11institutetext: Higgs Centre for Theoretical Physics, School of Physics and
Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland

# Charged and Neutral $\bar{B}_{u,d,s}\to\gamma$ Form Factors from
Light Cone Sum Rules at NLO

Tadeusz Janowski 1 Ben Pullin 1 Roman Zwicky<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the first analytic ${\cal O}(\alpha_{s})$-computation at
twist-$1$,$2$ of the $\bar{B}_{u,d,s}\to\gamma$ form factors within the
framework of sum rules on the light-cone. These form factors describe the
_charged_ decay $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$, contribute to the
flavour changing _neutral_ currents $\bar{B}_{d,s}\to\gamma\ell^{+}\ell^{-}$
and serve as inputs to more complicated processes. We provide a fit in terms
of a $z$-expansion with correlation matrix and extrapolate the form factors to
the kinematic endpoint by using the $g_{BB^{*}\gamma}$ couplings as a
constraint. Analytic results are available in terms of multiple polylogarithms
as ancillary files. We give binned predictions for the
$\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ branching ratio along with the
associated correlation matrix. By comparing with three SCET-computations we
extract the inverse moment $B$-meson distribution amplitude parameter
$\lambda_{B}=360(110)\textrm{\small MeV}$. The uncertainty thereof could be
improved by a more dedicated analysis. In passing, we extend the photon
distribution amplitude to include quark mass corrections with a prescription
for the magnetic vacuum susceptibility, $\chi_{q}$, compatible with the twist-
expansion. The values $\chi_{q}=3.21(15)\textrm{\small GeV}^{-2}$ and
$\chi_{s}=3.79(17)\textrm{\small GeV}^{-2}$ are obtained.

††preprint: CP3-Origins-2020-14 DNRF90

##  1 Introduction

In this work we provide the (photon) _on-shell_ form factors contributing to
semileptonic $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ and flavour changing
neutral decay (FCNC) $\bar{B}_{d,s}\to\gamma\ell^{+}\ell^{-}$ with the method
of light-cone sum rules (LCSR). This is the first complete next-to leading
order (NLO) computation in $\alpha_{s}$ including twist-$1$,$2$ and leading
order (LO) twist-$3$ and partial twist-$4$. The dynamics, contrary to naive
expectation, are more involved than for $\bar{B}\to
V(\rho,\omega,K^{*},\phi)$BB98b ; BZ04b . This comes as the point-like photon,
of leading twist-$1$, has no counterpart for the mesons and yet the photon is
to be described by a photon distribution amplitude (DA) (analogous to the
vector meson case and cf. Tab. 11 for a dictionary). The photon DA can be
interpreted as $\rho_{0},\omega_{0},\phi$-$\gamma$ conversion and is
normalised by the QCD magnetic vacuum susceptibility $\chi$, for which we
propose a prescription to include quark mass effects in Sec. 4.2.1. The
twist-$1$ part leading in the heavy quark scaling $F^{\bar{B}\to\gamma}\propto
m_{b}^{-1/2}$, differing from $F^{\bar{B}\to V}\propto m_{b}^{-3/2}$, which we
discuss in the hadronic picture in the large-$N_{c}$ limit using a dispersion
in Sec. 2.3.

In the literature the _charged_ $\bar{B}_{u}\to\gamma$ form factors have been
considered in LCSR at leading order (LO) in $\alpha_{s}$ in
Khodjamirian:1995uc ; Ali:1995uy ; Eilam:1995zv  and a partial (no twist-$1$)
NLO contribution in Ball:2003fq . The decay has received considerable
attention in QCD-factorisation and soft-collinear effective theory (SCET), in
part, as a toy model for factorisation and for its prospect to extract
parameters of the $B$-meson’s distribution amplitude parameters
Korchemsky:1999qb ; Lunghi:2002ju ; DescotesGenon:2002mw ; Bosch:2003fc ;
Beneke:2011nf ; Braun:2012kp ; Wang:2016qii ; Beneke:2018wjp ; Wang:2018wfj ;
Galda:2020epp ; Shen:2020hsp . In Sec. 5.4 we extract the $B$-meson DA
parameters $\lambda_{B}$ by comparing to a SCET evaluation in three
representative models. SCET-investigations include Korchemsky:1999qb ;
Lunghi:2002ju ; DescotesGenon:2002mw ; Bosch:2003fc  at leading power and
Beneke:2011nf  at next-to-leading log and partial sub-leading powers.
Subleading powers, (partially) corresponding to the photon DA, were accounted
for by dispersion techniques Braun:2012kp ; Wang:2016qii ; Beneke:2018wjp .
Other approaches include perturbative QCD Shen:2018abs  and a hybrid approach
of QCD factorisation and LCSR Wang:2018wfj  to which we compare in Sec. 4.3. A
first lattice form factor evaluation has been reported for
$D_{(s)},K,\pi\to\gamma$ in Desiderio:2020oej  and further evaluations of
$D_{s}(K)\to\gamma$ are in progress Kane:2019jtj .

The _neutral_ $\bar{B}_{d,s}\to\gamma$ form factors describe flavour changing
neutral currents which are generic probes for physics beyond the Standard
Model. The activity in this particular corner has been revived, since it has
been pointed out Dettori:2016zff  that, in a certain kinematic region,
$\bar{B}_{s}\to\gamma\mu^{+}\mu^{-}$ may be extracted from the very same
dataset as $\bar{B}_{s}\to\mu^{+}\mu^{-}$ at LHC experiments Aaij:2017vad ;
Aaboud:2018mst ; CMS:2019qnb  (cf. also Aditya:2012im ). This triggered a
number of phenomenological investigations GRZ17 ; Kozachuk:2017mdk ;
Beneke:2020fot ; Carvunis:2021jga . Previously these form factors had been
considered, in LCSR at LO Aliev:1996ud , in Kruger:2002gf  using aspects of
heavy quark symmetry (with input from Beyer:2001zn ), quark model computations
MN04 ; Kozachuk:2017mdk  and more recently in SCET Beneke:2020fot . In
addition, in this decay, the dipole operator
$m_{b}\bar{q}_{L}\sigma\\!\cdot\\!Fb$ necessitates photon _off-shell_ form
factors. These have been investigated for the first time, at LO with QCD sum
rules Albrecht:2019zul , utilising the on-shell form factors of this paper as
an input.

The paper is organised as follows. In Sec. 2 the form factors are defined and
their heavy quark scaling is discussed in a hadronic picture. In Sec. 3 the
correlation functions entering the computation, Ward identities and contact
terms (CTs ) are discussed extensively. The evaluation at LO and NLO for the
perturbative and soft parts is documented in Sec. 4. The numerics Sec. 5
contains our main practical results in terms of fits, plots, the extraction of
the inverse moment of the $B$-meson DA and the binned prediction for the
$\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ rate in Secs. 5.1, 5.2, 5.3 and 5.4
respectively. The paper ends with summary and conclusions in Sec. 6.
Appendices include conventions and comments on form factor definitions (Apps.
A), gauge invariance of the amplitude (App. B), coordinate space equation of
motion (including their non-trivial renormalisation) (App. C), a compendium of
photon DAs (App. D), results of correlation functions (App. E) and
discontinuities of multiple-polylogarithms in (App. F).

##  2 Generalities on the $\bar{B}\to\gamma$ Form Factors

The section provides the method-independent information on the $B\to\gamma$
form factors (FFs), such as the definition and the heavy quark scaling.

###  2.1 Definition of the Form Factors

Amongst the dimension six operators of the $b\to q$ effective Hamiltonian
there are the standard vector and tensor operators. Note that the scalar
operator vanishes by parity conservation of QCD and the pseudoscalar one only
contributes for an off-shell photon (cf. reference Albrecht:2019zul  for a
complete discussion). We consider, in addition, an operator which is redundant
by the equation of motion (EOM) but that will prove useful as a check and is
of interest per se Hambrock:2013zya ; BSZ15 . In complete generality, with
conventions as in GRZ17 ; Albrecht:2019zul , extended to include the charged
case, we define the FFs111It is important to keep in mind that the results
quoted throughout are for the $\bar{B}$-meson. For example, by applying a
charge $C$\- or $CP$-transformation, one infers that the vector (but not the
axial) parts change sign $V,T_{\perp}\to-V,T_{\perp}$ (assuming
$C|\bar{B}\rangle=|B\rangle$ and or $CP|\bar{B}\rangle=|B\rangle$) when
passing from the $\bar{B}$\- to the $B$-FFs. The adaption to generic phases
under $C$ and $CP$ is straightforward.

$\displaystyle\langle\gamma(k,\epsilon)|O^{V}_{\mu}|\bar{B}_{q}(p_{B})\rangle$
$\displaystyle\;=\;$ $\displaystyle s_{e}(P^{\perp}_{\mu}\,V_{\perp}(q^{2})$
$\displaystyle\;-\;$ $\displaystyle
P^{\parallel}_{\mu}\,(V_{\parallel}(q^{2})+Q_{\bar{B}_{q}}f^{(pt)}_{B_{q}})-P^{\textrm{Low}}_{\mu}Q_{\bar{B}_{q}}f^{(pt)}_{B_{q}})~{},$
$\displaystyle\langle\gamma(k,\epsilon)|O^{T}_{\mu}|\bar{B}_{q}(p_{B})\rangle$
$\displaystyle\;=\;$ $\displaystyle s_{e}(P^{\perp}_{\mu}\,T_{\perp}(q^{2})$
$\displaystyle\;-\;$ $\displaystyle
P^{\parallel}_{\mu}\,T_{\parallel}(q^{2}))~{},$ (1)
$\displaystyle\langle\gamma(k,\epsilon)|O^{\cal
D}_{\mu}|\bar{B}_{q}(p_{B})\rangle$ $\displaystyle\;=\;$ $\displaystyle
s_{e}(P_{\perp}^{\mu}\,{\cal D}_{\perp}(q^{2})$ $\displaystyle\;-\;$
$\displaystyle P_{\parallel}^{\mu}\,({\cal
D}_{\parallel}(q^{2})+\hat{m}_{-}\,Q_{\bar{B}_{q}}f^{(pt)}_{B_{q}})-P^{\textrm{Low}}_{\mu}\hat{m}_{-}\,Q_{\bar{B}_{q}}f^{(pt)}_{B_{q}})\;,$

where $\hat{m}_{\pm}\equiv m_{\pm}/m_{B_{q}}$, $m_{\pm}\equiv(m_{b}\pm m_{q})$
and $Q_{\bar{B}_{q}}=Q_{b}-Q_{q}$ is the charge of the $\bar{B}_{q}$-meson.
Comparison with literature-conventions are deferred to the next section.
Hereafter,

$f^{(pt)}_{B_{q}}=\frac{m_{B_{q}}f_{B_{q}}}{k\cdot
p_{B}}=\frac{2f_{B_{q}}/m_{B_{q}}}{1-q^{2}/m_{B_{q}}^{2}}$ (2)

denotes the point-like (or scalar QED) contribution, proportional to the decay
constant, and consists of the infrared (IR) sensitive Low-term, which captures
the gauge variance of the matrix element on the LHS (for charged
$\bar{B}_{q}$) and assures that the FFs themselves are gauge invariant (cf.
Sec. 3.2 and App. B.1). The gauge variance of the matrix element is cancelled
in the full $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ amplitude when the photon-
emission from the lepton is added cf. App. B.2. The same terms appears in
$P_{\parallel}^{\mu}$-structure whose subtraction from the FF was stressed in
Carrasco:2015xwa ; Desiderio:2020oej  cf. Sec. 2.2. Furthermore, $s_{e}$ is
the sign-convention of the covariant derivative (cf. App. A), $p_{B}=q+k$, on-
shell momenta $p_{B}^{2}=m_{B}^{2}$, $k^{2}=0$, $f_{B_{q}}$ the decay constant

$\langle
0|\bar{q}\gamma^{\mu}\gamma_{5}b|\bar{B}_{q}(p_{B})\rangle=ip_{B}^{\mu}f_{B_{q}}\;,$
(3)

local operators (describing the $b\to q$ transition)

$O^{V}_{\mu}\equiv-\frac{1}{e}m_{B_{q}}\bar{q}\gamma_{\mu}(1-\gamma_{5})b\;,\quad
O^{T}_{\mu}\equiv\frac{1}{e}\bar{q}iq^{\nu}\sigma_{\mu\nu}(1+\gamma_{5})b\;,\quad
O^{\cal
D}_{\mu}\equiv\frac{1}{e}\bar{q}(i\\!\overset{\leftrightarrow}{D})_{\mu}(1+\gamma_{5})b\;,$
(4)

and Lorentz structures ($\varepsilon_{0123}=1$ convention,
$e=\sqrt{4\pi\alpha}>0$ and
$P^{\perp,\parallel,\textrm{Low}}_{\mu}\equiv\epsilon^{*\rho}P^{\perp,\parallel,\textrm{Low}}_{\mu\rho}$)

$P^{\perp}_{\mu\rho}\equiv\varepsilon_{\mu\rho\beta\gamma}(p_{B})^{\beta}k^{\gamma}\;,\quad
P^{\parallel}_{\mu\rho}\equiv i\,(p_{B}\cdot
k\,g_{\mu\rho}-k_{\mu}\,(p_{B})_{\rho})\;,\quad
P^{\textrm{Low}}_{\mu\rho}\equiv i(p_{B})_{\mu}(p_{B})_{\rho}\;.$ (5)

The Lorentz structures are transverse, $q\cdot P^{\perp,\parallel}=0$, due to
the $\lambda=\pm 1$-helicity of the photon. As usual the Lorenz gauge,
$k\cdot\epsilon(\lambda,k)=0$, is assumed and $\epsilon\to\epsilon+k$ is the
residual gauge invariance encoded as $P^{\perp,\parallel}|_{\epsilon\to k}=0$.
By writing the Lorentz decomposition of
$\langle\gamma(k,\epsilon)|\,\bar{q}i\sigma^{\mu\nu}(1\\!+\\!\gamma_{5})b\,|\bar{B}_{q}(p_{B})\rangle$
and using the algebraic relation (A.2) the FF relation

$T_{\perp}^{B\to\gamma}(0)=T_{\parallel}^{B\to\gamma}(0)\;,$ (6)

can be shown to hold, in complete analogy to the $B\to V$ FFs, $T_{1}^{B\to
V}(0)=T_{2}^{B\to V}(0)$.

A short comment on the $\perp,\parallel$ subscripts. Interpreting the FF as a
$\bar{B}\to 1^{-(+)}\gamma$ transition, $\perp(\parallel)$ correspond to
perpendicular (parallel) $3$-dimensional polarisation vectors of the photon
and the $1^{-(+)}$ state (as can be seen from (5)). In the $B\to V\ell\ell$
literature, e.g. Kou:2018nap , the $\perp,\parallel$ combinations are known as
the transversity amplitudes and are useful because of definite CP-parities.

####  2.1.1 Form Factor Conventions in the Literature

We wish to explain why we have adopted a different notation w.r.t. to some of
the literature. There are three reasons: firstly, in the off-shell case there
are $7$ FFs Albrecht:2019zul  for which the $F_{A,V}$ notation becomes
inefficient. Second, the subtraction of the point-like term in the charged
case following Carrasco:2015xwa ; Desiderio:2020oej  (cf. Fig. 9 for
comparison). Third, in GRZ17  the sign of the FFs was chosen such that the
$\bar{B}_{d,s}\to\gamma$ FFs are positive, and since the spectator quark is
dominant this implies that the $\bar{B}_{u}\to\gamma$ FFs are negative. The
latter explains the sign difference

$V^{\bar{B}_{u}\to\gamma}_{\perp}=-F^{\bar{B}_{u}\to\gamma}_{V}\;,\quad
V^{\bar{B}_{u}\to\gamma}_{\parallel}+Q_{\bar{B}_{u}}f^{(pt)}_{\bar{B}_{u}}=-F^{\bar{B}_{u}\to\gamma}_{A}\;,$
(7)

w.r.t. to the SCET-literature with focus on the charged transition. Compared
with Desiderio:2020oej  we find

$V^{D_{s}\to\gamma}_{\perp}=-F^{D_{s}\to\gamma}_{V}\;,\quad
V^{D_{s}\to\gamma}_{\parallel}=-F^{D_{s}\to\gamma}_{A}\;,$ (8)

assuming their translation to the SCET-literature. Essentially these are the
same conventions as in (7) with the point-like part separated, which hopefully
will become the new standard. When comparing with MN04 ; Kozachuk:2017mdk  we
believe the following map should be used

$V^{\bar{B}_{d,s}\to\gamma}_{\perp,\parallel}=F^{\bar{B}_{d,s}\to\gamma}_{V,A}\;,\quad
T^{\bar{B}_{d,s}\to\gamma}_{\perp,\parallel}=FT^{\bar{B}_{d,s}\to\gamma}_{V,A}\;.$
(9)

Note that in comparing analytic results one needs to keep in mind that the
signs of $\chi$ and $s_{e}$ (and $s_{g}$) are author-dependent as well.

###  2.2 $B\to\gamma$ Form Factor in the context of QED-corrections to
leptonic decays

Since the $B\to\gamma$ FFs are the real emission corrections to the leptonic
decay $B^{+}\to\ell^{+}\nu$ or part of the FCNC $B_{d,s}\to\ell^{+}\ell^{-}$,
we consider it worthwhile to clarify a few aspects with regards to QED-
corrections to leptonic decays: the point-like term indicated in (2.1), the
absence $\ln m_{\ell}$-terms, the importance of the kinematic endpoint and
extrapolations. The arguments below go beyond $B$-physics since infrared
effects are rather universal.

In the charged case the photon couples to the charged $B$-meson, there is a
point-like or monopole contribution which can be identified by matching the
axial current $A_{5\mu}=\bar{q}\gamma_{\mu}\gamma_{5}b$ to a meson current
$A_{5\mu}\to-f_{B}D_{\mu}\bar{B}^{\dagger}$, with $\langle
0|B^{\dagger}(x)|\bar{B}\rangle=e^{-ip_{B}\cdot x}$ such that (3) holds. One
then easily finds

$\langle\gamma|A_{5\mu}|{\bar{B}}\rangle=f_{B}(-ies_{e}Q_{\bar{B}})\epsilon^{*\rho}\left(g_{\rho\mu}+\frac{(2p_{B}-k)_{\rho}q_{\mu}}{2k\cdot
p_{B}-k^{2}}\right)\;,$ (10)

where the metric term comes from the photon field in the covariant derivative.
Upon using $k^{2}=0$ and $\epsilon^{*}\cdot k=0$ one recovers the form in
(2.1).

It seems worthwhile to point out that leptonic decays are free from hard-
collinear logs of the form ${\cal O}(\alpha\ln m_{\ell})$, as these have been
shown to vanish from structure-dependent corrections by gauge invariance (cf.
section 3.4 in Isidori:2020acz ). Thus they can only arise from the point-like
contribution which is itself free from $\ln m_{\ell}$-terms. This can be seen
as follows. In the real emission the collinear terms would arise from the Low-
term which is proportional to the helicity-suppressed LO term and therefore
the logs are of the form $m^{2}_{\ell}\ln m_{\ell}$ at worst. Finally, since
real and virtual point-like contributions cancel each other in the photon-
inclusive rate the virtual corrections must be free from such terms
altogether. This completes the argument.

In the actual measurement of $B^{+}\to\ell^{+}\nu$ the undetected photon is
soft and $q^{2}$ is very close to the kinematic endpoint, which is beyond the
limit of the direct LCSR computation. This underlines the importance of the
extrapolation as discussed in Sec. 5.1.2 (based on the $g_{BB*\gamma}$
couplings Pullin:2021ebn ). A more direct sum rule analysis or the lattice
approaches mentioned earlier are of course an alternative thereto. We will
report on the impact of $B_{s}\to\gamma\ell^{+}\ell^{-}$ as a background to
$B_{d}\to\ell^{+}\ell^{-}$ elsewhere.

###  2.3 Heavy Quark Scaling of Form Factors

The heavy quark scaling, $m_{b}\to\infty$, of the FFs has been of wide
interest in connection with factorisation theorems. In some cases, such as for
the $B$-meson decay constant $f_{B_{q}}\sim m_{b}^{-1/2}$ (3), the scaling can
be traced back to the (relativistic) normalisation fo the $B$-meson state. For
the $B\to V$ FFs the same reasoning would lead to $m_{b}^{-1/2}$ but is
altered to $m_{b}^{-3/2}$ by the dynamics of a fast moving $s$-quark, released
by the heavy $b$, combining with a slow-moving spectator quark $\bar{q}$.
However, the $\bar{B}\to\gamma$ FFs scale like $m_{b}^{-1/2}$ because the
perturbative photon does not need to hadronise from two asymmetric
quarks.222Note, that the photon DA part, of the $B\to\gamma$ FF, scales like
$m_{b}^{-3/2}$, as the computation is formally analogous to the one of $B\to
V$ (cf. Tab. 11).

The question we would like to pose here is whether this scaling can be
understood from a hadronic picture. In the heuristic discussion put forward
below we argue that the scaling can be understood from a certain number of low
lying vector resonances coupling to the strange part of the electromagnetic
current. For this purpose it is useful to consider the photon off-shell FF at
first, $k^{2}\neq 0$, and write a dispersion relation in the photon momentum

$T^{*}_{\perp}(q^{2},k^{2})=\frac{1}{\pi}\int_{\textrm{cut}}\\!\\!\\!du\frac{\textrm{Im}_{u}[T^{*}_{\perp}(q^{2},u)]}{u-k^{2}-i0}+\textrm{s.t.}\;,$
(11)

where “s.t.” stands for subtraction terms and by
$T_{\perp}(q^{2})=\lim_{k^{2}\to 0}T^{*}_{\perp}(q^{2},k^{2})$ the on-shell FF
is recovered. It is convenient to assume the limit of large colours,
$N_{c}\to\infty$, for which the spectrum is entirely described by vector
mesons of zero width and the above dispersion representation assumes the
form333The dispersion integral needs one subtraction, cf. App. A.2.
Albrecht:2019zul , but in what follows we will ignore since it is consistent
with our assumption that very large terms in the sum do not affect the leading
power behaviour.

$T^{\bar{B}_{s}\to\gamma}_{\perp}(q^{2})=(\sum_{V_{n}=\phi,\phi^{\prime},\dots}\frac{f^{\textrm{e}m}_{V_{n}}}{m_{V_{n}}}\,T_{1}^{\bar{B}_{s}\to{V_{n}}}(q^{2})+\textrm{s.t.})+{\cal
O}(Q_{b})\;,$ (12)

where $f^{\textrm{em}}_{\phi}=Q_{s}f_{\phi}$ etc, with $f_{\phi}$ the standard
decay constant cf. Albrecht:2019zul  and we have separated the $Q_{b}$-term
which is suppressed by one power in $1/m_{b}$ (cf. Tab. 3). Up to a certain
number of resonances $\bar{n}$, where $m_{V_{\bar{n}}}\ll m_{B}$ holds, every
single term in (12) scales like $m_{b}^{-3/2}$. It is difficult to say
anything precise for the state above $\bar{n}$ since by assumption the LC-OPE
technique does not apply to this case and we cannot infer the behaviour in $n$
of the form factor. However, it would be rather surprising if this
contribution was leading in the heavy quark limit since the finite sum
contribution gives a result compatible with no heavy quark suppression other
than kinematic factors. How does the sum of terms from $n=1$ to $n=\bar{n}$
affect the overall scaling? In the limit of large colours the Regge ansatz
($n=1,2,3,\dots$) $m_{V_{n}}^{2}\propto n$ with $f_{V_{n}}\propto n^{0}$ is
phenomenologically successful in that it reproduces the correct logarithmic
behaviour of the vacuum polarisation Shifman:2000jv . From explicit
computation we know that $T_{1}^{\bar{B}_{s}\to V_{n}}\propto n^{0}$ thus
every summand scales like $n^{-1/2}m_{b}^{-3/2}$. If we further assume that
every summand is of the same sign then the sum (12) behaves like

$T^{\bar{B}_{s}\to\gamma}_{\perp}(q^{2})\propto\sum^{{\bar{n}}}_{n=1}\frac{f^{\textrm{e}m}_{V_{n}}}{m_{V_{n}}}\,T_{1}^{\bar{B}_{s}\to{V_{n}}}(q^{2})\propto\,m_{b}^{-3/2}\sqrt{\bar{n}}\;.$
(13)

It remains to determine how $\bar{n}$ scales with $m_{b}$. Let us imagine to
stretch $m_{b}$ by a factor of $\lambda$, $m_{b}\to\lambda m_{b}$, then
$m_{V_{\bar{n}}}\to\lambda m_{V_{\bar{n}}}$ (but $\Lambda_{\textrm{QCD}}$ and
$m_{V_{\bar{n}}}/m_{B}$ fixed), then implies $\bar{n}\to\lambda^{2}\bar{n}$
and thus $\bar{n}\propto m_{b}^{2}$. Hence from (13) we finally conclude our
anticipated result of

$T^{\bar{B}_{s}\to\gamma}_{\perp}(q^{2})\propto m_{b}^{-1/2}\;.$ (14)

The argumentation for the other three FFs is completely analogous. This is a
satisfactory result in that it matches our explicit results but remains
heuristic mainly since we are not able to make a precise statement about the
$n>\bar{n}$-terms other than that they are unlikely to change the scaling (cf.
above).

##  3 The $\bar{B}\to\gamma$ Form Factors from LCSR

The method of QCD sum rules SVZ79I ; SVZ79II  was originally established for
two-point correlation functions with a local OPE. LCSR were introduced
Balitsky:1997wi  to describe $\Sigma^{+}\to p\gamma$ FFs. It has been
understood that heavy-to-light FFs are to be described by LCSR rather than
$3$-point functions with local OPE Braun:1997kw . We refer the reader to
Colangelo:2000dp  for a review on practical aspects of sum rules.

###  3.1 Definition of the Correlation Function appropriate for the Sum Rule

In the LCSR the $B$-meson is interpolated by a current, with standard choice

$J_{B_{q}}\equiv(m_{b}+m_{q})\bar{b}i\gamma_{5}q\;,\quad\langle\bar{B}_{q}|J_{B_{q}}|0\rangle=m_{B_{q}}^{2}f_{B_{q}}\;,$
(15)

and the FF is extracted from the correlation function

$\displaystyle\Pi^{\Gamma}_{\mu\rho}(p_{B},q)\equiv\;$ $\displaystyle
i\int_{x}e^{-ip_{B}\cdot
x}\langle\gamma(k,\rho)|TJ_{B_{q}}(x)O_{\mu}^{\Gamma}(0)|0\rangle$ (16)
$\displaystyle=\;$ $\displaystyle
s_{e}\left(P^{\perp}_{\mu\rho}\,\Pi^{\Gamma}_{\perp}(p_{B}^{2},q^{2})-P^{\parallel}_{\mu\rho}\,\Pi^{\Gamma}_{\parallel}(p_{B}^{2},q^{2})+(p_{B})_{\mu}(p_{B})_{\rho}X^{\Gamma}+g_{\mu\rho}k\cdot
p_{B}Z^{\Gamma}+{\cal O}(k_{\rho})\right)\;.$

Above and hereafter we use the shorthand $\int_{x}=\int d^{4}x$ and
$\Gamma=V,T,{\cal D}$ is the index associated with the operators (4). The mass
dimensions of the scalar functions are: $[\Pi^{V,T,{\cal D}}]=1$. The index
$\rho$ is to be contracted by $\epsilon^{*\,\rho}$ and the structures
$X^{\Gamma}$ and $Y^{\Gamma}$ are related to the CTs and can be unambiguously
identified from the QED- and or axial-Ward identity (WI), to be discussed in
Sec. 3.2.

The LCSR is then obtained by evaluating (16), in the light-cone OPE, and
equating it to a dispersion representation. For $\Pi^{V}_{\perp}$ the LCSR
reads

$\Pi^{V}_{\perp}(p_{B}^{2},q^{2})=\frac{1}{\pi}\int_{0}^{\infty}ds\frac{\textrm{Im}[\Pi^{V}_{\perp}(s,q^{2})]}{s-p_{B}^{2}-i0}=\frac{V_{\perp}(q^{2})m_{B_{q}}^{2}f_{B_{q}}}{m_{B_{q}}^{2}-p_{B}^{2}-i0}+\dots\;,$
(17)

where the narrow width approximation for the $B$-meson has been assumed and
the ellipses correspond to heavier resonances and multiparticle states. The
FFs are then extracted via the standard procedures of Borel transformation and
approximating the heavier states by the perturbative integral, which is
exponentially suppressed due to the Borel transform

$V_{\perp}(q^{2})=\frac{1}{m_{B_{q}}^{2}f_{B_{q}}}\int_{m_{b}^{2}}^{s_{0}}ds\,e^{(m_{B_{q}}^{2}-s)/M^{2}}\rho_{\perp}^{V}(s,q^{2})\;.$
(18)

Above $\pi\rho_{\perp}^{V}(s,q^{2})=\textrm{Im}[\Pi^{V}_{\perp}(s,q^{2})]$ and
$M^{2}$ is the Borel mass, with some minimal detail given in App. A.1. If we
were able to compute $\rho_{\perp}^{V}$ exactly then $V_{\perp}(q^{2})$,
obtained from (18), would be independent of the Borel mass and its variation
serves as a quality measure of the sum rule. The other FFs are obtained in
exact analogy. Hereafter we will abbreviate $m_{B_{q}}=m_{B}$ and
$f_{B_{q}}=f_{B}$ unless otherwise stated.

###  3.2 QED and Axial Ward Identity

This section can be skipped by the casual reader as it deals with the
technical issue of how to project on the FFs and the appearance of the
$Q_{\bar{B}}$-terms in (2.1).

Generally on-shell matrix elements can be extracted as pole residues from
correlation functions. However for three-point functions, with a current
insertion for the photon, this procedure is complicated by CTs which manifest
themselves in the axial- and QED-WI. Whereas these matters have been discussed
previously in the literature Khodjamirian:2001ga , our presentation is more
complete in that we discuss the derivative and tensor case as well and work
purely at the level of correlation functions.

A general decomposition of the correlator (16) with electromagnetic current
insertion reads

$\displaystyle\Pi_{\mu\rho}^{\Gamma}(p_{B},q)$ $\displaystyle\equiv$
$\displaystyle\\!\\!s_{e}e\int_{x,y}e^{-ip_{B}\cdot x}e^{ik\cdot y}\langle
0|Tj_{\rho}(y)J_{B}(x)O_{\mu}^{\Gamma}(0)|0\rangle=s_{e}\left(P^{\perp}_{\mu\rho}\,\Pi^{\Gamma}_{\perp}\right.-P^{\parallel}_{\mu\rho}\,\Pi^{\Gamma}_{\parallel}+$
(19)
$\displaystyle\\!\\!\left.(p_{B})_{\mu}(p_{B})_{\rho}X^{\Gamma}+k_{\mu}(p_{B})_{\rho}Y^{\Gamma}+g_{\mu\rho}(k\\!\cdot\\!p_{B})Z^{\Gamma}+k_{\mu}k_{\rho}W^{\Gamma}+(p_{B})_{\mu}k_{\rho}U^{\Gamma}\right)\;,$

where the $p_{B}^{2}$ and $q^{2}$ dependence on the right-hand-side (RHS) is
suppressed for brevity. Compared to (16) the three structures $Y^{\Gamma}$,
$W^{\Gamma}$ and $U^{\Gamma}$ are introduced. The $U$\- and $W$-terms are
irrelevant after contraction with $\epsilon(k)^{\rho}$, as they vanish, and
will hereafter be ignored. It is the resolution of the ambiguity between
$Y$-$Z$ and $\Pi^{\Gamma}_{\parallel}$ that needs addressing.

One can derive a QED WI directly from the path-integral by varying
$q(x)\to(1+\omega(x)iQ_{q})q(x)$ and demanding invariance under $\omega(x)$,
similar to the derivation of the EOM given in App. C.1. For the QED WI we find

$\displaystyle k^{\rho}\Pi_{\mu\rho}^{V}$ $\displaystyle\;=\;$ $\displaystyle
s_{e}Q_{\bar{B}}\left(C^{A}_{\mu}(q)-C^{A}_{\mu}(p_{B})\right)m_{B}\;,$
$\displaystyle k^{\rho}\Pi_{\mu\rho}^{D}$ $\displaystyle\;=\;$ $\displaystyle
s_{e}\left(Q_{\bar{B}}\left(C^{A}_{\mu}(q)-C^{A}_{\mu}(p_{B})\right)m_{-}-\overline{Q}k_{\mu}C^{P}(p_{B}^{2})\right)\;,$
$\displaystyle k^{\rho}\Pi_{\mu\rho}^{T}$ $\displaystyle\;=\;$ $\displaystyle
0\;,$ (20)

and for the axial WI ($q(x)\to(1+\omega(x)i\gamma_{5})q(x)$) one obtains

$\displaystyle q^{\mu}\Pi_{\mu\rho}^{V}$ $\displaystyle\;=\;$ $\displaystyle
s_{e}\left(\Pi^{P}_{\rho}+Q_{\bar{B}}{C}^{A}_{\rho}(p_{B})\right)m_{B}\;,$
$\displaystyle q^{\mu}\Pi_{\mu\rho}^{D}$ $\displaystyle\;=\;$ $\displaystyle
s_{e}\left((\Pi^{P}_{\rho}+Q_{\bar{B}}{C}^{A}_{\rho}(p_{B}))m_{-}-\overline{Q}q_{\mu}C^{P}(p_{B}^{2})\right)\;,$
$\displaystyle q^{\mu}\Pi_{\mu\rho}^{T}$ $\displaystyle\;=\;$ $\displaystyle
0\;,$ (21)

where $\overline{Q}\equiv(Q_{b}+Q_{q})$ and the other terms on the RHS are

$\displaystyle\Pi^{P}_{\rho}(p_{B},k)$ $\displaystyle\;=\;$ $\displaystyle
i\int_{x,y}e^{-ip_{B}\cdot x}e^{ik\cdot y}\langle
0|Tj_{\rho}(y)J_{B_{q}}(x)\bar{q}\gamma_{5}b(0)|0\rangle\;,$ $\displaystyle
C^{A}_{\mu}(p)$ $\displaystyle\;=\;$ $\displaystyle i\int_{x}e^{-ip\cdot
x}\langle
0|TJ_{B_{q}}(x)\bar{q}\gamma_{\mu}\gamma_{5}(0)b|0\rangle=p_{\mu}C^{A}(p^{2})\;,$
$\displaystyle C^{P}(p^{2})$ $\displaystyle\;=\;$ $\displaystyle
i\int_{x}e^{-ip\cdot x}\langle
0|TJ_{B_{q}}(x)\bar{q}\gamma_{5}b(0)|0\rangle\;,$ (22)

and
$p^{2}C^{A}(p^{2})=(m_{b}+m_{q})C^{P}(p^{2})+m_{b}\langle\bar{b}b\rangle+m_{q}\langle\bar{q}q\rangle$.
Moreover, we have used

$\displaystyle m_{-}\int_{x}e^{-ip\cdot x}\langle
0|TJ_{B_{q}}(x)q\gamma_{\mu}\gamma_{5}(0)b|0\rangle=\int_{x}e^{-ip\cdot
x}\langle
0|TJ_{B_{q}}(x)\bar{q}(i\\!\overset{\leftrightarrow}{D})_{\mu}(0)\gamma_{5}b|0\rangle\;,$
$\displaystyle\int_{y}e^{ik\cdot y}\langle
0|Tj_{\rho}(y)(\bar{q}q+\bar{b}b)(0)|0\rangle=0\;.$ (23)

where the former follows from (C.3) upon omission of the QED current and the
latter follows from gauge invariance. Note that for the tensor part all lower
point correlation function vanish since there are not enough vectors to impose
the antisymmetry in the Lorentz vectors. The correction term proportional to
the sum of charges is due to the derivative term in the $O^{D}_{\mu}$-operator
cf. (C.3). Consistency,
$q^{\mu}k^{\rho}\Pi^{V}_{\mu\rho}|_{\mbox{\eqref{eq:GWI}}}=q^{\mu}k^{\rho}\Pi^{V}_{\mu\rho}|_{\mbox{\eqref{eq:AWI}}}$,
follows upon using
$k^{\rho}\Pi^{P}_{\rho}=Q_{\bar{B}}(C^{A}(q^{2})-C^{A}(p_{B}^{2}))$.

From the decomposition (19) and the QED WI one infers that

$\displaystyle X^{T}=0\;,$ $\displaystyle Y^{T}=0\;,$ $\displaystyle\quad
Z^{T}=0\;,$ (24) $\displaystyle X^{V}=\frac{Q_{\bar{B}}}{k\cdot
p_{B}}(C^{A}(q^{2})-C^{A}(p_{B}^{2}))m_{B}\;,\quad$ $\displaystyle
Y^{V}+Z^{V}$ $\displaystyle\;=\;$ $\displaystyle-\frac{Q_{\bar{B}}}{k\cdot
p_{B}}\,C^{A}(q^{2})m_{B}\;,$ $\displaystyle X^{D}=\frac{Q_{\bar{B}}}{k\cdot
p_{B}}(C^{A}(q^{2})-C^{A}(p_{B}^{2}))m_{-}\;,\quad$ $\displaystyle
Y^{D}+Z^{D}$ $\displaystyle\;=\;$ $\displaystyle-\frac{Q_{\bar{B}}}{k\cdot
p_{B}}\,C^{A}(q^{2})m_{-}+\frac{\overline{Q}}{k\cdot
p_{B}}C^{P}(p_{B}^{2})\;.$

The following three points are worth clarifying:

  * •

The choices $k\\!\cdot\\!p_{B}(Y^{V},Z^{V})=(0,-Q_{\bar{B}}\,C^{A}(q^{2}))$
and
$k\\!\cdot\\!p_{B}(Y^{D},Z^{D})=(0,-Q_{\bar{B}}\,C^{A}(q^{2})+\overline{Q}C_{P}(p_{B}^{2}))$
are compatible with the WI (24) and would not contribute to the $\|$-FFs.

  * •

The $X^{V}$-term is the analogue of the Low-term in (2.1). This can be seen by
rewriting the correlator

$\displaystyle\epsilon^{*\rho}\Pi_{\mu\rho}^{V}(p_{B},q)$ $\displaystyle\;=\;$
$\displaystyle-
s_{e}\big{(}P^{\parallel}_{\mu}\,\Pi^{V}_{\parallel}(p_{B}^{2},q^{2})\;+\;iP^{\textrm{Low}}_{\mu}\frac{Q_{\bar{B}}}{k\cdot
p_{B}}(C^{A}(q^{2})-C^{A}(p_{B}^{2}))m_{B}+\dots\;\big{)}$
$\displaystyle\;=\;$ $\displaystyle-
s_{e}\frac{m_{B}^{2}f_{B}}{m_{B}^{2}-p_{B}^{2}}\big{(}P^{\parallel}_{\mu}\,(V_{\parallel}(q^{2})+Q_{\bar{B}}f^{(pt)}_{B_{q}})\;+\;P^{\textrm{Low}}_{\mu}Q_{\bar{B}}f^{(pt)}_{B_{q}}+\dots\big{)}\;,$
(25)

using the dispersion representation (17) and

$C^{A}(p_{B}^{2})=\frac{im_{B}^{2}f_{B}^{2}}{m_{B}^{2}-p_{B}^{2}}+\dots\;.$
(26)

This derivation becomes exact, reproducing (2.1), when applying the LSZ-limit

${\lim_{p_{B}^{2}\to m_{B}^{2}}(p_{B}^{2}-m_{B}^{2})}$ to the correlation
function.

  * •

The analogous $X^{\cal D}$-term can be treated in the very same way. The
missing information is the matrix element

$\langle 0|O^{D}_{\mu}|\bar{B}_{q}\rangle=f^{D}_{B}(p_{B})_{\mu}\;,$ (27)

which is related to $f_{B}$ by the EOM (C.1) implying $f^{D}_{B}=-f_{B}$. When
factors are arranged one reproduces the CT in (2.1).

In practice, $Y^{\Gamma}=0$, meaning that

$\epsilon^{*\rho}\Pi_{\mu\rho}^{\Gamma}=s_{e}((i\epsilon^{*}\\!\cdot\\!p_{B})\Pi^{\Gamma}_{\parallel}k_{\mu}+A\epsilon_{\mu}^{*}+B(p_{B})_{\mu})\;,$
(28)

is the correct recipe for projecting on $\Pi^{\Gamma}_{\parallel}$. It should
be added that the entire discussion of these CTs can be avoided if one were to
consider directly a physical amplitude, as then invariance under
$\epsilon\to\epsilon+k$ holds cf. App. B.2. The final recipe to obtain
$V_{\parallel}$, with the point-like term subtracted, is to use

$\pi\rho_{\parallel}^{V}(s,q^{2})=\textrm{Im}[\Pi^{V}_{\parallel}(s,q^{2})-iX^{V}(s,q^{2})]\;,$
(29)

in (18), and similarly for $\cal{D}_{\|}$. The expression $iX^{V}$ corresponds
to $Q_{\bar{B}}f^{(pt)}_{B_{q}}$. From (24) and the corresponding WI below
(3.2) one can match to the standard pseudoscalar correlator Pullin:2021ebn .
In the notation of this reference the imaginary part of $iX^{V}$ reads

$\textrm{Im}[iX^{V}(s,q^{2})]=\pi\frac{2Q_{\bar{B}}}{s-q^{2}}\,m_{B}(m_{b}+m_{q})^{2}\frac{\rho_{f_{B}}(s)}{s}\;.$
(30)

For convenience we quote $\rho_{f_{B}}(s)|_{\textrm{PT-
LO}}=\frac{N_{c}}{8\pi^{2}}s((1-\frac{m_{b}^{2}}{s})^{2}+{\cal O}(m_{q}))$
which we have explicitly verified to emerge in our computation as in (30).

##  4 The Computation

The aim of this section is to describe the computation of the correlation
function (16) which contain both perturbative and light-cone OPE
contributions, as discussed in Secs. 4.1 and 4.2 respectively. These
computations have been performed using the packages of FeynCalc FeynCalc1 ;
FeynCalc2 , LiteRed litered:2013 , Kira kira:2017 , pySecDec pySecDec1 ;
pySecDec2  and PolyLogTools polylogtools .

First we discuss the treatment of $\gamma_{5}$ in dimensional regularisation
(DR). The emergence of DR as the standard tool for multi-loop calculations has
led to a number of prescriptions for ${\gamma_{5}}$ to allow for its
consistent use. In this work we adopt two different prescriptions depending on
the number of ${\gamma_{5}}$’s that appear in the traces. Our case includes
either one or two ${\gamma_{5}}$’s. For a single $\gamma_{5}$ we adopt the so-
called Larin scheme Larin:1993tq , whereby

${\gamma_{5}}=\frac{1}{4!}i\,\varepsilon_{\alpha\beta\gamma\delta}\,\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\delta},$
(31)

and the Levi-Civita tensor $\varepsilon_{\alpha\beta\gamma\delta}$ is
interpreted as a $d$-dimensional object, in the sense that
$\varepsilon_{\alpha\beta\gamma\delta}\varepsilon^{\alpha\beta\gamma\delta}=d(d-1)(d-2)(d-3)$
for example. This scheme is equivalent to the ’t Hooft-Veltman scheme
tHooft:1972tcz , up to ${\cal O}(d-4)$, when (31) is applied to the bare as
well as the counterterm graph. Breaking of $d$-dimensional Lorentz invariance,
as only $4$ indices are present in (31), has to be restored by a finite
renormalisation $Z^{P,s}_{5}=1-8\,C_{F}\frac{\alpha_{s}}{4\pi}+{\cal
O}(\alpha_{s}^{2})$ at NLO Larin:1993tq . In practice this amounts to the NLO
replacement

$J_{B_{q}}\rightarrow
J^{\textrm{Larin}}_{B_{q}}\equiv-Z^{P,s}_{5}(m_{b}+m_{q})/4!\,\varepsilon_{\alpha\beta\gamma\delta}\;\bar{b}\,\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\delta}q\;,$
(32)

in (16). When two $\gamma_{5}$’s are present we use naive DR, whereby
$\gamma_{5}$ is treated as an anticommuting object with regards to all
$\gamma_{\mu}$ matrices ($\mu=1...d$). We have checked that this is equivalent
to replacing each one with (31).

As the handling of the $\gamma_{5}$ is delicate it is good practice to subject
it to consistency tests. To this end we have checked the EOM at LO and NLO
which constitutes a non-trivial test (cf. Sec. 4.4 for more detail). We have
also verified the axial WI at LO; however since we project directly on scalar
function at NLO the same test is not easily performed, but given the EOM test
we can be confident of our results. A further test is that the FF relation (6)
is explicitly obeyed, which is particularly valuable as it tests
$\gamma_{5}$’s of even and odd number against each other.

Before going into some detail on the calculation let us comment on light quark
mass $m_{q}$-contributions which can be relevant for the strange quark. These
effects are included everywhere except at NLO twist-$1$ which would
necessitate master integrals beyond the current state of the art. These
effects are rather small and the inclusion of $m_{s}$-effects in the magnetic
vacuum susceptibility, that we propose, is a more significant effect.

###  4.1 Perturbation Theory and the Local OPE: Twist-1,4

In the perturbative and local OPE contribution the photon is treated as a
perturbative vertex with

$\langle\gamma(k,\lambda)|A_{\mu}(x)|0\rangle=\epsilon^{*}_{\mu}(k,\lambda)e^{ik\cdot
x}\;.$ (33)

In the literature this has been referred to as the hard contribution
Ball:2003fq . On grounds of the $1/m_{b}$ power behaviour, the perturbative
part and the condensate part are counted as twist-$1$ and -$4$ respectively.
We will make this clearer when discussing the light-cone OPE where the notion
of twist arises. It is helpful to distinguish the diagrams where the photon is
emitted from the $q$\- or the $b$-quarks, proportional to the $Q_{q}$\- and
$Q_{b}$-charge respectively, which we refer to as A- and B-type. It seems
worthwhile to point out that the soft contributions are all proportional to
$Q_{q}$.

####  4.1.1 Leading Order

The LO diagrams shown in Fig. 1 are straight forward to compute. Besides the
perturbative graph we include the quark condensate but neglect the gluon
condensate since $-m_{b}\langle\bar{q}q\rangle\gg\langle G^{2}\rangle$.
Explicit analytic results for the correlation function can be found in App. E.
A relevant and interesting aspect is that the A-type quark condensate diagram
behaves as $Q_{q}\langle\bar{q}q\rangle/k^{2}$ and diverges for an on-shell
photon. This can be seen as a heuristic motivation for the photon DA which
replaces these diagrams.

Figure 1: Perturbative diagrams of the A- and B-type as well as the B-type
quark condensate diagram. The absence of the A-type quark condensate diagram,
due its singular $1/k^{2}$-behaviour, is commented on in the main text.
Throughout, single and double line denote light $q=u,d,s$\- and a heavy
$b$-quark respectively.

####  4.1.2 Next-to-leading Order

The most labour-intensive task of this work is the computation of the
radiative corrections to the perturbative part. This computation is performed
off-shell, $k^{2}\neq 0$, as it provides a regularisation to IR-divergences
and we can make use of the NLO basis of master integrals DiVita:2017xlr . With
three external momenta and one internal mass, intended for $X^{0}\rightarrow
W^{+}W^{-}$ with $X^{0}=\\{H,Z,\gamma^{*}\\}$, they are state-of-the-art if
the method of differential equations in massive 2-loop integrals.

Figure 2: Radiative corrections to the perturbative contribution to the form
factors. Diagrams $A_{1-6})$ and $B_{1-6})$ are proportional to the charges
$Q_{q}$ and $Q_{b}$ respectively.

The diagrams are depicted in Fig. 2 with the additional derivative diagrams in
Fig. 3. After evaluating the traces in FeynCalc FeynCalc1 ; FeynCalc2  scalar
products of momenta are efficiently replaced with propagator denominators via
the LiteRed litered:2013  package, and then reduced to master integrals by
Kira kira:2017 . The master integrals (MIs) are expressed in terms of
multiple-polylogarithms (MPLs), cf. App. F. The $k^{2}\to 0$ limit is taken
and thereafter the discontinuities for the dispersion representation (17) are
found using the PolyLogTools polylogtools  package. We extended the basis of
MIs in DiVita:2017xlr  to include those that are given by the swapping of
external legs. Whereas technically this leads to an over-complete basis, this
was necessary in order to perform both the renormalisation and the $k^{2}\to
0$ limit efficiently where cancellation of the IR-divergent $\ln k^{2}$-terms
takes place. We have included these MIs as an ancillary file to the arXiv
version.444Of course these additional integrals follow by exchanging the
external momenta, namely $p_{B}^{2}\leftrightarrow q^{2}$ or
$p_{1}^{2}\leftrightarrow p_{2}^{2}$, in the MPLs of their original basis
counterparts. However, this would lead to lengthy results and make the
explicit cancellation of ultraviolet-(UV) and IR-divergences ($k^{2}\to 0$
limit) difficult as the MPLs contain many hidden relations. Hence we found it
easier to construct the additional basis elements using the procedure in
DiVita:2017xlr . All integrals, including the new MIs, have been numerically
verified against pySecDec pySecDec1 ; pySecDec2  and PolyLogTools polylogtools
.

Following this executive summary we give some more detail below. With the help
of LiteRed, scalar products of loop and external momenta appearing in the
numerators of the integrals depicted in the diagrams of Fig. 2 are written in
terms of propagator denominators such that all integrals are reduced to the
following form

$\text{MI}[T,\\{n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7}\\}]=\int
d^{d}\ell_{1}\int
d^{d}\ell_{2}\frac{1}{D_{T1}^{n_{1}}\,D_{T2}^{n_{2}}\,D_{T3}^{n_{3}}\,D_{T4}^{n_{4}}\,D_{T5}^{n_{5}}\,D_{T6}^{n_{6}}\,D_{T7}^{n_{7}}}\;,$
(34)

with $n_{i}\in\mathbb{Z}$ and $T\in\\{A,B\\}$. For the A-type basis,
corresponding to diagrams proportional to the light-quark charge $Q_{q}$, the
propagator denominators are

$\displaystyle D_{A1}$ $\displaystyle=\ell_{1}^{2},$ $\displaystyle D_{A2}$
$\displaystyle=\ell_{2}^{2},$ $\displaystyle D_{A3}$
$\displaystyle=(\ell_{1}+p_{B})^{2}-m_{b}^{2},\qquad
D_{A4}=(\ell_{2}+p_{B})^{2}-m_{b}^{2},$ $\displaystyle D_{A5}$
$\displaystyle=(\ell_{1}+k)^{2},$ $\displaystyle D_{A6}$
$\displaystyle=(\ell_{2}+k)^{2},$ $\displaystyle D_{A7}$
$\displaystyle=(\ell_{1}-\ell_{2})^{2},$ (35)

and for the B-type basis, corresponding to diagrams proportional to the heavy-
quark charge $Q_{b}$, we have,

$\displaystyle D_{B1}$ $\displaystyle=\ell_{1}^{2}-m_{b}^{2}\;,$
$\displaystyle D_{B2}$ $\displaystyle=\ell_{2}^{2}-m_{b}^{2}\;,$
$\displaystyle D_{B3}$ $\displaystyle=(\ell_{1}-p_{B}+k)^{2}\;,$
$\displaystyle D_{B4}$ $\displaystyle=(\ell_{2}-p_{B}+k)^{2}\;,$
$\displaystyle D_{B5}$ $\displaystyle=(\ell_{1}+k)^{2}-m_{b}^{2}\;,$
$\displaystyle D_{B6}$ $\displaystyle=(\ell_{2}+k)^{2}-m_{b}^{2}\;,$
$\displaystyle D_{B7}$ $\displaystyle=(\ell_{1}-\ell_{2})^{2}\;,$ (36)

with the usual $i0$-prescription implied. We note that the A-type and B-type
diagrams correspond to the $(a)$-$(b)$ and $(c)$-$(d)$ topologies in
DiVita:2017xlr . The corresponding MPLs are of weight $1$ to $4$ as is
appropriate for a 2-loop computation. The additional MIs that we compute
are:555We note that the MIs for the B-type basis, including the additional MIs
(4.1.2), were also recently computed in Ma:2021cxg  in the context of mixed
QCD-EW corrections to the $HW^{+}W^{-}$ vertex.

$\displaystyle\text{MI}[B,\\{0,2,1,0,1,1,1\\}]\;,\qquad\text{MI}[B,\\{0,1,2,0,1,1,1\\}]\;,\qquad\text{MI}[B,\\{0,1,1,0,1,1,1\\}]\;,$
$\displaystyle\text{MI}[B,\\{0,2,2,0,1,1,1\\}]\;,\qquad\text{MI}[B,\\{1,0,1,1,2,2,0\\}]\;,\qquad\text{MI}[B,\\{0,0,1,1,2,2,0\\}]\;.$
(37)

The bare $\Pi^{\parallel,\perp}$ are subject to renormalisation. Besides the
standard renormalisation of the $b$-quark mass in connection with the self-
energy graphs, one need to renormalise the operators of the correlation
function.666There is no need to discuss the renormalisation of overall local
$1/\epsilon^{2}_{\textrm{UV}}$ and $1/\epsilon_{\textrm{UV}}$ terms due to the
three operators coinciding as they are free of discontinuities in $p_{B}^{2}$.
Such terms do not contain any physical information and consequently do not
contribute to the dispersion integral. The electromagnetic current does not
renormalise, $Z_{J_{b}}=Z_{m_{b}}$ and the renormalisation of the weak
operators (4) is discussed in App. C.2. In particular the renormalisation of
$O^{D}_{\mu}$ provides a rather non-trivial test of the computation. After
this we perform the $k^{2}\to 0$ limit and take the discontinuities aided by
PolyLogTools. In order to do this efficiently we use shuffle algebra
relations, such as

$G(a,b,z)\,G(c,z)=G(a,b,c,z)+G(a,c,b,z)+G(c,a,b,z),$ (38)

to reorganise the result into a linear sum of MPLs. The task of computing the
discontinuity of the correlation function then becomes one of computing the
discontinuity of each individual MPL with details given in App. F. It should
be emphasised that the $Q_{b}$-type MIs are much more involved than the
$Q_{q}$-ones, which is also mirrored in their alphabet length of $18$ and $10$
respectively DiVita:2017xlr .

The final step in the computation of the perturbative densities is to
translate the MPLs into the more familiar classical polylogs.777Alternatively
one could use the HandyG package Naterop:2019xaf  to perform the numerical
evaluation. In doing so we move from complicated functions with relatively
simple branch cut structures, to simple functions with complicated branch cut
structures. Consequently, the representation of a given MPL in terms of
polylogs depends on the regions in which its arguments lie, meaning that a
single expression may not be sufficient to span all possible values of the
arguments. An explicit example of this can be seen in Eq.$(4.2)$ of
Frellesvig:2016ske . We have numerically verified that the perturbative
densities written in terms of classical polylogs are equivalent to those given
in terms of MPLs in the region of interest, $q^{2}<m_{b}^{2}<p_{B}^{2}$.

Figure 3: Additional diagrams required when computing the correlation function
$\Pi_{\mu}^{\mathcal{D}}$.

###  4.2 The light-cone OPE diagrams of Twist-$2$,$3$ and $4$

In phenomenology the light-cone OPE has its roots in deep inelastic
scattering, which is an inclusive measurement where the operators on the
light-cone are the parton distribution functions $\langle
P(p)|\bar{q}(z)\Gamma q(0)|P(p)\rangle$. The Fourier transform of the light-
cone direction $p^{-}z^{+}$ describes the probability of finding a constituent
parton of a certain momentum fraction. The application of the light-cone OPE
to hard exclusive processes CZ1984  is a different matter, as the light-cone
operators correspond to DAs, e.g. $\langle 0|\bar{d}(z)\Gamma
u(0)|\pi^{+}(p)\rangle$ and the Fourier transform in the light-cone direction,
denoted by the variable $u$ in this work, describes the amplitude for each
parton having momentum fraction $u$. We refer the reader to the technical
review Braun:2003rp  where the connection to conformal symmetry is presented
in depth and detail.

In a concrete setting an OPE is associated with factorisation. The light-cone
OPE corresponds to collinear factorisation and is often schematically
presented (in momentum space) as

$\Pi(q^{2},m_{b})=\sum_{i}T^{(i)}_{H}(q^{2},m_{b},\mu_{F})\otimes\phi_{i}(\mu_{F})\;,$
(39)

where $\Pi$ is the correlation function, $T^{(i)}_{H}$ a perturbatively
calculable hard scattering kernel, $\phi_{i}$ a DA, $\otimes$ denotes the
integration over the collinear momentum fractions (suppressed in the formula),
$\mu_{F}$ is the factorisation scale and the sum runs over DAs of increasing
twist. A necessary condition for factorisation to hold is that the
$\mu_{F}$-dependence cancels to the given order in a computation. This
corresponds to the absorption of collinear IR-divergences of the hard kernel
into the DAs. We will discuss this explicitly in Sec. 4.2.3.

####  4.2.1 Quark Mass Correction to the Photon Distribution Amplitude

This section serves to illustrate the leading twist-$2$ photon DA and we
present a method to incorporate quark mass corrections, which are sizeable in
the case of the strange quark. The leading twist-2 photon DA is given by (D.4)

$\langle\gamma(k,\epsilon)|\bar{q}(0)\sigma_{\alpha\beta}q(z)|0\rangle\;=\;is_{e}eQ_{q}\langle\bar{q}q\rangle\chi\epsilon^{*}_{[\alpha}k_{\beta]}\int_{0}^{1}due^{i\bar{u}kz}\phi_{\gamma}(u,\mu)+\textrm{higher
twist}\;,$ (40)

where the Wilson line is omitted for brevity and $\chi$ is known as the
magnetic vacuum susceptibility, which serves as a normalisation of the photon
DA in the sense that $f^{\perp}_{\gamma}=\langle\bar{q}q\rangle\chi$ is the
analogue of the transverse decay constant $f_{\rho}^{T}$ for a $\rho$-meson.
Its value without the inclusion of quark mass effects (and thus no quark
subscript) is $\chi\approx 3\,\mbox{GeV}$, at the scale $\mu=1\,\mbox{GeV}$,
known from QCD sum rules Ioffe:1983ju ; Ball:2002ps  and lattice computations
Bali:2012jv ; Bali:2020bcn .

For finite quark mass $m_{q}$ the _local_ matrix element is UV-divergent. More
precisely, it mixes with the identity (additive renormalisation) which is
symptomatic of any OPE. Thus including finite quark mass corrections requires
a specific prescription which we propose. Our starting point is the non-local
matrix element for an off-shell photon $\gamma^{*}(k^{2}\neq 0)$

$\displaystyle\langle\gamma^{*}(k,\epsilon)|\bar{q}(0)\sigma_{\alpha\beta}q(x)|0\rangle$
$\displaystyle\;=\;$ $\displaystyle-ie\epsilon_{\mu}^{*}\int e^{iky}\langle
0|Tj^{\mu}(y)\bar{q}(0)\sigma_{\alpha\beta}q(x)|0\rangle=$ (41)
$\displaystyle\;=\;$ $\displaystyle
is_{e}eQ_{q}\epsilon^{*}_{[\alpha}k_{\beta]}\int_{0}^{1}due^{i\bar{u}kx}\Phi_{\gamma^{*}}(u,k^{2},x^{2})\;,$

where we parameterise ($\mu$-dependence suppressed)

$\displaystyle\Phi_{\gamma^{*}}(u,k^{2},x^{2})=\langle\bar{q}q\rangle\chi\phi_{\gamma}(u,k^{2})+\frac{N_{c}m_{q}}{4\pi^{2}}\phi^{(m_{q})}_{\gamma^{*}}(k^{2},x^{2})\;,$

and remind the reader that, by definition, $\chi$ does not include quark mass
corrections. The LO contribution is straightforward to compute

$\displaystyle\phi^{(m_{q})}_{\gamma^{*}}(k^{2},x^{2})$ $\displaystyle\;=\;$
$\displaystyle-2K_{0}\left(\sqrt{(m_{q}^{2}-u\bar{u}k^{2})(-x^{2}+i0)}\,\right)+{\cal
O}(\alpha_{s})$ (42) $\displaystyle\;=\;$
$\displaystyle\ln\left(\frac{m_{q}^{2}-\bar{u}uk^{2}}{4\mu^{2}}\right)+\ln(-x^{2}+i0)\mu^{2}+2\gamma_{E}+{\cal
O}(\alpha_{s},x^{2})\;,$

where $K_{0}$ is the modified Bessel function of the second kind. The
$\ln(-x^{2}+i0)$-term signals the UV-divergence which is regulated by
$x^{2}\neq 0$. Upon taking the local limit from the start and using DR
$d=4-2\epsilon$, we identity at LO $1/\epsilon_{\textrm{UV}}+\ln
4\pi-\gamma_{E}\leftrightarrow-\ln\frac{(-x^{2}+i0)\mu^{2}}{4}-2\gamma_{E}$.

Before discussing the LO-prescription to remove the UV-divergence, let us
digress on the sign of the $\ln(-x^{2}+i0)$-term. Some time ago Ioffe and
Smilga Ioffe:1983ju  analysed the local matrix element using a UV cut-off
$\Lambda$ and a constituent quark mass. Taking into consideration their sign
convention of $\chi<0$, we expect an increase of $\Lambda^{2}\leftrightarrow
1/x^{2}$ to effectively enhance $\chi$ (recall $\langle\bar{q}q\rangle<0$ for
$m_{q}>0$). The same remark applies to the evaluation of this quantity in the
background of a constant magnetic field via its Dirac eigenvalues cf. App. B
in Bali:2012jv . Hence our sign is in accordance with these treatments.888The
relative sign, between the $\chi\langle\bar{q}q\rangle\phi_{\gamma}$, and the
$\ln(-x^{2})$-term differs from Balitsky:1997wi  (eq A.2) (and Ball:2002ps  eq
(2.20,3.17)), when $\chi>0$ is assumed as stated in those papers. This could
be due to $\chi<0$ convention used in previous publications Balitsky:1986st .
Note that there is a typo in that paper. Namely
$g_{\phi}^{2}/4\pi=9m_{\phi}^{2}/(4\pi f_{\phi}^{2})\approx 11.7$ and not the
inverse of what is quoted in that reference.

We would like to motivative the LO-prescription. The Fourier transform of
$\ln(-x^{2})$

$\mu^{d-4}\int
d^{d}xe^{ipx}\ln(-x^{2}\mu^{2})=i(4\pi)^{2}\frac{1}{p^{4}}+{\cal O}(\epsilon)$
(43)

corresponds to a zero momentum insertion approximation. We have checked by
explicit computation that this leads to the same result when one uses a mass-
insertion approximation in perturbation theory.999 This is not a good choice
per se as it is IR-divergent. Moreover, if one were to use the
$\ln(-x^{2}\mu^{2})$-term directly, via (43), this would lead to an IR-
divergence. Of course these issues are resolved if one drops the
$\ln(-x^{2}\mu^{2})$-term and uses perturbation theory with quark masses in
the denominator. In particular, the IR-divergence becomes regular $m_{q}\ln
m_{q}$. Hence at LO in $\alpha_{s}$ dropping the $\ln(-x^{2}\mu^{2})$-term is
the correct prescription in order to avoid double counting. Whether dropping
solely the $\ln(-x^{2}\mu^{2})$-term is the correct prescription for higher
order or not is an open question and deserves further exploration.

Now that the UV-divergent part is separated it remains to be seen what we can
do with the DA piece

$\phi^{(m_{q})}_{\gamma^{*}}(k^{2})=\phi^{(m_{q})}_{\gamma^{*}}(k^{2},x^{2})|_{\textrm{twist
--2}}=\ln(-\bar{u}uk^{2}/(4\mu^{2}))+2\gamma_{E}\;.$ (44)

If one could trust the $k^{2}\to 0$ limit then one could simply add it to the
twist-$2$ DA $\Phi^{(\phi_{\gamma})}_{\gamma}$ with
$\phi^{(m_{q})}_{\gamma^{*}}(0)$ in place. Clearly this is not the case since
perturbation theory is not applicable for $k^{2}\to 0$ as illustrated by the
singularity. A well-known solution to this problem is to invoke a dispersion
relation Ball:2002ps . We can build on their work for determining $\chi$ in
the massless quark limit and add our contribution. The only relevant
conceptual difference is that we have to use a once-subtracted dispersion
relation because of the logarithmic divergence.

It was found in Ball:2002ps  that the sum rules for higher moments are
unstable and we thus focus on the zeroth moment $\chi_{q}$ (recall
$f^{\perp}_{\gamma q}=\langle\bar{q}q\rangle\chi_{q}$). The once subtracted
dispersion representation reads

$f^{\perp}_{\gamma^{*}q}(k^{2})=\int_{0}^{1}du\Phi^{(m_{q})}_{\gamma^{*}}(u,k^{2})=f^{\perp}_{\gamma^{*}q}(k_{0}^{2})+\int_{\textrm{cut}}^{\infty}\frac{dr\,\rho^{f^{\perp}_{\gamma^{*}q}}(r)}{r-k^{2}-i0}R(r)\;,$
(45)

with $R(r)=\frac{k^{2}-k_{0}^{2}}{r-k_{0}^{2}}$ and
$\pi\rho^{f^{\perp}_{\gamma^{*}s}}=\textrm{Im}f^{\perp}_{\gamma^{*}s}$.

Assuming the quark content
($|\rho_{0}[\omega]\rangle\simeq(|\bar{u}u\rangle\mp|\bar{d}d\rangle)/\sqrt{2}$
and $|\phi\rangle\simeq|\bar{s}s\rangle$) and adopting the zero width
approximation we may rewrite the dispersion relation as follows

$\displaystyle f^{\perp}_{\gamma^{*}s}(k^{2})$ $\displaystyle\;=\;$
$\displaystyle
f^{\perp}_{\gamma^{*}s}(k_{0}^{2})+\frac{m_{\phi}f_{\phi}f_{\phi}^{T}}{m_{\phi}^{2}-k^{2}-i0}R(m_{\phi}^{2})+\int_{r^{(\phi)}_{0}}^{\infty}\frac{dr\,\rho^{f^{\perp}_{\gamma^{*}s}}(r)}{r-k^{2}-i0}R(r)\;,$
$\displaystyle f^{\perp}_{\gamma^{*}q}(k^{2})$ $\displaystyle\;=\;$
$\displaystyle
f^{\perp}_{\gamma^{*}q}(k_{0}^{2})+\frac{m_{\rho}f_{\rho}f_{\rho}^{T}}{m_{\rho}^{2}-k^{2}-i0}R(m_{\rho}^{2})+\int_{r^{(\rho)}_{0}}^{\infty}\frac{dr\,\rho^{f^{\perp}_{\gamma^{*}q}}(r)}{r-k^{2}-i0}R(r)\;,$
(46)

where $r_{0}$ is the hadronic threshold due to higher states and we have
adopted the isospin limit $f^{\perp}_{\gamma^{*}q}\equiv
f^{\perp}_{\gamma^{*}u}\approx f^{\perp}_{\gamma^{*}d}$. Now, in order to
evaluate the RHS, besides the decay constants
$(f_{\rho},f_{\rho}^{T})=(0.220(5),0.160(7))\,\mbox{GeV}$,
$(f_{\phi},f_{\phi}^{T})=(0.233(4),0.191(4))\,\mbox{GeV}$ BSZ15  and continuum
thresholds $(r_{0}^{(\rho)},r_{0}^{(\phi)})=(1.5(1),2.0(1))\,\mbox{GeV}^{2}$
Ball:2002ps , one needs the subtraction constant and the spectral density
$\rho(r)$. Both of these can be obtained from the local OPE
$f^{\perp}_{\gamma^{*}s}(k^{2})$. Using the results in App. B in Ball:2002ps
and adding our $m_{q}$-correction we find

$f^{\perp}_{\gamma^{*}q}(k^{2})=\frac{2\langle\bar{q}q\rangle}{k^{2}}\left(-1+C_{F}\frac{\alpha_{s}}{\pi}(1-\ln\frac{-k^{2}}{\mu^{2}})-\frac{1}{3}\frac{m_{0}^{2}}{k^{2}}\right)+\frac{N_{c}m_{q}}{4\pi^{2}}\left(2(\gamma_{E}-1)-\ln
4+\ln\frac{-k^{2}}{\mu^{2}}\right)\;,$ (47)

where $\langle\bar{q}\sigma\\!\cdot\\!Gq\rangle=m_{0}^{2}\langle qq\rangle$
with $m_{0}^{2}=0.8(2)\,\mbox{GeV}^{2}$. The spectral function

$\rho^{f^{\perp}_{\gamma^{*}s}}(s)=\frac{2C_{F}\alpha_{s}}{\pi}\frac{\langle\bar{q}q\rangle}{s}-\frac{N_{c}m_{q}}{4\pi^{2}}\;,$
(48)

is obtained from (47). Using $\chi_{q}=f^{\perp}_{\gamma
q}/\langle\bar{q}q\rangle$ and adopting the notation $f^{\perp}_{\gamma
q}=f^{\perp}_{\gamma^{*}q}(0)$ with reference to (4.2.1) we find (with
$\mu=1\,\mbox{GeV}$)

$\displaystyle f_{\gamma q}^{\perp}$ $\displaystyle\;=\;$
$\displaystyle-48(3)\,\mbox{MeV}\quad$ $\displaystyle\Leftrightarrow$
$\displaystyle\quad\chi_{q}$ $\displaystyle\;=\;$ $\displaystyle
3.21(15)\,\mbox{GeV}^{-2}\;,\quad$ $\displaystyle f_{\gamma s}^{\perp}$
$\displaystyle\;=\;$ $\displaystyle-61(3)\,\mbox{MeV}\;,\quad$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad\chi_{s}$
$\displaystyle\;=\;$ $\displaystyle 3.79(17)\,\mbox{GeV}^{-2}\;,$ (49)

where we have kept $-k_{0}^{2}>3\,\mbox{GeV}^{2}$ in the deep Euclidean region
as appropriate for a short distance expansion. In addition we quote an
$SU(3)$-ratio

$r_{SU(3)}=\frac{f_{\gamma s}^{\perp}}{f_{\gamma q}^{\perp}}=1.28(4)\;.$ (50)

All results are highly stable with respect to the subtraction point
$k_{0}^{2}$ which is a sign of consistency in both the formula and its input,
as the individual parts show sizeable variation (for
$k_{0}^{2}\in[2,10]\,\mbox{GeV}^{2}$ the results vary less than a percent).
The value of (50) corresponds to a typical $SU(3)$-ratio in the non-
perturbative realm. The $m_{s}$-contribution in perturbation theory can be
estimated to be of the order of $m_{s}/m_{b}\approx 0.02$ and can therefore be
dropped.

We wish to emphasise that whereas $\chi_{q}(f_{\gamma q})$ merely reproduces
the result in Ball:2002ps  (within minor variation of the input),
$\chi_{s}(f_{\gamma s})$ is the first determination of this quantity. The
quantity $\chi_{s}$ in Bali:2012jv ; Bali:2020bcn  should not be compared to
the $\chi_{s}$ above as it differs in the renormalisation prescription. We re-
emphasise that our prescription emerges naturally from the twist expansion.
The fact that a correction to a non-perturbative matrix element can be
computed with perturbative methods is somewhat exceptional and so let us
comment. A key difference to QCD condensates or matrix elements is that
$f^{\perp}_{\gamma q}$ is of mass dimension one which is taken by the quark
mass. In QCD this would involve ${\cal O}(\Lambda_{\textrm{QCD}}^{n})$-terms.
The remaining logarithmic divergence $\ln\Lambda_{\textrm{QCD}}$ can be
absorbed into the perturbative part (twist-$1$).

####  4.2.2 Leading Order

The LO graphs, corresponding to the light-cone DA contributions to the FFs are
depicted in Fig. 4. The expansion is performed up to twist-$4$, including
$3$-particle DAs, with the definition of the DAs given in App. D. As mentioned
above the $Q_{q}\langle\bar{q}q\rangle$-term is part of this expansion cf. Eq.
(D.1) to see this explicitly.

Figure 4: LO contributions in the DAs. $A_{1})$ $2$-particle DA $A_{2})$
$3$-particle DA. The variable $u\in[0,1]$ and $\bar{u}=1-u$ are the momentum
fractions carried by quarks. The point $vx$, $v\in[0,1]$, corresponds to the
gluon in the background field gauge propagator (D.3). The $\alpha_{i}$ denote
the momentum fractions of the quarks and gluon with
$\alpha_{1}\\!+\\!\alpha_{2}\\!+\\!\alpha_{3}\\!=\\!1$.

####  4.2.3 Next-to-leading Order

Further to the LO graphs we have also computed the twist-$2$ ${\cal
O}(\alpha_{s})$ corrections in the photon DA $\phi_{\gamma}(u)$. The relevant
diagrams are depicted in Fig. 5, with additional diagrams for
$\Pi^{D}_{\mu\rho}$ shown in Fig. 6, and the total results given in App. E.3
in terms of Passarino-Veltman (PV) functions. A noticeable feature is that in
the Feynman gauge the box diagram $A_{3})$ vanishes.

Figure 5: $\mathrm{O}(\alpha_{s})$ corrections to twist-$2$ DA contribution.
$A_{1}$) vertex-correction $A_{2}$) $J_{B}$-correction $A_{3}$) box-diagram
$A_{4-6}$) $b$\- and $q$-quark self energy corrections.

For the radiative corrections there are issues with regard to IR- and UV-
divergences to be discussed. A particularly clear discussion of these matters
can be found in B83 . First, the necessary conditions for collinear
factorisation to hold are (i) that there are no divergences on integration
over the collinear momentum fraction $u$ and (ii) that the collinear IR-
divergences can be absorbed into the bare DA, $\phi^{\textrm{bare}}_{\gamma}$,
which is not observable. We find that our integrals converge when integrated
over $u$ therefore satisfying the first condition. The collinear IR-
divergences are absorbed into the LO DA. For this to be possible the
divergences have to assume the following form
$T_{H,1}^{\textrm{div}}=c\,T_{H,0}\otimes V_{0}$. In this equation
$T_{H,0(1)}$ are the LO and NLO hard scattering amplitudes, $V_{0}$ is the LO
Efremov-Radyushkin-Brodsky-Lepage evolution kernel Efremov:1978rn ;
Efremov:1979qk ; Lepage:1980fj  and $c$ is a constant or simply the
counterterm B83 . One may use the property that the eigenfunctions and
eigenvalues of $V_{0}^{\perp}$ are known Shifman:1980dk

$V^{\perp}_{0}\otimes\varphi_{n}=\int_{0}^{1}V^{\perp}_{0}(u,v)\varphi_{n}(v)=-\gamma_{n}^{(0)\perp}\varphi_{n}(u)\;,$
(51)

with

$\varphi_{n}(u)=6u\bar{u}\,C^{(3/2)}_{n}(2u-1)\;,\quad\gamma_{n}^{(0)\perp}=2C_{F}\left(1+4\sum_{j=2}^{n+1}\frac{1}{j}\right)\;,$
(52)

where $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ and $C^{(3/2)}_{n}(x)$ is the
$n^{\textrm{th}}$ Gegenbauer polynomials of degree $3/2$. The divergencies can
then be absorbed into

$\phi_{\gamma}^{\textrm{bare}}=6u\bar{u}\sum_{n\geq
1}(a_{n}^{\perp})^{\textrm{bare}}C^{(3/2)}_{n}(2u-1)\;,\quad$ (53)

where

$(a_{n}^{\perp})^{\textrm{bare}}=(\mu/\mu_{0})^{2\epsilon}a_{n}^{\perp}(\mu)\left(1+c\frac{g^{2}}{(4\pi)^{2}}\gamma_{n}^{(0)\perp}\frac{1}{\epsilon}\right)\;.$
(54)

Requiring

$0=\frac{d}{d\ln\mu}(a_{n}^{\perp})^{\textrm{bare}}=a_{s}\gamma_{n}^{(0)\perp}(-1+2c)\;,\quad
a_{s}=g^{2}/(4\pi)^{2}\;,$ (55)

determines $c=\frac{1}{2}$. This is indeed the constant that we find in our
explicit computation and thus shows $\mu_{F}$-independence and cancellation of
the collinear IR-divergences. This is not a new result as the computation is
equivalent to the $\phi^{\perp}$-amplitude in $\bar{B}\to V$ FFs where it was
verified at the twist-$2$ level in BB98b ; BZ04b  and for the
$V^{\bar{B}_{u}\to\gamma}$ in Ball:2003fq .101010The difference to these
references is that we give explicit results. In BB98b  results were given with
Dirac traces to be evaluated from where the cancellation can easily be
verified.

Figure 6: Additional diagrams originating from the emission of the photon from
the $\mathcal{O}_{\mu}^{D}$-operator. In principle there are two additional
graphs, one for perturbation theory and for
$\overline{Q}\langle\bar{q}q\rangle$, where the photon is emitted from the
covariant derivative of the $O^{D}_{\mu}$-vertex. However, these graphs turn
out to be proportional to $\epsilon^{*}_{\mu}$ and are orthogonal to our
projection prescription (28), and we therefore do not include them. Further
not that if they were truly missing the EOM would not close.

Obtaining the dispersion representation (17) of the twist-$2$ NLO contribution
is not straightforward from the representation given in (E.10), which we write
schematically

$\Pi_{\perp}(p_{B}^{2})=\int_{0}^{1}du\pi_{\perp}(u,p_{B}^{2})\;,$ (56)

with $\pi_{\perp}(u,p_{B}^{2})=T_{\perp}(u,p_{B}^{2})\phi_{\gamma}(u)$.
Whereas the function $\Pi_{\perp}(p_{B}^{2})$ has a cut starting at
$m_{b}^{2}$, $\pi_{\perp}(u,p_{B}^{2})$ has additional cuts in the
$(u,p_{B}^{2})$-variables. Two natural choices present themselves. Firstly,
one commits to a certain form of the twist-$2$ photon DA and integrates over
$du$ in (56) such that the (poly)logs are functions of the external kinematic
variables only and one can then obtain the discontinuity in the standard way.
The other possibility is that one uses the fact that a FF’s discontinuity is
equivalent to its imaginary part. One may write, working with a single
subtraction term at $p_{B}^{2}=\bar{s}$ in order to regulate the logarithmic
UV-divergence,

$\displaystyle\Pi^{V}_{\perp}(p_{B}^{2})-\Pi^{V}_{\perp}(\bar{s})$
$\displaystyle\;=\;$
$\displaystyle\frac{(p_{B}^{2}-\bar{s})}{\pi}\int_{m_{b}^{2}}^{\infty}\frac{ds}{s-\bar{s}}\frac{\textrm{Im}[\int_{0}^{1}du\pi^{V}_{\perp}(u,s)]}{s-p_{B}^{2}-i0}$
(57) $\displaystyle\;\stackrel{{\scriptstyle p_{B}^{2}<m_{b}^{2}}}{{=}}\;$
$\displaystyle\textrm{Im}\left[\frac{(p_{B}^{2}-\bar{s})}{\pi}\int_{\Gamma_{1}}\frac{ds}{s-\bar{s}}\frac{\int_{\Gamma_{2}}du\pi_{\perp}^{V}(u,s)}{s-p_{B}^{2}}\right]\;,$

where $\Gamma_{1,2}$ are paths distorted into the upper complex planes
$\mathbb{C}_{s,u}$ to avoid singularities. We have validated this procedure
numerically.

In the final dispersion integral, after Borel transformation, the recipe reads

$V_{\perp}(q^{2})=\frac{1}{f_{B}m_{B}^{2}}\textrm{Im}\left[\frac{1}{\pi}\int_{\Gamma_{m_{b}^{2},s_{0}}}dse^{(m_{B}^{2}-s)/M^{2}}\int_{\Gamma_{2}}du\pi_{\perp}^{V}(u,s,q^{2})\right]\;,$
(58)

where the integration endpoint of $\Gamma_{1}$, which is $\infty$, is moved to
the continuum threshold $s_{0}$. This integral is numerically very stable.

###  4.3 Comparison of Form Factor computation with the Literature

The twist-$1$ and $2$ LO results agree with Khodjamirian:1995uc ; Ali:1995uy .
At twist-$3$ and $4$ we can compare with Ball:2003fq  and find that the
$\langle\bar{q}q\rangle$, ${\cal U}$, $\mathbb{A}$, $S_{\gamma}$ &
$T_{4\gamma}$ contributions are missing and our result in $\tilde{S}$ is
larger by a factor of six. At NLO solely the twist-$2$ result has been
reported in Ball:2003fq ; Wang:2018wfj . In the first reference it is
presented only in numeric form and the agreement is reasonable taking into
account the slightly different numerical input. In Wang:2018wfj  analytic
results are given but there is a sign difference between the twist-$1$ and the
twist-$2$ term (cf. their figure 5) in contrast to our result.111111We would
like to thank Yuming Wang for confirming this sign error. To continue the
discussion at this point it is useful to already take into account some of our
numerical results of the proceeding section. Correcting for the mentioned
sign, and taking into account the global sign difference (7), focusing on
twist-$1$ and $2$ one gets
$V^{\bar{B}_{u}\to\gamma}_{\perp}|_{\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Wang:2018wfj}{\@@citephrase{(}}{\@@citephrase{)}}}}}\approx-(0.28_{t=1}+0.08_{t=2})=-0.36$
whereas we find
$V^{\bar{B}_{u}\to\gamma}_{\perp}\approx-(0.124_{t=1}+0.082_{t=2})=-0.206$
(cf. Tab. 6). The discrepancy is too large in view of our ${\cal
O}(10\%)$-uncertainty. In addition the value $V_{\perp}=-0.36$ is too large in
view of the Belle exclusion limit (cf. Sec. 5.3). The breakdown suggests that
the culprit is the twist-$1$ contribution for which two sources can be
identified. First, the twist-$1$ result in Wang:2018wfj  is only to leading
power in $1/m_{b}$ and this suggests that the non-leading part is sizeable.
Second, the not too well-known $B$-meson DA parameters and the assumption of a
specific model for the DA itself (cf. the discussion in Sec. 5.4). To
conclude, our analysis suggests that a posteriori the hybrid approach in
Wang:2018wfj  is numerically not sound because it does not include the full
next-leading power corrections in the $1/m_{b}$-expansion.

###  4.4 Equations of Motion as a Test of the Computation

Generally EOM give rise to relations between correlation functions, cf. App. C
which often involve CTs. For simple local FFs these CTs are absent and imply,
in the case at hand, the following relations

$\displaystyle T_{\perp}(q^{2})+{\cal D}_{\perp}(q^{2})$ $\displaystyle\;=\;$
$\displaystyle\frac{(m_{b}+m_{q})}{m_{B}}V_{\perp}(q^{2})\;,$ $\displaystyle
T_{\parallel}(q^{2})+{\cal D}_{\parallel}(q^{2})$ $\displaystyle\;=\;$
$\displaystyle\frac{(m_{b}-m_{q})}{m_{B}}V_{\parallel}(q^{2})\;.$ (59)

We wish to stress that these relations are completely general and thus have to
be obeyed by any approach. We further note that the point-like part
$f^{(pt)}_{B_{q}}$ (2.1) cancels between the LHS and RHS in the
$\parallel$-equation.

Reassuringly, we find that (4.4) holds in all parameters for which we have
complete results: perturbation theory (twist-$1$),
$\\{\chi\langle\bar{q}q\rangle,a_{2}(\gamma),a_{4}(\gamma)\\}$ (twist-$2$),
$f_{3\gamma}$ at LO in $1/m_{b}$ (twist-$3$) and $Q_{b}\langle\bar{q}q\rangle$
(twist-$4$ local OPE). For twist-$1$ and $2$ this includes the NLO correction
in $\alpha_{s}$. There are some comments to be made on the other cases. The
parameter $f_{3\gamma}$ is defined from a $3$-particle matrix element of
twist-$3$ but could mix into a $4$-particle matrix element. This effect is
$1/m_{b}$-suppressed, which we have checked analytically, and fortunately
numerically negligible. The only true twist-$4$ term that we include is
$Q_{b}\langle\bar{q}q\rangle$ and now turn to the reason why, which has to our
knowledge not been thoroughly discussed in the literature.

It concerns the consistent inclusion of twist-$4$ $3$-particle DA parameters.
Our finding is that this demands $4$-particle DAs and is based on the
following two observations. Firstly, $4$-particle DAs of the form
$\bar{q}(x)G_{\alpha\beta}(vx)G_{\gamma\delta}(ux)\sigma_{\rho\sigma}q(0)$,
which do appear in the light-cone expansion of the propagator cf. eq. A.16 in
Balitsky:1987bk , allow for twist-$4$ contributions. Secondly, these terms
would mix via the EOM, generalisations of Eq. 4.40 Ball:2002ps , into the
$3$-particle DAs of twist-$4$.121212For example, for vector meson DAs the
$3$-particle parameter $\zeta_{3}\omega^{T}$, of twist-$3$, mixes into the
$2$-particle DA $h_{3}$ twist-$4$, e.g. Ball:1998ff , due to the EOM. In the
same way $3$\- and $4$-particle DAs are coupled through the EOM and lower
twist parameter can always mix into a higher twist parameter. The point is
that the EOM at the FF-level need to close in each hadronic parameter
separately, to be used in the numerics. In fact we find that of the
$3$-particle twist-$4$ parameters only $\zeta_{2}^{+}$ closes and
$\\{\kappa,\kappa_{+},\zeta_{1},\zeta_{1}^{+},\zeta_{2}\\}$ including the
$\langle\bar{q}Fq\rangle$-type contributions (D.25) do not.131313 For the
charged transitions further twist-$4$ contributions are needed, namely the
photon parton of the photon DA connecting to the charged lepton. Strictly
speaking, it is then better to consider the amplitude rather than the FF
(beyond the rather trivial issue of the $Q_{\bar{B}}$-CTs in (2.1)).  As there
is no argument as to whether any of the FF-combination are to be preferred
over others we are led to conclude that one needs to drop those parameters
altogether. Fortunately quite a few of them are zero (Tab. 1) and the
numerical impact of the others is well below $10\%$ (Tab. 6). Whereas the
numbers in the table seem higher for the charged $\bar{B}_{u}$, footnote 13 is
relevant in that it probably means that the numbers are overestimated. Note
that for the photon DA the omission of twist-$4$ parameters for the
$3$-particle case then implies that the $2$-particle twist-$4$ DAs are
effectively zero as their only contribution comes from the EOM with the
$3$-particle DA of twist-$4$. This is different to the vector mesons where the
$m_{V}^{2}$-contributions appear as $2$-particle ones of twist-$4$ and are
sizeable for the $\phi$-meson for example BSZ15 .

##  5 Numerics

In this section we turn to the numerics for which we first discuss the input,
the sum rule parameters, correlations of parameters due to the EOM and the
fit-ansatz in Sec. 5.1. In Sec. 5.2 FF-plots, fit parameters and values for
$q^{2}=0$ and are given in Fig. 7, Tab. 5 Tab. 4 respectively The correlation
matrix can be found in an ancillary file to the arXiv version. The prediction
of the $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ rate with bins and correlation
matrix is given in Sec. 5.3 In Sec. 5.4 we present the extraction of the
$B$-meson DA parameter $\lambda_{B}$ by comparing to a set of three SCET-
computations.

###  5.1 Input Parameters, Sum Rule Parameters and Fits

The input parameters, including the values for the photon DAs, are collected
in Tab. 1. An important aspect is the choice of the mass scheme. The
$\overline{\text{MS}}$\- and the pole-scheme give rise to large effects in
either higher twist or radiative corrections which suggests that neither is
optimal (cf. Fig. 13 and Tab. 10). The kinetic scheme, originally developed
for the OPE in inclusive decays Bigi:1994em , can be regraded as a compromise
of these two schemes and we have found that it does indeed give rise to stable
results. From Fig. 12 it is seen that this leads to stability in
$\mu_{\textrm{kin}}$ with uncertainties in the $1$-$2\%$ range. To our
knowledge this is the first time this scheme is used in the context of LCSR
and we also adapt it in the closely related work on the
$g_{BB^{*}\gamma}$-type on-shell couplings Pullin:2021ebn . The value of the
kinetic mass, shown in Tab. 1, has been obtained using the two-loop relation
between the $\overline{\text{MS}}$ and kinetic mass given in Gambino:2017vkx .
The uncertainty on the kinetic mass has been estimated by adding the error on
the $\overline{\text{MS}}$ mass and the one from the conversion formula
(variation of scale $\mu_{m}\in[m_{b}/2,2m_{b}]$) in quadrature. In line with
other applications we set the scale of the kinetic scheme to
$\mu_{\textrm{kin}}\\!=(\\!1\pm 0.4)\,\mbox{GeV}$.

Running coupling parameters
---
$\alpha_{s}(m_{Z})\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $m_{Z}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}$ |  |  |  |
$0.1176(20)$ | $91.19\,{\textrm{GeV}}$ |  |  |  |
Scales
$\mu_{F}^{B_{d,u}}=\mu_{\text{cond}}^{B_{d,u}}=\mu_{\alpha_{s}}^{B_{d,u}}$ | $\mu_{F}^{B_{s}}=\mu_{\text{cond}}^{B_{s}}=\mu_{\alpha_{s}}^{B_{s}}$ | $\mu_{\text{UV}}$ | $\mu_{\textrm{kin}}$
$2.24(1.00)\,{\textrm{GeV}}$ | $2.44(1.00)\,{\textrm{GeV}}$ | $4.78(1.00)\,{\textrm{GeV}}$ | $1.00(40)\,{\textrm{GeV}}$
$J^{P}=0^{-}\mbox{ Meson masses~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}$
$m_{B_{d}}$ | $m_{B_{u}}$ | $m_{B_{s}}$ |  |  |
$5.280\,{\textrm{GeV}}$ | $5.280\,{\textrm{GeV}}$ | $5.367\,{\textrm{GeV}}$ |  |  |
$J^{P}=1^{\pm}\mbox{ Meson masses~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}$
$m_{B^{*}_{d}}$ | $m_{B^{*}_{u}}$ | $m_{B^{*}_{s}}$ | $m_{B_{1d}}$ | $m_{B_{1u}}$ | $m_{B_{1s}}$
$5.325\,{\textrm{GeV}}$ | $5.325\,{\textrm{GeV}}$ | $5.415\,{\textrm{GeV}}$ | $5.726\,{\textrm{GeV}}$ | $5.726\,{\textrm{GeV}}$ | $5.829\,{\textrm{GeV}}$
Quark masses PDG
$m_{s{|2\\!\,{\textrm{GeV}}}}$ | $m_{b}(m_{b})$ | $m_{b}^{\textrm{pole}}$ | $m_{b}^{kin}(1\,{\textrm{GeV}})^{\dagger}$ |  |
$92.9(7)\,{\textrm{MeV}}$ | $4.18(4)\,{\textrm{GeV}}$ | $4.78(6)\,{\textrm{GeV}}$ | $4.53(6)\,{\textrm{GeV}}$ |  |
Condensates
$\langle\bar{q}q\rangle_{|2\\!\,{\textrm{GeV}}}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Bali:2012jv}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\langle\bar{s}s\rangle\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{McNeile:2012xh}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\langle G^{2}\rangle\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{SVZ79I,SVZ79II}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $m_{0}^{2}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ioffe:2002ee}{\@@citephrase{(}}{\@@citephrase{)}}}}$ |  |
$-(269(2)\,{\textrm{MeV}})^{3}$ | $1.08(16)\langle\bar{q}q\rangle$ | $0.012(4)\,{\textrm{GeV}}^{4}$ | $0.8(2)\,{\textrm{GeV}}^{2}$ |  |
Photon distribution amplitude parameters
$f_{\gamma q{|1\\!\,{\textrm{GeV}}}}^{\perp}\mbox{~{}\eqref{eq:fga}}$ | $f_{\gamma s{|1\\!\,{\textrm{GeV}}}}^{\perp}\mbox{~{}\eqref{eq:fga}}$ | $f_{3\gamma{|1\\!\,{\textrm{GeV}}}}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\omega_{\gamma{|1\\!\,{\textrm{GeV}}}}^{A}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\omega_{\gamma{|1\\!\,{\textrm{GeV}}}}^{V}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ |
$-48(3)\,{\textrm{MeV}}$ | $-61(3)\,{\textrm{MeV}}$ | $-0.004(2)^{*}\,{\textrm{GeV}}^{2}$ | $3.8(1.8)^{*}$ | $-2.1(1.0)^{*}$ |
$\kappa_{|1\\!\,{\textrm{GeV}}}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\kappa^{+}_{|1\\!\,{\textrm{GeV}}}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\zeta_{1{|1\\!\,{\textrm{GeV}}}}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\zeta_{1{|1\\!\,{\textrm{GeV}}}}^{+}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\zeta_{2{|1\\!\,{\textrm{GeV}}}}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$ | $\zeta_{2{|1\\!\,{\textrm{GeV}}}}^{+}\mbox{~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ball:2002ps}{\@@citephrase{(}}{\@@citephrase{)}}}}$
$0.2(2)$ | $0(0)$ | $0.4(4)$ | $0(0)$ | $0.3(3)$ | $0(0)$

Table 1: Summary of input parameters. An asterisk, *, indicates that the
quantity has been evaluated in the vector meson dominance approximation.
Recall that $f_{\gamma q}=\chi_{q}{\left\langle\bar{q}q\right\rangle}$ where
$\chi_{q}$ is the magnetic vacuum susceptibility of the quark condensate. The
Gegenbauer moments of the photon DA, absent in this table, are discussed after
(D.20) and set to $a_{2}(\gamma)=0(0.1)$ and $a_{4}(\gamma)=0(0)$. For the
meson masses we have not indicated an error as they are negligible for this
work. Although we do not use the twist-$4$ in the numerics, for reasons
discussed in Sec. 4.4 we include their values in the table for completeness.
The values of the $\kappa$ and $\zeta$-parameters are taken from the paper
where the photon DA was first introduced Balitsky:1997wi , and estimated those
matrix elements to be correct within a factor of two and note that the ones
set to zero are systematically smaller. We have
$\mu_{\alpha_{s}}^{2}=\mu_{F}^{2}\\!=\\!m_{B}^{2}\\!-\\!(m_{b}^{pole})^{2}$
and the renormalisation scale of the tensor current is set to
$\mu_{\text{UV}}=m_{b}^{pole}$.

The sum rule specific parameters, given in Tab. 2, are determined by imposing
a series of consistency conditions. Whilst we refrain from assigning a $q^{2}$
dependence to the SR parameters we consider the constraints set out below to
apply for $q^{2}\in[0,7]\,\mbox{GeV}^{2}$. For the effective threshold $s_{0}$
it is required that the daughter SR, which is formally exact,

$m_{B}^{2}=M^{4}\frac{d}{dM^{2}}\ln\int_{m_{b}^{2}}^{s_{0}}ds\;e^{-s/M^{2}}\rho(s,q^{2})\;,$
(60)

reproduces the known $B$-meson mass to within $1$-$2\%$. The $SU(3)$ ratio of
effective thresholds should approximately reproduce that of the meson masses
$s_{0}^{B_{s}}/s_{0}^{B_{d,u}}\approx m_{B_{s}}^{2}/m_{B_{d,u}}^{2}$, which we
use as an additional loose constraint. The Borel mass is obtained by
considering two competing factors. A large Borel mass leads to faster
convergence of the LC-OPE at the cost of greater contamination from the
continuum states, whereas a small value of the Borel mass achieves exactly the
opposite. We balance these two factors by requiring both that the continuum
contributes no more than $30\%$ and that the highest twist contribution
remains below $10\%$ of the result. Varying the Borel mass to $\pm
2\,\mbox{GeV}$ results in $1$-$2\%$ changes in the FF-value which is rather
satisfactory. Next, we turn to correlations imposed by the EOM.

| $\bar{B}_{d}$ | $\bar{B}_{u}$ | $\bar{B}_{s}$
---|---|---|---
$\\!\\!\\{s_{0},M^{2}\\}^{V_{\perp},T_{\perp},T_{\parallel}}$ | 35.2(2.0), 8.0(2.0) | 35.2(2.0), 8.0(2.0) | 36.2(2.0), 8.0(2.0)
$\\{s_{0},M^{2}\\}^{V_{\parallel}}$ | 35.8(2.0), 8.5(2.0) | 35.8(2.0), 8.5(2.0) | 36.8(2.0), 9.5(2.0)
$\\!\\!\\{s_{0},M^{2}\\}^{f_{B}}$ | 34.3(2.0), 5.6(2.0) | 34.3(2.0), 5.6(2.0) | 35.5(2.0), 6.4(2.0)
Table 2: Summary of the sum rule parameters. The reason for not correlating
the continuum thresholds for $T_{\parallel}$ and $V_{\parallel}$ is given in
Sec. 5.1.1.

####  5.1.1 Equations of Motion as Constraints on Sum Rule Parameters

Besides providing a non-trivial check of both the computation and the
formalism of the OPE and DAs itself cf. Sec. 4.4, the EOM serve an additional
purpose in that they correlate the vector and tensor FF as the derivative FF
turns out to be small. This observation was first made for the $B\to V$-FFs,
with $V=\rho,\phi,...$ a light meson, in Hambrock:2013zya  and more
systematically exploited in BSZ15 . Numerically, the derivative FFs in (4.4)
are suppressed by an order of magnitude as compared to the vector and tensor
ones. This means that the sum rule specific parameters, of Borel mass and
continuum threshold, ought to be approximately equal $s_{0}^{V}\simeq
s_{0}^{T}$ and $M_{V}\simeq M_{T}$; otherwise this would imply an
unprecedented violation of quark-hadron duality for the derivative FF.

Does this still hold for the $B\to\gamma$ FFs? Inspecting Tab. 5 we note that
this pattern persists for the $\perp$-direction but not for the
$\parallel$-direction. As the computation is a 1-to-1 formal map with the
$B\to V$ FFs for higher twist-$2$ and above (cf. Tab. 11) this suggests that
there is something special going on in the perturbative diagrams (twist-$1$
and $Q_{b}\langle qq\rangle$).

FF |  $Q_{q}$ PT${}^{\text{ LO [NLO]}}$ |  $Q_{b}$ PT${}^{\text{ LO [NLO]}}$ | $\chi\langle\bar{q}q\rangle^{\text{ LO [NLO]}}$ | $f_{3\gamma}$ | $Q_{b}\langle\bar{q}q\rangle$ | $Q_{q}\langle\bar{q}q\rangle$
---|---|---|---|---|---|---
$V_{\perp}$ | $m_{b}^{-1/2[-1/2]}$ | $m_{b}^{-3/2[-3/2]}$ | $m_{b}^{-3/2[-1/2]}$ | $m_{b}^{-3/2}$ | $m_{b}^{-3/2}$ | $m_{b}^{-5/2}$
$V_{\parallel}$ | $m_{b}^{-1/2[-1/2]}$ | $m_{b}^{-3/2[-3/2]}$ | $m_{b}^{-3/2[-1/2]}$ | $m_{b}^{-3/2}$ | $m_{b}^{-3/2}$ | $m_{b}^{-5/2}$
$T_{\perp}$ | $m_{b}^{-1/2[-1/2]}$ | $m_{b}^{-3/2[-3/2]}$ | $m_{b}^{-3/2[-1/2]}$ | $m_{b}^{-5/2}$ | $m_{b}^{-3/2}$ | $m_{b}^{-5/2}$
$T_{\parallel}$ | $m_{b}^{-1/2[-1/2]}$ | $m_{b}^{-3/2[-3/2]}$ | $m_{b}^{-3/2[-1/2]}$ | $m_{b}^{-5/2}$ | $m_{b}^{-3/2}$ | $m_{b}^{-5/2}$
$D_{\perp}$ | $m_{b}^{-3/2[-1/2]}$ | $m_{b}^{-5/2[-3/2]}$ | 0 [$m_{b}^{-3/2}$] | $m_{b}^{-5/2}$ | 0 | 0
$D_{\parallel}$ | $m_{b}^{-3/2[-1/2]}$ | $m_{b}^{-3/2[-3/2]}$ | 0 [$m_{b}^{-3/2}$] | $m_{b}^{-5/2}$ | $m_{b}^{-3/2}$ | $m_{b}^{-3/2}$
Table 3: Heavy quark scaling of the form factors according to the rules in
(61). The breaking of the hierarchy $V,T\gg{\cal D}$ at NLO is compatible with
the findings in Beneke:2000wa . The $T,V_{\perp,\parallel}\propto
Q_{q}m_{b}^{-1/2}$ and $T,V_{\perp,\parallel}\propto Q_{b}m_{b}^{-3/2}$ are
discussed from a dispersion relation approach in Secs. 2.3 (Eq. (14)) and
5.1.1 respectively.

In order to understand the origin let us briefly recall how the FF-hierarchy
can be understood in $B\to V$. We proceed by investigating the heavy quark
limit of the FF. Whereas sum rules do not lend themselves to a heavy quark
expansion, since some $m_{b}$ dependence is hidden in hadronic parameters, it
is nevertheless possible to extract the leading $1/m_{b}$-behaviour by making
the following well-known substitutions

$m_{B}\to m_{b}+\bar{\Lambda}\;,\quad s_{0}\to(m_{b}+\omega_{0})^{2}\;,\quad
M^{2}\to 2m_{b}\tau\;,$ (61)

where $\bar{\Lambda},\omega_{0}$ and $\tau$ are all parameters of ${\cal
O}(\Lambda_{\textrm{QCD}})$. For $B\to V$ it was observed that the derivative
FFs are $1/m_{b}$-suppressed at ${\cal O}(\alpha_{s}^{0})$ Hambrock:2013zya ;
BSZ15 . This pattern is broken at NLO where they show the same
scaling.141414This is without doubt related to the large energy limit FF
relations Charles:1998dr , subsequently systematically integrated into QCD
factorisation Beneke:2000wa  and soft-collinear effective theory Bauer:2000yr
. Such corrections were dubbed symmetry breaking corrections in Beneke:2000wa
and investigated at NNLO in Beneke:2005gs . The FF relations in Beneke:2000wa
are compatible with our finding from the EOM at LO in $1/m_{b}$. The
$B\to\gamma$ scaling are collected in Tab. 3. The twist-$2$ parameter
$\chi\langle\bar{q}q\rangle$, of course, confirms the earlier findings,
whereas for twist-$1$ it is found that the hierarchy is obeyed in $Q_{q}$ but
not $Q_{b}$. The former is in complete accord with the heavy quark scaling
argument, presented in Sec. 2.3 where the $Q_{q}$-part is inferred as a sum of
$B\to V_{n}$ FFs. As for the latter we can offer a similar argumentation as in
Secs. 2.3, Eq. (14), by using the dispersion representation in the limit of
large colours

$T^{\bar{B}_{s}\to\gamma}_{\perp}(q^{2})|_{Q_{b}}=\sum_{V_{n}=\Upsilon,\Upsilon^{\prime},\dots}\frac{f^{\textrm{e}m}_{V_{n}}}{m_{V_{n}}}\,T_{1}^{\bar{B}_{s}\to
V_{n}}(q^{2})+\textrm{s.t.}\;,$ (62)

where we focus on the resonances for simplicity. The heavy quark scaling

$(m_{V_{n}},f^{\textrm{e}m}_{V_{n}},T_{1}^{V_{n}\to\bar{B}_{s}}(0))\propto(m_{b},m_{b}^{-1/2},m_{b}^{0})\;,$
(63)

of the hadronic parameters are all trivial. In fact each generic term in (62)
scales like $m_{b}^{-3/2}$. Contrary to the case for the $Q_{q}$-charge, there
is no obvious significant alteration for the sum when scaling in $m_{b}$ and
we conclude $T^{\bar{B}_{s}\to\gamma}_{\perp}(q^{2})|_{Q_{b}}\propto
m_{b}^{-3/2}$, which is indeed consistent with our findings cf. Tab. 3. This
explains the scaling but also makes it clear that the $Q_{b}$-part has nothing
to do with light vector mesons and the ideas behind the large energy limit are
thus not applicable.151515Charmonium states can be neglected since $\langle
0|\bar{b}\gamma_{\mu}b|\psi_{\bar{c}c}\rangle$ is rather small. Hence there is
no reason to believe that such a hierarchy is in place and the pattern in Tab.
5 is now qualitatively understood. We consider it instructive to give a
numeric breakdown for the LO contributions up to twist-$2$

$\displaystyle V^{\bar{B}_{q}\to\gamma}_{\parallel}(0)$
$\displaystyle\;\simeq\;$ $\displaystyle(-0.01\,Q_{b}$ $\displaystyle\;-\;$
$\displaystyle 0.09\,Q_{q})|_{\text{PT}}$ $\displaystyle\;-\;$ $\displaystyle
0.11\,Q_{q}|_{\text{twist-2}}$ $\displaystyle\;+\;$
$\displaystyle\textrm{higher twist}\;,$ $\displaystyle
T^{\bar{B}_{q}\to\gamma}_{\parallel}(0)$ $\displaystyle\;\simeq\;$
$\displaystyle(-0.03\,Q_{b}$ $\displaystyle\;-\;$ $\displaystyle
0.17\,Q_{q})|_{\text{PT}}$ $\displaystyle\;-\;$ $\displaystyle
0.09\,Q_{q}|_{\text{twist-2}}$ $\displaystyle\;+\;$
$\displaystyle\textrm{higher twist}\;,$ $\displaystyle{\cal
D}^{\bar{B}_{q}\to\gamma}_{\parallel}(0)$ $\displaystyle\;\simeq\;$
$\displaystyle(\phantom{-}0.02\,Q_{b}$ $\displaystyle\;+\;$ $\displaystyle
0.10\,Q_{q})|_{\text{PT}}$ $\displaystyle\;+\;$
$\displaystyle\phantom{0.0}0\,Q_{q}|_{\text{twist-2}}$ $\displaystyle\;+\;$
$\displaystyle\textrm{higher twist}\;,$ $\displaystyle
V^{\bar{B}_{q}\to\gamma}_{\perp}(0)$ $\displaystyle\;\simeq\;$
$\displaystyle(-0.03\,Q_{b}$ $\displaystyle\;-\;$ $\displaystyle
0.17\,Q_{q})|_{\text{PT}}$ $\displaystyle\;-\;$ $\displaystyle
0.11\,Q_{q}|_{\text{twist-2}}$ $\displaystyle\;+\;$
$\displaystyle\textrm{higher twist}\;,$ $\displaystyle
T^{\bar{B}_{q}\to\gamma}_{\perp}(0)$ $\displaystyle\;\simeq\;$
$\displaystyle(-0.03\,Q_{b}$ $\displaystyle\;-\;$ $\displaystyle
0.17\,Q_{q})|_{\text{PT}}$ $\displaystyle\;-\;$ $\displaystyle
0.09\,Q_{q}|_{\text{twist-2}}$ $\displaystyle\;+\;$
$\displaystyle\textrm{higher twist}\;,$ $\displaystyle{\cal
D}^{\bar{B}_{q}\to\gamma}_{\perp}(0)$ $\displaystyle\;\simeq\;$
$\displaystyle(\phantom{-}0.00\,Q_{b}$ $\displaystyle\;+\;$ $\displaystyle
0.03\,Q_{q})|_{\text{PT}}$ $\displaystyle\;+\;$
$\displaystyle\phantom{0.0}0\,Q_{q}|_{\text{twist-2}}$ $\displaystyle\;+\;$
$\displaystyle\textrm{higher twist}\;,$ (64)

where $q=u,d$. Above we see that whilst the derivative FF is small in the
$\perp$-direction it provides a considerable contribution in the
$\|$-direction, and is the reason we allow the central values of the continuum
thresholds of $V_{\parallel}^{\bar{B}\to\gamma}$ and
$T_{\parallel}^{\bar{B}\to\gamma}$ to differ, in contrast to the other FFs.
Moreover the numerical suppression of ${\cal D}^{\perp}$ in perturbation
theory might or might not be accidental. In view of these findings we apply a
moderate (smaller than in $B\to V$ BSZ15 ) correlation of the tensor versus
vector thresholds. Further to the correlations induced by the EOM, the
effective thresholds of the FFs and the decay constant are correlated, which
is warranted as both thresholds are associated with the $B$-meson state

$\text{corr}(s_{0}^{V^{q}_{\|[\perp]}},s_{0}^{T^{q}_{\|[\perp]}})=0.5\;,\quad\text{corr}(s_{0}^{\Gamma^{q}_{\|[\perp]}},s_{0}^{f_{B}})=0.5\;,$
(65)

where $q=u,d,s$ and $\Gamma=V,T$.

####  5.1.2 Fit-ansatz

A generic FF, say $F_{n}$, is fitted to a $z$-expansion ansatz, following the
same setup as in previous work BSZ15 ; Albrecht:2019zul , where the leading
pole is factored out

$F_{n}^{\bar{B}\to\gamma}(q^{2})=\frac{1}{1-q^{2}/m_{R}^{2}}\left(\alpha_{n0}+\sum_{k=1}^{N}\alpha_{nk}(z(q^{2})-z(0))^{k}\right)\;.$
(66)

The variable $z$ describes the following map into the complex plane:
$z(t)=\frac{\sqrt{t_{+}-t}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-t}+\sqrt{t_{+}-t_{0}}}$
($t_{0}\equiv t_{+}(1-\sqrt{1-t_{-}/t_{+}})$, $t_{\pm}\equiv(m_{B_{q}}\pm
m_{\rho})^{2}$ and $m_{\rho}=770\,\mbox{MeV}$). It is noted that $t_{\pm}\to
m_{B_{q}}\pm 2m_{\pi},m_{B_{s}}\pm 2m_{K}$ is formally correct but we use
$t_{\pm}$ as above with no impact on our results in the physical region. We
note that whilst $t_{+}$ corresponds to the multiparticle production
threshold, the choice of $t_{0}$ (and therefore $t_{-}$) is arbitrary and does
not impact our results. The masses of the lowest lying $J^{P}=1^{-(+)}$
states, relevant to the $\perp(\parallel)$-projection, are
$m_{R}=m_{B^{*}_{q}(B_{1q})}$.

Relations between the FFs provide constraints on the fit parameters. For
example (6) implies

$T^{\bar{B}\to\gamma}_{\perp}(0)=T^{\bar{B}\to\gamma}_{\parallel}(0)\;\;\Leftrightarrow\;\;\alpha_{T_{\perp}0}=\alpha_{T_{\parallel}0}\;.$
(67)

Further constraints arise from the dispersion representation. Considering the
vector FF for concreteness,

$V^{\bar{B}\to\gamma}_{\parallel[\perp]}(q^{2})=\frac{1}{\pi}\int^{\infty}_{\textrm{cut}}dt\frac{\textrm{Im}[V^{\bar{B}\to\gamma}_{\parallel[\perp]}(t)]}{t-q^{2}-i0}=\frac{r^{V}_{\parallel[\perp]}}{1-q^{2}/m_{B^{*}[B_{1}]}^{2}}+\dots\;,$
(68)

one can infer the residue of the lowest-lying resonance161616For $B\to\pi$ a
two pole ansatz fitted to the $f_{+}(q^{2})$ FF in LCSR within the
$0<q^{2}<14\,\mbox{\small GeV}^{2}$-range reproduces a residue, consistent
with the best determination of the these hadronic parameters BZ04
($g_{DD^{*}\pi}$ from experiment and heavy quark scaling).

$\displaystyle
r^{(V,T)}_{\perp}\;=\;\left(\frac{m_{B}f_{B^{*}}}{m_{B^{*}}},f^{T}_{B^{*}}\right)g_{BB^{*}\gamma}\;,\quad
r^{(V,T)}_{\parallel}\;=\;\left(\frac{m_{B}f_{B_{1}}}{m_{B_{1}}},f^{T}_{B_{1}}\right)g_{BB_{1}\gamma}\;,$
(69)

with some more detail in Pullin:2021ebn  along with the definition of the
decay constants $f_{B^{*},B_{1}}^{(T)}$ and the on-shell matrix elements
$g_{BB^{*}(B_{1})\gamma}$ describing the $B^{*}(B_{1})\to B\gamma$ transition.
This relation enforces the constraint

$r_{V_{\perp}}=\alpha_{V_{\perp}0}+\sum_{k=1}^{N}\alpha_{V_{\perp}k}(z(m_{B^{*}_{q}}^{2})-z(0))^{k}\;,$
(70)

and similar constraints for other FFs. The main idea of the fit is that the
four FFs with $15$ $q^{2}$-points (ranging from $[0,14]\,\mbox{GeV}^{2})$ and
$N_{\text{samples}}=2500$ samples are subjected to a single fit providing
correlations between the fit parameters of the FFs. This is the same Markov
Chain Monte Carlo methodology as used in BSZ15 . Data points are generated
from normal distributions correlated as in (65), with the exception of
$\\{\mu_{F},\mu_{\text{cond}},\mu_{\alpha_{s}},\mu_{\text{kin}},M^{2},M_{f_{B}}^{2}\\}$
for which a skew normal distribution is applied. The latter is necessary since
clearly not all parameters have a symmetric validity-range. The skew normal
distribution is defined by three parameters, which we take to be the mean, the
standard deviation and a _shape parameter_ , $\alpha_{\text{shape}}$. The
latter is chosen such that the probability of generating a sample below a
certain cut-off is less than $10^{-4}$. For the scales we impose
$\\{\mu_{F},\mu_{\text{cond}},\mu_{\alpha_{s}}\\}>0.8\,\mbox{GeV}$ and
$\mu_{\text{kin}}>0.2\,\mbox{GeV}$ whilst for the Borel parameters we require
$M^{2}>4\,\mbox{GeV}^{2}$ and $M^{2}_{f_{B}}>2\,\mbox{GeV}^{2}$, below which
the twist expansion is non-convergent. The fitted FFs reproduce the mean
values of generated FF data sets to within $\approx 0.5\%$ over the fitted
region. The values of the fit parameters are given in Tab. 4, with some more
description in the caption. Covariance and correlation matrices along with the
central values of the fit-parameters are given as ancillary file appended to
the arXiv version of this paper. The JSON formatted file is ordered according
to charge, statistical quantity and FF(s) of interest and can be readily used
in Mathematica. For example, after loading the data

| $V_{\perp}^{\bar{B}_{d}\to\gamma}$ | $V_{\perp}^{\bar{B}_{s}\to\gamma}$ | $V_{\perp}^{\bar{B}_{u}\to\gamma}$ | $T_{\perp}^{\bar{B}_{d}\to\gamma}$ | $T_{\perp}^{\bar{B}_{s}\to\gamma}$ | $T_{\perp}^{\bar{B}_{u}\to\gamma}$
---|---|---|---|---|---|---
$\alpha_{0}$ | $\phantom{\text{-}}0.133(17)$ | $\phantom{\text{-}}0.158(18)$ | $\text{-}0.213(26)$ | $\phantom{\text{-}}0.133(13)$ | $\phantom{\text{-}}0.151(13)$ | $\text{-}0.213(19)$
$\alpha_{1}$ | $\text{-}0.123(66)$ | $\text{-}0.459(200)$ | $\phantom{\text{-}}0.278(155)$ | $\text{-}0.093(362)$ | $\text{-}0.284(357)$ | $\phantom{\text{-}}0.240(545)$
$\alpha_{2}$ | $\phantom{\text{-}}1.280(913)$ | $\phantom{\text{-}}1.798(1.303)$ | $\text{-}1.760(1.486)$ | $\phantom{\text{-}}1.075(3.277)$ | $\phantom{\text{-}}2.489(3.356)$ | $\text{-}1.346(4.707)$
$\alpha_{3}$ | $\phantom{\text{-}}3.373(2.485)$ | $\phantom{\text{-}}6.603(3.283)$ | $\text{-}4.940(4.016)$ | $\phantom{\text{-}}2.764(6.493)$ | $\phantom{\text{-}}7.288(6.915)$ | $\text{-}3.848(9.281)$
| $V_{\|}^{\bar{B}_{d}\to\gamma}$ | $V_{\|}^{\bar{B}_{s}\to\gamma}$ | $V_{\|}^{\bar{B}_{u}\to\gamma}$ | $T_{\|}^{\bar{B}_{d}\to\gamma}$ | $T_{\|}^{\bar{B}_{s}\to\gamma}$ | $T_{\|}^{\bar{B}_{u}\to\gamma}$
$\alpha_{0}$ | $\phantom{\text{-}}0.081(9)$ | $\phantom{\text{-}}0.104(9)$ | $\text{-}0.151(18)$ | $\phantom{\text{-}}0.133(13)$ | $\phantom{\text{-}}0.151(13)$ | $\text{-}0.213(19)$
$\alpha_{1}$ | $\text{-}0.138(53)$ | $\text{-}0.177(63)$ | $\phantom{\text{-}}0.312(129)$ | $\text{-}0.071(377)$ | $\text{-}0.237(359)$ | $\phantom{\text{-}}0.230(535)$
$\alpha_{2}$ | $\phantom{\text{-}}0.803(769)$ | $\phantom{\text{-}}1.882(649)$ | $\text{-}1.749(1.260)$ | $\phantom{\text{-}}1.241(3.207)$ | $\phantom{\text{-}}1.841(3.139)$ | $\text{-}1.397(4.340)$
$\alpha_{3}$ | $\phantom{\text{-}}1.986(1.408)$ | $\phantom{\text{-}}3.951(1.220)$ | $\text{-}4.263(2.314)$ | $\phantom{\text{-}}2.746(4.768)$ | $\phantom{\text{-}}4.180(4.684)$ | $\text{-}3.472(6.397)$

Table 4: Fit parameters. To obtain the fits we generate $2500$ random samples
of the input parameters, normally distributed such that for a given parameter
the mean of the total sample set corresponds to the central value and the
standard deviation to the uncertainty. The fit parameters $\alpha_{n}$ are
then produced by fitting the data set generated by evaluating the form factors
at integer values of $q^{2}\in[0,14]\,\mbox{GeV}^{2}$ for each sample to the
ansatz (66) with $N=3$.

data$=$Import[…/BtoGam_FF.json]

all information can be accessed via the OptionValue command. For example, the
central value of $\alpha_{T_{\perp}0}^{\bar{B}_{u}}$ is obtained with

OptionValue[data, "Bu"$\to$"central"$\to$"Tperp"$\to$"a0"] .

Similarly the correlation between $\alpha_{V_{\|}0}^{\bar{B}_{u}}$ and
$\alpha_{T_{\|}2}^{\bar{B}_{u}}$ can be accessed via

OptionValue[data, "Bu"$\to$"correlation"$\to$"VparaTpara"$\to$"a0a2"] .

The uncertainty and covariance matrix are accessed in an analogous manner.

###  5.2 Results and Plots

The central values of the FFs (2.1) at $q^{2}=0$ and the residues (69) are
given in Tab. 5. Plots of the neutral $\bar{B}_{d}$-, $\bar{B}_{s}$\- and
charged $\bar{B}_{u}$-modes are shown in Fig. 7. Together with the fit-values
given in Tab. 4 of the previous section and the correlation matrix given in an
ancillary file this constitutes the main practical results of our paper.

A breakdown of contributions, split according to twist and DAs, is given in
Tab. 6. We remind the reader that all twist-$4$ contributions, other than the
$Q_{b}{\left\langle\bar{q}q\right\rangle}$ condensates, are dropped as they
require the inclusion of $4$-particle DAs cf. Sec. 4.4. This effect which is
fortunately well below $10\%$ as can inferred from Tab. 5.

| $V_{\perp}(0)$ | $V_{\|}(0)$ | $T_{\perp,\|}(0)$ |
---|---|---|---|---
$\\!\\!\bar{B}_{d}$ | $0.130(17)$ | $0.079(9)\phantom{0}$ | $0.129(17)$ | $-$
$\bar{B}_{s}$ | $0.160(18)$ | $0.105(10)$ | $0.153(16)$ | $-$
$\bar{B}_{u}$ | $-0.209(26)$ | $-0.148(18)$ | $-0.209(26)$ | $-$
| $r_{\perp}^{V}$ | $r_{\parallel}^{V}$ | $r_{\perp}^{T}$ | $r_{\parallel}^{T}$
$\\!\\!\bar{B}_{d}$ | $\phantom{-}0.179(19)$ | ${0.076}(16)$ | ${0.171}(20)$ | ${0.104}(15)$
$\bar{B}_{s}$ | $\phantom{-}0.235(25)$ | ${0.114}(18)$ | ${0.224}(24)$ | ${0.146}(17)$
$\bar{B}_{u}$ | $-0.300(34)$ | ${-0.159}(31)$ | ${-0.284}(33)$ | ${-0.199}(28)$
Table 5: Values of the form factors (2.1) at $q^{2}=0$ along with the residues
(69) evaluated in the kinetic scheme, cf. Pullin:2021ebn . Values for the
threshold and Borel mass can be found in Tab. 2. All other input parameters
are given in Tab. 1.

Whereas formally the fit-parameter $\alpha_{0}$ equals the FF-value at
$q^{2}=0$, on comparison with Tab. 5 we see a deviation of ${\cal O}(2\%)$
between the two. This comes as the FFs are evaluated analytically whereas the
fit-values follow from the Monte Carlo Markov-chain analysis
($N_{\text{samples}}=2500$ random samples) and the deviations are due to
slightly asymmetric distribution of the uncertainty.

From the plots one infers the typical behaviour of a FF. Its value at
$q^{2}=0$ is below the normalisation of a current, e.g. the electromagnetic
pion FF $f_{+}^{\pi\to\pi}(0)=1$ and it raises as a result of hadronic poles
(or multiquark thresholds in partonic QCD). One sees that the constraint from
the first pole is well in line with the raise reproduced in our computation
for $q^{2}\approx 14\,\mbox{GeV}^{2}$ and would become progressively worse. In
the tensor case the $\perp$\- and $\parallel$-directions only show significant
deviations for large $q^{2}$, with the coincidence at $q^{2}=0$ enforced by
(6). For the vector FFs this pattern would be reproduced if the derivative FFs
were small. However, this is not the case in the $\|$-direction, as the vector
form factor differ more notably (cf. (5.1.1) and the discussion of Sec.
5.1.1). Furthermore, we find that for both the twist-2 and leading $1/m_{b}$
twist-1 contribution (proportional to $Q_{q}$ only) the $\perp$\- and
$\parallel$-directions give identical results. Hence any degree of asymmetry
between $\perp$ and $\parallel$ must arise from sub-leading perturbative
and/or higher twist contributions. Our decision to drop hadronic parameters
that do not close under the EOM from the analysis (cf. Sec. 4.4) therefore
acts to increase the symmetry between the two directions.

twist | pa | DA | $V_{\perp}^{\bar{B}_{u}}(0)$ | $V_{\perp}^{\bar{B}_{d}}(0)$ | $V_{\|}^{\bar{B}_{u}}(0)$ | $V_{\|}^{\bar{B}_{d}}(0)$ | $T_{\perp,\|}^{\bar{B}_{u}}(0)$ | $T_{\perp,\|}^{\bar{B}_{d}}(0)$ | $T_{\perp,\|}^{\bar{B}_{s}}(0)$
---|---|---|---|---|---|---|---|---|---
1 | - | $\textrm{PT}\,{\cal O}{(\alpha_{s}^{0})}$ | $-0.104$ | $0.065$ | $-0.054$ | $0.032$ | $-0.105$ | $0.065$ | $0.086$
1 | - | $\textrm{PT}\,{\cal O}{(\alpha_{s})}$ | $-0.024$ | $0.019$ | $-0.010$ | $0.006$ | $-0.027$ | $0.021$ | $0.021$
2 | 2 | $\phi_{\gamma}(u)\,{\cal O}{(\alpha_{s}^{0})}$ | $-0.072$ | $0.036$ | $-0.073$ | $0.037$ | $-0.062$ | $0.031$ | $0.037$
2 | 2 | $\phi_{\gamma}(u)\,{\cal O}{(\alpha_{s})}$ | $-0.010$ | $0.005$ | $-0.011$ | $0.006$ | $-0.011$ | $0.005$ | $0.007$
3 | 2 | $\Psi_{(a)}(u)$ | $-0.002$ | $0.001$ | – | – | $-0.001$ | $<\\!10^{-3}$ | $-0.003$
3 | 2 | $\Psi_{(v)}^{(1)}(u)$ | – | – | $-0.003$ | $0.001$ | $-0.002$ | $0.001$ | $-0.001$
3 | 3 | $\mathcal{A}(\underline{\alpha})$ | – | – | – | – | $-0.002$ | $0.001$ | $<\\!10^{-3}$
3 | 3 | $\mathcal{V}(\underline{\alpha})$ | – | – | – | – | $-0.001$ | $<\\!10^{-3}$ | $<\\!10^{-3}$
4 | 2 | $h_{\gamma}^{(2)}(u)$ | – | – | $0.011$ | $-0.005$ | $0.004$ | $-0.002$ | $-0.002$
4 | 2 | $\mathbb{A}(u)$ | $0.015$ | $-0.007$ | $0.014$ | $-0.007$ | $0.011$ | $-0.005$ | $-0.005$
4 | 3 | $S(\underline{\alpha})\\!+\\!\tilde{S}(\underline{\alpha})$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $-0.001$ | $<\\!10^{-3}$ | $<\\!10^{-3}$
4 | 3 | $\sum\limits_{{i=1}}^{4}T_{i}^{(1)}(\underline{\alpha})$ | $-0.002$ | $0.001$ | $-0.002$ | $0.001$ | $-0.004$ | $0.002$ | $0.002$
4 | 3 | $S_{\gamma}$ | $0.003$ | $0.003$ | $-0.002$ | $-0.002$ | $0.003$ | $0.001$ | $0.003$
4 | 3 | $T_{4}^{\gamma}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$ | $<\\!10^{-3}$
4 | - | $Q_{q}\left\langle\bar{q}q\right\rangle$ | – | – | $0.010$ | $0.002$ | – | – | –
4 | - | $Q_{b}\left\langle\bar{q}q\right\rangle$ | $0.004$ | $0.004$ | $0.003$ | $-0.003$ | $0.003$ | $0.003$ | $0.003$
|  | Total∗ | $-0.194$ | $0.127$ | $-0.118$ | $0.067$ | $-0.197$ | $0.127$ | $0.151$

Table 6: A breakdown of contributions for a representative selection of form
factors according to associated twist, “pa” = number of partons and DAs. The
asterisk in total is a reminder that it includes twist-$4$ contributions that
do not close under the EOM (cf. Sec. 4.4). The definitions of the DAs can be
found in App. D. Results are given in the kinetic scheme with the kinetic
scale $\mu_{\textrm{kin}}=1\,\mbox{GeV}$.

Figure 7: Form factors for the $\bar{B}_{d,s,u}\to\gamma$ transition where the
data points (with solid bars as respective uncertainties) have been generated
by a $N_{\text{samples}}=2500$ sample and fitted to the ansatz (66). The plots
are reproduced from the fit-values in Tab. 4 and the uncertainty bands stem
from the correlation matrix given as an ancillary file cf. Sec. 5.1 for
further details. Whereas we consider the LCSR computation valid up to
$q^{2}<14\,\mbox{GeV}^{2}$, the ansatz (66) with the pole residue from
Pullin:2021ebn  allows one to extend the form factor beyond this region. The
sign of the charged form factors is reversed so that they are positive. We
refer the reader to App. 2.1.1 for discussion of conventions.

####  5.2.1 Comparison with the Literature

In this section we compare our FFs to some of the results in the literature.
Comparison with SCET computations of these FFs Beneke:2020fot  is not
straightforward as the $B$-meson DA parameters are not known with a lot of
certainty. It is therefore advisable to turn the tables and use our
predictions to set bounds as done in Sec. 5.4. Comments on comparison in terms
of analytic computations can be found at the beginning of App. E. Lattice
results are available for $D_{d,s}\to\gamma$ and $K,\pi\to\gamma$ modes in
Desiderio:2020oej  and in preparation by another group Kane:2019jtj . At low
photon recoil such computations are of importance for QED-corrections. Whereas
we have chosen not to present $D\to\gamma$ FFs, the pole residues of the
vector FFs, $g_{DD^{*}\gamma}$, are computed in our other work Pullin:2021ebn
and are consistent with experiment in the measured modes. A notable aspect is
that the shape in Desiderio:2020oej  does not seem to be compatible with the
$D^{*}$-pole playing a significant role close to the kinematic endpoint of
$D\to\gamma\ell\bar{\nu}$ ($q^{2}=m_{D}^{2}$).

We can, however, compare to the quark model computations MN04 ;
Kozachuk:2017mdk  (with
$T^{\bar{B}_{d}\to\gamma}_{\perp,\parallel}=F^{\bar{B}_{d}\to\gamma}_{T_{V,A}}$
dictionary). Two representative plots of the neutral modes are given in Fig.
8. There is rather good agreement at low $q^{2}$ but we can clearly see that
our FFs are larger for higher $q^{2}$ which ought have a significant impact on
phenomenology.

Figure 8: Representative comparison plots for neutral modes. Our plots
(solid,blue/green) , denoted by JPZ, compared against the quark model
computation MN04  (short-dashed,light grey) from 2004 and an improved version
Kozachuk:2017mdk  (dashed,black) with the modified pole version. Note that
there are $\bar{B}_{s}\to\gamma$ results in the 2004 work. Our results are
larger and, in particular, show an earlier rise. In view of this the numerical
agreement of the central value at $V^{\bar{B}_{d}\to\gamma}_{\perp}(0)$ seems
somewhat accidental.

For illustrative purposes we have added Fig. 9 comparing the form factor with
and without the point-like contribution added.

Figure 9: Plot comparing the $V_{\parallel}^{\bar{B}_{u}\to\gamma}$ form
factor, as defined in (2.1), with and without the inclusion of the point-like
contribution $Q_{\bar{B}}f^{(pt)}_{\bar{B}_{u}}$. The value of the decay
constant $f_{B}=190\,\mbox{MeV}$ is taken from Pullin:2021ebn  (cf. references
therein). The point-like term provides a significant contribution, most
notably towards the $q^{2}=m_{B}^{2}$ pole.

###  5.3 Predictions for the $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ Rate

The $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ decay proceeds via a charged
current and depends on the local FF only. Neglecting the lepton mass,
anticipating a measurement in the electron mode, the differential rate after
integrating over the lepton angle is given by

$\frac{d\Gamma}{dE_{\gamma}}=\frac{G^{2}_{F}\alpha|V_{\textrm{ub}}|^{2}}{6\pi^{2}}m_{B}E_{\gamma}^{3}\left(1-\frac{2E_{\gamma}}{m_{B}}\right)\left(|V_{\perp}|^{2}+|V_{\parallel}|^{2}\right)\;,$
(71)

where $E_{\gamma}=(m_{B}^{2}-q^{2})/(2m_{B})\in[0,\frac{m_{B}}{2}]$ is the
photon energy in the $B$-meson restframe, such that
$\frac{d\Gamma}{dq^{2}}=-\frac{1}{2m_{B}}\frac{d\Gamma}{dE_{\gamma}}$. We
refer the reader to App. B.3 for an explanation of why the point-like part,
proportional to $f_{B}$, disappears in the $m_{\ell}\to 0$ limit.

Plots of the branching fraction are shown, in both the $q^{2}$ and the
$E_{\gamma}$ variables, in Fig. 10. The $q^{2}$ and $E_{\gamma}$ covariance
matrices between the bins of Tab. 7 are respectively given by

$\text{cov}_{\text{bin}}^{q^{2}}\\!=\\!\\!\left(\\!\begin{array}[]{cccc}0.0534&0.0502&0.0459&0.0393\\\
0.0502&0.0471&0.0431&0.0369\\\ 0.0459&0.0431&0.0394&0.0337\\\
0.0393&0.0369&0.0337&0.0289\\\
\end{array}\\!\right),\quad\text{cov}_{\text{bin}}^{E_{\gamma}}\\!=\\!\left(\\!\\!\begin{array}[]{cccc}0.5537&0.4521&0.3188&0.1787\\\
0.4521&0.3736&0.2675&0.1523\\\ 0.3188&0.2675&0.1954&0.1133\\\
0.1787&0.1523&0.1133&0.0669\\\ \end{array}\\!\right).$ (72)

The $\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ rate will, finally, be measured at
Belle II Kou:2018nap . A $90\%$ confidence level (CL), ${\cal
B}(\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu})_{E_{\gamma}>1\,\mbox{GeV}}<3.0\cdot
10^{-6}\;@90\%\textrm{ CL}$, with Belle data has been reported two years ago
Gelb:2018end . This limit is about a factor of two above our predictions of
$1.60(37)\cdot 10^{-6}$ given in Tab. 7.

Figure 10: Plots showing the differentiated branching fraction with respect to
the di-lepton momentum transfer $q^{2}$ (left) and the photon energy
$E_{\gamma}$ (right). The solid blue line indicates the central value with the
correlated uncertainty (shaded region) on both the fit-parameters and the CKM
matrix element $|V_{\textrm{ub}}|$. The contribution to the total uncertainty
from the remaining input parameters is negligible. In the evaluation of the
branching ratio we use $G_{F}=1.166378\cdot 10^{-5}\,\mbox{GeV}^{-2}$,
$|V_{\textrm{ub}}|=3.82(24)\cdot 10^{-3}$, $\tau_{B}=1.638\cdot 10^{-12}s$ PDG
and the fine structure constant at the $B$-meson scale $\alpha=1/129$.

$q^{2}$ bin $[\,\mbox{\small GeV}^{2}]$ | $[0,6]\,\mbox{\small GeV}^{2}$ | $[2,6]\,\mbox{\small GeV}^{2}$ | $[3,6]\,\mbox{\small GeV}^{2}$ | $[4,6]\,\mbox{\small GeV}^{2}$
---|---|---|---|---
$10^{6}\times\mathcal{B}(\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu})$ | 0.243(53) | 0.214(47) | 0.180(39) | 0.132(29)
$E_{\gamma}$ bin $[\,\mbox{\small GeV}]$ | $[0.5\,\mbox{\small GeV},\frac{m_{B}}{2}]$ | $[1.0\,\mbox{\small GeV},\frac{m_{B}}{2}]$ | $[1.5\,\mbox{\small GeV},\frac{m_{B}}{2}]$ | $[2.0\,\mbox{\small GeV},\frac{m_{B}}{2}]$
$10^{6}\times\mathcal{B}(\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu})$ | 2.224(554) | 1.604(374) | 0.883(195) | 0.305(67)

Table 7: Integrated branching fraction for various $q^{2}$ and $E_{\gamma}$
bins. The photon energies to the left are denoted by
$E_{\gamma}^{\textrm{min}}$ elsewhere in the text. The last three branching
fraction predictions are overlaid (rescaled by $(3.70/3.82)^{2}$ since
$|V_{\textrm{ub}}|_{\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Beneke:2018wjp}{\@@citephrase{(}}{\@@citephrase{)}}}
}}=3.70(16)\cdot 10^{-3}$) in Fig. 11 in order to extract $\lambda_{B}$.

###  5.4 Extraction of the Inverse Moment $\lambda_{B}$ of the $B$-meson DA

As previously mentioned, besides being a toy model for factorisation,
$\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ is of interest to extract the
$B$-meson DA parameters from experiment. Here, we replace experiment by our
computation which comes with the additional bonus that the ${\cal O}(8\%)$
uncertainty of the CKM matrix element $|V_{\textrm{ub}}|=3.82(24)\cdot
10^{-3}$ PDG  is eliminated. This is also useful as $|V_{\textrm{ub}}|$
suffers from some tension between exclusive and inclusive $b\to u$ transitions
PDG  .

Let us turn to some minimal definitions in order to clarify the context. The
$B$-meson DA of leading twist-$2$, $\phi^{B}_{+}$ was originally introduced in
Grozin:1996pq . Its inverse moment

$\frac{1}{\lambda_{B}(\mu)}=\int_{0}^{\infty}d\omega\frac{\phi^{B}_{+}(\omega,\mu)}{\omega}\;,$
(73)

is a genuinely unknown non-perturbative parameter $\lambda_{B}={\cal
O}(\Lambda_{\textrm{QCD}})$. Above $\mu$ is the renormalisation scale (or
factorisation scale in processes), assumed to be $\mu=1\,\mbox{\small GeV}$ if
not stated otherwise. Mainly due to its non-local character it has so far
evaded a direct first principle determination. In computations using the heavy
quark limit it is often the leading uncertainty. More precisely, at NLO there
are two further non-perturbative parameters that appear, the inverse
logarithmic moments $\hat{\sigma}_{1,2}$ e.g. DescotesGenon:2002mw ;
Beneke:2011nf

$\hat{\sigma}_{n}(\mu)=\lambda_{B}(\mu)\int_{0}^{\infty}\frac{d\omega}{\omega}\ln^{n}\frac{\mu_{0}}{\omega}\phi_{+}(\omega,\mu)\;,\quad\mu_{0}=\lambda_{B}e^{-\gamma_{E}}\;,$
(74)

and one should be aware of different conventions due to the choice of
$\mu_{0}$ resulting in different values of $\sigma_{n}$. Matters are further
complicated by the fact that $\lambda_{B}$ and $\hat{\sigma}_{1,2}$ are
defined at $\mu=1\,\mbox{\small GeV}$ but in computations they are evaluated
at the hard-collinear scale $\mu={\cal
O}(\sqrt{\Lambda_{\textrm{QCD}}m_{b}})\approx 2.2\,\mbox{\small GeV}$. The
evolution of the $B$-meson DA parameters involves solving an integro-
differential equation Lange:2003ff ; Braun:2019wyx ; Galda:2020epp  already at
${\cal O}(\alpha_{s})$ with no autonomous evolution of the parameters (unlike
for light-meson DAs where the conformal symmetry allows for a simpler picture,
cf. Sec. 4.2.3). In essence a separate (indirect) determination of these
parameters is not feasible. Thus actual studies one is forced to use a model-
ansatz for $\phi^{B}_{+}$. This is done in reference Beneke:2018wjp  for a
$3$-parameter family, which permits an analytic evolution at ${\cal
O}(\alpha_{s})$. This $3$-parameter model is further constrained to three
$2$-parameter models which give rise to the following ranges in
$\hat{\sigma}_{1,2}$171717The ranges
$(\sigma_{1},\sigma_{2})_{\mu_{0}=1{\mbox{\tiny\text{
GeV}}}}=(1.5(1),3(2))\,\mbox{\small GeV}$ (the $\hat{\sigma}_{1}$-value comes
from non-local sum rule Braun:2003wx  and the $\hat{\sigma}_{2}$-value is the
one adopted in Beneke:2011nf ), translate into ranges of
$(\hat{\sigma}_{1},\hat{\sigma}_{2})\approx(-0.1(1.0),-4.4(3.2)(2.0))\,\mbox{\small
GeV}$ when $\lambda_{B}=360\,\mbox{\small MeV}$ is assumed.

Model I $\displaystyle-0.31<\;$ $\displaystyle\hat{\sigma}_{1}$
$\displaystyle<0.0\;,\quad$ $\displaystyle\phantom{-}0.74<\;$
$\displaystyle\hat{\sigma}_{2}$ $\displaystyle<1.64\;,\quad$
$\displaystyle\textrm{\color[rgb]{0,0,0.5} [blue]}\;,$ Model II
$\displaystyle-0.31<\;$ $\displaystyle\hat{\sigma}_{1}$
$\displaystyle<0.69\;,\quad$ $\displaystyle\phantom{-}0.74<\;$
$\displaystyle\hat{\sigma}_{2}$ $\displaystyle<5.42\;,\quad$
$\displaystyle\textrm{\color[rgb]{0,0.6,0} [green]}\;,$ Model III
$\displaystyle-0.69<\;$ $\displaystyle\hat{\sigma}_{1}$
$\displaystyle<0\;,\quad$ $\displaystyle\phantom{-}1.16<\;$
$\displaystyle\hat{\sigma}_{2}$ $\displaystyle<1.64\;,\quad$
$\displaystyle\textrm{\color[rgb]{1,0,0} [red]}\;.$ (75)

Figure 11: Plots taken from Beneke:2018wjp  (coloured curves) overlaid with
our predictions for the integrated branching ratios (cf. Tab. 7) scaled to
account for the fact that in Beneke:2018wjp
$|V_{\textrm{ub}}|_{\textrm{excl}}=3.70(16)\cdot 10^{-3}$ is used. This
results an effect of a factor of $0.938$ on the branching fraction. Solid
black lines represent our central values with uncertainty bands indicated by
dashed lines. The colour coding of the plots corresponds to the different
models cf. (5.4) and Beneke:2018wjp  of course.

The variation of the model and the model parameter effectively accounts for
the lack of knowledge of $\phi^{B}_{+}$.

$E_{\gamma}^{\text{min}}$ | $\lambda_{B}^{\text{Model I}}$ | $\lambda_{B}^{\text{Model II}}$ | $\lambda_{B}^{\text{Model III}}$ | $\lambda_{B}^{\text{Model Avg.}}$
---|---|---|---|---
$1.0\,\mbox{GeV}$ | $355(60)\,\mbox{MeV}$ | $295(60)\,\mbox{MeV}$ | $415(60)\,\mbox{MeV}$ | $350(120)\,\mbox{MeV}$
$1.5\,\mbox{GeV}$ | $365(60)\,\mbox{MeV}$ | $310(60)\,\mbox{MeV}$ | $415(60)\,\mbox{MeV}$ | $360(110)\,\mbox{MeV}$
$2.0\,\mbox{GeV}$ | $375(60)\,\mbox{MeV}$ | $315(60)\,\mbox{MeV}$ | $420(60)\,\mbox{MeV}$ | $370(110)\,\mbox{MeV}$
Table 8: Values of $\lambda_{B}$ read off from Fig. 11 for the three
$\phi_{+}^{B}$ models, rounded to $5\,\mbox{MeV}$. The rightmost column
contains the mean value over the models for each minimum photon energy cut,
rounded to the nearest $10\,\mbox{MeV}$ and the
$E_{\gamma}^{\text{min}}=1.5\,\mbox{GeV}$ is selected as the our final value
in (76).

In Fig. 11 we have overlaid our branching fraction predictions, shown in Tab.
7, with the plots from Beneke:2018wjp . From Tab. 8 one sees that the values
predicted for a given model are largely stable on the minimum photon energy
cut. There is, as expected, some dependence on the specific model of
$\phi_{+}^{B}$. However, averaging over the ranges of the three models (cf.
right hand column of Tab. 8), one again sees the consistency between the
different photon cuts, which suggests good agreement in the shape for these
values.

Whereas $E_{\gamma}^{\textrm{min}}=2\,\mbox{\small GeV}$ is arguably the best
result because of the range of validity of the approaches, it still seems
preferable to take an average of the three values for
$\lambda_{B}^{\text{Model Avg.}}$ to deduce our final value in this work

$\lambda_{B}=360(110){\,\mbox{MeV}}\;.$ (76)

We now turn to discussing the value obtained in relation to other
determinations. The previously mentioned Belle measurement from 2018 sets a
bound of $\lambda_{B}>240{\,\mbox{\small MeV}}$ $@90\%\textrm{ CL}$ using
Beneke:2018wjp  as the reference input. A direct determination via a QCD sum
rule, which however uses non-local quark condensates which are not free from
model-dependence, was performed in 2003 and gave
$\lambda_{B}=460(100){\,\mbox{\small MeV}}$ Braun:2003wx  (and the previously
quoted value for $\hat{\sigma}_{1}$). An update Khodjamirian:2020hob  using
improved numerical input and two models for the $B$-meson DA yields
$\lambda_{B}=383(153){\,\mbox{\small MeV}}$, a value closer to ours. The same
strategy as here has been applied at LO Ball:2003fq ; Khodjamirian:2005ea  and
NLO Wang:2015vgv ; Gao:2019lta  with various specific model functions for
$\phi_{+}^{B}$. The $B\to\gamma$ Ball:2003fq  LO-analysis has been done in the
heavy quark limit and resulting in $\lambda_{B}=600{\,\mbox{\small MeV}}$ with
no given error. The LO $B\to\pi$ analysis with an exponential model gave
$\lambda_{B}=460(160){\,\mbox{\small MeV}}$ Khodjamirian:2005ea . At NLO there
are the works on $B\to\pi$ Wang:2015vgv  and $B\to\rho$ Gao:2019lta  with
$\lambda_{B}=354^{+30}_{-38}{\,\mbox{\small MeV}}$ and
$343^{+64}_{-79}{\,\mbox{\small MeV}}$ respectively for which we quote the
exponential model values (cf. those references for other model
determinations).

$\lambda_{B}[\,\mbox{\small MeV}]$ | ref. | year | method | B-DA model
---|---|---|---|---
$460(100)$ | Braun:2003wx  | ’03 | QCD SR (non-local condensate) | –
$383(153)$ | Khodjamirian:2020hob  | ’20 | idem | –
$354^{+30}_{-38}$ | Wang:2015vgv  | ’15 | B-LCSR vs LCSR $B\to\pi$ BZ04b ; Duplancic:2008ix  | exponential (81) Wang:2015vgv
$368^{+42}_{-32}$ | Wang:2015vgv  | ’15 | idem | Model-II (81) Wang:2015vgv
$389^{+35}_{-28}$ | Wang:2015vgv  | ’15 | idem | Model-II (81) Wang:2015vgv
$303^{+35}_{-28}$ | Wang:2015vgv  | ’15 | idem | Model-II (81) Wang:2015vgv
$343^{+64}_{-79}$ | Gao:2019lta  | ’19 | B-LCSR vs LCSR $B\to\rho$ BSZ15  | exponential
$370^{+69}_{-86}$ | Gao:2019lta  | ’19 | idem | local duality
600 | Ball:2003fq  | ’03 | SCET DescotesGenon:2002mw  vs LCSR $B\to\gamma$ ($m_{b}\to\infty$) | –
365(60) | this work | ’21 | LCSR vs SCET Beneke:2018wjp  | Model I (5.4a) Beneke:2018wjp
310(60) | this work | ’21 | idem | Model II (5.4b) Beneke:2018wjp
415(60) | this work | ’21 | idem | Model III (5.4c) Beneke:2018wjp

Table 9: Comparison-table of $\lambda_{B}$ with respect to the literature. The
issue of the model-dependence of the indirect determinations by matching two
FF computations is clearly visible. B-LCSR stands for LCSR with a $B$-meson
DA. Cf. final paragraph of the section for further comments.

To conclude, from Tab. 9 it is seen that the restriction to a single model
does reduce the error considerably and underlines the importance of the QCD SR
determinations. In summary, the picture looks consistent but a dedicated heavy
quark analysis of the NLO $B\to\gamma$ analysis must ultimately be the most
reliable method as it will not involve any photon DAs. We anticipate returning
to this issue in a forthcoming publication.

##  6 Summary and Discussions

In this work we have computed the $\bar{B}_{u,d,s}\to\gamma$ form factors at
NLO for twist-$1$,$2$ and LO in twist-$3$,$4$ with light-cone sum rules. This
involved the evaluation of the diagrams in Fig. 2 which was the technically
most challenging part of this work and involved the use of a series of tools
FeynCalc1 ; FeynCalc2 ; litered:2013 ; kira:2017 ; polylogtools  and state-of-
the-art master integrals DiVita:2017xlr . The form factors are provided as
fits to a $z$-expansion ansatz (66) with fit-parameters given in Tab. 4. The
correlation matrix can be found in an ancillary file to the arXiv version.
Form factor values at $q^{2}=0$ and plots are presented in Tab. 5 and Fig. 7
respectively. This constitutes our main practical results. Analytic results of
the correlation functions in terms of Goncharov functions are given as an
ancillary file (Mathematica notebook) to the arXiv version. An important
technical aside is the choice of the $m_{b}$ mass-scheme for which we have
adapted the kinetic scheme, which shows good scale-stability (cf. Fig. 12)
unlike the pole- or $\overline{\text{MS}}$-scheme for which either the
$\alpha_{s}$\- or twist-corrections are unnaturally large.

Besides the form factors themselves, we have established a few valuable
theoretical results and advances. In Sec. 2.3, starting from a dispersion
representation, we motivated the $m_{b}^{-1/2}$ heavy quark scaling of the
$\bar{B}\to\gamma$ form factors from a purely hadronic picture. Additionally,
in Sec. 4.2.1, we proposed a scheme to add (strange) quark mass corrections to
the magnetic vacuum susceptibility by separating the perturbative part. The
following values have been obtained at ${\cal O}(\alpha_{s}^{0})$:
$\chi_{q}=3.21(15)\textrm{\small GeV}^{-2}$ and
$\chi_{s}=3.79(17)\textrm{\small GeV}^{-2}$. Moreover, the equations of motion
(4.4) were verified to hold in all hadronic parameters for which completeness
can be expected. In doing so it became clear that in order to include the
twist-$4$ $3$-particle distribution amplitude parameters, given in Tab. 12,
one needs to include $4$-particle DAs of twist-$4$ which have not yet been
classified. For $\bar{B}\to\gamma$ form factors the impact is fortunately
rather small.

The _charged_ form factors describe the
$\bar{B}_{u}\to\gamma\ell^{-}\bar{\nu}$ decay, anticipated to be measured at
Belle II Kou:2018nap , for which our differential rate prediction, binned and
including correlation matrix, are given in Sec. 5.3. Comparing to SCET-
computations, of three models of the $B$-meson distribution amplitude, we have
extracted its inverse moment (73), $\lambda_{B}=360(110)\,\mbox{\small
MeV}$.181818 It seems worthwhile to mention that comparing theory computations
with each other eliminates the sizeable $|V_{\textrm{ub}}|$-uncertainty.
Moreover, this analysis is to be regarded as a crude first approach and can be
improved by doing a fully differential analysis, including the two logarithmic
moments $\hat{\sigma}_{1,2}$ into a $3$-parameter fit, which we might turn to
in a future publication. The _neutral_ form factors are the input to the
flavour changing neutral current process
$\bar{B}_{d,s}\to\gamma\ell^{+}\ell^{-}$, which will come into focus by LHCb
in the muon-mode LHCb:2021awg ; LHCb:2021vsc ; LHCbtalk ; Ferreres-
Sole:2021qxv , alongside off-shell form factors Albrecht:2019zul  and genuine
long distance contributions GRZ17 ; Kozachuk:2017mdk ; Beneke:2020fot . The
scope of applications of the $\bar{B}\to\gamma$ form factors does not end here
as they can serve as inputs, to related processes, in terms of subtraction
constants of dispersion relations. Examples are, the weak annihilation process
in $\bar{B}\to V\gamma$ e.g. Lyon:2013gba ,
$B_{s}\to\mu\mu\,\textrm{axion/dark photon}$ for which generic photon off-
shell $B\to\gamma^{*}$ form factors are needed (cf. App. A Albrecht:2019zul
for how dispersion relations can bridge the $k^{2}\approx
m^{2}_{\rho,\omega,\phi}$ resonance region) and
$\bar{B}_{u}\to\ell^{+}\ell^{-}\ell^{{}^{\prime}-}\bar{\nu}$ (requiring
$q^{2},k^{2}\neq 0$) as recently investigated in Bharucha:2021zay ;
Beneke:2021rjf . Our form factor predictions should be helpful in reducing the
theory error in all of these modes and enhance the new physics sensitivity.

### Acknowledgments

We are grateful to Gunnar Bali, Martin Beneke, Christoph Bobeth, Vladmir
Braun, Matteo Di Carlo, Paolo Gambino, Einan Gardi, Yao Ji, Chris Sachrajda
and Yuming Wang for useful discussions and to James Gratrex for careful
reading of the manuscript. We are indebted to Thomas Gehrmann and Roberto
Bonciani for guidance in the master integral literature, Claude Duhr and James
Matthews for help with Polylog Tools and Stefano Di Vita, Pierpaolo Mastrolia,
Amedeo Primo & Ulrich Schubert for useful exchange and guidance with regard to
DiVita:2017xlr . BP is also grateful to Calum Milloy for useful discussions
and guidance regarding the method of differential equations and the subtleties
therein. RZ acknowledges Damir Becirevic for the kind hospitality at IJCLab,
Pôle Théorie Orsay-Paris when this work was in its initial stages. RZ is
supported by an STFC Consolidated Grant, ST/P0000630/1. BP is supported by an
STFC Training Grant ST/N504051/1.

### Ancillary files

We include a number of ancillary files. A file with the renormalised NLO
correlation functions for twist-$1$ in terms of MPLs (corr_PT_NLO.m); the new
set of master integrals (MI_vzzb.m) discussed in Sec. 4.1.2 which is linearly
dependent on the basis DiVita:2017xlr  but of value to make cancellations
explicit. The $z$-expansion fits, given as a JSON file (BtoGam_FF.json), with
some description in Sec. 5.1.2 and the notebook (FF-plots_from_json.nb) that
exemplifies its implementation.

##  Appendix A Conventions and Additional Plots

We define the covariant derivative as

$D_{\mu}=\partial_{\mu}+s_{e}ieQ_{f}A_{\mu}+s_{g}ig_{s}A^{a}_{\mu}T^{a}$ (A.1)

where $e>0$ ($\alpha=e^{2}/(4\pi))$, $Q_{e}=-1$, $Q_{u,c,t}=2/3$ and
$Q_{b,d,s}=-1/3$. This leads to a Feynman rule
$-s_{e}ieQ_{f}\not{A}-s_{g}ig_{s}\not{A^{a}}T^{a}$. Note that the sign of the
FFs (2.1) depends on $s_{e}$ which we therefore prefer to keep explicit. The
sign of the gluon term $s_{g}$ is also kept which appears in the light-cone
propagator in the background field (D.3) as well as the $3$-particle DAs. The
final results are independent of $s_{g}$ and the rate is of course independent
of $s_{e}$, but attention must be paid when compared or combined with other
work. Using the $g=\textrm{diag}(1,-1,-1,-1)$ metric the following relation
holds,

$\sigma^{\alpha\beta}\gamma_{5}=-\frac{i}{2}\varepsilon^{\alpha\beta\gamma\delta}\sigma_{\gamma\delta}\;,$
(A.2)

with $\varepsilon_{0123}=1$ and
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. Moreover we have
$T^{a}T^{a}=C_{F}\mathbf{1}$ with $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ and
$\alpha_{s}=g^{2}/(4\pi)$ as usual.

###  A.1 Borel Transformation

The Borel transform is defined as

$f_{\hat{\mathcal{B}}}(M_{\hat{\mathcal{B}}}^{2})={\hat{\mathcal{B}}}f(Q^{2})=\lim_{Q^{2}\to\infty,n\to\infty}\dfrac{(Q^{2})^{n+1}}{n!}\left(-\dfrac{d}{dQ^{2}}\right)^{n+1}f(Q^{2})$
(A.3)

where $Q^{2}=-q^{2}$ is a Euclidean momentum and
$Q^{2}/n=M_{\hat{\mathcal{B}}}^{2}$ defines the so-called Borel mass.

For the perturbative part and the condensates, which are poles at tree level,
one needs the following Borel transformation

$\displaystyle{\hat{\mathcal{B}}}\dfrac{1}{(Q^{2}+m^{2})^{a}}$
$\displaystyle=\dfrac{1}{\Gamma(a)(M_{\hat{\mathcal{B}}}^{2})^{a}}e^{-m^{2}/M_{\hat{\mathcal{B}}}^{2}}\,.$
(A.4)

The DA-part can be reduced to a form $du\frac{1}{\Delta^{n}}$ ($n\geq 1$) for
which simple rules for Borel transformation are given in App. A in BSZ15 . Of
course one could integrate over the DA-parameters $du$ or
$D\underline{\alpha}$, then subject the result to a dispersion relation and
use (A.4). The advantage of using the former method is that one does not need
to commit to a specific form/truncation of the DAs.

###  A.2 Plots of Scale Dependence and Twist-contributions

In this appendix we collect additional plots and tables.

Figure 12: Plots of a representative selection of form factors, evaluated at
$q^{2}=0$. The plots serve to highlight the reduction in the kinetic scale
dependence when radiative corrections to twist-1 and -2 are included (dashed
red line) as compared to the LO result (solid blue line).

The merit of the NLO versus a LO analysis shows in the mass scale dependence
plots in Fig. 12. The advantages of the kinetic-scheme over the pole- and
$\overline{\text{MS}}$\- schemes become apparent in Fig. 13 and Tab. 10,
respectively.

twist | DA | $V_{\|}^{\bar{B}_{u}\to\gamma}(0)$ | $V_{\perp}^{\bar{B}_{d}\to\gamma}(0)$ | $T_{\perp,\|}^{\bar{B}_{u}\to\gamma}(0)$
---|---|---|---|---
|  | Kin | $\overline{\text{MS}}$ | Kin | $\overline{\text{MS}}$ | Kin | $\overline{\text{MS}}$
1 | $\textrm{PT}\,{\cal O}{(\alpha_{s}^{0})}$ | $45.7\%$ | $70.0\%$ | $51.3\%$ | $86.2\%$ | $51.4\%$ | $79.5\%$
1 | $\textrm{PT}\,{\cal O}{(\alpha_{s})}$ | $8.4\%$ | $-6.3\%$ | $14.9\%$ | $-18.0\%$ | $16.5\%$ | $-10.6\%$