December  22, 2022

Precision calculations of $B_{d, s}  \to \pi, K$  decay form factors

in soft-collinear effective theory

Bo-Yan Cui$^a$,
Yong-Kang Huang$^a$ [correspondence author<EMAIL_ADDRESS>Yue-Long Shen$^b$,
Chao Wang$^c$,
Yu-Ming Wang$^a$ [correspondence author<EMAIL_ADDRESS>
$^a$   School of Physics, Nankai University, Weijin Road 94, 300071 Tianjin, China

$^b$ College of Information Science and Engineering, Ocean  University of China,

Qingdao 266100,  Shandong, China

$^c$ College of Mathematics and Physics, Huaiyin Institute of Technology,

Huaian 223003, Jiangsu, China

We improve QCD calculations of the semileptonic $B_{d, s}  \to \pi, K$  decay form factors at large hadronic recoil
by implementing the next-to-leading-logarithmic resummation for the obtained leading-power light-cone sum rules
in the soft-collinear effective theory (SCET) framework and by computing for the first time the non-vanishing spectator-quark mass correction
dictating the SU(3)-flavour symmetry breaking effects between  these fundamental quantities at the one-loop accuracy.
Additionally, we endeavour to investigate a variety of the subleading-power contributions to these heavy-to-light form factors
at ${\cal O}(\alpha_s^0)$ with the same methodology, by including the higher-order terms in the heavy-quark expansion of the hard-collinear quark propagator, by evaluating the desired effective matrix element of  the next-to-leading-order term
$(\bar \xi_{\rm hc} \, W_{\rm hc}) \,
\gamma_{\mu} \,   \left [  i \,  {\slashed D}_{\top} /  \left (  2 \, m_b \right ) \right ] \,   h_v$
in the ${\rm SCET_{I}}$ representation of the weak transition current,
by taking into account the off-light-cone contributions of the two-body heavy-quark effective theory (HQET) matrix elements
as well as  the three-particle higher-twist corrections from the subleading bottom-meson light-cone distribution amplitudes (LCDAs),
and by computing the twist-five and twist-six four-body higher-twist effects with the aid of the factorization approximation.
Having at our disposal the SCET sum rules for the exclusive $B$-meson decay form factors under discussion,
we further explore in detail  numerical implications of the newly computed subleading-power corrections
by employing the three-parameter model for both the leading-twist and higher-twist $B$-meson distribution amplitudes.
Taking advantage of the customary  Bourrely-Caprini-Lellouch (BCL) parametrization for the complete set
of the semileptonic $B_{d, s}  \to \pi, K$  form factors, we then determine the correlated numerical results for the interesting
series coefficients, by carrying out the simultaneous fit of the exclusive $B$-meson decay form factors to
both the achieved SCET sum rule predictions at small momentum transfer ($q^2$) and the available lattice QCD results at large momentum transfer.
Subsequently, we perform a comprehensive phenomenological analysis of the full angular observables,
the lepton-flavour university ratios and the lepton polarization asymmetries for the flavour-changing charged-current
$B \to \pi \ell \bar \nu_{\ell}$  and $B_s \to K \ell \bar \nu_{\ell}$ decays  (with  $\ell= \mu, \, \tau$)
together with the differential $q^2$-distribution for the exclusive rare $B \to K \nu_{\ell} \bar \nu_{\ell}$ decays
in the Standard Model.

§ INTRODUCTION

Precision calculations of the semileptonic $B_{d, s}  \to \pi, K$  decay form factors   are of paramount importance
for exploring the celebrated Cabibbo-Kobayashi-Maskawa (CKM) mechanism in the Standard Model (SM)
and for sharpening our understanding towards diverse facets of the strong interaction dynamics
encoded in the exclusive heavy-hadron decay processes.
Particularly, the longstanding discrepancy between the dedicated $|V_{ub}|$ determinations from
the exclusive $B \to \pi \ell \bar \nu_{\ell}$ decays
and the inclusive $B \to X_{u} \ell \bar \nu_{\ell}$ processes [1]  has been continually triggering
the enormous theoretical efforts of determining such heavy-to-light $B$-meson form factors
with an ever-increasing accuracy. In the small hadronic recoil region,
the first-principles calculations for a rich variety of the non-perturbative matrix elements
appearing in the theory descriptions of the semileptonic heavy-meson decays  $B \to \pi \ell \bar \nu_{\ell}$
[2, 3, 4, 5, 6, 7]
and $B_s \to K \ell \bar \nu_{\ell}$ decays [4, 8, 9, 10, 11, 12]
and of the exclusive electroweak penguin $B \to K \nu_{\ell} \bar \nu_{\ell}$ transitions [13, 14, 15]
have been pursued with the numerical lattice gauge theory by different groups
(see  also  [17, 18, 16] for an overview).
Moreover, numerous analytical QCD frameworks with distinct approximations have been constructed to address
the semileptonic $B$-meson decay form factors at large hadronic recoil systematically based upon the heavy quark expansion techniques.

Employing the perturbative QCD factorization theorem for the vacuum-to-pseudoscalar-meson correlation function
and implementing further the parton-hadron duality ansätz enables us to derive the desired light-cone sum rules (LCSR)
for the heavy-to-light $B_{d, s}  \to \pi, K$  form factors at the leading-order (LO) accuracy [19, 20]
and in the next-to-leading-order (NLO) approximation [21, 22, 23, 24, 25]
(see also [27] for the twist-two ${\cal O}(\alpha_s^2 \, \beta_0)$ correction to the vector form factor
$f_{B \to \pi}^{+}(q^2)$), taking advantage of the light-meson light-cone distribution amplitudes (LCDAs) with  definite collinear twists
[28, 29] as the fundamental  non-perturbative ingredients.
Alternatively, the light-cone QCD sum rules for the exclusive  bottom-meson decay form factors
can be derived from the vacuum-to-$B$-meson correlation function with the pseudoscalar meson state
interpolated by an appropriate partonic current [30, 31]
(see [32, 33] for an equivalent and independent formulation
in the soft-collinear effective theory (SCET) framework),
following the analogous theory prescriptions as described above.
An attractive advantage of this alternative version of QCD sum rules on the light-cone
consists in the very appearance of the universal $B$-meson distribution amplitudes in heavy quark effective theory (HQET)
for the obtained expressions of all the bottom-meson decay form factors,  independent of the particular light-hadron
in the final states. Along the same vein, both the higher-order perturbative corrections and the subleading-power contributions
to the heavy-to-light $B$-meson decay form factors [34, 35, 36, 37, 38, 39], the heavy-to-heavy  $B$-meson decay form factors [40, 37, 41],
as well as the semileptonic  heavy-baryon decay form factors [42, 43, 44] have been computed from the LCSR method with the heavy-hadron distribution amplitudes.

Yet another theory framework to evaluate the exclusive heavy-hadron decay matrix elements has been developed
to  regularize the emerged end-point divergences in the conventional collinear factorization formalism
by introducing the intrinsic transverse momenta of the associated soft and collinear partons participating
the short-distance scattering process [45, 46, 47]
(see, however, [48] for additional discussions),
motivated from the theory of the on-shell Sudakov form factor [49]
and the asymptotic behaviour of elastic meson-meson scattering at high energy [50].
Applying such transverse-momentum-dependent (TMD) factorization approach further  allows for the higher-order QCD computations
of a large number of exclusive hadronic matrix elements [51, 52, 53, 54, 55, 56, 57, 58, 59]
including both the leading-twist and higher-twist contributions simultaneously.
We mention in passing that constructing the factorization-compatible definitions of the  TMD wavefunctions
free of both the rapidity divergence and the pinch singularity  becomes tremendously delicate,
demanding the introduction  of an  intricate soft substraction function
defined with the non-dipolar off-light-cone Wilson lines [60, 61, 62].

Inspired by the encouraging experimental progresses on measuring the differential $B \to \pi \ell \nu$ decay rates
from the BaBar [63, 64], Belle [65, 66]
and Belle II [67] Collaborations,
on the first observation of the semileptonic $B_s \to K \ell \bar \nu_{\ell}$ decays
at the LHCb experiment [68], and on the anticipated discovery of the flavour-changing neutral current (FCNC)
$B \to K \nu_{\ell} \bar \nu_{\ell}$ decay process at Belle II [69],
we aim at improving further theory predictions for the semileptonic $B_{d, s}  \to \pi, K$  decay form factors
from the LCSR technique with the HQET $B$-meson distribution amplitudes as previously achieved in [34, 36],
by incorporating a various types of  newly computed  subleading-power contributions
into the next-to-leading-logarithmic (NLL) resummation improved leading-power effects
in the heavy quark expansion.
More specifically, the major new  ingredients of the present paper can be summarized in the following.

* Applying the method of the SCET sum rules we evaluate the non-vanishing spectator-quark mass corrections
to the exclusive $B_{d, s}  \to \pi, K$  transition form factors at large hadronic recoil at ${\cal O}(\alpha_s)$,
which will be demonstrated to generate the  SU(3)-flavour symmetry breaking effect
not suppressed in the heavy quark expansion and to preserve the large-recoil symmetry relations
for the soft contributions to the heavy-to-light bottom-meson  decay  form factors.
We then proceed to perform the complete NLL summation of the enhanced logarithms of $m_b/\Lambda_{\rm QCD}$
appearing in the leading-power factorization formulae for the vacuum-to-$B$-meson correlation functions
defined with an interpolating current for the energetic pseudoscalar meson,
by employing the standard renormalization-group (RG) formalism.

* We compute for the first time the subleading-power terms from the heavy quark expansion of the hard-collinear quark propagator
including further two distinct sources of the light-quark mass corrections at tree level
by taking advantage of the classical equations of motion for both the soft light quark and the effective heavy quark
as well as the two-particle and three-particle light-cone bottom-meson distribution amplitudes in HQET.
Moreover, we will verify explicitly that such particular power-suppressed contributions can bring about
the notable symmetry-breaking corrections to the semileptonic $B_{d, s}  \to \pi, K$ form factors.

* We construct the SCET sum rules for the subleading matrix element of the effective heavy-to-light current
$\langle M(p) | (\bar \xi_{\rm hc} \, W_{\rm hc}) \,
\gamma_{\mu} \,   \left [  i \,  {\slashed D}_{\top} /  \left (  2 \, m_b \right ) \right ] \,   h_v | \bar B_{q^{\prime}}(v)\rangle$
from the  ${\rm SCET_{I}}$  expansion of the  flavour-changing weak current $\bar q \, \Gamma_i \, b$ at the LO accuracy,
by employing the established relations between the relevant light-ray HQET operators [70, 71, 72].

* We derive the tree-level sum rules for the twist-five and twist-six four-body higher-twist corrections
to the exclusive $B_{d, s}  \to \pi, K$ form factors with the factorization ansätz,
which allows for expressing these subleading-twist  distribution amplitudes in terms of the quark condensate
and the appropriate two-particle HQET distribution amplitudes (see for instance [73, 74] for further discussions).

* We update the previous theory predictions for the complete set of  the semileptonic $B_{d, s}  \to \pi, K$ form factors
in the entire kinematic regime by interpolating  between the improved SCET sum rule computations at small momentum transfer
and the numerical lattice QCD determinations at large momentum transfer with the conventional  Bourrely-Caprini-Lellouch (BCL) parametrization [75, 76, 77].
Our numerical explorations will evidently  reveal that
implementing the obtained LCSR constraints of the heavy-to-light bottom-meson
form factors in the theory analysis is indeed beneficial for pinning down
the yielding uncertainties of the extracted  series coefficients.

The remainder of this paper is structured as follows.
We will set up the computational framework in Section <ref>
by defining the exclusive heavy-to-light $B$-meson decay form factors of our interest,
by summarizing the general strategies of constructing the desired SCET sum rules
with the bottom-meson distribution amplitudes,
and by presenting the manifest expressions of the short-distance matching coefficients
entering the perturbative factorization formulae for the considered vacuum-to-$B$-meson correlation functions
at the leading-power accuracy. In particular, we derive the spectator-quark mass corrections
to the perturbative hard-collinear functions  at ${\cal O}(\alpha_s)$
and demonstrate further the factorization-scale independence of the obtained expressions for the correlation functions.
Applying the RG evolution equations for both the hard matching coefficients
and the two-particle twist-two and twist-three $B$-meson distribution amplitudes enables us to carry out
the NLL summation of the parametrically enhanced logarithms in the factorized expressions of the vacuum-to-$B$-meson
correlation functions.
We turn to investigate four different classes of the next-to-leading-power (NLP) corrections to
the exclusive $B_{d, s}  \to \pi, K$  transition form factors
at tree level on the basis of  the SCET sum rules
in Section <ref>, by invoking the appropriate operator identities
between the two-body and three-body light-cone HQET operators
and by employing the higher-twist $B$-meson distribution amplitudes up to the twist-six accuracy.
Numerical explorations of the resulting SCET sum rules for the $B_{d, s}  \to \pi, K$ form factors
at large hadronic recoil including both the updated leading-power contributions in the  $\Lambda_{\rm QCD}/m_b$ expansion
with the non-vanishing light-quark masses and the newly derived subleading power corrections
will be displayed in Section <ref>.
We then proceed to perform the simultaneous fit of the  customary BCL parametrization for
these transition form factors to  the obtained LCSR predictions and the available lattice QCD results,
yielding the correlated numerical values of the nonperturbative form-factor parameters.
Taking advantage of the improved determinations of the bottom-meson decay form factors in the full kinematic region,
we also provide the SM predictions for a variety  of  phenomenologically interesting observables
for the semileptonic $B \to \pi \ell \bar \nu_{\ell}$  and $B_s \to K \ell \bar \nu_{\ell}$ decay processes,
such as  the differential branching fractions, the lepton-flavour universality ratios
the forward-backward asymmetries as well as the lepton polarization asymmetries,
and for the  theoretically cleanest electroweak penguin $B \to K \nu_{\ell} \bar \nu_{\ell}$ decays
in this section.
We will conclude in Section <ref> with a summary of our major observations
and perspectives on the future developments.

§ THE NLL LCSR FOR THE EXCLUSIVE $B_{D, S}  \TO \PI, K$ FORM FACTORS  AT LEADING POWER

We  employ the standard  definitions of the semileptonic heavy-to-light decay form factors
according to the Lorentz decompositions of the following bilinear quark current matrix elements

\begin{eqnarray}
\langle M(p)|  \bar q \gamma_{\mu} b| \bar B (p_B)\rangle
&=& f_{B M}^{+}(q^2) \, \left [ p_B + p -\frac{m_B^2-m_{M}^2}{q^2} q  \right ]_{\mu}
+  f_{B M}^{0}(q^2) \, \frac{m_B^2-m_{M}^2}{q^2} q_{\mu} \,,
\nonumber \\
\langle M(p)|  \bar q \sigma_{\mu \nu} q^{\nu} b| \bar B (p_B)\rangle
&=& \frac{i \, f_{B M}^{T}(q^2)}{m_B+m_M} \,
\left [ q^2 \, (2 \, p + q)_{\mu} - \left (m_B^2-m_{M}^2 \right ) q_{\mu} \right ] \,,
\label{form-factor definition}
\end{eqnarray}

where $m_M$ and $p$ correspond to the mass and the four-momentum of the light pseudoscalar meson,
and $q$ stands for the transfer momentum of the flavour-changing weak current.
Applying the procedure displayed in [32, 33, 34, 36],
the SCET sum rules for these form factors can be constructed with the vacuum-to-$B$-meson correlation function

\begin{eqnarray}
\begin{aligned}
\Pi_{\mu} (n \cdot p, \bar n \cdot p) & =  \int d^4 x \, e^{i p \cdot x} \langle 0 | {\rm T} \left \{
\bar q^{\prime}(x)\, \slashed {n} \gamma_5 \,q(x),\, \bar q(0)\, \Gamma_\mu \,b(0) \right  \} | \bar B (p_B) \rangle
\\
& =  \left\{
\begin{array}{l}
\Pi(n \cdot p, \bar n \cdot p)\, n_\mu + \tilde \Pi (n \cdot p, \bar n \cdot p)\, \bar n_\mu,
\hspace{2.0 cm} \Gamma_\mu = \gamma_\mu \vspace{0.5 cm} \\
\left ( - {i \over 2}  \right) \, \Pi_{\rm T}(n \cdot p, \bar n \cdot p)\,
\left [ \bar n \cdot q \, n_\mu - n \cdot q \, \bar n_\mu \right ] ,
\hspace{1.16cm} \Gamma_\mu = \sigma_{\mu \nu} q^\nu
\end{array}
\hspace{0.5 cm} \right.
\label{correlation_function}
\end{aligned}
\end{eqnarray}

where the local QCD current $\bar q^{\prime}(x)\, \slashed {n} \gamma_5 \,q(x)$ interpolates the  pseudoscalar meson.
We further introduce the two light-cone vectors $n_{\mu}$ and $\bar n_{\mu}$ satisfying the constraints
$n^2=\bar n^2=0$ and $n \cdot \bar n=2$, which allow us to write down the four-velocity vector of the heavy bottom-meson
$v_{\mu} = p_B /m_B = (n_{\mu} + \bar n_{\mu})/2$.
In order to facilitate the derivation of the soft-collinear factorization formulae for
the three emerged invariant functions $\Pi$, $\tilde \Pi$ and $\Pi_{\rm T}$,
we adopt the following power counting  scheme for the  interpolating-current momentum  and the light-quark masses

\begin{eqnarray}
n \cdot p \sim {\cal O}(m_b) \,, \qquad \bar n \cdot p \sim {\cal O}(\Lambda_{\rm QCD}) \,,
\qquad m_q \sim m_{q^{\prime}} \sim {\cal O}(\Lambda_{\rm QCD}) \,.
\label{power-counting scheme}
\end{eqnarray}

Performing the two-step matching program ${\rm QCD} \to {\rm SCET_{I}} \to {\rm SCET_{II}}$
for the correlation function (<ref>) in sequence leads to  the familiar factorization formulae
at leading power in the heavy quark expansion [34, 36]

\begin{eqnarray}
\Pi &=& {\cal F}_B(\mu) \, m_B \sum \limits_{k=\pm} \,
{\cal C}^{(k)}(n \cdot p, \mu) \, \int_0^{\infty} {d \omega \over \omega- \bar n \cdot p - i 0}~
{\cal J}^{(k)}\left({\mu^2 \over n \cdot p \, \omega},{\omega \over \bar n \cdot p}\right) \,
\phi_B^{k}(\omega,\mu)  \,, \nonumber \\
\tilde \Pi &=& {\cal F}_B(\mu) \, m_B \sum \limits_{k=\pm} \,
{\cal \widetilde{C}}^{(k)}(n \cdot p, \mu) \, \int_0^{\infty} {d \omega \over \omega- \bar n \cdot p - i 0}~
{\cal \widetilde{J}}^{(k)}\left({\mu^2 \over n \cdot p \, \omega},{\omega \over \bar n \cdot p}\right) \,
\phi_B^{k}(\omega,\mu)  \,, \nonumber \\
\Pi_{\rm T} &=&  {\cal F}_B(\mu) \, m_B \sum \limits_{k=\pm} \,
{\cal C}^{(k)}_{\rm T}(n \cdot p, \mu, \nu) \, \int_0^{\infty} {d \omega \over \omega- \bar n \cdot p - i 0}~
{\cal J}^{(k)}_{\rm T}\left({\mu^2 \over n \cdot p \, \omega},{\omega \over \bar n \cdot p}\right) \,
\phi_B^{k}(\omega,\mu) \,.
\label{LP factorization formulae for the correlators}
\end{eqnarray}

Including the light spectator-quark mass corrections in the factorized expressions  apparently cannot affect
the hard coefficients from the first-step matching  ${\rm QCD} \to {\rm SCET_{I}}$
in the leading-power approximation,
which can be actually determined from the perturbative matching of the heavy-to-light current
by integrating out the hard fluctuation modes with virtualities of order $m_b^2$ [78, 79].
However, the non-vanishing spectator-quark mass can indeed result in the leading-power contributions
to the hard-collinear functions from the ${\rm SCET_{I}} \to {\rm SCET_{II}}$ matching
with the adopted power-counting scheme (<ref>)

\begin{eqnarray}
{\cal \widetilde{J}}^{(+)}  &=& {\alpha_s \, C_F \over 4 \pi}
\bigg  \{ \left [  r \, \left (1 -  {\bar n \cdot p \over \omega} \right )  + {m_q + 2 \,m_{q^{\prime}} \over \omega} \right ]
\ln \left (1- {\omega \over \bar n \cdot p} \right )
- {m_{q^{\prime}} \over \omega} \,   \ln^2 \left (1- {\omega \over \bar n \cdot p} \right )
{\omega -  \bar n \cdot p \over \omega}
\nonumber \\
&& - {m_{q^{\prime}} \over \omega} \, \left [ 2 \, \ln { \mu^2 \over  n \cdot p (\omega- \bar n \cdot p) } + 5 \right ] \,
\left [  \ln \left (1- {\omega \over \bar n \cdot p} \right )  \,
{\omega -  \bar n \cdot p \over \omega} - 1 \right ]
\bigg \} \,,
\nonumber \\
{\cal J}^{(+)}_{\rm T}  &=&  {\alpha_s \, C_F \over 4 \pi}
\bigg  \{ \left [ - \, \left (1 -  {\bar n \cdot p \over \omega} \right ) + {m_q + 2 \,m_{q^{\prime}} \over \omega} \right ]
\ln \left (1- {\omega \over \bar n \cdot p} \right )
- {m_{q^{\prime}} \over \omega} \,   \ln^2 \left (1- {\omega \over \bar n \cdot p} \right )
{\omega -  \bar n \cdot p \over \omega}
\nonumber \\
&& - {m_{q^{\prime}} \over \omega} \, \left [ 2 \, \ln { \mu^2 \over  n \cdot p (\omega- \bar n \cdot p) } + 5 \right ] \,
\left [  \ln \left (1- {\omega \over \bar n \cdot p} \right )  \,
{\omega -  \bar n \cdot p \over \omega} - 1 \right ]
\bigg \} \,,
\label{spectataor-quark mass corrections}
\end{eqnarray}

where we have introduced the dimensionless kinematic variable $r=n \cdot p /m_b$.
It is straightforward to verify the factorization-scale independence of
the derived factorization formulae by employing the RG evolution equation for
the twist-three $B_{q^{\prime}}$-meson  distribution amplitude $\phi_B^{-}(\omega, \mu)$
in the absence of the three-particle LCDA contribution  at one loop [80]

\begin{eqnarray}
{d \over d \ln \mu} \phi_B^{-}(\omega, \mu) &=&
- {\alpha_s \over 4 \pi}  \, \int_0^{\infty} d \omega^{\prime} \,
\left [ \gamma_{+}^{(1)}(\omega, \omega^{\prime}, \mu)
- \Gamma_{\rm cusp}^{(0)} \, \frac{\theta(\omega^{\prime} - \omega)}{\omega^{\prime} } \right ]
\phi_B^{-}(\omega, \mu) \,
\nonumber \\
&& -  {\alpha_s \over 4 \pi}  \, \int_0^{\infty} d \omega^{\prime} \, \Gamma_{\rm cusp}^{(0)} \,
\left [ {m_{q^{\prime}} \, \theta(\omega^{\prime} - \omega) \over \omega^{\prime 2}}  \right ]_{\oplus}  \,
\phi_B^{+}(\omega, \mu) + {\cal O}(\alpha_s^2) \,,
\end{eqnarray}

where the perturbative kernel $\gamma_{+}^{(1)}$ and the cusp anomalous dimension $\Gamma_{\rm cusp}^{(0)}$
can be written as

\begin{eqnarray}
&& \gamma_{+}^{(1)} = \left \{  \left ( \Gamma_{\rm cusp}^{(0)} \, \ln {\mu \over \omega} -2  \right ) \, \delta( \omega - \omega^{\prime} )
- \Gamma_{\rm cusp}^{(0)} \, \left [ \frac{\omega \,  \theta(\omega^{\prime}-\omega)}{\omega^{\prime} \, (\omega^{\prime}-\omega)}
+ \frac{\theta(\omega-\omega^{\prime})}{(\omega-\omega^{\prime})}  \right ]_{\oplus} \right \} \,  \, C_F \,,
\nonumber \\
&& \Gamma_{\rm cusp}^{(0)} = 4 \, C_F  \,,
\end{eqnarray}

with the $\oplus$-function defined by

\begin{eqnarray}
\int_0^{\infty} \, d \omega^{\prime} \, \left [ f(\omega,\omega^{\prime}) \right ]_{\oplus} \, g(\omega^{\prime})
= \int_0^{\infty} \, d \omega^{\prime} \, f(\omega,\omega^{\prime})  \,
\left [ g(\omega^{\prime}) - g(\omega) \right ] \,.
\end{eqnarray}

Importantly, the yielding spectator-quark mass corrections to the hard-collinear matching coefficients
appear to be universal for the two vacuum-to-$B$-meson correlation functions
with distinct weak transition currents.
This interesting pattern can be attributed to the very fact that only the one-loop correction to
the light-pseudoscalar-meson vertex diagram can bring about the non-vanishing spectator-quark mass effect
at leading power, thus validating the earlier speculation on the differences between
the  heavy-to-light form factors for exclusive $B$-meson and $B_s$-meson decays [81].
The remaining short-distance functions in the factorized correlation functions
(<ref>) have been determined in [36]
at the one-loop accuracy.

Inspecting the obtained soft-collinear factorization formulae for the considered invariant functions
indicates that there is no common choice of the factorization scale to get rid of the parametrically
large logarithms of $m_b/\Lambda_{\rm QCD}$.
Adopting the factorization scale of order $\sqrt{m_b \, \Lambda_{\rm QCD}}$, we are then required  to
perform an all-order summation of such enhanced logarithms entering in both the hard matching coefficients
and the two-particle bottom-meson distribution amplitudes.
Taking advantage of the momentum-space RG equations for ${\cal \widetilde{C}}^{(-)}$,
${\cal C}^{(-)}_{\rm T}$ and the HQET decay constant ${\cal F}_B$
(expressible in terms of the QCD decay constant $f_B$ and the matching coefficient $K(\mu)$ [82])
allows us to  derive the desired scale dependence of these quantities in the following form

\begin{eqnarray}
&& {\cal F}_B (\mu)  =  \hat{U}_2(\mu_{h2}, \, \mu) \,\,  {\cal F}_B (\mu_{h2})  \,,
\qquad
{\cal \widetilde{C}}^{(-)}(n \cdot p, \mu) = \hat{U}_1(n \cdot p, \, \mu_{h1}, \, \mu)
\,\,  {\cal \widetilde{C}}^{(-)}(n \cdot p, \mu_{h1}) \,,
\nonumber \\
&& {\cal C}^{(-)}_{\rm T}(n \cdot p, \mu, \nu)  =  U_1(n \cdot p, \, \mu_{h1}, \, \mu)
\,\,  \hat{U}_3(\nu_h, \nu)  \, {\cal C}_{\rm T}^{(-)}(n \cdot p, \mu_{h1}, \nu_h) \,.
\end{eqnarray}

The manifest expressions of the QCD evolution function $\hat{U}_1(n \cdot p, \, \mu_{h1}, \, \mu)$
can be obtained from the expanded result of $U_1(E_{\gamma}, \, \mu_{h}, \, \mu)$ presented in [83]
with the replacement $E_{\gamma} \to n \cdot p /2$.
The two additional  RG functions $\hat{U}_2$ and $\hat{U}_3$ at the NLL accuracy are given by

\begin{eqnarray}
\hat{U}_2(\mu_{h2}, \, \mu) &=& z_2^{- \frac{\gamma_K^{(0)}}{2 \,\beta_0}} \bigg [1+ \frac{\alpha_s(\mu_{h2})}{4 \pi}  \,
\left (  {\gamma_K^{(1)} \over 2 \, \beta_0} - {\gamma_K^{(0)} \, \beta_1 \over 2 \, \beta_0^2 } \right ) (1-z_2)
+{\cal O}(\alpha_s^2) \bigg ]  \,,
\nonumber  \\
\hat{U}_3(\nu_h, \nu) &=& z_3^{- \frac{\gamma_{\rm T}^{(0)}}{2 \,\beta_0}} \bigg [1+ \frac{\alpha_s(\nu_{h})}{4 \pi}  \,
\left (  {\gamma_{\rm T}^{(1)} \over 2 \, \beta_0} - {\gamma_{\rm T}^{(0)} \, \beta_1 \over 2 \, \beta_0^2 } \right ) (1-z_3)
+{\cal O}(\alpha_s^2) \bigg ]  \,,
\end{eqnarray}

where the necessary anomalous dimensions for the HQET heavy-to-light current
and for the QCD tensor current are  [84, 85, 86]

\begin{eqnarray}
\gamma_K^{(0)} =  3 \, C_F  \,,  & \qquad &
\gamma_K^{(1)} =  C_F \, \left [ {127 \over 6}  + {14 \, \pi^4 \over 9} - {5 \over 3} \, n_f \right ]  \,,
\nonumber \\
\gamma_{\rm T}^{(0)} =  -2 \, C_F  \,,   & \qquad &
\gamma_{\rm T}^{(1)} =   C_F \, \left [ 19 \, C_F  -  {257 \over 9} \, C_A  + {26 \over 9} \, n_f \,\right ]  \,,
\end{eqnarray}

with the conventions $z_2=\alpha_s(\mu)/\alpha_s(\mu_{h2})$ and $z_3=\alpha_s(\nu)/\alpha_s(\nu_{h})$.
Applying the dual-space representations for the $B$-meson distribution amplitudes constructed in [87, 88]

\begin{eqnarray}
\phi_B^{+}(\omega, \mu) &=& \int_0^{+\infty} d s  \, \sqrt{\omega \, s } \, J_1 (2 \,\sqrt{\omega \, s } ) \, \eta_{+}(s, \mu) \,,
\nonumber \\
\phi_B^{-}(\omega, \mu) &=& \int_0^{+\infty} d s  \, \sqrt{\omega \, s } \, J_0 (2 \,\sqrt{\omega \, s } ) \,
\left [  \eta_{+}(s, \mu) + \eta_{3}^{(0)}(s, \mu)\right ] \,,
\end{eqnarray}

both the twist-two and twist-three coefficient functions $\eta_{+}(s, \mu)$ and $\eta_{3}^{(0)}(s, \mu)$ are observed to
possess the autonomous scale dependence [89]

\begin{eqnarray}
\eta_{+}(s, \mu) &=& U_{\phi}^{\rm tw2}(s, \mu, \mu_0) \, \eta_{+}(s, \mu_0) \,,
\nonumber \\
\eta_{3}^{(0)}(s, \mu) &=& U_{\phi}^{\rm tw3}(s, \mu, \mu_0) \, \eta_{3}^{(0)}(s, \mu_0) \,.
\end{eqnarray}

The analytical results of the two evolution factors $U_{\phi}^{\rm tw2}(s, \mu, \mu_0)$ and $ U_{\phi}^{\rm tw3}(s, \mu, \mu_0)$
can be further written as [72, 89]

\begin{eqnarray}
U_{\phi}^{\rm tw2}(s, \mu, \mu_0) &=&  {\rm exp} \left \{ -{ \Gamma_{\rm cusp}^{(0)}  \over 4 \, \beta_0^2} \,
\left [ \ln z_0 - 1 +{1 \over z_0}  \right ] - {\beta_1 \over 2 \beta_0^2} \, \ln^2 z_0
+ \left ( {\Gamma_{\rm cusp}^{(1)} \over \Gamma_{\rm cusp}^{(0)}} - {\beta_1 \over 2 \beta_0} \right ) \,
\left [ z_0 - 1 - \ln z_0 \right ]  \right \}
\nonumber \\
&& \times \, \left ( s \, e^{2 \, \gamma_E} \, \mu_0 \right )^{\Gamma_{\rm cusp}^{(0)} \, \ln z_0 / (2 \, \beta_0)} \,\,
z_0^{\gamma_{\rm tw2}^{(0)} / (2 \, \beta_0)}\,,
\nonumber \\
U_{\phi}^{\rm tw3}(s, \mu, \mu_0) &=& z_0^{\gamma_{\rm tw3}^{(0)} / (2 \, \beta_0)} \, U_{\phi}^{\rm tw2}(s, \mu, \mu_0)  \,,
\end{eqnarray}

where  the dimensionless quantity $z_0=\alpha_s(\mu) /  \alpha_s(\mu_0)$
and the newly appeared  anomalous dimensions $\Gamma_{\rm cusp}^{(0)}$, $\Gamma_{\rm cusp}^{(1)}$,
$\gamma_{\rm tw2}^{(0)}$ and  $\gamma_{\rm tw3}^{(0)}$ are explicitly given by

\begin{eqnarray}
\Gamma_{\rm cusp}^{(0)} = 4 \, C_F  \,,
\hspace{0.5 cm}
\Gamma_{\rm cusp}^{(1)} =  C_F \, \left [ {268 \over 3}  - 4 \, \pi^2  - {40 \over 9} \, n_f \right ] \,,
\hspace{0.5 cm}
\gamma_{\rm tw2}^{(0)} = -2 \, C_F \,,
\hspace{0.5 cm}
\gamma_{\rm tw3}^{(0)} = 2 \, N_c \,.
\end{eqnarray}

In order to construct  the SCET sum rules for the exclusive heavy-to-light form factors,
we  proceed to derive the hadronic dispersion relations for the vacuum-to-$B$-meson correlation functions
with the aid of the conventional parameterizations for the bottom-meson decay matrix elements
collected in  (<ref>)

\begin{eqnarray}
\Pi_{\mu, {\rm V}} (n \cdot p, \bar n \cdot p)  &=& \left ( {1 \over 2} \right ) \,
\frac{f_{P} \, m_B}{m_{M}^2/ n \cdot p - \bar n \cdot p}
\bigg \{  \bar n_{\mu} \, \left [ \frac{n \cdot p}{m_B} \, f_{BM}^{+} (q^2) + f_{BM}^{0} (q^2)  \right ]
\nonumber \\
&& +  \,  n_{\mu} \, \frac{m_B}{n \cdot p-m_B}  \, \,
\left [ \frac{n \cdot p}{m_B} \, f_{BM}^{+} (q^2) -  f_{B M}^{0} (q^2)  \right ] \bigg \} \, \nonumber \\
&& + \, \int_{\omega_s}^{+\infty}   \, \frac{d \omega^{\prime} }{\omega^{\prime} - \bar n \cdot p - i 0} \,
\left [ \rho_{V, 1}^{h}(\omega^{\prime}, n \cdot p)  \, n_{\mu} \,
+\rho_{V, 2}^{h}(\omega^{\prime}, n \cdot p)  \, \bar{n}_{\mu}  \right ] \,,
\nonumber \\
\Pi_{\mu, {\rm T}} (n \cdot p, \bar n \cdot p)  &=&
\left ( - {i \over 2}  \right) \,  \left [ \bar n \cdot q \, n_\mu - n \cdot q \, \bar n_\mu  \right ]  \,
\bigg \{  \frac{f_{M} \, n \cdot p }{m_{M}^2/ n \cdot p - \bar n \cdot p}
\, \left [ {m_B \over m_B+m_M} \,  f_{B M}^{T} (q^2) \right ] \nonumber \\
&& + \, \int_{\omega_s}^{+\infty}   \, \frac{d \omega^{\prime} }{\omega^{\prime} - \bar n \cdot p - i 0} \,\,
\rho_{\rm T}^{h}(\omega^{\prime}, n \cdot p) \bigg \}  \,,
\label{hadronic dispersion relations}
\end{eqnarray}

where we adopt  the standard definition for  the decay constant of the pseudoscalar meson [90]

\begin{eqnarray}
\langle 0 |\bar q^{\prime}\, \slashed {n} \gamma_5 \,q | M(p)\rangle = i \, n \cdot p \, f_M.
\end{eqnarray}

Evidently,  $\Pi_{\mu, {\rm V}}$ and $\Pi_{\mu, {\rm T}}$ correspond to $\Gamma_\mu = \gamma_\mu$
and $\Gamma_\mu = \sigma_{\mu \nu} q^\nu$ for the particular spin structure of the weak current
$\bar q(0)\, \Gamma_\mu \,b(0)$ in the definition (<ref>), respectively.
Matching the spectral representations of the NLL resummation improved factorization formulae
with the obtained hadronic dispersion relation (<ref>)
and implementing further the Borel transformation in the variable $\bar n \cdot p \to \omega_M$,
we can readily derive the NLL sum rules for the semileptonic $B_{d, s}  \to \pi, K$  decay form factors
in the leading-power approximation

\begin{eqnarray}
&& f_{M} \,\, {\rm exp} \left [- {m_{M}^2 \over n \cdot p \,\, \omega_M} \right ] \,\,
\left \{ \frac{n \cdot p} {m_B} \, f_{B M, \, \rm LP}^{+}(q^2)
\,, \,\,\,   f_{B M, \, \rm LP}^{0}(q^2)  \right \}  \,  \nonumber \\
&& =   \left [ \hat{U}_2(\mu_{h2}, \mu) \, \mathcal{F}_B(\mu_{h2}) \right ]
\,\, \int_0^{\omega_s} \,\, d \omega^{\prime} \, e^{-\omega^{\prime}/\omega_M} \,
\nonumber \\
&&  \hspace{0.4 cm} \times \, \bigg \{ \widetilde{\bf \Phi}_{B, \, \rm {eff}}^{+} (\omega^{\prime}, \mu)
+  \, \left [  \hat{U}_1(n \cdot p, \mu_{h1}, \mu) \,\, \widetilde{\cal{C}}^{(-)}(n \cdot p, \mu_{h1}) \right ]\,
\widetilde{\bf \Phi}_{B, \, \rm {eff}}^{-} (\omega^{\prime}, \mu) \nonumber \\
&& \hspace{1.0 cm} \pm \, { n \cdot p - m_B \over m_B} \,
\left [{\bf \Phi}_{B, \, \rm {eff}}^{+} (\omega^{\prime}, \mu)
+ {\cal C}^{(-)}(n \cdot p, \mu_{h1}) \, {\bf \Phi}_{B, \, \rm {eff}}^{-} (\omega^{\prime}, \mu)   \right ]  \bigg \} \,,
\label{NLL sum rules at LP I}
\\
%\nonumber \\
&& f_{M} \,\,  {\rm exp} \left [- {m_{M}^2 \over n \cdot p \,\, \omega_M} \right ]  \,\,
\frac{n \cdot p} {m_B+m_M} \, f_{B M, \, \rm LP}^{T}(q^2) \,  \nonumber \\
&& =   \left [ \hat{U}_2(\mu_{h2}, \mu) \, {\cal F}_B(\mu_{h2}) \right ]
\,\, \int_0^{\omega_s} \,\, d \omega^{\prime} \, e^{-\omega^{\prime}/\omega_M} \,  \nonumber \\
&&  \hspace{0.4 cm} \times \bigg \{ \widehat{\bf \Phi}_{B, \, \rm {eff}}^{+} (\omega^{\prime}, \mu)
+  \,\left [ \hat{U}_1(n \cdot p, \mu_{h1}, \mu) \,\, \hat{U}_3(\nu_{h}, \nu) \,\,
{\cal C}^{(-)}_{\rm T}(n \cdot p, \mu_{h1}, \nu_{h})  \right ] \,
\widetilde{\bf \Phi}_{B, \, \rm {eff}}^{-} (\omega^{\prime}, \mu)  \bigg \} \,.
\hspace{0.8 cm}
\label{NLL sum rules at LP II}
\end{eqnarray}

For brevity we have introduced the effective bottom-meson “distribution amplitudes"
absorbing the hard-collinear strong interaction dynamics into the standard HQET soft functions

\begin{eqnarray}
\widetilde{\bf \Phi}_{B, \, \rm {eff}}^{+} &=&
{\alpha_s \, C_F \over 4 \, \pi} \,
\bigg \{   r \, \int_{\omega^{\prime}}^{\infty} \, d \omega \,
{\phi_B^{+}(\omega, \mu) \over \omega}
- \left ( m_q + 2 \, m_{q^{\prime}} \right ) \, \int_{\omega^{\prime}}^{\infty} \, d \omega \,
\ln \left ( {\omega -  \omega^{\prime} \over \omega^{\prime}} \right )   \,\,
{d \over d \omega} \, {\phi_B^{+}(\omega, \mu) \over \omega}
\nonumber \\
&&  \hspace{-0.8 cm}
- \, 2 \, m_{q^{\prime}}  \, \int_{0}^{\infty} \, {d \omega \over \omega } \,
\left [ \theta(\omega - \omega^{\prime}) \, \left ( \ln{\mu^2  \over n \cdot p \, \omega^{\prime} }
+ {5 \over 2} \right ) + \theta(\omega^{\prime} - \omega) \,
\ln \left ( {\omega^{\prime}  - \omega \over \omega^{\prime} } \right ) \right ]  \,
\, {d \phi_B^{+}(\omega, \mu)  \over d \omega}   \bigg \},
\nonumber \\
\widetilde{\bf \Phi}_{B, \, \rm {eff}}^{-} &=&
\phi_{B}^{-}(\omega^{\prime}, \mu)
+  \frac{\alpha_s \, C_F}{4 \, \pi} \,\, \bigg \{ \int_0^{\omega^{\prime}} \,\, d \omega \,\,\,
\left [ {2 \over \omega - \omega^{\prime}}  \,\,\, \left (\ln {\mu^2 \over n \cdot p \, \omega^{\prime}}
- 2 \, \ln {\omega^{\prime} - \omega \over \omega^{\prime}} \right )\right ]_{\oplus} \,
\phi_{B}^{-}(\omega, \mu)
\nonumber \\
&&  \hspace{-1.5 cm}
- \int_{\omega^{\prime}}^{\infty} \,\, d \omega \,\,\,
\bigg [ \ln^2 {\mu^2 \over n \cdot p \, \omega^{\prime}}
- \left ( 2 \, \ln {\mu^2 \over n \cdot p \, \omega^{\prime}}  + 3 \right ) \,\,
\ln {\omega - \omega^{\prime} \over \omega^{\prime}}
+ \, 2 \,\, \ln {\omega \over \omega^{\prime}}   + {\pi^2 \over 6} - 1 \bigg ]
\, {d \phi_{B}^{-}(\omega, \mu) \over d \omega}  \bigg \},
\nonumber \\
{\bf \Phi}_{B, \, \rm {eff}}^{+} &=&  {\alpha_s \, C_F \over 4 \, \pi} \,
\int_{\omega^{\prime}}^{\infty} \, d \omega \, {\phi_B^{+}(\omega, \mu) \over \omega}  \,,
\hspace{3.0 cm}
{\bf \Phi}_{B, \, \rm {eff}}^{-}  = \phi_B^{-}(\omega, \mu),
\\
\widehat{\bf \Phi}_{B, \, \rm {eff}}^{+}  &=&  {\alpha_s \, C_F \over 4 \, \pi} \,
\bigg \{  - \int_{\omega^{\prime}}^{\infty} \, d \omega \,
{\phi_B^{+}(\omega, \mu) \over \omega}
- \left ( m_q + 2 \, m_{q^{\prime}} \right ) \, \int_{\omega^{\prime}}^{\infty} \, d \omega \,
\ln \left ( {\omega -  \omega^{\prime} \over \omega^{\prime}} \right )   \,\,
{d \over d \omega} \, {\phi_B^{+}(\omega, \mu) \over \omega}
\nonumber \\
&&  \hspace{-0.8 cm}
- \, 2 \, m_{q^{\prime}}  \, \int_{0}^{\infty} \, {d \omega \over \omega } \,
\left [ \theta(\omega - \omega^{\prime}) \, \left ( \ln{\mu^2  \over n \cdot p \, \omega^{\prime} }
+ {5 \over 2} \right ) + \theta(\omega^{\prime} - \omega) \,
\ln \left ( {\omega^{\prime}  - \omega \over \omega^{\prime} } \right ) \right ]  \,
\, {d \phi_B^{+}(\omega, \mu)  \over d \omega}   \bigg \}.
\nonumber
\end{eqnarray}

It remains interesting to point out that the newly computed spectator-quark mass corrections
preserve the so-called large-recoil symmetry relations for the soft contributions to the exclusive
$B_{d, s}  \to \pi, K$ form factors at leading power in the heavy quark expansion
(see [91] for further discussions).
Bearing in  mind the scaling behaviour of the light-cone variable
$\omega^{\prime} \sim \omega_s \sim {\cal O}(\Lambda_{\rm QCD}^2/m_b)$
in the established sum rules (<ref>) and (<ref>),
we can immediately observe that $\ln \left [(\omega-\omega^{\prime}) / \omega^{\prime} \right ]$
entering in the nonperturbative functions $\widetilde{\bf \Phi}_{B, \, \rm {eff}}^{+}$
and $\widehat{\bf \Phi}_{B, \, \rm {eff}}^{+}$
must be counted as the large logarithm $\ln \left ( m_b / \Lambda_{\rm QCD} \right )$ in the heavy quark limit.
The very appearance of such logarithmic term  in the yielding SCET sum rules further implies that
evaluating the spectator-quark mass contribution with the perturbative factorization technique straightforwardly
will  give rise to  the soft-collinear convolution integrals with unwanted end-point singularities
(see [92, 93, 94, 95, 96, 97]
for the interesting progress on exploring the end-point dynamics
and tackling the rapidity logarithms in the different contexts).

§ THE  LCSR FOR THE EXCLUSIVE $B_{D, S}  \TO \PI, K$ FORM FACTORS BEYOND LEADING POWER

We are now in a position to investigate the power-suppressed corrections to the exclusive bottom-meson decay form factors
from a variety of distinct sources  by applying the LCSR method with
the  higher-twist HQET distribution amplitudes.
To this end, we will need to construct the subleading-power factorization formulae for the vacuum-to-$B$-meson
correlation functions and take advantage of the non-trivial  identities for the two-body and three-body light-ray HQET operators
due to the classical equations of motion [70, 71, 72].

§.§ The NLP contribution form the hard-collinear propagator

The subleading power two-particle and three-particle corrections
to the vacuum-to-bottom-meson correlation function (<ref>)
at the tree-level accuracy,
where the square box indicates an insertion of the weak vertex $\bar{q} \, \Gamma_\mu  \, b$
and the waveline stands for the interpolating current
$\bar q^{\prime} \, \slashed {n} \, \gamma_5 \, q$ for the light pseudoscalar meson.

The first class of the subleading power contribution arises from retaining the higher-order terms
in the heavy quark expansion of the hard-collinear quark propagator as depicted in Figure <ref>.
Applying the computational strategy discussed in [98, 99],
we proceed  to write down the two-particle contribution to the QCD correlation function (<ref>)
at the tree-level accuracy

\begin{eqnarray}
\Pi_{\mu, {\rm V}, \rm NLP}^{\rm hc}  (n \cdot p, \bar n \cdot p) & = &  i \int d^4x \int \frac{d^4 k}{(2 \pi)^4}
{\rm exp} \left (i \, k \cdot x \right )  \, \frac{1} {(p-k)^2 - m_q^2 +i0} \,
\nonumber \\
&& \times \, \left \langle 0 \left | \overline{q}^{\prime}(x) \, \slashed{n} \, \gamma_5 \, ( \slashed{p} - \slashed{k} + m_q ) \,
\gamma_\mu \, h_v(0) \right | \bar B(p_B) \right \rangle \,.
\label{2P NLP correction at LO}
\end{eqnarray}

Implementing an expansion in powers of $\Lambda_{\rm QCD}/m_b$ for  the  hard-collinear quark propagator
in (<ref>) immediately leads to

\begin{eqnarray}
\frac{(\slashed{p} - \slashed{k}) + m_q} {(p-k)^2 - m_q^2 + i0}
&=&\underbrace{\frac{1}{\bar n \cdot (p-k)} \, {\slashed{\bar n}  \over 2}}
+ \underbrace{{1 \over (p-k)^2 } \, \left [ \bar n \cdot p \,  {\slashed{n}  \over 2}
- \slashed{k} + \frac{n \cdot k \,\,  \bar n \cdot p}{\bar n \cdot (p-k)} \,  {\slashed{\bar n}  \over 2}  \right ]}
\nonumber \\
&& \hspace{1.0 cm} {\rm LP}  \hspace{4.2 cm} {\rm NLP}
\nonumber \\
&& + \underbrace{{1 \over (p-k)^2 } \,
\left [ m_q + {m_q^2 - m_{q^{\prime}}^2 \over \bar n \cdot (p-k)} \,  {\slashed{\bar n}  \over 2}   \right ]}
+ \, ...,
\nonumber \\
&& \hspace{1.5 cm} {m_{q^{(\prime)}} \,\, \rm  NLP}
\label{NLP terms of hard-collinear propagator}
\end{eqnarray}

where the abbreviation “LP" represents the  leading-power effect discussed in Section <ref>.
We can further cast the subleading-power terms displayed in the first line of (<ref>)
in the form

\begin{eqnarray}
\Pi_{\mu, {\rm V}, \rm NLP}^{\rm hc, \, I} (n \cdot p, \bar n \cdot p)
& = &  \int d^4x \int \frac{d^4 k}{(2 \pi)^4} \,
{\rm exp} \left (i \, k \cdot x \right )  \, \frac{1} {(p-k)^2 + i0} \,
\nonumber \\
&& \times \, \bigg \{ {\partial \over \partial x_{\rho}} \,
\left \langle 0 \left | \overline{q}^{\prime}(x) \, \slashed{n} \, \gamma_5 \, \gamma_{\rho} \,
\gamma_\mu \, h_v(0) \right | \bar B(p_B) \right \rangle
\nonumber \\
&& \hspace{0.2 cm} - { \bar n \cdot p \over \bar n \cdot (p-k)} \,
(2 \, v_{\rho} - \bar n_{\rho}) \,
{\partial \over \partial x_{\rho}} \,
\left \langle 0 \left | \overline{q}^{\prime}(x) \, \slashed{n} \, \gamma_5 \, {\slashed{\bar n} \over 2} \,
\gamma_\mu \, h_v(0) \right | \bar B(p_B) \right \rangle  \bigg \},
\hspace{1.0 cm}
\end{eqnarray}

by employing the standard technique of the integration by parts (IBP) as well as the precise relation between
the four-vectors  $n_{\alpha} = 2 \, v_{\alpha} - \bar n_{\alpha}$.
Taking advantage of the well-known operator identities due to the HQET equations of motion

\begin{eqnarray}
v_{\rho} \,  {\partial \over \partial x_{\rho}} \,
\left [  \bar q^{\prime}(x) \,\Gamma \,  h_v(0) \right ]
&=& i \, \int_0^1 d u \, \bar u \, \bar q^{\prime}(x) \,g_s \, G_{\alpha \beta}(u x) \,
x^{\alpha} \, v^{\beta} \, \Gamma \,  h_v(0)
+ (v \cdot \partial) \,  \left [  \bar q^{\prime}(x) \,\Gamma \,  h_v(0) \right ]  \,,
\hspace{0.8 cm}
\label{the first HQET EOMs}
\\
{\partial \over \partial x_{\rho}} \bar q^{\prime}(x) \, \gamma_{\rho} \, \Gamma \,   h_v(0)
&=& - i \, \int_0^1 d u \, u \,\, \bar q^{\prime}(x) \,g_s \, G^{\lambda \rho}(u x) \,
x_{\lambda} \, \gamma_{\rho} \, \Gamma \,  h_v(0)
+ i \, m_{q^{\prime}} \, \bar q^{\prime}(x) \Gamma \,  h_v(0)  \,,
\label{the second HQET EOMs}
\end{eqnarray}

where the total translation operator $\partial_{\rho}$ acting on an arbitrary composite operator
${\cal O}(x_1, ..., x_n)$ with $n$ space-time arguments is defined by

\begin{eqnarray}
\partial_{\rho} \, {\cal O}(x_1, ..., x_n)
= {\partial \over \partial y^{\rho}} \,
{\cal O}(x_1+y, ..., x_n+y) \bigg|_{y=0}\,,
\end{eqnarray}

we can then readily derive the factorized expressions for the  effective NLP  matrix elements
of our interest at tree level

\begin{eqnarray}
\Pi_{\rm NLP}^{\rm hc, \, I}
& = &  \left [ { 2 \, {\cal F}_B(\mu) \, m_B \over n \cdot p } \right ] \,
\bigg \{ \int_0^{\infty} d \omega_1 \, \int_0^{\infty} d \omega_2 \, \int_0^{1}  d u \,
\frac{u \, \Phi_4(\omega_1, \omega_2, \mu) + \Psi_4(\omega_1, \omega_2, \mu)}
{\left [ \bar n \cdot p - \omega_1 - u \, \omega_2 \right ]^2} \,
\nonumber \\
&& + \int_0^{\infty} d \omega \, \left ( \bar{\Lambda}  - {\omega + m_{q^{\prime}} \over 2}  \right ) \,
\frac{\phi_B^{+}(\omega, \mu)}{\left [ \bar n \cdot p - \omega \right ]} \bigg \}
+  {\cal O}(\alpha_s)\,,
\hspace{1.0 cm} \\
\tilde{\Pi}_{\rm NLP}^{\rm hc, \, I}
& = &  \left [ - { 2 \, {\cal F}_B(\mu) \, m_B \over n \cdot p } \right ] \,
\bigg \{ \int_0^{\infty} d \omega_1 \, \int_0^{\infty} d \omega_2 \, \int_0^{1}  d u \,
\frac{\bar u \, (\bar n \cdot p + \omega_1 + u \, \omega_2 ) }
{\left [ \bar n \cdot p - \omega_1 - u \, \omega_2 \right ]^3}
\,  \Psi_5(\omega_1, \omega_2, \mu)
\nonumber \\
&& + \int_0^{\infty} d \omega \, \left ( \bar{\Lambda}  - {\omega \over 2}  \right ) \,
\frac{\omega \, \phi_B^{+}(\omega, \mu)}{\left [ \bar n \cdot p - \omega \right ]^2} \bigg \}
+  {\cal O}(\alpha_s) \,.
\end{eqnarray}

The hadronic parameter $\bar{\Lambda}$ characterizing the “effective mass" of the bottom-meson state in HQET
can be defined in an explicitly covariant and gauge invariant manner [100]

\begin{eqnarray}
\bar \Lambda \equiv \frac{\langle 0 | \bar q \,\,  i \, v \cdot \overleftarrow{D} \, \Gamma \,   h_v | \bar B_q (v) \rangle}
{\langle 0 | \bar q \,  \Gamma \,   h_v | \bar B_q (v) \rangle} \,.
\label{def: effective bottom-meson mass}
\end{eqnarray}

Additionally, we have adopted the systematic parametrization of the three-body light-cone HQET matrix element
at the twist-six accuracy [72]

\begin{eqnarray}
&& \langle 0 | \bar q_{\alpha}(\tau_1 \, \bar n) \, g_s \, G_{\mu \nu}(\tau_2 \, \bar n) \,
h_{v \, \beta}(0) | \bar B_v \rangle \nonumber \\
&& = {{\cal F}_B(\mu) \, m_B \over 4} \,
\bigg [ (1 + \slashed{v}) \, \bigg \{ (v_{\mu} \gamma_{\nu} - v_{\nu} \gamma_{\mu})  \,
\left [\hat{\Psi}_A(\tau_1, \tau_2, \mu) - \hat{\Psi}_V(\tau_1, \tau_2, \mu) \right ]
- i \, \sigma_{\mu \nu} \, \hat{\Psi}_V(\tau_1, \tau_2, \mu) \nonumber  \\
&& \hspace{0.4 cm}
- (\bar n_{\mu} \, v_{\nu} - \bar n_{\nu} \, v_{\mu} ) \, \hat{X}_A(\tau_1, \tau_2, \mu)
+ (\bar n_{\mu} \, \gamma_{\nu} - \bar n_{\nu} \, \gamma_{\mu} ) \,
\left [ \hat{W}(\tau_1, \tau_2, \mu)  + \hat{Y}_A(\tau_1, \tau_2, \mu)   \right ] \nonumber \\
&& \hspace{0.4 cm} + \, i \, \epsilon_{\mu \nu \alpha \beta} \,
\bar n^{\alpha} \, v^{\beta}  \, \gamma_5 \, \hat{\tilde{X}}_A(\tau_1, \tau_2, \mu)
- \, i \, \epsilon_{\mu \nu \alpha \beta} \,
\bar n^{\alpha} \, \gamma^{\beta}  \, \gamma_5 \, \hat{\tilde{Y}}_A(\tau_1, \tau_2, \mu)  \nonumber \\
&&  \hspace{0.4 cm}  - \, ( \bar n_{\mu} \, v_{\nu} -  \bar n_{\nu} \, v_{\mu} ) \,
\slashed{\bar  n } \, \hat{W}(\tau_1, \tau_2, \mu)
+ \, ( \bar n_{\mu} \, \gamma_{\nu} - \bar n_{\nu} \, \gamma_{\mu} ) \,
\slashed{\bar  n } \, \hat{Z}(\tau_1, \tau_2, \mu)   \bigg \}  \, \gamma_5 \bigg ]_{\beta \, \alpha}  \,.
\label{def: 3-particle HQET  DAs}
\end{eqnarray}

The momentum-space distribution amplitudes can be obtained by carrying out the Fourier transformation
in the two light-cone variables $\tau_{1, 2}$  [72, 101]

\begin{eqnarray}
&& \Psi_{X}(\omega_1, \omega_2, \mu) =  \int_{-\infty}^{+\infty} {d \tau_1 \over 2 \pi}
\, \int_{-\infty}^{+\infty} {d \tau_2 \over 2 \pi} \,\,
{\rm exp} \left [ i (\omega_1 \, \tau_1 + \omega_2 \, \tau_2) \right ]  \,
\hat{\Psi}_{X}(\tau_1 - i 0, \tau_2 - i 0, \mu)  \,,
\nonumber \\
&& \Psi_{X}  \in  \left \{\Psi_V, \, \Psi_A, \, X_A, \, Y_A, \, \tilde{X}_A, \, \tilde{Y}_A, \, W, \, Z\right \},
\,\,
\hat{\Psi}_{X}  \in  \left \{\hat{\Psi}_V, \, \hat{\Psi}_A, \, \hat{X}_A, \, \hat{Y}_A, \, \hat{\tilde{X}}_A, \,
\hat{\tilde{Y}}_A, \, \hat{W}, \, \hat{Z}\right \} \,.
\hspace{1.0 cm}
\end{eqnarray}

To facilitate the construction of the perturbative factorization formulae,
it turns out to be more advantageous to introduce  the three-particle HQET distribution amplitudes  with the definite collinear twist
by virtue of the appearing invariant functions  (see  [102, 103] for further discussions
on the comparison between dynamical twist and  geometric twist )

\begin{eqnarray}
\Phi_3 &=& \Phi_A - \Phi_V \,,
\qquad  \hspace{3.8 cm}
\Phi_4 = \Phi_A + \Phi_V \,,
\nonumber \\
\Psi_4  &=& \Psi_A + X_A \,,
\qquad  \hspace{3.8 cm}
\tilde{\Psi}_4  = \Psi_V - \tilde{X}_A \,,
\nonumber \\
\tilde{\Phi}_5 &=&  \Psi_A + \Psi_V + 2 \, Y_A - 2 \, \tilde{Y}_A + 2 \, W \,,
\qquad
\Psi_5 =  - \Psi_A + X_A - 2 \, Y_A  \,,
\nonumber \\
\tilde{\Psi}_5 &=&  - \Psi_V -  \tilde{X}_A + 2 \, \tilde{Y}_A  \,,
\qquad  \hspace{2.2 cm}
\Phi_6 =  \Phi_A - \Phi_V + 2 \, Y_A  + 2 \, W
+ 2 \, \tilde{Y}_A - 4 \, Z \,.
\hspace{1.0 cm}
\end{eqnarray}

Applying the standard factorization method, we can compute the power-suppressed contributions
presented in the second line of (<ref>)
due to the non-vanishing quark masses

\begin{eqnarray}
\Pi_{\rm NLP}^{\rm hc, \, II}
& = &  \left [ - { {\cal F}_B(\mu) \, m_B \over n \cdot p } \right ] \,
\int_0^{\infty} d \omega \,
\frac{m_q} {\bar n \cdot p - \omega }  \, \phi_B^{-}(\omega, \mu) +  {\cal O}(\alpha_s)  \,,
\\
\tilde{\Pi}_{\rm NLP}^{\rm hc, \, II}
& = &  \left [ - { {\cal F}_B(\mu) \, m_B \over n \cdot p } \right ] \,
\int_0^{\infty} d \omega \,
\frac{\left (m_q^2  - m_{q^{\prime}}^2 \right  )} {\left [ \bar n \cdot p - \omega \right ]^2}  \,
\, \phi_B^{-}(\omega, \mu) +  {\cal O}(\alpha_s) \,.
\end{eqnarray}

In contrast to the NLP active-quark mass corrections,  the newly identified spectator-quark mass contributions
cannot generate the large-recoil symmetry breaking effects at ${\cal O}(\alpha_s^0)$.

Along the same vein, we can derive the soft-collinear factorization formulae
for the particular subleading-power corrections to the correlation function
$\Pi_{\mu, {\rm T}}$ from the higher-order terms in the heavy quark expansion of the hard-collinear quark propagator

\begin{eqnarray}
\Pi_{\rm T, \, NLP}^{\rm hc, \, I}
=   \left ( \tilde{\Pi}_{\rm NLP}^{\rm hc, \, I} - \Pi_{\rm NLP}^{\rm hc, \, I} \right )  + {\cal O}(\alpha_s)\,,
\qquad
\Pi_{\rm T, \, NLP}^{\rm hc, \, II}
=  \left (  \tilde{\Pi}_{\rm NLP}^{\rm hc, \, II} - \Pi_{\rm NLP}^{\rm hc, \, II} \right ) + {\cal O}(\alpha_s) \,.
\label{relation of the hc propagator corrections}
\end{eqnarray}

These interesting constraints can be attributed to the classical HQET equation of motion

\begin{eqnarray}
\left ( \sigma_{\mu \nu} \, q^{\nu} \right ) \, \mathds{1}  \, h_v
= \left ( \sigma_{\mu \nu} \, q^{\nu} \right ) \, \slashed{v}  \, h_v
= \left ( - {i \over 2}  \right) \, \left [ \bar n \cdot q \, n_\mu - n \cdot q \, \bar n_\mu \right ] \,
\left ( {\slashed {n} \over 2 } - {\slashed {\bar n} \over 2 }  \right )  \, h_v \,.
\end{eqnarray}

Including the non-Eikonal gluonic interaction with the bottom-quark field
in the higher-order corrections to the vacuum-to-bottom-meson correlation functions will generally
invalidate such symmetry relations.

Expressing the established NLP factorization formulae in the dispersion forms
and equating the achieved spectral representations for the correlation functions
with the corresponding  hadronic dispersion relations (<ref>)
allows us to construct the desired  LCSR for the power-suppressed contributions
from the expanded hard-collinear quark propagator

\begin{eqnarray}
&& f_{M} \,\, {\rm exp} \left [- {m_{M}^2 \over n \cdot p \,\, \omega_M} \right ] \,\,
\left \{ \frac{n \cdot p} {m_B} \, f_{B M, \, \rm NLP}^{+, \, \rm hc}(q^2)
\,, \,\,\,   f_{B M, \, \rm NLP}^{0, \, \rm hc}(q^2) \,,
\,\,\, \frac{n \cdot p} {m_B + m_M} \, f_{B M, \, \rm NLP}^{T, \, \rm hc}(q^2) \right \}  \,
\nonumber \\
&& =   { \mathcal{F}_B(\mu) \over n \cdot p} \,
\bigg [ \int_0^{\omega_s}  \, d \omega_1 \,\, \int_0^{\infty}  \, {d \omega_2 \over \omega_2}
\,\, \bigg \{ e^ {-{\omega_1 + \omega_2 \over \omega_M} }  \,
\varrho_{\rm NLP}^{\rm hc, \, I}(\omega_1, \omega_2,  \mu) \, \theta(\omega_s - \omega_1 - \omega_2)
\nonumber \\
&& \hspace{0.5 cm}
+  \, \left [ \left (  e^ {-{\omega_1 \over \omega_M} } -   e^ {-{\omega_s  \over \omega_M} } \right ) \,
\theta(\omega_1 + \omega_2 -\omega_s)
+ \left ( e^ {-{\omega_1 \over \omega_M}} -   e^ {-{\omega_1 + \omega_2 \over \omega_M} } \right ) \,
\theta(\omega_s - \omega_1 - \omega_2 )  \right ]
\nonumber \\
&& \hspace{0.5 cm}
\times \, \varrho_{\rm NLP}^{\rm hc, \, II}(\omega_1, \omega_2,  \mu)
+  e^ {-{\omega_1  \over \omega_M} }  \,  \varrho_{\rm NLP}^{\rm hc, \, III}(\omega_1, \omega_2,  \mu)  \bigg \}
+ \int_0^{\omega_s}  \, d \omega  \,\, e^ {-{\omega \over \omega_M} }  \,\,
\tilde{\varrho}_{\rm NLP}^{\rm hc}(\omega, \mu) \bigg ]
+  {\cal O}(\alpha_s) \,.
\hspace{1.0 cm}
\label{NLP LCSR from the hc propagator}
\end{eqnarray}

The yielding expressions for the emerged coefficient functions $\varrho_{\rm NLP}^{\rm hc, \, (I, II, III)}$
and $\tilde{\varrho}_{\rm NLP}^{\rm hc}$ can be explicitly written as

\begin{eqnarray}
\varrho_{\rm NLP}^{\rm hc, \, I}
&=& 2 \, \left \{ {\omega_1 + \omega_2 \over \omega_2} \, \Psi_5(\omega_1, \omega_2, \mu) \,
- \, \kappa_{i} \, \left (\Phi_4 + \Psi_4)(\omega_1, \omega_2, \mu \right )  \right \} \,,
\\
\varrho_{\rm NLP}^{\rm hc, \, II}
&=&  -2 \, \left ( {\omega_M \over \omega_2} \right ) \,
\left [ \Psi_5(\omega_1, \omega_2, \mu) + \kappa_i \, \Phi_4(\omega_1, \omega_2, \mu)  \right ] \,,
\\
\varrho_{\rm NLP}^{\rm hc, \, III}
&=& 2 \, \left \{ \left ({d \over d \omega_1} - {1 \over \omega_2 } \right ) \,
\left [ \omega_1 \, \Psi_5(\omega_1, \omega_2, \mu) \right ] \,
- \, \kappa_{i}\, \Phi_4(\omega_1, \omega_2, \mu)  \right \} \,,
\\
\tilde{\varrho}_{\rm NLP}^{\rm hc} &=&  \left [ 1 - \left ( {\omega_s - \omega \over \omega} \right ) \,
{m_q^2 - m_{q^{\prime}}^2 \over \omega_s \, \omega_M} \,
+ \left(\omega - 2 \, \bar \Lambda - {m_q^2 - m_{q^{\prime}}^2 \over \omega_s}  \right ) \,
{d \over d \omega}   \right ] \,
\left [ \omega \, \phi_B^{-}(\omega, \mu) \right ]
\nonumber  \\
&& + \, \kappa_i \, \left [ \left (\omega - 2 \, \bar \Lambda \right ) \, \phi_B^{+}(\omega, \mu)  +
\left (m_q + m_{q^{\prime}} \right ) \, \phi_B^{-}(\omega, \mu) \right ] \,,
\end{eqnarray}

where the non-universal $\kappa_i$-factors are responsible for  the symmetry-breaking effects

\begin{eqnarray}
\kappa_{+} = - \kappa_{0} = {(n \cdot p - m_B) / m_B}  \,,
\qquad \kappa_{T} = -1 \,.
\end{eqnarray}

Importantly, we have further verified that the newly derived  sum rules (<ref>)
for the light-quark-mass insensitive NLP corrections from the heavy quark expansion of
the hard-collinear propagator are consistent with the previous computations
for the semileptonic $B \to D \ell \bar \nu_{\ell}$ form factors accomplished  in [41].

§.§ The NLP contribution from the subleading effective current

We now proceed to determine the second class of the  subleading-power correction
to the exclusive $B_{d, s}  \to \pi, K$   form factors
arising from the peculiar higher-order term
in the ${\rm SCET}_{\rm I}$ representation of the heavy-to-light transition current
$\bar q \, \Gamma_\mu \,b$ [104]

\begin{eqnarray}
J^{\rm (A2)} \supset
(\bar \xi_{\rm hc} \, W_{\rm hc}) \,\,
\Gamma \,\, \left ( { i \, \overrightarrow{\slashed D}_{\top} \over 2 \, m_b} \right )
\, \,   h_v   + ... \,,
\qquad
D_{\top}^{\mu} &\equiv&  D^{\mu} - (v\cdot D)  \, v^{\mu} \,,
\label{NLP SCET current}
\end{eqnarray}

which  corresponds to the standard ${\rm QCD} \to {\rm HQET}$ matching for the bottom-quark field

\begin{eqnarray}
b(x) =  {\rm exp}\left ( - i \, m_b \, v \cdot x  \right ) \,
\bigg [ 1 + { i \, \overrightarrow{{\slashed D}}_{\top}  \over 2 \, m_b}
+  { (v\cdot \overrightarrow{D}) \, \overrightarrow{{\slashed D}}_{\top}  \over 4 \, m_b^2}
- { \overrightarrow{{\slashed D}}_{\top} \, \overrightarrow{{\slashed D}}_{\top} \over 8 \, m_b^2}
+ {\cal O} \left ({1 \over m_b^3} \right ) \bigg ] \, h_v(x) \,.
\end{eqnarray}

It is then straightforward to express the resulting NLP contribution to the vacuum-to-$B$-meson
correlation function (<ref>) in the following form

\begin{eqnarray}
\Pi_{\mu, {\rm V}, \rm NLP}^{\rm (A2)}   (n \cdot p, \bar n \cdot p) & = &
- \left ( {n \cdot p \over 4 \, m_b} \right ) \, \int d^4x \int \frac{d^4 k}{(2 \pi)^4}
{\rm exp} \left (i \, k \cdot x \right )  \, \frac{1} {(p-k)^2 - m_q^2 +i0} \,
\nonumber \\
&& \times \, \left \langle 0 \left | \overline{q}^{\prime}(x) \, \slashed{n} \, \gamma_5 \, \slashed{\bar n}  \,
\gamma_\mu \, \overrightarrow{{\slashed D}}_{\top} \, h_v(0) \right | \bar B(p_B) \right \rangle
\nonumber \\
&=&  - \left ( {n \cdot p \over 2 \, m_b} \right ) \, \int d^4x \int \frac{d^4 k}{(2 \pi)^4}
{\rm exp} \left (i \, k \cdot x \right )  \, \frac{\bar n_{\mu} } {(p-k)^2 - m_q^2 +i0} \,
\nonumber \\
&& \times \, \left \langle 0 \left | \overline{q}^{\prime}(x) \,
\left ( \overrightarrow{{\slashed D}}_{\top} \,\,  \slashed{n} \,
+  \,2 \, \bar n \cdot \overrightarrow{D}_{\top} \right ) \,
\gamma_5 \,  h_v(0) \right | \bar B(p_B) \right \rangle \,,
\label{NLP SCET current correction at LO}
\end{eqnarray}

where we have employed the lowest-order equation of motion of the effective heavy-quark field

\begin{eqnarray}
i \, v \cdot \overrightarrow{D} \, h_v = 0 \,.
\end{eqnarray}

This evidently permits the  replacement
$i\, \overrightarrow{\slashed D}_{\top} \, h_v \to i \, \overrightarrow{\slashed D} \, h_v$
in the effective weak current at the NLP accuracy.
Taking advantage of an additional HQET operator identity [70, 71, 72]

\begin{eqnarray}
\bar q^{\prime}(x) \, \Gamma \, {\overrightarrow{D}}_{\rho} \,  h_v(0)
&=& \partial_{\rho} \left [ \bar q^{\prime}(x) \, \Gamma \, h_v(0) \right ] \,
+ \, i \, \int_0^1 d u \, \bar u \,\, \bar q^{\prime}(x) \,g_s \, G_{\lambda \rho}(u x) \,
x^{\lambda} \, \Gamma \,  h_v(0) \nonumber \\
&& - \, {\partial \over \partial x^{\rho}} \bar q^{\prime}(x) \, \Gamma \, h_v(0)  \,,
\end{eqnarray}

in combination with the two operator relations displayed in (<ref>)
and (<ref>),
we can construct the perturbative factorization formula for the non-local hadronic matrix element
in (<ref>)

\begin{eqnarray}
\Pi_{\rm NLP}^{\rm (A2)}    & = &   {\cal O}(\alpha_s) \,,
\\
\tilde{\Pi}_{\rm NLP}^{\rm (A2)}   & = &
\left [ {\mathcal{F}_B(\mu)  \, m_B  \over 2 \, m_b} \right ] \,
\bigg \{  \int_0^{\infty} \, d \omega_1 \, \int_0^{\infty} \, d \omega_2 \,
{2 \, (\Psi_4 + \Phi_4)(\omega_1, \omega_2, \mu) \over (\omega_1 - \bar n \cdot p) \, (\omega_1 + \omega_2 - \bar n \cdot p) } \,
\nonumber \\
&& \hspace{-0.5 cm}
+ \, \int_0^{\infty} \, { d \omega \over \omega - \bar n \cdot p} \,
\left [ \left(\omega - 2 \, \bar \Lambda \right ) \, \phi_B^{+}(\omega, \mu)
+ \left (\omega - \bar \Lambda + m_{q^{\prime}} \right )  \, \phi_B^{-}(\omega, \mu)  \right ] \bigg \}
+ {\cal O}(\alpha_s)  \,.
\label{factorization formulae for the NLP SCET current correction}
\end{eqnarray}

Applying the analogous computational strategy,  we can further  compute the yielding NLP correction to
the correlation function $\Pi_{\mu, {\rm T}}$ at tree level

\begin{eqnarray}
\Pi_{\rm T, \, NLP}^{\rm (A2)}
=   - \left ( \tilde{\Pi}_{\rm NLP}^{\rm (A2)} - \Pi_{\rm NLP}^{\rm (A2)} \right )  + {\cal O}(\alpha_s) \,.
\label{relation of the NLP SCET corrections}
\end{eqnarray}

We remark in passing that  such an interesting constraint (<ref>)
differs from the previously established relations (<ref>)
for the NLP corrections from the HQET expansion of the hard-collinear quark propagator
by an overall factor of “$-1$".
This observation can be traced back to the anti-commutation relation

$\left \{ \slashed{v},   \overrightarrow{\slashed D}_{\top} \right \} \, h_v = 0$,

thus ensuring an exact algebraic identity

\begin{eqnarray}
\left ( \sigma_{\mu \nu} \, q^{\nu} \right ) \, \left ( { i \, \overrightarrow{\slashed D}_{\top} \over 2 \, m_b} \right )  \, h_v
&=& \left (  {i \over 2}  \right) \, \left [ \bar n \cdot q \, n_\mu - n \cdot q \, \bar n_\mu \right ] \,
\left ( {\slashed {n} \over 2 } - {\slashed {\bar n} \over 2 }  \right )  \,
\left ( { i \, \overrightarrow{\slashed D}_{\top} \over 2 \, m_b} \right )  \, h_v \,.
\end{eqnarray}

Matching the spectral representations for the established soft-collinear factorization formulae
and (<ref>)
with the corresponding hadronic dispersion relations (<ref>)
enables us to derive the final expressions for the  NLP sum rules of the $B_{d, s}  \to \pi, K$ decay form factors

\begin{eqnarray}
&& f_{M} \,\, {\rm exp} \left [- {m_{M}^2 \over n \cdot p \,\, \omega_M} \right ] \,\,
\left \{ \frac{n \cdot p} {m_B} \, f_{B M, \, \rm NLP}^{+, \, \rm (A2)}(q^2)
\,, \,\,\,   f_{B M, \, \rm NLP}^{0, \, \rm (A2)}(q^2) \,,
\,\,\, \blue{\left [ - \frac{n \cdot p} {m_B + m_M} \right ]} \, f_{B M, \, \rm NLP}^{T, \, \rm (A2)}(q^2) \right \}  \,
\nonumber \\
&& =   { \mathcal{F}_B(\mu) \over n \cdot p} \,
\bigg [ \int_0^{\omega_s}  \, d \omega_1 \,\, \int_0^{\infty}  \, {d \omega_2 \over \omega_2}
\,\, \bigg \{ e^ {-{\omega_1 + \omega_2 \over \omega_M} }  \, \varrho_{\rm NLP}^{\rm (A2), \, I}(\omega_1, \omega_2,  \mu) \,
\theta(\omega_s - \omega_1 - \omega_2 )
\nonumber \\
&& \hspace{1.8 cm}
+  \, e^ {-{\omega_1  \over \omega_M} }  \,  \varrho_{\rm NLP}^{\rm (A2), \, II}(\omega_1, \omega_2,  \mu) \bigg \} \,\,
+ \int_0^{\omega_s}  \, d \omega  \,\, e^ {-{\omega \over \omega_M} }  \,\,
\tilde{\varrho}_{\rm NLP}^{\rm (A2)}(\omega, \mu) \bigg ]
+  {\cal O}(\alpha_s) \,.
\label{NLP LCSR from the subleading SCET current}
\end{eqnarray}

The non-perturbative coefficient functions $\varrho_{\rm NLP}^{\rm (A2), \, (I, II)}$
and $\tilde{\varrho}_{\rm NLP}^{\rm (A2)}$ can be expressed in terms of the two-particle and three-particle
HQET distribution amplitudes

\begin{eqnarray}
&& \varrho_{\rm NLP}^{\rm (A2), \, I}
= - {n \cdot p \over m_b} \, \left [  \left (\Phi_4 + \Psi_4)(\omega_1, \omega_2, \mu \right ) \right ]  \,,
\qquad
\varrho_{\rm NLP}^{\rm (A2), \, II}
=  {n \cdot p \over m_b} \,  \left (\Phi_4 + \Psi_4)(\omega_1, \omega_2, \mu \right ) \,,
\\
&&  \tilde{\varrho}_{\rm NLP}^{\rm (A2)} =  {n \cdot p \over 2 \, m_b} \,
\left [ \left (\omega - 2 \, \bar \Lambda \right ) \, \phi_B^{+}(\omega, \mu)  +
\left (\omega -  \bar \Lambda + m_{q^{\prime}} \right )  \, \phi_B^{-}(\omega, \mu) \right ] \,.
\hspace{1.0 cm}
\end{eqnarray}

Remarkably, the considered NLP corrections to the heavy-to-light form factors
from the effective matrix elements of the subleading ${\rm SCET}_{\rm I}$ current $J^{\rm (A2)}$
preserve the well-known symmetry relation between the vector and scalar form factors,
but violate the large-recoil symmetry of the vector and tensor form factors already at ${\cal O}(\alpha_s^0)$.

§.§ The NLP contribution form the higher-twist two-particle and three-particle LCDAs

As emphasized repeatedly in [105, 106], the systematic and consistent  description of
the higher-twist corrections to exclusive hard reactions in QCD
will require us to  simultaneously take into account the transverse-momentum dependence of the valence (anti)-quarks in the leading
Fock-state wavefunction and the subleading distribution amplitudes of the non-minimal partonic configuration
with additional gluons and/or quark-antiquark pairs.
Including the off-light-cone corrections to the renormalized two-body  non-local HQET matrix element
at the ${\cal O}(x^2)$ accuracy discussed in [72]

\begin{eqnarray}
&& \langle  0 | \left (\bar q^{\prime} \, Y_s \right)_{\beta} (x) \,
\left (Y_s^{\dag} \, h_v \right )_{\alpha}(0)| \bar B(v)\rangle \nonumber \\
&& = - \frac{i \, {\cal F}_B(\mu) \, m_B}{4}  \,
\int_0^{\infty} \, d \omega \,\, {\rm exp} \left [ {- i \, \omega \, v \cdot x} \right ] \,
\bigg \{  \frac{1+ \slashed{v}}{2} \, \,
\bigg [ 2 \, \left ( \phi_{B}^{+}(\omega, \mu) + \blue{x^2 \, g_B^{+}(\omega, \mu)}  \right ) \nonumber \\
&& \hspace{0.5 cm} - {1 \over v \cdot x}  \,
\left  [ \left ( \phi_{B}^{+}(\omega, \mu) - \phi_{B}^{-}(\omega, \mu)  \right )
+ \blue{ x^2 \, \left ( g_{B}^{+}(\omega, \mu) - g_{B}^{-}(\omega, \mu)  \right ) }  \right ]  \,
\slashed{x} \bigg ] \, \gamma_5 \bigg \}_{\alpha \beta}\,,
\label{def: off-light-cone two-particle B-meson DAs}
\end{eqnarray}

we can proceed to construct the tree-level factorization formulae for the two-particle higher-twist corrections
to the three invariant functions $\Pi$, $\tilde \Pi$ and $\Pi_{\rm T}$ [36]

\begin{eqnarray}
\Pi_{\rm NLP}^{\rm 2PHT}  & = &   {\cal O}(\alpha_s) \,,
\\
\tilde{\Pi}_{\rm NLP}^{\rm 2PHT}   & = &
\left [ {2 \, \mathcal{F}_B(\mu)  \, m_B  \over n \cdot p} \right ] \,
\bigg \{  \int_0^{\infty} \, d \omega_1 \, \int_0^{\infty} \, d \omega_2 \, \int_0^{1} \, d u  \,\,
{\bar u \, \Psi_5(\omega_1, \omega_2, \mu) \over  \left [ \bar n \cdot p - \omega_1 - u \, \omega_2 \right ]^2 } \,
\nonumber \\
&& %\hspace{-2.5 cm}
- \, 2  \, \int_0^{\infty} \, d \omega \,\,
{\hat{g}_B^{-}(\omega, \mu)  \over  \left [ \bar n \cdot p - \omega \right ]^2  } \, \bigg \}
+ {\cal O}(\alpha_s)  \,,
\\
\Pi_{\rm T, \, NLP}^{\rm 2PHT}
&=&    \left ( \tilde{\Pi}_{\rm NLP}^{\rm 2PHT} - \Pi_{\rm NLP}^{\rm 2PHT} \right )  + {\cal O}(\alpha_s) \,.
\label{factorization formulae for the 2P higher-twist corrections}
\end{eqnarray}

In the above derivation we have employed the nontrivial constraint on the momentum-space distribution amplitudes
due to the HQET equations of motion

\begin{eqnarray}
-2 \, {d^2 g_B^{-}(\omega, \mu)  \over d \omega^2} &=&
\left [ {3 \over 2} + (\omega - \bar \Lambda) \, {d \over d \omega}   \right ] \, \phi_B^{-}(\omega, \mu)
- \left ( {1 \over 2}  \right ) \, \phi_B^{+}(\omega, \mu)
\nonumber \\
&& + \int_0^{\infty} \, {d \omega_2 \over \omega_2 } \,
\left ( {d \over d \omega} - {1 \over \omega_2} \right ) \, \Psi_5(\omega, \omega_2, \mu)
+ \int_0^{\omega} \, {d \omega_2 \over \omega_2^2 } \, \Psi_5(\omega-\omega_2, \omega_2, \mu) \,.
\hspace{1.0 cm}
\label{the LO EOM for gBm}
\end{eqnarray}

This allows us to decompose the higher-twist LCDA $g_B^{-}(\omega, \mu)$ into the “genuine" twist-five
three-particle distribution amplitude $\Psi_5(\omega_1, \omega_2, \mu)$
and the lower-twist “Wandzura-Wilczek" contribution labelled as $\hat{g}_B^{-}(\omega, \mu)$,
which can be explicitly expressed in terms of the customary two-particle $B$-meson distribution amplitudes

\begin{eqnarray}
\hat{g}_B^{-}(\omega, \mu) =  \left ( {1 \over 4} \right ) \, \int_{\omega}^{\infty}  \, d \rho \,
\left  \{ (\rho - \omega) \, \left [ \phi_B^{+}(\rho) -  \phi_B^{-}(\rho) \right ]
- 2 \, (\bar \Lambda - \rho)  \, \phi_B^{-}(\rho)  \right \}  \,.
\label{def: WW part of gBminus}
\end{eqnarray}

Applying further the light-cone expansion for the massive-quark propagator in the background gluon field
up to the gluon field strength terms without the covariant derivatives  [107, 20]
(see also [108, 109] for an alternative representation)

\begin{eqnarray}
S(x, 0, m_q) &\equiv& \langle 0 | {\rm T} \, \{q(x), \, \bar q (0)  \} | 0\rangle
\nonumber \\
& \supset  &  i \, g_s \, \int_{-\infty}^{+\infty} \,\, {d^4 \ell \over (2 \pi)^4} \,
{\rm exp} \left [ {- i \, \ell \cdot x} \right ] \,
\int_0^1 \, d u \, \left  [ {u \, x_{\mu} \, \gamma_{\nu} \over \ell^2 - m_q^2}
- \frac{(\slashed{\ell} + m_q) \, \sigma_{\mu \nu}}{2 \, (\ell^2 - m_q^2)^2}  \right ]
\, G^{\mu \nu}(u \, x) \,,
\hspace{1.0 cm}
\end{eqnarray}

we can continue to  write down the tree-level factorized expressions for the yielding three-particle higher-twist corrections
to the vacuum-to-$B$-meson correlation functions (<ref>)
displayed in Figure <ref>(b) [36]

\begin{eqnarray}
\Pi_{\rm NLP}^{\rm 3PHT}    & = &
\left [  { {\cal F}_B(\mu) \, m_B \over n \cdot p } \right ] \,
\int_0^{\infty} d \omega_1 \, \int_0^{\infty} d \omega_2 \, \int_0^{1}  d u \,
\frac{1} {\left [ \bar n \cdot p - \omega_1 - u \, \omega_2 \right ]^2} \,
\nonumber \\
&& \times \, \left \{  2 \,  \bar u \, \Phi_4(\omega_1, \omega_2, \mu)
+ {m_q \over n \cdot p} \left (\Psi_5 - \tilde{\Psi}_5 \right )(\omega_1, \omega_2, \mu)  \right \}
+ {\cal O}(\alpha_s)  \,,
\\
\tilde{\Pi}_{\rm NLP}^{\rm 3PHT}   & = &
\left [  { {\cal F}_B(\mu) \, m_B \over n \cdot p } \right ] \,
\int_0^{\infty} d \omega_1 \, \int_0^{\infty} d \omega_2 \, \int_0^{1}  d u \,
\frac{1} {\left [ \bar n \cdot p - \omega_1 - u \, \omega_2 \right ]^2} \,
\nonumber \\
&& \times \, \left \{ \left [ (2 \, u -1) \, \Psi_5 - \tilde{\Psi}_5 \right ] (\omega_1, \omega_2, \mu)
- {2 \, m_q \over n \cdot p} \, \Phi_6(\omega_1, \omega_2, \mu)  \right \}
+ {\cal O}(\alpha_s)   \,,
\hspace{1.0 cm}  \\
\Pi_{\rm T, \, NLP}^{\rm 3PHT}
&=&    \left ( \tilde{\Pi}_{\rm NLP}^{\rm 3PHT} - \Pi_{\rm NLP}^{\rm 3PHT} \right )  + {\cal O}(\alpha_s) \,.
\label{factorization formulae for the3P higher-twist corrections}
\end{eqnarray}

Adding together the obtained two-particle and three-particle subleading twist corrections
in the dispersion form and implementing the standard continuum subtraction procedure with the parton-hadron duality ansätz
leads to the following sum rules for the NLP contributions to the semileptonic $B_{d, s}  \to \pi, K$  form factors

\begin{eqnarray}
&& f_{M} \,\, {\rm exp} \left [- {m_{M}^2 \over n \cdot p \,\, \omega_M} \right ] \,\,
\left \{ \frac{n \cdot p} {m_B} \, f_{B M, \, \rm NLP}^{+, \, \rm HT}(q^2)
\,, \,\,\,   f_{B M, \, \rm NLP}^{0, \, \rm HT}(q^2) \,,
\,\,\, \frac{n \cdot p} {m_B + m_M} \, f_{B M, \, \rm NLP}^{T, \, \rm HT}(q^2) \right \}  \,
\nonumber \\
&& =   { \mathcal{F}_B(\mu) \over n \cdot p} \,
\bigg [ \int_0^{\omega_s}  \, d \omega_1 \,\, \int_0^{\infty}  \, {d \omega_2 \over \omega_2}
\,\, \bigg \{ e^ {-{\omega_1 + \omega_2 \over \omega_M} }  \,
\varrho_{\rm NLP}^{\rm HT, \, I}(\omega_1, \omega_2,  \mu) \,
\theta(\omega_s - \omega_1 - \omega_2 )
\nonumber \\
&& \hspace{0.5 cm}
+  \, \left [ \left (  e^ {-{\omega_1 \over \omega_M} } -   e^ {-{\omega_s  \over \omega_M} } \right ) \,
\theta(\omega_1 + \omega_2 -\omega_s)
+ \left ( e^ {-{\omega_1 \over \omega_M}} -   e^ {-{\omega_1 + \omega_2 \over \omega_M} } \right ) \,
\theta(\omega_s - \omega_1 - \omega_2 )  \right ]
\nonumber \\
&& \hspace{0.5 cm}
\times \, \varrho_{\rm NLP}^{\rm HT, \, II}(\omega_1, \omega_2,  \mu)
+  e^ {-{\omega_1  \over \omega_M} }  \,  \varrho_{\rm NLP}^{\rm HT, \, III}(\omega_1, \omega_2,  \mu) \bigg \} \,\,
+ \int_0^{\omega_s}  \, d \omega  \,\, e^ {-{\omega \over \omega_M} }  \,\,
\tilde{\varrho}_{\rm NLP}^{\rm HT}(\omega, \mu) \bigg ]
+  {\cal O}(\alpha_s) \,.
\hspace{1.0 cm}
\label{NLP LCSR from the 2P and 3P higher-twist effects}
\end{eqnarray}

The explicit expressions for the  coefficient functions $\varrho_{\rm NLP}^{\rm HT, \, (I, II, III)}$
and $\tilde{\varrho}_{\rm NLP}^{\rm HT}$ are given by

\begin{eqnarray}
\varrho_{\rm NLP}^{\rm HT, \, I} &=&  \left [ - \left (1 +  {m_q \over n \cdot p} \, \kappa_i \right ) \,
\left (\Psi_5 - \tilde{\Psi}_5 \right )(\omega_1, \omega_2, \mu) \right ]
+ {2 \, m_q \over n \cdot p}  \, \Phi_6(\omega_1, \omega_2, \mu),
\\
\varrho_{\rm NLP}^{\rm HT, \, III} &=&  \left [ 2 \,  \kappa_i \, \Phi_4(\omega_1, \omega_2, \mu)
+ \left (1 +  {m_q \over n \cdot p} \, \kappa_i \right ) \,
\left (\Psi_5 - \tilde{\Psi}_5 \right )(\omega_1, \omega_2, \mu) \right ]
- {2 \, m_q \over n \cdot p}  \, \Phi_6(\omega_1, \omega_2, \mu),
\hspace{0.70 cm} \\
\varrho_{\rm NLP}^{\rm HT, \, II} &=& -2 \,  \left ( {\omega_M \over \omega_2} \right ) \,
\kappa_i \,  \Phi_4(\omega_1, \omega_2, \mu)  \,,
\qquad  \hspace{2.0 cm}
\tilde{\varrho}_{\rm NLP}^{\rm HT} =  - 4 \, {d \hat{g}_B^{-}(\omega, \mu) \over d \omega}.
\end{eqnarray}

Interestingly, the obtained tree-level sum rules for the two-particle higher-twist contributions are independent of  the non-universal $\kappa_i$-factors, thus maintaining the large-recoil symmetry relations of the considered bottom-meson decay form factors.
According to our power-counting  scheme for the intrinsic LCSR parameters
$\omega_s \sim \omega_M \sim {\cal O}(\Lambda_{\rm QCD}^2/m_b)$,
we can immediately determine the desired scaling behaviours of the two-particle and three-particle higher-twist corrections
at ${\cal O}(\alpha_s^0)$ in the heavy quark limit [36]

\begin{eqnarray}
f_{B M, \, \rm NLP}^{+, \, \rm HT} \sim f_{B M, \, \rm NLP}^{0, \, \rm HT} \sim f_{B M, \, \rm NLP}^{T, \, \rm HT}
\sim {\cal O} ((\Lambda_{\rm QCD}/m_b)^{5/2}) \,,
\end{eqnarray}

which  turn out to be  suppressed by one power of $\Lambda_{\rm QCD}/m_b$ in comparison with
the leading-power SCET sum rules (<ref>) and (<ref>).
It is however important to  emphasize that evaluating  the (currently unknown) higher-order  radiative corrections to
the three-particle twist-three $B$-meson LCDA contributions can actually bring about
the unsuppressed symmetry-preserving effects for the exclusive $B_{d, s}  \to \pi, K$ form factors
in the heavy quark expansion.
This observation can be understood from the very fact that  the two-particle twist-three distribution amplitude
$\phi_B^{-}(\omega, \mu)$ appearing in the tree-level sum rules  can be generated by the one-loop renormalization
of the three-particle $B$-meson LCDA  $\Phi_3(\omega_1, \omega_2, \mu)$ [110, 89]
(see  [111] for further discussions in the SCET framework).

§.§ The NLP contribution form the higher-twist four-particle effects

Diagrammatic representations of the twist-five and twist-six four-particle corrections
to the considered vacuum-to-bottom-meson correlation functions (<ref>)
at the ${\cal O}(\alpha_s)$ accuracy.

We are now in a position to compute the NLP corrections to the heavy-to-light bottom-meson decay form factors
from the twist-five and twist-six four-particle LCDA contributions in the factorization approximation,
following the computational prescriptions for the electromagnetic pion form factor at intermediate momentum transfer [112],
the exclusive photon-pion transition form factor  $\gamma^{\ast} \gamma \to \pi$  [73]
and  the radiative leptonic $B \to \gamma \ell \bar \nu_{\ell}$ decay amplitude [74].
Evaluating the lowest-order Feynman diagrams in Figure <ref>
straightforwardly leads to the factorized expressions for such non-leading Fock-state corrections

\begin{eqnarray}
\Pi_{\rm NLP}^{\rm 4P}   & = & {\cal O}(\alpha_s^2)
%{2 \, \pi \over 3} \, \frac{\alpha_s(\mu)  \, C_F  \, \langle \bar q q \rangle}{(n \cdot p)^2 \, \bar n \cdot p} \,
%{\cal F}_B(\mu) \, m_B \, \int_0^{\infty} \frac{d \omega}{\omega - \bar n \cdot p} \,
%\bigg \{ \left ({3 \over 2} +  {\bar n \cdot p \over \bar n \cdot p - \omega} \right ) \, \phi_B^{-}(\omega, \mu)
%\nonumber \\
%&& + \, \frac{\bar n \cdot p \, (\bar n \cdot p - \omega) }{\omega^2}  \,
%\left [ 2 + {2 \, \bar n \cdot p - \omega \over \omega} \,
%\ln \left ({\bar n \cdot p - \omega \over \bar n \cdot p }\right )  \right ] \,
%(\phi_B^{-} - \phi_B^{+})(\omega, \mu)   \, \bigg \}\,,
\\
\tilde{\Pi}_{\rm NLP}^{\rm 4P}  & = &
{2 \, \pi \over 3} \, \frac{\alpha_s(\mu)  \, C_F  \, \langle \bar q^{\prime} q^{\prime} \rangle}{n \cdot p \, \bar n \cdot p} \,
{\cal F}_B(\mu) \, m_B \, \int_0^{\infty} \frac{d \omega}{\omega - \bar n \cdot p} \,
{\phi_B^{+}(\omega, \mu)   \over \omega} \,
%\bigg \{ % \frac{(\bar n \cdot p)^2}{m_b \, (\bar n \cdot p - \omega)}\, \phi_B^{-}(\omega, \mu)
\nonumber \\
&& \times \,  \left \{  %{\bar n \cdot p \over \omega  - \bar n \cdot p }
{2 \, \bar n \cdot p \over \omega} \,
\left [ 1 + {\bar n \cdot p - \omega \over \omega}  \, \ln {\bar n \cdot p - \omega  \over \bar n \cdot p} \right ]
- 1 +  {\langle \bar q q \rangle \over \langle \bar q^{\prime} q^{\prime} \rangle } \,
{\omega \over \omega - \bar n \cdot p }    \right \}  \,
%\bigg \} \,,
+ {\cal O}(\alpha_s^2)\,,
\hspace{0.8 cm}
\\
\Pi_{\rm T, \, NLP}^{\rm 4P}   & = &
\left (\tilde{\Pi}_{\rm NLP}^{\rm 4P} - \Pi_{\rm NLP}^{\rm 4P} \right )
%{2 \, \pi \over 3} \, \frac{\alpha_s(\mu)  \, C_F  \, \langle \bar q q \rangle}{n \cdot p \, \bar n \cdot p} \,
%{\cal F}_B(\mu) \, m_B \, \int_0^{\infty} \frac{d \omega}{\omega - \bar n \cdot p} \,
% {\phi_B^{+}(\omega, \mu)   \over \omega}  \,
%\nonumber \\
%&&  \times \,  \left \{ {\bar n \cdot p \over \omega - \bar n \cdot p}
%+ {2 \, \bar n \cdot p \over \omega} \,
%\left [ 1 + {\bar n \cdot p - \omega \over \omega}  \, \ln {\bar n \cdot p - \omega  \over \bar n \cdot p} \right ]
%\right \}
+ {\cal O}(\alpha_s^2) \,.
\end{eqnarray}

It is perhaps worth mentioning that the  determined  four-particle contributions
to the vacuum-to-bottom-meson correlation functions (<ref>) from
the particular diagram (e) in Figure <ref>
can be most conveniently  computed  with the familiar background-field expansion of the quark propagator
on the light-cone [107]

\begin{eqnarray}
\langle 0 | {\rm T} \, \{q(x), \, \bar q (0)  \} | 0\rangle
& \supset & \frac{\Gamma({d / 2}-1)}{8 \, \pi^{d/2} \, (-x^2)^{d/2-1}}  \,
\int_0^1 d u \, u \, \bar u \, \slashed{x} \, x_{\nu}  \, D^{\mu} \, g_s \, G^{\mu \nu}(u x)
\nonumber \\
&& + \, \frac{\Gamma({d / 2}-2)}{16 \, \pi^{d/2} \, (-x^2)^{d/2-2}}  \,
\int_0^1 d u \, \left ( u \, \bar u  - {1 \over 2} \right ) \,
D^{\mu} \, g_s \, G^{\mu \nu}(u x) \, \gamma^{\nu} \,,
\end{eqnarray}

together with  the classical equation of motion in QCD

\begin{eqnarray}
D^{\mu} \, G_{\mu \nu}^{a} = - i\, g_s \, \sum_{q} \, \bar q \, \gamma_{\nu} \, T^a \, q.
\end{eqnarray}

Additionally, our explicit calculations of the three diagrams (a), (b) and (c) in Figure <ref>
indicate that they can only  generate the yet higher-power corrections in the heavy quark expansion
when compared with the dominating  contributions from the  diagrams (d) and (e).
These enlightening pattern differs drastically from the counterpart NLP contributions to
the two helicity form factors  of $B \to \gamma \ell \bar \nu_{\ell}$,
due to the longitudinally polarized pseudoscalar-meson current in the former
and the transversely polarized on-shell photon state in the latter.
Moreover, the remaining  diagram (f) in Figure <ref>
turns out to be insensitive to both the hard and hard-collinear QCD dynamics.

We can proceed to work out the dispersion representations for the four-particle corrections
to the invariant functions $\Pi$, $\tilde \Pi$ and $\Pi_{\rm T}$ at ${\cal O}(\alpha_s)$
and further derive the NLP sum rules for the yielding twist-five and twist-six contributions with the standard strategy

\begin{eqnarray}
&& f_{M} \,\, {\rm exp} \left [- {m_{M}^2 \over n \cdot p \,\, \omega_M} \right ] \,\,
\left \{ \frac{n \cdot p} {m_B} \, f_{B M, \, \rm NLP}^{+, \, \rm 4P}(q^2)
\,, \,\,\,   f_{B M, \, \rm NLP}^{0, \, \rm 4P}(q^2) \,,
\,\,\, \frac{n \cdot p} {m_B + m_M} \, f_{B M, \, \rm NLP}^{T, \, \rm 4P}(q^2) \right \}  \,
\nonumber \\
&& =  {2 \, \pi \over 3} \, \frac{\alpha_s(\mu)  \, C_F  \, \langle \bar q^{\prime} q^{\prime}  \rangle}{n \cdot p}  \,
{\cal F}_B(\mu) \,
\bigg \{  \int_{\omega_s}^{+\infty} \,  {d \omega  \over \omega^2}  \,
\left [  {2 \, \omega_M \over \omega} \, \left (  {\rm exp} \left (- {\omega_s \over  \omega_M} \right )  -1 \right )
+ 1 - {\langle \bar q  q  \rangle \over \langle \bar q^{\prime} q^{\prime}  \rangle}   \right ]
\phi_B^{+}(\omega, \mu)
\nonumber \\
&& \hspace{0.5 cm}
+ \, \int_0^{\omega_s} \,  {d \omega  \over \omega^2}  \,
\left [ \left ( 1 - { 2 \, \omega_M \over \omega}
- {\langle \bar q  q  \rangle \over \langle \bar q^{\prime} q^{\prime}  \rangle} \right )
+ \left (  1 + {2 \, \omega_M \over \omega}
+ {\langle \bar q  q  \rangle \over \langle \bar q^{\prime} q^{\prime}  \rangle} \,
\left (1 + {\omega \over \omega_M} \right ) \right ) \,
{\rm exp} \left (- {\omega \over  \omega_M} \right )   \right ] \,
\nonumber \\
&& \hspace{0.5 cm}
\,\, \times \, \phi_B^{+}(\omega, \mu)  \,
+  \, {\langle \bar q  q  \rangle \over \langle \bar q^{\prime} q^{\prime}  \rangle} \,
\left [ {\rm exp} \left (- {\omega_s \over  \omega_M} \right ) \,
{\phi_B^{+}(\omega_s, \mu)  \over \omega_s} \right ]
\bigg \}   + {\cal O}(\alpha_s^2) \,.
\label{LCSR for the 4P corrections}
\end{eqnarray}

In contrast with the previously determined subleading-twist corrections to
the exclusive  $B \to \gamma \ell \bar \nu_{\ell}$  decay form factors [74],
these non-valence  Fock-state contributions  appear to preserve  the (classical) large-recoil symmetry relations
between the semileptonic $B_{d, s}  \to \pi, K$  form factors,
according to the newly established LCSR (<ref>).

Collecting the  different pieces together, we can now summarize the eventual NLP sum rules
for the exclusive bottom-meson decay form factors at small momentum transfer

\begin{eqnarray}
f_{B M, \, \rm NLP}^{i}  =  f_{B M, \, \rm NLP}^{i, \, \rm hc}
+  f_{B M, \, \rm NLP}^{i, \, \rm (A2)}
+ f_{B M, \, \rm NLP}^{i, \, \rm HT}
+  f_{B M, \, \rm NLP}^{i, \, \rm 4P} \,,
\hspace{0.5 cm}
(i=+, \, -, \, T)
\label{NLP sum rules: summary}
\end{eqnarray}

where the analytical expressions for the individual terms on the right-handed side
have been displayed in (<ref>),
and (<ref>), respectively.

§ NUMERICAL ANALYSIS

Having at our disposal the improved sum rules for the exclusive $B_{d, s}  \to \pi, K$ form factors
at large hadronic recoil including both the leading-power spectator-quark mass corrections at ${\cal O}(\alpha_s)$
and the newly derived NLP contributions from four distinct dynamical sources,
we are now prepared to explore their numerical implications on a variety of the phenomenological observables
for the semileptonic $B \to \pi \ell \bar \nu_{\ell}$  and $B_s \to K \ell \bar \nu_{\ell}$ decays  (with  $\ell= \mu, \, \tau$)
as well as the theoretically cleanest electroweak penguin $B \to K \nu_{\ell} \bar \nu_{\ell}$ decay processes.
To achieve this goal, we will first specify the essential theory inputs (for instance, the electroweak SM parameters,
the bottom-quark mass,  the pseudoscalar-meson decay constants,
both the leading-twist and higher-twist HQET distribution amplitudes,
the intrinsic sum rule parameters) appearing in the obtained expressions for the heavy-to-light bottom-meson form factors.
In particular, we will extrapolate the updated LCSR predictions of the considered form factors to the entire kinematic region
by performing the numerical fits of the series coefficients in the conventional BCL expansions [75, 76, 77],
taking into account further the available lattice QCD results at large momentum transfer.
An emphasis will be placed on the very impacts of the newly achieved LCSR predictions
on pinning down the theory uncertainties  of the exclusive $B_{d, s}  \to \pi, K$ form factors
by  carrying out the analogous BCL  fits merely to the numerical lattice QCD determinations
in the lower-recoil region.

§.§ Theory inputs

$G_F$                                             $1.166379 \times 10^{-5} \,\, {\rm GeV}^{-2} $
[90]
$\alpha_s^{(5)}(m_Z)$                              $0.1179 \pm 0.0009$                                [90]

$m_{\mu}$                                          $105.658$ MeV                                      [90]
$m_{\tau}$                                         $1776.86 \pm 0.12$ MeV                             [90]

$\overline{m}_b (\overline{m}_b)$                  $4.203 \pm 0.011$  GeV                             [90]
$m_b^{\rm PS} (2 \, {\rm GeV})$                    $4.532^{+0.013}_{-0.039}$  GeV                     [113]

$m_{B_d}$                                          $5279.66 \pm 0.12$ MeV                              [90]
$\tau_{B_d}$                                       $(1.519 \pm 0.004)$ ps                              [90]

$m_{B_s}$                                          $5366.92 \pm 0.10$ MeV                              [90]
$\tau_{B_s}$                                       $(1.527 \pm 0.011)$ ps                              [90]

$f_{B_d}|_{N_f = 2+1+1}$                           $190.0 \pm 1.3$ MeV                                [16]
$f_{B_s}|_{N_f = 2+1+1}$                           $230.3 \pm 1.3$ MeV                                 [16]

$m_{B^{\ast} (1-)}$                               $5324.70 \pm 0.21$ MeV                              [90]
$m_{B_s^{\ast} (0+)}$                             $5415.4^{+1.8}_{-1.5}$ MeV                           [90]

$m_{B^{\ast} (0+)}$                               $5627 \pm 35$ MeV                                [114]
$m_{B_s^{\ast} (0+)}$                             $5718 \pm 35$ MeV                                 [114]

$\overline{m}_u (2 \, {\rm GeV})$                  $2.20 \pm 0.08$  MeV                             [90]
$\overline{m}_d (2 \, {\rm GeV})$                  $4.69 \pm 0.05$  MeV                             [90]

$\overline{m}_s (2 \, {\rm GeV})$                  $93.1 \pm 0.6$  MeV                             [90]

$m_{\pi}$                  $139.57$  MeV                             [90]
$f_{\pi}$               $130.2 \pm 1.2$  MeV                      [90]

$m_{K}$                  $493.677$  MeV                             [90]
$f_{K}$               $155.7 \pm 0.3$  MeV                      [90]

$\lambda_{B_d}(\mu_0)$                            $(350 \pm 150)$ MeV                                 [115, 99]
$\{0.7, \, 6.0\}$

$\lambda_E^2(\mu_0)/\lambda_H^2(\mu_0)$           $0.50 \pm 0.10$                                    [74]
$ \{\widehat{\sigma}_{B_{d, s}}^{(1)}(\mu_0), \,
\widehat{\sigma}_{B_{d, s}}^{(2)}(\mu_0)\}$              $\{0.0, \, \pi^2/6\}$                                [115]

$2 \, \lambda_E^2(\mu_0) + \lambda_H^2(\mu_0)$           $(0.25 \pm 0.15) \, {\rm GeV^2}$             [74]
$\{-0.7, \, -6.0\}$

$\lambda_{B_s}(\mu_0)$                            $(400 \pm 150)$ MeV                                 [115]

$\lambda$      $0.2250 \pm   0.0006$         [90]        $A$
$ 0.826^{+0.018}_{-0.015}$        [90]

$\bar \rho$     $ 0.159 \pm 0.010$           [90]       $\bar \eta$
$0.348 \pm 0.01$     [90]

$s_{0}^{\pi}$              $\left \{ (0.70 \pm 0.05)  \, {\rm GeV^2} \right \} $                       [36, 31]
$M^2$                    $(1.25 \pm 0.25) \, {\rm GeV^2} $                       [36, 31]

$s_{0}^{K}$                $\left \{ (1.05 \pm 0.05)  \, {\rm GeV^2} \right \} $                       [36, 31]

$ \langle \bar q  q \rangle  (2 \, {\rm GeV})$              $ -(286 \pm 23) \, {\rm MeV^3}$                       [16]
$ \langle \bar s s \rangle : \langle \bar q  q \rangle $                   $0.8 \pm 0.1$                       [116, 117]

Numerical values of the theory input parameters employed in the LCSR determinations
of the exclusive $B_{d, s}  \to \pi, K$ form factors as well as  the subsequent phenomenological analysis
for the semileptonic bottom-meson decay observables.

We summarize explicitly the numerical values of the necessary SM inputs and  the  hadronic parameters in
Table <ref>.
We will adopt the three-loop evolution of the strong coupling constant $\alpha_s(\mu)$ in the ${\rm \overline{MS}}$ scheme
by taking the determined  interval $\alpha_s^{(5)}(m_Z)$ from [90]
and employing the perturbative matching scales $\mu_{4} =4.8 \, {\rm GeV}$ and $\mu_{4} =1.3 \, {\rm GeV}$
for crossing  $n_f=4$ and $n_f=3$, respectively [98, 115].
In addition, the bottom-quark mass entering the short-distance coefficient functions of the obtained SCET sum rules
is generally understood to be the pole mass on account of the on-shell kinematics.
Converting the precisely known ${\rm \overline{MS}}$ mass to the counterpart  pole scheme will, however, bring about
the numerical results sensitive to the truncation order of  the perturbative matching relation
due to the existence of an infrared renormalon [118, 119].
Consequently, we will take advantage of  the potential-subtracted (PS) renormalization scheme [120]
for the bottom-quark mass (see [121] for an overview of several popular definitions for the heavy-quark mass)
and then perform the scheme conversion  of the hard functions from the pole mass
to the PS mass scheme.
In addition, we will employ the four-flavour lattice-computation results [16]
for the leptonic decay constants of bottom mesons
in the ${\rm SU}(2)$ isospin-symmetric limit
(see [122]  for additional discussions on the strong-isospin breaking corrections).
Following the theory prescription displayed in [90],
the adopted pion decay constant $f_{\pi}$ corresponds to  the three-flavour FLAG 2021 average
[16]
with the uncertainty increased by including the $0.7 \, \%$ charm sea-quark contribution.

We now turn to discuss the acceptable phenomenological models for the two-particle and three-particle
bottom-meson distribution amplitudes in HQET, fulfilling the nontrivial constraints from the classical equations of motion
and the expected asymptotic behaviours at small quark and gluon momenta from the conformal symmetry analysis.
For definiteness, we will employ the newly proposed three-parameter ansätz for the two-particle distribution amplitudes
in  coordinate space [74]
(see [123] for an alternative parametrization in terms of an expansion in associated Laguerre polynomials)

\begin{eqnarray}
\eta_{+}(s, \mu_0)= {}_{1}F_{1}(\alpha; \beta; -s \, \omega_0) \,,
\qquad
\eta_{3}^{(0)}(s, \mu_0)= - {\lambda_E^2 - \lambda_H^2  \over 18} \, s^2 \,
\left [  {}_{1}F_{1}(\alpha+2; \beta+2; -s \, \omega_0) \right ]\,,
\hspace{1.0 cm}
\label{three-parameter model for the 2P DAs}
\end{eqnarray}

which allows us to construct the analytical solutions to the Lange-Neubert evolution equations in the one-loop approximation
[41]

\begin{eqnarray}
\phi_B^{+}(\omega, \mu) &=& \hat{U}_{\phi}^{\rm tw2}(\mu, \mu_0) \,\, {1 \over \omega^{\kappa_s+1}} \,\,
{\Gamma(\beta) \over \Gamma(\alpha) } \, {\cal G}(\omega, \alpha, \beta;  0, 2, 1)\,,
\\
\phi_B^{-}(\omega, \mu) &=& \hat{U}_{\phi}^{\rm tw2}(\mu, \mu_0) \,\, {1 \over \omega^{\kappa_s+1}} \,\,
{\Gamma(\beta) \over \Gamma(\alpha) } \, {\cal G}(\omega, \alpha, \beta;  0, 1, 1)
\nonumber \\
&& + \, \hat{U}_{\phi}^{\rm tw3}(\mu, \mu_0) \,\,
\left [ - {\lambda_E^2(\mu) - \lambda_H^2(\mu)  \over 18 }  \right ]
\,\, {1 \over \omega^{\kappa_s+3}} \,\, %{\beta \, (\beta+1)  \over \alpha \, (\alpha+1)} \,\,
{\Gamma(\beta+2) \over \Gamma(\alpha+2) } \,\,
\bigg \{ {\cal G}(\omega, \alpha, \beta;  0, 3, 3)
\\
&& + \, \left (\beta - \alpha \right ) \,
\left [ {\omega \over \omega_0} \, {\cal G}(\omega, \alpha, \beta;  0, 2, 2)
- \beta \, {\omega  \over \omega_0} \,  {\cal G}(\omega, \alpha, \beta;  1, 2, 2)
- {\cal G}(\omega, \alpha, \beta;  1, 3, 3) \right ] \bigg \}  \,.
\nonumber
\end{eqnarray}

The manifest expressions for the two evolution functions $\hat{U}_{\phi}^{\rm tw2}$
and $\hat{U}_{\phi}^{\rm tw3}$ in momentum space can be further written as

\begin{eqnarray}
\hat{U}_{\phi}^{\rm tw2}(\mu, \mu_0) &=&  {\rm exp} \bigg  \{ -{ \Gamma_{\rm cusp}^{(0)}  \over 4 \, \beta_0^2} \,
\bigg  [ {4 \, \pi \over \alpha_s(\mu_0)} \,
\left ( \ln z_0 - 1 +{1 \over z_0}  \right ) - {\beta_1 \over 2 \beta_0^2} \, \ln^2 z_0
\nonumber \\
&& + \left ( {\Gamma_{\rm cusp}^{(1)} \over \Gamma_{\rm cusp}^{(0)}} - {\beta_1 \over 2 \beta_0} \right ) \,
\left [ z_0 - 1 - \ln z_0 \right ] \bigg  ]  \bigg  \}
\, \left ( e^{2 \, \gamma_E} \, \mu_0 \right )^{\Gamma_{\rm cusp}^{(0)} \, \ln z_0 / (2 \, \beta_0)} \,\,
z_0^{\gamma_{\rm tw2}^{(0)} / (2 \, \beta_0)} \,,
\hspace{1.0 cm}
\\
\hat{U}_{\phi}^{\rm tw3}(\mu, \mu_0) &=& z_0^{\gamma_{\rm tw3}^{(0)} / (2 \, \beta_0)} \,
\hat{U}_{\phi}^{\rm tw2}(\mu, \mu_0)  \,.
\end{eqnarray}

Moreover, we have introduced the following conventions for the expansion coefficient $\kappa_s$
as well as the Meijer ${\cal G}$-function [124]

\begin{eqnarray}
\kappa_s = {\Gamma_{\rm cusp}^{(0)}  \over 2 \, \beta_0}  \,
\ln {\alpha_s(\mu) \over \alpha_s(\mu_0)} \,,
\qquad
{\cal G}(\omega, \alpha, \beta;  l, m, n)
= G^{21}_{23}\bigg({}^{1, \, \beta+l}_{\kappa_s + m, \, \alpha, \, \kappa_s+n} \, \bigg| \frac{\omega}{\omega_0}\, \bigg )  \,.
\end{eqnarray}

The appearing HQET parameters $\lambda_E^2$ and  $\lambda_H^2$ can be defined by the effective matrix elements
of the chromo-electric and chromo-magnetic operators [125]

\begin{eqnarray}
&& \langle 0 | \bar q^{\prime} (0) \, g_s \, G_{\mu \nu} \, \Gamma \, h_v(0)| \bar B(v) \rangle
\nonumber \\
&& = - {{\cal F}_{B}(\mu) \, m_{B} \over 6} \,
{\rm Tr} \left \{ \gamma_5 \,\, \Gamma \,\,
\left (  {1 + \slashed v \over 2} \right ) \,
\left [ \lambda_H^2  \, \left (  i \, \sigma_{\mu \nu} \right )
+ (\lambda_H^2 - \lambda_E^2) \,
\left ( v_{\mu} \, \gamma_{\nu} - v_{\nu} \, \gamma_{\mu} \right ) \right ]  \right \}   \,.
\end{eqnarray}

Solving the  RG evolution equations for these two hadronic quantities  $\lambda_E^2$ and  $\lambda_H^2$ at one loop

\begin{eqnarray}
{d \over d \ln \mu} \,
\left(
\begin{array}{c}
\lambda_E^2(\mu) \\
\lambda_H^2(\mu) \\
\end{array}
\right)
+  {\alpha_s(\mu) \over 4 \pi} \, \gamma_{\rm EH}^{(0)}  \,
\left(
\begin{array}{c}
\lambda_E^2(\mu) \\
\lambda_H^2(\mu) \\
\end{array}
\right) = 0
\,,
\label{1-loop evolution equations of lambdaE and lamdbaH}
\end{eqnarray}

with the available anomalous dimension matrix from [126, 127]

\begin{eqnarray}
\gamma_{\rm EH}^{(0)} =
\left(
\begin{array}{cc}
{8 \over 3} \, C_F + {3 \over 2}  \, N_c \,\,  &  \,\,  {4 \over 3} \, C_F - {3 \over 2}  \, N_c \\
{4 \over 3} \, C_F - {3 \over 2}  \, N_c \,\,  & \,\,  {8 \over 3} \, C_F + {5 \over 2}  \, N_c \\
\end{array}
\right)
\,,
\end{eqnarray}

we can readily determine their renormalization-scale dependencies by diagonalizing the renormalization kernel
at the leading-logarithmic accuracy.
The method of two-point QCD sum rules has been applied to estimate these HQET parameters repeatedly,
yielding the numerical predictions significantly different each other even with the sizeable theory uncertainties

\begin{eqnarray}
\left \{  \lambda_E^2({1\, \rm GeV}),  \,\, \lambda_H^2({1\, \rm GeV}) \right \}
= \left\{
\begin{array}{l}
\left \{  (0.11 \pm 0.06) \, {\rm GeV^2}, \,\, (0.18 \pm 0.07) \, {\rm GeV^2}  \right \}  \,,
\qquad \hspace{0.5 cm} \text{[125]}
\vspace{0.5 cm} \\
\left \{  (0.03 \pm 0.02) \, {\rm GeV^2}, \,\, (0.06 \pm 0.03) \, {\rm GeV^2}  \right \}    \,,
\qquad  \hspace{0.5 cm}
\text{[127]}
\vspace{0.5 cm} \\
\left \{  (0.01 \pm 0.01) \, {\rm GeV^2}, \,\, (0.15 \pm 0.05) \, {\rm GeV^2}  \right \}    \,.
\qquad  \hspace{0.5 cm}
\text{[128]}
\end{array}
\hspace{0.8 cm} \right.
\label{sum rule results  for lambdaE and lambdaH}
\end{eqnarray}

The substantial discrepancies of the obtained  numerical results between [125]
and [127] can be traced back to the remarkable perturbative corrections
to the short-distance Wilson coefficients for the dimension-five quark-gluon mixed condensate
$\langle \bar q \, \sigma_{\mu \nu} \, G^{\mu \nu} q \rangle$
and to the further inclusion of the particular higher-order power corrections at tree level
from  the dimension-six vacuum condensates in the factorization approximation in the latter.
On the other hand, the authors of [128] proposed to
employ an alternative diagonal correlation function of the two three-body HQET currents
(instead of the non-diagonal current-current correlator implemented in [125, 127])
for constructing the desired sum rules of the essential ingredients  $\lambda_E^2$ and  $\lambda_H^2$,
in an attempt to pin down  the systematic uncertainty from the parton-hadron duality.
However, such attractive benefits are unfortunately  achieved at the price of worsening
the operator-product-expansion (OPE) convergence in the partonic computation of the new correlation function
and enhancing the intricate contributions from the continuum and higher excited states
(see [128] for more detailed discussions).
In the absence of the satisfactory determinations of these two HQET quantities,
we will therefore take the numerical intervals for the two combinations $2 \, \lambda_E^2 + \lambda_H^2$
and $\lambda_E^2/\lambda_H^2$ displayed in Table <ref>,
covering the allowed ranges of the previously obtained results from [125, 127]
and simultaneously satisfying the  derived (conservative) upper bounds from [128]
due to the positive definite spectral densities.

Applying the customary  definitions of the inverse logarithmic moments for
the leading-twist bottom-meson distribution amplitude [74, 115, 98]

\begin{eqnarray}
\frac{1}{\lambda_{B}(\mu)} &=& \int_0^{\infty} \, {d \omega \over \omega} \, \phi_B^{+}(\omega, \mu)  \,,
\qquad
%\nonumber \\
\frac{\widehat{\sigma}_{B}^{(n)} (\mu)}{\lambda_{B}(\mu)} =
\int_{0}^{\infty} \, {d \omega \over \omega} \,
\left [ \ln \left ( {\lambda_{B}(\mu_0) \over \omega} \right )
- \gamma_E \right ]^{n} \, \phi_{B}^{+}(\omega, \mu) \,,
\hspace{1.0 cm}
\end{eqnarray}

we can immediately determine these fundamental nonperturbative quantities in terms of
the three shape parameters in our model (<ref>)

\begin{eqnarray}
\lambda_{B}(\mu_0) &=& \left ( {\alpha-1 \over \beta-1} \right ) \, \omega_0 \,,
\qquad
\widehat{\sigma}_{B}^{(1)} (\mu_0) =  \psi (\beta-1) - \psi (\alpha-1)
+ \ln  \left ( {\alpha-1 \over \beta-1} \right ) \,,
\nonumber \\
\widehat{\sigma}_{B}^{(2)} (\mu_0) &=& \left [  \widehat{\sigma}_{B}^{(1)} (\mu_0)  \right ]^2 +
\psi^{(1)}(\alpha-1) - \psi^{(1)}(\beta-1)  + {\pi^2 \over 6} \,,
\end{eqnarray}

where $\psi(z)$ stands for the digamma function defined by the logarithmic derivative of the $\Gamma$-function.
Adopting the one-loop RG equation for the twist-two bottom-meson distribution amplitude
enables us to derive the leading-logarithmic evolutions for the first few momentums

\begin{eqnarray}
\lambda_{B}(\mu) &=& \lambda_{B}(\mu_0) \,
\left \{ 1 + {\alpha_s(\mu_0) \, C_F \over  \pi} \, \ln {\mu \over \mu_0}  \,
\left [  \widehat{\sigma}_{B}^{(1)} (\mu_0) +  \ln {\sqrt{\mu \mu_0} \, e^{\gamma_E} \over \lambda_{B}(\mu_0)}
- \left ( {1 \over 2} \right )   \right ] \right \}
+ {\cal O}(\alpha_s^2) \,, \hspace{1.0 cm}
\\
\widehat{\sigma}_{B}^{(1)} (\mu) &=&  \widehat{\sigma}_{B}^{(1)} (\mu_0)
\, \left \{ 1 +   {\alpha_s(\mu_0) \, C_F \over  \pi} \, \ln {\mu \over \mu_0}  \,
\left [ \widehat{\sigma}_{B}^{(1)} (\mu_0)  -
\frac{\widehat{\sigma}_{B}^{(2)} (\mu_0) }{\widehat{\sigma}_{B}^{(1)} (\mu_0) }  \right ]  \right \}
+ {\cal O}(\alpha_s^2) \,,
\\
\widehat{\sigma}_{B}^{(2)} (\mu) &=&  \widehat{\sigma}_{B}^{(2)} (\mu_0)
\, \left \{ 1 +   {\alpha_s(\mu_0) \, C_F \over  \pi} \, \ln {\mu \over \mu_0}  \,
\left [  \widehat{\sigma}_{B}^{(1)} (\mu_0)  -
\frac{\widehat{\sigma}_{B}^{(3)} (\mu_0) - 4 \, \zeta_3 }{\widehat{\sigma}_{B}^{(2)} (\mu_0) }   \right ]  \right \}
+ {\cal O}(\alpha_s^2) \,.
\end{eqnarray}

In spite of  the enormous efforts undertaken to determine the first inverse moment $\lambda_B^{-1}(\mu)$
with  different theory prescriptions, we are still unable to accomplish the robust computations of
this fundamental parameter  from the first field-theoretical principles at present
(see however [129, 130] for interesting discussions from the lattice QCD perspectives).
Consequently,  we prefer to employ the conservative interval $\lambda_{B_d}(\mu_0) = (350 \pm 150) \, {\rm MeV}$
accommodating the indirect extractions from the radiative leptonic $B \to \gamma \ell \bar \nu$ decay rates
[132, 131, 133] in the subsequent numerical analysis
and assign further ${\cal O}(15  \, \%)$ SU(3)-flavour symmetry breaking effect for the ratio $\lambda_{B_s}/\lambda_{B_d}$
(characterizing the typical splitting between the constituent down-quark and strange-quark masses).
We mention in passing that these two HQET  parameters have been recently computed from the traditional QCD sum rule approach
by investigating the appropriate correlation function of an effective light-cone heavy-to-light current
and an interpolating current for the pseudoscalar heavy-meson state [134],
following closely the theory strategy suggested in [101, 125].
The yielding numerical predictions $\lambda_{B_d}(\mu_0) = (383 \pm 153) \, {\rm MeV}$
and $\lambda_{B_s}/\lambda_{B_d} = 1.19 \pm 0.14$ [134]
are apparently in the same ballpark as the corresponding intervals displayed in Table <ref>.

The general  ansätz for the higher twist distribution amplitudes
incorporating both the anticipated low-momentum behaviour
and the tree-level equations-of-motion constraints
can be constructed by introducing a unique   profile function  [74, 98, 99]

\begin{eqnarray}
\Phi_3(\omega_1, \omega_2, \mu_0) &=&
- \left ( {1 \over 2} \right ) \, \varkappa(\mu_0) \,
\left [ \lambda_E^2 (\mu_0) - \lambda_H^2 (\mu_0) \right ] \,
\omega_1 \, \omega_2^2 \, f^{\prime}(\omega_1 + \omega_2) \,,
\\
\Phi_4(\omega_1, \omega_2, \mu_0) &=&
\left ( {1 \over 2} \right )  \, \varkappa(\mu_0) \, \left [ \lambda_E^2 (\mu_0) + \lambda_H^2 (\mu_0) \right ] \,
\omega_2^2 \, f(\omega_1 + \omega_2) \,,
\\
\Psi_4(\omega_1, \omega_2, \mu_0) &=&
\varkappa(\mu_0) \,  \lambda_E^2 (\mu_0) \, \omega_1 \,  \omega_2 \, f(\omega_1 + \omega_2) \,,
\\
\tilde{\Psi}_4(\omega_1, \omega_2, \mu_0) &=&
\varkappa(\mu_0) \,  \lambda_H^2(\mu_0) \, \omega_1 \,  \omega_2 \, f(\omega_1 + \omega_2) \,,
\\
\Psi_5(\omega_1,\omega_2, \mu_0) &=&
- \varkappa(\mu_0) \,  \lambda_E^2(\mu_0) \, \omega_2  \, \int_{\omega_1 + \omega_2}^\infty d \eta \, f(\eta),
\\
\Phi_6(\omega_1, \omega_2, \mu_0) &=&
\varkappa(\mu_0) \, \left [ \lambda_E^2 (\mu_0) - \lambda_H^2 (\mu_0) \right ] \,
\int_{\omega_1 + \omega_2}^\infty d \eta_1 \,  \int_{\eta_1}^{\infty} \, d \eta_2 \, f(\eta_2)\,.
\end{eqnarray}

We further collect the explicit expressions of the two-particle subleading-twist HQET distribution amplitudes
from  the off-light-cone corrections for completeness

\begin{eqnarray}
g_B^{+}(\omega, \mu) &=& \hat{g}_B^{+}(\omega, \mu)
- {1 \over 2} \, \int_0^{\omega} \, d \omega_1 \, \int_0^1 d u \,
{\bar u \over u} \,\, \psi_4 \left (\omega, {\omega - \omega_1 \over u}, \mu \right ) \,,
\nonumber \\
g_B^{-}(\omega, \mu) &=& \hat{g}_B^{-}(\omega, \mu)
- {1 \over 2} \, \int_0^{\omega} \, d \omega_1 \, \int_0^1 d u \,
{\bar u \over u} \,\, \psi_5 \left (\omega, {\omega - \omega_1 \over u}, \mu \right ) \,,
\label{full expressions of gBpm}
\end{eqnarray}

where the twist-five “Wandzura-Wilczek" term $ \hat{g}_B^{-}(\omega, \mu)$

has been presented in (<ref>) and

\begin{eqnarray}
\hat{g}_B^{-}(\omega, \mu) =  \left ( {1 \over 4} \right ) \, \int_{\omega}^{\infty}  \, d \rho \,
\left  \{ (\rho - \omega) \, \left [ \phi_B^{-}(\rho) -  \phi_B^{+}(\rho) \right ]
- 2 \, (\bar \Lambda - \rho)  \, \phi_B^{+}(\rho)  \right \}  \,.
\label{def: WW part of gBplus}
\end{eqnarray}

The particular three-parameter model for the twist-two coordinate-space  LCDA $\eta_{+}(s, \mu_0)$
in (<ref>) implies the following nonperturbative function $f(\omega)$
and the normalization constant $\varkappa(\mu_0)$

\begin{eqnarray}
f(\omega) &=& {\Gamma(\beta) \over \Gamma(\alpha)}  \,
U \left (\beta-\alpha, 3-\alpha, {\omega \over \omega_0} \right ) \,
{1 \over \omega_0^2} \,\, {\rm exp} \left ( - {\omega \over \omega_0} \right ) \,,
\nonumber \\
\varkappa^{-1}(\mu_0) &=& \left ( {1 \over 2} \right ) \, \int_{0}^{\infty} d \omega \, \omega^3 \, f(\omega)
= \left [ {3 \, \alpha \, (\alpha-1) \over  \beta \, (\beta-1) } \right ] \, \omega_0^2  \,.
\end{eqnarray}

Furthermore, we will take the “effective mass" of the bottom-meson state entering the NLP sum rules
(<ref>) and  (<ref>)
as $\bar \Lambda = m_{B_q} - m_{b} +{\cal O}(\Lambda_{\rm QCD}^2/m_b)$ with $m_{b} = (4.8 \pm 0.1) \, {\rm GeV}$
numerically (see [135] for the yet higher-order correction to this essential mass relation and
[136, 137] for further discussions on the scheme dependence of this hadronic quantity).

Additionally, we will vary the hard-matching scales $\mu_{h1}$ and  $\mu_{h2}$
appearing in the NLL SCET sum rules for the heavy-to-light bottom-meson transition  form factors (<ref>)
and (<ref>) in the interval $[m_b/2, \, 2\, m_b]$ with the central value $m_b$,
as widely implemented in the exclusive heavy-hadron decay phenomenologies [74, 115, 98].
The renormalization  scale $\nu$ of the QCD tensor current will be taken as the hard scale of order of the $b$-quark mass.
By contrast, the factorization scale $\mu$ will be treated as the hard-collinear scale
in the  range of $(1.5 \pm 0.5) \, {\rm GeV}$.

Following the standard procedure outlined in [34],
the determinations of two intrinsic  LCSR parameters $\omega_M$ and $\omega_s$ can be routinely achieved by
imposing the necessary constraints on the smallness of the continuum contributions in the dispersion integrals
of the three invariant functions $\Pi$, $\tilde{\Pi}$ and $\Pi_{\rm T}$
and on the stability of the obtained sum rules against the variation of the Borel mass $\omega_M$.
Proceeding with this practical prescription leads to the following intervals

\begin{eqnarray}
M^2 &=& n \cdot p \, \omega_M = (1.25 \pm 0.25) \, {\rm GeV^2}  \,,
\qquad
s_0^{\pi} =  n \cdot p \, \omega_s = (0.70 \pm 0.05) \, {\rm GeV^2} \,,
\nonumber \\
s_0^{K} &=&  n \cdot p \, \omega_s = (1.05 \pm 0.05) \, {\rm GeV^2} \,,
\end{eqnarray}

which are in excellent agreement with the numerical results employed in the LCSR computations
of the pion-photon form factor [138] as well as the pion electromagnetic form factor [112],
and in the two-point QCD sum rules for the $K$-meson decay constant [139].

§.§ Numerical  predictions for the $B_{d, s}  \to \pi, K$ form factors

We are now ready  to explore the phenomenological impacts of the NLL resummation improved leading-power contributions
(including further the light spectator-quark mass effects) and  the newly derived NLP corrections at the tree-level accuracy
on the semileptonic  $B_{d, s}  \to \pi, K$ decay form factors at large hadronic recoil.
In order to develop a transparent understanding of the dynamical patterns dictating these intricate form factors,
we display explicitly the yielding  leading-power contributions to the complete set of
the exclusive bottom-meson decay  form factors at the NLL accuracy,
the NLP contribution from expanding the hard-collinear quark propagator in the small parameter $\Lambda_{\rm QCD}/m_b$,
the NLP contribution from the power suppressed term in the ${\rm SCET}_{\rm I}$ representation of the weak transition current,
the subleading-twist contribution from the two-particle and three-particle HQET distribution amplitudes,
together with  the twist-five and twist-six four-particle bottom-meson distribution amplitudes in Figures
and <ref>
in the kinematic range $0 \leq q^2 \leq 8 \, {\rm GeV^2}$.
In particular, we have included the perturbative uncertainties for the individual pieces under discussion
from varying both  the hard and hard-collinear matching scales as indicated by the separate error bands.
The NLL resummation improved sum rules on the light-cone are indeed  beneficial for pinning down
the obtained theory uncertainties when compared with  the counterpart leading-logarithmic computations.
Generally, the perturbative QCD corrections to the short-distance matching coefficients in the SCET sum rules
can  shift the corresponding leading-power contributions  by an amount of ${\cal O} (30 \, \%)$ numerically.
It is evident from Figures <ref>
and <ref>
that the most prominent subleading-power corrections to the heavy-to-light
bottom-meson decay form factors  arise from  the peculiar  higher-twist contributions of the two-particle and three-particle
HQET  distribution amplitudes  at ${\cal O}(\alpha_s^0)$ (more precisely from the two-particle twist-five
off-light-cone correction as already noticed in [38, 41])
yielding consistently $(25-30) \, \%$ reduction of the corresponding leading-power LCSR predictions at NLL
for $q^2 \in \left [0, \,  8 \right] \, {\rm GeV^2}$.
By contrast, the  estimated  four-particle twist-five and twist-six corrections in the factorization approximation
can only bring about insignificant impacts on the leading-power contributions to the exclusive $B_{d, s}  \to \pi, K$ form factors:
$(2-4) \, \%$ numerically, which can be attributed 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>).
This interesting observation indicates that the higher-twist four-particle contributions will actually be   suppressed
by an extra power of $\Lambda_{\rm QCD}^2/s_0$ (rather than the additional powers of $\Lambda_{\rm QCD}/m_b$
in the heavy quark expansion) in analogy to the previous discussions [73, 131] in  different contexts.
Moreover, the newly determined subleading-power contributions from the hard-collinear quark propagator
shown in (<ref>) can further generate the sizeable
destructive interferences (as large as ${\cal O} (20 \, \%)$  numerically) with the counterpart leading-power contributions.
It remains important to emphasize that the obtained hierarchy relations for the exclusive bottom-meson decay form factors
at maximal recoil

\begin{eqnarray}
f_{B K}^{+}(0) > f_{B_s K}^{+}(0) >  f_{B \pi}^{+}(0) \,,
\qquad
f_{B K}^{T}(0) > f_{B_s K}^{T}(0) >  f_{B \pi}^{T}(0)\,,
\end{eqnarray}

coincide precisely with the emerged patterns predicted by the two independent LCSR computations
with the final-state pseudoscalar-meson distribution amplitudes [140, 141]
(see  [142] for the alternative estimates with the TMD factorization approach and
[143] for the quantitative analysis in the framework of Dyson-Schwinger equations).

Breakdown of the distinct dynamical mechanisms contributing to
the semileptonic $B \to \pi \ell \bar \nu_{\ell}$ form factors (left panel)
and to the electroweak penguin  $B \to K \nu_{\ell} \bar \nu_{\ell}$ decay form factors (right panel)
in the kinematic region $0 \leq q^2 \leq 8 \, {\rm GeV^2}$
from the updated LCSR computations with the HQET bottom-meson distribution amplitudes,
where the perturbative uncertainties due to the variations of both the hard and hard-collinear matching scales
are indicated by the individual error bands.

Breakdown of the distinct dynamical mechanisms contributing to the semileptonic
$B_s \to K \ell \bar \nu_{\ell}$ form factors  in the kinematic region $0 \leq q^2 \leq 8 \, {\rm GeV^2}$
from the updated LCSR computations with the HQET bottom-meson distribution amplitudes,
where the perturbative uncertainties due to the variations of both the hard and hard-collinear matching scales
are indicated by the individual error bands.

Theory predictions for the SU(3)-flavour symmetry breaking effects between
the semileptonic $B \to \pi$ and $B \to K$ form factors from the updated SCET sum rules
with the bottom-meson distribution amplitudes at the twist-six accuracy,
where the perturbative uncertainties due to the variations of both the hard and hard-collinear matching scales
are indicated by the individual error bands.

We are now in a position to investigate the SU(3)-flavour symmetry breaking effects between
the exclusive $B \to \pi$ and $B \to K$ form factors, on the basis of the established SCET sum rules
with the bottom-meson distribution amplitudes up to the twist-six accuracy,
by introducing further the following  quantities [36, 38]

\begin{eqnarray}
{\cal R}_{\rm SU(3)}^{i}(q^2) = \left[  f_{B \to K}^{i}(q^2) \right ] : \left [ f_{B \to \pi}^{i}(q^2) \right ] \,,
\qquad
({\rm with } \,\, i = +, \, 0, \, T )\,.
\end{eqnarray}

In our theoretical framework such flavour-symmetry violations arise from the apparent discrepancies
in the light-quark masses, in  the light-flavour hadron masses,
in the leptonic decay constants $f_{\pi}$ and $f_{K}$,
in the threshold parameters for the pion and kaon channels,
and finally in the nonperturbative quark-condensate densities
$\langle \bar u u \rangle$ and $\langle \bar s s \rangle$.
It can be observed from Figure <ref>
that the leading-power LCSR predictions
for the SU(3)-flavour symmetry breaking effects give rise to  numerically
${\cal O} (30 \, \%)$ for the two ratios ${\cal R}_{\rm SU(3)}^{+, \, 0}$
and ${\cal O} (40 \, \%)$ for the tensor-form-factor ratio ${\cal R}_{\rm SU(3)}^{T}$
in the large recoil region.
In particular,  the newly determined subleading-power contributions from the same LCSR method
can only bring about the insignificant numerical impacts on the SU(3)-flavour symmetry violating effects
(see [38] for the analogous observation
for the exclusive semileptonic $B \to V \ell \bar \nu_{\ell}$ form factors).
It is worthwhile mentioning that  we have not taken into account the remaining sources
of the SU(3)-flavour symmetry breaking effects due to the electromagnetic corrections
and the systematic uncertainties owing to the parton-hadron duality approximation.

Theory predictions for the large-recoil symmetry breaking corrections  between
the exclusive  $B \to \pi$ form factors from the updated SCET sum rules
with the bottom-meson distribution amplitudes at the twist-six accuracy.
For a comparison, we further display the numerical results from the QCD factorization approach
with the so-called physical form factor scheme [91]
by taking the light-meson LCSR computation of the vector form factor $f_{B \pi}^{+}(q^2)$ [141]
as the fundamental theory input.

We proceed to explore the celebrated large-recoil symmetry breaking effects between
the exclusive heavy-to-light bottom-meson decay form factors
due to the higher-order perturbative corrections
and the intricate subleading-power corrections in the $\Lambda_{\rm QCD}/m_b$ expansion.
In order to facilitate the straightforward comparison with the theory predictions
from the QCD factorization approach, it proves convenient to investigate the two particular
form-factor ratios for the semileptonic $B \to \pi \ell \bar \nu_{\ell}$
and $B \to \pi \ell \bar {\ell}$  decay processes [91, 36]

\begin{eqnarray}
{\cal R}_{B \pi}^{0 +}(q^2) = \left(  {m_B  \over n \cdot p} \right ) \,  \frac{f_{B \pi}^{0}(q^2)}{f_{B \pi}^{+}(q^2)} \,,
\qquad
{\cal R}_{B \pi}^{T +}(q^2) =  \left ( {m_B  \over m_B + m_{\pi}} \right ) \, \frac{f_{B \pi}^{T}(q^2)}{f_{B \pi}^{+}(q^2)} \,.
\end{eqnarray}

It is interesting to notice that the NLL resummation improved LCSR predictions for the two quantities  ${\cal R}_{B \pi}^{0 +}$
and ${\cal R}_{B \pi}^{T +}$ are in reasonable agreement with the QCD factorization computations at the leading-power accuracy.
On the contrary, confronting  our numerical  results for the form-factor ratio ${\cal R}_{B \pi}^{T +}$
from the bottom-meson LCSR method including four different classes of the  NLP corrections
with the counterpart predictions from QCD factorization reveal an intriguing  tension
on both the magnitude and sign of the large-recoil symmetry breaking corrections
as displayed in Figure <ref>.
Inspecting the individual terms in the obtained subleading-power sum rules (<ref>)
indicates that the emerged discrepancies between the two different QCD calculations
stem from the newly determined NLP contribution of the power-suppressed ${\rm SCET_{I}}$ current $J^{\rm (A2)}$
with the  LCSR  method as collected in (<ref>) explicitly,
which has not been included in the numerical exploration of the current QCD factorization framework [91].
As a consequence, it becomes more and more demanding to construct the appropriate perturbative factorization formula
for this NLP  matrix element directly with the modern effective-field-theory technique.

Apparently, we can only establish the soft-collinear factorization formulae for the desired vacuum-to-$B$-meson
correlation functions (<ref>) at small and intermediate momentum transfers,
$0 \leq q^2 \leq q_{\rm cut}^2$, where the practical choice of $q_{\rm cut}^2$ varies between $8 \, {\rm GeV^2}$
and $10 \, {\rm GeV^2}$ numerically.
We are therefore required to extrapolate the bottom-meson LCSR computations of
the semileptonic $B_{d, s}  \to \pi, K$ form factors towards the large momentum transfer $q^2$
by applying the $z$-series parametrization based upon the positivity and analyticity properties
of these transition form factors [144, 145, 146].
Adopting the conformal transformation [75, 77, 76]

\begin{eqnarray}
z(q^2, t_0)  = \frac{\sqrt{t_{+} - q^2} - \sqrt{t_{+} - t_0}}
{\sqrt{t_{+} - q^2} +\sqrt{t_{+} - t_0}} \,,
\label{def: z parameter}
\end{eqnarray}

with the threshold parameter $t_{+} \equiv \left [ m_B + m_{\pi (K)} \right ]^2$  for the exclusive semileptonic
$b \to u (s)$ transitions,

enables us to map the complex cut $q^2$-plane onto the unit disk $|z(q^2, t_0)|  \leq 1$.
On the other hand, the free parameter $t_0 < t_{+}$ corresponds to the value of $q^2$
mapping onto the origin in the $z$-plane, namely $z(t_0, t_0)=0$.
In order to maximally reduce the interval of $z$ obtained after mapping (<ref>)
of the semileptonic domain $q^2 \in \left [0, \, (m_{B_{q^{\prime}}}- m_M)^2 \right ]$,
the auxiliary parameter $t_0$ can be customarily taken as

\begin{eqnarray}
t_0 = t_{+} - \sqrt{t_{+} \, (t_{+} - t_{-})} \,,
\qquad t_{-} = (m_{B_{q^{\prime}}}- m_M)^2 \,.
\end{eqnarray}

following closely the comprehensive discussions presented in  [16].
Taking into account further the asymptotic behaviours of the vector form factors
near threshold due to the angular momentum conservation  implies the simplified series expansion
originally proposed in [77] (see [147, 148] for an alternative parametrization)

\begin{eqnarray}
f_{B_{q^{\prime}} M}^{+}(q^2) &=&   \frac{1}{1-q^2/m_{B_{q}^{\ast}}^2 }  \,
\sum_{k=0}^{N-1} \, b_{k}^{+} \,
\left [  z(q^2, t_0)^k  - (-1)^{k - N} \, {k \over N } \, z(q^2, t_0)^N  \right ]    \,,
\label{z-parametrization for the vector form factor}
\end{eqnarray}

where the lowest  bottom-meson $B_{d}^{\ast}$ ($B_{s}^{\ast}$) is expected to be the only resonance
of the $J^P=1^{-}$ channel below the $B \pi$ ($B K$) production region.
It is evident from the  BCL parametrization (<ref>) that
the well-known scaling behaviour $f_{B_{q^{\prime}} M}^{+}(q^2) \sim 1/q^2$ at $|q^2| \to \infty$
from the perturbative QCD analysis [149] is indeed fulfilled.
For the practical purpose,  we will truncate the $z$-series expansion at $N=3$
in the subsequent fitting program.

Along the same vein, we can proceed to parameterize the remaining  form factors
by adjusting the overall pole functions appropriately and by dropping out the near-threshold constraints
for the scalar transition  form factors

\begin{eqnarray}
f_{B_{(s)} \pi (K)}^{0}(q^2) &=&  \, \sum_{k=0}^{N-1} \, b_{k}^{0} \,\, z(q^2, t_0)^k    \,,
\qquad
f_{B K}^{0}(q^2) =  \frac{1}{1-q^2/m_{B_{s}^{\ast}  (0+)}^2} \, \sum_{k=0}^{N-1} \, b_{k}^{0} \,\, z(q^2, t_0)^k \,,
\nonumber  \\
f_{B_{q^{\prime}} M}^{T}(q^2) &=&  \frac{1}{1-q^2/m_{B_{q}^{\ast}}^2} \,
\sum_{k=0}^{N-1} \, b_{k}^{T} \,
\left [  z(q^2, t_0)^k  - (-1)^{k - N} \, {k \over N } \, z(q^2, t_0)^N  \right ]    \,.
\label{z-parametrization for the scalar and tensor form factors}
\end{eqnarray}

The very disappearance of the pole factors in the above-mentioned parameterizations
of the two particular form factors $f_{B \pi}^{0}(q^2)$ and $f_{B_{s} K}^{0}(q^2)$
can be attributed to the fact that
the low-lying bottom-resonance in the $J^P=0^{+}$ channel
with the predicted mass  $m_{B^{\ast} (0+)} = (5627 \pm 35) \, {\rm MeV}$ [114]
turns out to be located above the $B \pi$ production threshold
$m_B + m_{\pi} = (5419 \pm 0.12) \, {\rm MeV}$ [90].
By contrast, the low-lying bottom-meson resonance
in the BCL parametrization (<ref>)
for the scalar form factor $f_{B K}^{0}(q^2)$,
with the estimated mass $m_{B_s^{\ast} (0+)} = (5718  \pm 35) \, {\rm MeV}$
from the heavy-hadron chiral perturbation theory [114],
appears to be somewhat  below the particle-pair production threshold
$m_B + m_{K} = (5773 \pm 0.12) \, {\rm MeV}$ [90].
For convenience, we have also summarized the relevant resonance masses employed
in our $z$-parametrization fits in Table <ref>.

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

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

$b_0^+$     $0.404(13)$     $1$     $0.276$     $-0.446$     $0.291$     $0.164$     $0.583$     $0.266$     $-0.24$

$b_1^+$     $-0.618(63)$           $1$     $-0.374$     $0.067$     $0.485$     $0.234$     $0.693$     $-0.193$

$b_2^+$     $-0.473(215)$               $1$     $0.108$     $0.157$     $-0.185$     $-0.185$     $0.592$

$b_0^0$     $0.496(19)$                    $1$     $-0.226$     $0.318$     $0.06$     $0.018$

$b_1^0$     $-1.537(56)$                         $1$     $0.04$     $0.431$     $0.165$

$b_0^T$     $0.396(15)$                              $1$     $0.178$     $-0.423$

$b_1^T$     $-0.553(73)$                                    $1$     $-0.307$

$b_2^T$     $-0.248(235)$                                         $1$

Theory predictions for the correlated $z$-series coefficients
in the vector, scalar and tensor $B \to \pi$ 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, 3, 7]
with the preferred truncation $N=3$.

Theory predictions for the complete set of the semileptonic $B \to \pi$ 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 QCD (from [4, 3, 7]) data points.
We further display the yielding numerical results of these form factors by performing an alternative $z$-series fit
against the “only lattice QCD" data points [4, 3, 7]
exclusively for a comparison.

We are now prepared to determine the BCL series coefficients $b_k^{+, \, 0, \, T}$
by performing the binned $\chi^2$ fit of the updated LCSR predictions for the bottom-meson decay  form factors
at three distinct kinematic points, namely $q^2= \left \{-4.0, \, 0, \, 4.0  \right \} \, {\rm GeV^2}$,
in combination with the available lattice data points in the higher-$q^2$ region.
Enforcing the  kinematic constraint between the vector and scalar form factors
$f_{B M}^{+}(0) = f_{BM}^{0}(0) $  allows  us further  to derive the following exact relations
between the expansion coefficients of our interest

\begin{eqnarray}
b_2^{0}  &=& 12.78 \, \left (  b_0^{+} -   b_0^{0}  \right ) + 3.482 \, b_1^{+} + 1.186 \, b_2^{+}
- 3.575 \, b_1^{0} \,, \qquad ({\rm for} \,\, B \to \pi)
\nonumber \\
b_2^{0} &=&  24.06  \, \left ( b_0^{+} -   b_0^{0}  \right ) + 4.837 \, b_1^{+} + 1.136 \, b_2^{+}
- 4.905 \, b_1^{0} \,, \qquad ({\rm for} \,\, B_s \to K)
\nonumber \\
b_2^{0} &=&  48.59  \, \left ( b_0^{+} -   b_0^{0}  \right ) + 6.923 \, b_1^{+} + 1.096 \, b_2^{+}
- 6.971 \, b_1^{0} \,. \qquad ({\rm for} \,\, B \to K)
\label{exact relations of z-series parameters}
\end{eqnarray}

With regard to the lattice QCD results for the semileptonic $B \to \pi$ form factors,
we can  straightforwardly employ the synthetic data points of $f_{B \pi}^{+}(q^2)$ and $f_{B \pi}^{0}(q^2)$
at three representative values of $q^2 = \left \{19.0, \, 22.6, \, 25.1  \right \} \, {\rm GeV^2}$
with the full correlation matrices from the RBC/UKQCD Collaboration [4],
by adopting $N_f=2+1$-flavour gauge-field ensembles
with the domain-wall fermion action and Iwasaki gluon action.
However, the FNAL/MILC Collaboration do not provide the yielding data points for the  exclusive $B \to \pi$ form factors
explicitly  in their publications [3, 7],
which present the outcome of the combined BCL fit to their data points with the truncation $N=3$ instead.
Consequently, we will take advantage of the BCL fit results to generate the correlated synthetic data points
of the three $B \to \pi$ form factors in the kinematic region $19.0 \, {\rm GeV}^2 \leq q^2 \leq  26.4 \, {\rm GeV}^2$.
Carrying out the simultaneous fit of the conventional  BCL parameterizations
(<ref>) and
to our LCSR pseudo data points as well as the lattice QCD data points gives rise  to
the desired intervals of the $z$-series coefficients and their correlation matrix
for the semileptonic $B \to \pi$ form factors displayed in Table <ref>.
Furthermore, we observe that this numerical fit yields a minimal $\chi^2=7.93$
for $14$ degrees of freedom in the fitting program,
which corresponds to an excellent $p$-value of $89 \, \%$ in turn.
It has been  verified manifestly  that the fitted BCL coefficients  fulfill
the very dispersive bounds [77, 150] derived from the correlation functions
of two flavour-changing currents with the aid of unitarity and crossing symmetry
(see [151] for further discussions on the scaling behaviour of the sum of coefficients
$\sum_{k=0}^{N} \left (b_k^{+} \right )^2$  in the heavy quark limit).
In order to develop a transparent understanding towards the eventually predicted momentum-transfer dependence
from interpolating the LCSR and lattice QCD results,
we display the obtained  numerical predictions for  the  three exclusive $B \to \pi$ form factors
versus  $z$ (left panel) and versus $q^2$ (right panel) in the entire kinematic region
in Figure <ref>,
where the counterpart predictions of these form factors  from implementing an alternative $z$-expansion fit
of  the “only lattice QCD" data points [4, 3, 7]
exclusively are further shown for the convenience.
It is evident from this comparative exploration that including the newly derived LCSR data points
at small momentum transfer in our fitting procedure will indeed be highly beneficial for
improving the theory precision for all three $B \to \pi$ form factors
in the kinematic regime  $0.10 \leq z \leq 0.31$
(namely, $q^2 \in \left [ -4.0 ,   15.5 \right ]  \, {\rm GeV^2}$) significantly.
This interesting  observation can be actually understood from the very fact that
extrapolating the current lattice QCD results
towards the lower $q^2$ region solely will bring about the more pronounced uncertainties
for the form-factor shapes  in comparison with the direct LCSR computations
as already discussed in [4, 3, 7].

Additionally, it is instructive to compare our form-factor predictions from the combined BCL fit
with the theoretical expectations from the heavy quark spin symmetry and the current algebra method.
In the zero-recoil limit we can derive an interesting relation between the vector and scalar form factors
up to the accuracy of ${\cal O}(\Lambda_{\rm QCD}^2/m_b^2)$ [152]

\begin{eqnarray}
\lim_{q^2 \to m_{B_{q^{\prime}}}^2 } \, \frac{f_{B_{q^{\prime}} M}^{+}(q^2)}{f_{B_{q^{\prime}} M}^{0}(q^2)}
= \left ( \frac{f_{B_q^{\ast}}} {f_{B_{q^{\prime}}}} \right ) \, \left (1- {m_M^2 \over m_{B_{q^{\prime}}}^2}  \right )\,
\left ( \frac{\hat{g}_{\rm eff}}{ 1 - q^2 /  m_{B_q^{\ast}}^2} \right )   \,,
\label{soft-pion relation of fplus and fzero}
\end{eqnarray}

where the static coupling $\hat{g}_{\rm eff}$ entering the Lagrangian density of
the heavy-hadron chiral perturbation theory is independent of the heavy quark mass [135].
We will adopt the nonperturbative determination of this  low-energy constant from the NLO LCSR computations
with the pion distribution amplitudes $\hat{g}_{\rm eff}=0.30 \pm 0.02$ [153]
(see also [154, 155] for the earlier analyses in the same framework).
Moreover, we will employ the updated numerical results for the leptonic decay constants
of the heavy-light vector mesons [117]

\begin{eqnarray}
f_{B^{\ast}} = 210^{+10}_{-12} \, {\rm MeV} \,,
\qquad
f_{B_{s}^{\ast}} = 251^{+14}_{-16} \, {\rm MeV}  \,,
\end{eqnarray}

based upon the standard method of the two-point QCD sum rules.
We present the obtained numerical predictions for the form-factor ratio
$\left [ \left (1 - q^2 /  m_{B_q^{\ast}}^2 \right ) \, f_{B_{q^{\prime}} M}^{+}(q^2) \right ]  :
\left [ f_{B_{q^{\prime}} M}^{0}(q^2) \right ]$ from the combined $z$-expansion fitting program
in the lower-recoil region  in Figure <ref>,
where the complementary predictions with uncertainties from the combination of heavy quark and chiral
symmetries are further displayed explicitly for a comparison.

Theory predictions of the particular form-factor ratio
$\left [ \left (1 - q^2 /  m_{B_q^{\ast}}^2 \right ) \, f_{B_{q^{\prime}} M}^{+}(q^2) \right ]  :
\left [ f_{B_{q^{\prime}} M}^{0}(q^2) \right ]$ for the exclusive semileptonic $B_{d, s}  \to \pi, K$  decays
determined from the combined $z$-expansion fit
against the LCSR and lattice QCD data points (blue bands) and from the theoretical expectations
of the heavy quark spin symmetry and the current algebra technique
at NLO in the $\Lambda_{\rm}/m_b$ expansion [152] (grey bands).

The heavy quark spin symmetry relation between the vector and tensor form factors in the low recoil region
can be  derived by exploring  an exact operator identity on account of
the QCD equations of motion for the quark fields

\begin{eqnarray}
i \, \partial^{\nu} \left (\bar q \, i \, \sigma_{\mu \nu} \, b \right )
= i \,  \, \partial_{\mu} \left (\bar q  \, b \right )  \,
- (m_b + m_q) \, \bar q \, \gamma_{\mu} \, b  \,
- 2 \, \left ( \bar q \, i \, \overleftarrow{D}_{\mu} \, b \right ) \,,
\label{QCD EOM for the tensor current}
\end{eqnarray}

and by employing the Lorentz decomposition for the subleading heavy-to-light HQET matrix element
of the dimension-four operator [156, 157]

\begin{eqnarray}
\langle M(p) | \bar q \, i \, \overleftarrow{D}_{\mu} \, h_v | \bar B(p_B)\rangle
= \delta_{+}(q^2) \, (2 \, p  + q)_{\mu} + \delta_{-}(q^2) \, q_{\mu}  \,.
\label{effective dimension-four HQET matrix element}
\end{eqnarray}

Performing the conventional matching procedure ${\rm QCD} \to {\rm  HQET}$
for the emerged flavour-changing weak currents in the previous identity (<ref>)

\begin{eqnarray}
\bar q \, \gamma_{\mu} \, b  & \to &  C_0^{(v)}(\mu) \,\,  \bar q \, \gamma_{\mu} \, h_v
\, + \, C_1^{(v)}(\mu) \,\,  \bar q \, v_{\mu} \, h_v
\, + \, \left ( {1 \over 2 \, m_b} \right ) \, \bar q \, \gamma_{\mu} \,  i \, \slashed{D}  \, h_v
+ ...\,,
\nonumber \\