TUM-HEP-1330/21

QCD factorization for the $B\to\gamma\ell\nu_{\ell}$ decay beyond leading
power

Bo-Yan Cuia,b , Yue-Long Shenc , Chao Wangd and Yan-Bing Weie,f

a Department of Physics, Chongqing University of Science and Technology,
Chongqing 401331, P.R. China

b School of Physics, Nankai University, Tianjin 300071, P.R. China
c College of Information Science and Engineering, Ocean University of China,
Songling Road 238, Qingdao, Shandong 266100, P.R. China
d Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian,
Jiangsu 223001, P.R. China
e Faculty of Science, Beijing University of Technology, Beijing 100124, P.R.
China
fPhysik Department T31, James-Franck-Straße 1, Technische Universität München,
D-85748 Garching, Germany
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
The radiative leptonic $B\to\gamma\ell\nu_{\ell}$ decay serves as an ideal
platform to determine the $B$-meson inverse moment which is a fundamental
nonperturbative parameter for the $B$ meson. In this paper, we explore precise
QCD contributions to this decay with an energetic photon. We reproduce the
next-to-next-to-leading-logarithmic resummation formula for the decay
amplitude at leading power in $\Lambda_{\rm QCD}/m_{b}$. Employing operator
identities, we calculate subleading-power contributions from the expansion of
the hard-collinear propagator of the internal up quark and the heavy-quark
expansion of the bottom quark. We update the contributions from the hadronic
structure of the photon to the $B\to\gamma\ell\nu_{\ell}$ process with the
dispersion technique. Together with other yet known power corrections,
phenomenological applications including the partial branching fraction and
ratio of the branching fractions of the radiative $B$ decay are investigated.

###### Contents

  1. 1 Introduction
  2. 2 Theoretical overview of the $B\to\gamma\ell\nu_{\ell}$ decay
  3. 3 Form factors at LP
    1. 3.1 Resummation improved formula
      1. 3.1.1 NNLL resummation for $C$ and $\tilde{f}_{B}$
      2. 3.1.2 Partial NNLL resummation for $\phi_{+}$
  4. 4 Subleading-power contributions
    1. 4.1 The $\xi^{\rm fac}_{\rm ht}$ and $\xi^{\rm fac}_{\rm hqe}$ terms
    2. 4.2 Contribution from the hard-collinear propagator
    3. 4.3 Contribution from the resolved photon
  5. 5 Numerical analysis
    1. 5.1 Theoretical inputs
    2. 5.2 Predictions for the form factors
    3. 5.3 Phenomenological applications
  6. 6 Conclusion
  7. A $B$-meson LCDA
  8. B NNLL resummation
    1. B.1 Perturbative corrections up to NNLO
    2. B.2 Anomalous dimensions
    3. B.3 The evolution function $U(E_{h},\mu_{i},\mu)$
    4. B.4 Partial NNLL resummation for $\phi_{+}$
    5. B.5 Expansion of evolution factors
  9. C Photon LCDA

##  1 Introduction

The radiative leptonic $B\to\gamma\ell\nu_{\ell}$ decay is of great interest
for its offering a clean probe for determining $\lambda_{B}$, the inverse
moment of $B$-meson light-cone distribution amplitude (LCDA), which is a
crucial nonperturbative parameter for the $B$ meson in the heavy-quark
effective theory (HQET). Employing the Standard-Model prediction for the
radiative $B$ decay [1], the Belle collaboration has reported an upper limit
for the inverse moment $\lambda_{B}>0.24$ GeV at 90% CL [2]. To extract a more
practical value for the inverse moment, more accurate experimental data and
theoretical calculations are indispensable. With the ongoing super $B$ factory
Belle-II collaboration that will provide greatly improved data shortly, it is
urgent to re-investigate the radiative leptonic $B$ decay from the theoretical
point of view.

Factorization properties of the $B\to\gamma\ell\nu_{\ell}$ decay have been
explored at leading power (LP) in $\Lambda_{\rm QCD}/m_{b}$ with distinct QCD
techniques including the soft-collinear effective theory (SCET) [3, 4, 5, 6,
7], and the LP contribution to the decay with the short-distance hard-
scattering kernel up to the next-to-next-to-leading-order (NNLO) accuracy in
$\alpha_{s}$ was considered recently [8, 9]. Also next-to-leading-power (NLP)
corrections in $\Lambda_{\rm QCD}/m_{b}$ to the $B\to\gamma\ell\nu_{\ell}$
decay were discussed extensively [10, 11, 12, 13, 1, 14]. The NLP
contributions include the local contribution from subleading terms of the
hard-collinear quark propagator and the correction from photon emitted off the
$b$ quark both at leading order (LO) in $\alpha_{s}$ [10], the soft
contribution with the light-cone sum rules (LCSR) approach up to next-to-
leading order (NLO) in $\alpha_{s}$ [11, 12], the tree-level contribution with
higher-twist $B$-meson LCDAs [12, 1], the LO correction from subleading terms
of the heavy-quark expansion of the $b$ quark [1], and the resolved photon
contribution up to NLO [13].

Motivated by the recently derived LP factorization formula for the
$B\to\gamma\ell\nu_{\ell}$ decay and various NLP corrections to the decay, we
attempt to improve the theoretical predictions for the leptonic radiative $B$
decay. The main new ingredients of the present work are summarized as follows:

  * •

We calculate the “non-local” NLP correction from the hard-collinear quark
propagator at the tree level by using operator-level identities. The operator
identities are indispensable for non-local operators whose fields do not have
light-like separation. Further, employing operator-level identities, we re-
derive the LO contribution from subleading terms in the heavy-quark expansion
of the $b$ quark field.

  * •

We update the resolved photon corrections in the framework of LCSR with photon
LCDAs. Employing the proper momentum power-counting scheme, we calculate the
leading-twist photon-LCDA contribution to the NLO accuracy. Also, we calculate
the LO corrections from higher-twist photon LCDAs with the correct definition
of these LCDAs.

We find that the newly calculated resolved photon contribution and the
correction from the hard-collinear quark propagator are also numerically
important (see Section 5).

The rest of this paper is organized as follows: We first give a prefatory
overview of the $B\to\gamma\ell\nu_{\ell}$ process in Section 2. In the
following section, we collect the NNLL-resummation formula for the decay
amplitude at LP. Then we calculate various subleading-power corrections in
Section 4. Further in Section 5, numerical impacts of higher-order
contributions in $\alpha_{s}$ and higher-power contributions in $\Lambda_{\rm
QCD}/m_{b}$ are discussed. Finally, a summary of the main conclusion of this
work is given in Section 6.

##  2 Theoretical overview of the $B\to\gamma\ell\nu_{\ell}$ decay

The $B\to\gamma\ell\nu_{\ell}$ decay amplitude can be defined by the QCD
matrix element

$\displaystyle\mathcal{A}(B^{-}\to\gamma\ell\bar{\nu})=\frac{G_{F}\,V_{ub}}{\sqrt{2}}\langle\gamma\,\ell\,\bar{\nu}|\,[\bar{\ell}\gamma_{\mu}(1-\gamma_{5})\nu]\,[\bar{u}\gamma^{\mu}(1-\gamma_{5})b]\,|B^{-}\rangle\;.$
(1)

We will work in the $B$-meson rest frame in which the momentum of the $B$
meson is $m_{B}v$ with the four vector $v_{\mu}=(1,0,0,0)$. With the photon
having momentum $p$, the momentum transfer between the $B$ meson and the
photon $q=m_{B}v-p$ represents the sum of momenta of the final state leptons.
Since there exist light-like final state particles, it is convenient to employ
the light-cone coordinate by introducing two light-cone vectors $n_{\mu}$ and
$\bar{n}_{\mu}$ satisfying $n^{2}=\bar{n}^{2}=0$ and $n\cdot\bar{n}=2$. In
this coordinate we have $v_{\mu}=(n_{\mu}+\bar{n}_{\mu})/2$, and a given
vector could be decomposed as $k_{\mu}=n\cdot k\,\bar{n}_{\mu}/2+\bar{n}\cdot
k\,n_{\mu}/2+k_{\perp\mu}$. Setting the photon propagating along the $\bar{n}$
direction, the relevant momenta are expressed as

$\displaystyle p_{\mu}=$ $\displaystyle~{}\frac{n\cdot
p}{2}\,\bar{n}_{\mu}\equiv E_{\gamma}\bar{n}_{\mu}\,,$ $\displaystyle
q_{\mu}=$ $\displaystyle~{}\frac{n\cdot
q}{2}\,\bar{n}_{\mu}+\frac{\bar{n}\cdot q}{2}\,n_{\mu}\,,$ (2)

with $E_{\gamma}\leq m_{B}/2$ the energy of the photon.

Taking advantage of the Ward identity, the radiative $B$ decay is determined
by two form factors, $F_{V}$ and $\hat{F}_{A}$, defined through the vacuum-
to-$B$-meson correlation function

$\displaystyle\int d^{4}x\,e^{ip\cdot x}\,\langle 0|\,{\rm T}\\{\,j^{\nu}_{\rm
em}(x),[\bar{u}\gamma^{\mu}(1-\gamma_{5})b](0)\\}\,|B^{-}\rangle=v\cdot
p\,\big{(}\,g^{\mu\nu}\,\hat{F}_{A}-i\,\epsilon^{\mu\nu
nv}\,F_{V}\big{)}+\cdots\,,$ (3)

where the electroweak current $j^{\nu}_{\rm
em}=\sum_{q}Q_{q}\,\bar{q}\gamma^{\nu}q$ and $\epsilon^{0123}=1$. Note that
the form factors for the other Lorentz structures, which are not crucial for
the radiative leptonic $B$ decay, are represented by the ellipsis (see Ref.
[12] for details). The differential decay width of the
$B\to\gamma\ell\nu_{\ell}$ decay is

$\displaystyle\frac{d\,\Gamma}{d\,E_{\gamma}}=$
$\displaystyle~{}\frac{\alpha_{\rm
em}\,G_{F}^{2}\,|V_{ub}|^{2}}{6\,\pi^{2}}\,m_{B}\,E_{\gamma}^{3}\,\left(1-\frac{2\,E_{\gamma}}{m_{B}}\right)\,\Big{(}\big{|}F_{V}\big{|}^{2}+\big{|}F_{A}\big{|}^{2}\Big{)}\,,$
$\displaystyle F_{A}(E_{\gamma})=$
$\displaystyle~{}\hat{F}_{A}(E_{\gamma})+\frac{Q_{\ell}\,f_{B}}{E_{\gamma}}\,,$
(4)

where the $Q_{\ell}$ term in the second equation corresponds to the
contribution from the photon emitted off the charged lepton. We have neglected
masses of the final state leptons.

For an energetic photon with $E_{\gamma}\sim\mathcal{O}(m_{b})$, the form
factors are expressed as [10]

$\displaystyle F_{V}(E_{\gamma})=$
$\displaystyle~{}\frac{Q_{u}\,m_{B}}{2E_{\gamma}}\,R(E_{\gamma})+\Big{[}\xi(E_{\gamma})+\Delta\xi(E_{\gamma})\Big{]}\,,$
$\displaystyle\hat{F}_{A}(E_{\gamma})=$
$\displaystyle~{}\frac{Q_{u}\,m_{B}}{2E_{\gamma}}\,R(E_{\gamma})+\Big{[}\xi(E_{\gamma})-\Delta\xi(E_{\gamma})\Big{]}\,.$
(5)

The first terms on the right-hand side of the equations stand for the LP
contribution which preserves the large recoil symmetry relation. And the power
corrections are included in the terms in square brackets, where we have
separated the symmetry-breaking term $\Delta\xi(E_{\gamma})$ from the
symmetry-preserving term $\xi(E_{\gamma})$.

##  3 Form factors at LP

The form factors are determined by a single function $R(E_{\gamma})$ at LP in
the heavy-quark expansion. This function is factorized into convolution of
short-distance hard-scattering kernel $T$ and long-distance $B$-meson LCDA
$\phi_{+}$

$R(E_{\gamma})=\tilde{f}_{B}(\mu)\int_{0}^{\infty}\frac{d\omega}{\omega}\,T(E_{\gamma},\omega,\mu)\,\phi_{+}(\omega,\mu)\,,$
(6)

where the leading-twist LCDA $\phi_{+}$ is defined in Appendix A and $\omega$
corresponds to the $n$-direction momentum component of the light quark in the
$B$ meson.

Since the hard-scattering kernel $T$ contains contributions of two well
separated perturbative scales, which are the hard scale of
$\mathcal{O}(m_{b})$ and the hard-collinear scale of
$\mathcal{O}(\sqrt{m_{b}\Lambda_{\rm QCD}})$, it could be further factorized

$T(E_{\gamma},\omega,\mu)=C(E_{\gamma},\mu)\,J(E_{\gamma}\omega,\mu)\,.$ (7)

Here the hard function $C$ and the jet function $J$ receiving respectively
contributions from the hard-scale and the hard-collinear-scale dynamics can be
calculated by matching from QCD onto SCETI and from SCETI onto SCETII
individually [5, 6]. The static $B$-meson decay constant $\tilde{f}_{B}$,
which arises form the matrix element of HQET operator, is related to the QCD
$B$-meson decay constant $f_{B}$ by

$\displaystyle\tilde{f}_{B}(\mu)=K^{-1}(\mu)\,f_{B}\,.$ (8)

The factor $K$ is the short-distance coefficient for the matching of the
corresponding QCD weak current onto the HQET operator. The functions
$C(E_{\gamma},\mu)$, $J(E_{\gamma}\omega,\mu)$ and $K(\mu)$ can be calculated
perturbatively, and the coefficients up to NNLO accuracy are calculated in
[15, 16, 17, 18, 19, 20] and are collected in the Appendix B for convenience.

###  3.1 Resummation improved formula

To improve the convergence behavior of the perturbative expansion of
coefficient functions, large-logarithmic resummation must be performed for
these functions. Choosing the factorization scale to be of the order of the
hard-collinear scale $\sqrt{\Lambda_{\rm QCD}m_{b}}$, the jet function
$J(E_{\gamma}\omega,\mu)$ is free of large logarithms. Employing the standard
RG-equation approach, we fulfill the large-logarithmic resummation to
functions $C(E_{\gamma},\mu)$ and $\tilde{f}_{B}(\mu)$ to the NNLL accuracy
and the LCDA $\phi_{+}(\omega,\mu)$ to a partial NNLL accuracy.

It is not easy to perform the large-logarithmic resummation for the $B$-meson
LCDA in the momentum space even to the leading-logarithmic (LL) accuracy,
because the RG equation of the momentum-space $\phi_{+}(\omega,\mu)$ is an
integro-differential equation. To resum the large logarithms, we will perform
a Laplace transformation to the LCDA [21, 8]

$\displaystyle\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu)=\int_{0}^{\infty}\frac{d\omega}{\omega}\,\Big{(}\frac{\omega}{\widehat{\omega}}\Big{)}^{-\eta}\,\phi_{+}(\omega,\mu)\,.$
(9)

In the Laplace space, the convolution of the jet function with the LCDA is
relatively trivial

$\displaystyle\int_{0}^{\infty}\frac{d\omega}{\omega}\,J(E_{\gamma}\omega,\mu)\,\phi_{+}(\omega,\mu)={\mathcal{J}}(\partial_{\eta},\mu)\,\bigg{(}\frac{2E_{\gamma}\,\widehat{\omega}}{\mu^{2}}\bigg{)}^{\eta}\,\tilde{\phi}_{+}(-\eta,\widehat{\omega},\mu)\Big{|}_{\eta\to
0}\,,$ (10)

where ${\mathcal{J}}(L_{p},\mu)\equiv J(E_{\gamma}\omega,\mu)$ with
$L_{p}=\ln(2E_{\gamma}\,\omega/\mu^{2})$. Although the LCDA $\tilde{\phi}_{+}$
depends on the reference parameter $\widehat{\omega}$, the convolution is
independent of this parameter. We can in principle set $\widehat{\omega}$ to
be any value of $\mathcal{O}(\Lambda_{\rm QCD})$. The resummation improved
function $R(E_{\gamma})$ is expressed as

$R(E_{\gamma})=U_{1}(E_{\gamma},\mu_{h1},\mu)\,C(E_{\gamma},\mu_{h1})\,U_{2}(\mu_{h2},\mu)\,\tilde{f}_{B}(\mu_{h2})\Big{[}{\mathcal{J}}(\partial_{\eta},\mu)\,\tilde{\phi}_{+,U}(\eta,\mu_{s},\mu)\Big{]}\Big{|}_{\eta=0}\,.$
(11)

The scales $\mu_{h1}$ and $\mu_{h2}$ are both hard scales of
$\mathcal{O}(m_{b})$ and $\mu_{s}$ is a soft scale of
$\mathcal{O}(\Lambda_{\rm QCD})$. These scales ensure that functions
$C(E_{\gamma},\mu_{h1})$, $\tilde{f}_{B}(\mu_{h2})$ and
$\phi_{+}(\omega,\mu_{s})$ are free of large logarithms. The factors $U_{1}$
(see (14a)) and $U_{2}$ (see (14b)) are separately the evolution factors of
$C(E_{\gamma},\mu)$ and $\tilde{f}_{B}(\mu)$. The function
$\tilde{\phi}_{+,U}$ (see (20)) corresponds to the partial NNLL resummed
$B$-meson LCDA in the Laplace space, which will be defined later.

####  3.1.1 NNLL resummation for $C$ and $\tilde{f}_{B}$

The evolution factors $U_{1}(E_{\gamma},\mu_{h1},\mu)$ and
$U_{2}(\mu_{h2},\mu)$ satisfy the same RG equations as $C(E_{\gamma},\mu)$ and
$\tilde{f}_{B}(\mu)$, respectively,

$\displaystyle\frac{d}{d\ln\mu}\,U_{1}(E_{\gamma},\mu_{h1},\mu)=$
$\displaystyle~{}\Big{[}\Gamma_{\rm
cusp}(\alpha_{s})\,\ln\frac{2\,E_{\gamma}}{\mu}+\gamma_{H}(\alpha_{s})\Big{]}\,U_{1}(E_{\gamma},\mu_{h1},\mu)\,,$
$\displaystyle\frac{d}{d\ln\mu}\,U_{2}(\mu_{h2},\mu)=$
$\displaystyle~{}\gamma_{hl}(\alpha_{s})\,U_{2}(\mu_{h2},\mu)\,,$ (12)

with initial conditions $U_{1}(E_{\gamma},\mu_{h1},\mu_{h1})=1$ and
$U_{2}(\mu_{h2},\mu_{h2})=1$. The anomalous dimensions $\Gamma_{\rm cusp}$,
$\gamma_{H}$ and $\gamma_{hl}$ are collected in Appendix B.

For later convenience, we first investigate the RG equation for a general
function $U$, which depends on anomalous dimensions $\Gamma_{\rm cusp}$ and
$\gamma_{\rm gen}$,

$\displaystyle\frac{d}{d\ln\mu}\,U(E_{h},\mu_{i},\mu)=$
$\displaystyle~{}\Big{[}\Gamma_{\rm
cusp}(\alpha_{s})\,\ln\frac{E_{h}}{\mu}+\gamma_{\rm
gen}(\alpha_{s})\Big{]}\,U(E_{h},\mu_{i},\mu)\,,$ (13)

with initial condition $U(E_{h},\mu_{i},\mu_{i})=1$. The solution to the
general RG equation is collected in Appendix B. The scale $\mu$ is the
factorization scale and scale $\mu_{i}$ is an initial scale which could be a
hard scale of $\mathcal{O}(m_{b})$ or a soft scale of
$\mathcal{O}(\Lambda_{\rm QCD})$ or another scale. Taking certain $E_{h}$ and
anomalous dimensions, this equation can reproduce the RG equation satisfied by
the evolution factor $U_{1}$ or $U_{2}$. Then the two evolution factors can be
obtained specifically

$\displaystyle U_{1}(E_{\gamma},\mu_{h1},\mu)=$
$\displaystyle~{}U(2E_{\gamma},\mu_{h1},\mu)\Big{|}_{\gamma_{\rm
gen}\to\gamma_{H}}\,,$ (14a) $\displaystyle U_{2}(\mu_{h2},\mu)=$
$\displaystyle~{}U(\mu_{h2},\mu_{h2},\mu)\Big{|}_{\Gamma_{\rm cusp}\to
0,\,\gamma_{\rm gen}\to\gamma_{hl}}\,.$ (14b)

Note that $U_{2}$ is independent of the cusp anomalous dimension. Then $U_{2}$
can be derived from $U$ by setting $\Gamma_{\rm cusp}$ to 0 and choosing any
nonzero value for $E_{h}$, which is chosen to be $\mu_{h2}$ here.

####  3.1.2 Partial NNLL resummation for $\phi_{+}$

In the Laplace space, it is relatively straightforward to solve the RG
equation of the LCDA, which is [8]

$\displaystyle~{}\left(\frac{d}{d\ln\mu}+\Gamma_{\rm
cusp}(\alpha_{s})\frac{\partial}{\partial\eta}\right)\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu)$
$\displaystyle=$ $\displaystyle~{}\left\\{\Gamma_{\rm
cusp}(\alpha_{s})\left[\ln\frac{\widehat{\omega}}{\mu}+\tilde{h}(-\eta)-\psi(1+\eta)-\psi(1-\eta)-2\gamma_{E}\right]-\gamma(\alpha_{s})\right\\}\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu)\,,$
(15)

where $\psi$ is the digamma function. The two-loop anomalous dimension
$\gamma$ and the two-loop level $\tilde{h}$ were calculated in Ref. [22], see
the Appendix B for details. The explicit solution to the RG equation (15) at
$\alpha_{s}$ accuracy can be found in Ref. [8]. We would like to derive the
formal $\alpha^{2}_{s}$-level solution, which could be employed to fulfill the
NNLL resummation to the LCDA once the three-loop anomalous dimensions of
$\gamma$ and $\tilde{h}$ are known. Since any combination of
$\eta+a_{\Gamma}(\mu_{s})$ is a solution to (15) with the right-hand-side
terms setting to zero, we separate the RG-evolution factor into
$\eta$-independent $U_{\phi 1}$ and $\eta$-dependent $U_{\phi 2}$ parts, where

$\displaystyle
a_{\Gamma}(\mu_{s})=-\int_{\alpha_{s}(\mu_{s})}^{\alpha_{s}(\mu)}d\alpha\,\frac{\Gamma_{\rm
cusp}(\alpha)}{\beta(\alpha)}$ (16)

with $\beta$ the QCD beta function. And the combination $U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)\,U_{\phi
2}(\eta,\mu_{s},\mu)\,\tilde{\phi}_{+}(\eta+a_{\Gamma}(\mu_{s}),\widehat{\omega},\mu_{s})$
will be the solution to the RG equation satisfied by $\tilde{\phi}_{+}$.

The factor $U_{\phi 1}$ satisfies the RG equation

$\displaystyle\frac{d}{d\ln\mu}\,U_{\phi 1}(\widehat{\omega},\mu_{s},\mu)=$
$\displaystyle~{}\left\\{\Gamma_{\rm
cusp}(\alpha_{s})\,\ln\frac{\widehat{\omega}}{\mu}-\Big{[}2\gamma_{E}\,\Gamma_{\rm
cusp}(\alpha_{s})+\gamma(\alpha_{s})\Big{]}\right\\}U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)\,,$ (17)

with initial condition $U_{\phi 1}(\widehat{\omega},\mu_{s},\mu_{s})=1$. The
equation is closely related to the general RG equation (13), and we have

$\displaystyle U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)=U(\widehat{\omega},\mu_{s},\mu)\Big{|}_{\gamma_{\rm
gen}\to-(\gamma+2\gamma_{E}\Gamma_{\rm cusp})}\,.$ (18)

The factor $U_{\phi 2}$ is expressed as

$\displaystyle U_{\phi 2}(\eta,\mu_{s},\mu)=$
$\displaystyle~{}\exp\Big{[}\int\limits_{\alpha_{s}(\mu_{s})}^{\alpha_{s}(\mu)}\frac{d\alpha}{\beta(\alpha)}\,\Gamma_{\rm
cusp}(\alpha)\,\tilde{h}(\alpha,-\eta-
a_{\Gamma}(\mu_{\alpha}))\Big{]}\,\frac{\Gamma\big{(}1+\eta+a_{\Gamma}(\mu_{s})\big{)}\,\Gamma(1-\eta)}{\Gamma\big{(}1-\eta-
a_{\Gamma}(\mu_{s})\big{)}\,\Gamma(1+\eta)}\,,$ (19)

with $\mu_{\alpha}$ satisfing $\alpha_{s}(\mu_{\alpha})\equiv\alpha$.

Having the evolution factors $U_{\phi 1}$ and $U_{\phi 2}$ at hand, the large
logarithms appearing in the LCDA $\tilde{\phi}_{+}$ can be resummed directly.
The evolution function $\tilde{\phi}_{+,U}$ is then

$\displaystyle\tilde{\phi}_{+,U}(\eta,\mu_{s},\mu)=$
$\displaystyle~{}\bigg{(}\frac{2E_{\gamma}\,\widehat{\omega}}{\mu^{2}}\bigg{)}^{\eta}\,U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)\,U_{\phi
2}(-\eta,\mu_{s},\mu)\,\tilde{\phi}_{+}(-\eta+a_{\Gamma}(\mu_{s}),\widehat{\omega},\mu_{s})\,.$
(20)

We add the factor $(2E_{\gamma}\,\widehat{\omega}/\mu^{2})^{\eta}$ into the
definition of $\tilde{\phi}_{+,U}$ to ensure that it is independent of the
reference parameter $\widehat{\omega}$. To expand the evolution function
$\tilde{\phi}_{+,U}$ in the strong coupling constant, the functions depending
on $a_{\Gamma}(\mu_{s})$ should be expanded in $\alpha_{s}(\mu_{s})/(4\pi)$.
These expansion coefficients are collected in Appendix B. The NNLL resummation
of the LCDA could be performed if the three-loop anomalous dimensions of
$\gamma$ and $\tilde{h}$ are available. While in this paper, we will set the
three-loop anomalous dimensions of $\gamma$ and $\tilde{h}$ to zero.

Plugging the evolution factors $U_{1}$ ((14a)), $U_{2}$ ((14b)) and
$\tilde{\phi}_{+,U}$ ((20)) into (11), we obtain the partial NNLL-resummation
improved LP function $R(E_{\gamma})$. The complete NNLL-resummation factors
$U_{1}(E_{\gamma},\mu_{h1},\mu)$ and $U_{2}(\mu_{h2},\mu)$ correspond to the
evolution of the hard function from a hard scale $\mu_{h1}$ to the
factorization scale $\mu$ and the evolution of $\tilde{f}_{B}$ from a hard
scale $\mu_{h2}$ to $\mu$, respectively. The factor
$\tilde{\phi}_{+,U}(\eta,\mu_{s},\mu)$ represents the partial NNLL evolution
of the $B$-meson LCDA from a soft scale $\mu_{s}$ to $\mu$.

##  4 Subleading-power contributions

In this section, we will proceed to show the NLP corrections originated from
various sources

$\displaystyle\xi(E_{\gamma})$ $\displaystyle=$ $\displaystyle\xi^{\rm
fac}(E_{\gamma})+\xi^{\rm hp}(E_{\gamma})\,,$
$\displaystyle\Delta\xi(E_{\gamma})$ $\displaystyle=$
$\displaystyle\Delta\xi^{\rm fac}(E_{\gamma})+\Delta\xi^{\rm
hp}(E_{\gamma})\,.$ (21)

The superscript “hp” stands for non-factorizable contributions from the
hadronic structure of the photon, which suffer from end-point singularities.
The LO factorizable contributions calculated in this work, represented by the
superscript “fac”, can be expressed as

$\displaystyle\xi^{\rm fac}(E_{\gamma})$ $\displaystyle=$
$\displaystyle\xi^{\rm fac}_{\rm ht}(E_{\gamma})+\xi^{\rm fac}_{\rm
hqe}(E_{\gamma})+\xi^{\rm fac}_{\rm hc}(E_{\gamma})\,,$
$\displaystyle\Delta\xi^{\rm fac}(E_{\gamma})$ $\displaystyle=$
$\displaystyle\Delta\xi^{\rm fac}_{\rm b}(E_{\gamma})+\Delta\xi^{\rm fac}_{\rm
hc}(E_{\gamma})\,.$ (22)

The subscripts “ht”, “hqe”, “b”, and “hc” correspond to contributions from
higher-twist $B$-meson LCDAs up to the twist six, subleading terms in the
heavy-quark expansion of the $b$-quark field, the photon emitted off the $b$
quark, and subleading terms in the hard-collinear propagator, respectively.
The contribution arising from the photon emitted off the $b$ quark breaks the
large-recoil symmetry relation between the form factors. The heavy-quark-
expansion and the higher-twist-LCDA mechanisms only generate symmetry-
preserving contributions at LO. The factorizable symmetry-breaking terms are
well-known [10]

$\displaystyle\Delta\xi^{\rm fac}_{\rm
b}(E_{\gamma})=\frac{Q_{b}\,\tilde{f}_{B}\,m_{B}}{2\,m_{b}\,E_{\gamma}}\,,$
$\displaystyle\Delta\xi^{\rm fac}_{\rm
hc}(E_{\gamma})=\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{(2\,E_{\gamma})^{2}}\,.$
(23)

Below we calculate the other contributions $\xi^{\rm fac}_{\rm ht}$ ((25)),
$\xi^{\rm fac}_{\rm hqe}$ ((29)), $\xi^{\rm fac}_{\rm hc}$ ((32)), $\xi^{\rm
hp}$ ((4.3)), and $\Delta\xi^{\rm hp}$ ((4.3)).

###  4.1 The $\xi^{\rm fac}_{\rm ht}$ and $\xi^{\rm fac}_{\rm hqe}$ terms

The higher-twist LCDA contribution can be calculated with the help of the
light-cone expansion of the quark propagator in the background gluon field
[23]

$\displaystyle\langle 0|{\rm T}\,\\{\bar{q}(x),q(0)\\}|0\rangle\supset
ig_{s}\int{d^{4}\ell\over(2\pi)^{4}}{e^{-i\,\ell\cdot
x}\over\ell^{2}+i0}\,\int_{0}^{1}du\left[u\,x_{\mu}\gamma_{\nu}-\frac{\not{\ell}\,\,\sigma_{\mu\nu}}{2\left(\ell^{2}+i0\right)}\right]G^{\mu\nu}(u\,x)\,,$
(24)

where $G_{\mu\nu}=i/g_{s}[D_{\mu},D_{\nu}]$ with
$D_{\mu}=\partial_{\mu}-ig_{s}A_{\mu}$. The corresponding contribution is [1]

$\displaystyle\xi^{\rm fac}_{\rm ht}(E_{\gamma})$ $\displaystyle=$
$\displaystyle\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{(2E_{\gamma})^{2}}\Big{[}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\xi\int_{0}^{1}du\,\frac{\psi_{4}(\omega,\xi)-\tilde{\psi}_{4}(\omega,\xi)}{(\omega+u\,\xi)^{2}}$
(25)
$\displaystyle-4\int_{0}^{\infty}d\omega\,\frac{\hat{g}_{+}(\omega)}{\omega^{2}}\Big{]}\,.$

The $B$-meson LCDAs for $\psi_{4}(\omega,\xi),\tilde{\psi}_{4}(\omega,\xi)$
and $\hat{g}_{+}(\omega)$ are defined in Appendix A.

Taking into account subleading terms in the matching of the $b$ quark from QCD
onto HQET, the heavy-to-light current at the tree level is

$\displaystyle\bar{q}\,\gamma_{\mu}(1-\gamma_{5})\,b=\bar{q}\,\gamma_{\mu}\,(1-\gamma_{5})\,h_{v}+\frac{1}{2m_{b}}\,\bar{q}\,\gamma_{\mu}\,(1-\gamma_{5})\,i\not\\!\\!D_{\rm
T}\,h_{v}+\mathcal{O}(\frac{1}{m_{b}^{2}})\,,$ (26)

where $D_{\rm T,\mu}=D_{\mu}-v\cdot D\,v_{\mu}$. Calculating the NLP
contribution from the second term on the right-hand side of the equation, we
meet with the matrix element of the operator
$\bar{q}(x)\,\Gamma\,D_{\rho}\,h_{v}(0)$, which can be handled by using the
operator relation

$\displaystyle\bar{q}(x)\,[x,0]\,\Gamma\,D_{\rho}\,h_{v}(0)=$
$\displaystyle~{}\partial_{\rho}\,\Big{[}\bar{q}(x)\,[x,0]\,\Gamma\,h_{v}(0)\Big{]}-\Big{[}\frac{\partial}{\partial
x^{\rho}}\bar{q}(x)\Big{]}\,[x,0]\,\Gamma\,h_{v}(0)$
$\displaystyle+i\int^{1}_{0}du\,\bar{u}\,\bar{q}(x)\,[x,ux]\,x^{\lambda}\,G_{\lambda\rho}(ux)\,[ux,0]\,\Gamma\,h_{v}(0)\,,$
(27)

with the Wilson line $[z_{2},z_{1}]={\rm
P}\,\exp\Big{[}ig_{s}\int^{z_{2}}_{z_{1}}dz^{\mu}A_{\mu}(z)\Big{]}$ ensuring
the gauge invariance of the operators. The first term and the third term on
the right-hand side, which correspond to two-particle and three-particle LCDA
contributions, respectively, can be computed straightforwardly. While the
second term on the right-hand side of the equation provides us with
troublesome operators $[\not\\!\partial\bar{q}(x)]\,\Gamma\,h_{v}(0)$ and
$[n\cdot\partial\,\bar{q}(x)]\,\Gamma\,h_{v}(0)$, whose matrix elements are
not well defined since the quark fields are not separated by a light-like
distance. These operators can be translated to desirable light-ray operators
by applying operator identities

$\displaystyle\big{[}\frac{\partial}{\partial
x^{\rho}}\bar{q}(x)\big{]}\,\gamma^{\rho}\,[x,0]\,\Gamma\,h_{v}(0)=$
$\displaystyle-i\int^{1}_{0}du\,u\,\bar{q}(x)\,[x,ux]\,x^{\lambda}\,G_{\lambda\rho}(ux)\,\gamma^{\rho}\,[ux,0]\,\Gamma\,h_{v}(0)\,,$
(28a) $\displaystyle\big{[}\frac{\partial}{\partial
x^{\rho}}\bar{q}(x)\big{]}\,v^{\rho}\,[x,0]\,\Gamma\,h_{v}(0)=$
$\displaystyle~{}v\cdot\partial\,\big{[}\bar{q}(x)\,[x,0]\,\Gamma\,h_{v}(0)\big{]}$
$\displaystyle+i\int^{1}_{0}du\,\bar{u}\,\bar{q}(x)\,[x,ux]\,x^{\lambda}\,G_{\lambda\rho}(ux)\,v^{\rho}\,[ux,0]\,\Gamma\,h_{v}(0)\,,$
(28b)

which are derived from equation of motions $\not\\!\\!D\,q(x)=0$ and $v\cdot
D\,h_{v}=0$. Then the rest of the calculations are relatively easy, and
interested readers could refer to Ref. [24] for details. Finally, one obtains
for the $b$-quark expansion contribution [1]

$\displaystyle\xi^{\rm fac}_{\rm hqe}(E_{\gamma})$ $\displaystyle=$
$\displaystyle~{}\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{4\,m_{b}\,E_{\gamma}}\,\bigg{\\{}\frac{\bar{\Lambda}}{\lambda_{B}}+\int_{0}^{\infty}d\omega\,\Big{[}\int_{0}^{\omega}d\eta\,\frac{\phi_{+}(\eta)-\phi_{-}(\eta)}{\omega}-\phi_{+}(\omega)\Big{]}$
(29) $\displaystyle+$ $\displaystyle
2\int_{0}^{\infty}d\omega\int_{0}^{\infty}\frac{d\xi}{\xi}\,\Big{[}\frac{1}{\omega}+\frac{1}{\xi}\,\ln\frac{\omega}{\omega+\xi}\Big{]}\,\phi_{3}(\omega,\xi)\bigg{\\}}\,,$

where $\bar{\Lambda}=m_{B}-m_{b}$ is the “effective mass” of the $B$ meson in
the HQET and the inverse moment $\lambda_{B}$ will be introduced later in
Section 5.

The contributions $\xi^{\rm fac}_{\rm ht}$ and $\xi^{\rm fac}_{\rm hqe}$ can
also be extracted from the $B\to\gamma\gamma$ decay [24, 25]. The $P_{7}$
operator in the effective Hamiltonian, which contributes to the
$B\to\gamma\gamma$ decay at LP, is

$\displaystyle P_{7}=\Big{(}-\frac{e}{16\pi^{2}}\,E_{\gamma}\,m_{b}\,A^{\rm
em}_{\mu}\Big{)}\,[\bar{q}\,\gamma_{\perp}^{\mu}\,(1-\gamma_{5})\not\\!n\,b]\,,$
(30)

with $A^{\rm em}$ the photon field. If the operator
$[\bar{q}\,\gamma_{\perp}^{\mu}\,(1-\gamma_{5})\not\\!n\,h_{v}]$ can be
simplified to $2[\bar{q}\,\gamma_{\perp}^{\mu}\,(1-\gamma_{5})\,h_{v}]$ which
is the heavy-to-light current in the $B\to\gamma\ell\nu_{\ell}$ decay, we can
just get the corresponding contribution to the $B\to\gamma\ell\nu_{\ell}$
decay from the $B\to\gamma\gamma$ result. This is actually the case for
$\xi^{\rm fac}_{\rm ht}$ and $\xi^{\rm fac}_{\rm hqe}$ contributions at the
tree level, because the hard-collinear propagator provides us with the
$(\cdots)\not\\!\bar{n}$ structure which turns $\not\\!n\,h_{v}$ to
$2\not\\!v\,h_{v}=2h_{v}$.

###  4.2 Contribution from the hard-collinear propagator

Subleading terms in the hard-collinear propagator will contribute to the decay
form factors at NLP [26, 27, 28, 29, 30]. Employing $k^{2}=0$ with $k$ the
momentum of the light quark in the $B$ meson, the hard-collinear propagator
reads

$\displaystyle\frac{i\,(p\hskip-5.29999pt/-k\hskip-5.29999pt/)}{(p-k)^{2}}=-\frac{i}{\bar{n}\cdot
k}\,\frac{\not\\!\bar{n}}{2}+\Big{[}\frac{i\,n\cdot
k}{2\,E_{\gamma}\,\bar{n}\cdot
k}\,\frac{\not\\!\bar{n}}{2}+\frac{i\not\\!k_{\perp}}{2\,E_{\gamma}\,\bar{n}\cdot
k}+\frac{i}{2\,E_{\gamma}}\,\frac{\not\\!n}{2}\Big{]}\,.$ (31)

The first term on the right-hand side is the LP term. The terms in the square
brackets generate NLP contributions with the last term contributing to
$\Delta\xi^{\rm fac}_{\rm hc}$. The first two terms in the square brackets
provide us with a non-local symmetry-preserving contribution at LO

$\displaystyle\xi^{\rm fac}_{\rm hc}(E_{\gamma})$ $\displaystyle=$
$\displaystyle~{}\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{2\,E_{\gamma}^{2}}\,\Big{[}\int_{0}^{\infty}d\omega\int_{0}^{\infty}\frac{d\xi}{\xi}\,\Big{(}\frac{1}{\xi}\,\ln\frac{\omega}{\omega+\xi}+\frac{1}{\omega}\Big{)}\,\psi_{4}(\omega,\xi)-\frac{\bar{\Lambda}}{\lambda_{B}}+\frac{1}{2}\Big{]}\,.$
(32)

As in the calculation of the contribution $\xi^{\rm fac}_{\rm hqe}$, we have
used the operator identities in (28b) to compute $\xi^{\rm fac}_{\rm hc}$.

The symmetry-preserving contribution $\xi^{\rm fac}_{\rm hc}$ can also be
obtained from results in the $B\to\gamma\gamma$ decay [24]. Taking a closer
look at terms in (31), we find that the $\not\\!k_{\perp}$ term does not
contribute to the $B\to\gamma\ell\nu_{\ell}$ and the $B\to\gamma\gamma$ decays
at LO. For the contribution from the $n\cdot k$ term, the operator $P_{7}$ can
be simplified to the heavy-to-light current
$[\bar{q}\,\gamma_{\perp}^{\mu}\,(1-\gamma_{5})\,b]$. Then we conclude that
the non-local symmetry-preserving contributions in the two processes are the
same up to a trivial pre-factor.

###  4.3 Contribution from the resolved photon

The subleading SCETII operator ${\cal
O}\supset[\bar{q}_{s}\,h_{v}]_{s}\,\,[\bar{\xi}\,\xi]_{c}$, which contains
collinear fields, provides power-suppressed contribution to the
$B\to\gamma\ell\nu_{\ell}$ decay [7]. The collinear part of its matrix element
$\langle\gamma(p)|[\bar{\xi}\,\xi]_{c}|0\rangle$, which defines the photon
LCDA, represents the contribution from the hadronic structure of the photon.
While end-point divergences will appear when trying to derive the
factorization formula for this contribution. In this paper we employ the LCSR
with photon LCDA to calculate the resolved-photon contribution.

The photon-to-vacuum correlation function with an interpolating current (with
momentum $p+q$) for the $B$ meson is constructed as

$\displaystyle\Pi^{\rm hp}_{\mu}(p,q)=\int d^{4}x\,e^{iq\cdot
x}\,\langle\,\gamma(p)\,|\,T\\{\bar{u}(x)\,\gamma_{\mu\perp}\,(1-\gamma_{5})\,b(x),\;\bar{b}(0)\,\gamma_{5}\,u(0)\\}\,|0\,\rangle\,,$
(33)

with $q^{2}<m^{2}_{b}$. We first calculate the correlation function in the
region $(p+q)^{2}<m^{2}_{b}$ and then apply the dispersion relation to
analytically continue the result to the physical region. We also employ the
power-counting schemes $(p+q)^{2}-m^{2}_{b}\sim\mathcal{O}(m^{2}_{b})$ to
avoid the end-point divergence and $q^{2}-m^{2}_{b}\sim\mathcal{O}(m^{2}_{b})$
inherited from $E_{\gamma}\sim\mathcal{O}(m_{b})$.

First, we consider the leading twist-2 photon LCDA contribution to the form
factors. The hard-collinear factorization formula for the correlation function
is

$\displaystyle\Pi^{\rm hp}_{\mu,\rm LT}(p,q)=-Q_{u}\,g_{\rm em}\,n\cdot
p\,\chi(\nu)\,\langle\bar{q}q\rangle(\nu)\,\varepsilon^{*\alpha}\,(g_{\mu\alpha}^{\perp}-i\epsilon_{\mu\alpha
nv})\int_{0}^{1}du\,\phi_{\gamma}(u,\nu)\,H_{\gamma}(p,q,u,\nu)\,,$ (34)

with $\chi$ the magnetic susceptibility of the quark condensate,
$\langle\bar{q}q\rangle$ the quark condensate, $\varepsilon^{\alpha}$ the
polarization vector of the photon, and $u$ the momentum fraction carried by
the quark in the photon. Here the scale $\nu$ stands for the factorization
scale for the photon-to-vacuum correlation function. The long-distance photon
LCDA $\phi_{\gamma}$ is defined in Appendix C. The hard matching coefficient
$H_{\gamma}$ is recalculated up to NLO with NDR scheme of $\gamma_{5}$ [31]

$\displaystyle H_{\gamma}^{(0)}(p,q,u,\nu)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\,\frac{\bar{n}\cdot
q}{u(p+q)^{2}+\bar{u}\,q^{2}-m_{b}^{2}+i0}$ (35) $\displaystyle
H_{\gamma}^{(1)}(p,q,u,\nu)$ $\displaystyle=$ $\displaystyle
H_{\gamma}^{(0)}(p,q,u,\nu)\,\bigg{\\{}-2\,\bigg{[}{\overline{r}_{2}\over
r_{1}-r_{2}}\,\ln{\overline{r}_{1}\over\overline{r}_{2}}+{\overline{r}_{3}\over
r_{1}-r_{3}}\,\ln{\overline{r}_{1}\over\overline{r}_{3}}+{3\over\overline{r}_{1}}\bigg{]}\,\ln{\nu^{2}\over
m_{b}^{2}}$ (36)
$\displaystyle+2\,\bigg{[}\Big{(}\frac{\overline{r}_{2}}{r_{1}-r_{2}}+\frac{\overline{r}_{3}}{r_{1}-r_{3}}\Big{)}\,\Big{(}{\rm
Li}_{2}(r_{1})+\ln^{2}\overline{r}_{1}\Big{)}-\frac{\overline{r}_{2}}{r_{1}-r_{2}}\,\Big{(}{\rm
Li}_{2}(r_{2})+\ln^{2}\overline{r}_{2}\Big{)}$
$\displaystyle-\frac{\overline{r}_{3}}{r_{1}-r_{3}}\,\Big{(}{\rm
Li}_{2}(r_{3})+\ln^{2}\overline{r}_{3}\Big{)}\bigg{]}+\bigg{[}\frac{\overline{r}_{1}}{r_{1}}\Big{(}\frac{1-3r_{1}}{r_{1}-r_{2}}-\frac{2}{r_{1}-r_{3}}\Big{)}\,+\frac{\overline{r}^{2}_{1}}{r_{1}^{2}}-4\bigg{]}\,\ln\overline{r}_{1}$
$\displaystyle-\frac{\overline{r}_{2}\,(1-3\,r_{2})}{r_{2}\,(r_{1}-r_{2})}\,\ln\overline{r}_{2}+\frac{2\,\overline{r}_{3}}{r_{3}\,(r_{1}-r_{3})}\,\ln\overline{r}_{3}+\frac{1}{r_{1}}-\frac{8}{\overline{r}_{1}}-3\bigg{\\}}\,,$

where $r_{1}=(u\,p+q)^{2}/m_{b}^{2}$, $r_{2}=q^{2}/m_{b}^{2}$,
$r_{3}=(p+q)^{2}/m_{b}^{2}$, and $\overline{r}_{i}\equiv 1-r_{i}$. Choosing
the factorization scale $\nu$ of $\mathcal{O}(m_{b})$, the combined factor
$[\chi(\nu)\langle\bar{q}q\rangle(\nu)\phi_{\gamma}(u,\nu)]$ need to be
evolved from a non-perturbative scale $\nu_{0}$ to $\nu$ by applying the RG-
equation approach, see Ref. [13] for details.

Employing the standard LCSR approach, which includes applications of the
dispersion relation, the quark-hadron duality, and the Borel transformation,
the NLL-resummation improved resolved-photon contribution with twist-2 photon
LCDA is

$\displaystyle\xi^{\rm hp}_{\rm LT}(E_{\gamma})$ $\displaystyle=$
$\displaystyle
Q_{u}\,\frac{m_{b}+m_{u}}{2\,f_{B}}\,\chi(\nu)\,\langle\bar{q}q\rangle(\nu)\int_{0}^{s_{0}}\,ds\,{\rm
exp}\left[-{s-m_{B}^{2}\over M^{2}}\right]\,\rho(s,q^{2},\nu)\,.$ (37)

Here $M^{2}$ is the Borel mass and $s_{0}$ represents the integral threshold.
In the sum rules, $\xi^{\rm hp}_{\rm LT}$ is independent of $(p+q)^{2}$ but
still depends on $q^{2}$, thus the sum rules is valid only in the region
$q^{2}<m^{2}_{b}$ (or $E_{\gamma}>(m^{2}_{B}-m^{2}_{b})/(2m_{B})$). The newly
derived spectral density is written as

$\displaystyle\rho(s,q^{2},\nu)=\rho^{(0)}(s,q^{2},\nu)+\frac{\alpha_{s}(\nu)\,C_{F}}{4\pi}\,\rho^{(1)}(s,q^{2},\nu)\,,$
(38)

where

$\displaystyle m_{b}^{2}\,\rho^{(0)}(s,q^{2},\nu)$ $\displaystyle=$
$\displaystyle-\frac{1}{w_{sr}}\,\phi_{\gamma}\Big{(}\frac{\overline{r}_{2}}{w_{sr}},\nu\Big{)}\,\theta(\hat{s})\,,$
(39) $\displaystyle m_{b}^{2}\,\rho^{(1)}(s,q^{2},\nu)$ $\displaystyle=$
$\displaystyle-{2\over w_{sr}}\,\ln{\nu^{2}\over
m_{b}^{2}}\,\bigg{\\{}\,\int_{0}^{1}\,du\,\Big{[}{1-2u\over
u\bar{u}}\,\theta(u_{sr})+{1\over\bar{u}}\,\theta(\hat{s})\,\Big{]}\,\phi_{\gamma}(u,\nu)$
(40)
$\displaystyle+\int_{0}^{1}\,du\,\bigg{[}\ln{u^{2}_{sr}\over\hat{s}\,\overline{r}_{2}}\,\theta(u_{sr})-\,\ln\bigg{|}{u_{sr}\over\hat{s}}\bigg{|}\,\theta(\hat{s})\bigg{]}\,\phi_{\gamma}^{\prime}(u,\nu)+\,{3\over
w_{sr}}\,\theta(\hat{s})\,\phi_{\gamma}^{\prime}\Big{(}{\overline{r}_{2}\over
w_{sr}},\nu\Big{)}\,\bigg{\\}}$
$\displaystyle+\int_{0}^{1}du\,\phi_{\gamma}(u,\nu)\,\bigg{\\{}\theta(u_{sr})\bigg{[}\frac{2\,(1-2u)}{u\,\bar{u}\,w_{sr}}\,\ln\frac{u^{2}_{sr}}{u_{sr}+1}+\frac{6\,\ln
u_{sr}}{(u_{sr}+1)^{2}}+\frac{4}{u_{sr}}\,\ln(u_{sr}+1)$
$\displaystyle+\Big{(}\frac{2\,(2u-1)}{u\,\bar{u}\,w_{sr}}+\frac{3\,\overline{r}_{2}}{u\,w_{sr}}+1\Big{)}\,\frac{1}{u_{sr}+1}\bigg{]}-2\,\theta(\hat{s})\Big{(}\frac{1}{\hat{s}+1}+\ln\frac{\hat{s}+1}{\hat{s}^{2}}\Big{)}\Big{(}{\cal
P}\frac{1}{u_{sr}}+\frac{1}{\bar{u}\,w_{sr}}\Big{)}\bigg{\\}}$
$\displaystyle+{1\over
w_{sr}}\int_{0}^{1}\,du\,\phi_{\gamma}^{\prime}(u,\nu)\,\theta(u_{sr})\,\bigg{[}4\,\ln^{2}u_{sr}+\ln\frac{u^{2}_{sr}}{u_{sr}+1}+\frac{1-6\ln
u_{sr}}{u_{sr}+1}-{4\over 3}\,\pi^{2}\bigg{]}$
$\displaystyle+\frac{2}{w_{sr}}\,\bigg{[}{\rm Li}_{2}(1-\overline{r}_{2})-{\rm
Li}_{2}(-\hat{s})-\ln(\hat{s}+1)\,\ln\hat{s}+\ln^{2}\overline{r}_{2}+\ln^{2}\hat{s}+\frac{3\,\overline{r}_{2}-2}{2(1-\overline{r}_{2})}\,\ln\overline{r}_{2}$
$\displaystyle-\frac{\ln\hat{s}}{\hat{s}+1}+\frac{3}{2}-\frac{7}{6}\pi^{2}\bigg{]}\,\theta(\hat{s})\,\phi_{\gamma}\Big{(}\frac{\overline{r}_{2}}{w_{sr}},\nu\Big{)}-8\,\frac{1}{w_{sr}^{2}}\,\theta(\hat{s})\,\phi^{\prime}_{\gamma}\Big{(}\frac{\overline{r}_{2}}{w_{sr}},\nu\Big{)}\,,$

with $\hat{s}=s/m^{2}_{b}-1$, $w_{sr}=\hat{s}+\overline{r}_{2}$, and
$u_{sr}=u\,\hat{s}-\bar{u}\,\overline{r}_{2}$.

The two-particle and three-particle photon LCDAs up to twist four also
generate NLP contributions at LO. Note that there are a few misprints in
definitions of higher-twist photon LCDAs in Ref. [13], we therefore
recalculate corrections from the higher-twist photon LCDAs which are updated
in Appendix C. The corresponding NLP contributions at the tree level are

$\displaystyle\xi^{\rm hp}_{\rm 2PHT}=$ $\displaystyle-
Q_{u}\,\frac{m_{b}+m_{u}}{4\,f_{B}\,m_{B}}\,\frac{1}{m_{b}^{2}-q^{2}}\,\exp\Big{(}\frac{m_{B}^{2}}{M^{2}}\Big{)}\,$
$\displaystyle\times\bigg{\\{}\bigg{[}\exp\Big{(}-\frac{s_{0}}{M^{2}}\Big{)}\,\rho^{\rm
a}_{\rm
2PHT}(\kappa(s_{0}),\nu)+\frac{1}{M^{2}}\int_{m_{b}^{2}}^{s_{0}}ds\,\exp\Big{(}-\frac{s}{M^{2}}\Big{)}\,\rho^{\rm
a}_{\rm 2PHT}(\kappa(s),\nu)\bigg{]}$
$\displaystyle-\frac{1}{2}\bigg{[}\exp\Big{(}-\frac{s_{0}}{M^{2}}\Big{)}\,\Big{(}\frac{1}{m_{b}^{2}-q^{2}}+\frac{1}{M^{2}\,\kappa_{0}}+\frac{1}{\kappa_{0}}\frac{d}{d\kappa_{0}}\,\Big{)}\rho^{\rm
b}_{\rm 2PHT}(\kappa_{0},\nu)\Big{|}_{\kappa_{0}\to\kappa(s_{0})}$
$\displaystyle+\frac{1}{M^{4}}\int_{m_{b}^{2}}^{s_{0}}ds\,\exp\Big{(}-\frac{s}{M^{2}}\Big{)}\,\frac{1}{\kappa(s)}\,\rho^{\rm
b}_{\rm 2PHT}(\kappa(s),\nu)\bigg{]}\bigg{\\}}\,,$ (41) $\displaystyle\xi^{\rm
hp}_{\rm 3PHT}=$
$\displaystyle-\frac{m_{b}+m_{u}}{f_{B}\,m_{B}}\,\frac{1}{m_{b}^{2}-q^{2}}\,\exp\Big{(}\frac{m_{B}^{2}}{M^{2}}\Big{)}\,\langle\bar{q}q\rangle(\nu)\,$
$\displaystyle\times\bigg{\\{}\exp\Big{(}-\frac{s_{0}}{M^{2}}\Big{)}\int_{0}^{\kappa(s_{0})}d\alpha_{q}\int_{\kappa_{0q}}^{1-\alpha_{q}}\frac{d\alpha_{g}}{\alpha_{g}}\Big{[}Q_{u}\,\rho^{G}_{\rm
3PHT}+Q_{b}\,\rho^{\gamma}_{\rm
3PHT}\Big{]}\Big{(}\alpha_{q},\alpha_{g},\frac{\kappa_{0q}}{\alpha_{g}},\nu\Big{)}$
$\displaystyle+\frac{1}{M^{2}}\int_{m_{b}^{2}}^{s_{0}}ds\int_{0}^{\kappa(s)}d\alpha_{q}\int_{\kappa_{q}}^{1-\alpha_{q}}\frac{d\alpha_{g}}{\alpha_{g}}\,\exp\Big{(}-\frac{s}{M^{2}}\Big{)}\,$
$\displaystyle\times\Big{[}Q_{u}\,\rho^{G}_{\rm
3PHT}+Q_{b}\,\rho^{\gamma}_{\rm
3PHT}\Big{]}\Big{(}\alpha_{q},\alpha_{g},\frac{\kappa_{0q}}{\alpha_{g}},\nu\Big{)}\Big{|}_{\kappa_{0q}\to\kappa(s_{0})-\alpha_{q},\,\kappa_{q}\to\kappa(s)-\alpha_{q}}\bigg{\\}}\,,$
(42) $\displaystyle\Delta\xi^{\rm hp}_{\rm 2PHT}=$ $\displaystyle~{}\xi^{\rm
hp}_{\rm 2PHT}\Big{|}_{\scriptsize\begin{array}[]{l}\rho^{\rm a}_{\rm
2PHT}\to\Delta\rho^{\rm a}_{\rm 2PHT}\\\ \rho^{\rm b}_{\rm
2PHT}\to\Delta\rho^{\rm b}_{\rm 2PHT}\end{array}}\,,\qquad\Delta\xi^{\rm
hp}_{\rm 3PHT}=\xi^{\rm hp}_{\rm
3PHT}\Big{|}_{\scriptsize\begin{array}[]{l}\rho^{G}_{\rm
3PHT}\to\Delta\rho^{G}_{\rm 3PHT}\\\ \rho^{\gamma}_{\rm
3PHT}\to\Delta\rho^{\gamma}_{\rm 3PHT}\end{array}}\,,$ (47)

where $\kappa(s)=(m_{b}^{2}-q^{2})/(s-q^{2})$, and the spectral densities are

$\displaystyle\rho^{\rm a}_{\rm 2PHT}(u,\nu)$ $\displaystyle=$ $\displaystyle
m_{b}\,f_{3\gamma}(\nu)\,\Big{[}\psi^{(a)}-2\,\bar{\psi}^{(v)}\Big{]}(u,\nu)-\langle\bar{q}q\rangle(\nu)\,\Big{[}\mathbb{A}+\bar{h}_{\gamma}\Big{]}(u,\nu)\,,$
$\displaystyle\Delta\rho^{\rm a}_{\rm 2PHT}(u,\nu)$ $\displaystyle=$
$\displaystyle
m_{b}\,f_{3\gamma}(\nu)\,\Big{[}\psi^{(a)}+2\,\bar{\psi}^{(v)}\Big{]}(u,\nu)+\langle\bar{q}q\rangle(\nu)\,\bar{h}_{\gamma}(u,\nu)\,,$
$\displaystyle\rho^{\rm b}_{\rm 2PHT}(u,\nu)$ $\displaystyle=$ $\displaystyle
2\,m_{b}^{2}\,\langle\bar{q}q\rangle(\nu)\,\Big{[}\mathbb{A}+\bar{h}_{\gamma}\Big{]}(u,\nu)\,,$
$\displaystyle\Delta\rho^{\rm b}_{\rm 2PHT}(u,\nu)$ $\displaystyle=$
$\displaystyle-2\,m_{b}^{2}\,\langle\bar{q}q\rangle(\nu)\,\bar{h}_{\gamma}(u,\nu)\,,$
(48) $\displaystyle\rho^{G}_{\rm 3PHT}(\alpha_{q},\alpha_{g},z,\nu)$
$\displaystyle=$ $\displaystyle
z\,\Big{(}T_{1}-2\,T_{2}-T_{3}+T_{4}\Big{)}(\underline{\alpha};\nu)+\bar{z}\,\Big{(}S-\tilde{S}+T_{1}+T_{2}+T_{3}\Big{)}(\underline{\alpha};\nu)\,,$
$\displaystyle\Delta\rho^{G}_{\rm 3PHT}(\alpha_{q},\alpha_{g},z,\nu)$
$\displaystyle=$ $\displaystyle
z\,\Big{(}S+\tilde{S}+T_{2}\Big{)}(\underline{\alpha};\nu)-\bar{z}\,\Big{(}2\,T_{2}+T_{4}\Big{)}(\underline{\alpha};\nu)$
$\displaystyle\rho^{\gamma}_{\rm 3PHT}(\alpha_{q},\alpha_{g},z,\nu)$
$\displaystyle=$
$\displaystyle\bar{z}\,S_{\gamma}(\underline{\alpha};\nu)+z\,T_{4}^{\gamma}(\underline{\alpha};\nu)\,,$
$\displaystyle\Delta\rho^{\gamma}_{\rm 3PHT}(\alpha_{q},\alpha_{g},z,\nu)$
$\displaystyle=$ $\displaystyle
z\,S_{\gamma}(\underline{\alpha};\nu)-\bar{z}\,T_{4}^{\gamma}(\underline{\alpha};\nu)\,,$
(49)

with $\bar{z}\equiv 1-z$,
$\underline{\alpha}\equiv\\{\alpha_{q},1-\alpha_{q}-\alpha_{g},\alpha_{g}\\}$
and $f_{3\gamma}$ a non-perturbative parameter. The two-particle LCDAs
$\psi^{(a)}$ and $\bar{\psi}^{(v)}$ are twist-3 LCDAs, and all the other LCDAs
are of twist four. One can then conclude that, at the tree level, the three-
particle twist-3 photon LCDAs do not contribute to the
$B\to\gamma\ell\nu_{\ell}$ decay at NLP.

In summary, the resolved-photon contribution is

$\displaystyle\xi^{\rm hp}(E_{\gamma})$ $\displaystyle=$
$\displaystyle\xi^{\rm hp}_{\rm LT}(E_{\gamma})+\xi^{\rm hp}_{\rm
2PHT}(E_{\gamma})+\xi^{\rm hp}_{\rm 3PHT}(E_{\gamma})\,,$
$\displaystyle\Delta\xi^{\rm hp}(E_{\gamma})$ $\displaystyle=$
$\displaystyle\Delta\xi^{\rm hp}_{\rm 2PHT}(E_{\gamma})+\Delta\xi^{\rm
hp}_{\rm 3PHT}(E_{\gamma})\,.$ (50)

It is clear that the leading twist-2 photon LCDA only generates symmetry-
preserving contributions, while the higher-twist photon LCDAs contribute both
to the symmetry-preserving and symmetry-breaking terms.

##  5 Numerical analysis

We are now ready to show the phenomenological impact of higher-order and
higher-power corrections to the $B\to\gamma\ell\nu_{\ell}$ decay. In doing
that, we first specify the theoretical inputs used in this work.

###  5.1 Theoretical inputs

We employ the Mathematica package RunDec [32] for the five-loop running of
strong coupling constant with the initial condition $\alpha_{s}(m_{Z})=0.1181$
and $m_{Z}=91.1876$ GeV in the context of the number of active flavor
$n_{f}=5$, and also adopt the threshold values $\mu_{4}=5.0$ GeV and
$\mu_{3}=1.2$ GeV for crossing $n_{f}=4$ and $n_{f}=3$, respectively. The
package HPL [33, 34] is used to calculate harmonic polylogarithms in the hard
function.

Parameter | Value | Ref. | Parameter | Value | Ref.
---|---|---|---|---|---
$f_{B}$ |  $0.190\pm 0.0013$ GeV | [35] | $m_{b}^{\rm pole}$ |  $4.8\pm 0.1$ GeV | [24]
$\overline{m}_{b}(\overline{m}_{b})$ |  $4.198\pm 0.012$ GeV | [35] | $\overline{m}_{u}(2\rm{GeV})$ |  $2.32\pm 0.10$ MeV | [36]
$\mu$ |  $1.5\pm 0.5$ GeV |  | $\mu_{h1},\mu_{h2}$ | $[\overline{m}_{b}/2,2\overline{m}_{b}]$ |
$\mu_{s}$ | 1.0 GeV |  | $\widehat{\omega}$ |  $0.197^{+0.199}_{-0.099}$ GeV |
$\lambda_{B}(\mu_{s})$ |  $0.350\pm 0.150$ GeV | [37] | $\sigma_{B}^{(2)}(\mu_{s})$ | $1.5\pm 1.0$ |
$\sigma_{B}^{(3)}(\mu_{s})$ | $0.0\pm 6.0$ |  | $\sigma_{B}^{(4)}(\mu_{s})$ | $0.0\pm 6.0$ |
$(\lambda_{E}^{2}/\lambda_{H}^{2})(\mu_{s})$ | $0.50\pm 0.10$ | [1] | $(2\lambda_{E}^{2}+\lambda_{H}^{2})(\mu_{s})$ |  $0.25\pm 0.15$ GeV2 | [1]
Resolved-photon parameters
$\nu$ | $[\overline{m}_{b}/2,2\overline{m}_{b}]$ |  | $\nu_{0}$ | 1.0 GeV |
$\langle\bar{q}q\rangle(\nu_{0})$ | $(-266\pm 21~{}\rm{MeV})^{3}$ | [36] | $\chi(\nu_{0})$ |  $3.15\pm 0.3$ GeV-2 | [38]
$f_{3\gamma}(\nu_{0})$ |  $-(4\pm 2)\times 10^{-3}$ GeV-2 | [38] | $a_{2}(\nu_{0})$ | $0.07\pm 0.07$ | [38]
$\omega^{A}_{\gamma}(\nu_{0})$ | $-2.1\pm 1.0$ | [38] | $\omega^{V}_{\gamma}(\nu_{0})$ | $3.8\pm 1.8$ | [38]
$\kappa(\nu_{0})$ | $0.2\pm 0.2$ | [38] | $\kappa^{+}(\nu_{0})$ | 0 | [38]
$\zeta_{1}(\nu_{0})$ | $0.4\pm 0.4$ | [38] | $\zeta_{1}^{+}(\nu_{0}),\,\zeta_{2}^{+}(\nu_{0})$ | 0 | [38]
$s_{0}$ |  $37.5\pm 2.5$ GeV2 | [13] | $M^{2}$ |  $18.0\pm 3.0$ GeV2 | [13]
Table 1: Summary of the numerical values for input parameters implemented in
the phenomenological analysis of the $B\to\gamma\ell\nu_{\ell}$ decay.

The leading-twist $B$-meson LCDA serves as the most important non-perturbative
input. Thanks to the smallness of $a_{\Gamma}$, we can expand the LCDA around
zero [8]

$\displaystyle\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu_{s})=\frac{1}{\lambda_{B}(\mu_{s})}\,\Big{[}1+\sum_{n\geq
2}\frac{\eta^{n}}{n!}\,\sigma_{B}^{(n)}(\mu_{s})\Big{]}\,,$ (51)

and truncate the expansion at a large $n=4$. The inverse(-logarithmic) moments
of the $B$-meson LCDA are defined as

$\displaystyle\frac{1}{\lambda_{B}(\mu)}=$
$\displaystyle\int_{0}^{\infty}\,\frac{d\omega}{\omega}\,\phi_{+}(\omega,\mu)=\tilde{\phi}_{+}(0,\widehat{\omega},\mu)\,,$
$\displaystyle\sigma_{B}^{(n)}(\mu)=$
$\displaystyle~{}\lambda_{B}(\mu)\int_{0}^{\infty}\,\frac{d\omega}{\omega}\,\ln^{n}\frac{\widehat{\omega}}{\omega}\,\phi_{+}(\omega,\mu)=\lambda_{B}(\mu)\,\Big{(}\frac{d}{d\eta}\Big{)}^{n}\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu)\Big{|}_{\eta\to
0}\,.$ (52)

In the expansion of $\tilde{\phi}_{+}$, we have choose certain value for
$\widehat{\omega}$ such that $\sigma_{B}^{(1)}(\mu_{s})\equiv 0$ 111By
introducing
$\widehat{\sigma}_{B}^{(n)}(\mu)=\lambda_{B}(\mu)\int_{0}^{\infty}\,\frac{d\omega}{\omega}\,\ln^{n}\frac{\lambda_{B}(\mu)e^{-\gamma_{E}}}{\omega}\,\phi_{+}(\omega,\mu)\,,$
(53) the parameter $\widehat{\omega}$ is determined though
$\widehat{\omega}=\lambda_{B}(\mu_{s})\,\exp[-\widehat{\sigma}_{B}^{(1)}(\mu_{s})-\gamma_{E}]$.
From $\lambda_{B}(\mu_{s})=0.35$ GeV and
$\widehat{\sigma}_{B}^{(1)}(\mu_{s})=0.0\pm 0.7$, we obtain
$\widehat{\omega}=0.197^{+0.199}_{-0.099}$ GeV. . The values for parameters of
the $B$-meson LCDA are summarized in Tab. 1.

The higher-twist $B$-meson LCDAs fulfill certain asymptotic behaviours and are
also constrained by the equation of motions. Applying general forms of the
$B$-meson LCDAs proposed in Ref. [1], which take into consideration the above
constraints, the NLP contributions involving the $B$-meson LCDAs could be
remarkably simplified

$\displaystyle\xi^{\rm ht}(E_{\gamma})=$
$\displaystyle-\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{2\,E_{\gamma}^{2}}\,\Big{[}\frac{2\,(\lambda_{E}^{2}+2\lambda_{H}^{2})}{6\bar{\Lambda}^{2}+2\lambda_{E}^{2}+\lambda_{H}^{2}}+\frac{1}{2}\Big{]}\,,$
$\displaystyle\xi^{\rm hqe}(E_{\gamma})=$
$\displaystyle~{}\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{4\,m_{b}\,E_{\gamma}}\,\Big{[}\frac{\bar{\Lambda}}{\lambda_{B}}-2+\frac{4\,(\lambda_{E}^{2}-\lambda_{H}^{2})}{6\bar{\Lambda}^{2}+2\lambda_{E}^{2}+\lambda_{H}^{2}}\Big{]}\,,$
$\displaystyle\xi^{\rm hc}(E_{\gamma})=$
$\displaystyle~{}\frac{Q_{u}\,\tilde{f}_{B}\,m_{B}}{2\,E_{\gamma}^{2}}\,\Big{[}\frac{\lambda_{E}^{2}(12-\pi^{2})}{6\bar{\Lambda}^{2}+2\lambda_{E}^{2}+\lambda_{H}^{2}}-\frac{\bar{\Lambda}}{\lambda_{B}}+\frac{1}{2}\Big{]}\,,$
(54)

where the factorization-scale dependence of the parameters $\tilde{f}_{B}$,
$\lambda_{B}$, $\bar{\Lambda}$, and $\lambda^{2}_{E,H}$ should be understood
222 The HQET parameters $\lambda^{2}_{E}$ and $\lambda^{2}_{H}$ are defined by
the hadronic matrix element of the three-body effective local operator [39,
40] $\displaystyle~{}\langle
0|\,\bar{q}(0)\,g_{s}\,G_{\mu\nu}(0)\,\Gamma\,h_{v}(0)\,|\bar{B}(v)\rangle$
$\displaystyle=$ $\displaystyle-{\tilde{f}_{B}(\mu)\,m_{B}\over 6}\,{\rm
Tr}\left\\{\gamma_{5}\,\Gamma\,{1+\not{v}\over
2}\,\Big{[}\lambda_{H}^{2}(\mu)\,\left(i\,\sigma_{\mu\nu}\right)+[\lambda_{H}^{2}(\mu)-\lambda_{E}^{2}(\mu)]\,\left(v_{\mu}\,\gamma_{\nu}-v_{\nu}\,\gamma_{\mu}\right)\Big{]}\right\\}\,.$
(55) . We also apply the LL resummation to $\lambda^{2}_{E,H}$ employing RG
equations of them [41, 42] (see Ref. [24] for details). The values for
$\lambda^{2}_{E,H}$ at the intrinsic scale are collected in Tab. 1, and we use
the relation $\bar{\Lambda}=m_{B}-m^{\rm pole}_{b}$ to determining
$\bar{\Lambda}$ with $m^{\rm pole}_{b}$ the pole mass of the $b$ quark.

For the sake of calculating the resolved photon corrections, phenomenological
models for the photon LCDAs have to be introduced. The series expansion of the
leading-twist photon LCDA in terms of the Gegenbauer polynomials and the
conformal expansions of the chiral-even twist-three and chiral-odd twist-four
LCDAs can be found in Ref. [13]. The relevant parameters are collected in Tab.
1.

In addition, the factorization scale $\mu$ is varied in the range $\mu=(1.5\pm
0.5)$ GeV. We also vary the hard scales $\mu_{h1}$ and $\mu_{h2}$ in the
interval $[\overline{m}_{b}/2,2\overline{m}_{b}]$ around the defaulted value
$\overline{m}_{b}$. We adopt the $B$-meson decay constant from the lattice
gauge simulations collected in the PDG [36]. The up and bottom quark masses
are chosen in the context of $\overline{\rm MS}$ scheme from the PDG [36]. For
the sake of convenience, we list the parameters in Tab. 1.

###  5.2 Predictions for the form factors

Figure 1: The $E_{\gamma}$ dependence (left plot) and the $\lambda_{B}$
dependence (right plot) of the form factors where
$\tilde{R}(E_{\gamma})=Q_{u}\,m_{B}/(2E_{\gamma})\,R(E_{\gamma})$.

First, we focus on the LP contribution to the form factors, which includes
radiative corrections. Fixing the photon energy to $E_{\gamma}=2.2$ GeV, we
show the $\lambda_{B}$ and $\sigma^{(n)}_{B}$ dependence of the function
$R(E_{\gamma})$

$\displaystyle R^{\rm LL}$ $\displaystyle=$
$\displaystyle\frac{f_{B}}{\lambda_{B}(\mu_{s})}\,\Big{[}0.859+4.15\times
10^{-3}\sigma_{B}^{(2)}-1.36\times 10^{-4}\sigma_{B}^{(3)}+3.34\times
10^{-6}\sigma_{B}^{(4)}+\cdots\Big{]}\,,$ $\displaystyle R^{\rm NLL}$
$\displaystyle=$
$\displaystyle\frac{f_{B}}{\lambda_{B}(\mu_{s})}\,\Big{[}0.696+3.03\times
10^{-2}\sigma_{B}^{(2)}-3.02\times 10^{-3}\sigma_{B}^{(3)}+1.50\times
10^{-4}\sigma_{B}^{(4)}+\cdots\Big{]}\,,$ $\displaystyle R^{\rm NNLL}$
$\displaystyle=$
$\displaystyle\frac{f_{B}}{\lambda_{B}(\mu_{s})}\,\Big{[}0.642+4.36\times
10^{-2}\sigma_{B}^{(2)}+3.72\times 10^{-4}\sigma_{B}^{(3)}+3.64\times
10^{-4}\sigma_{B}^{(4)}+\cdots\Big{]}\,,$ (56)

where all the other parameters are taken their central values. Our results are
comparable to that of Ref. [8]. With $\sigma^{(n)}_{B}=0$, one finds that the
NLL correction reduces the LL result by about 25%, and the NNLL result is
around 10% smaller than the NLL result. It is also clear that the coefficients
of $\sigma^{(2)}_{B}$ are much larger than the coefficients of
$\sigma^{(3,4)}_{B}$ in all three cases. With a large value for
$\sigma^{(2)}_{B}$, this term can generate sizeable correction to the form
factors. Employing the simple exponential model of $\phi_{+}$

$\displaystyle\phi_{+}(\omega,\mu)=\frac{\omega}{\omega^{2}_{0}}\,\exp\Big{(}-\frac{\omega}{\omega_{0}}\Big{)}\,,\qquad\omega_{0}=\lambda_{B}(\mu)\,,$
(57)

one roughly obtain $\sigma^{(2)}_{B}\in[1.6,2.3]$. We will utilize a
relatively conserved interval $\sigma^{(2)}_{B}\in[0.5,2.5]$ to include also
the values derived in Ref. [8]. Setting all the other parameters to their
central values, we show the $E_{\gamma}$ and the $\lambda_{B}$ dependences of
the form factors $F_{V/A}$ with the LL (dotted black curve), NLL (dashed black
curve), and NNLL (solid black curve) resummation accuracy in Fig. 1. We can
see that the form factors decrease with the increase of $E_{\gamma}$ or
$\lambda_{B}$. One also observes apparently the convergent tend of the
radiative corrections.

In Fig. 1, we also show the numerical impacts of the NLP contributions with
the factorizable and the resolved-photon corrections separated. It is
remarkable that symmetry-preserving factorizable NLP term $\xi^{\rm fac}$
(dashed blue lines) can introduce about a 50% decrease to the LP contribution
$\tilde{R}^{\rm NNLL}$ when $E_{\gamma}$ is around 1.5 GeV. While
approximately half of this decrease will be compensated by the resolved-photon
mechanism so that the total symmetry-preserving correction $\xi$ (solid blue
curves) reduces $\tilde{R}^{\rm NNLL}$ by around 20% which is of the same
order as the radiative correction. This also suggests that the factorizable
and resolved-photon contributions should be calculated consistently. The
symmetry-breaking $\Delta\xi$ term (solid red curves), which mainly from the
factorizable dynamics $\Delta\xi^{\rm fac}$ (dashed red lines), results in
corrections to $\tilde{R}^{\rm NNLL}$ by an amount of $\mathcal{O}(10\%)$.

Figure 2: The $E_{\gamma}$ dependence (left plot) and the $\lambda_{B}$
dependence (right plot) of the NLP factorizable form factors $\xi^{\rm fac}$
and $\Delta\xi^{\rm fac}$.

We illustrate in detail the effects of various factorizable NLP mechanisms in
Fig. 2. The symmetry-preserving hard-collinear quark expansion $\xi^{\rm
fac}_{\rm hc}$ (dashed blue curves) and higher-twist LCDS $\xi^{\rm fac}_{\rm
ht}$ (dotted blue curves) patterns are the most significant ones, which can
bring about 25% and $15\%\sim 30\%$ reductions to the LP form factors,
respectively. The symmetry-breaking contributions are dominated by the hard-
collinear quark expansion term $\Delta\xi^{\rm fac}_{\rm hc}$ (dashed red
curves), which introduces about a 15% correction to $\tilde{R}^{\rm NNLL}$. It
is surprising that the hard-collinear quark expansion dynamics alone can
decrease the form factor $F_{A}$ at LP by about 40%. The effects of heavy-
quark-expansion correction $\xi^{\rm fac}_{\rm hqe}$ (dot-dashed blue curves)
and the purely symmetry-breaking correction $\Delta\xi^{\rm fac}_{\rm b}$
(dotted red curves) are tiny.

Figure 3: The $E_{\gamma}$ dependence of the resolved-photon form factors
$\xi^{\rm hp}$ and $\Delta\xi^{\rm hp}$.

The consequences of the hadronic photon contributions to the form factors are
shown in Fig. 3. The effects of the symmetry-breaking mechanisms (red curves)
are negligible. Among the symmetry-preserving terms, only the ones
corresponding to two-particle photon LCDAs are numerically significant. It is
found that the leading-twist hadronic photon contribution $\xi^{\rm hp}_{\rm
LT}$ (dashed blue curves) introduces about a 50% enhancement to the LP form
factor. Meanwhile, the two-particle higher-twist photon LCDA contribution
(dotted blue curves) can decrease the LP form factors by an amount of
$\mathcal{O}(10\%)$.

Figure 4: The $E_{\gamma}$ dependence (left plot) and the $\lambda_{B}$ dependence (right plot) of the form factors $F_{V}$ and $F_{A}$. | Total | $\mu$ | $\mu_{h1}$ | $\sigma_{B}^{(2)}$ | $2\lambda_{E}^{2}+\lambda_{H}^{2}$ | $\widehat{\omega}$ | $\langle\bar{q}q\rangle$ | $s_{0}$ | $M^{2}$
---|---|---|---|---|---|---|---|---|---
$F_{V}$ | $0.387^{+0.078}_{-0.077}$ | ${}^{+0.000}_{-0.021}$ | ${}^{+0.047}_{-0.028}$ | ${}^{+0.029}_{-0.029}$ | ${}^{+0.021}_{-0.017}$ | ${}^{+0.020}_{-0.000}$ | ${}^{+0.036}_{-0.031}$ | ${}^{+0.011}_{-0.013}$ | ${}^{+0.013}_{-0.019}$
$F_{A}$ | $0.142^{+0.078}_{-0.081}$ | ${}^{+0.000}_{-0.021}$ | ${}^{+0.047}_{-0.028}$ | ${}^{+0.029}_{-0.029}$ | ${}^{+0.021}_{-0.017}$ | ${}^{+0.020}_{-0.000}$ | ${}^{+0.030}_{-0.026}$ | ${}^{+0.010}_{-0.013}$ | ${}^{+0.017}_{-0.027}$
Table 2: Theoretical predictions for the form factors with $E_{\gamma}=1.5$
GeV and $\lambda_{B}$ fixed at $0.35$ GeV. The total uncertainties, obtained
by adding all separate uncertainties in quadrature, as well as the numerically
important individual uncertainties are displayed.

We show our final predictions for the form factors $F_{A}$ (red curves) and
$F_{V}$ (blue curves) as well as their difference (black curves) in Fig. 4.
The theoretical uncertainties, which are obtained by adding in quadrature the
separate uncertainties caused by individual variations of all input
parameters, are indicated by the bands. The difference between the form
factors $F_{A}-F_{V}$ is always negative because it is dominant by two terms
$-2\Delta\xi^{\rm fac}_{\rm hc}$ and $Q_{\ell}f_{B}/E_{\gamma}$, as one can
see from the above analysis. It decreases fast with $E_{\gamma}$ decreases in
the small-$E_{\gamma}$ region since $\Delta\xi^{\rm fac}_{\rm hc}$ is
proportional to $1/E^{2}_{\gamma}$. In the region where $\lambda_{B}$ is large
and $E_{\gamma}$ is small, this difference will compensate the positive form
factor $F_{V}$ so that $F_{A}$ can be negligible, see the right plot. Fixing
$E_{\gamma}=1.5$ GeV and $\lambda_{B}=0.35$ GeV, we also show our predictions
for the form factors with uncertainties in Tab. 2.

###  5.3 Phenomenological applications

Figure 5: The $\lambda_{B}$ dependence of partial branching fractions
$\Delta\mathcal{BR}(1.5~{}{\rm GeV})$ (left plot) and
$\Delta\mathcal{BR}(2.0~{}{\rm GeV})$ (right plot).

Armed with our predictions for the $B\to\gamma\ell\nu_{\ell}$ form factors, we
proceed to discuss the constraints on the inverse moment $\lambda_{B}$ taking
advantage of the future measurements of the partial branching fractions with a
photon-energy cut to get rid of the soft photon radiation. With the help of
the differential decay width (4), one derives the partial branching fraction
with a phase-space cut on the photon energy

$\displaystyle\Delta{\cal BR}(E_{\rm cut})=\tau_{B}\,\int_{E_{\rm
cut}}^{m_{B}/2}\,dE_{\gamma}\,{d\,\Gamma\over d\,E_{\gamma}}\,,$ (58)

where $\tau_{B}$ is the lifetime of the $B$ meson.

The newly improved observation from the Belle collaboration shows that the
partial branching fraction for $B^{+}\to\gamma\ell^{+}\nu_{\ell}$ with
$E_{\gamma}>1$ GeV is $\Delta{\cal BR}(1~{}{\rm GeV})=(1.4\pm 1.0\pm
0.4)\times 10^{-6}$ and the Bayesian upper limit is $\Delta{\cal BR}(1~{}{\rm
GeV})<3.0\times 10^{-6}$ at 90% confidence level [2]. Because that the
$1/E_{\gamma}$ power expansion becomes not reliable in the small-$E_{\gamma}$
region [1], experimental data with a larger photon energy cut, like 1.5 or 2.0
GeV, is needed for a precise determination of the inverse moment. We show the
$\lambda_{B}$ dependence of the partial branching fractions of the radiative
leptonic decay in Fig. 5 for $E_{\rm cut}>1.5$ GeV and $E_{\rm cut}>2$ GeV. It
can be observed that the partial branching fraction grows with the decrease of
$\lambda_{B}$, which is mainly due to the reciprocal dependence of the LP form
factors on the inverse moment. We also find that the NLP effects (red curves)
generate about $40\%\sim 50\%$ decreases to the partial branching fractions at
LP (black curves).

Figure 6: The $\lambda_{B}$ dependence of $\mathcal{R}(2.0~{}{\rm
GeV},1.5~{}{\rm GeV})$ and the $E_{\rm cut}$ dependence of $\mathcal{R}(E_{\rm
cut},1.5~{}{\rm GeV})$.

To eliminate the theoretical uncertainties originating from our prediction, we
define the ratio of partial branching fractions

$\mathcal{R}(E_{\gamma 1},E_{\gamma 2})=\frac{\Delta{\cal BR}(E_{\gamma
1})}{\Delta{\cal BR}(E_{\gamma 2})}\,.$ (59)

We show our predictions for this ratio in Fig. 6. One can conclude from the
left plot that the uncertainty of $\mathcal{R}(2.0~{}{\rm GeV},1.5~{}{\rm
GeV})$ is of $\mathcal{O}(5\%)$ and also the ratio is almost independent of
$\lambda_{B}$. The effect of the NLP corrections to the $\mathcal{R}$ is
significant, it introduces about a 10% increase to the ratio at LP. In the
right plot, we also illustrate the $E_{\rm cut}$ dependence of
$\mathcal{R}(E_{\rm cut},1.5~{}{\rm GeV})$, which equals 1 at $E_{\rm
cut}=1.5$ GeV and 0 at $E_{\rm cut}=m_{B}/2$. We hope that our prediction
could be checked by future experiments.

##  6 Conclusion

In this paper, we reconsidered the $B\to\gamma\ell\nu_{\ell}$ decay for photon
with large energy $E_{\gamma}\sim\mathcal{O}(m_{b})$. First, we formally re-
derived the NNLL-resummation improved factorization formula for the form
factors at LP in the Laplace space. In the derivation, we chose the
factorization scale to be a hard-collinear scale such that the jet function is
free of large logarithms. Employing this resummation formula, we performed
complete NNLL resummations to the hard function and the static $B$-meson decay
constant. While because of the yet unknown three-loop anomalous dimension of
the leading-twist $B$-meson LCDA, we could only fulfill a partial NNLL
resummation to the LCDA.

Second, we calculated various NLP contributions to the decay form factors. We
computed the symmetry-preserving NLP correction from hard-collinear quark
propagator $\xi^{\rm fac}_{\rm hc}$ at the tree level by using operator-level
identities from the equation of motions of quarks. The operator identities are
essential when we meet operators, such as
$[\not\\!\partial\bar{q}(x)]\,\Gamma\,h_{v}(0)$ and
$[v\cdot\partial\,\bar{q}(x)]\,\Gamma\,h_{v}(0)$, whose fields are not
separated by light-like distances. Applying the operator identities, we
reproduced the LO contribution from the subleading term of the $b$-quark field
in the heavy-quark expansion $\xi^{\rm fac}_{\rm hqe}$. In the framework of
the LCSR, we updated the resolved photon contributions $\xi^{\rm hp}$ and
$\Delta\xi^{\rm hp}$, which includes leading-twist photon LCDA contribution at
the one-loop level as well as LO correction from higher-twist photon LCDAs,
with the corrected definitions of photon LCDAs. For completeness, we also
collected symmetry-breaking NLP corrections from the photon emission off the
$b$ quark $\Delta\xi^{\rm fac}_{\rm b}$ and hard-collinear quark propagator
$\Delta\xi^{\rm fac}_{\rm hc}$ [10] and symmetry-preserving correction from
the higher-twist $B$-meson LCDAs $\xi^{\rm fac}_{\rm ht}$ [1].

Third, employing the above derived LP and NLP contributions, we investigated
their numerical impacts on the form factors. The corrections from the NLO
contribution in $\alpha_{s}$ and NLP contribution in $\Lambda_{\rm QCD}/m_{b}$
to the form factors are rough of the same size, the former generates about
$25\%$ decrease to both $F_{V}$ and $F_{A}$, and the later brings about a
$10\%$ ($30\%$) decrease to the form factor $F_{V}$ ($F_{A}$). Among all the
NLP contributions, the resolved photon correction from leading twist-2 photon
LCDA is numerically most important, it increases the LP form factors by around
50%. The correction from the hard-collinear quark expansion introduces a
30-40% reduction to the form factor $F_{A}$. The symmetry-preserving
contribution from the higher-twist LCDAs decreases the form factors by an
amount of $\mathcal{O}(20)\%$. All the other mechanisms are numerically not
that significant.

Last but not least, we analyzed the partial branching faction $\Delta{\cal
BR}$ as well as the ratio of partial branching factions $\mathcal{R}$ of the
$B\to\gamma\ell\nu_{\ell}$ decay. For our results to be reliable, we
introduced a large photon-energy cut $E_{\gamma}>1.5$ GeV or $E_{\gamma}>2.0$
GeV to the $B\to\gamma\ell\nu_{\ell}$ decay. We found that the partial
branching faction depends strongly on the inverse moment $\lambda_{B}$. While
the newly introduced ratio $\mathcal{R}$ is almost independent of
$\lambda_{B}$. The uncertainties of the partial branching factions will to a
large extent cancel in the ratio, and this leads to a small, about 5%,
uncertainty to the ratio $\mathcal{R}$.

The various NLP contributions to the form factors could be re-analyzed in the
framework of SCET, where different dynamics receive definite operator-level
definitions. This will deepen our knowledge of the $B\to\gamma\ell\nu_{\ell}$
decay. The NLO corrections to the various subleading-power contributions,
which could numerically be of the same size as the NNLO corrections at LP, are
of great importance. These corrections are also dispensable for a better
understanding of exclusive $B$-meson decays.

### Acknowledgements

The research of Y.L.S. is supported by the National Natural Science Foundation
of China with Grant No. 12175218 and the Natural Science Foundation of
Shandong with Grant No. ZR2020MA093. C.W is supported in part by the National
Natural Science Foundation of China with Grant No.12105112 and the Natural
Science Foundation of Jiangsu Education Committee with Grant No. 21KJB140027.
Y.B.W. is supported in part by the Alexander-von-Humboldt Stiftung.

##  Appendix A $B$-meson LCDA

The definitions of the two-particle and three-particle $B$-meson LCDAs are
[40]

$\displaystyle\langle\,0\,|\,\bar{q}_{\alpha}(x)\,[x,0]\,h_{v\beta}(0)\,|\,\bar{B}(v)\,\rangle$
$\displaystyle=$
$\displaystyle-\frac{i\tilde{f}_{B}(\mu)\,m_{B}}{4}\,\bigg{[}\frac{1+\\!\not\\!v}{2}\,\,\bigg{\\{}2\,\Big{(}\Phi_{+}(v\cdot
x)+x^{2}\,G_{+}(v\cdot x)\Big{)}$ $\displaystyle-{\not\\!x\over v\cdot
x}\,\Big{[}\Big{(}\Phi_{+}(v\cdot x)-\Phi_{-}(v\cdot
x)\Big{)}+x^{2}\,\Big{(}G_{+}(v\cdot x)-G_{-}(v\cdot
x)\Big{)}\Big{]}\bigg{\\}}\,\gamma_{5}\,\bigg{]}_{\beta\alpha}\,,$ (60)
$\displaystyle\langle\,0\,|\,\bar{q}_{\alpha}(t_{1}\,\bar{n})\,[t_{1}\bar{n},t_{2}\bar{n}]\,g_{s}\,G_{\mu\nu}(t_{2}\,\bar{n})\,[t_{2}\bar{n},0]\,h_{v\,\beta}(0)\,|\,\bar{B}(v)\,\rangle$
$\displaystyle=$ $\displaystyle~{}{\tilde{f}_{B}(\mu)\,m_{B}\over
2}\,\bigg{[}\frac{1+\\!\not\\!v}{2}\,\bigg{\\{}(v_{\mu}\,\gamma_{\nu}-v_{\nu}\,\gamma_{\mu})\,\Big{(}\Psi_{A}-\Psi_{V}\Big{)}-i\,\sigma_{\mu\nu}\,\Psi_{V}-(\bar{n}_{\mu}\,v_{\nu}-\bar{n}_{\nu}\,v_{\mu})\,X_{A}$
$\displaystyle+(\bar{n}_{\mu}\,\gamma_{\nu}-\bar{n}_{\nu}\,\gamma_{\mu})\,\Big{(}W+Y_{A}\Big{)}+\,i\,\epsilon_{\mu\nu\alpha\beta}\,\bar{n}^{\alpha}\,v^{\beta}\,\gamma_{5}\,\tilde{X}_{A}-\,i\,\epsilon_{\mu\nu\alpha\beta}\,\bar{n}^{\alpha}\,\gamma^{\beta}\,\gamma_{5}\,\tilde{Y}_{A}$
$\displaystyle-\,(\bar{n}_{\mu}\,v_{\nu}-\bar{n}_{\nu}\,v_{\mu})\not\\!{\bar{n}}\,W+\,(\bar{n}_{\mu}\,\gamma_{\nu}-\bar{n}_{\nu}\,\gamma_{\mu})\not\\!\bar{n}\,Z\bigg{\\}}\,\gamma_{5}\,\bigg{]}_{\beta\,\alpha}(t_{1},t_{2})\,,$
(61)

where $G_{\mu\nu}=i/g_{s}[D_{\mu},D_{\nu}]$ (with
$D_{\mu}=\partial_{\mu}-ig_{s}A_{\mu}$). The momentum-space two-particle and
three-particle LCDAs are related to the position-space LCDAs by

$\displaystyle\Phi_{\rm 2P}(t)=$
$\displaystyle\int_{0}^{\infty}d\omega\,e^{-i\omega t}\,\phi_{\rm
2P}(\omega)\,,$ $\displaystyle\Phi_{\rm 3P}(t_{1},t_{2})=$
$\displaystyle\int_{0}^{\infty}d\omega_{1}\,d\omega_{2}\,e^{-i\omega_{1}t_{1}-i\omega_{2}t_{2}}\,\phi_{\rm
3P}(\omega_{1},\omega_{2})\,,$ (62)

where uppercase letters and lowercase letters represent position-space LCDAs
and momentum-space LCDAs, respectively. The three-particle LCDAs can be
expressed by the LCDAs with definite twists

$\displaystyle\Phi_{3}=$ $\displaystyle~{}\Psi_{A}-\Psi_{V}\,,$
$\displaystyle\Psi_{A}=$ $\displaystyle~{}\frac{1}{2}\,(\Phi_{3}+\Phi_{4})\,,$
$\displaystyle\Phi_{4}=$ $\displaystyle~{}\Psi_{A}+\Psi_{V}\,,$
$\displaystyle\Psi_{V}=$
$\displaystyle~{}\frac{1}{2}\,(-\Phi_{3}+\Phi_{4})\,,$
$\displaystyle\Psi_{4}=$ $\displaystyle~{}\Psi_{A}+X_{A}\,,$ $\displaystyle
X_{A}=$ $\displaystyle~{}\frac{1}{2}\,(-\Phi_{3}-\Phi_{4}+2\,\Psi_{4})\,,$
$\displaystyle\tilde{\Psi}_{4}=$ $\displaystyle~{}\Psi_{V}-\tilde{X}_{A}\,,$
$\displaystyle\tilde{X}_{A}=$
$\displaystyle~{}\frac{1}{2}\,(-\Phi_{3}+\Phi_{4}-2\,\tilde{\Psi}_{4})\,,$
$\displaystyle\Phi_{5}=$
$\displaystyle~{}\Psi_{A}+\Psi_{V}+2\,Y_{A}-2\,\tilde{Y}_{A}+2\,W\,,$
$\displaystyle Y_{A}=$
$\displaystyle~{}\frac{1}{2}\,(-\Phi_{3}-\Phi_{4}+\Psi_{4}-\Psi_{5})\,,$
$\displaystyle\Psi_{5}=$ $\displaystyle-\Psi_{A}+X_{A}-2\,Y_{A}\,,$
$\displaystyle\tilde{Y}_{A}=$
$\displaystyle~{}\frac{1}{2}\,(-\Phi_{3}+\Phi_{4}-\tilde{\Psi}_{4}+\tilde{\Psi}_{5})\,,$
$\displaystyle\tilde{\Psi}_{5}=$
$\displaystyle-\Psi_{V}-\tilde{X}_{A}+2\,\tilde{Y}_{A}\,,$ $\displaystyle W=$
$\displaystyle~{}\frac{1}{2}\,(\Phi_{4}-\Psi_{4}-\tilde{\Psi}_{4}+\Phi_{5}+\Psi_{5}+\tilde{\Psi}_{5})\,,$
$\displaystyle\Phi_{6}=$
$\displaystyle~{}\Psi_{A}-\Psi_{V}+2\,Y_{A}+2\,\tilde{Y}_{A}+2\,W-4\,Z\,,$
$\displaystyle Z=$
$\displaystyle~{}\frac{1}{4}\,(-\Phi_{3}+\Phi_{4}-2\,\tilde{\Psi}_{4}+\Phi_{5}+2\,\tilde{\Psi}_{5}-\Phi_{6})\,.$
(63)

For the higher-twist LCDA $G_{+}$, we also separate the generic two-particle
part $\hat{G}_{+}$ from the three-particle part

$\displaystyle
G_{+}(t)=\hat{G}_{+}(t)-\frac{1}{2}\int_{0}^{1}du\,\bar{u}\,\Psi_{4}(t,ut)\,.$
(64)

##  Appendix B NNLL resummation

###  B.1 Perturbative corrections up to NNLO

The functions $C(E_{\gamma},\mu)$, $J(E_{\gamma}\omega,\mu)$ and $K(\mu)$ can
be calculated perturbatively. The hard function is expanded in $\alpha_{s}$ as

$\displaystyle
C(E_{\gamma},\mu)=\sum_{n=0}C^{(n)}(\mu)\,\Big{(}\frac{\alpha_{s}(\mu)}{4\pi}\Big{)}^{n}\,,$
(65)

and the coefficients up to NNLO accuracy are [15, 16, 17, 18]

$\displaystyle C^{(0)}(\mu)$ $\displaystyle=$ $\displaystyle 1\,,\hskip
56.9055ptC^{(1)}(\mu)=C_{F}\Big{[}-\frac{L^{2}}{2}+L\,f_{-1}(u)+f_{0}(u)\Big{]}\Big{|}_{u=2E_{\gamma}/m_{b}}\,,$
$\displaystyle C^{(2)}(\mu)$ $\displaystyle=$
$\displaystyle\bigg{\\{}F_{1,0}^{(2)}(u)+C_{F}^{2}\Big{[}-\frac{5}{24}L^{4}+\frac{5}{6}L^{3}f_{-1}(u)+\frac{L^{2}}{2}\big{(}3f_{0}(u)-f_{-1}(u)\big{)}+L\big{(}2f_{1}(u)$
(66) $\displaystyle-$ $\displaystyle
f_{-1}(u)f_{0}(u)\big{)}-f_{-1}(u)f_{1}(u)+f_{2}(u)\Big{]}$ $\displaystyle+$
$\displaystyle
C_{F}\bigg{[}\frac{17}{18}L^{3}-\frac{25}{6}L^{2}f_{-1}(u)-\frac{L}{9}(\pi^{2}+87f_{0}(u))-\frac{31}{3}f_{1}(u)+\frac{2}{9}\zeta_{3}$
$\displaystyle+$ $\displaystyle
n_{f}\Big{(}-\frac{L^{3}}{9}+\frac{L^{2}}{3}f_{-1}(u)+\frac{2}{3}Lf_{0}(u)+\frac{2}{3}f_{1}(u)\Big{)}\bigg{]}\bigg{\\}}\Big{|}_{u=2E_{\gamma}/m_{b}},$

with $L\equiv\ln(\mu^{2}/m_{b}^{2})$ and $n_{f}$ the number of light fermion
flavours. The function $F_{1,0}^{(2)}(u)$ is

$\displaystyle F_{1,0}^{(2)}(u)=$
$\displaystyle~{}C_{F}^{2}\Big{[}\frac{L^{4}}{3}+\frac{4}{3}L^{3}\,j_{-3}(u)+2L^{2}\,j_{-2}(u)+2L\,j_{-1}(u)+j_{0}(u)\Big{]}$
$\displaystyle+3C_{F}\Big{[}-\frac{11}{9}L^{3}+2L^{2}\,k_{-2}(u)+2L\,k_{-1}(u)+k_{0}(u)\Big{]}$
$\displaystyle+C_{F}\frac{n_{f}}{2}\Big{[}\frac{4}{9}L^{3}+2L^{2}\,p_{-2}(u)+2L\,p_{-1}(u)+p_{0}(u)\Big{]}$
$\displaystyle+C_{F}\frac{1}{2}\Big{[}\frac{16}{9}L^{3}+2L^{2}\,q_{-2}(u)+2L\,q_{-1}(u)+q_{0}(u)\Big{]}\,,$
(67)

where the coefficient functions $f_{i}(u)$, $j_{i}(u)$, $k_{i}(u)$, $p_{i}(u)$
and $q_{i}(u)$ can be found in Ref. [17]. The jet function and the function
$K$ could be expanded similarly. The corresponding coefficients up to NNLO
level for the jet function are [19]

$\displaystyle J^{(0)}(\mu)=$ $\displaystyle~{}1\,,\hskip
56.9055ptJ^{(1)}(\mu)=C_{F}\,\Big{(}L_{p}^{2}-\frac{\pi^{2}}{6}-1\Big{)}\,,$
$\displaystyle J^{(2)}(\mu)=$
$\displaystyle~{}C_{F}\,\bigg{[}\frac{2}{3}\,L_{p}^{4}+\frac{2n_{f}-33}{9}\,L_{p}^{3}-\frac{10n_{f}+11\pi^{2}-189}{9}\,L_{p}^{2}$
$\displaystyle+\frac{38n_{f}+1278\zeta_{3}+36\pi^{2}-915}{27}\,L_{p}+n_{f}\,\Big{(}\frac{4\zeta_{3}}{9}+\frac{5\pi^{2}}{54}+\frac{7}{81}\Big{)}$
$\displaystyle-\frac{16\zeta_{3}}{3}-\frac{149\pi^{4}}{540}-\frac{119\pi^{2}}{36}+\frac{601}{54}\bigg{]}\,,$
(68)

with $L_{p}=\ln(2E_{\gamma}\,\omega/\mu^{2})$, and for the function $K$ are
[20]

$\displaystyle K^{(0)}(\mu)$ $\displaystyle=$ $\displaystyle 1\,,\hskip
56.9055ptK^{(1)}(\mu)=-C_{F}\,\Big{(}\frac{3}{2}\ln\frac{\mu^{2}}{m_{b}^{2}}+2\Big{)}\,,$
$\displaystyle K^{(2)}(\mu)$ $\displaystyle=$ $\displaystyle-
C_{F}\,\Big{[}\frac{9}{8}\,C_{F}\,\ln^{2}\frac{\mu^{2}}{m_{b}^{2}}+\Big{(}3\,C_{F}+\frac{127}{12}+\frac{7\pi^{2}}{9}+\frac{5}{6}n_{f}\Big{)}\ln\frac{\mu^{2}}{m_{b}^{2}}$
(69) $\displaystyle+$
$\displaystyle\frac{683}{48}+\frac{17\pi^{2}}{6}+\frac{11\zeta_{3}}{3}+\frac{2\ln
2}{3}\pi^{2}\Big{]}\,.$

The expansion coefficients for the function $K^{-1}(\mu)$, which are actually
needed in the factorization formula, can be obtained from $K^{(i)}$
straightforwardly.

###  B.2 Anomalous dimensions

The evolution factors $U_{1}(E_{\gamma},\mu_{h1},\mu)$ and
$U_{2}(\mu_{h2},\mu)$ satisfy the same RG-equation as $C(E_{\gamma},\mu)$ and
$\tilde{f}_{B}(\mu)$, respectively,

$\displaystyle\frac{d}{d\ln\mu}\,U_{1}(E_{\gamma},\mu_{h1},\mu)=$
$\displaystyle~{}\Big{[}\Gamma_{\rm
cusp}(\alpha_{s})\,\ln\frac{2\,E_{\gamma}}{\mu}+\gamma_{H}(\alpha_{s})\Big{]}\,U_{1}(E_{\gamma},\mu_{h1},\mu)\,,$
$\displaystyle\frac{d}{d\ln\mu}\,U_{2}(\mu_{h2},\mu)=$
$\displaystyle~{}\gamma_{hl}(\alpha_{s})\,U_{2}(\mu_{h2},\mu)\,,$ (70)

with initial condition $U_{1}(E_{\gamma},\mu_{h1},\mu_{h1})=1$ and
$U_{2}(\mu_{h2},\mu_{h2})=1$. The anomalous dimensions and the QCD beta
function are expanded as

$\displaystyle~{}\gamma_{H}(\alpha_{s})=\sum_{n=0}\gamma_{n,H}\,\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{n+1}\,,$
$\displaystyle~{}\beta(\alpha_{s})=\mu\frac{d\alpha_{s}}{d\mu}=-2\alpha_{s}\sum_{n=0}\beta_{n}\,\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{n+1}\,.$
(71)

The other anomalous dimensions are expanded similarly to $\gamma_{H}$.

We collect the beta function [43]

$\displaystyle\beta_{0}=$
$\displaystyle~{}11-\frac{2}{3}n_{f},\qquad\beta_{1}=102-\frac{38}{3}n_{f},\qquad\beta_{2}=\frac{2857}{2}-\frac{5033}{18}n_{f}+\frac{325}{54}n_{f}^{2},$
$\displaystyle\beta_{3}=$
$\displaystyle~{}\frac{149753}{6}+3564\zeta_{3}-\Big{(}\frac{1078361}{162}+\frac{6508}{27}\,\zeta_{3}\Big{)}n_{f}+\Big{(}\frac{50065}{162}+\frac{6472}{81}\,\zeta_{3}\Big{)}n^{2}_{f}+\frac{1093}{729}n^{3}_{f}\,,$
(72)

the cusp anomalous dimension [44]

$\displaystyle\Gamma_{0}=$
$\displaystyle~{}4C_{F}\,,\quad\Gamma_{1}=C_{F}\Big{[}\frac{268}{3}-4\pi^{2}-\frac{40}{9}n_{f}\Big{]}\,,$
$\displaystyle\Gamma_{2}=$
$\displaystyle~{}C_{F}\big{[}1470-\frac{536\pi^{2}}{3}+\frac{44\pi^{4}}{5}+264\zeta_{3}+n_{f}\Big{(}-\frac{1276}{9}+\frac{80\pi^{2}}{9}-\frac{208}{3}\zeta_{3}\Big{)}-\frac{16}{27}n_{f}^{2}\big{]}\,,$
$\displaystyle\Gamma_{3}=$
$\displaystyle~{}C_{F}\big{[}\frac{84278}{3}-\frac{44200\pi^{2}}{9}+\frac{2706\pi^{4}}{5}-\frac{2504\pi^{6}}{105}+20944\zeta_{3}-10824\zeta_{5}-528\pi^{2}\zeta_{3}-432\zeta_{3}^{2}$
$\displaystyle+n_{f}\big{(}-\frac{344345}{81}+\frac{12800\pi^{2}}{27}-\frac{616\pi^{4}}{45}-\frac{153920\zeta_{3}}{27}+\frac{19504\zeta_{5}}{9}+\frac{416\pi^{2}\zeta_{3}}{3}\big{)}$
$\displaystyle+n_{f}^{2}\big{(}\frac{17875}{243}-\frac{304\pi^{2}}{81}-\frac{104\pi^{4}}{135}+\frac{4160\zeta_{3}}{27}\big{)}+n_{f}^{3}\big{(}-\frac{32}{81}+\frac{64\zeta_{3}}{27}\Big{)}\big{]}$
$\displaystyle-\frac{160\pi^{2}}{3}-\frac{496\pi^{6}}{189}+\frac{320\zeta_{3}}{3}-960\zeta_{3}^{2}+\frac{8800\zeta_{5}}{3}+\frac{n_{f}}{27}\big{(}160\pi^{2}-320\zeta_{3}-1600\zeta_{5}\big{)}\,,$
(73)

where we have used $\Gamma_{n}\equiv\Gamma_{n,\rm cusp}$, the anomalous
dimension of the hard function [45]

$\displaystyle\gamma_{0,H}=$
$\displaystyle-5C_{F}\,,\quad\gamma_{1,H}=C_{F}\big{[}-\frac{1585}{18}-\frac{5\pi^{2}}{6}+34\zeta_{3}+n_{f}\,\big{(}\frac{125}{27}+\frac{\pi^{2}}{3}\big{)}\big{]}\,,$
$\displaystyle\gamma_{2,H}=$
$\displaystyle~{}C_{F}\big{[}-\frac{307753}{324}+\frac{2941\pi^{2}}{54}-\frac{2221\pi^{4}}{270}+\frac{13858\zeta_{3}}{9}-\frac{2860\zeta_{5}}{3}-\frac{1544\pi^{2}\zeta_{3}}{27}$
$\displaystyle+n_{f}\,\big{(}\frac{19805}{243}+\frac{781\pi^{2}}{81}+\frac{17\pi^{4}}{405}-\frac{212\zeta_{3}}{27}\big{)}+n_{f}^{2}\,\big{(}\frac{2633}{729}-\frac{10\pi^{2}}{27}-\frac{8\zeta_{3}}{27}\big{)}\big{]}\,,$
(74)

the anomalous dimension of HQET decay constant $\tilde{f}_{B}(\mu)$ [46]

$\displaystyle\gamma_{0,hl}=$
$\displaystyle~{}3C_{F}\,,\quad\gamma_{1,hl}=-C_{F}\big{[}-\frac{127}{6}-\frac{14\pi^{2}}{9}+\frac{5}{3}n_{f}\big{]}\,,$
$\displaystyle\gamma_{2,hl}=$ $\displaystyle-
C_{F}\big{[}\frac{61}{4}-\frac{686\pi^{2}}{27}-\frac{380\pi^{4}}{81}+\frac{178\zeta_{3}}{3}+n_{f}\Big{(}\frac{172}{27}+\frac{196\pi^{2}}{81}+\frac{332\zeta_{3}}{9}\Big{)}+n_{f}^{2}\,\frac{35}{27}\big{]}\,,$
(75)

and the two-loop anomalous dimension $\gamma$ of $B$-meson LCDA [19]

$\displaystyle\gamma_{0}=-2C_{F}\,,\quad\gamma_{1}=C_{F}\big{[}\frac{206}{9}-\frac{53\pi^{2}}{18}-50\zeta_{3}+n_{f}\big{(}-\frac{32}{27}+\frac{5\pi^{2}}{9}\big{)}\big{]}\,.$
(76)

###  B.3 The evolution function $U(E_{h},\mu_{i},\mu)$

The solution to the general RG equation is

$\displaystyle U(E_{h},\mu_{i},\mu)=$
$\displaystyle~{}\exp\Big{[}S(\mu_{i})-a_{\gamma_{\rm
gen}}(\mu_{i})\Big{]}\,\Big{(}\frac{E_{h}}{\mu_{i}}\Big{)}^{-a_{\Gamma}(\mu_{i})}\,,$
(77)

where we have introduced the following functions

$\displaystyle~{}a_{\gamma_{\rm gen}}(\mu_{i})\equiv a_{\gamma_{\rm
gen}}(\mu_{i},\mu)=-\int_{\alpha_{s}(\mu_{i})}^{\alpha_{s}(\mu)}d\alpha\,\frac{\gamma_{\rm
gen}(\alpha)}{\beta(\alpha)}\,,$ $\displaystyle~{}S(\mu_{i})\equiv
S(\mu_{i},\mu)=-\int_{\alpha_{s}(\mu_{i})}^{\alpha_{s}(\mu)}d\alpha\,\frac{\Gamma_{\rm
cusp}(\alpha)}{\beta(\alpha)}\int_{\alpha_{s}(\mu_{i})}^{\alpha}\frac{d\hat{\alpha}}{\beta(\hat{\alpha})}\,,$
(78)

and a similar definition for $a_{\Gamma}\equiv a_{\Gamma_{\rm cusp}}$. In
practice, the evolution factor $U$ should be expanded in the QCD coupling
constant. The solution to the next-to-leading logarithmic (NLL) resummation
accuracy can be found in Ref. [10]. To derive the NNLL-resummation formula,
the functions $S$ 333 The function $S$ starts at order $\alpha^{-1}_{s}$.
Since the order $\alpha^{-1}_{s}$ term in the natural exponential function
should not be expanded, it is convenient to include this term into $S^{(0)}$,
see Appendix B. , $a_{\gamma_{\rm gen}}$ and $a_{\Gamma}$ need to be expanded
in $\alpha_{s}(\mu_{i})/(4\pi)$ up to the two-loop level, and the expansion
coefficients are collected in Appendix B. With these coefficients at hand, it
is straightforward to obtain

$\displaystyle U(E_{h},\mu_{i},\mu)=$
$\displaystyle~{}\exp\Big{[}S^{(0)}-a_{\gamma_{\rm
gen}}^{(0)}\Big{]}\,\Big{(}\frac{E_{h}}{\mu_{i}}\Big{)}^{-a_{\Gamma}^{(0)}}\Big{[}1+U^{(1)}\,\frac{\alpha_{s}(\mu_{i})}{4\pi}+U^{(2)}\,\Big{(}\frac{\alpha_{s}(\mu_{i})}{4\pi}\Big{)}^{2}+\mathcal{O}(\alpha^{3}_{s})\Big{]}\,,$
(79)

where

$\displaystyle U^{(1)}$ $\displaystyle=$
$\displaystyle-\ln\frac{E_{h}}{\mu_{i}}\,a_{\Gamma}^{(1)}+S^{(1)}-a_{\gamma_{\rm
gen}}^{(1)}\,,$ $\displaystyle U^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln^{2}\frac{E_{h}}{\mu_{i}}\,\Big{(}a_{\Gamma}^{(1)}\Big{)}^{2}+\ln\frac{E_{h}}{\mu_{i}}\,\Big{[}-a_{\Gamma}^{(2)}-S^{(1)}\,a_{\Gamma}^{(1)}+a_{\Gamma}^{(1)}\,a_{\gamma_{\rm
gen}}^{(1)}\Big{]}$ (80) $\displaystyle+$ $\displaystyle
S^{(2)}-a_{\gamma_{\rm
gen}}^{(2)}+\frac{1}{2}\,\Big{(}S^{(1)}\Big{)}^{2}-S^{(1)}\,a_{\gamma_{\rm
gen}}^{(1)}+\frac{1}{2}\,\Big{(}a_{\gamma_{\rm gen}}^{(1)}\Big{)}^{2}\,.$

The $\mu$ and $\mu_{i}$ dependence of the factors $S^{(i)}$,
$a_{\Gamma}^{(i)}$ and $a_{\gamma_{\rm gen}}^{(i)}$ should be understood.

###  B.4 Partial NNLL resummation for $\phi_{+}$

To obtain the RG equation of the LCDA $\tilde{\phi}_{+}$, it is convenient to
introduce a Mellin transformation 444 The explicit relation between LCDAs in
the momentum space and Mellin space is
$\displaystyle\varphi_{+}(j)=\frac{-i}{2\pi}\,\Gamma(-j)\int^{\infty}_{0}d\omega\,\Big{(}\frac{\omega}{\mu
e^{\gamma_{E}}}\Big{)}^{j}\,\phi_{+}(\omega)\,.$ (81) The relation is not
necessary in this work.  and derive the RG equation for the Mellin-space LCDA
$\varphi_{+}$ [47],

$\displaystyle~{}\Big{(}\frac{d}{d\ln\mu}-\Gamma_{\rm
cusp}(\alpha_{s})\,\frac{\partial}{\partial j}\Big{)}\,\varphi_{+}(j,\mu)$
$\displaystyle=-\bigg{\\{}\Gamma_{\rm
cusp}(\alpha_{s})\,\Big{[}\tilde{h}(1)-\tilde{h}(1+j)+\psi(j+2)-\psi(2)\Big{]}+\gamma_{+}(\alpha_{s})+\gamma_{hl}(\alpha_{s})+j\bigg{\\}}\,\varphi_{+}(j,\mu)\,,$
$\displaystyle~{}\tilde{h}(j)\equiv\tilde{h}(\alpha_{s},j)=\int^{1}_{0}dx\,\frac{\bar{x}^{j}}{x}\,h(\alpha_{s},\bar{x})\,,$
(82)

where $h(\alpha_{s},x)$ needs to be expanded in $\alpha_{s}/(4\pi)$ in the
same way as the hard function, $\psi$ is the digamma function and
$\bar{x}\equiv 1-x$. The two-loop anomalous dimension of $\gamma_{+}$ and the
two-loop level $h^{(1)}$ were calculated in Ref. [22]. Actually the explicit
form of $\gamma_{+}$ is not necessary in this work, and $h^{(1)}$ is

$\displaystyle h^{(1)}(x)=\ln x\,\Big{[}\beta_{0}+2C_{F}\,\Big{(}\ln
x-\frac{1+x}{x}\,\ln\bar{x}-\frac{3}{2}\Big{)}\Big{]}\,,$ (83)

where $\beta_{0}$ is the one-loop QCD $\beta$-function (see Appendix B).
Employing the trivial relation between the Mellin-space LCDA and the Laplace-
space LCDA [24]

$\displaystyle\varphi_{+}(j,\mu)=$
$\displaystyle~{}\frac{-i}{2\pi}\,\Gamma(-j)\,\Big{(}\frac{\hat{\omega}}{\mu
e^{\gamma_{E}}}\Big{)}^{j+1}\,(\mu
e^{\gamma_{E}})\,\tilde{\phi}_{+}(-j-1,\widehat{\omega},\mu)\,,$ (84)

one obtains the RG equation of $\tilde{\phi}_{+}$ easily [8]

$\displaystyle~{}\left(\frac{d}{d\ln\mu}+\Gamma_{\rm
cusp}(\alpha_{s})\frac{\partial}{\partial\eta}\right)\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu)$
$\displaystyle=$ $\displaystyle~{}\left\\{\Gamma_{\rm
cusp}(\alpha_{s})\left[\ln\frac{\widehat{\omega}}{\mu}+\tilde{h}(-\eta)-\psi(1+\eta)-\psi(1-\eta)-2\gamma_{E}\right]-\gamma(\alpha_{s})\right\\}\tilde{\phi}_{+}(\eta,\widehat{\omega},\mu)\,,$
(85)

where

$\displaystyle\gamma(\alpha_{s})=\gamma_{+}(\alpha_{s})+\gamma_{hl}(\alpha_{s})+\Gamma_{\rm
cusp}(\alpha_{s})\,\left[\tilde{h}(1)-1\right]\,.$ (86)

The two-loop anomalous dimension of $\gamma$ is determined by that of
$\gamma_{+}$ and $h^{(1)}$. The explicit expression for $\gamma$ is collected
in Appendix B.

The explicit solution to the RG equation (15) at $\alpha_{s}$ accuracy can be
found in Ref. [8]. We would like to derive the formal $\alpha^{2}_{s}$-level
solution, which could be employed to fulfill the NNLL resummation to the LCDA
once the three-loop anomalous dimensions of $\gamma_{+}$ and $h$ are known.
Since any combination of $\eta+a_{\Gamma}(\mu_{s},\mu)$ is a solution to (15)
with the right-hand-side terms set to zero, we separate the RG-evolution
factor into $\eta$-independent $U_{\phi 1}$ and $\eta$-dependent $U_{\phi 2}$
parts. And the combination $U_{\phi 1}(\widehat{\omega},\mu_{s},\mu)\,U_{\phi
2}(\eta,\mu_{s},\mu)\,\tilde{\phi}_{+}(\eta+a_{\Gamma}(\mu_{s}),\widehat{\omega},\mu_{s})$
will be the solution to the RG equation satisfied by $\tilde{\phi}_{+}$.

The factor $U_{\phi 1}$ satisfies the RG equation

$\displaystyle\frac{d}{d\ln\mu}\,U_{\phi 1}(\widehat{\omega},\mu_{s},\mu)=$
$\displaystyle~{}\left\\{\Gamma_{\rm
cusp}(\alpha_{s})\,\ln\frac{\widehat{\omega}}{\mu}-\Big{[}2\gamma_{E}\,\Gamma_{\rm
cusp}(\alpha_{s})+\gamma(\alpha_{s})\Big{]}\right\\}U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)\,,$ (87)

with initial condition $U_{\phi 1}(\widehat{\omega},\mu_{s},\mu_{s})=1$. The
equation is closely related to the general RG equation (13), and we have

$\displaystyle U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)=U(\widehat{\omega},\mu_{s},\mu)\Big{|}_{\gamma_{\rm
gen}\to-(\gamma+2\gamma_{E}\Gamma_{\rm cusp})}\,.$ (88)

The factor $U_{\phi 2}$ is expressed as

$\displaystyle U_{\phi 2}(\eta,\mu_{s},\mu)=$
$\displaystyle~{}\exp\bigg{[}G(\eta,\mu_{s},\mu)+\int^{a_{\Gamma}(\mu_{s})}_{0}d\hat{a}_{\Gamma}\,\Big{(}\psi(1+\eta+\hat{a}_{\Gamma})+\psi(1-\eta-\hat{a}_{\Gamma})\Big{)}\bigg{]}$
$\displaystyle=$
$\displaystyle~{}\exp\Big{[}G(\eta,\mu_{s},\mu)\Big{]}\,R_{\Gamma}(\eta,\mu_{s},\mu)\,,$
$\displaystyle G(\eta,\mu_{s},\mu)=$
$\displaystyle~{}\int\limits_{\alpha_{s}(\mu_{s})}^{\alpha_{s}(\mu)}\frac{d\alpha}{\beta(\alpha)}\,\Gamma_{\rm
cusp}(\alpha)\,\tilde{h}(\alpha,-\eta-a_{\Gamma}(\mu_{\alpha}))\,,$
$\displaystyle R_{\Gamma}(\eta,\mu_{s},\mu)=$
$\displaystyle~{}\frac{\Gamma\big{(}1+\eta+a_{\Gamma}(\mu_{s})\big{)}\,\Gamma(1-\eta)}{\Gamma\big{(}1-\eta-
a_{\Gamma}(\mu_{s})\big{)}\,\Gamma(1+\eta)}\,,$ (89)

with $\mu_{\alpha}$ satisfing $\alpha_{s}(\mu_{\alpha})\equiv\alpha$.

Having the evolution factors $U_{\phi 1}$ and $U_{\phi 2}$ at hand, the large
logarithms appearing in the LCDA $\tilde{\phi}_{+}$ can be resummed directly.
The evolution function $\tilde{\phi}_{+,U}$ is then

$\displaystyle\tilde{\phi}_{+,U}(\eta,\mu_{s},\mu)=$
$\displaystyle~{}\bigg{(}\frac{2E_{\gamma}\,\hat{\omega}}{\mu^{2}}\bigg{)}^{\eta}\,U_{\phi
1}(\widehat{\omega},\mu_{s},\mu)\,U_{\phi
2}(-\eta,\mu_{s},\mu)\,\tilde{\phi}_{+}(-\eta+a_{\Gamma}(\mu_{s}),\widehat{\omega},\mu_{s})\,.$
(90)

###  B.5 Expansion of evolution factors

For completeness, we recollect the definitions of functions entering the
evolution factors

$\displaystyle~{}a_{\gamma_{\rm
gen}}(\mu_{i},\mu)=-\int_{\alpha_{s}(\mu_{i})}^{\alpha_{s}(\mu)}d\alpha\,\frac{\gamma_{\rm
gen}(\alpha)}{\beta(\alpha)}$ $\displaystyle=$
$\displaystyle~{}\sum_{n=0}a_{\gamma_{\rm
gen}}^{(n)}\,\Big{(}\frac{\alpha_{s}(\mu_{i})}{4\pi}\Big{)}^{n}\,,$
$\displaystyle~{}S(\mu_{i},\mu)=-\int_{\alpha_{s}(\mu_{i})}^{\alpha_{s}(\mu)}d\alpha\,\frac{\Gamma_{\rm
cusp}(\alpha)}{\beta(\alpha)}\int_{\alpha_{s}(\mu_{i})}^{\alpha}\frac{d\hat{\alpha}}{\beta(\hat{\alpha})}$
$\displaystyle=$
$\displaystyle~{}\sum_{n=0}S^{(n)}\,\Big{(}\frac{\alpha_{s}(\mu_{i})}{4\pi}\Big{)}^{n}\,,$
$\displaystyle~{}G(\eta,\mu_{i},\mu)=\int\limits_{\alpha_{s}(\mu_{i})}^{\alpha_{s}(\mu)}\frac{d\alpha}{\beta(\alpha)}\,\Gamma_{\rm
cusp}(\alpha)\,\tilde{h}(\alpha,-\eta-a_{\Gamma}(\mu_{\alpha}))$
$\displaystyle=$
$\displaystyle~{}\sum_{n=0}G^{(n)}\,\Big{(}\frac{\alpha_{s}(\mu_{i})}{4\pi}\Big{)}^{n}\,,$
$\displaystyle~{}R_{\Gamma}(\eta,\mu_{i},\mu)=\frac{\Gamma\big{(}1+\eta+a_{\Gamma}(\mu_{i})\big{)}\,\Gamma(1-\eta)}{\Gamma\big{(}1-\eta-
a_{\Gamma}(\mu_{i})\big{)}\,\Gamma(1+\eta)}$ $\displaystyle=$
$\displaystyle~{}\sum_{n=0}R_{\Gamma}^{(n)}\,\Big{(}\frac{\alpha_{s}(\mu_{i})}{4\pi}\Big{)}^{n}\,,$
$\displaystyle~{}\tilde{\phi}_{+}(\eta+a_{\Gamma}(\mu_{i}),\hat{\omega},\mu_{i})$
$\displaystyle=$
$\displaystyle~{}\sum_{n=0}\tilde{\phi}_{+}^{(n)}\,\Big{(}\frac{\alpha_{s}(\mu_{i})}{4\pi}\Big{)}^{n}\,.$
(91)

The scale $\mu_{i}$ is an initial scale, which should be $\mu_{h1}$,
$\mu_{h2}$ and $\mu_{s}$ for evolution factors
$U_{1}(E_{\gamma},\mu_{h1},\mu)$, $U_{2}(\mu_{h2},\mu)$ and
$\tilde{\phi}_{+,U}(\eta,\mu_{s},\mu)$, respectively.

The expansion coefficients for $a_{\gamma_{\rm gen}}$ and $S$ are

$\displaystyle a_{\gamma_{\rm gen}}^{(0)}$ $\displaystyle=$
$\displaystyle~{}\frac{{\gamma_{0,\rm gen}}}{2\beta_{0}}\,\ln r\,,\quad
a_{\gamma_{\rm gen}}^{(1)}=\frac{1}{2\beta_{0}}\,\Big{(}{\gamma_{1,\rm
gen}}-\frac{{\gamma_{0,\rm gen}}\,\beta_{1}}{\beta_{0}}\Big{)}\,(r-1)\,,$
$\displaystyle a_{\gamma_{\rm gen}}^{(2)}$ $\displaystyle=$
$\displaystyle~{}\frac{1}{4\beta_{0}}\,\Big{(}{\gamma_{2,\rm
gen}}-\frac{{\gamma_{1,\rm gen}}\,\beta_{1}}{\beta_{0}}+\frac{{\gamma_{0,\rm
gen}}(\beta_{1}^{2}-\beta_{0}\,\beta_{2})}{\beta_{0}^{2}}\Big{)}\,(r^{2}-1)\,,$
(92) $\displaystyle S^{(0)}$ $\displaystyle=$
$\displaystyle~{}\frac{\Gamma_{0}}{4\beta_{0}^{2}}\,\Big{[}\frac{4\pi}{\alpha_{s}(\mu_{i})}\,(1-\ln
r-\frac{1}{r})+\frac{\beta_{1}}{2\beta_{0}}\ln^{2}r-\Big{(}\frac{\Gamma_{1}}{\Gamma_{0}}-\frac{\beta_{1}}{\beta_{0}}\Big{)}(r-1-\ln
r)\Big{]}\,,$ $\displaystyle S^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{0}}{8\beta_{0}^{2}}\,\Big{[}\frac{\Gamma_{2}}{\Gamma_{0}}(r-1)^{2}+\frac{\beta_{2}}{\beta_{0}}(1-r^{2}+2\ln
r)-\frac{\Gamma_{1}\beta_{1}}{\Gamma_{0}\beta_{0}}(3-4r+r^{2}+2r\ln r)$
$\displaystyle+$
$\displaystyle\frac{\beta_{1}^{2}}{\beta_{0}^{2}}(1-r)(1-r-2\ln r)\Big{]}\,,$
$\displaystyle S^{(2)}$ $\displaystyle=$
$\displaystyle~{}\frac{\Gamma_{0}}{48\beta_{0}^{3}}\,\Big{[}-\frac{2\Gamma_{3}\beta_{0}}{\Gamma_{0}}(r-1)^{2}(2r+1)-\beta_{3}\,(1+3r^{2}-4r^{3}+6\ln
r)$ (93) $\displaystyle+$
$\displaystyle\frac{\Gamma_{2}\beta_{1}}{\Gamma_{0}}\,(5-9r^{2}+4r^{3}+6r^{2}\ln
r)+\frac{\beta_{2}\beta_{1}}{\beta_{0}}\,(-7+12r+3r^{2}-8r^{3}+6(2-r^{2})\ln
r)$ $\displaystyle+$
$\displaystyle\frac{4\Gamma_{1}\beta_{2}}{\Gamma_{0}}\,(2-3r+r^{3})+\frac{\Gamma_{1}\beta_{1}^{2}}{\Gamma_{0}\beta_{0}}\,(-11+12r+3r^{2}-4r^{3}-6r^{2}\ln
r)$ $\displaystyle+$
$\displaystyle\frac{2\beta_{1}^{3}}{\beta_{0}^{2}}\,(4-6r+2r^{3}+3(r^{2}-1)\ln
r)\Big{]}\,,$

with $r=\alpha_{s}(\mu)/\alpha_{s}(\mu_{i})$. Note that $S^{(0)}$ also
includes an order $\alpha^{-1}_{s}$ term. The expansion coefficients for $G$
are

$\displaystyle G^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{0}}{2\beta_{0}}\,r\int_{0}^{1}dx\,\frac{x^{-\eta}}{1-x}\,h^{(1)}(x)\,\frac{1-r^{-l_{x}-1}}{l_{x}+1}\,,$
$\displaystyle G^{(2)}$ $\displaystyle=$
$\displaystyle~{}\frac{\Gamma_{0}}{2\beta_{0}}\,r^{2}\int_{0}^{1}dx\,\frac{x^{-\eta}}{1-x}\,\bigg{\\{}h^{(1)}(x)\,\bigg{[}\bigg{(}\frac{\beta_{1}}{\beta_{0}}-\Big{(}\frac{\Gamma_{1}}{\Gamma_{0}}-\frac{\beta_{1}}{\beta_{0}}\Big{)}\,l_{x}\bigg{)}\,\frac{1-r^{-l_{x}-2}}{l_{x}+2}$
(94) $\displaystyle-$
$\displaystyle\Big{(}\frac{\Gamma_{1}}{\Gamma_{0}}-\frac{\beta_{1}}{\beta_{0}}\Big{)}\,\frac{l_{x}\,\big{(}1-r^{-l_{x}-1}\big{)}}{l_{x}+1}\bigg{]}-h^{(2)}(x)\,\frac{1-r^{-l_{x}-2}}{l_{x}+2}\bigg{\\}}\,,$

where $l_{x}\equiv\Gamma_{0}/(2\beta_{0})\ln x$. The expansion coefficients
for $R_{\Gamma}$ and $\tilde{\phi}_{+}$ are

$\displaystyle R^{(0)}_{\Gamma}=$
$\displaystyle~{}\frac{\Gamma\big{(}1+\eta+a^{(0)}_{\Gamma}\big{)}\,\Gamma(1-\eta)}{\Gamma\big{(}1-\eta-a^{(0)}_{\Gamma}\big{)}\,\Gamma(1+\eta)}\,,\qquad
R^{(1)}_{\Gamma}=a^{(1)}_{\Gamma}\,\Big{[}\psi(1+\eta+a^{(0)}_{\Gamma}\big{)}+\psi(1-\eta-a^{(0)}_{\Gamma}\big{)}\Big{]}\,R^{(0)}_{\Gamma}\,,$
$\displaystyle R^{(2)}_{\Gamma}=$
$\displaystyle~{}\frac{a^{(2)}_{\Gamma}}{a^{(1)}_{\Gamma}}\,R^{(1)}_{\Gamma}+\frac{1}{2}\,\bigg{\\{}a^{(1)2}_{\Gamma}\,\Big{[}\psi^{(1)}(1+\eta+a^{(0)}_{\Gamma}\big{)}-\psi^{(1)}(1-\eta-a^{(0)}_{\Gamma}\big{)}\Big{]}\,+\Big{(}\frac{R^{(1)}_{\Gamma}}{R^{(0)}_{\Gamma}}\Big{)}^{2}\bigg{\\}}R^{(0)}_{\Gamma}\,,$
(95) $\displaystyle\tilde{\phi}_{+}^{(0)}=$
$\displaystyle~{}\tilde{\phi}_{+}(\eta+a^{(0)}_{\Gamma})\,,\quad\tilde{\phi}_{+}^{(1)}=a^{(1)}_{\Gamma}\,\tilde{\phi}^{\prime}_{+}(\eta+a^{(0)}_{\Gamma})\,,\quad\tilde{\phi}_{+}^{(2)}=~{}a^{(2)}_{\Gamma}\,\tilde{\phi}^{\prime}_{+}(\eta+a^{(0)}_{\Gamma})+\frac{a^{(1)2}_{\Gamma}}{2}\,\tilde{\phi}^{\prime\prime}_{+}(\eta+a^{(0)}_{\Gamma})\,.$
(96)

The expansion coefficients of $a_{\Gamma}\equiv a_{\Gamma_{\rm cusp}}$ can be
obtained from that of $a_{\gamma_{\rm gen}}$ by a replacement of the anomalous
dimension $\gamma_{\rm gen}\to\Gamma_{\rm cusp}$.

##  Appendix C Photon LCDA

We summarize the operator-level definitions of the two-particle and three-
particle photon distribution amplitudes up to and including the twist-four
accuracy, following the systematic classification detailed in [38], and taking
this opportunity to correct several misprints in the previous expressions
displayed in Appendix B of [13].

$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,\sigma_{\alpha\beta}\,\,q(0)|0\rangle$
$\displaystyle=-i\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,(p_{\beta}\,\epsilon_{\alpha}^{\ast}-p_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int_{0}^{1}\,du\,e^{i\,u\,p\cdot
x}\,\left[\chi(\nu)\,\phi_{\gamma}(u,\nu)+{x^{2}\over
16}\,\mathbb{A}(u,\nu)\right]$ $\displaystyle\hskip 8.5359pt-\,{i\over
2}\,g_{\rm em}\,Q_{q}\,{\langle\bar{q}q\rangle(\nu)\over q\cdot
x}\,(x_{\beta}\,\epsilon_{\alpha}^{\ast}-x_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int_{0}^{1}\,du\,e^{i\,u\,p\cdot
x}\,h_{\gamma}(u,\nu)\,,$ (97)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,\gamma_{\alpha}\,\,q(0)|0\rangle=g_{\rm
em}\,Q_{q}\,f_{3\gamma}(\nu)\,\epsilon_{\alpha}^{\ast}\,\int_{0}^{1}\,du\,e^{i\,u\,p\cdot
x}\,\psi^{(v)}(u,\nu)\,,$ (98)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,\gamma_{\alpha}\,\gamma_{5}\,\,q(0)|0\rangle$
$\displaystyle={g_{\rm em}\over
4}\,Q_{q}\,f_{3\gamma}(\nu)\,\varepsilon_{\alpha\beta\rho\tau}\,p^{\rho}\,x^{\tau}\,\epsilon^{\ast\,\beta}\,\int_{0}^{1}\,du\,e^{i\,u\,p\cdot
x}\,\,\psi^{(a)}(u,\nu)\,,$ (99)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,g_{s}\,G_{\alpha\beta}(v\,x)\,\,q(0)|0\rangle$
$\displaystyle=-i\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,(p_{\beta}\,\epsilon_{\alpha}^{\ast}-p_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,S(\underline{\alpha};\nu)\,,$ (100)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,g_{s}\,\widetilde{G}_{\alpha\beta}(v\,x)\,i\,\gamma_{5}\,\,q(0)|0\rangle$
$\displaystyle=i\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,(p_{\beta}\,\epsilon_{\alpha}^{\ast}-p_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,\widetilde{S}(\underline{\alpha};\nu)\,,$ (101)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,g_{s}\,\widetilde{G}_{\alpha\beta}(v\,x)\,\gamma_{\rho}\,\gamma_{5}\,\,q(0)|0\rangle$
$\displaystyle=-g_{\rm
em}\,Q_{q}\,f_{3\gamma}(\nu)\,p_{\rho}\,(p_{\beta}\,\epsilon_{\alpha}^{\ast}-p_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,A(\underline{\alpha};\nu)\,,$ (102)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,g_{s}\,G_{\alpha\beta}(v\,x)\,i\,\gamma_{\rho}\,\,q(0)|0\rangle$
$\displaystyle=-g_{\rm
em}\,Q_{q}\,f_{3\gamma}(\nu)\,p_{\rho}\,(p_{\beta}\,\epsilon_{\alpha}^{\ast}-p_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,V(\underline{\alpha};\nu)\,,$ (103)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,g_{\rm
em}\,Q_{q}\,F_{\alpha\beta}(v\,x)\,\,q(0)|0\rangle$ $\displaystyle=-i\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,(p_{\beta}\,\epsilon_{\alpha}^{\ast}-p_{\alpha}\,\epsilon_{\beta}^{\ast})\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,S_{\gamma}(\underline{\alpha};\nu)\,,$ (104)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,\sigma_{\rho\tau}\,\,g_{s}\,G_{\alpha\beta}(v\,x)\,\,q(0)|0\rangle$
$\displaystyle=\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,\left[p_{\rho}\,\epsilon_{\alpha}^{\ast}\,g_{\tau\beta}^{\perp}-p_{\tau}\,\epsilon_{\alpha}^{\ast}\,g_{\rho\beta}^{\perp}-(\alpha\leftrightarrow\beta)\right]\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,T_{1}(\underline{\alpha};\nu)$ $\displaystyle\hskip 11.38092pt+\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,\left[p_{\alpha}\,\epsilon_{\rho}^{\ast}\,g_{\tau\beta}^{\perp}-p_{\beta}\,\epsilon_{\rho}^{\ast}\,g_{\tau\alpha}^{\perp}-(\rho\leftrightarrow\tau)\right]\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,T_{2}(\underline{\alpha};\nu)$ $\displaystyle\hskip 11.38092pt+\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,\frac{(p_{\alpha}\,x_{\beta}-p_{\beta}\,x_{\alpha})(p_{\rho}\,\epsilon_{\tau}^{\ast}-p_{\tau}\,\epsilon_{\rho}^{\ast})}{p\cdot
x}\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,T_{3}(\underline{\alpha};\nu)$ $\displaystyle\hskip 11.38092pt+\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,\frac{(p_{\rho}\,x_{\tau}-p_{\tau}\,x_{\rho})(p_{\alpha}\,\epsilon_{\beta}^{\ast}-p_{\beta}\,\epsilon_{\alpha}^{\ast})}{p\cdot
x}\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,T_{4}(\underline{\alpha};\nu)\,,$ (105)
$\displaystyle\langle\gamma(p)|\bar{q}(x)\,W_{c}(x,0)\,\,\sigma_{\rho\tau}\,\,g_{\rm
em}\,Q_{q}\,F_{\alpha\beta}(v\,x)\,\,q(0)|0\rangle$ $\displaystyle=\,g_{\rm
em}\,Q_{q}\,\langle\bar{q}q\rangle(\nu)\,\frac{(p_{\rho}\,x_{\tau}-p_{\tau}\,x_{\rho})(p_{\alpha}\,\epsilon_{\beta}^{\ast}-p_{\beta}\,\epsilon_{\alpha}^{\ast})}{p\cdot
x}\,\int[{\cal
D}\underline{\alpha}]\,e^{i\,(\alpha_{q}+\,v\,\alpha_{g})\,p\cdot
x}\,T_{4}^{\gamma}(\underline{\alpha};\nu)+\cdots.$ (106)

Here, we adopt the following convention for the dual gluon-field strength
tensor

$\displaystyle\widetilde{G}_{\alpha\beta}={1\over
2}\,\varepsilon_{\alpha\beta\rho\tau}\,G^{\rho\tau}\,,\qquad\int[\mathcal{D}\underline{\alpha}]=\int
d\alpha_{q}\int d\alpha_{\bar{q}}\int
d\alpha_{g}\,\delta(1-\alpha_{q}-\alpha_{\bar{q}}-\alpha_{g})\,,$ (107)

with $\underline{\alpha}=\\{\alpha_{q},\alpha_{\bar{q}},\alpha_{g}\\}$, and
$\alpha_{q},~{}\alpha_{\bar{q}},~{}\alpha_{g}$ are momentum fractions carried
by the quark, anti-quark and gluon in the photon, respectively. It is
convenient to define $\bar{\psi}$ and $\bar{h}_{\gamma}$ by

$\displaystyle\bar{\psi}^{(v)}(u,\nu)=2\int_{0}^{u}d\hat{u}\,\psi^{(v)}(\hat{u},\nu)\,,\quad\bar{h}_{\gamma}(u,\nu)=-4\int_{0}^{u}d\hat{u}\,(u-\hat{u})\,h_{\gamma}(\hat{u},\nu)\,.$
(108)

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