aainstitutetext: Theoretical Physics Department, CERN,
1211 Geneva 23, Switzerlandbbinstitutetext: Dipartimento di Fisica, Università
di Torino & INFN, Sezione di Torino,
Via Pietro Giuria 1, I-10125 Turin, Italyccinstitutetext: Max Planck Institute
for Physics,
Föhringer Ring 6, 80805 München, Germanyddinstitutetext: Technische
Universität München, Physik-Department,
James-Franck-Strasse 1, 85748 Garching, Germanyeeinstitutetext: IIS G. Plana,
Piazza Generale di Robilant 5, I-10141 Turin, Italy

# QED effects in inclusive semi-leptonic $B$ decays

Dante Bigi a Marzia Bordone b,c,d Paolo Gambino c
Ulrich Haisch e and Andrea Piccione
bigid@iol.it,marzia.bordone<EMAIL_ADDRESS>haisch<EMAIL_ADDRESS>
###### Abstract

We perform a detailed analysis of QED corrections to the total decay width and
the electron energy spectrum of the inclusive semi-leptonic $B\to X_{c}e\nu$
decay. Our calculation includes short-distance electroweak corrections, the
complete ${\cal O}(\alpha)$ partonic terms and leading-logarithmic QED effects
up to ${\cal O}(\Lambda^{3}_{\rm QCD}/m_{b}^{3})$. In the case of the
branching ratio and the electron energy moments we perform a detailed
numerical comparison of our results against those obtained with the Monte
Carlo (MC) tool PHOTOS. While the comparison indicates good overall agreement,
our computation contains QED effects not included in PHOTOS and should
therefore provide a better description of photon radiation to $B\to X_{c}e\nu$
as measured by the $B$-factories. The calculations performed here represent
the first steps in the construction of a fully differential higher-order QED
MC generator for inclusive semi-leptonic $B$ decays.

††preprint: CERN-TH-2023-145, MPP-2023-186

##  1 Introduction

After the calculation of the ${\cal O}(\alpha_{s}^{3})$ corrections to the
total semi-leptonic decay width Fael  _et al._ (2021), the inclusive
determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix element $|V_{cb}|$
has an uncertainty of about $1.2\%$ Bordone  _et al._ (2021). At this level of
precision a careful study of QED and electroweak (EW) effects is imperative,
not only for the inclusive semi-leptonic decay width but also for the moments
of the kinematic distributions that enter the Heavy Quark Expansion (HQE)
global fits of $B\to X_{c}\ell\nu$. At present, the inclusive extraction of
$|V_{cb}|$ includes only the short-distance EW logarithmic corrections derived
in the classic paper by Sirlin Sirlin (1982), while photon radiation is
generally subtracted from the measured moments in the experimental analyses.
The subtraction is performed using the Monte Carlo (MC) code PHOTOS Barberio
and Was (1994) which includes soft and collinear photon radiation from charged
leptons and final-state hadrons but lacks hard and structure-dependent
radiative effects as well as virtual corrections, which are sometimes enhanced
by a factor $\pi^{2}$.

The study of inclusive semi-leptonic $B\to X_{c}\ell\nu$ decays is based on an
Operator Product Expansion (OPE) that holds for sufficiently inclusive
observables. In particular, the theoretical predictions for the total decay
width and the first few moments of the lepton energy and the hadronic
invariant mass can be formulated as a double-series expansion in the strong
coupling constant $\alpha_{s}$ and the ratio $\Lambda_{\rm QCD}/m_{b}$
involving the QCD scale and the bottom-quark mass. The power corrections start
at quadratic order and are parametrised in terms of $B$-meson matrix elements
of local operators. Information on these non-perturbative matrix elements is
obtained from the normalised moments, and it is then employed in the
calculation of the total decay width from which $|V_{cb}|$ is extracted (cf.
for instance Bordone  _et al._ (2021); Bernlochner  _et al._ (2022) for recent
analyses). At the $B$-factories the experimental measurements of the moments
have been performed applying a lower cut on the lepton energy $E_{\ell}$ or
the invariant mass of the final-state lepton pair $q^{2}$.

When QED interactions enter the game, things become more involved. The first
important observation in this context concerns the definition of the final-
state observables. If only QCD is considered, the leptonic and hadronic final
states are completely independent and cleanly separate. Gluon radiation is
fully included in the hadronic final state so that the hadronic invariant mass
squared $M_{X}^{2}$ and the hadronic energy $E_{X}$ are defined in a
completely inclusive way. This is consistently done in the theoretical
calculations of the inclusive semi-leptonic decay width Fael  _et al._ (2021)
and the moments Pak and Czarnecki (2008); Biswas and Melnikov (2010); Fael
_et al._ (2022), as well as in the corresponding experimental measurements.
Photon radiation, however, affects both charged leptons and hadrons making a
completely inclusive definition of their respective properties impossible.
Moreover, the measurements of $E_{\ell}$ and $q^{2}$ always depend on the
details of the experimental analysis. At the $B$-factories, for example, the
momentum of an electron and hence its energy $E_{e}$ is measured most
precisely in the drift chambers from the curvature of its trajectory in the
magnetic field Abashian  _et al._ (2002); Aubert  _et al._ (2002). The
reconstruction of the true value of $E_{e}$ involves modelling the photon
radiation due to the interaction with the magnetic field and the material in
the detector components Brun  _et al._ (1994); Agostinelli  _et al._ (2003) as
well as bremsstrahlungs recovery Abashian  _et al._ (2002); Aubert  _et al._
(2002). Given the detector geometry and the way electrons are reconstructed,
most of the prompt photons emitted by the electron in the form of final-state
radiation however go undetected and are not corrected for in the $E_{e}$
measurement. We will therefore work under the assumption that the true
electron energy determined by the $B$-factory experiments corresponds to the
energy before the electron enters the detector, and after all final-state
radiation associated to the hard process has taken place. As a consequence,
all these photons are effectively summed over in the experimental measurement
which can be qualified as totally inclusive with respect to photons. While
this is certainly an approximation, we believe it is closest to the
experimental setup. The same assumption is also adopted in the precision QED
calculations of the electron energy spectrum in muon decay van Ritbergen and
Stuart (2000); Arbuzov  _et al._ (2002); Arbuzov and Melnikov (2002); Arbuzov
(2003); Anastasiou  _et al._ (2007); Pak and Czarnecki (2008) when compared to
the experimental measurements like those performed at MuLan Webber  _et al._
(2011); Tishchenko  _et al._ (2013).

A second, related observation concerns the OPE, which in the presence of QED
interactions is no longer valid in general. While it still holds in the case
of the total decay width, the OPE fails for quantities that describe exclusive
properties of the leptonic final state (such as $E_{\ell}$ or $q^{2}$) or the
hadronic final state (such as $M_{X}^{2}$ or $E_{X}$). This is because a
photon can connect the charged lepton with the hadronic initial or final
states. In the context of our work, the breakdown of the OPE is signalled by
the presence of collinear logarithms of the form
$\alpha/\pi\ln\left(m_{b}^{2}/m_{e}^{2}\right)$ that appear in the electron
energy spectrum and its moments. These logarithmically enhanced QED effects
arise already at leading power in the HQE, i.e. the partonic level, and are
expected to provide the bulk of the complete ${\cal O}(\alpha)$ contributions.
In the absence of an OPE, we generally expect power corrections to start
already at ${\cal O}(\alpha\hskip 0.35565pt\Lambda_{\rm QCD}/m_{b})$. However,
the logarithmically enhanced corrections related to photons collinear to the
electron represent an exception: since they factorise, they can be computed
using the structure function method directly from the lowest order, and this
holds for the power-suppressed contributions as well.

In this article, we focus our attention on the QED corrections to the total
decay width of $B\to X_{c}e\nu$, the electron energy spectrum and its moments.
These observables play a crucial role in the HQE global fits that are used to
determine $|V_{cb}|$. Our aim is to validate the subtraction procedure
performed by the experiments and to improve it by calculating QED effects not
included in PHOTOS. To this purpose, we identify two types of large QED
corrections, namely collinear terms proportional to
$\ln\left(m_{b}^{2}/m_{e}^{2}\right)$ and threshold corrections enhanced by a
factor $\pi^{2}$. For the electron energy spectrum of $b\to ce\nu$ we describe
the analytic calculation of these contributions in Section 2 and Section 3,
respectively. In Section 4 we then perform a complete ${\cal O}(\alpha)$
calculation of the electron energy spectrum and the total decay width in $b\to
ce\nu$. A comparison of the obtained results shows that the logarithmically
enhanced ($\pi^{2}$-enhanced) terms indeed provide the dominant part of the
full ${\cal O}(\alpha)$ corrections in the case of the electron energy
spectrum (total decay width). In our comparison with experiment presented in
Section 5, we therefore include the full short-distance EW corrections (beyond
the logarithm computed in Sirlin (1982)), the complete ${\cal O}(\alpha)$
terms, and the leading-logarithmic (LL) QED effects in the power corrections
of ${\cal O}(\Lambda^{2}_{\rm QCD}/m_{b}^{2})$ and ${\cal O}(\Lambda^{3}_{\rm
QCD}/m_{b}^{3})$. Section 6 contains our conclusions and an outlook, while
lengthy analytic expressions and technical details are relegated to a few
appendices.

##  2 LL effects in the electron energy spectrum

In this section, we describe the calculation of the LL QED corrections to the
electron energy spectrum of the $B\to X_{c}e\nu$ transition. These corrections
are proportional to logarithms of the ratio of the bottom-quark and the
electron mass. We will first detail the general formalism and then apply it to
the partonic case. The same formalism can also be used to compute the LL QED
corrections to the power-suppressed contributions in the electron energy
moments of $B\to X_{c}e\nu$. The analytic expressions for these corrections at
${\cal O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm
QCD}^{3}/m_{b}^{3})$ are relegated to Appendix A, while in Appendix B we
extend the formalism presented in this section to the next-to-leading-
logarithmic (NLL) order and apply it to the leading-power computation of $B\to
X_{c}e\nu$.

The LL QED corrections to the electron energy spectrum of $B\to X_{c}e\nu$ can
be easily computed with the structure function method Lipatov (1974);
Altarelli and Parisi (1977). Further details on this approach can be found for
instance in Arbuzov  _et al._ (2002); Arbuzov and Melnikov (2002); Arbuzov
(2003). The main ingredient to calculate these effects is the leading-order
(LO) electron-electron splitting function

$P_{ee}^{(0)}(z)=\left[\frac{1+z^{2}}{1-z}\right]_{+}=\lim_{\Delta\,\to\,0}\left[\frac{1+z^{2}}{1-z}\,\theta(1-z-\Delta)+\left(\frac{3}{2}+2\ln\Delta\right)\delta(1-z)\right]\,,$
(1)

which is a kind of plus distribution and encodes the LL QED correction to the
probability of finding an electron with energy fraction $z$ in the initial
electron. Here $\theta(z)$ and $\delta(z)$ are the Heaviside step function and
the Dirac delta function, respectively. The LO electron-electron splitting
function is normalised such that

$\int_{0}^{1}\\!dz\,P_{ee}^{(0)}(z)=0\,,$ (2)

which guarantees that in the total decay width $\Gamma$ logarithmic
corrections are not present as required by the Kinoshita-Lee-Nauenberg (KLN)
theorem Kinoshita (1962); Lee and Nauenberg (1964). In practice, the LL
corrections to the electron energy spectrum are obtained by evaluating the
following convolution

$\left(\frac{d\Gamma}{dy}\right)^{(1)}_{\rm
LL}=\frac{\alpha}{2\pi}\,\bar{L}_{b/e}\int_{y}^{1-\rho}\,\frac{dx}{x}\,P_{ee}^{(0)}\left(\frac{y}{x}\right)\,\left(\frac{d\Gamma}{dx}\right)^{(0)}\,,$
(3)

where $y=2E_{e}/m_{b}$ with $E_{e}$ the electron energy and
$\rho=m_{c}^{2}/m_{b}^{2}$ with $m_{b}$ and $m_{c}$ the bottom- and charm-
quark mass, respectively. We have furthermore introduced the abbreviation

$\bar{L}_{b/e}=\ln\bigg{(}\frac{m_{b}^{2}}{m_{e}^{2}}\bigg{)}-1\,.$ (4)

and $\left(d\Gamma/dy\right)^{(0)}$ denotes the electron energy spectrum
calculated at LO assuming a massless electron.

To give a simple example of the general formalism let us apply (3) to the case
of the partonic electron energy spectrum. In this case one has

$\left(\frac{d\Gamma}{dy}\right)^{(0)}=\Gamma^{(0)}\,f^{(0)}(y)\,,\qquad\Gamma^{(0)}=\frac{G_{F}^{2}|V_{cb}|^{2}}{192\pi^{2}}\hskip
0.7113ptm_{b}^{5}\,,$ (5)

where $G_{F}=1.1663787\cdot 10^{-5}\,{\rm GeV}^{-2}$ is the Fermi constant as
extracted from muon decay, $V_{cb}$ is the relevant CKM matrix element and
Manohar and Wise (2000)

$f^{(0)}(y)=\Bigg{[}\,2\left(3-2y\right)y^{2}-6y^{2}\rho-\frac{6y^{2}\rho^{2}}{(1-y)^{2}}+\frac{2\left(3-y\right)y^{2}\rho^{3}}{(1-y)^{3}}\,\Bigg{]}\;\theta(1-y-\rho)\,.$
(6)

Inserting the latter result into (3) and combining it with (5) we obtain

$\frac{d\Gamma}{dy}=\Gamma^{(0)}\,f(y)\,,\qquad
f(y)=f^{(0)}(y)+\frac{\alpha}{2\pi}\,\bar{L}_{b/e}\,f^{(1)}_{\rm LL}(y)\,,$
(7)

where

$\displaystyle\begin{split}f^{(1)}_{\rm
LL}(y)&=\Bigg{\\{}\,\frac{5}{3}+\frac{4}{3}\left(3-6y+4y^{2}\right)y+4\left(3-2y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\\\\[5.69054pt]
&\phantom{xxx}-\left[9+3y-12y^{2}+8y^{3}+12\left(1-y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho}{1-y}\\\\[5.69054pt]
&\phantom{xxx}+\left[9+2y^{3}+6\left(1-2y-y^{2}\right)\ln\left(\frac{1-y}{\rho}\right)-12\hskip
0.7113pty^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{2}}{(1-y)^{2}}\hskip
17.07164pt\\\\[5.69054pt]
&\phantom{xxx}-\left[\frac{5}{3}+y+4y^{2}-\frac{2}{3}y^{3}+2\left(1-3y-3y^{2}+y^{3}\right)\ln\left(\frac{1-y}{\rho}\right)\right.\\\\[5.69054pt]
&\phantom{xxxxxi}\left.-4\left(3-y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{3}}{(1-y)^{3}}\Bigg{\\}}\;\theta(1-y-\rho)\,.\end{split}$
(8)

Notice that in the limit $\rho\to 0$ our result for $f^{(1)}_{\rm LL}(y)$
agrees with (12) of Arbuzov  _et al._ (2002) which contains the LL QED terms
for muon decay. This agreement is expected because in the fully massless case
the electron energy spectrum of $b\to ue\nu$ and $\mu\to e\nu\nu$ are the
same.

The total decay width for $b\to ce\nu$ is obtained by integrating (7) over
$y\in[0,1-\rho]$. Using the relations

$\int_{0}^{1-\rho}\\!dy\,f^{(0)}(y)=g(\rho)\,,\qquad\int_{0}^{1-\rho}\\!dy\,f^{(1)}_{\rm
LL}(y)=0\,,$ (9)

where

$g(\rho)=1-8\rho-12\rho^{2}\ln\rho+8\rho^{3}-\rho^{4}\,,$ (10)

one finds

$\Gamma=\Gamma^{(0)}g(\rho)\,,$ (11)

which implies that the total decay width of $b\to ce\nu$ does not receive LL
QED effects. Notice that this cancellation of mass singularities is required
by the KLN theorem. In fact, the same cancellation occurs for the LL and the
NLL QED corrections calculated in Appendix A and Appendix B, respectively.
This feature is related to the existence of an OPE at ${\cal O}(\alpha)$ for
the total semi-leptonic decay width and represents a non-trivial cross-check
of our calculation of logarithmically enhanced QED corrections to the electron
energy spectrum of $B\to X_{c}e\nu$.

Figure 1:  The red curve corresponds to $f(y)$ defined in (7) while the black
curve represents the LO contribution $f^{(0)}(y)$ as given in (6). A kinetic
bottom- and $\overline{\rm MS}$ charm-quark mass is employed and final states
containing electrons are considered.

While the absence of logarithmically enhanced QED corrections in the total
decay width of $B\to X_{c}e\nu$ is guaranteed by the KLN theorem, such effects
appear in general in other observables. The impact of the LL QED corrections
to the electron energy spectrum in the partonic $b\to ce\nu$ transition is
illustrated in Figure 1. The shown results have been obtained for a kinetic
bottom-quark mass of $m_{b}^{\rm kin}(1\,{\rm GeV})=4.57\,{\rm GeV}$ and a
$\overline{\rm MS}$ charm-quark mass of $\overline{m}_{c}(2\,{\rm
GeV})=1.09\,{\rm GeV}$, corresponding to the central values of the recent HQE
global fit Bordone  _et al._ (2021). We furthermore have employed
$m_{e}=511\,{\rm keV}$ and $\alpha(m_{b})=1/133$. The same input will be used
hereafter. From the figure it is evident that the LL QED corrections change
the shape of the electron energy spectrum of $b\to ce\nu$ in a non-trivial
way. In the tail region, i.e. for $y\to 0$, the spectrum receives a positive
shift while close to the endpoint, i.e. for $y\to 1-\rho$, the LL QED
corrections tend to zero. This can be understood analytically from the
following limiting behaviours:

$\lim_{y\,\to\,0}f^{(1)}_{\rm
LL}(y)=\frac{5}{3}-9\rho+9\rho^{2}-\frac{5}{3}\rho^{3}-2\left(3-\rho\right)\rho^{2}\ln\rho\,,\qquad\lim_{y\,\to\,1-\rho}f^{(1)}_{\rm
LL}(y)=0\,.$ (12)

In the peak region, i.e. in the vicinity of $y=1-\sqrt{\rho}$, the electron
energy distribution is reduced by the LL QED effects, as expected from the
second relation in (9) and (12). We add that the approximation of the
logarithmically enhanced QED corrections to the electron energy spectrum that
has been proposed in Atwood and Marciano (1990) does not accurately capture
the functional form of the exact expression $f^{(1)}_{\rm LL}(y)$. In
particular, the tail is not described correctly by the exponentiated
radiation-damping factor introduced in Atwood and Marciano (1990) because this
approximation misses hard-collinear photon contributions that populate the
low-energy spectrum.

##  3 Threshold corrections in the total decay width

In this section, we explain how to compute a certain class of
$\pi^{2}$-enhanced QED corrections first identified in Ginsberg (1968); Atwood
and Marciano (1990) in the case of exclusive semi-leptonic $K$ and $B$ decays.
In the modern literature discussing QED corrections to meson decays these
terms are often referred to as Coulomb corrections (see for instance Isidori
(2008); Kubis and Schmidt (2010); de Boer  _et al._ (2018); Calí  _et al._
(2019); Mishra and Mahajan (2021); Bansal  _et al._ (2022)). For what concerns
the $b\to ce\nu$ process these terms arise from the virtual corrections where
the photon is exchanged between the charm-quark and the electron line in the
final state. The corresponding Feynman diagram is shown on the right in Figure
2. Interfering the relevant virtual matrix element with the Born-level
contribution and normalising the result to the Born-level matrix element
squared, we find

$\begin{split}\Delta_{ce}&=\frac{2\hskip 0.35565pt{\rm Re}\left({\cal M}_{\rm
Born}\,{\cal M}_{\text{virtual}}^{ce\hskip 0.7113pt\ast}\right)}{|{\cal
M}_{\rm Born}|^{2}}\\\\[5.69054pt] &=-\frac{\alpha}{\pi}\hskip
1.42262ptQ_{c}\hskip 0.35565ptQ_{e}\hskip 1.42262pt{\rm
Re}\,\Bigg{\\{}1+\frac{1}{q^{2}-m_{c}^{2}-m_{e}^{2}}\bigg{[}\left(q^{2}-m_{c}^{2}\right)B_{0}(m_{c}^{2},0,m_{c}^{2})\\\\[5.69054pt]
&\hskip
99.58464pt+\left(q^{2}-m_{e}^{2}\right)B_{0}(m_{e}^{2},0,m_{e}^{2})-\left(m_{c}^{2}+m_{e}^{2}\right)B_{0}(q^{2},m_{c}^{2},m_{e}^{2})\bigg{]}\\\\[5.69054pt]
&\hskip
71.13188pt\phantom{xi}-\left(q^{2}-m_{c}^{2}-m_{e}^{2}\right)C_{0}(m_{c}^{2},q^{2},m_{e}^{2},0,m_{c}^{2},m_{e}^{2})\Bigg{\\}}\,.\end{split}$
(13)

Here $Q_{c}=2/3$ and $Q_{e}=-1$ are the electric charges of the charm quark
and the electron in units of $e$, while
$q^{2}=\left(p_{b}-p_{\nu}\right)^{2}=\left(p_{c}+p_{e}\right)^{2}$ with
$p_{b}$, $p_{c}$, $p_{e}$ and $p_{\nu}$ denoting the four-momentum of the
bottom quark, charm quark, electron and neutrino, respectively. The correction
(13) has been expressed in terms of the following one-loop Passarino-Veltman
(PV) integrals:

$\begin{split}B_{0}\big{(}p_{1}^{2},m_{0}^{2},m_{1}^{2}\big{)}&=\frac{\mu^{4-d}}{i\hskip
0.35565pt\pi^{d/2}\hskip
0.7113ptr_{\Gamma}}\int\\!\frac{d^{d}l}{\Pi_{i=0,1}\hskip
0.7113ptP(l+p_{i},m_{i})}\,,\\\\[5.69054pt]
C_{0}\big{(}p_{1}^{2},(p_{1}-p_{2})^{2},p_{2}^{2},m_{0}^{2},m_{1}^{2},m_{2}^{2}\big{)}&=\frac{\mu^{4-d}}{i\hskip
0.35565pt\pi^{d/2}\hskip
0.7113ptr_{\Gamma}}\int\\!\frac{d^{d}l}{\Pi_{i=0,1,2}\hskip
0.7113ptP(l+p_{i},m_{i})}\,.\end{split}$ (14)

Here $\mu$ is the renormalisation scale that keeps track of the correct
dimension of the integrals in $d=4-2\epsilon$ space-time dimensions,
$r_{\Gamma}=\Gamma^{2}(1-\epsilon)\hskip
0.7113pt\Gamma(1+\epsilon)/\Gamma(1-2\epsilon)$ with $\Gamma(z)$ denoting the
Euler gamma function, $P(k,m)=k^{2}-m^{2}$ and $p_{0}=0$. The definitions (14)
resemble those of the LoopTools package Hahn and Perez-Victoria (1999).

The $\pi^{2}$-enhanced QED terms now arise from the result (13) in the
threshold region, i.e. when $q^{2}\to(m_{c}+m_{e})^{2}$. In this limit the
three-point one-loop PV integral
$C_{0}(m_{c}^{2},q^{2},m_{e}^{2},0,m_{c}^{2},m_{e}^{2})$ develops a
discontinuity. Explicitly, one has

$\text{disc}\,{\rm
Re}\left(C_{0}(m_{c}^{2},q^{2},m_{e}^{2},0,m_{c}^{2},m_{e}^{2})\right)=-\frac{\pi^{2}}{3\hskip
0.7113pt\beta_{ce}}\hskip 1.42262pt\frac{1}{p_{c}\cdot p_{e}}\,,$ (15)

with

$\beta_{ce}=\sqrt{1-\frac{m_{c}^{2}\hskip 1.42262ptm_{e}^{2}}{(p_{c}\cdot
p_{e})^{2}}}\,,$ (16)

the relative velocity between the charm quark and the electron in the rest
frame of either particle. Notice that the observed $1/\beta_{ce}$ factor
signals the presence of a Coulomb pole that describes the universal long-
distance electromagnetic interactions between pairs of charged non-
relativistic final-state particles. In the limit of vanishing electron mass,
one has $\beta_{ce}\to 1$ and the discontinuity of (13) is hence simply given
by

$\text{disc}\,\Delta_{ce}=-\frac{2\pi^{2}}{3}\frac{\alpha}{\pi}\hskip
1.42262ptQ_{c}\hskip 0.35565ptQ_{e}=\frac{4\pi\alpha}{9}\,.$ (17)

To obtain the $\pi^{2}$-enhanced QED correction to the total decay width of
$b\to ce\nu$ one now has to integrate (17) over the relevant three-particle
phase space. Since in the massless electron limit, $\text{disc}\,\Delta_{ce}$
is independent of kinematics this integration is however trivial. Instead of
(11) one therefore obtains

$\Gamma=\Gamma^{(0)}g(\rho)\left(1+\frac{4\pi\alpha}{9}\right)=\Gamma^{(0)}g(\rho)\,\Big{(}1+1.05\%\Big{)}\,,$
(18)

where the explicit form of the function $g(\rho)$ can be found in (10), which
is also an excellent approximation to the massive electron case. Notice that
we do not resum the Coulomb corrections as done for instance in Isidori
(2008); de Boer  _et al._ (2018); Calí  _et al._ (2019) since in our case the
associated Sommerfeld enhancement Sommerfeld (1931) is insignificant. Beyond
the leading power term computed above, one expects power corrections enhanced
by $\pi^{2}$, starting at ${\cal O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ for the
total semi-leptonic width. These power corrections are sensitive to the
spectator quark and may lead to an observable difference between the $B^{0}$
and $B^{+}$ semi-leptonic decay widths.

##  4 Complete ${\cal O}(\alpha)$ calculation

In the following, we describe the different ingredients of our calculation of
the ${\cal O}(\alpha)$ corrections to $b\to ce\nu$. We first provide a
comprehensive discussion of the anatomy of the computation, the virtual and
real corrections and their combination to obtain ultraviolate (UV) and
infrared (IR) finite results. While our approach is general and can be used to
calculate the ${\cal O}(\alpha)$ corrections to any observable in $b\to
ce\nu$, we present in this article only numerical results for the total decay
width and the electron energy spectrum. These results are then compared to the
LL and threshold corrections computed earlier. Finally, the relevant
quantities used in the experimental analyses are introduced and a first
numerical analysis is performed.

###  4.1 The calculation in a nutshell

The calculation of semi-leptonic $B\to X_{c}e\nu$ decay properties is
characterised by a large hierarchy of scales, i.e. $M_{W}^{2}\gg m_{b}^{2}$,
and is most conveniently performed in the framework of an effective field
theory (EFT). If power-suppressed terms of ${\cal O}(m_{b}^{2}/M_{W}^{2})$ are
neglected, the relevant interactions are encoded by the weak Hamiltonian

${\cal H}=-\frac{4\hskip 0.35565ptG_{F}}{\sqrt{2}}\,C(\mu)\hskip
0.7113ptQ\,,\qquad Q=\big{(}\bar{c}\hskip 1.42262pt\gamma_{\mu}P_{L}\hskip
0.7113ptb\big{)}\big{(}\bar{e}\hskip 1.42262pt\gamma^{\mu}P_{L}\hskip
0.7113pt\nu\big{)}\,,$ (19)

where $P_{L}$ projects onto fermionic fields with left-handed chirality. The
Wilson coefficient $C(\mu)$ contains the short-distance contributions from
scales above $\mu$. In the case of $B\to X_{c}e\nu$ and at leading power in
$\Lambda_{\rm QCD}/m_{b}$, the matrix element of the dimension-six operator
$Q$ is computed using perturbation theory for the semi-leptonic decay of a
free bottom quark. In QCD the hadronic current present in $Q$ is a (partially)
conserved current and therefore the anomalous dimension of $Q$ vanishes.
However, if photonic corrections are considered, $Q$ develops a non-zero
anomalous dimension and as a result its Wilson coefficient depends non-
trivially on the renormalisation scale $\mu$, but no additional operator
appears. While the LL dependence on $\mu$ has been computed more than 40 years
ago by Sirlin in Sirlin (1982), the complete ${\cal O}(\alpha)$ matching
corrections to (19) have only been found rather recently in Brod and Gorbahn
(2008); Gorbahn  _et al._ (2023). We perform an independent calculation, whose
basic steps are outlined in Appendix E, and obtain

$C(\mu)=1+\frac{\alpha}{2\pi}\left[\hskip
0.7113pt\ln\left(\frac{M_{Z}^{2}}{\mu^{2}}\right)-\frac{11}{6}\hskip
0.7113pt\right]\,.$ (20)

This result holds for the conventional Fermi constant $G_{F}=1.1663787\cdot
10^{-5}\,{\rm GeV}^{-2}$ as measured in muon decay and agrees with Brod and
Gorbahn (2008); Gorbahn  _et al._ (2023). Identifying the renormalisation
scale $\mu$ with $m_{b}=m_{b}^{\rm kin}(1\,{\rm GeV})$ and using
$M_{Z}=91.1876\,{\rm GeV}$ the EW correction to the Wilson coefficient amounts
to a positive shit of about $0.5\%$. We note that the rational term in (20) is
renormalisation scheme dependent and that the quoted value corresponds to the
use of dimensional regularisation in $d=4-2\epsilon$ space-time dimensions
with a naive anti-commuting $\gamma_{5}$ (NDR). At ${\cal O}(\alpha)$ the
scale and scheme dependence of $C(\mu)$ is cancelled by that of the ${\cal
O}(\alpha)$ corrections to the matrix element $\left\langle
Q(\mu)\right\rangle$ computed in the EFT using the same renormalisation
prescription. The basic steps of this computation are detailed in the
following.

We begin the calculation of the ${\cal O}(\alpha)$ corrections to the matrix
element of the four-fermion operator $Q$ by parametrising the differential
decay width of $b\to ce\nu$ including ${\cal O}(\alpha)$ effects as follows

$d\Gamma=\frac{\big{|}C(\mu)\big{|}^{2}}{4\hskip 0.35565ptm_{b}}\Big{[}B\hskip
0.7113ptd\phi_{3}+\frac{\alpha}{\pi}\,\big{(}V\hskip
0.35565ptd\phi_{3}+R\hskip 0.7113ptd\phi_{4}\big{)}\Big{]}\,.$ (21)

Here $B$, $V$ and $R$ encode the Born-level, virtual and real corrections,
respectively, $d\phi_{3}$ and $d\phi_{4}$ represent the relevant three- and
four-body phase space and $C(\mu)$ is the Wilson coefficient given in (20). In
terms of the dimensionless kinematic parameters

$y_{ij}=\frac{2p_{i}\cdot p_{j}}{m_{b}^{2}}\,.$ (22)

the Born-level contribution is given by

$B=N_{B}\hskip 1.42262pty_{b\nu}\hskip 1.42262pty_{ce}\,,\qquad N_{B}=32\hskip
0.7113ptG_{F}^{2}\hskip 0.7113pt|V_{cb}|^{2}\hskip 1.42262ptm_{b}^{4}\,.$ (23)

In order to regulate the UV divergences that appear in $V$ we use the NDR
scheme. The finite parts of $V$ will depend on this scheme choice, but as
already mentioned this scheme dependence will cancel against the scheme
dependence of the Wilson coefficient (20). The same statement is true for the
scale dependence. The logarithmic IR divergences associated to soft photon
emissions are isolated and regulated by using a fictitious photon mass
$m_{\gamma}$. These IR-singular terms appear in both the virtual and the real
${\cal O}(\alpha)$ contributions but have to cancel in their combination
making the final result manifestly independent of the photon mass in the
$m_{\gamma}\to 0$ limit. This regulator independence provides an important
check of the numerical implementation of our calculation. For what concerns
collinear divergences, they are naturally regulated by the electron mass,
which we always keep finite.

####  4.1.1 Virtual corrections

Examples of Feynman diagrams that describe the virtual ${\cal O}(\alpha)$
corrections to $b\to ce\nu$ are displayed in Figure 2. External leg
corrections are not shown in the figure but included in the results given
below. We decompose the virtual contributions as follows

$V=V_{\rm UV}+V_{\rm IR}+V_{\rm fin}\,.$ (24)

where

$\displaystyle V_{\rm UV}$
$\displaystyle=\frac{B}{4}\left(-Q_{b}^{2}-Q_{c}^{2}-Q_{e}^{2}+2\hskip
0.35565ptQ_{b}\hskip 0.7113ptQ_{c}+2\hskip 0.35565ptQ_{b}\hskip
0.7113ptQ_{e}-8\hskip 0.35565ptQ_{c}\hskip
0.7113ptQ_{e}\right)\left[\frac{1}{\epsilon}+\ln\left(\frac{\mu^{2}}{m_{b}^{2}}\right)\right]\,,$
(25) $\displaystyle V_{\rm IR}$
$\displaystyle=\frac{B}{2}\left(Q_{b}^{2}\hskip 0.7113ptJ_{b}+Q_{c}^{2}\hskip
0.7113ptJ_{c}+Q_{e}^{2}\hskip 0.7113ptJ_{e}+2\hskip 0.7113ptQ_{b}\hskip
0.7113ptQ_{c}\hskip 0.7113ptI_{bc}^{1}+2\hskip 0.7113ptQ_{b}\hskip
0.7113ptQ_{e}\hskip 0.7113ptI_{be}^{1}+2\hskip 0.7113ptQ_{c}\hskip
0.7113ptQ_{e}\hskip 0.7113ptI_{ce}^{1}\right)\,,$ (26) $\displaystyle V_{\rm
fin}$
$\displaystyle=N_{B}\,\Bigg{\\{}\Bigg{[}\left\\{\frac{Q_{b}^{2}}{4}\left(1+2\hskip
0.35565ptI_{b}^{0}\right)+\left(b\rightarrow c\right)+\left(b\rightarrow
e\right)\right\\}$ $\displaystyle\hskip 49.79231pt-\left\\{Q_{b}\hskip
0.7113ptQ_{c}\left(\frac{5}{4}+\frac{I^{0}_{bc}}{2}-m_{b}^{2}\left(I_{bc}^{x}+\rho\hskip
0.7113ptI_{bc}^{y}\right)\right)+\left(c\rightarrow e,\rho\rightarrow
r\right)\right\\}$ $\displaystyle\hskip 49.79231pt+Q_{c}\hskip
0.7113ptQ_{e}\left(\frac{3}{2}+2\hskip
0.35565ptI^{0}_{ce}-m_{b}^{2}\left(\rho\hskip 0.7113ptI_{ce}^{x}+r\hskip
0.7113ptI_{ce}^{y}\right)\right)\Bigg{]}\,y_{b\nu}\hskip 1.42262pty_{ce}$
$\displaystyle\hskip 42.67912pt+\Bigg{\\{}Q_{b}\hskip
0.7113ptQ_{c}\,\bigg{[}\,\frac{1}{4}\left(y_{be}\hskip
0.7113pty_{c\nu}-y_{bc}\hskip 0.7113pty_{e\nu}\right)+m_{b}^{2}\hskip
1.42262pty_{ce}\left(y_{bc}\hskip 0.7113pty_{b\nu}-y_{c\nu}\right)I_{bc}^{x}$
$\displaystyle\hskip 92.47145pt+m_{b}^{2}\hskip
1.42262pty_{b\nu}\left(y_{bc}\hskip 0.7113pty_{ce}-\rho\hskip
0.7113pty_{be}\right)I_{bc}^{y}$ $\displaystyle\hskip
92.47145pt-\frac{m_{b}^{2}}{2}\hskip
1.42262pt\Big{(}y_{bc}\left(y_{b\nu}\hskip 0.7113pty_{ce}+y_{be}\hskip
0.7113pty_{c\nu}\right)-y_{e\nu}\hskip 0.7113pty_{bc}^{2}$
$\displaystyle\hskip 123.76965pt-2\hskip 1.42262pt\big{(}y_{ce}\hskip
0.7113pty_{c\nu}+\rho\left(y_{be}\hskip 0.7113pty_{b\nu}-2\hskip
0.7113pty_{e\nu}\right)\big{)}\hskip 0.7113ptI_{bc}^{xy}\Big{)}\bigg{]}$
$\displaystyle\hskip 64.01869pt+\left(c\leftrightarrow e,\rho\rightarrow
r\right)\Bigg{\\}}+Q_{c}\hskip 0.7113ptQ_{e}\hskip 1.42262ptm_{b}^{2}\hskip
1.42262pty_{b\nu}\left(y_{ce}^{2}-2\hskip 0.35565pt\rho\hskip
0.7113ptr\right)\left(I_{ce}^{x}+I_{ce}^{y}\right)\Bigg{\\}}\,.$

Here $Q_{b}=-1/3$, $Q_{c}=2/3$ and $Q_{e}=-1$ denote the electric charges of
the particle involved and we have introduced the abbreviation
$r=m_{e}^{2}/m_{b}^{2}$. The explicit expressions for the integrals $J_{i}$,
$I_{i}^{0}$, $I_{ij}^{0}$, $I_{ij}^{x}$, $I_{ij}^{y}$, $I_{ij}^{xy}$ and
$I_{ij}^{1}$ can be found in Appendix C. Notice that both $V_{\rm UV}$ and
$V_{\rm IR}$ are proportional to the Born-level contribution (23). This is a
consequence of the well-known factorisation properties of UV and IR
divergences. In contrast, only parts of the finite contributions $V_{\rm fin}$
factorise. To illustrate this property we have decomposed $V_{\rm fin}$ into a
term proportional to $y_{b\nu}\hskip 1.42262pty_{ce}$ and terms that are not
directly proportional to the Born kinematics. We finally notice that the
$1/\epsilon$ poles present in $V_{\rm UV}$ are cancelled by the $\overline{\rm
MS}$ renormalisation of the four-fermion operator introduced in (19).

Figure 2:  Virtual QED corrections to the $b\to ce\nu$ transition. The blue
squares indicate an insertion of the four-fermion operator $Q$ introduced in
(19) that induces the decay.

####  4.1.2 Real corrections

To determine the real contributions to the differential decay width (21)
requires a treatment of the four-body phase space. In our work, we divide
$d\phi_{4}$ into hard and soft regions using the improved phase space slicing
method introduced in Kilgore and Giele (1997). One has schematically

$\begin{split}R\hskip
1.42262ptd\phi_{4}&=\left[R\left(1-\theta_{\gamma}\right)+R\hskip
1.42262pt\theta_{\gamma}\right]d\phi_{4}\\\\[5.69054pt] &=R_{\rm
hard}\left(1-\theta_{\gamma}\right)d\phi_{4}+\theta_{\gamma}\,\big{[}T_{1}(\theta_{\gamma})+T_{2}(\theta_{\gamma})+T_{3}(\theta_{\gamma})\big{]}\,,\end{split}$
(28)

where $R$ is the exact real contribution to (21) and the slicing of the phase
space is indicated by $\theta_{\gamma}$. In our computation, we employ
$\theta_{\gamma}=\theta(E_{\rm min}-E_{\gamma})$ which evaluates to $0$ in the
hard region, i.e. for $E_{\gamma}>E_{\rm min}$, while it returns $1$ in the
soft region, i.e. for $E_{\gamma}<E_{\rm min}$. Here $E_{\gamma}$ denotes the
energy of the photon in the bottom-quark rest frame and $E_{\rm min}$ is a
resolution parameter that determines whether the photon is resolved or
unresolved. Notice that since the first term in the second line of (28)
encodes hard photon emission the photon mass can be set to zero when
evaluating the corresponding matrix element squared. The subtraction terms in
(28) can be written as

$T_{1}(\theta_{\gamma})=R_{\rm soft}\hskip 1.42262ptd\phi^{\rm
soft}_{4}\,,\quad T_{2}(\theta_{\gamma})=\left(R-R_{\rm
soft}\right)d\phi_{4}\,,\quad T_{3}(\theta_{\gamma})=R_{\rm soft}\hskip
1.42262pt\big{(}d\phi_{4}-d\phi^{\rm soft}_{4}\big{)}\,,$ (29)

with

$d\phi^{\rm soft}_{4}=d\phi_{3}\hskip 1.42262ptd\phi_{\gamma}\,,\qquad\quad
d\phi_{\gamma}=\frac{1}{8\pi^{2}}\sqrt{E_{\gamma}^{2}-m_{\gamma}^{2}}\hskip
2.13394ptdE_{\gamma}\hskip 1.42262ptd\cos\theta\,.$ (30)

Here $\theta$ is the angle that the photon makes with the $z$ direction and
$d\phi_{3}$ denotes the three-body phase space of the Born-level process $b\to
ce\nu$. Notice that $T_{1}(\theta_{\gamma})$ contains the soft divergences
needed to cancel the singularities of the virtual contribution, while
$T_{2}(\theta_{\gamma})$ and $T_{3}(\theta_{\gamma})$ are finite and vanish as
the domain of support of $\theta_{\gamma}$ is taken to zero Eynck  _et al._
(2002). In our case this means that for sufficiently small $E_{\rm min}$ only
the term $T_{1}(\theta_{\gamma})$ needs to be included in the calculation.

From the above discussion it follows that in order to determine the real
${\cal O}(\alpha)$ corrections to the partonic $b\to ce\nu$ process we need to
calculate $R_{\rm hard}$ and $R_{\rm soft}$. The relevant Feynman graphs are
depicted in Figure 3. For the hard real contribution we obtain

$\displaystyle\begin{split}R_{\rm hard}&=\frac{16\hskip
0.35565pt\pi^{2}N_{B}}{m_{b}^{2}}\,\Bigg{\\{}\frac{Q_{b}^{2}\hskip
1.42262pty_{ce}\big{(}y_{b\gamma}\hskip
1.42262pty_{\nu\gamma}-2\left(y_{b\nu}-y_{\nu\gamma}\right)\big{)}}{2\hskip
0.7113pty_{b\gamma}^{2}}+\frac{Q_{c}^{2}\hskip
1.42262pty_{b\nu}\big{(}y_{c\gamma}\hskip 1.42262pty_{e\gamma}-2\hskip
0.35565pt\rho\left(y_{ce}+y_{e\gamma}\right)\big{)}}{2\hskip
0.7113pty_{c\gamma}^{2}}\\\\[5.69054pt] &\hskip
58.32823pt+\frac{Q_{e}^{2}\hskip 1.42262pty_{b\nu}\big{(}y_{c\gamma}\hskip
1.42262pty_{e\gamma}-2\hskip
0.7113ptr\left(y_{ce}+y_{c\gamma}\right)\big{)}}{2\hskip
0.7113pty_{e\gamma}^{2}}-Q_{b}\hskip
1.42262ptQ_{c}\left[\,\frac{y_{b\nu}\left(y_{be}+y_{ce}\right)}{2\hskip
0.7113pty_{b\gamma}}\right.\\\\[5.69054pt] &\hskip
72.55461pt\left.-\frac{y_{ce}\left(y_{b\nu}+y_{c\nu}\right)}{2\hskip
0.7113pty_{c\gamma}}-\frac{y_{bc}\hskip
0.7113pt\big{(}y_{b\nu}\left(2y_{ce}+y_{e\gamma})-y_{ce}\hskip
0.7113pty_{\nu\gamma}\right)\big{)}}{2\hskip 0.7113pty_{b\gamma}\hskip
0.7113pty_{c\gamma}}\right]\\\\[5.69054pt] &\hskip 58.32823pt-Q_{b}\hskip
1.42262ptQ_{e}\left[\,\frac{y_{b\nu}\left(y_{bc}+y_{ce}\right)}{2\hskip
0.7113pty_{b\gamma}}-\frac{y_{ce}\left(y_{b\nu}+y_{e\nu}\right)}{2\hskip
0.7113pty_{e\gamma}}\right.\\\\[5.69054pt] &\hskip
99.58464pt\left.-\frac{y_{be}\hskip
0.7113pt\big{(}y_{b\nu}\left(2y_{ce}+y_{c\gamma})-2\hskip 0.7113pty_{ce}\hskip
0.7113pty_{\nu\gamma}\right)\big{)}}{2\hskip 0.7113pty_{b\gamma}\hskip
0.7113pty_{c\gamma}}\right]\\\\[5.69054pt] &\hskip
58.32823pt-\frac{Q_{c}\hskip 1.42262ptQ_{e}\hskip 1.42262pty_{b\nu}\hskip
1.42262pt\big{(}\left(y_{ce}+y_{c\gamma}\right)\left(y_{ce}+y_{e\gamma}\right)-\rho\hskip
0.7113pty_{e\gamma}-r\hskip 0.7113pty_{c\gamma}\big{)}}{y_{c\gamma}\hskip
0.7113pty_{e\gamma}}\Bigg{\\}}\,.\end{split}$ (31)

Figure 3:  Real emission QED contributions to the $b\to ce\nu$ process. The
colour coding resembles that used in Figure 2.

In the case of the soft real contribution that appears in
$T_{1}(\theta_{\gamma})$ we instead find

$\displaystyle\begin{split}R_{\rm soft}&=-\frac{16\hskip
0.35565pt\pi^{2}B}{m_{b}^{2}}\left[\frac{Q_{b}^{2}}{\left(\lambda-
y_{b\gamma}\right)^{2}}+\frac{Q_{c}^{2}\hskip
1.42262pt\rho}{\left(\lambda+y_{c\gamma}\right)^{2}}+\frac{Q_{e}^{2}\hskip
1.42262ptr}{\left(\lambda+y_{e\gamma}\right)^{2}}\right.\\\\[5.69054pt]
&\left.\hskip 59.75095pt+\frac{Q_{b}\hskip 0.7113ptQ_{c}\hskip
1.42262pty_{bc}}{\left(\lambda-
y_{b\gamma}\right)\left(\lambda+y_{c\gamma}\right)}+\frac{Q_{b}\hskip
0.7113ptQ_{e}\hskip 1.42262pty_{be}}{\left(\lambda-
y_{b\gamma}\right)\left(\lambda+y_{e\gamma}\right)}+\frac{Q_{c}\hskip
0.7113ptQ_{e}\hskip
1.42262pty_{ce}}{\left(\lambda+y_{c\gamma}\right)\left(\lambda+y_{e\gamma}\right)}\right]\,,\hskip
17.07164pt\end{split}$ (32)

where $\lambda=m_{\gamma}^{2}/m_{b}^{2}$. This result is simple enough that
one can integrate it over the $d\phi_{\gamma}$ part of the soft phase space
(30) and obtain an analytic result. We find

$\begin{split}\int\\!\theta_{\gamma}\hskip 0.7113ptR_{\rm soft}\hskip
0.7113ptd\phi_{\gamma}&=\frac{B}{2}\hskip 1.42262pt\Big{[}\hskip 0.7113pt-
Q_{b}^{2}\hskip 1.42262ptI_{R}(p_{b})-Q_{c}^{2}\hskip
1.42262ptI_{R}(p_{c})-Q_{e}^{2}\hskip 1.42262ptI_{R}(p_{e})\\\\[5.69054pt]
&\hskip 33.43207pt-Q_{b}\hskip 1.42262ptQ_{c}\hskip
1.42262ptI_{R}(p_{b},p_{c})-Q_{b}\hskip 1.42262ptQ_{e}\hskip
1.42262ptI_{R}(p_{b},p_{e})+Q_{c}\hskip 1.42262ptQ_{e}\hskip
1.42262ptI_{R}(p_{c},p_{e})\hskip 0.7113pt\Big{]}\,.\end{split}$ (33)

The soft integrals $I_{R}(p_{i})$ and $I_{R}(p_{i},p_{j})$ are given in
Appendix D. Note that (32) and (33) are both proportional to the Born-level
contribution (23). This is again a result of the factorisation of IR physics
and a necessary condition so that soft singularities in $R_{\rm soft}$ can
cancel similar contribution in the virtual contributions $V_{\rm IR}$.

####  4.1.3 Implementation and validation

Figure 4:  Dependence of the complete ${\cal O}(\alpha)$ correction to the
total decay width, i.e. $\Delta\Gamma^{(1)}(m_{b})$ in (37), on
$E_{\mathrm{min}}$. The error bars represent the numerical integration error
provided by CUBA.

In the following, we briefly discuss the numerical implementation of the
virtual and real corrections, their combination and the validation of our MC
code. The three- and four-particle phase spaces $d\phi_{3}$ and $d\phi_{4}$
play an important role in the calculation of the differential decay width
(21). We adopt the four-body phase-space parameterisations given in Asatrian
_et al._ (2012). The multi-dimensional sampling necessary to integrate the
virtual and real contribution over their respective phase spaces is performed
with the help of the CUBA library Hahn (2005). To validate that soft
singularities cancel between (26) and (33) and to check the irrelevance of the
subtraction terms $T_{2}(\theta_{\gamma})$ and $T_{3}(\theta_{\gamma})$ in
(28), we have computed the electron energy spectrum and the total decay width
for few different values of the resolution parameter in the range $E_{\rm
min}\in[10^{-5},10^{-3}]\,{\rm GeV}$. Within the statistical uncertainties
given by CUBA we find agreement between the different results. In the case of
the complete ${\cal O}(\alpha)$ correction to the total decay width this
feature is illustrated in Figure 4. Hereafter we employ the value $E_{\rm
min}=10^{-4}\,{\rm GeV}$ as our standard choice. We have furthermore
numerically verified that the total decay width of $b\to c\ell\nu$ does not
contain logarithmic mass singularities by employing $m_{\ell}=511\,{\rm keV}$
or $m_{\ell}=106\,{\rm MeV}$. These two choices correspond to the case of an
electron or a muon in the final state. As a final cross-check of our MC
implementation we have computed the ${\cal O}(\alpha)$ corrections to the
total decay width of $\mu\to e\nu\nu$ and the ${\cal O}(\alpha_{s})$
corrections to the total decay width of $b\to ce\nu$, reproducing the classic
results Berman (1958); Kinoshita and Sirlin (1959); Nir (1989) with a
precision far better than $1\%$.

###  4.2 Numerical results

In Figure 5 we display the complete ${\cal O}(\alpha)$ corrections (green
curve and band) and the corresponding LL approximation (red curve) computed in
Section 2. The green curve corresponds to an interpolation obtained by
considering 40 different bins that cover the full physical region of
$y\in[2\hskip 0.7113pt\sqrt{r},1-\rho+r]$, while the green band reflects the
associated numerical integration uncertainties. Relative to the total ${\cal
O}(\alpha)$ corrections these uncertainties typically amount to around $1\%$,
except close to the zero of the depicted green curve. For the purpose of this
comparison, we have factored the Wilson coefficient out and set the
renormalisation scale $\mu$ equal to $m_{b}=m_{b}^{\rm kin}(1\,{\rm GeV})$. We
observe a relatively good agreement between the LL terms and the complete
${\cal O}(\alpha)$ corrections to the electron energy spectrum of the partonic
$b\to ce\nu$ transition, especially in the hard part of the spectrum, where
the LL approximation is expected to work best, and where the differences
amount to around $10\%$ to $20\%$. Writing

$f^{(1)}(y)=\frac{\bar{L}_{b/e}}{2}\,f^{(1)}_{\mathrm{LL}}(y)+\Delta
f^{(1)}(y)\,,$ (34)

with $f^{(1)}_{\mathrm{LL}}(y)$ given in (8), we can use our numerical results
for $f^{(1)}(y)$ to obtain a simple approximate expression for $\Delta
f^{(1)}(y)$. Employing $\rho=0.057$ and $r=1.25\cdot 10^{-8}$ and identifying
again the renormalisation scale $\mu$ with $m_{b}=m_{b}^{\rm kin}(1\,{\rm
GeV})$, we find

$\displaystyle\Delta f^{(1)}(y)$ $\displaystyle=\Bigg{[}-2.04264+119.012\hskip
1.42262pty-476.678\hskip 1.42262pty^{2}+2034.14\hskip 1.42262pty^{3}$ (35)
$\displaystyle\phantom{xxx}-4402.22\hskip 1.42262pty^{4}+4505.93\hskip
1.42262pty^{5}-1807.38\hskip 1.42262pty^{6}$
$\displaystyle\phantom{xxx}-66.8251\left(y-y_{\rm max}\right)\ln\left(y_{\rm
max}-y\right)\,\Bigg{]}\,\theta(y_{\rm max}-y)\,,$

where $y_{\rm max}=1-\rho+r$. This formula encodes the exact non-LL terms for
the input parameters listed above with a relative accuracy of better than
$1\%$. It is worth noting that in Section 2 we have used $m_{b}$ as the hard
scale in the logarithm $\bar{L}_{b/e}$ as defined in (4). This is a somewhat
arbitrary choice because the hard scale is in fact of the order of the energy
released, i.e. of ${\cal O}(m_{b}-m_{c})$, and using a scale lower than
$m_{b}$ in the LL QED effects might thus be more appropriate. To investigate
this aspect, we also display in Figure 5 the electron energy spectrum obtained
using $\bar{L}_{c/e}$ instead of $\bar{L}_{b/e}$ in the LL QED prediction
(dotted red curve). We observe a better agreement near the endpoint but not
elsewhere, suggesting that the terms beyond the LLs cannot be accounted for by
a rescaling. Hereafter we hence evaluate all LL QED corrections with our
standard choice $\bar{L}_{b/e}$.

By direct integration over the full phase space, we also obtain a value of the
${\cal O}(\alpha)$ effects in the total decay width of the partonic $b\to
ce\nu$ process,

$\Gamma=\Gamma^{(0)}g(\rho)\,\big{|}C(\mu)\big{|}^{2}\;\Big{[}\,1+\Delta\Gamma^{(1)}(\mu)\,\Big{]}\,,$
(36)

where $\Gamma^{(0)}$ and $g(\rho)$ are defined in (5) and (10), respectively.
The correction $\Delta\Gamma^{(1)}(\mu)$ represents the ${\cal O}(\alpha)$
contribution to the matrix element of the operator introduced in (19)
evaluated at the scale $\mu$. For the input parameters used before, we find

$\Delta\Gamma^{(1)}(\mu)=\frac{\alpha}{\pi}\left[\hskip
0.7113pt\ln\left(\frac{\mu^{2}}{m_{b}^{2}}\right)+5.516\hskip
0.7113pt(14)\hskip 0.7113pt\right]\,,$ (37)

where the coefficient of the logarithm is exact while the quoted numerical
coefficient has as indicated an uncertainty of around $0.3\%$ which is
associated to our MC phase-space integration. Combining (20), (36) and (37),
one finds to ${\cal O}(\alpha)$ that

$\begin{split}\frac{\Gamma}{\Gamma^{(0)}g(\rho)}&=1+\frac{\alpha}{\pi}\left[\hskip
0.7113pt\ln\left(\frac{M_{Z}^{2}}{m_{b}^{2}}\right)-\frac{11}{6}+5.516\hskip
0.7113pt(14)\hskip 0.7113pt\right]\\\\[11.38109pt]
&=1+1.43\%-0.44\%+1.32\%=1+2.31\%\,,\end{split}$ (38)

where in the second line we have dropped the quoted uncertainty but given the
numerical results of the individual ${\cal O}(\alpha)$ terms as well as their
sum. The first observation to make is that the renormalisation scale
dependence has cancelled between the ${\cal O}(\alpha)$ corrections to the
Wilson coefficient and the virtual contributions to the matrix element
$\big{(}$cf. (20) and (25)$\big{)}$ leaving behind the EW logarithm first
computed in Sirlin (1982). In fact, it is interesting to note that this
logarithm represents about $60\%$ of the total ${\cal O}(\alpha)$ correction
in (38). Comparing the result (18) with (38) one furthermore observes that the
$\pi^{2}$-enhanced terms calculated in Section 3 provide about $80\%$ of
$\Delta\Gamma^{(1)}(m_{b})$, i.e. the complete ${\cal O}(\alpha)$ contribution
to the matrix element of (19). Hence, the complete ${\cal O}(\alpha)$
correction to the total decay width of $b\to ce\nu$ is well approximated by
the sum of the EW logarithm and the $\pi^{2}$-enhanced threshold effects,
which are both scale- and scheme-independent.

Figure 5:  Comparison of the complete ${\cal O}(\alpha)$ corrections (34) to
the electron energy spectrum (green curve) in $b\to ce\nu$ and the
corresponding LL approximation (red curve). In the former case also the
uncertainty of our numerical phase-space integration is indicated (green
band). The LL approximation using $\bar{L}_{c/e}$ instead of $\bar{L}_{b/e}$
is displayed as well (dotted red curve). See the main text for additional
details.

The relevant quantities used in the experimental analyses are the branching
ratio of $B\to X_{c}e\nu$, the electron energy spectrum and its moments with a
lower cut $E_{\rm cut}$ on the electron energy in the rest frame of the
decaying $B$ meson. A quantity employed in the extrapolation of the semi-
leptonic branching ratio with a cut $E_{e}>E_{\rm cut}$ to the total semi-
leptonic branching ratio is

$R^{\ast}(E_{\rm cut})=\frac{\displaystyle\int_{E_{\rm cut}}^{E_{\rm
max}}dE_{e}\,\frac{d\Gamma}{dE_{e}}}{\displaystyle\int_{m_{e}}^{E_{\rm
max}}dE_{e}\,\frac{d\Gamma}{dE_{e}}}\,,$ (39)

where $E_{\rm max}=m_{b}/2\left(1-\rho+r\right)$ denotes the maximum of the
electron energy available in the partonic $b\to ce\nu$ transition. The moments
of the electron energy spectrum are defined as

$\left\langle E_{e}^{n}\right\rangle_{E_{e}>E_{\rm
cut}}=\frac{\displaystyle\int_{E_{\rm cut}}^{E_{\rm
max}}dE_{e}\,E_{e}^{n}\,\frac{d\Gamma}{dE_{e}}}{\displaystyle\int_{E_{\rm
cut}}^{E_{\rm max}}dE_{e}\,\frac{d\Gamma}{dE_{e}}}\,.$ (40)

In the $B$-factory analyses Aubert  _et al._ (2004); Urquijo  _et al._ (2007);
Aubert  _et al._ (2010) the central moments of the electron energy spectrum
are considered. The first three central moments are given by

$\begin{split}\ell_{1}(E_{\rm cut})&=\left\langle
E_{e}\right\rangle_{E_{e}>E_{\rm cut}}\,,\\\\[8.53581pt] \ell_{2}(E_{\rm
cut})&=\Big{\langle}\big{(}E-\langle
E_{e}\rangle\big{)}^{2}\Big{\rangle}_{E_{e}>E_{\rm cut}}\,,\\\\[5.69054pt]
\ell_{3}(E_{\rm cut})&=\Big{\langle}\big{(}E-\langle
E_{e}\rangle\big{)}^{3}\Big{\rangle}_{E_{e}>E_{\rm cut}}\,.\end{split}$ (41)

$E_{\rm cut}$ | $R^{\ast\hskip 0.7113pt{\rm LO}}$ | $R^{\ast}$ | $\Delta R^{\ast\hskip 0.7113pt{\rm LL}}$ | $\Delta R^{\ast\hskip 0.7113pt(1)}$
---|---|---|---|---
$0.6$ | $0.9556$ | $0.9516$ | $-0.0150$ | $-0.0120\hskip 0.7113pt(3)$
$0.8$ | $0.9005$ | $0.8919$ | $-0.0211$ | $-0.0176\hskip 0.7113pt(3)$
$1.0$ | $0.8171$ | $0.8017$ | $-0.0266$ | $-0.0230\hskip 0.7113pt(1)$
$1.2$ | $0.7041$ | $0.6798$ | $-0.0304$ | $-0.0270\hskip 0.7113pt(1)$
$1.5$ | $0.4824$ | $0.4431$ | $-0.0306$ | $-0.0282\hskip 0.7113pt(1)$
$E_{\rm cut}$ | $\ell_{1}^{\rm LO}$ | $\ell_{1}$ | $\Delta\ell_{1}^{\rm LL}$ | $\Delta\ell_{1}^{(1)}$
$m_{e}$ | $1.4187$ | $1.3867$ | $-0.0389$ | $-0.0342\hskip 0.7113pt(1)$
$0.6$ | $1.4639$ | $1.4345$ | $-0.0229$ | $-0.0219\hskip 0.7113pt(5)$
$0.8$ | $1.5101$ | $1.4831$ | $-0.0181$ | $-0.0176\hskip 0.7113pt(4)$
$1.0$ | $1.5718$ | $1.5478$ | $-0.0138$ | $-0.0136\hskip 0.7113pt(3)$
$1.2$ | $1.6469$ | $1.6272$ | $-0.0102$ | $-0.0101\hskip 0.7113pt(2)$
$1.5$ | $1.7803$ | $1.7721$ | $-0.0058$ | $-0.0059\hskip 0.7113pt(1)$
$E_{\rm cut}$ | $\ell_{2}^{\rm LO}$ | $\ell_{2}$ | $\Delta\ell_{2}^{\rm LL}$ | $\Delta\ell_{2}^{(1)}$
$m_{e}$ | $0.1832$ | $0.1836$ | $+0.0163$ | $+0.0122\hskip 0.7113pt(2)$
$0.6$ | $0.1449$ | $0.1452$ | $+0.0024$ | $+0.0019\hskip 0.7113pt(2)$
$0.8$ | $0.1165$ | $0.1171$ | $+0.0006$ | $+0.0004\hskip 0.7113pt(1)$
$1.0$ | $0.0870$ | $0.0883$ | $-0.00024$ | $-0.00033\hskip 0.7113pt(7)$
$1.2$ | $0.0597$ | $0.0619$ | $-0.00048$ | $-0.00054\hskip 0.7113pt(3)$
$1.5$ | $0.0272$ | $0.0308$ | $-0.00035$ | $-0.00038\hskip 0.7113pt(1)$
$E_{\rm cut}$ | $\ell_{3}^{\rm LO}$ | $\ell_{3}$ | $\Delta\ell_{3}^{\rm LL}$ | $\Delta\ell_{3}^{(1)}$
$m_{e}$ | $-0.0416$ | $-0.0325$ | $-0.0065$ | $-0.0036\hskip 0.7113pt(2)$
$0.6$ | $-0.0184$ | $-0.0108$ | $+0.00238$ | $+0.00236\hskip 0.7113pt(8)$
$0.8$ | $-0.0090$ | $-0.0027$ | $+0.00186$ | $+0.00182\hskip 0.7113pt(5)$
$1.0$ | $-0.0031$ | $\phantom{+}0.0017$ | $+0.00114$ | $+0.00112\hskip 0.7113pt(3)$
$1.2$ | $-0.00029$ | $\phantom{+}0.0032$ | $+0.00058$ | $+0.00057\hskip 0.7113pt(2)$
$1.5$ | $\phantom{+}0.00057$ | $\phantom{+}0.0023$ | $+0.000136$ | $+0.000137\hskip 0.7113pt(5)$
Table 1:  Comparison of the LL and the complete ${\cal O}(\alpha)$
contributions to the observables defined in (39) and (41) for different values
of the lower cut $E_{\rm cut}$ on the electron energy. All quantities are
given in units of ${\rm GeV}$ to the appropriate power. The second column
shows the LO results while the third column corresponds to the state-of-the-
art QCD predictions used in the recent HQE global fit Bordone  _et al._
(2021). The fourth and fifth columns present the absolute shifts due to the LL
and the complete ${\cal O}(\alpha)$ corrections, respectively. Only the
leading power terms are included in these ${\cal O}(\alpha)$ predictions.
Additional explanation can be found in the main text.

Notice that all the observables introduced in (39), (40) and (41) are ratios
and as such they are insensitive to the overall normalisation. This means in
particular that they neither depend on the CKM matrix element $V_{cb}$ nor on
the Wilson coefficient $C(\mu)$. They are also renormalisation scale
independent and, to an excellent approximation, insensitive to the
$\pi^{2}$-enhanced threshold correction. Since they do not cancel out like in
the total decay width, the LL contributions should therefore represent the
leading source of QED corrections to (39), (40) and (41), and it can be
expected that in the case of the higher moments these corrections may be
sizeable. To illustrate the latter point let us consider the central moments
defined in (41) in the absence of a cut on the electron energy, i.e.
$E_{e}>m_{e}$. Relative to the LO results we find that the LL QED corrections
to the first three central moments are

$\begin{split}\delta\ell_{1}^{\rm
LL}(m_{e})&=-\frac{2}{3}\,\frac{\alpha}{\pi}\,\bar{L}_{b,e}=-2.74\%\,,\\\\[5.69054pt]
\delta\ell_{2}^{\rm
LL}(m_{e})&=2.16\,\frac{\alpha}{\pi}\,\bar{L}_{b,e}=+8.9\%\,,\\\\[5.69054pt]
\delta\ell_{3}^{\rm
LL}(m_{e})&=3.78\,\frac{\alpha}{\pi}\,\bar{L}_{b,e}=+15.6\%\,.\end{split}$
(42)

Notice that the result for $\delta\ell_{1}^{\rm LL}(m_{e})$ is independent of
$\rho$ because, in the absence of a lower cut on the electron energy, the
moments of a convolution factorise and the LL correction to the first moment
is proportional to the lowest order result. The results for
$\delta\ell_{2}^{\rm LL}(m_{e})$ and $\delta\ell_{3}^{\rm LL}(m_{e})$ instead
depend on $\rho$ and we have employed the input parameters discussed before in
our numerics. For the corresponding complete ${\cal O}(\alpha)$ corrections we
instead obtain the relative shifts

$\begin{split}\delta\ell_{1}^{(1)}(m_{e})&=-10.08\hskip 0.7113pt(3)\hskip
0.7113pt\,\frac{\alpha}{\pi}=-2.412\hskip 0.7113pt(7)\%\,,\\\\[5.69054pt]
\delta\ell_{2}^{(1)}(m_{e})&=27.8\hskip 0.7113pt(5)\hskip
0.7113pt\,\frac{\alpha}{\pi}=+6.7\hskip 0.7113pt(1)\%\,,\\\\[5.69054pt]
\delta\ell_{3}^{(1)}(m_{e})&=36\hskip
0.7113pt(2)\,\frac{\alpha}{\pi}=+8.7\hskip 0.7113pt(5)\%\,.\end{split}$ (43)

As indicated these predictions have relative uncertainties due to our
numerical phase-space integration of around $0.3\%$, $1.8\%$ and $5.6\%$,
respectively. From (42) and (43) one observes that while for the first central
moment the LL and the full ${\cal O}(\alpha)$ predictions agree quite well, in
the case of the second and third central moments the complete ${\cal
O}(\alpha)$ results are markedly smaller than their LL counterparts. This is
because in the higher central moments there are strong cancellations among the
various contributing linear moments, which subdominant contributions tend to
disrupt, leading to larger deviations from the LL approximation.

A comprehensive comparison of the LL and the complete ${\cal O}(\alpha)$
contributions to the observables defined in (39) and (41) is presented in
Table 1. The numbers given in the table show that the LL approximation
captures the bulk of the full ${\cal O}(\alpha)$ corrections to
$R^{\ast}(E_{\rm cut})$, $\ell_{1}(E_{\rm cut})$, $\ell_{2}(E_{\rm cut})$ and
$\ell_{3}(E_{\rm cut})$. In fact, the LL approximation tends to improve with
increasing $E_{\rm cut}$ values. Also notice that the considered QED
corrections are comparable in magnitude to the higher-order QCD and power
corrections that have been included in the recent HQE global fit Bordone  _et
al._ (2021). We add that the latter feature probably only applies to
$R^{\ast}(E_{\rm cut})$ and to the moments of the electron energy spectrum
(41), where perturbative QCD and power corrections are relatively suppressed.

##  5 Comparison with BaBar implementation of QED effects

As already explained in the introduction, both BaBar and Belle subtract photon
radiation associated to charged final-state particles using PHOTOS, because
such effects were not included in the HQE theoretical predictions that were
available at that time. In fact, the BaBar papers Aubert  _et al._ (2004,
2010) report explicitly the size of the PHOTOS correction that is applied for
each observable. In the publication Aubert  _et al._ (2010), the BaBar
collaboration updated its earlier measurement Aubert  _et al._ (2004) using
more recent values of the branching ratios of some of the background
processes. This more recent paper reports the size of the PHOTOS corrections
only for $E_{\rm cut}=\\{0.6,1.5\\}\,{\rm GeV}$, finding results which are
very close to those given previously in Aubert  _et al._ (2004). Since the
earlier work Aubert  _et al._ (2004) provides numbers for five different cuts
$E_{\rm cut}=\\{0.6,0.8,1.0,1.2,1.5\\}\,{\rm GeV}$ we will use them for our
comparison of the branching ratio of $B\to X_{c}e\nu$ $\big{(}$${\rm BR}_{\rm
incl}(E_{\rm cut})$$\big{)}$ and the first three central moments.

To obtain QED correction factors for the distributions of the inclusive $B\to
X_{c}e\nu$ decay, the BaBar collaboration has generated spectra for various
exclusive decay channels such as $B\to De\nu$, $B\to D^{\ast}e\nu$ and $B\to
D^{\ast\ast}e\nu$ without and with up to two photon emissions utilising
PHOTOS, appropriately weighting each exclusive sample. BaBar then subtracts
the obtained corrections from the measured or uncorrected values of ${\rm
BR}_{\rm incl}(E_{\rm cut})$, $\ell_{1}(E_{\rm cut})$, $\ell_{2}(E_{\rm cut})$
and $\ell_{3}(E_{\rm cut})$ to derive the corrected or QCD values of the
branching ratio and the central moments. Notice that the procedure that BaBar
uses to estimate the size of the QED corrections to $B\to X_{c}e\nu$ is only
able to capture the soft and collinear photonic corrections to the matrix
element of the operator $Q$ in specific channels, while it clearly misses
structure-dependent contributions, EW corrections that are associated to
short-distance physics encoded in the Wilson coefficient, as well as hard
photon emission and virtual photon corrections such as the $\pi^{2}$-enhanced
threshold corrections discussed in Section 3.

Any comparison with the QED correction factors obtained by BaBar requires a
choice of how to split the complete QED corrections to the differential decay
width (21) into an effective coefficient $A_{\rm EW}$ and a function $f(y)$
that describes the normalised electron energy spectrum in $B\to X_{c}e\nu$. We
choose the following decomposition

$\frac{d\Gamma}{dy}=\Gamma^{(0)}\,A_{\rm EW}\,f(y)\,,\qquad A_{\rm
EW}=1+\frac{\alpha}{\pi}\hskip
0.7113pt\ln\left(\frac{M_{Z}^{2}}{m_{b}^{2}}\right)=1.0143\,,$ (44)

with

$\displaystyle\begin{split}f(y)&=f^{(0)}(y)+\frac{\alpha}{\pi}\,\left[f^{(1)}(y)-\frac{11}{6}\hskip
0.7113pt\right]+\left(\frac{\alpha}{2\pi}\right)^{2}\frac{\bar{L}_{b/e}^{2}}{2}\,f^{(2)}_{\rm
NLL}(y)\\\\[5.69054pt]
&+\sum_{i=\pi,G}\frac{\mu_{i}^{2}}{m_{b}^{2}}\left[f^{(0)}_{i}(y)+\frac{\alpha}{2\pi}\,\bar{L}_{b/e}\,f^{(1)}_{i,{\rm
LL}}(y)\right]+\sum_{j=D,SL}\frac{\rho_{j}^{3}}{m_{b}^{3}}\left[f^{(0)}_{j}(y)+\frac{\alpha}{2\pi}\,\bar{L}_{b/e}\,f^{(1)}_{j,{\rm
LL}}(y)\right]\,.\hskip 11.38109pt\end{split}$ (45)

Here $\Gamma^{(0)}$ is given in (5) and the logarithmic ${\cal O}(\alpha)$
correction in $A_{\rm EW}$ is the scheme-independent part of the Wilson
coefficient (20) with the renormalisation scale $\mu$ set to $m_{b}$. The
numerical value of $A_{\rm EW}$ quoted in (44) corresponds to $m_{b}^{\rm
kin}(1\,{\rm GeV})=4.57\,{\rm GeV}$ and $M_{Z}=91.1876\,{\rm GeV}$. It is the
value of $A_{\rm EW}$ that is typically used in the literature (see for
example Bordone  _et al._ (2021)). The functions $f^{(0)}(y)$ and $f^{(1)}(y)$
are given in (6) and (35), respectively, and the term $-11/6$ is the scheme-
dependent part of the Wilson coefficient (20). Since it is contained in $f(y)$
the function is formally scheme independent up to ${\cal O}(\alpha)$. The
function $f^{(2)}_{\rm NLL}(y)$ describes the NLL QED effects to the partonic
electron energy spectrum and is evaluated in Appendix B, while the
contributions in the second line contain the LL QED contributions to the power
corrections that are detailed in Appendix A. In our numerical study, we will
use the values $\mu_{\pi}^{2}=0.477\,{\rm GeV}^{2}$, $\mu_{G}^{2}=0.306\,{\rm
GeV}^{2}$, $\rho_{D}^{3}=0.185\,{\rm GeV}^{3}$ and $\rho_{SL}^{3}=-0.130\,{\rm
GeV}^{3}$, corresponding to the central values of the HQE parameters obtained
in the recent global fit Bordone  _et al._ (2021).

Figure 6:  Comparison of the absolute shift of the QED corrections to ${\rm
BR}_{\rm incl}(E_{\rm cut})$ as a function of the lower cut $E_{\rm cut}$ on
the electron energy. The black curve corresponds to the correction obtained by
BaBar in Aubert  _et al._ (2004) using PHOTOS, while the red (green) curve
corresponds to our QED prediction including the LL terms (all QED corrections)
given in (45). The grey band represents the systematic uncertainty on the
PHOTOS bremsstrahlungs corrections that BaBar quotes in Aubert  _et al._
(2004), while the black error bars correspond to the total uncertainties of
the QED corrected BaBar results. For further explanations consult the main
text.

In Figure 6 we present a comparison of the absolute shift of the QED
corrections to the inclusive branching ratio of $B\to X_{c}e\nu$ imposing a
lower cut $E_{e}>E_{\rm cut}$ on the electron energy. The black curve
corresponds to the correction obtained by BaBar in Aubert  _et al._ (2004)
using PHOTOS and the grey band represents the associated systematic
uncertainties. The black error bars furthermore correspond to the total
uncertainties of the QED corrected measurements. Our LL prediction contains
all terms in (45) that are proportional to $\alpha/\pi\hskip
0.7113pt\bar{L}_{b/e}$ and is indicated by the red line. The green curve
instead corresponds to our best QED prediction considering all contributions
to (45). We observe that the BaBar results and our LL predictions show
relative deviations of around $40\%$. This is an expected feature because
PHOTOS as any other QED parton shower MC should capture the bulk of the
logarithmically enhanced photonic effects, even though it works with point-
like hadrons. However, the absence of virtual effects and in particular of the
$\pi^{2}$-enhanced threshold corrections in PHOTOS explains why our full QED
predictions are visibly above the BaBar results. Also notice that for $E_{\rm
cut}\lesssim 1.0\,{\rm GeV}$ the differences between the BaBar numbers and our
best QED predictions are larger than the systematic uncertainties on the
PHOTOS bremsstrahlungs corrections that BaBar provides in the publication
Aubert  _et al._ (2004). The found differences are however always safely
within the total experimental uncertainties.

In Table 2 we present a more detailed breakdown of the individual relative QED
corrections to ${\rm BR}_{\rm incl}(E_{\rm cut})$. All theory predictions are
normalised to the state-of-the-art QCD predictions of the branching ratio that
have been obtained recently in Bordone  _et al._ (2021). One first observes
that compared to the partonic LL QED effects the corresponding NLL corrections
are very small. Similar observations have been made in the case of muon decay
in Arbuzov  _et al._ (2002); Arbuzov and Melnikov (2002); Arbuzov (2003).
Likewise, the power-suppressed LL QED corrections of ${\cal O}(\Lambda_{\rm
QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm QCD}^{3}/m_{b}^{3})$ have only
a minor impact on ${\rm BR}_{\rm incl}(E_{\rm cut})$. In fact, the calculated
${\cal O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm
QCD}^{3}/m_{b}^{3})$ effects slightly improve the agreement with the BaBar
corrections. The non-logarithmic ${\cal O}(\alpha)$ effects calculated in
Section 4 and encoded by $\Delta f^{(1)}(y)$ are instead numerically quite
relevant and tend to reduce the partonic LL QED effects in magnitude. Notice
that this reduction would be larger by around $0.4\%$ if the constant $-11/6$
had been included in $A_{\rm EW}$ and not in $f(y)$ $\big{(}$cf. (44) and
(45)$\big{)}$. As a result when using our best QED calculation to correct the
BaBar measurements we obtain ${\rm BR}_{\rm incl}(E_{\rm cut})$ values that
are on average larger by about $0.2\sigma$ than the QED corrected values for
${\rm BR}_{\rm incl}(E_{\rm cut})$ given in Aubert  _et al._ (2004). Since for
low $E_{\rm cut}$ the values of $\delta{\rm BR}_{\rm incl}$ are about $0.8\%$
above the $\delta{\rm BR}^{\rm BaBar}_{\rm incl}$ numbers, we expect that our
improved calculation of QED effects will decrease the corrected inclusive
branching ratio and therefore decrease $|V_{cb}|$ by roughly $0.4\%$ compared
to the determination performed in Bordone  _et al._ (2021). Making this
statement more precise, however, would require a full HQE global fit which is
beyond the scope of this work.

$E_{\rm cut}$ | $\delta{\rm BR}^{\rm BaBar}_{\rm incl}$ | $\delta{\rm BR}^{\rm LL}_{\rm incl}$ | $\delta{\rm BR}^{\rm NLL}_{\rm incl}$ | $\delta{\rm BR}^{\alpha}_{\rm incl}$ | $\delta{\rm BR}^{1/m_{b}^{2}}_{\rm incl}$ | $\delta{\rm BR}_{\rm incl}$ | $\sigma$
---|---|---|---|---|---|---|---
$0.6$ | $-1.26\%$ | $-1.92\%$ | $-1.95\%$ | $-0.54\%$ | $-0.50\%$ | $-0.45\%$ | $+0.34$
$0.8$ | $-1.87\%$ | $-2.88\%$ | $-2.91\%$ | $-1.36\%$ | $-1.29\%$ | $-1.22\%$ | $+0.30$
$1.0$ | $-2.66\%$ | $-4.03\%$ | $-4.04\%$ | $-2.38\%$ | $-2.26\%$ | $-2.15\%$ | $+0.25$
$1.2$ | $-3.56\%$ | $-5.43\%$ | $-5.41\%$ | $-3.65\%$ | $-3.43\%$ | $-3.27\%$ | $+0.14$
$1.5$ | $-5.22\%$ | $-8.41\%$ | $-8.26\%$ | $-6.37\%$ | $-5.73\%$ | $-5.39\%$ | $-0.09$
Table 2:  Relative size of the QED corrections to ${\rm BR}_{\rm incl}(E_{\rm
cut})$. The values of $E_{\rm cut}$ are given in units of ${\rm GeV}$. The
entries in the column $\delta{\rm BR}^{\rm BaBar}_{\rm incl}$ are the
corrections obtained by BaBar in Aubert  _et al._ (2004), while the numbers
for $\delta{\rm BR}^{\rm LL}_{\rm incl}$, $\delta{\rm BR}^{\rm NLL}_{\rm
incl}$ and $\delta{\rm BR}^{\alpha}_{\rm incl}$ successively include the LL,
NLL and complete ${\cal O}(\alpha)$ corrections to the $b\to ce\nu$ branching
ratio. The $\delta{\rm BR}^{1/m_{b}^{2}}_{\rm incl}$ numbers include all
partonic QED effects as well as the LL QED corrections to the ${\cal
O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ power corrections. The entries in the
column $\delta{\rm BR}_{\rm incl}$ represent our best predictions and include
besides all partonic QED effects the power-suppressed LL QED corrections up to
${\cal O}(\Lambda_{\rm QCD}^{3}/m_{b}^{3})$ $\big{(}$see (45)$\big{)}$. The
relative shifts in standard deviations ($\sigma$) that we obtain when using
our best QED calculation to correct the BaBar measurements are given in the
last column. See main text for additional details.

Figure 7:  Comparison of the absolute shift of the QED corrections to
$\ell_{1}(E_{\rm cut})$, $\ell_{2}(E_{\rm cut})$ and $\ell_{3}(E_{\rm cut})$
as a function of the lower cut $E_{\rm cut}$ on the electron energy. The
colour coding resembles that used in Figure 6.

The absolute shift of the QED corrections to $\ell_{1}(E_{\rm cut})$,
$\ell_{2}(E_{\rm cut})$ and $\ell_{3}(E_{\rm cut})$ is shown in the three
panels in Figure 7. In order to not spoil the strong cancellations between the
quantum corrections to the numerator and the denominator that enter the
normalised central moments Biswas and Melnikov (2010); Gambino (2011) we
perform a double-series expansion in $\alpha$ and $\Lambda_{\rm QCD}/m_{b}$
when calculating the ratios (41). In this expansion we keep all the terms up
to the order indicated by the superscript following the notation introduced in
(45). We add that we have verified that the expanded and unexpanded results of
the central moments are numerically quite close together. The black curves
correspond to the QED corrections estimated by BaBar in Aubert  _et al._
(2004) with the help of PHOTOS, while the red (green) lines represent our LL
$\big{(}$full ${\cal O}(\alpha)$$\big{)}$ predictions. The grey bands
represent the systematic uncertainties that are associated to the experimental
subtraction procedure of QED corrections performed in Aubert  _et al._ (2004),
while the black error bars correspond to the total uncertainties of the BaBar
measurements. From all three plots it is evident that the LL QED corrections
describe the BaBar corrections pretty well and that the numerical impact of
the non-LL ${\cal O}(\alpha)$ corrections is notably smaller in the case of
$\ell_{1}(E_{\rm cut})$, $\ell_{2}(E_{\rm cut})$ and $\ell_{3}(E_{\rm cut})$
than for ${\rm BR}_{\rm incl}(E_{\rm cut})$. Still the inclusion of the term
$\alpha/\pi\left(\Delta f^{(1)}(y)-11/6\right)$ in the calculation of the
central moments in general improves the agreement between the BaBar and our
QED corrections. Also notice that in the case of $\ell_{2}(E_{\rm cut})$ the
differences between the BaBar numbers and our best QED predictions are within
the systematic uncertainty band of the PHOTOS bremsstrahlungs corrections,
while this is not the case for $\ell_{1}(E_{\rm cut})$ and $\ell_{3}(E_{\rm
cut})$. Given that the systematic uncertainties associated to the subtraction
of QED effects are always a subdominant component in the total experimental
uncertainties, our absolute shifts $\Delta\ell_{1}(E_{\rm cut})$,
$\Delta\ell_{2}(E_{\rm cut})$ and $\Delta\ell_{3}(E_{\rm cut})$ are, however,
always fully compatible with the combined errors quoted by BaBar.

$E_{\rm cut}$ | $\delta\ell_{1}^{\rm BaBar}$ | $\delta\ell_{1}^{\rm LL}$ | $\delta\ell_{1}^{\rm NLL}$ | $\delta\ell_{1}^{\alpha}$ | $\delta\ell_{1}^{1/m_{b}^{2}}$ | $\delta\ell_{1}$ | $\sigma$
---|---|---|---|---|---|---|---
$0.6$ | $-1.29\%$ | $-1.60\%$ | $-1.58\%$ | $-1.48\%$ | $-1.45\%$ | $-1.42\%$ | $-0.22$
$0.8$ | $-1.01\%$ | $-1.22\%$ | $-1.20\%$ | $-1.16\%$ | $-1.13\%$ | $-1.10\%$ | $-0.31$
$1.0$ | $-0.74\%$ | $-0.89\%$ | $-0.88\%$ | $-0.87\%$ | $-0.84\%$ | $-0.82\%$ | $-0.40$
$1.2$ | $-0.53\%$ | $-0.63\%$ | $-0.62\%$ | $-0.62\%$ | $-0.59\%$ | $-0.58\%$ | $-0.32$
$1.5$ | $-0.29\%$ | $-0.33\%$ | $-0.32\%$ | $-0.34\%$ | $-0.31\%$ | $-0.30\%$ | $-0.13$
$E_{\rm cut}$ | $\delta\ell_{2}^{\rm BaBar}$ | $\delta\ell_{2}^{\rm LL}$ | $\delta\ell_{2}^{\rm NLL}$ | $\delta\ell_{2}^{\alpha}$ | $\delta\ell_{2}^{1/m_{b}^{2}}$ | $\delta\ell_{2}$ | $\sigma$
$0.6$ | $+0.31\%$ | $+1.65\%$ | $+1.43\%$ | $+0.91\%$ | $+0.48\%$ | $+0.50\%$ | $+0.07$
$0.8$ | $-0.34\%$ | $+0.50\%$ | $+0.34\%$ | $+0.04\%$ | $-0.40\%$ | $-0.33\%$ | $+0.01$
$1.0$ | $-1.00\%$ | $-0.27\%$ | $-0.38\%$ | $-0.60\%$ | $-1.08\%$ | $-0.95\%$ | $+0.04$
$1.2$ | $-1.27\%$ | $-0.78\%$ | $-0.85\%$ | $-1.05\%$ | $-1.60\%$ | $-1.42\%$ | $-0.08$
$1.5$ | $-1.91\%$ | $-1.15\%$ | $-1.18\%$ | $-1.40\%$ | $-2.24\%$ | $-1.93\%$ | $-0.01$
$E_{\rm cut}$ | $\delta\ell_{3}^{\rm BaBar}$ | $\delta\ell_{3}^{\rm LL}$ | $\delta\ell_{3}^{\rm NLL}$ | $\delta\ell_{3}^{\alpha}$ | $\delta\ell_{3}^{1/m_{b}^{2}}$ | $\delta\ell_{3}$ | $\sigma$
$0.6$ | $-17.3\%$ | $-22.1\%$ | $-23.1\%$ | $-22.7\%$ | $-22.6\%$ | $-22.5\%$ | $+0.35$
$0.8$ | $-42.8\%$ | $-68.9\%$ | $-69.2\%$ | $-66.7\%$ | $-62.9\%$ | $-62.9\%$ | $+0.45$
$1.0$ | $+63.9\%$ | $+67.3\%$ | $+66.3\%$ | $+63.9\%$ | $+56.2\%$ | $+57.6\%$ | $+0.34$
$1.2$ | $+11.7\%$ | $+18.1\%$ | $+17.6\%$ | $+17.1\%$ | $+13.3\%$ | $+14.7\%$ | $+0.28$
$1.5$ | $-0.47\%$ | $+5.92\%$ | $+5.69\%$ | $+5.61\%$ | $+2.10\%$ | $+4.69\%$ | $+0.23$
Table 3:  Relative size of the QED corrections to the first three central
moments of the electron energy spectrum (41). The notation resembles that used
in Table 2.

In Table 3 we present a complete breakdown of the individual relative QED
corrections to $\ell_{1}(E_{\rm cut})$, $\ell_{2}(E_{\rm cut})$ and
$\ell_{3}(E_{\rm cut})$. All theory predictions are again normalised to the
state-of-the-art QCD results for the central moments Bordone  _et al._ (2021).
One first notices that while in the case of $\ell_{1}(E_{\rm cut})$ and
$\ell_{2}(E_{\rm cut})$ the relative corrections are at the level of $1\%$ or
below, for $\ell_{3}(E_{\rm cut})$ the $\delta\ell_{3}(E_{\rm cut})$ values
typically exceed $10\%$. This feature can be traced back to the smallness of
the QCD prediction for $\ell_{3}(E_{\rm cut})$ (cf. Table 1) with
$\ell_{3}(0.9\,{\rm GeV})\simeq 0$ leading to a sign flip of
$\delta\ell_{3}(E_{\rm cut})$ between $E_{\rm cut}=0.8\,{\rm GeV}$ and $E_{\rm
cut}=1.0\,{\rm GeV}$. In the case of $\ell_{1}(E_{\rm cut})$, one furthermore
sees that successively including the NLL, the complete ${\cal O}(\alpha)$
corrections as well as the LL QED terms of the ${\cal O}(\Lambda_{\rm
QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm QCD}^{3}/m_{b}^{3})$ power
corrections steadily improves the agreement between our and the BaBar numbers.
As a result, our best QED predictions differ from the BaBar results by no more
than $0.13\%$ for $\ell_{1}(E_{\rm cut})$. In the case of $\ell_{2}(E_{\rm
cut})$ and $\ell_{3}(E_{\rm cut})$, we find that the LL QED corrections of
${\cal O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm
QCD}^{3}/m_{b}^{3})$ are larger than the partonic ${\cal O}(\alpha)$
corrections, and that the inclusion of the former contributions in many cases
help to improve the agreement between our and the PHOTOS calculation of QED
effect. These features are particularly apparent in the case of
$\ell_{2}(E_{\rm cut})$. Numerically, we obtain differences below $0.19\%$ and
below $21\%$ for the second and third central moment, respectively. Taking
into account the total experimental uncertainties the found deviations however
amount to always less than $0.5\sigma$.

##  6 Conclusions and outlook

In order to match the increasing experimental accuracy in semi-leptonic,
radiative and rare $B$ decays, a reliable assessment of EW and QED corrections
is desirable in general, in some cases even mandatory. In recent years, such
calculations have received considerable theoretical attention de Boer  _et
al._ (2018); Calí  _et al._ (2019); Mishra and Mahajan (2021); Bansal  _et
al._ (2022); Gambino and Haisch (2000, 2001); Bobeth  _et al._ (2004); Huber
_et al._ (2006, 2008); Bobeth  _et al._ (2014); Huber  _et al._ (2015);
Bordone  _et al._ (2016); Beneke  _et al._ (2018, 2019); Isidori  _et al._
(2020, 2022); Cornella  _et al._ (2023); Huang  _et al._ (2023); Choudhury
_et al._ (2023); Gorbahn  _et al._ (2023). This article represents a first
step towards obtaining precision predictions for inclusive semi-leptonic $B\to
X_{c}\ell\nu$ decays including EW and QED corrections that can be directly
confronted with existing and future $B$-factory measurements. To this purpose,
we have identified two types of large QED corrections, namely collinear terms
proportional to $\ln\left(m_{b}^{2}/m_{\ell}^{2}\right)$ and threshold
corrections enhanced by a factor $\pi^{2}$. In order to show that these terms
indeed provide the dominant part of the QED corrections, we have performed a
complete ${\cal O}(\alpha)$ calculation of the electron energy spectrum and
the total decay width of the partonic process $b\to ce\nu$. The relevant
virtual and real matrix elements have been calculated analytically while the
phase-space integration is performed numerically, using an improved phase
space slicing method to handle soft singularities. This in principle allows
for a fully-differential calculation of any IR-safe observable in $b\to
ce\nu$. Motivated by the observation that the logarithmically enhanced
corrections represent the dominant ${\cal O}(\alpha)$ effects in the electron
energy spectrum of $b\to ce\nu$, we have furthermore calculated the NLL QED
corrections in the partonic case and the LL QED corrections in the case of the
power corrections of ${\cal O}(\Lambda^{2}_{\rm QCD}/m_{b}^{2})$ and ${\cal
O}(\Lambda^{3}_{\rm QCD}/m_{b}^{3})$ to the first few moments of the spectrum.
These calculations can be performed analytically using splitting functions. We
have also completed the computation of the short-distance Wilson coefficient
at ${\cal O}(\alpha)$ finding agreement with Brod and Gorbahn (2008); Gorbahn
_et al._ (2023).

Including all the EW and QED corrections mentioned above, we have then
performed a detailed comparison of our results against those obtained by BaBar
in Aubert  _et al._ (2004) using the PHOTOS package (see Calí  _et al._
(2019); Bobeth  _et al._ (2014); Huber  _et al._ (2015); Isidori  _et al._
(2022) for similar comparisons). This MC code is the standard tool used in the
relevant experimental $B$-factory analyses to subtract photon radiation from
the measured distributions. While PHOTOS includes the real and virtual QED
corrections that are needed to obtain an exact description of the soft-photon
region of the phase space, other effects such as the interference between
initial- and final-state photons and threshold corrections enhanced by
$\pi^{2}$ are not incorporated in the generator. Our comprehensive comparison
to PHOTOS shows that in the case of the branching ratio the missing pieces
lead to a systematic shift upwards by around $0.8\%$ for low cuts on the
electron energy. Naively, this will decrease the value of $|V_{cb}|$ by
roughly $0.4\%$ compared to the inclusive determination performed in Bordone
_et al._ (2021). For what concerns the first and second central moment of the
electron energy spectrum, we find differences of below $0.13\%$ and below
$0.19\%$, respectively. In the case of the third central moment of the
electron energy spectrum, the maximal deviations between our and the PHOTOS
results can reach up to around $21\%$. We also find that the differences
between the BaBar numbers and our best QED predictions are often larger than
the systematic uncertainties of the PHOTOS bremsstrahlungs corrections
provided in Aubert  _et al._ (2004). Taking into account the total
experimental uncertainties the found deviations however amount to always less
than $0.5\sigma$. Interestingly, the inclusion of power corrections tends to
improve the agreement between our and the PHOTOS calculation of QED effects in
the branching ratio and the electron energy moments. Since our computation
contains QED effects not included in PHOTOS, we believe that our calculation
provides a better description of QED radiation in $B\to X_{c}e\nu$ than
PHOTOS.

The calculations and studies performed in this work can be extended in various
ways. In the case of the inclusive semi-leptonic $B\to X_{c}\ell\nu$ decays,
it would be worthwhile to generalise the computations presented here to the
moments of the hadronic and leptonic invariant mass distribution, since these
moments play an important role in the HQE global fits Bordone  _et al._
(2021); Bernlochner  _et al._ (2022) as well. Similar statements apply to the
forward-backward asymmetry Turczyk (2016) that has not been measured at BaBar
and Belle, but should be accessible at Belle II. The calculation techniques
employed in our article can also be applied to exclusive semi-leptonic $B$
decays. While the impact of QED effects on $B\to D\ell\nu$ have been studied
by several groups de Boer  _et al._ (2018); Calí  _et al._ (2019); Bansal  _et
al._ (2022), to our knowledge an analogous analysis for the $B\to
D^{\ast}\ell\nu$ decay does not exist. Such a calculation is also motivated by
the measurements of the lepton flavour universality ratios $R(D)$ and
$R(D^{\ast})$ that show long-standing deviations from their respective SM
predictions at the level of $3\sigma$ Amhis  _et al._ (2023). In summary, our
calculation and its planned extension to exclusive semi-leptonic $b\to
c\ell\nu$ decays sets the theoretical foundations to build a new MC generator
specifically designed to be used by the $B$-factory collaborations. Such a
tool will be a crucial ingredient in any future extraction of $|V_{cb}|$ that
aims at a precision close to $1\%$.

###### Acknowledgements.

We are grateful to Martin Beneke, Florian Bernlochner, Rhorry Gauld, Martin
Gorbahn, Gino Isidori, Martin Jung, Marcello Rotondo and Christoph Schwanda
for useful discussions and explanations. The research of PG is supported in
part by the Italian Ministry of Research (MIUR) under grant PRIN 20172LNEEZ.
He also received partial support from the Excellence Cluster ORIGINS which is
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2094 – 390783311 as well
as from the Sino-German Collaborative Research Center TRR110 “Symmetries and
the Emergence of Structure in QCD” (DFG Project-ID 196253076, NSFC Grant No.
12070131001, - TRR 110).

##  Appendix A LL effects beyond leading power

In this appendix, we give the analytic results for LL QED corrections to the
electron energy spectrum beyond leading power. At the LO the ${\cal
O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm
QCD}^{3}/m_{b}^{3})$ corrections to the electron energy spectrum can be
parameterised as

$\frac{d\Gamma}{dy}=\Gamma^{(0)}\left[f^{(0)}(y)+\frac{\mu_{\pi}^{2}}{m_{b}^{2}}\,f^{(0)}_{\pi}(y)+\frac{\mu_{G}^{2}}{m_{b}^{2}}\,f^{(0)}_{G}(y)+\frac{\rho_{D}^{3}}{m_{b}^{3}}\,f^{(0)}_{D}(y)+\frac{\rho_{LS}^{3}}{m_{b}^{3}}\,f^{(0)}_{LS}(y)\right]\,,$
(46)

where the expression for $\Gamma^{(0)}$ and $f^{(0)}(y)$ can be found in (5)
and (6), respectively. In the case of a massless electron, the remaining LO
functions appearing in (46) are Bigi  _et al._ (1993); Manohar and Wise
(1994); Voloshin (1995); Gremm and Kapustin (1997)

$\displaystyle f^{(0)}_{\pi}(y)$ $\displaystyle=\left[-\frac{10}{3}\hskip
1.42262pty^{3}-\frac{2\left(5-2y\right)y^{3}\rho^{2}}{\left(1-y\right)^{4}}+\frac{4\left(10-5y+y^{2}\right)y^{3}\rho^{3}}{3\left(1-y\right)^{5}}\right]\theta(1-y-\rho)\,,$
(47) $\displaystyle f^{(0)}_{G}(y)$
$\displaystyle=\left[\frac{2}{3}\left(6+5y\right)y^{2}-\frac{4\left(3-2y\right)y^{2}\rho}{\left(1-y\right)^{2}}\right.$
$\displaystyle\left.\phantom{xxx}-\frac{6\left(2-y\right)y^{2}\rho^{2}}{\left(1-y\right)^{3}}+\frac{10\left(6-4y+y^{2}\right)y^{2}\rho^{3}}{3\left(1-y\right)^{4}}\right]\theta(1-y-\rho)\,,$
$\displaystyle f^{(0)}_{D}(y)$
$\displaystyle=\left[-\frac{2\left(24-11y-y^{2}\right)y^{2}}{3\left(1-y\right)}+\frac{16\left(3-3y+y^{2}\right)y^{2}\rho}{3\left(1-y\right)^{3}}\right.$
$\displaystyle\left.\phantom{xxx}-\frac{2\left(24-5y-11y^{2}+4y^{3}\right)y^{2}\rho^{2}}{3\left(1-y\right)^{5}}+\frac{8\left(6-y\right)y^{2}\rho^{3}}{3\left(1-y\right)^{6}}\right]\theta(1-y-\rho)$
(49)
$\displaystyle\phantom{xx}-\frac{2\left(1+\rho\right)^{2}\left(1-\rho\right)^{4}}{3\rho^{2}}\hskip
1.42262pt\delta(1-y-\rho)\,,$ $\displaystyle f^{(0)}_{SL}(y)$
$\displaystyle=\left[-\frac{2}{3}\hskip
1.42262pty^{3}+\frac{2\left(12-13y+4y^{2}\right)y^{2}\rho^{2}}{\left(1-y\right)^{4}}\right.$
$\displaystyle\left.\phantom{xxx}-\frac{8\left(9-10y+5y^{2}-y^{3}\right)y^{2}\rho^{3}}{3\left(1-y\right)^{5}}\right]\theta(1-y-\rho)\,.$

Notice that the function $f^{(0)}_{D}(y)$ has a singularity at the endpoint of
the spectrum but the moments of the spectrum are well-defined.

By inserting the above results into (3) it is straightforward to derive the LL
QED corrections to the electron energy spectrum at ${\cal O}(\Lambda_{\rm
QCD}^{2}/m_{b}^{2})$ and ${\cal O}(\Lambda_{\rm QCD}^{3}/m_{b}^{3})$. We
obtain

$\displaystyle f_{\pi,{\rm LL}}^{(1)}(y)$
$\displaystyle=\Bigg{\\{}\left[-\frac{2}{9}\left(5-y+23y^{2}-74y^{3}+70y^{4}-20y^{5}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.-\frac{20}{3}\left(1-y\right)^{2}y^{3}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{1}{\left(1-y\right)^{2}}$
$\displaystyle\phantom{xxx}+\frac{2}{3}\left(9-18y+12y^{2}-28y^{3}+29y^{4}-10y^{5}\right)\frac{\rho}{\left(1-y\right)^{3}}$
$\displaystyle\phantom{xxx}-\left[\frac{1}{3}\left(18-42y-30y^{2}-38y^{3}+4y^{4}+y^{5}\right)\right.$
$\displaystyle\phantom{xxxxxi}+\hskip
1.42262pt4\left(1-4y+6y^{2}+y^{3}-y^{4}\right)\ln\left(\frac{1-y}{\rho}\right)$
(51) $\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt4\left(5-2y\right)y^{3}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{2}}{\left(1-y\right)^{4}}$
$\displaystyle\phantom{xxx}+\left[\frac{2}{9}\left(5-10y-58y^{2}-47y^{3}+16y^{4}-2y^{5}\right)\right.$
$\displaystyle\phantom{xxxxxi}+\hskip
1.42262pt\frac{4}{3}\left(1-5y+10y^{2}+10y^{3}-5y^{4}+y^{5}\right)\ln\left(\frac{1-y}{\rho}\right)$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt\frac{8}{3}\left(10-5y+y^{2}\right)y^{3}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{3}}{\left(1-y\right)^{5}}\Bigg{\\}}\;\theta(1-y-\rho)\,,$
$\displaystyle f_{G,{\rm LL}}^{(1)}(y)$
$\displaystyle=\Bigg{\\{}\left[-\frac{1}{9}\left(5-23y+24y^{2}+100y^{3}-40y^{4}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt\frac{4}{3}\left(6+5y\right)\left(1-y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{1}{1-y}$
$\displaystyle\phantom{xxx}+\left[\frac{1}{3}\left(9-39y-18y^{2}+20y^{3}\right)\left(1-y\right)-\left(4+12y^{2}-8y^{3}\right)\ln\left(\frac{1-y}{\rho}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.-\hskip
1.42262pt8\left(3-2y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho}{\left(1-y\right)^{2}}$
$\displaystyle\phantom{xxx}-\left[3-29y-4y^{2}+\frac{8}{3}\hskip
1.42262pty^{3}-\frac{5}{3}\hskip
1.42262pty^{4}-6\left(1-3y-y^{2}+y^{3}\right)\ln\left(\frac{1-y}{\rho}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt12\left(2-y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{2}}{\left(1-y\right)^{3}}$
$\displaystyle\phantom{xxx}+\left[\frac{5}{9}\left(1-28y-15y^{2}+2y^{3}+y^{4}\right)-\frac{10}{3}\left(1-4y-6y^{2}+4y^{3}-y^{4}\right)\ln\left(\frac{1-y}{\rho}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt\frac{20}{3}\left(6-4y+y^{2}\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{3}}{\left(1-y\right)^{4}}\Bigg{\\}}\;\theta(1-y-\rho)\,,$
$\displaystyle f_{D,{\rm LL}}^{(1)}(y)$
$\displaystyle=\Bigg{\\{}\frac{2\left(1+y^{2}\right)}{3\left(1-y\right)\rho^{2}}-\frac{2\left(1-2y-y^{2}\right)}{3\left(1-y\right)^{2}\rho}$
$\displaystyle+\left[\frac{2}{9}\left(34-153y+255y^{2}-158y^{3}+12y^{4}+18y^{5}-4y^{6}\right)\right.$
$\displaystyle\phantom{xxxxxi}-\hskip
1.42262pt8\left(1+y^{2}\right)\left(1-y\right)^{2}\ln\left(\frac{1-y}{\rho}\right)$
$\displaystyle\phantom{xxxxxi}\left.-\hskip
1.42262pt\frac{4}{3}\left(24-11y-y^{2}\right)\left(1-y\right)^{2}y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{1}{\left(1-y\right)^{3}}$
$\displaystyle\phantom{xxx}-\left[\frac{2}{3}\left(6-29y+110y^{2}-112y^{3}+28y^{4}+3y^{5}-2y^{6}\right)\right.$
$\displaystyle\phantom{xxxxxi}-\hskip
1.42262pt\frac{16}{3}\left(1+3y-3y^{2}+y^{3}\right)\left(1-y\right)y\ln\left(\frac{1-y}{\rho}\right)$
$\displaystyle\phantom{xxxxxi}\left.-\hskip
1.42262pt\frac{32}{3}\left(3-3y+y^{2}\right)\left(1-y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho}{\left(1-y\right)^{4}}$
(53)
$\displaystyle\phantom{xxx}-\left[\frac{1}{3}\left(14-70y-152y^{2}+86y^{3}-32y^{4}+11y^{5}-y^{6}\right)\right.$
$\displaystyle\phantom{xxxxxi}+\hskip
1.42262pt\frac{4}{3}\left(2-10y+44y^{2}-25y^{3}-y^{4}+2y^{5}\right)\ln\left(\frac{1-y}{\rho}\right)$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt\frac{4}{3}\left(24-5y-11y^{2}+4y^{3}\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{2}}{\left(1-y\right)^{5}}$
$\displaystyle\phantom{xxx}+\left[\frac{2}{9}\left(5-45y-159y^{2}+5y^{3}+15y^{4}-6y^{5}+y^{6}\right)+\frac{16}{3}\left(6-y\right)y^{2}\ln\left(\frac{1-y}{\rho}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt\frac{16}{3}\left(6-y\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{3}}{\left(1-y\right)^{6}}\Bigg{\\}}\;\theta(1-y-\rho)$
$\displaystyle\phantom{xxx}-\frac{2\left(1+\rho\right)^{2}\left(1-\rho\right)^{4}}{3\rho^{2}}\int_{y}^{1-\rho}\frac{dx}{x}\,P_{ee}^{(0)}\left(\frac{y}{x}\right)\delta(1-x-\rho)\,,$
$\displaystyle f_{LS,{\rm LL}}^{(1)}(y)$
$\displaystyle=\Bigg{\\{}\left[\frac{2}{9}\left(8-7y+8y^{2}+4y^{3}-14y^{4}+4y^{5}\right)\right.$
$\displaystyle\phantom{xxxxxi}\left.-\hskip
1.42262pt\frac{4}{3}\left(1-y\right)^{2}y^{3}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{1}{\left(1-y\right)^{2}}$
$\displaystyle\phantom{xxx}+\frac{2}{3}\left(9+15y-29y^{2}+13y^{3}-2y^{4}\right)y\,\frac{\rho}{\left(1-y\right)^{3}}$
$\displaystyle\phantom{xxx}-\left[\frac{1}{3}\left(78+48y-58y^{2}+26y^{3}-7y^{4}\right)y+\left(8-32y+20y^{3}-8y^{4}\right)\ln\left(\frac{1-y}{\rho}\right)\right.\hskip
28.45274pt$ $\displaystyle\phantom{xxxxxi}\left.-\hskip
1.42262pt4\left(12-13y+4y^{2}\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{2}}{\left(1-y\right)^{4}}$
$\displaystyle\phantom{xxx}-\left[\frac{2}{9}\left(8-97y-19y^{2}+4y^{3}+13y^{4}-5y^{5}\right)\right.$
$\displaystyle\phantom{xxxxxi}-\hskip
1.42262pt\frac{8}{3}\left(1-5y-8y^{2}+10y^{3}-5y^{4}+y^{5}\right)\ln\left(\frac{1-y}{\rho}\right)$
$\displaystyle\phantom{xxxxxi}\left.+\hskip
1.42262pt\frac{16}{3}\left(9-10y+5y^{2}-y^{3}\right)y^{2}\ln\left(\frac{1-y-\rho}{y}\right)\right]\frac{\rho^{3}}{\left(1-y\right)^{5}}\Bigg{\\}}\;\theta(1-y-\rho)\,.$

The result for $f_{D,{\rm LL}}^{(1)}(y)$ still contains the convolution with a
delta function

$\delta_{D}(y)=\int_{y}^{1-\rho}\frac{dx}{x}\,P_{ee}^{(0)}\left(\frac{y}{x}\right)\delta(1-x-\rho)=\frac{P_{ee}^{(0)}\Big{(}\frac{y}{1-\rho}\Big{)}}{1-\rho}\,,$
(55)

where $P_{ee}^{(0)}(z)$ is given in (1). The corrected spectrum $f_{D,{\rm
LL}}^{(1)}(y)$ has still a singularity regulated by the plus distribution. The
moments of the spectrum however are regular and can be expressed as

$\begin{split}\int_{y_{\rm
cut}}^{1-\rho}dy\,y^{i}\,\delta_{D}(y)&=\Delta_{D}^{i}+\int_{0}^{y_{\rm
cut}}dy\,y^{i}\left[\frac{y^{2}+\left(1-\rho\right)^{2}}{\left(y-1+\rho\right)\left(1-\rho\right)^{2}}\right]\,,\end{split}$
(56)

where we have written the integral over $y$ as the difference of two
integrals, one ranging over $y\in[0,1-\rho]$ and a second one ranging over
$y\in[0,y_{\rm cut}]$. While the latter does not include the endpoint of the
spectrum and can be trivially performed, the first integral can be solved
using the definition of the plus distribution (1) leading to

$\Delta_{D}^{0}=0\,,\quad\Delta_{D}^{1}=-\frac{4}{3}\left(1-\rho\right)\,,\quad\Delta_{D}^{2}=-\frac{25}{12}\left(1-\rho\right)^{2}\,,\quad\Delta_{D}^{3}=-\frac{157}{60}\left(1-\rho\right)^{3}\,.\quad$
(57)

Notice that $\Delta_{D}^{0}=0$ is a necessary condition to satisfy the KLN
theorem.

##  Appendix B NLL effects in the partonic case

The NLL QED corrections to the electron energy spectrum in the inclusive $B\to
X_{c}e\nu$ decay can be obtained by the following convolution

$\left(\frac{d\Gamma}{dy}\right)^{(2)}_{\rm
NLL}=\left(\frac{\alpha}{2\pi}\right)^{2}\,\frac{\bar{L}_{b/e}^{2}}{2}\hskip
1.42262pt\left\\{\,\int_{y}^{1-\rho}\,\frac{dx}{x}\hskip
0.7113pt\left[P_{ee}^{(1)}\left(\frac{y}{x}\right)+\frac{2}{3}\hskip
1.42262ptP_{ee}^{(0)}\left(\frac{y}{x}\right)\right]\,\left(\frac{d\Gamma}{dx}\right)^{(0)}\,\right\\}\,.$
(58)

The expression for the LO electron-electron splitting function
$P_{ee}^{(0)}(z)$ has already been given in (1) and the next-to-leading-order
(NLO) equivalent takes the following form

$\displaystyle\begin{split}P_{ee}^{(1)}(z)&=\lim_{\Delta\,\to\,0}\Bigg{\\{}\,2\left[\frac{1+z^{2}}{1-z}\left(2\ln(1-z)-\ln
z+\frac{3}{2}\right)+\frac{1+z}{2}\hskip 1.42262pt\ln
z+z-1\right]\,\theta(1-z-\Delta)\\\\[5.69054pt] &\hskip
42.67912pt\,+\left[\left(\frac{3}{2}+2\ln\Delta\right)^{2}-\frac{2}{3}\hskip
0.7113pt\pi^{2}\right]\delta(1-z)\,\Bigg{\\}}\,.\end{split}$ (59)

Notice that since we are interested in the total NLL QED effects including
both real and virtual $e^{+}e^{-}$ pairs that result from a photon splitting,
we have included in (58) only the non-singlet part of the pair corrections. As
explained for instance in the articles Arbuzov  _et al._ (2002); Arbuzov and
Melnikov (2002); Arbuzov (2003), in this way double counting of real
$e^{+}e^{-}$ pairs is avoided.

Extending (7) to the NLL we then write

$\frac{d\Gamma}{dy}=\Gamma^{(0)}\hskip 0.7113ptf(y)\,,\qquad
f(y)=f^{(0)}(y)+\frac{\alpha}{2\pi}\,\bar{L}_{b/e}\,f^{(1)}_{\rm
LL}(y)+\left(\frac{\alpha}{2\pi}\right)^{2}\frac{\bar{L}_{b/e}^{2}}{2}\,f^{(2)}_{\rm
NLL}(y)\,,$ (60)

where the expression for $f^{(0)}(y)$ and $f^{(1)}_{\rm LL}(y)$ can be found
in (6) and (8), respectively, and we decompose the function $f^{(2)}_{\rm
NLL}(y)$ as follows

$\displaystyle f^{(2)}_{\rm
NLL}(y)=\Phi(y)\,\theta(1-y-\rho)+\frac{2}{3}\hskip 0.7113ptf^{(1)}_{\rm
LL}(y)\,.$ (61)

Here $\Phi(y)$ is a rather complicated analytic function involving various
logarithms and dilogarithms that depend on both $y$ and $\rho$. Instead of
giving the explicit form of $\Phi(y)$ we provide an approximate result. Using
$\rho=0.057$, we obtain

$\displaystyle\begin{split}\Phi(y)&=\frac{320605}{2}+2335446\hskip
1.42262pty-\frac{31925665}{4}\hskip 1.42262pty^{2}+8569202\hskip
1.42262pty^{3}-\frac{13352367}{4}\hskip 1.42262pty^{4}+\frac{427559}{2}\hskip
1.42262pty^{5}\\\\[5.69054pt] &\phantom{xx}+\frac{83233}{2}\hskip
1.42262pty^{6}-\frac{5}{4}\ln y+\left(y_{\rm
max}-y\right)\Bigg{[}\frac{6927}{2}+\frac{776600}{3}\hskip
1.42262pt\left(y_{\rm max}-y\right)\\\\[5.69054pt] &\hskip
22.76228pt+\frac{3229267}{2}\hskip 1.42262pt\left(y_{\rm
max}-y\right)^{2}+\frac{4356808}{3}\hskip 1.42262pt\left(y_{\rm
max}-y\right)^{3}\Bigg{]}\ln(y_{\rm max}-y)\,,\end{split}$ (62)

with $y_{\rm max}=1-\rho=0.943$. This approximation works to a relative
accuracy of better than $1\%$ for the considered parameters.

##  Appendix C Virtual integrals

In this appendix, we collect the analytic expressions for the integrals that
appear in the virtual corrections (26) and (LABEL:eq:Vfin). Many of the
integrals given below can be evaluated following the discussion in Aquila  _et
al._ (2005) but some integrals related to photon exchange between the final-
state particles need a dedicated treatment.

After introducing the Feynman parameter $x$, the diagrams describing the
external leg corrections depend on

$M_{i}^{2}=x^{2}\hskip 0.7113ptm_{i}^{2}+m_{\gamma}^{2}\left(1-x\right)\,,$
(63)

where $m_{i}$ denotes the mass of the corresponding internal fermion. The loop
integrals relevant for the self-energy corrections are

$\displaystyle J_{i}$
$\displaystyle=2\int_{0}^{1}\\!dx\,x\left(1-x^{2}\right)\frac{m_{i}^{2}}{M_{i}^{2}}=-\ln\left(\frac{m_{\gamma}^{2}}{m^{2}_{i}}\right)-1\,,$
(64) $\displaystyle I^{0}_{i}$
$\displaystyle=\int_{0}^{1}\\!dx\left(1-x\right)\ln\left(\frac{M_{i}^{2}}{m_{b}^{2}}\right)=\frac{1}{2}\left[\ln\left(\frac{m_{i}^{2}}{m_{b}^{2}}\right)-3\right]\,,$
(65)

where the final results correspond to the lowest order in an expansion for
small photon mass.

The structure of the loop integrals that arise from the Feynman diagrams where
the photon connects two different fermions are more complicated. After
introducing the Feynman parameters $x$ and $y$, these integrals depend on

$M_{ij}^{2}=\left(x\hskip 0.7113ptp_{i}\pm y\hskip
0.7113ptp_{j}\right)^{2}+m_{\gamma}^{2}\left(1-x-y\right)\,,$ (66)

where the index pair can take the following values $ij=\\{bc,be,ce\\}$ while
$p_{i}$ and $p_{j}$ denote the associated four-momenta. The $\pm$ sign in the
definition (66) encapsulates the different kinematic configurations of the
relevant graphs. In the case of $ij=\\{bc,be\\}$ where the photon is exchanged
between the initial and final state one has to take the $+$ sign, while for
$ij=\\{ce\\}$ corresponding to pure final-state interactions one has to choose
the $-$ sign. To write the results for the vertex integrals in a compact form,
we introduce the following set of variables

$\omega_{ij}=-2\hskip 0.7113ptp_{i}\cdot
p_{j}\,,\quad\sigma_{ij}=\sqrt{\omega_{ij}-4\hskip 0.7113ptm_{i}^{2}\hskip
0.35565ptm_{j}^{2}}\,,\quad
z_{ij}^{\pm}=-\frac{\omega_{ij}\pm\sigma_{ij}}{2m_{i}^{2}}\,,\quad\Sigma_{ij}=\omega_{ij}+m_{i}^{2}+m_{j}^{2}\,.$
(67)

In terms of the abbreviations introduced in (67) we obtain

$\displaystyle I^{0}_{ij}$
$\displaystyle=\int_{0}^{1}\\!dx\int_{0}^{1-x}\\!dy\,\ln\left(\frac{M_{ij}^{4}}{m_{b}^{4}}\right)$
$\displaystyle=I^{0}_{i}+I^{0}_{j}+\frac{1}{2\hskip
0.7113pt\Sigma_{ij}}\left[\sigma_{ij}\ln\left(\frac{z_{ij}^{+}}{z_{ij}^{-}}\right)+\left(m^{2}_{i}-m^{2}_{j}\right)\ln\left(\frac{m_{i}^{2}}{m_{j}^{2}}\right)\right]\,,$
$\displaystyle I^{x}_{ij}$
$\displaystyle=\int_{0}^{1}\\!dx\int_{0}^{1-x}\\!dy\,\frac{x}{M_{ij}^{2}}=-\frac{1}{2\hskip
0.7113pt\Sigma_{ij}}\left[\frac{\omega_{ij}+2\hskip
0.35565ptm_{j}^{2}}{\sigma_{ij}}\hskip
0.7113pt\ln\left(\frac{z_{ij}^{+}}{z_{ij}^{-}}\right)-\ln\left(\frac{m_{i}^{2}}{m_{j}^{2}}\right)\right]\,,$
(69) $\displaystyle I^{y}_{ij}$
$\displaystyle=\int_{0}^{1}\\!dx\int_{0}^{1-x}\\!dy\,\frac{y}{M_{ij}^{2}}=-\frac{1}{2\hskip
0.7113pt\Sigma_{ij}}\left[\frac{\omega_{ij}+2\hskip
0.35565ptm_{i}^{2}}{\sigma_{ij}}\hskip
0.7113pt\ln\left(\frac{z_{ij}^{+}}{z_{ij}^{-}}\right)+\ln\left(\frac{m_{i}^{2}}{m_{j}^{2}}\right)\right]\,,$
(70) $\displaystyle I^{xy}_{ij}$
$\displaystyle=\int_{0}^{1}\\!dx\int_{0}^{1-x}\\!dy\,\frac{x\hskip
0.35565pty}{M_{ij}^{2}}$
$\displaystyle=\frac{\big{(}m_{i}^{2}-m_{j}^{2}\big{)}^{2}-\big{(}m_{i}^{2}+m_{j}^{2}\big{)}\hskip
1.42262pt\Sigma_{ij}}{4\hskip 0.35565pt\sigma_{ij}\hskip
0.7113pt\Sigma_{ij}^{2}}\,\ln\left(\frac{z_{ij}^{+}}{z_{ij}^{-}}\right)+\frac{m_{i}^{2}-m_{j}^{2}}{4\hskip
0.35565pt\Sigma_{ij}^{2}}\ln\left(\frac{m_{i}^{2}}{m_{j}^{2}}\right)-\frac{1}{2\hskip
0.35565pt\Sigma_{ij}}\,.$

These results hold in the cases $ij=\\{bc,be\\}$ while for $ij=\\{ce\\}$ one
has to use the replacement $\omega_{ij}\to-\omega_{ij}$ in the above
expressions as well as in $z_{ij}^{\pm}$. Notice that we have again taken the
limit $m_{\gamma}\to 0$ to obtain the final expressions.

The expression for the integral $I_{ij}^{1}$ is the most complicated one. It
also depends in a non-trivial way on whether one considers $ij=\\{bc,be\\}$ or
$ij=\\{ce\\}$. In the case $ij=\\{bc,be\\}$ we obtain

$I^{1}_{ij}=\omega_{ij}\int_{0}^{1}\\!dx\int_{0}^{1-x}\\!dy\,\frac{1}{M_{ij}^{2}}=\frac{\omega_{ij}}{2\hskip
0.7113pt\sigma_{ij}}\left[\ln\left(\frac{m_{\gamma}^{2}}{m_{i}\hskip
0.7113ptm_{j}}\right)\ln\left(\frac{z_{ij}^{+}}{z_{ij}^{-}}\right)+2\hskip
0.7113pt\Delta I_{ij}^{1}\right]\,,$ (72)

with

$\Delta
I_{ij}^{1}=\ln\big{(}z_{ij}^{-}\big{)}\ln\left(\frac{\sqrt{z_{ij}^{-}}\hskip
2.13394pt\bar{z}_{ij}^{+}}{\sqrt{z_{ij}^{+}}\hskip
2.13394pt\bar{z}_{ij}^{-}}\right)+{\rm Li}_{2}\left(1-\frac{z_{ij}^{+}\hskip
2.13394pt\bar{z}_{ij}^{-}}{z_{ij}^{-}\hskip
2.13394pt\bar{z}_{ij}^{+}}\right)-{\rm
Li}_{2}\left(1-\frac{\bar{z}_{ij}^{-}}{\bar{z}_{ij}^{+}}\right)\,.$ (73)

Here ${\rm Li}_{2}(z)$ is the usual dilogarithm and we have introduced the
abbreviations $\bar{z}_{ij}^{\pm}=1-z_{ij}^{\pm}$. The result (72) again
applies in the limit $m_{\gamma}\to 0$.

For $ij=\\{ce\\}$ the integral $I_{ij}^{1}$ cannot be obtained by a suitable
substitution from (72), because it develops an imaginary part in the physical
region and therefore has to be computed separately. Using the techniques
described in Appendix B of Aquila  _et al._ (2005) it is however relatively
straightforward to compute the integral $I_{ij}^{1}$ for the case of virtual
photon exchange between the charged final-state particles. We find

$I_{ce}^{1}=\frac{\omega_{ce}}{2\hskip 0.35565pt\sigma_{ce}}\hskip
2.13394pt\Bigg{\\{}\left[\ln\left(\frac{y_{+}\hskip
1.42262pt\bar{y}_{-}}{y_{-}\hskip 1.42262pt\bar{y}_{+}}\right)-2\hskip
0.35565pti\hskip
0.35565pt\pi\right]\ln\left(\frac{m_{\gamma}^{2}}{q^{2}}\right)+\Delta
I_{ce}^{1}\Bigg{\\}}\,,$ (74)

where as before
$q^{2}=\left(p_{b}-p_{\nu}\right)^{2}=\left(p_{c}+p_{e}\right)^{2}$ and

$y_{\pm}=\frac{q^{2}+m_{c}^{2}-m_{e}^{2}\pm\sigma_{ce}}{2\hskip
0.7113ptq^{2}}\,,\qquad\bar{y}_{\pm}=1-y_{\pm}\,.$ (75)

The real and imaginary parts of $\Delta I_{ce}^{1}$ are given by

$\displaystyle\begin{split}{\rm Re}\left(\Delta I_{ce}^{1}\right)&=-2\hskip
0.7113pt{\rm Li}_{2}\left(-\frac{y_{-}}{y_{+}-y_{-}}\right)-2\hskip
0.7113pt{\rm
Li}_{2}\left(-\frac{\bar{y}_{+}}{y_{+}-y_{-}}\right)+2\ln\left(y_{+}-y_{-}\right)\ln\left(\frac{y_{-}\hskip
1.42262pt\bar{y}_{+}}{y_{+}-y_{-}}\right)\\\\[5.69054pt]
&\phantom{xx}+\frac{1}{2}\ln^{2}\left(\frac{\bar{y}_{-}}{\bar{y}_{+}}\right)+\frac{1}{2}\ln^{2}\left(\frac{y_{-}}{y_{+}}\right)-\ln^{2}\left(\bar{y}_{-}\right)-\ln^{2}\left(y_{+}\right)+\frac{4\hskip
0.35565pt\pi^{2}}{3}\,,\\\\[11.38109pt] {\rm Im}\left(\Delta
I_{ce}^{1}\right)&=4\hskip 0.35565pti\hskip
0.35565pt\pi\ln\left(y_{+}-y_{-}\right)\,.\end{split}$ (76)

Notice that the dilogarithms in (76) have all been chosen to be real by using
the identity ${\rm Li}_{2}\left(z\right)=-{\rm
Li}_{2}\left(1-z\right)-\ln\left(z\right)\ln\left(1-z\right)+\pi^{2}/6$ and
that only the real part of $I_{ce}^{1}$ enters our calculation. The
contribution of the $\pi^{2}$ term in ${\rm Re}\left(\Delta I_{ce}^{1}\right)$
reproduces the result (17) obtained in Section 3.

We finally remark that only the virtual integrals (64), (72) and (74) develop
logarithmic singularities for $m_{\gamma}\to 0$. These divergences are
precisely the soft singularities which are needed to cancel the analogous
divergent terms in the soft integrals to be discussed next.

##  Appendix D Soft integrals

The soft integrals that enter (33) are given by the following expressions

$\displaystyle I_{R}(p_{i})$
$\displaystyle=-\ln\left(\frac{m_{\gamma}^{2}}{4\hskip
0.35565ptE_{\mathrm{min}}^{2}}\right)+\frac{p_{i}^{0}}{|\vec{p}_{i}|}\,\ln\left(\frac{p_{i}^{0}-|\vec{p}_{i}|}{p_{i}^{0}+|\vec{p}_{i}|}\right)\,,$
(77) $\displaystyle I_{R}(p_{i},p_{j})$ $\displaystyle=-\frac{2p_{i}\cdot
p_{j}}{\sigma_{ij}}\left[\ln\left(\frac{m_{\gamma}^{2}}{4E_{\mathrm{min}}^{2}}\right)\ln\left(\frac{z^{+}_{ij}}{z^{-}_{ij}}\right)+2\hskip
0.35565pt\Delta I_{R}(p_{i},p_{j})\right]\,,$ (78)

where

$\displaystyle\begin{split}\Delta I_{R}(p_{i},p_{j})&={\rm
Li}_{2}\left(1+z_{ij}^{-}\,\frac{p_{i}^{0}+|\vec{p}_{i}|}{v_{ij}}\right)+{\rm
Li}_{2}\left(1+z_{ij}^{-}\,\frac{p_{i}^{0}-|\vec{p}_{i}|}{v_{ij}}\right)-{\rm
Li}_{2}\left(1+\frac{p_{j}^{0}+|\vec{p}_{j}|}{v_{ij}}\right)\hskip
17.07164pt\\\\[5.69054pt] &\phantom{xx}-{\rm
Li}_{2}\left(1+\frac{p_{j}^{0}-|\vec{p}_{j}|}{v_{ij}}\right)+\frac{1}{4}\left[\ln^{2}\Bigg{(}\frac{p_{i}^{0}-|\vec{p}_{i}|}{p_{i}^{0}+|\vec{p}_{i}|}\Bigg{)}-\ln^{2}\Bigg{(}\frac{p_{j}^{0}-|\vec{p}_{j}|}{p_{j}^{0}+|\vec{p}_{j}|}\Bigg{)}\right]\,,\end{split}$
(79)

with

$v_{ij}=-\frac{1}{2}\,\frac{(z_{ij}^{-})^{2}\hskip
0.7113ptm_{i}^{2}-m_{j}^{2}}{z_{ij}^{-}\hskip 1.42262ptp_{i}^{0}-\hskip
0.7113ptp_{j}^{0}}\,.$ (80)

Here $p_{i}^{0}$ and $|\vec{p}_{i}|$ denote the energy component and the
magnitude of the three-momentum of $p_{i}$, respectively. Notice that both
(77) and (78) contain soft singularities of the form
$\ln\left(m_{\gamma}^{2}/(4E_{\rm min}^{2})\right)$ that depend on the phase-
space slicing parameter $E_{\rm min}$. For sufficiently small $E_{\rm min}$
the sum over virtual and real contributions however becomes independent of
this resolution parameter.

##  Appendix E EW corrections to the Wilson coefficient

The EW corrections to low-energy charged-current (CC) processes have been
studied long ago both for the semi-leptonic Sirlin (1982) and purely leptonic
case Sirlin (1980). However, in these early works only the scheme-independent
logarithmic terms have been computed while scheme-dependent finite one-loop
corrections have not been considered. These corrections have subsequently been
calculated in the leptonic, semi-leptonic and non-leptonic case in the
articles Brod and Gorbahn (2008); Gorbahn  _et al._ (2023); Gambino and Haisch
(2000, 2001); Hill and Tomalak (2020). Below we present a comprehensive
discussion of the calculation of EW corrections to all three types of CC
processes. This is accomplished by splitting the corrections into universal
and non-universal parts and providing the explicit electric charge dependence
for the latter.

As we are interested in computing the $O(\alpha)$ contributions to the Wilson
coefficient $C(\mu)$ entering (19), we can neglect all external momenta as
well as all quark and lepton masses because they are all much smaller than the
weak scale. In this approximation the Born-level matrix elements of a generic
CC process is given by the matrix element of the local operator

${\cal M}_{\rm Born}^{\rm CC}=\frac{4\hskip 0.35565ptG_{F}}{\sqrt{2}}\hskip
1.42262pt\langle Q_{\rm CC}\rangle\,,\qquad Q_{\rm
CC}=\big{(}\bar{\psi}_{u}\hskip 0.35565pt\gamma_{\mu}P_{L}\hskip
0.35565pt\psi_{d}\big{)}\big{(}\bar{\psi}_{d^{\prime}}\hskip
0.35565pt\gamma^{\mu}P_{L}\hskip 0.35565pt\psi_{u^{\prime}}\big{)}\,.$ (81)

Here we have removed possible CKM factors and $\psi_{u^{(\prime)}}$
$\big{(}$$\psi_{d^{(\prime)}}$$\big{)}$ denotes the component of a fermionic
$SU(2)_{L}$ doublet field with weak isospin $T_{3}=1/2$ ($T_{3}=-1/2$). Notice
that we have expressed the above Born-level result in terms of the Fermi
constant $G_{F}$ as extracted from muon decay. In this way, as we will see in
a moment, most of the EW radiative corrections cancel, including all terms
that depend on the top-quark and the Higgs-boson mass. In the limit of
vanishing external momenta, all ${\cal O}(G_{F}\hskip 0.7113pt\alpha)$
corrections are proportional to $\langle Q_{\rm CC}\rangle$ and can thus be
expressed as ${\cal O}(\alpha)$ contribution to its Wilson coefficient $C_{\rm
CC}(\mu)$. In other words, there is no mixing with other operators.

The virtual EW corrections to a given low-energy CC process can schematically
be written as

${\cal M}_{\rm virtual}^{\rm CC}={\cal M}_{\rm Born}^{\rm
CC}\,\left[\delta_{\rm box}+\delta_{\rm vertex}+\frac{{\rm Re}\hskip
0.35565pt\big{(}A_{WW}(M_{W}^{2})\big{)}-A_{WW}(0)}{M_{W}^{2}}-\frac{2\hskip
0.35565pt\delta g}{g}\right]\,,$ (82)

where the contributions $\delta_{\rm box}$ and $\delta_{\rm vertex}$ encode
the relevant box and vertex corrections, $A_{WW}(q^{2})$ is the unrenormalised
two-point function of the $W$ boson, and ${\delta g}$ is the counterterm of
the $SU(2)_{L}$ coupling constant $g$, whose precise definition is irrelevant
for our purposes. The $W$-boson mass is renormalised on-shell, as indicated by
its counterterm ${\rm Re}\hskip 0.35565pt\big{(}A_{WW}(M_{W}^{2})\big{)}$. The
sum of these four contributions is UV finite. We perform the calculation in
the Feynman gauge.

Figure 8:  Examples of Feynman diagrams that contribute to the short-distance
EW corrections encoded by the Wilson coefficient $C(\mu)$. Only contributions
involving $Z$-boson exchange are shown but there are also photon loops.

In the case of the $b\to ce\nu$ transition the term $\delta_{\rm box}$ in (82)
receives contributions from Feynman diagrams like the one shown on the left-
hand side in Figure 8. Using dimensional regularisation for both UV and IR
singularities and keeping the electric charges $Q_{i}$ of the external
fermions explicit, we obtain

$\delta_{\rm
box}=\frac{g^{2}}{16\pi^{2}}\;\Bigg{\\{}\left[-\frac{1}{\epsilon}+\ln\left(\frac{M_{Z}^{2}}{\mu^{2}}\right)\right]\chi\hskip
0.7113pts_{w}^{2}+\zeta\hskip 0.7113pts_{w}^{2}-\left[5-\frac{5}{2\hskip
0.35565pts_{w}^{2}}\right]\ln\left(\frac{M_{Z}^{2}}{M_{W}^{2}}\right)\Bigg{\\}}\,.$
(83)

Here $s_{w}$ denotes the sine of the weak mixing angle and $M_{Z}$ is the mass
of the $Z$ boson. The $1/\epsilon$ pole is a IR divergence originating from
the photon boxes, while the $Z$-boson boxes are finite. The process-dependent
factors $\chi$ and $\zeta$ are given by

$\begin{split}\chi&=Q_{u}\big{(}Q_{u^{\prime}}-4\hskip
0.35565ptQ_{d^{\prime}}\big{)}+Q_{d}\big{(}Q_{d^{\prime}}-4\hskip
0.35565ptQ_{u^{\prime}}\big{)}=\begin{cases}\,1\,,&\text{leptonic}\,,\\\
\,3\,,&\text{semi-leptonic}\,,\\\ \,\frac{7}{3}\,,&\text{non-leptonic}\,,\\\
\end{cases}\\\\[11.38109pt] \zeta&=Q_{u}\left(\frac{Q_{u^{\prime}}}{2}+5\hskip
0.35565ptQ_{d^{\prime}}\right)+Q_{d}\left(\frac{Q_{d^{\prime}}}{2}+5\hskip
0.35565ptQ_{u^{\prime}}\right)=\begin{cases}\,\frac{1}{2}\,,&\text{leptonic}\,,\\\\[2.84526pt]
\,-\frac{19}{6}\,,&\text{semi-leptonic}\,,\\\\[2.84526pt]
\,-\frac{35}{18}\,,&\text{non-leptonic}\,.\\\ \end{cases}\end{split}$ (84)

To reduce products of strings of three Dirac matrices resulting from one-loop
boxes involving the exchange of a $W$ boson and a photon, we have employed the
relation

$\big{(}\bar{\psi}_{u}\hskip
0.35565pt\gamma_{\mu}\gamma_{\nu}\gamma_{\lambda}P_{L}\hskip
0.35565pt\psi_{d}\big{)}\big{(}\bar{\psi}_{d^{\prime}}\hskip
0.35565pt\gamma^{\mu}\gamma^{\nu}\gamma^{\lambda}P_{L}\hskip
0.35565pt\psi_{u^{\prime}}\big{)}=\left(16-4\hskip
0.35565pt\epsilon\right)Q_{\rm CC}+E_{\rm CC}\,,$ (85)

where $Q_{\rm CC}$ is the operator introduced in (81), $E_{\rm CC}$ is a so-
called evanescent operator and the particular choice of ${\cal O}(\epsilon)$
term in (85) corresponds to the NDR scheme definition Buras and Weisz (1990).
This scheme choice only affects $\zeta$, while $\chi$ is unaffected and hence
scheme independent. The scheme dependence of $\zeta$ however cancels in the
final physical results, provided that all parts of the calculation are
performed consistently using (85). Notice that in our calculation of the
matrix element of the operator $Q$ as detailed in Section 4.1 we also use the
NDR scheme and as a result the final result (44) for the electron energy
spectrum is scheme independent.

The term $\delta_{\rm vertex}$ in (82) receives contributions from vertex
diagrams and external leg corrections associated to wave function
renormalisation (WFR). An example of a possible vertex graph relevant in the
case of $b\to ce\nu$ is depicted on the right in Figure 8. In agreement with
Sirlin (1978), we find that $\delta_{\rm vertex}$ does not depend on the
fermion doublet under consideration and therefore represents a universal
process-independent contribution. In the case of a quark doublet the
renormalisation turns out to be more subtle. In the limit of vanishing
external quark masses, one can limit oneself to the renormalisation of the
left-handed quark fields. For down quarks, the WFR matrix can be decomposed as

$(Z_{d}^{L})_{ij}=\delta_{ij}+(\delta Z_{d}^{L})_{ij}\,,\qquad(\delta
Z_{d}^{L})_{ij}=A_{d}\hskip 1.42262pt\delta_{ij}+V_{ti}V_{tj}^{\ast}\hskip
1.42262ptB_{d}\,,$ (86)

where $A_{d}$ contains the light-quark contribution, while $B_{d}$ includes
the difference between the top- and the light-quark contribution that results
from the Glashow-Iliopoulos-Maiani (GIM) mechanism. The explicit form of these
coefficients can be found in (39) and (40) of Gambino  _et al._ (1999a). In
the case of a quark vertex like in $b\to ce\nu$, the counterterm contribution
corresponding to the renormalisation of the external bottom-quark line is
proportional to the Born-level matrix element (81) and takes the following
form

$\delta_{b,\rm ct}=\sum_{q=d,s,b}\,\frac{(\delta
Z_{d}^{L})_{bq}}{2}\,V_{cq}=\sum_{q=d,s,b}\,\frac{1}{2}\,\Big{[}A_{d}\hskip
1.42262pt\delta_{bq}+V_{tb}V_{tq}^{\ast}\hskip
1.42262ptB_{d}\Big{]}\,V_{cq}=\frac{A_{d}}{2}\,V_{cb}\,.$ (87)

Notice that in order to arrive at the final result we have used the unitarity
of the CKM matrix, i.e. $\sum_{q=d,s,b}V_{iq}V_{jq}^{\ast}=\delta_{ij}$. Due
to the GIM mechanism the counterterm (87) hence does not receive a
contribution from top-quark loops which is a necessary condition to obtain a
universal vertex correction $\delta_{\rm vertex}$. For what concerns the WFR
of the external charm-quark line, it proceeds in the same way as described
above but in that case $B_{u}=0$. The corresponding counterterm $\delta_{c,\rm
ct}=A_{u}/2\,V_{cb}$ therefore also receives contributions only from loops
involving massless quarks.

Since we have normalised the Born-level amplitude (81) in terms of $G_{F}$, to
calculate the ${\cal O}(\alpha)$ correction to the Wilson coefficient of
$Q_{\rm CC}$ relevant for the semi-leptonic or non-leptonic case one has to
subtract from (82) the corresponding leptonic matrix element. Thereby the
universal part of (83) cancels out. The same is true for the last two terms in
(82) as well as for the universal vertex correction $\delta_{\rm vertex}$. We
add that, in principle, also the CKM matrix has to be renormalised. However,
as we are assuming massless light quarks, we just need to remove the ill-
defined anti-hermitian part of the WFR constants Denner and Sack (1990);
Gambino  _et al._ (1999). Once this is done, the rest of calculation is
unaffected, meaning that it resembles the calculation that did not take CKM
renormalisation into account. Notice also that all the residual photonic
corrections to muon decay (those described by photonic corrections in the EFT)
are included in the conventional definition of $G_{F}$ Sirlin (1980) and do
not need to be subtracted. The final result of our full SM calculation
performed at vanishing external momenta and masses is therefore

${\cal M}_{\rm virtual}^{\rm CC}={\cal M}_{\rm Born}^{\rm
CC}\;\frac{\alpha}{4\pi}\left\\{\left(\chi-1\right)\left[-\frac{1}{\epsilon}+\ln\left(\frac{M_{Z}^{2}}{\mu^{2}}\right)\right]+\zeta-\frac{1}{2}\right\\}\,.$
(88)

In order to obtain the Wilson coefficient we still need to match the full SM
calculation with the EFT one, performed in the same scheme and at vanishing
momenta and masses. As the EFT photonic loops vanish under these conditions,
the only contribution from the EFT is the counterterm related to the operator
$\overline{\rm MS}$ renormalisation, a pure $1/\epsilon$ pole that cancels the
divergence in (88). We therefore obtain the following Wilson coefficients for
the CC processes of interest

$C_{\rm
CC}(\mu)=1+\frac{\alpha(\mu)}{4\pi}\begin{cases}\,\left[2\ln\left(\frac{M_{Z}^{2}}{\mu^{2}}\right)-\frac{11}{3}\right]\,,&\text{semi-
leptonic}\,,\\\\[8.53581pt]
\,\left[\frac{4}{3}\ln\left(\frac{M_{Z}^{2}}{\mu^{2}}\right)-\frac{22}{9}\right]\,,&\text{non-
leptonic}\,,\end{cases}$ (89)

where as indicated the electromagnetic coupling $\alpha$ is naturally also
evaluated at $\mu$. The first line leads to the result given in (44) and
agrees with Brod and Gorbahn (2008); Gorbahn  _et al._ (2023), while the
second line coincides with the expression for the Wilson coefficient derived
in Gambino and Haisch (2000, 2001); Brod and Gorbahn (2008).

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