \bar \nu_{\ell}$ decay rates
from  the BaBar [63, 64], Belle [65, 66]
and Belle II [67] Collaborations are further shown for a numerical comparison.
In addition, we collect simultaneously the obtained numerical results from  fitting the BCL $z$-series parameterizations
with  three distinct scenarios  of the input data points:
I) only synthetic lattice data points,
II) synthetic lattice data points $\oplus$ LCSR results,
III)  synthetic lattice data points $\oplus$ LCSR results $\oplus$ experimental data.
We can readily observe from Figure <ref>
that employing our improved  LCSR predictions at large hadronic recoil
in the BCL expansion  fit program  will be highly beneficial for pinning  down  the theory uncertainties
from the particular  fitting strategy with only the synthetic lattice data points.
Moreover, we discover the slight tension of the predicted large-recoil $B \to \pi  \mu \bar \nu_{\mu}$ decay distributions
between the scenarios II) and  III) fitting strategies with the BCL $z$-series parameterizations.
With regard to the counterpart exclusive $B_{s} \to K \ell \bar \nu_{\ell}$ decay channels,
taking into account the newly obtained LCSR data points in the numerical fit
will  bring about the moderate improvements on the resulting partial decay rates
determined from fitting against only the synthetic lattice data points,
as previously discussed in Section <ref>.
For convenience, we also collect here our theory predictions for the total branching fractions
of $B_{s} \to K \ell \bar \nu_{\ell}$ with the extracted interval of the CKM matrix element $|V_{ub}|$
shown in (<ref>)

\begin{eqnarray}
\mathcal{BR}(B_s \to K \mu  \bar \nu_{\mu})  &=& (1.200 \pm 0.128) \times 10^{-4} \,,
\nonumber \\
\mathcal{BR}(B_s \to K \tau \bar \nu_{\tau}) &=& (0.847 \pm 0.078)  \times 10^{-4} \,,
\end{eqnarray}

the former of which coincides well with the first experimental measurement from the LHCb Collaboration
$\mathcal{BR}(B_s \to K \mu  \bar \nu_{\mu}) =
\left [ 1.06 \pm 0.05 (\rm stat) \pm 0.08 (\rm syst)\right ]  \times 10^{-4}$  [68]
by employing the Cabibbo favored semileptonic $B_{s} \to D_s \ell \bar \nu_{\ell}$ decay process
as the normalization  channel.
Unfortunately, both the two  semitauonic bottom-meson decays
$B \to \pi \tau \bar \nu_{\tau}$ and $B_s \to K \tau \bar \nu_{\tau}$
have not been observed in the high luminosity  Belle II and LHCb experiments to date
(see however  the upper limit of  ${\cal BR} (B \to \pi \tau \bar \nu_{\tau}) < 2.5 \times 10^{-4}$
at the $90 \, \%$ confidence level from the Belle Collaboration [163]).

Observables      Lattice QCD     Lattice QCD $\oplus$ LCSR     LCSR      This work

3*${\cal R}_{\pi}$     $0.69 \pm 0.19$ [4]
$0.78 \pm 0.10$ [165]      $0.69^{+0.03}_{-0.05}$ [25]      $0.720 \pm 0.027 \big |_{N=3}$

$0.767 \pm 0.145$ [164]      $0.699 \pm 0.022$ [160]      $0.68^{+0.10}_{-0.09}$ [166]

$0.838 \pm 0.075$  [164]     $0.677 \pm 0.010$ [162]      $0.65^{+0.13}_{-0.11}$ [166]       $0.746 \pm 0.039 \big |_{N=4}$

3*${\cal R}_{K}$     $0.77 \pm 0.12$  [4]      $-$     $-$
$0.700 \pm 0.016\big |_{N=3}$

$ 0.695 \pm 0.050 $ [8]

$0.836 \pm 0.034$ [11]     $-$     $-$      $0.680 \pm 0.019\big |_{N=4}$

Theory predictions for the LFU ratios of the exclusive semileptonic
$B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$ decay processes from the combined BCL expansion fitting
against the synthetic lattice data points and the newly obtained  LCSR results.

In light of the increasing sensitivity of the semitauonic bottom-hadron decays
to the mysterious NP signature  due to the very large $\tau$-lepton  mass,
we proceed to investigate two particular lepton-flavour-universality (LFU) probing observables
for the exclusive $B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$ decays
independent of the CKM matrix element $|V_{ub}|$

\begin{eqnarray}
{\cal R}_{\pi(K)}  =  \frac{\Gamma \left (B_{(s)} \to \pi(K) \tau \bar \nu_{\tau} \right )}
{\Gamma \left (B_{(s)} \to \pi(K) \mu  \bar \nu_{\nu} \right )}
=  \frac{\int_{m_{\tau}^2}^{q_{\rm max}^2} \, d q^2 \,
{d \Gamma(B_{(s)} \to \pi(K) \tau \bar \nu_{\tau} ) / d q^2 }}
{\int_{m_{\mu}^2}^{q_{\rm max}^2} \, d q^2 \,
{d \Gamma(B_{(s)} \to \pi(K) \mu \bar \nu_{\mu} ) / d q^2 }} \,.
\end{eqnarray}

Apparently, precision predictions of such interesting LFU quantities would require a good knowledge
of both the vector and scalar form factors in the whole semileptonic regions.
Adopting the combined BCL $z$-series fit results with two distinct truncations $N \in \left \{ 3, \, 4 \right \}$
yields the desired numerical predictions for the two LFU observables
as summarized in Table <ref>.
Generally, our numerical results for ${\cal R}_{\pi(K)}$ are compatible with the previous theory determinations
based upon the lattice simulation and LCSR methods.
In addition, our BCL expansion  fit result of the LFU ratio ${\cal R}_{\pi}$
can evidently  accommodate the rather loose Belle measurement of
${\cal R}_{\pi} | _{\rm Belle \,\, 2016}= 1.05 \pm 0.51$ [163].
We further present the resulting predictions for the two LFU ratios
of the differential $B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$  decay distributions
in Figure <ref>,
which can be straightforwardly confronted with the counterpart numerical results
from the  RBC/UKQCD Collaborations [4],
HPQCD [8] and FNAL/MILC [11] Collaborations.

Theory predictions for the two interesting  LFU ratios ${\cal R}_{\pi(K)}$
of the differential $B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$  decay distributions
obtained from the combined BCL expansion fitting against
the synthetic lattice data points and the newly obtained bottom-meson LCSR results.

Applying the two-fold differential spectrum of the flavour-changing charged-current
$B_{q^{\prime}} \to M \ell \bar \nu_{\ell}$ decay process
displayed in  (<ref>),
we can  construct two angular observables sensitive to Beyond the Standard Model (BSM) physics
associated with  electroweak  symmetry breaking

\begin{eqnarray}
{\cal A}_{\rm FB}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}} (q^2)
&=&  \left [ \frac{d \Gamma(B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell})} {d q^2} \right ]^{-1} \,
\int_{-1}^{1} d \cos \theta_{\ell} \,\,  {\rm sgn}(\cos \theta_{\ell}) \,
\frac{d^2 \Gamma(B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell})}
{d q^2 \, d \cos \theta_{\ell} } \,,
\nonumber \\
&=& \left [{1 \over 2} \,  b_{\theta_{\ell}}(q^2) \right ] :
\left [ a_{\theta_{\ell}}(q^2) + {1 \over 3} \, c_{\theta_{\ell}}(q^2) \right ]
\\
{\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}} (q^2)
&=&  1 +{2 \over 3} \, \left [ \frac{d \Gamma(B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell})} {d q^2} \right ]^{-1} \,
{d^2 \over d (\cos \theta_{\ell})^2 } \, \frac{d^2 \Gamma(B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell})}
{d q^2 \, d \cos \theta_{\ell} }
\nonumber \\
&=&  \left [ a_{\theta_{\ell}}(q^2) + c_{\theta_{\ell}}(q^2) \right ] :
\left [ a_{\theta_{\ell}}(q^2) + {1 \over 3} \, c_{\theta_{\ell}}(q^2) \right ]  \,.
\hspace{-1.0 cm}
\label{definition: flat term}
\end{eqnarray}

Evidently, both the normalized forward-backward asymmetries ${\cal A}_{\rm FB}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$
and the $q^2$ differential flat terms ${\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$ [158]
will vanish in the massless lepton limit in the SM.
Another appropriate candidate for the potential  BSM probe can be introduced by investigating
the polarization asymmetry of the final-state lepton

\begin{eqnarray}
&& {\cal A}_{\lambda_{\ell}}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}} (q^2)
= \left [ \frac{d \Gamma(B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell})} {d q^2} \right ]^{-1} \,
\left [ \frac{d \Gamma^{\lambda_{\ell}= -1/2}}{d q^2}
- \frac{d \Gamma^{\lambda_{\ell}= + 1/2}}{d q^2} \right ]\left(B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell} \right )
\nonumber \\
&& =  1 - {2 \over 3} \, \left \{ \left[ 3 \,  \left ( a_{\theta_{\ell}}(q^2) + c_{\theta_{\ell}}(q^2) \right )
+ {2 \, m_{\ell}^2 \over q^2 -  m_{\ell}^2} \, c_{\theta_{\ell}}(q^2)  \right ]
: \left [ a_{\theta_{\ell}}(q^2) + {1 \over 3} \, c_{\theta_{\ell}}(q^2) \right ] \right \}  \,,
%\hspace{-1.0 cm}
\label{analytical result of polarization asymmetry}
\end{eqnarray}

which turns out to be sensitive to  helicity-violating NP interactions.
The  analytic structure of the above-mentioned expression of the lepton polarization fraction
can be actually understood from the $\ell$-helicity conservation of the semileptonic $b \to q \ell \bar \nu_{\ell}$ transition
in the massless lepton approximation in the SM.
In order to facilitate  the numerical comparisons with the future experimental measurements,
we collect our theory predictions for the three different classes of the
angular  observables ${\cal A}_{\rm FB}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}(q^2)$,
${\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}(q^2)$ and
${\cal A}_{\lambda_{\ell}}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}(q^2)$
in Figure <ref>.
It is perhaps worthwhile to mention that our predictions for the normalized differential  forward-backward asymmetries
are  in excellent agreement with  the available lattice QCD simulation results from both the RBC/UKQCD [4]
and  HPQCD (without  $B  \to \pi \ell \bar \nu_{\ell}$) [8] Collaborations.
On the other hand, the resulting predictions for the lepton polarization fractions
of $B_{s}  \to K \ell \bar \nu_{\ell}$ coincide well with the previous HPQCD [8]
and FNAL/MILC [11] determinations
(see also [167] for the numerical predictions with the vector and scalar $B_s \to K$ form factors
computed with the TMD factorization approach).
Moreover, we summarize our numerical predictions  for the aforementioned three distinct classes
of the integrated observables by employing the combined BCL expansion fit results of
the semileptonic heavy-to-light $B_{(s)} \to \pi (K)$ form factors
in Table <ref>,
where we further confront our results with the previous determinations from the lattice QCD
and LCSR techniques for convenience.
Generally, our newly obtained results of the integrated angular observables
for the semileptonic  $B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$ decays
are compatible with the available QCD determinations within the theory uncertainties,
but with the exceptions of the previously extracted intervals of the flat terms
${\cal F}_{\rm H}^{B  \to \pi \mu \bar \nu_{\mu}}$ and ${\cal F}_{\rm H}^{B  \to \pi \tau \bar \nu_{\tau}}$
from [160], which turn out to be approximately one fourth of our numerical predictions individually.
In order to better clarify such striking discrepancies, we can readily derive an exact but quite loose bound,
independent of the scalar form-factor ratio,
for the $q^2$ differential flat term by applying the explicit definition (<ref>)
as well as the kinematic constraint
$q^2 \in \left [m_{\ell}^2 \,,  (m_{B_{(s)} }  - m_{\pi (K)})^2 \right ]$.
For definiteness, we obtain

\begin{eqnarray}
{\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}} (q^2)
\geq   \frac{3}{1 + 2 \, y_{\ell, \, {\rm max}}} \,,
\qquad
y_{\ell, \, {\rm max}} \equiv  \left [  \frac{m_{B_{(s)} }  - m_{\pi (K)}} {m_{\ell}} \right ]^2 \,.
\label{an exact bound of FH}
\end{eqnarray}

Plugging the input values for the emerged hadron and lepton masses
summarized in Table <ref> into (<ref>)
immediately leads to the desired numerical bounds

\begin{eqnarray}
{\cal F}_{\rm H}^{B  \to \pi \mu \bar \nu_{\mu}}(q^2) \geq 6.337 \times 10^{-4},
& \qquad &
{\cal F}_{\rm H}^{B  \to \pi \tau \bar \nu_{\tau}}(q^2)  \geq  0.169   \,,
\nonumber \\
{\cal F}_{\rm H}^{B_s  \to K \mu \bar \nu_{\mu}}(q^2) \geq 7.049 \times 10^{-4},
& \qquad &
{\cal F}_{\rm H}^{B_s  \to K \tau \bar \nu_{\tau}} (q^2) \geq  0.187   \,,
\end{eqnarray}

which are well respected by our combined BCL $z$-fit results for the corresponding four observables
as collected in Table <ref>.

Theory predictions for the three distinct classes of the angular observables
${\cal A}_{\rm FB}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}(q^2)$,
${\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}(q^2)$ and
${\cal A}_{\lambda_{\ell}}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}(q^2)$
obtained from the combined BCL expansion fitting against
the synthetic lattice data points and  the newly obtained bottom-meson LCSR results.

0.85 !

Observables      Lattice QCD     Lattice QCD $\oplus$ LCSR       This work

3*${\cal A}_{\rm FB}^{B  \to \pi \mu \bar \nu_{\mu}}$

$\left (3.99 \pm 0.35 \right )  \times 10^{-3} \big |_{N=3}$

$\left (4.4 \pm 1.3 \right ) \times 10^{-3}$ [4]
$\left (4.8 \pm 0.3 \right ) \times 10^{-3}$ [160]

$ \left (3.72 \pm 0.51 \right )  \times 10^{-3}  \big |_{N=4}$

3*${\cal A}_{\rm FB}^{B  \to \pi \tau \bar \nu_{\tau}}$
$0.248 \pm 0.005  \big |_{N=3}$

$0.252 \pm 0.012$ [4]      $0.259 \pm 0.004$ [160]

$ 0.244 \pm 0.007  \big |_{N=4}$

3*${\cal A}_{\rm FB}^{B_s  \to K \mu \bar \nu_{\mu}}$
$\left (3.9 \pm 1.1 \right )  \times 10^{-3} $ [4]
$\left (4.49 \pm 0.21 \right )  \times 10^{-3} \big |_{N=3}$

$\left (6.6 \pm 1.0\right ) \times 10^{-3}$ [8]      $-$

$\left (3.21 \pm 0.97 \right ) \times 10^{-3}$ [11]
$ \left (5.14 \pm 0.33 \right ) \times 10^{-3}  \big |_{N=4}$

3*${\cal A}_{\rm FB}^{B_s  \to K \tau \bar \nu_{\tau}}$     $0.2650 \pm 0.0079$ [4]
$0.267 \pm 0.002 \big |_{N=3}$

$0.284 \pm 0.017$ [8]      $-$

$0.2536 \pm 0.0084$ [11]             $ 0.272 \pm 0.003  \big |_{N=4}$

3*${\cal F}_{\rm H}^{B  \to \pi \mu \bar \nu_{\mu}}$
$\left (8.04 \pm 0.72 \right )  \times 10^{-3} \big |_{N=3}$

$-$      $\left (2.4  \pm 0.1 \right )  \times 10^{-3}$  [160]

$ \left (7.52 \pm 1.02 \right )  \times 10^{-3} \big |_{N=4}$

3*${\cal F}_{\rm H}^{B  \to \pi \tau \bar \nu_{\tau}}$
$0.514 \pm 0.012 \big |_{N=3}$

$-$     $0.134 \pm 0.003$  [160]

$ 0.508 \pm 0.014   \big |_{N=4}$

3*${\cal F}_{\rm H}^{B_s  \to K \mu \bar \nu_{\mu}}$
$\left (9.10 \pm 0.43 \right )  \times 10^{-3} \big |_{N=3}$

$-$       $-$

$ \left (10.36 \pm 0.67 \right )  \times 10^{-3}   \big |_{N=4}$

3*${\cal F}_{\rm H}^{B_s  \to K \tau \bar \nu_{\tau}}$
$0.555 \pm 0.006   \big |_{N=3}$

$-$       $-$

$ 0.565 \pm 0.007  \big |_{N=4}$

3*${\cal A}_{\lambda_{\ell}}^{B  \to \pi \mu \bar \nu_{\mu}}$
$ 0.988 \pm 0.001  \big |_{N=3}$

$-$     $-$

$ 0.989 \pm 0.002  \big |_{N=4}$

3*${\cal A}_{\lambda_{\ell}}^{B  \to \pi \tau \bar \nu_{\tau}}$
$ 0.266  \pm 0.029  \big |_{N=3}$

$-$     $0.21 \pm 0.02$  [160]

$ 0.272  \pm 0.032   \big |_{N=4}$

3*${\cal A}_{\lambda_{\ell}}^{B_s  \to K \mu \bar \nu_{\mu}}$
$ 0.987 \pm 0.001  \big |_{N=3}$

$0.982^{+0.018}_{-0.079}$ [8]     $-$

$ 0.985 \pm 0.001   \big |_{N=4}$

3*${\cal A}_{\lambda_{\ell}}^{B_s  \to K \tau \bar \nu_{\tau}}$
$ 0.191 \pm 0.014  \big |_{N=3}$

$0.105 \pm 0.063$ [8]      $-$

$ 0.172 \pm 0.017  \big |_{N=4}$

Theory predictions for the three distinct classes of the integrated  observables
${\cal A}_{\rm FB}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$,
${\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$ and
${\cal A}_{\lambda_{\ell}}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$
obtained from the combined BCL $z$-series expansion fitting of the exclusive $B_{(s)} \to \pi (K)$ form factors against
the synthetic lattice data points and the newly obtained bottom-meson LCSR results.

§.§ Phenomenological analysis of the   $B \to K \nu_{\ell} \bar \nu_{\ell}$  observables

We are now in a position to explore  phenomenological implications of  the  newly determined $B \to K$ form factors
on the electroweak  penguin  $B \to K \nu_{\ell} \bar \nu_{\ell}$ decays,
which are expected to be observed with first $10 \, {\rm ab}^{-1}$ of the Belle II data [168]
(see the earlier experimental searches  by the BaBar [169],
Belle [170] and Belle II [171] Collaborations).

Importantly, the expected sensitivity of the total branching fraction for
$B \to K \nu_{\ell} \bar \nu_{\ell}$ (summing over neutrino flavours)
with $50 \, {\rm ab}^{-1}$  of integrated luminosity
has been estimated to be at the level of $10 \, \%$,
thus comparable to the current theoretical uncertainties of the  SM predictions [69].
It is straightforward to derive the differential decay width formula for the  theoretically
cleanest FCNC $B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}$  decay process of the neutral bottom meson
[172, 173]

\begin{eqnarray}
\frac{d \Gamma(B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell} )}
{d q^2 } &=&  {G_F^2 \, \alpha_{\rm em}^2 \over  256 \, \pi^5} \,
{\lambda^{3/2}(m_B^2, m_K^2, q^2) \over m_B^3 \, \sin^4 \theta_W } \,
|V_{tb} \, V_{ts}^{\ast}|^{2} \,
\left [ X_t \left ({m_t^2 \over m_W^2}, {m_H^2 \over m_W^2}, \sin \theta_W, \mu \right )   \right ]^2
\nonumber \\
&& \times \left |f_{BK}^{+}(q^2) \right |^2  \,,
\label{differential decay width of B0 to K0 nu nu}
\end{eqnarray}

where the CKM matrix elements $|V_{tb}|$ and $|V_{ts}^{\ast}|$ can be further evaluated from the four
Wolfenstein parameters collected in Table <ref> with the expanded matching relations
at the accuracy of ${\cal O}(\lambda^9)$ [174].
The short-distance Wilson coefficient $X_t$ can be expanded perturbatively in terms of the SM gauge couplings

\begin{eqnarray}
X_t = X_t^{(0)} + {\alpha_s \over 4 \pi} \,  X_t^{{\rm QCD}(1)} +
{\alpha_{\rm em} \over 4 \pi} \,  X_t^{{\rm EW} (1)} + ...  \,,
\end{eqnarray}

where the LO contribution $X_t^{(0)}$ [175],  the NLO QCD correction $X_t^{{\rm QCD}(1)}$
[176, 177, 178]
and the two-loop electroweak correction $ X_t^{{\rm EW} (1)}$ [179] are already known analytically.
By contrast, there exists an additional long-distance contribution to the counterpart charged channel
$B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}$ due to the double charged-current interaction
$B^{-} \to \tau (\to K^{-} \nu_{\tau}) \, \bar \nu_{\tau}$ at tree level as originally discussed in [180].
In the narrow $\tau$-lepton width approximation
($\Gamma_{\tau} \simeq 2.3 \times 10^{-3} \, {\rm eV}$ [90]),
we can readily derive the tree-level charged-current contribution to the exclusive rare
$B^{-} \to K^{-} \nu_{\tau} \bar \nu_{\tau}$ decay rate

\begin{eqnarray}
\frac{d \Gamma(B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell} )}
{d q^2 } \bigg|_{\rm LD} &=&  {G_F^4  \, |V_{ub} V_{us}^{\ast}|^2 \over 64 \, \pi^2 \, m_{B^{-}}^3} \,
\left | f_{B^{-}} \, f_{K^{-}}  \right |^2  \,
\frac{m_{\tau}^3}{\Gamma_{\tau}} \,
\left [ ( m_{B^{-}}^2 - m_{\tau}^2 ) \, ( m_{\tau}^2  - m_{K^{-}}^2 ) -   m_{\tau}^2 \, q^2  \right ],
\nonumber \\
\label{tree contribution to B to K nu nu}
\end{eqnarray}

where  the invariant mass distribution of the two invisible particles
satisfies the  constraint [90]

\begin{eqnarray}
0 \leq q^2 \leq  \frac{( m_{B^{-}}^2 - m_{\tau}^2 ) \, ( m_{\tau}^2  - m_{K^{-}}^2 )}{ m_{\tau}^2} \,.
\end{eqnarray}

At the first sight, this  new mechanism will be suppressed by an extra factor of $G_F^2$
in comparison with the customary penguin contribution presented in (<ref>).
However,  the very appearance of $1/\Gamma_{\tau}$ on the right-handed side of (<ref>),
due to the on-shell $\tau$-lepton enhancement, will be counted as ${\cal O}(G_F^{-2})$ parametrically,
thus compensating the observed suppression factor [181].
Moreover, the interference effect  between the tree and penguin amplitudes
turns out to be numerically negligible (estimated to be at the order of $10^{-11}$ [180])
on account of the extremely small  $\tau$-lepton width.
We display in Figure <ref>
the yielding results for the differential decay distributions of
$B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}$ and $B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}$
by employing  the form factors determined from  fitting the BCL $z$-series parameterizations
with two  distinct scenarios of the input data points:
I) only synthetic lattice data points,
II) synthetic lattice data points $\oplus$ LCSR results.

Theory predictions for the differential decay distributions of
$B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}$ (left panel) and $B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}$
(right panel)  by applying the form factors determined from the two distinct  scenarios of the BCL $z$-series
expansion fitting strategies.

In order to confront our numerical predictions  with the anticipated measurements from the high-luminosity Belle II experiment,
we introduce  the following three $q^2$-binned observables for the semileptonic $B \to K \nu_{\ell} \bar \nu_{\ell}$ decays
[36]

\begin{eqnarray}
&& \Delta {\cal BR}^{B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}}(q_1^2, \,  q_2^2)
= \tau_{B^{0}} \, \int_{q_1^2}^{q_2^2} \, d q^2 \,
\frac{d \Gamma(B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell} )} {d q^2 } \,,
\nonumber \\
&& \Delta {\cal BR}^{B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}}(q_1^2, \,  q_2^2)
= \tau_{B^{-}} \, \int_{q_1^2}^{q_2^2} \, d q^2 \,
\frac{d \Gamma(B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell} )} {d q^2 } \,,
\nonumber \\
&& {\cal R}_{K \pi}(q_1^2, \,  q_2^2) =  \left [ \int_{q_1^2}^{q_2^2} \, d q^2 \,
\frac{d \Gamma(B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell} )} {d q^2 } \right ] :
\left [ \int_{q_1^2}^{q_2^2} \, d q^2 \,
\frac{d \Gamma(B^{0} \to \pi^{-} \bar \nu_{\mu} \nu_{\mu} )} {d q^2 } \right ]  \,,
\label{definitions: observables for B to K nu nu}
\end{eqnarray}

where the ratio  of partially integrated differential branching
fractions  ${\cal R}_{K \pi}$  is expected to  suffer from the lower hadronic uncertainties
due to the  correlations between the exclusive $B \to \pi$ and $B \to K$ form factors.
We summarize our final  predictions for these three quantities with the choice of the $q^2$ intervals
following [69] in Table <ref>,
where we further display the previous theoretical determinations with the lattice simulation and LCSR methods
for convenience.

$[q_1^2, \, q_2^2]$     (${\rm in \,\, GeV}^2$)
$10^6 \times \Delta {\cal BR}^{B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}}(q_1^2, q_2^2)$
$10^6 \times \Delta {\cal BR}^{B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}} (q_1^2, q_2^2)$
$10^2 \times {\cal R}_{K \pi}(q_1^2, q_2^2)$

$[0.0, 1.0]$     $0.253^{+0.020}_{-0.019}$      $0.324^{+0.023}_{-0.021}$       $3.949 \pm 0.319$

$[1.0, 2.5]$     $0.381^{+0.028}_{-0.027}$      $0.482^{+0.033}_{-0.030}$       $3.932 \pm 0.303$

$[2.5, 4.0]$     $0.380^{+0.027}_{-0.025}$      $0.476^{+0.031}_{-0.029}$       $3.904 \pm 0.289$

$[4.0, 6.0]$     $0.502^{+0.034}_{-0.032}$      $0.622^{+0.039}_{-0.035}$       $3.859 \pm  0.278$

$[6.0, 8.0]$     $0.492^{+0.032}_{-0.030}$      $0.602^{+0.037}_{-0.034}$       $3.788 \pm  0.270$

$[8.0, 12.0]$     $0.924^{+0.058}_{-0.053}$      $1.112^{+0.066}_{-0.060}$       $3.626 \pm  0.262$

$[12.0, 16.0]$     $0.776^{+0.047}_{-0.043}$      $0.916^{+0.053}_{-0.048}$       $3.245 \pm  0.238$

$\left [16.0, \, (m_B-m_{K})^2 \right ]$     $0.607^{+0.036}_{-0.032}$      $0.705^{+0.040}_{-0.036}$       $1.918 \pm 0.129$

7*$\left [0.0, \, (m_B-m_{K})^2 \right ]$      $6.02^{+1.68}_{-1.76}$  [36]
$5.10 \pm 0.80$ [172]        $-$

$4.01 \pm 0.49$  [181]       $3.98 \pm 0.47$ [173]         $-$

$4.67 \pm 0.35$ [182]     $4.94 \pm 0.52$ [181]      $-$

$4.1^{+1.3}_{-1.0}$ [183]     $5.67 \pm 0.38$ [182]         $-$

$4.4 \pm 1.5$ [184]       $4.53 \pm 0.64$ [185]        $-$

$-$      $4.65 \pm 0.62$ [186]        $-$

$4.315^{+0.271}_{-0.248}$  (this work)      $5.239^{+0.311}_{-0.281}$  (this work)       $3.240 \pm 0.211$  (this work)

Theory predictions for the three partially integrated differential observables
$\Delta {\cal BR}^{B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}}$,
$\Delta {\cal BR}^{B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}}$,
and ${\cal R}_{K \pi}$ (see (<ref>) for their explicit definitions)
obtained from the combined BCL $z$-series expansion fitting of
the exclusive $B \to \pi, K$ form factors against
the synthetic lattice data points and the newly obtained bottom-meson LCSR results.

§ CONCLUSIONS

In the current paper we have carried out the improved computations of the semileptonic
$B_{d, s}  \to \pi, K$  decay form factors at large hadronic recoil,
which evidently belong to the most important hadronic quantities in heavy quark physics,
by employing the method of light-cone sum rules (LCSR) in soft-collinear effective theory (SCET)
with both the leading-twist and higher-twist bottom-meson distribution amplitudes.
In particular, we have computed for the first time the non-vanishing spectator-quark
mass corrections to these form factors at NLO in the strong coupling constant,
which appeared to preserve the approximated large-recoil symmetry relations for
the heavy-to-light transition form factors
and turned out to escape from an extra suppression of the powers of  $\Lambda_{\rm QCD}/m_b$
in heavy quark expansion.
Our explicit sum rules for these  spectator-quark mass corrections further implied that
evaluating such SU(3)-flavour symmetry breaking effects directly with the perturbative
factorization technique would result in the soft-collinear convolution integrals
with the notorious rapidity singularities.
Moreover, we have accomplished the complete NLL resummation for  the parametrically enhanced logarithms
of $m_b / \Lambda_{\rm QCD}$ entering in the factorized expressions of
the leading-power contributions to the considered  vacuum-to-bottom-meson correlation functions
displayed in (<ref>) by taking advantage of the standard renormalization-group
formalism in momentum space.
We then proceeded to explore the four distinct classes of the NLP contributions to
the exclusive $B_{d, s}  \to \pi, K$  form factors with the same LCSR technique at tree level:
I) the higher-order terms from  heavy quark expansion of the hard-collinear quark propagator,
II) the subleading power corrections from the effective matrix element of the ${\rm SCET_I}$ weak current
$(\bar \xi_{\rm hc} \, W_{\rm hc}) \, \gamma_{\mu} \,
\left [  i \,  {\slashed D}_{\top} /  \left (  2 \, m_b \right ) \right ] \,   h_v$,
III) the higher-twist corrections from the two-particle and three-particle
heavy quark effective theory (HQET) distribution amplitudes,
IV) the four-particle twist-five and twist-six contributions in the factorization approximation.
We have extensively used the nontrivial operator identities between the two-body and three-body light-cone HQET operators
due to the classical equations of motion in our constructions of the NLP sum rules.
Interestingly,  we observed that only the particular class-I NLP contribution
from the expanded hard-collinear propagator can generate the large-recoil symmetry violation effect
between the vector and scalar form factors,
while both the  class-I and class-II NLP corrections can yield the symmetry breaking effects
between the vector and tensor form factors at large hadronic recoil.

Having at our disposal the updated  SCET sum rules for the exclusive heavy-to-light
bottom-meson decay form factors, we turned to investigate the numerical implications
of the NLL resummation improved leading-power contributions and the newly obtained
four classes of the NLP corrections at tree level on the theory predictions for
the  semileptonic  $B_{d, s}  \to \pi, K$  form factors of our interest,
by employing the general three-parameter ansatz for the necessary HQET distribution amplitudes.
It has been explicitly shown that the most prominent subleading power contribution
arises from the two-particle twist-five off-light-cone  correction,
which can reduce the corresponding leading-power LCSR predictions  in the kinematic region
$0 \leq  q^2 \leq 8 \, {\rm GeV^2}$ by an amount of  $(25-30)\, \%$ numerically.
On the contrary, the yielding impacts from the four-particle higher-twist
bottom-meson distribution amplitudes have been demonstrated to be numerically  insignificant
in the factorization approximation owing to the smallness of the normalization constant
$|\langle \bar q q \rangle : (\lambda_B \, s_0)| \simeq 10 \, \%$
in the tree-level sum rules (<ref>).
In addition, the higher-order QCD corrections to the short-distance
matching coefficients appeared  in  the leading-power SCET sum rules
can bring about consistently ${\cal O} (30 \, \%)$ reductions
of the counterpart leading-order LCSR predictions.
It remains important to remark that our numerical results
for the semileptonic bottom-meson decay form factors indicate the following
hierarchy relations
$f_{B K}^{+ \, (T)}(0) > f_{B_s K}^{+ \, (T)}(0) >  f_{B \pi}^{+ \, (T)}(0)$
in the maximal recoil limit.
Remarkably, we predicted very sizeable SU(3)-flavour symmetry violating  effects
between the exclusive $B \to \pi$  and $B \to K$ form factors
based upon the established LCSR with the HQET distribution amplitudes:
numerically ${\cal O} (30 \, \%)$ for the two form-factor ratios
${\cal R}_{\rm SU(3)}^{+, \, 0}$   and  ${\cal O} (40 \, \%)$
for the particular ratio ${\cal R}_{\rm SU(3)}^{T}$.

Subsequently,  we extrapolated the bottom-meson LCSR computations for
the exclusive $B_{d, s}  \to \pi, K$   form factors
towards the large momentum transfer $q^2$ with the aid of
the Bourrely-Caprini-Lellouch (BCL) $z$-series parameterizations for these form factors.
It is interesting to note that including the newly obtained LCSR predictions at small momentum transfer
in our numerical fitting procedure turned out to be highly beneficial for
pinning down the theory uncertainties of all the three  $B \to \pi$ form factors
in the kinematic regime $0.10 \leq z(q^2) \leq 0.31$  significantly,
due to the yet non-negligible errors  of the lattice QCD results from the RBC/UKQCD Collaboration [4]
(numerically at the level of  $(8.4-14.3) \, \%$  for the vector form factor
and  $(7.6-13.6) \, \%$ for the scalar form factor).
Furthermore, we have confronted our combined BCL fit results of the $B \to \pi$ form factors
with the theoretical expectations from the particular low-recoil symmetry relations
(<ref>) and  (<ref>),
on account of the combination of the heavy quark spin symmetry and the so-called soft-pion approximation,
in Figures (<ref>) and
(<ref>) manifestly.
Additionally, the available high-precision lattice QCD results of
the semileptonic $B \to K$ form factors from both the FNAL/MILC Collaboration  [14]
and the HPQCD Collaboration [15] appeared to be in tension with
the strongly correlated predictions from the improved bottom-meson LCSR determinations,
when carrying out the numerical interpolations between the lattice simulation and LCSR results
with the standard BCL $z$-series expansions.
With regard to the flavour-changing charged-current $B_s \to K \ell \bar \nu_{\ell}$ form factors,
the yielding BCL fit results for the three coefficients $b_{0, 1, 2}^T$
in Table <ref> became more uncertain
in comparison with the counterpart results for the $z$-expansion parameters
of the tensor $B \to \pi$ form factor, due to the very absence  of
the lattice data points for $f_{B_s K}^{T}(q^2)$ in the lower recoil region.
Importantly, we indeed benefited from the combined BCL  $z$-expansion fit
to both the LCSR and lattice simulation results by  improving further
the theory  accuracy of the vector and scalar form factors $f_{B_s K}^{+, \, 0}(q^2)$,
numerically at the level of ${\cal O}(20 \, \%)$,
when compared with the counterpart BCL fitting procedure
with the “only lattice QCD" data points.
It has been verified that  our BCL  $z$-fit predictions for  the scalar form-factor ratios
of both the exclusive  $B \to K$ and $B_s \to K$ form factors at the zero-recoil limit
differ from the expected Isgur-Wise relations due to the combined heavy quark and chiral symmetries enormously,
thus supporting the previous lattice simulation results  from the  FNAL/MILC Collaboration [14].

Performing the simultaneous BCL expansion fit to the SCET sum rules, lattice QCD and experimental data points
enabled us to extract the desired CKM  matrix element $|V_{ub}|$
from the “golden"  exclusive process $B \to \pi \ell \bar \nu_{\ell}$ with the two distinct truncations

\begin{eqnarray}
\left |V_{ub} \right |_{B \to \pi \ell \bar \nu_{\ell}} = (3.76 \pm 0.13) \times \, 10^{-3} \,,
\qquad
({\rm BCL \,\, fit \,\, with}  \,\, N=3)
\nonumber \\
\left |V_{ub} \right |_{B \to \pi \ell \bar \nu_{\ell}} = (3.72 \pm 0.14) \times \, 10^{-3} \,.
\qquad
({\rm BCL \,\, fit \,\, with}  \,\, N=4)
\label{Vub value: N=3 and 4}
\end{eqnarray}

We are therefore led to conclude that  truncating the $z$-series expansions at the order $N=3$
in the numerical fit procedure can indeed be justified  for the practical purpose
due to the apparent stability against the truncation order.
Our numerical predictions for the  two  particular lepton-flavour-universality (LFU) ratios
for the exclusive $B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}$ decays
have been collected in Figure <ref>
and Table <ref>,
and we quote here the obtained results for such gold-plated quantities

\begin{eqnarray}
{\cal R}_{\pi} = 0.720 \pm 0.027  \,,
\qquad
{\cal R}_{K} = 0.700 \pm 0.016  \,,
\qquad
({\rm BCL \,\, fit \,\, with}  \,\, N=3)
\nonumber \\
{\cal R}_{\pi} = 0.746 \pm 0.039  \,,
\qquad
{\cal R}_{K} = 0.680 \pm 0.019  \,.
\qquad
({\rm BCL \,\, fit \,\, with}  \,\, N=4)
\label{Rpi and RK value: N=3 and 4}
\end{eqnarray}

The yielding results for the three different classes of the angular observables sensitive to
the potential Beyond the Standard Model (BSM) physics
${\cal A}_{\rm FB}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$,
${\cal F}_{\rm H}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$ and
${\cal A}_{\lambda_{\ell}}^{B_{(s)}  \to \pi (K) \ell \bar \nu_{\ell}}$
have been explicitly  displayed in Figure
and Table <ref>,
where the previous theory determinations from the lattice simulation and LCSR approaches
were further shown for a comparison.
We finally presented our numerical predictions for the three partially integrated differential observables
$\Delta {\cal BR}^{B^{0} \to K^{0} \nu_{\ell} \bar \nu_{\ell}}$,
$\Delta {\cal BR}^{B^{-} \to K^{-} \nu_{\ell} \bar \nu_{\ell}}$,
and ${\cal R}_{K \pi}$ (see (<ref>) for their explicit definitions)
for the exclusive rare $B \to K \nu_{\ell} \bar \nu_{\ell}$ decays
in Table <ref>.

Future developments of the theory predictions for the heavy-to-light bottom-meson decays
beyond our work can be pursued forward in a variety of directions.
First, it would be interesting to perform the full two-loop QCD computations
of the semileptonic  $B_{d, s}  \to \pi, K$  decay form factors  at large hadronic recoil,
by employing  the SCET sum rules framework, in order to reduce further the current perturbative uncertainties
displayed in Figures <ref>
and  <ref>.
Actually, the desired hard matching coefficients in the ${\rm SCET_I}$ representations
of the flavour-changing QCD currents $\bar q \, \Gamma_i \, b$ have been already
determined at the ${\cal O}(\alpha_s^2)$ accuracy.
The only missing ingredients for constructing the two-loop factorization formulae of
the very vacuum-to-bottom-meson correlation functions (<ref>)
consist in the yet unknown short-distance Wilson coefficients
in the second-step ${\rm SCET_I} \to {\rm SCET_{II}}$ matching procedure.
Second, investigating the subleading-power contributions to the exclusive $B_{d, s}  \to \pi, K$   form factors
systematically with the effective field theory  techniques
and then evaluating the resulting (non)-local soft-collinear matrix elements
with the appropriate nonperturbative QCD methods will be evidently in high demand
from both the conceptual and phenomenological perspectives.
As a matter of fact, it would be of utmost importance to achieve the analytical regularization
of the unwanted end-point divergences entering  in the factorized expressions of
the  ${\rm SCET_{I}}$ matrix elements for the two-body $A$-type currents.
Third, advancing our knowledge of the poorly constrained bottom-meson distribution amplitudes
in HQET with model-independent techniques (for instance, along the line of [129])
will be indispensable for enhancing further our predictive power
of the exclusive bottom-meson decay matrix elements in the theory frameworks
of both QCD factorization and light-cone sum rules.
In particular the yielding noticeable uncertainties due to the two-particle $B_{d, s}$-meson distribution amplitudes
has already become the major stumbling block for accomplishing precision calculations of a wide range of
the interesting physical observables accessible at the LHCb and Belle II experiments.
Fourth, the established strategies of evaluating the subleading-power  $B_{d, s}  \to \pi, K$  matrix elements
can be further extended to compute the NLP corrections to the analogous heavy-to-light
$B \to \rho, \omega, K^{\ast}$ decay form factors  (thus going well beyond our previous work [38])
and to explore the delicate strong interaction mechanisms dictating
the Cabibbo favored semileptonic $B_{(s)} \to D_{(s)}^{\ast} \ell \bar \nu_{\ell}$ decay processes.

§ ACKNOWLEDGEMENTS

We are grateful to Zi-Hao Mi for collaboration in the early stages of this project
and to Svenja Granderath and Yan-Bing Wei for illuminating discussions.
The research of Y.L.S. is supported by the  National Natural Science Foundation of China  with
Grant No. 12175218 and the Natural Science Foundation of Shandong with Grant No.  ZR2020MA093.
C.W is supported in part by the National Natural Science Foundation of China
with Grant No. 12105112 and  the Natural Science Foundation of
Jiangsu Education Committee with Grant No. 21KJB140027.
Y.M.W acknowledges support from

the  National Natural Science Foundation of China  with
Grant No. 11735010 and 12075125, and  the Natural Science Foundation of Tianjin
with Grant No. 19JCJQJC61100.

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