# Improved $K_{e3}$ radiative corrections sharpen the $K_{\mu 2}$–$K_{l3}$
discrepancy

Chien-Yeah Seng1 Daniel Galviz1 Mikhail Gorchtein2,3,4 Ulf-G. Meißner1,5,6
1Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for
Theoretical Physics,
Universität Bonn, 53115 Bonn, Germany  2Helmholtz Institute Mainz, D-55099
Mainz, Germany  3GSI Helmholtzzentrum für Schwerionenforschung, 64291
Darmstadt, Germany  4Johannes Gutenberg University, D-55099 Mainz, Germany
5Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center
for Hadron Physics, Forschungszentrum Jülich, 52425 Jülich, Germany  6Tbilisi
State University, 0186 Tbilisi, Georgia

(August 27, 2024)

###### Abstract

The measurements of $V_{us}$ in leptonic $(K_{\mu 2})$ and semileptonic
$(K_{l3})$ kaon decays exhibit a $3\sigma$ disagreement, which could originate
either from physics beyond the Standard Model or some large unidentified
Standard Model systematic effects. Clarifying this issue requires a careful
examination of all existing Standard Model inputs. Making use of a newly-
proposed computational framework and the most recent lattice QCD results, we
perform a comprehensive re-analysis of the electroweak radiative corrections
to the $K_{e3}$ decay rates that achieves an unprecedented level of precision
of $10^{-4}$, which improves the current best results by almost an order of
magnitude. No large systematic effects are found, which suggests that the
electroweak radiative corrections should be removed from the “list of
culprits” responsible for the $K_{\mu 2}$–$K_{l3}$ discrepancy.

##  I Introduction

Despite the discovery of the Higgs boson in the year 2012 Aad:2012tfa ;
Chatrchyan:2012ufa  that completed the particle spectrum in the Standard Model
(SM), there exist numerous observed phenomena in astrophysics, e.g. dark
matter, dark energy and the matter-antimatter asymmetry, that do not find
their explanations within this theory framework and thus call for physics
beyond the Standard Model (BSM). Unfortunately, all direct searches in high-
energy colliders have so far returned null results. On the other hand,
precision experiments have observed several interesting anomalies in flavor
physics that point towards the possible existence of BSM physics. This
research concerns one of these observed anomalies, namely the irregularities
in the top-row Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.

The unitarity of the CKM matrix is a rigorous SM prediction Cabibbo:1963yz ;
Kobayashi:1973fv . In particular, the top-row CKM unitarity (which is also
known as the Cabibbo unitarity) that involves the matrix elements $V_{ud}$ and
$V_{us}$ ($V_{ub}$ is negligible) has received the most attention because they
can be measured to high precision in hadron and nuclear beta decays. Recently,
a series of improvements in the theory Seng:2018yzq ; Seng:2018qru ;
Gorchtein:2018fxl ; Czarnecki:2019mwq  of the electroweak radiative
corrections (RC) in the extraction of $V_{ud}$ led to an apparent deviation of
the Cabibbo unitarity at a level of $3\sigma$ Zyla:2020zbs . However, in this
work we will not focus on $V_{ud}$, but rather on $V_{us}$ which possesses yet
another interesting anomaly by itself.

The matrix element $V_{us}$ can be extracted from leptonic ($K_{l2}$) or
semileptonic ($K_{l3}$) kaon decay, hyperon decay and tau decay, but the last
two processes are an order of magnitude less precise and will be ignored here.
From the leptonic kaon decay, the following ratio is obtained:

$\frac{|V_{us}|f_{K^{+}}}{|V_{ud}|f_{\pi^{+}}}=0.23871(20)\left[\frac{\Gamma_{K\rightarrow\mu\nu(\gamma)}}{\Gamma_{\pi\rightarrow\mu\nu(\gamma)}}\right]^{\frac{1}{2}}\leavevmode\nobreak\
,$ (1)

where $f_{K^{+}}$ and $f_{\pi^{+}}$ are the $K^{+}$ and the $\pi^{+}$ decay
constant, respectively, which require lattice QCD inputs. The theory
uncertainty on the right-hand side is less than $10^{-3}$, thanks to the
cancellation of the common electroweak RC to the leptonic kaon and pion decay
rate Marciano:2004uf ; Cirigliano:2011tm . With the $N_{f}=2+1+1$ FLAG average
of $f_{K^{+}}/f_{\pi^{+}}$ Aoki:2019cca  and the recent value of $V_{ud}$
obtained from superallowed beta decays Seng:2018yzq , the following result is
quoted in PDG 2020 Zyla:2020zbs :

$|V_{us}|=0.2252(5)\>\>(K_{\mu 2})$ (2)

Meanwhile, in the semileptonic kaon decay process $K\to\pi l^{+}\nu(\gamma)$
one does not measure a ratio, but obtains $V_{us}$ directly from the decay
rate, where the SM inputs include the electroweak RC, the $K\pi$ form factors
and the SU(2) isospin-breaking effects (we postpone the detailed discussions
to the main text). With the most recent theory inputs of these quantities, PDG
2020 quotes the following result:

$|V_{us}|=0.2231(4)_{\mathrm{exp+RC}}(6)_{\mathrm{lat}}\>\>(K_{l3})$ (3)

We observe a $\sim 3\sigma$ disagreement between the numbers in Eq.(2) and
(3), with a $\sim 1\%$ difference between the two central values. This
provides another interesting hint to the existence of BSM physics
Belfatto:2019swo ; Tan:2019yqp ; Grossman:2019bzp ; Coutinho:2019aiy ;
Cheung:2020vqm ; Crivellin:2020lzu ; Endo:2020tkb ; Capdevila:2020rrl ;
Kirk:2020wdk  which, to some extent, is even more promising than the top-row
CKM unitarity deficit. In fact, the extraction of $V_{us}$ is free from
complicated nuclear-structure uncertainties (except those that enter $V_{ud}$
in Eq. (1), whose effect on $V_{us}$ is subdominant to the existing
uncertainties). For instance, if the total uncertainty in Eqs.(2) and (3) is
reduced to $4\times 10^{-4}$ or below, with the central values unchanged, the
discrepancy will reach $5\sigma$ which is sufficient to claim an observation
of a BSM signal. Achieving this final goal requires a careful re-analysis of
all the SM inputs, not just to reduce their uncertainties but also to make
sure that no large unidentified SM corrections were missed in existing
analyses.

In this work, we study a particularly important SM correction to the kaon
semileptonic decay, namely the electroweak RC. Earlier studies of this topic
by Ginsberg Ginsberg:1966zz ; Ginsberg:1968pz ; Ginsberg:1969jh ;
Ginsberg:1970vy , Becherrawy Becherrawy:1970ah  and later by Bytev et al.
Bytev:2002nx  and Andre Andre:2004tk  assumed specific models for the strong
and electroweak interactions which made a rigorous analysis of the theory
uncertainties rather challenging. Another class of works, e.g. by García and
Maya Garcia:1981it  and by Juárez-León et al. JuarezLeon:2010tj ;
Torres:2012ge ; Neri:2015eba  put more emphasis on the so-called “model-
independent” piece in the long-distance electromagnetic corrections (i.e. the
convection term contribution, which we will explain in the main text) but were
unable to place any constrain on the “model-dependent” piece originating from
non-perturbative Quantum Chromodynamics (QCD) at the hadronic scale. So far,
the only approach that allows a systematic error analysis in every part of the
electroweak RC has been the chiral perturbation theory (ChPT) calculation by
Cirigliano et al. Cirigliano:2001mk ; Cirigliano:2004pv ; Cirigliano:2008wn ,
where the most general electroweak interactions between hadrons and dynamical
photons Urech:1994hd  and leptons Knecht:1999ag  are arranged according to
increasing powers of $p/\Lambda_{\chi}$, where $p$ is a typical small momentum
scale in such interactions and $\Lambda_{\chi}\simeq 4\pi F_{\pi}$ is the
chiral symmetry breaking scale, with $F_{\pi}=92.1$ MeV the pion decay
constant. Within this framework, the long-distance electromagnetic RC to
$K_{l3}$ decay is calculated to $\mathcal{O}(e^{2}p^{2})$, and the theory
uncertainty comes from two major sources: The unknown low-energy constants
(LECs) at $\mathcal{O}(e^{2}p^{2})$, and the neglected contributions of the
order $\mathcal{O}(e^{2}p^{4})$. Both uncertainties are estimated to be of the
order $10^{-3}$. At this point it seems formidable to make any further
progress within the same theory framework, because (1) the LECs are only
calculable within phenomenological models Ananthanarayan:2004qk ;
DescotesGenon:2005pw  with outcomes that are highly uncertain, and (2) to
reduce the higher-order corrections one needs to perform a full two-loop ChPT
calculation which is not only technically challenging but more importantly,
involves even more unknown LECs.

A series of preparatory works were done since early 2020 in order to
eventually overcome the difficulties mentioned above. First, a new theory
framework based on the hybridization of the classical Sirlin’s approach
Sirlin:1977sv  and modern ChPT was formulated Seng:2019lxf  in order to resum
the most important $\mathcal{O}(e^{2}p^{2n})$ effects while retaining the full
model-independent characteristics in the traditional ChPT approach. Next,
lattice QCD was introduced to study the part of the RC in semileptonic decays
that carries the largest hadronic uncertainties, namely the axial $\gamma
W$-box diagram. The first calculation was done on the pion Feng:2020zdc ,
which removed the dominant theory uncertainty in the semileptonic pion decay
and also confirmed the result of the previous dispersion-relation analysis of
the RC in free neutron Seng:2020wjq . Shortly after that, following the
suggestion in Ref.Seng:2020jtz  a new lattice calculation of the $K\pi$ axial
$\gamma W$-box in the flavor SU(3) limit was performed Ma:2021azh . Up to this
point, we finally have all the necessary ingredients and are in the position
to present a fully-updated numerical analysis of the electroweak RC in kaon
semileptonic decays that eventually reduces the existing theory uncertainty by
almost an order of magnitude, i.e. to the level of $10^{-4}$.

The main results in this study were presented in an earlier paper Seng:2021boy
, and here we will show all the details. We concentrate on the $K_{e3}$
channel and not $K_{\mu 3}$ throughout this study for reasons that will become
clear in the main text. The contents of this work are arranged as follows. In
Section II we introduce the basic notation and set up our theory framework. In
Sections III–VI we present our update of the contributions from the “virtual”
electroweak RC; in particular, we demonstrate in Section VI how the most
recent lattice QCD results are used to constrain the hadronic uncertainties in
the physical $K\pi$ axial $\gamma W$-box diagram. The contribution from the
real-photon emission process is calculated in Section VII. In Section VIII we
discuss how our new results should be interpreted in the ChPT language, and
show the numerical improvement against the existing calculations. Final
discussions and conclusions are provided in Section IX.

##  II Notation and setup

One of the most important avenues to extract $V_{us}$ is the inclusive kaon
semileptonic decay $K_{l3}$, i.e. the process
$K(p)\to\pi(p^{\prime})+l^{+}(p_{l})+\nu_{e}(p_{\nu})+n\gamma$, where
$l=e,\mu$, and $n\geq 0$ is the number of photons in the final state. It will
be evident later that the case $l=e$ allows for a much better control of the
theory uncertainties, so throughout this paper, we will concentrate on this
particular case. If all massless final-state particles are left unobserved,
the differential decay rate of the process is fully described by three
independent, dimensionless Lorentz-invariant variables111In the existing
literature $x$ is more often defined as $P^{2}$, which carries a dimension.:

$x\equiv\frac{P^{2}}{M_{K}^{2}},\>\>y\equiv\frac{2p\cdot
p_{e}}{M_{K}^{2}},\>\>z\equiv\frac{2p\cdot
p^{\prime}}{M_{K}^{2}}\leavevmode\nobreak\ ,$ (4)

where $P\equiv p-p^{\prime}-p_{e}$. Notice that $x$ is strictly zero
(neglecting neutrino mass) for $n=0$, but may take a non-zero value when
$n\geq 1$. We may have as well introduced the usual Mandelstam variables
$s\equiv(p^{\prime}+p_{e})^{2}$, $t\equiv(p-p^{\prime})^{2}$ and
$u\equiv(p-p_{e})^{2}$, but none of them is independent of $\\{x,y,z\\}$.

At $\mathcal{O}(G_{F}^{2})$ (where $G_{F}=1.1663787(6)\times 10^{-5}$ GeV-2 is
the Fermi constant extracted from muon decay Zyla:2020zbs ), only the $n=0$
process contributes to the $K_{e3}$ decay rate. Its corresponding tree-level
amplitude is given by:

$M_{0}=-\frac{G_{F}}{\sqrt{2}}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})v_{e}F^{K\pi}_{\mu}(p^{\prime},p)\leavevmode\nobreak\
,$ (5)

where the effects of the strong interaction are fully contained in the
following hadronic matrix element of the charged weak current:

$F_{\mu}^{K\pi}(p^{\prime},p)\equiv\left\langle\pi(p^{\prime})\right|(J^{W}_{\mu})^{\dagger}\left|K(p)\right\rangle=V_{us}^{*}\left[f_{+}^{K\pi}(t)(p+p^{\prime})_{\mu}+f_{-}^{K\pi}(t)(p-p^{\prime})_{\mu}\right]\leavevmode\nobreak\
.$ (6)

The equation above defines the charged weak form factors
$f_{\pm}^{K\pi}(t)$222We wish to remind the readers that our sign convention
for the form factors is $f_{+}^{K\pi}(0)<0$, which is also adopted in our
previous works, e.g.Seng:2019lxf ; Seng:2020jtz , but may be opposite to other
existing literature. This serves to be consistent with the sign convention of
the charged weak current $(J_{W}^{\mu})^{\dagger}$ derived from ChPT.. It is
also customary to define a third form factor:

$f_{0}^{K\pi}(t)\equiv
f_{+}^{K\pi}(t)+\frac{t}{M_{K}^{2}-M_{\pi}^{2}}f_{-}^{K\pi}(t)\leavevmode\nobreak\
,$ (7)

and call $f_{+}^{K\pi}(t)$ and $f_{0}^{K\pi}(t)$ the “vector” and “scalar”
form factor, respectively. From the definition above, it is obvious that
$f_{0}^{K\pi}(0)=f_{+}^{K\pi}(0)$, so another common step is to factor out
their $t=0$ value:

$\bar{f}_{+,0}(t)\equiv\frac{f_{+,0}^{K\pi}(t)}{f_{+}^{K\pi}(0)}\leavevmode\nobreak\
.$ (8)

There are several different ways to parameterize $\bar{f}_{+,0}(t)$, e.g.
Taylor expansion, monopole parameterization and dispersive parameterization.
The interested reader may consult Ref. Lazzeroni:2018glh  and references
therein for the details, and we will also come back to this point in Section
IV.

It is instructive to display explicitly the absolute square of the tree-level
amplitude above (upon summing over the lepton spin, as we will always do
throughout this work):

$|M_{0}|^{2}(x,y,z)=G_{F}^{2}F^{K\pi}_{\mu}(p^{\prime},p)(F_{\nu}^{K\pi}(p^{\prime},p))^{*}\mathrm{Tr}\left[\not{P}\gamma^{\mu}(\not{p}_{e}-m_{e})\gamma^{\nu}(1-\gamma_{5})\right]\leavevmode\nobreak\
.$ (9)

Here we purposely retain the $x$-dependence in the formula above despite the
fact that $x=0$ when $n=0$. The $x$-dependence becomes important later when we
discuss the squared amplitude of the bremsstrahlung process. The impact of the
form factors $f_{\pm}^{K\pi}$ on the tree-level decay rate relies heavily on
the leptonic trace in Eq.(9). Suppose we define:

$H(a,b)\equiv\mathrm{Tr}\left[\not{P}(\not{p}+a\not{p}^{\prime})(\not{p}_{e}-m_{e})(\not{p}+b\not{p}^{\prime})(1-\gamma_{5})\right]_{x=0}\leavevmode\nobreak\
,$ (10)

then a straightforward calculation shows:

$\displaystyle H(+1,+1)$ $\displaystyle=$
$\displaystyle-2M_{K}^{4}\left[4(y-1)(y+z-1)+4r_{\pi}-r_{e}(r_{\pi}+4y+3z-3)+r_{e}^{2}\right]$
$\displaystyle H(+1,-1)$ $\displaystyle=$ $\displaystyle H(-1,+1)$
$\displaystyle=$
$\displaystyle-2M_{K}^{4}r_{e}\left[-r_{e}+r_{\pi}+2y+z-3\right]$
$\displaystyle H(-1,-1)$ $\displaystyle=$
$\displaystyle-2M_{K}^{4}r_{e}\left[r_{e}-r_{\pi}+z-1\right]\leavevmode\nobreak\
,$ (11)

where $r_{\pi}\equiv M_{\pi}^{2}/M_{K}^{2}$ and $r_{e}\equiv
m_{e}^{2}/M_{K}^{2}$. We observe that only $H(+1,+1)$ is not explicitly
suppressed by the factor $r_{e}\approx 10^{-6}$. Following the notations in
Appendix A, the decay rate at $\mathcal{O}(G_{F}^{2})$ in given by:

$\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}=\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz|M_{0}|^{2}(0,y,z)\leavevmode\nobreak\
.$ (12)

From the argument above, it is apparent that only $f_{+}^{K\pi}(t)$, and not
$f_{-}^{K\pi}(t)$, is relevant in
$\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}$. Of course the actual value of
$\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}$ depends on the specific
parameterization of $\bar{f}_{+}(t)$ and the parameters therein, but the
impact of the different choices is generically of the order $0.1\%$. Since in
this paper $\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}$ serves only as a
normalization factor to the already-small RC, such a difference is completely
negligible.

The electroweak RC induces a shift of the tree-level decay rate:
$\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}\rightarrow\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}+\delta\Gamma_{K_{e3}}$.
We define the quantity:

$\delta_{K_{e3}}\equiv\frac{\delta\Gamma_{K_{e3}}}{\left(\Gamma_{K_{e3}}\right)_{\mathrm{tree}}}$
(13)

that represents the fractional correction to the decay rate, and we will
discuss its relation to the different quantities within the ChPT framework in
Section VIII. To match the precision level of current and near-future
experiments, we need a theoretical prediction of $\delta_{K_{e3}}$ up to
$\mathcal{O}(\alpha)$. At this level, the only two contributors are (1) the
$\mathcal{O}(G_{F}\alpha)$ electroweak RC to the $n=0$ decay amplitude, and
(2) the tree-level contribution from the $n=1$ process. We will spend the next
few sections discussing these two contributions.

##  III Virtual correction: analytic pieces

We start by discussing the virtual corrections, i.e. the
$\mathcal{O}(G_{F}\alpha)$ electroweak RC to the $n=0$ decay amplitude. It is
possible to express such corrections entirely in terms of perturbations to the
charged weak form factors, i.e333Using the on-shell condition, one can show
that other leptonic bilinear structures, such as
$i\epsilon^{\lambda\mu\nu\alpha}p_{\mu}p^{\prime}_{\nu}p_{e\alpha}\bar{u}_{\nu}\gamma_{\lambda}(1-\gamma_{5})v_{e}$,
are linear combinations of
$\bar{u}_{\nu}(\not{p}\pm\not{p}^{\prime})(1-\gamma_{5})v_{e}$. ,

$\delta
M_{\mathrm{vir}}=-\frac{G_{F}}{\sqrt{2}}V_{us}^{*}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})v_{e}\left[(p+p^{\prime})_{\mu}\delta
f_{+}^{K\pi}+(p-p^{\prime})_{\mu}\delta
f_{-}^{K\pi}\right]\leavevmode\nobreak\ .$ (14)

The only complication is that $\delta f_{\pm}^{K\pi}$ are complex functions of
two variables, e.g. $\\{y,z\\}$, rather than real functions of a single
variable $t$.

The virtual contribution to $\delta_{K_{e3}}$ at $\mathcal{O}(\alpha)$ arises
from the interference between $M_{0}$ and $\delta M_{\mathrm{vir}}$, i.e.
$|M_{0}|^{2}\to|M_{0}|^{2}+\delta|M|^{2}_{\mathrm{vir}}$, with
$\delta|M|^{2}_{\mathrm{vir}}\equiv 2\mathfrak{Re}\left\\{M_{0}^{*}\delta
M_{\mathrm{vir}}\right\\}$. Again, by restricting ourselves to $K_{e3}$, we
only need to know $\delta f_{+}^{K\pi}$ in order to determine the perturbation
to the $n=0$ squared amplitude:

$\delta|M|^{2}_{\mathrm{vir}}(y,z)=2|M_{0}|^{2}(0,y,z)\frac{\mathfrak{Re}\left\\{\delta
f_{+}^{K\pi}\right\\}}{f_{+}^{K\pi}(t)}+\mathcal{O}(r_{e})\leavevmode\nobreak\
.$ (15)

Based on the theory framework outlined in Refs.Seng:2019lxf ; Seng:2020jtz ,
the $\mathcal{O}(G_{F}\alpha)$ virtual corrections to the $n=0$ decay
amplitude can be summarized by the following equation:

$\displaystyle\delta M_{\mathrm{vir}}$ $\displaystyle=$
$\displaystyle\left[-\frac{\alpha}{2\pi}\left(\ln\frac{M_{W}^{2}}{M_{Z}^{2}}+\frac{1}{4}\ln\frac{M_{W}^{2}}{m_{e}^{2}}-\frac{1}{2}\ln\frac{m_{e}^{2}}{M_{\gamma}^{2}}+\frac{9}{8}+\frac{3}{4}a_{\mathrm{pQCD}}\right)+\frac{1}{2}\delta_{\mathrm{HO}}^{\mathrm{QED}}\right]M_{0}$
(16) $\displaystyle+\delta M_{2}+\delta M_{3}+\delta M_{\gamma
W}\leavevmode\nobreak\ .$

Let us briefly explain the notation above, all the details are given in
Ref.Seng:2019lxf . First, the terms in the square bracket come from the “weak”
RC including its $\mathcal{O}(\alpha_{s})$ perturbative QCD (pQCD) corrections
$a_{\mathrm{pQCD}}\approx 0.068$, the electron wavefunction renormalization,
and the resummation of the large QED logs represented by
$\mathrm{\delta}_{\mathrm{HO}}^{\mathrm{QED}}=0.0010(3)$ Erler:2002mv . An
infinitesimal photon mass $M_{\gamma}$ is introduced to regularize the
infrared (IR) divergence in the electron wavefunction renormalization. Next,
the quantities $\delta M_{2,3}$ represent the contributions from two separate
pieces of the electromagnetic RC to the charged weak form factors, known as
the “two-point function” and “three-point function”, respectively. Finally,
$\delta M_{\gamma W}$ represents the contribution from the $\gamma W$-box
diagram:

$\delta M_{\gamma
W}=-\frac{G_{F}e^{2}}{\sqrt{2}}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{\bar{u}_{\nu}\gamma^{\nu}(\not{q}^{\prime}\gamma^{\mu}-2p_{e}^{\mu})(1-\gamma_{5})v_{e}}{\left[(p_{e}-q^{\prime})^{2}-m_{e}^{2}\right]\left[q^{\prime
2}-M_{\gamma}^{2}\right]}T_{\mu\nu}^{K\pi}(q^{\prime};p^{\prime},p)\leavevmode\nobreak\
,$ (17)

where we have introduced the so-called “generalized Compton tensor”
$T_{\mu\nu}^{K\pi}$ which plays a central role in the upcoming analysis:

$T^{\mu\nu}_{K\pi}(q^{\prime};p^{\prime},p)\equiv\int
d^{4}x\,e^{iq^{\prime}\cdot
x}\left\langle\pi(p^{\prime})\right|T\\{J_{\mathrm{em}}^{\mu}(x)J_{W}^{\nu\dagger}(0)\\}\left|K(p)\right\rangle\leavevmode\nobreak\
,$ (18)

where $T\\{\ldots\\}$ denotes the conventional time-ordering. It satisfies the
following Ward identities:

$\displaystyle q^{\prime}_{\mu}T_{K\pi}^{\mu\nu}(q^{\prime};p^{\prime},p)$
$\displaystyle=$ $\displaystyle-iF_{K\pi}^{\nu}(p^{\prime},p)$ $\displaystyle
q_{\nu}T_{K\pi}^{\mu\nu}(q^{\prime};p^{\prime},p)$ $\displaystyle=$
$\displaystyle-
iF_{K\pi}^{\mu}(p^{\prime},p)-i\Gamma_{K\pi}^{\mu}(q^{\prime};p^{\prime},p)\leavevmode\nobreak\
,$ (19)

with $q\equiv p^{\prime}+q^{\prime}-p$, and

$\Gamma^{\mu}_{K\pi}(q^{\prime};p^{\prime},p)\equiv\int
d^{4}x\,e^{iq^{\prime}\cdot
x}\left\langle\pi(p^{\prime})\right|T\\{J_{\mathrm{em}}^{\mu}(x)\partial\cdot
J_{W}^{\dagger}(0)\\}\left|K(p)\right\rangle\leavevmode\nobreak\ .$ (20)

The first line in Eq.(19) is a consequence of the exact conservation of the
electromagnetic current, while the second line entails the partial
conservation of the charged weak current.

Using now the following Dirac matrix identity:

$\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}=g^{\mu\nu}\gamma^{\alpha}-g^{\mu\alpha}\gamma^{\nu}+g^{\nu\alpha}\gamma^{\mu}-i\epsilon^{\mu\nu\alpha\beta}\gamma_{\beta}\gamma_{5}\leavevmode\nobreak\
,$ (21)

(with $\epsilon^{0123}=-1$) one splits the $\gamma W$-box diagram into two
pieces: $\delta M_{\gamma W}=\delta M_{\gamma W}^{a}+\delta M_{\gamma W}^{b}$,
where the antisymmetric tensor is contained in the second term444We used to
label them as $\delta M_{\gamma W}^{V}$ and $\delta M_{\gamma W}^{A}$ in
Ref.Seng:2020jtz , but this may cause confusions with notations of box
diagrams in some literature when we further divide the contributions from the
vector and axial charged weak current in $T_{\mu\nu}^{K\pi}$, so here we adopt
an alternative labeling.. A great simplification is observed upon combining
$\delta M_{2}$ with $\delta M_{\gamma W}^{a}$ Seng:2020jtz :

$\displaystyle\delta M_{2}+\delta M_{\gamma W}^{a}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left[\ln\frac{M_{W}^{2}}{m_{e}^{2}}+\frac{3}{4}+\frac{1}{2}\tilde{a}_{g}^{\mathrm{res}}\right]M_{0}+\frac{G_{F}e^{2}}{\sqrt{2}}\bar{u}_{\nu}\gamma_{\lambda}(1-\gamma_{5})v_{e}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}$ (22)
$\displaystyle\times\frac{1}{(p_{e}-q^{\prime})^{2}-m_{e}^{2}}\left\\{\frac{2p_{e}\cdot
q^{\prime}q^{\prime\lambda}}{(q^{\prime
2}-M_{\gamma}^{2})^{2}}T^{\mu}_{K\pi\mu}(q^{\prime};p^{\prime},p)+\frac{2p_{e\mu}}{q^{\prime
2}-M_{\gamma}^{2}}T^{\mu\lambda}_{K\pi}(q^{\prime};p^{\prime},p)\right.$
$\displaystyle\left.-\frac{(p-p^{\prime})_{\mu}}{q^{\prime
2}-M_{\gamma}^{2}}T^{\lambda\mu}_{K\pi}(q^{\prime};p^{\prime},p)+\frac{i}{q^{\prime
2}-M_{\gamma}^{2}}\Gamma^{\lambda}_{K\pi}(q^{\prime};p^{\prime},p)\right\\}$
$\displaystyle\equiv$
$\displaystyle\frac{\alpha}{2\pi}\left[\ln\frac{M_{W}^{2}}{m_{e}^{2}}+\frac{3}{4}+\frac{1}{2}\tilde{a}_{g}^{\mathrm{res}}\right]M_{0}+\left(\delta
M_{2}+\delta M_{\gamma W}^{a}\right)_{\mathrm{int}}\leavevmode\nobreak\ .$

The terms in the square bracket in Eq.(22) are exactly known as they are
isolated from the full one-loop integral with the help of the Ward identities
in Eq.(19), as well as the operator product expansion (OPE) at leading-twist
in the $q^{\prime}\sim M_{W}$ region (see Eq.(6.1) in Ref.Seng:2019lxf ), and
$\tilde{a}_{g}^{\mathrm{res}}\approx 0.019$ entails the
$\mathcal{O}(\alpha_{s})$ pQCD corrections of such terms. The remaining
“integral” piece requires further theoretical analysis and will be treated in
the next section. Meanwhile, the other component of the $\gamma W$-box diagram
reads:

$\displaystyle\delta M_{\gamma W}^{b}$ $\displaystyle=$
$\displaystyle-i\frac{G_{F}e^{2}}{\sqrt{2}}\bar{u}_{\nu}\gamma_{\lambda}(1-\gamma_{5})v_{e}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{(p_{e}-q^{\prime})^{2}-m_{e}^{2}}\frac{1}{q^{\prime
2}}\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}T_{\mu\nu}^{K\pi}(q^{\prime};p^{\prime},p)\leavevmode\nobreak\
,$

which can be split into two pieces, as well: $\delta M_{\gamma W}^{b}=\delta
M_{\gamma W}^{b,V}+\delta M_{\gamma W}^{b,A}$, where $\delta M_{\gamma
W}^{b,V}$ ($\delta M_{\gamma W}^{b,A}$) picks up the contribution from the
vector (axial) charged weak current in the generalized Compton tensor
$T_{\mu\nu}^{K\pi}$.

At this point, we can combine the terms in the square brackets from Eqs.(16)
and (22). They are analytically known and do not require any further
treatment. Their contribution to $\delta f_{+}^{K\pi}$ is given by

$\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{I}}=\left\\{\frac{\alpha}{2\pi}\left[\ln\frac{M_{Z}^{2}}{m_{e}^{2}}-\frac{1}{4}\ln\frac{M_{W}^{2}}{m_{e}^{2}}+\frac{1}{2}\ln\frac{m_{e}^{2}}{M_{\gamma}^{2}}-\frac{3}{8}+\frac{1}{2}\tilde{a}_{g}\right]+\frac{1}{2}\delta_{\mathrm{HO}}^{\mathrm{QED}}\right\\}f_{+}^{K\pi}(t)\leavevmode\nobreak\
,$ (24)

where
$\tilde{a}_{g}=-(3/2)a_{\mathrm{pQCD}}+\tilde{a}_{g}^{\mathrm{res}}\approx-0.083$.
We use the subscript “I” to signify the fact that it carries an IR divergence.
We will see later that the remaining IR-divergent pieces in the virtual
corrections come from $\left(\delta M_{2}+\delta M_{\gamma
W}\right)_{\mathrm{int}}$ and $\delta M_{3}$, and will carry the subscript
“II” and “III”, respectively.

All the remaining $\mathcal{O}(G_{F}\alpha)$ electroweak RC to the $n=0$ decay
amplitude not included in Eq. (24) are fully contained in the following
quantities: $\left(\delta M_{2}+\delta M_{\gamma
W}^{a}\right)_{\mathrm{int}}$, $\delta M_{\gamma W}^{b,V}$, $\delta M_{3}$ and
$\delta M_{\gamma W}^{b,A}$. They will be studied in the next three sections.

##  IV  Virtual correction: $\left(\delta M_{2}+\delta M_{\gamma
W}^{a}\right)_{\mathrm{int}}$ and $\delta M_{\gamma W}^{b,V}$

Figure 1: Pole (left, middle) and seagull (right) contribution to
$T_{\mu\nu}^{K\pi}$ at low energy.

In this section we evaluate the loop integrals in $\left(\delta M_{2}+\delta
M_{\gamma W}^{a}\right)_{\mathrm{int}}$ and $\delta M_{\gamma W}^{b,V}$. The
first important observation is that these integrals cannot depend on physics
at large virtual momentum $q^{\prime}$ (so we could take
$M_{W}^{2}/(M_{W}^{2}-q^{\prime 2})\to 1$ in the integrand). In $\left(\delta
M_{2}+\delta M_{\gamma W}^{a}\right)_{\mathrm{int}}$, this is because the
numerators in the integrand contain explicit factors of $p_{e}$,
$p-p^{\prime}$ or quark masses (in $\Gamma^{\lambda}_{K\pi}$); whereas in
$\delta M_{\gamma W}^{b,V}$, it is because there is no extra antisymmetric
tensor coming from $\left(T_{\mu\nu}^{K\pi}\right)_{V}$, so the integral
vanishes when $q^{\prime}\gg(p-p^{\prime})$ or $p_{e}$ due to symmetry.
Therefore, these integrals are saturated by contributions from the
intermediate hadronic states at low energy.

All the information on the hadronic structure in these integrals is contained
in the generalized Compton tensor $T^{\mu\nu}_{K\pi}$ and the vector
$\Gamma_{K\pi}^{\mu}$. Within the former, we distinguish two types of
contributions shown in Fig.1: the pole term associated with a charged meson
(initial or final, depending on the reaction channel) propagator which leads
to a $1/q^{\prime}$ behavior in the soft photon limit, and the seagull term
which is regular in that limit. The pole term is model-independent and given
in terms of the meson weak and electromagnetic form factors, whereas the
seagull term, alongside the form factors, contains information about excited
states, and is generally model-dependent. It is common to single out the Born
part of the generalized Compton tensor, defined as the pole terms complemented
by a part of the seagull term that ensures that the Ward identities in Eq.(19)
are satisfied. In this way, the remaining, non-Born part is regular for
$q^{\prime}\to 0$ and also obeys Ward identities individually.

Guided by the order $\mathcal{O}(p^{2})$ result in chiral expansion for the
Compton tensor,

$\displaystyle\left(T^{\mu\nu}_{K^{0}\pi^{-}}\right)_{p^{2}}$ $\displaystyle=$
$\displaystyle
iV_{us}^{*}\left[\frac{(2p^{\prime}+q^{\prime})^{\mu}(p+p^{\prime}+q^{\prime})^{\nu}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}-g^{\mu\nu}\right]$
$\displaystyle\left(T^{\mu\nu}_{K^{+}\pi^{0}}\right)_{p^{2}}$ $\displaystyle=$
$\displaystyle-\frac{iV_{us}^{*}}{\sqrt{2}}\left[\frac{(2p-q^{\prime})^{\mu}(p+p^{\prime}-q^{\prime})^{\nu}}{(p-q^{\prime})^{2}-M_{K}^{2}}-g^{\mu\nu}\right]\leavevmode\nobreak\
,$ (25)

and

$\displaystyle\left(\Gamma_{K^{0}\pi^{-}}^{\mu}\right)_{p^{2}}$
$\displaystyle=$ $\displaystyle
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}(2p^{\prime}+q^{\prime})^{\mu}$
$\displaystyle\left(\Gamma_{K^{+}\pi^{0}}^{\mu}\right)_{p^{2}}$
$\displaystyle=$
$\displaystyle-\frac{V_{us}^{*}}{\sqrt{2}}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p-q^{\prime})^{2}-M_{K}^{2}}(2p-q^{\prime})^{\mu}\leavevmode\nobreak\
,$ (26)

we thus define the Born contributions for the two decay channels as

$\displaystyle T^{\mu\nu,\,\rm B}_{K^{0}\pi^{-}}$ $\displaystyle=$
$\displaystyle iV_{us}^{*}F_{\mathrm{em}}^{\pi^{-}}(q^{\prime
2})\left[\frac{(2p^{\prime}+q^{\prime})^{\mu}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}\left(f_{+}^{K^{0}\pi^{-}}(q^{2})(2p+q)^{\nu}-f_{-}^{K^{0}\pi^{-}}(q^{2})q^{\nu}\right)-g^{\mu\nu}f_{\rm
seagull}^{K^{0}\pi^{-}}\right]$ $\displaystyle T^{\mu\nu,\,\rm
B}_{K^{+}\pi^{0}}$ $\displaystyle=$ $\displaystyle
iV_{us}^{*}F_{\mathrm{em}}^{K^{+}}(q^{\prime
2})\left[\frac{(2p-q^{\prime})^{\mu}}{(p-q^{\prime})^{2}-M_{K}^{2}}\left(f_{+}^{K^{+}\pi^{0}}(q^{2})(2p^{\prime}-q)^{\nu}-f_{-}^{K^{+}\pi^{0}}(q^{2})q^{\nu}\right)-g^{\mu\nu}f_{\rm
seagull}^{K^{+}\pi^{0}}\right],$

where $F_{\mathrm{em}}^{\pi^{-}}(q^{\prime 2})$ and
$F_{\mathrm{em}}^{K^{+}}(q^{\prime 2})$ are the electromagnetic form factors
of the $\pi^{-}$ and the $K^{+}$, respectively555In principle the photon can
also couple to $K^{0}$ due to its non-zero charge radius, so
$F_{\mathrm{em}}^{K^{0}}(q^{\prime 2})\neq 0$ when $q^{\prime 2}\neq 0$.
However, a simple ChPT calculation at $\mathcal{O}(p^{4})$ indicates that
$|F_{\mathrm{em}}^{K^{0}}(q^{\prime 2})|<0.02$ when $|q^{\prime 2}|<0.1$ GeV2
(see, e.g., Ref.Shi:2020rkz ), so to our required precision it is completely
negligible. On the other hand, $F_{\mathrm{em}}^{\pi^{0}}(q^{\prime 2})$ is
exactly zero due to $G$-parity., which satisfy
$F_{\mathrm{em}}^{\pi^{-}}(0)=-1$ and $F_{\mathrm{em}}^{K^{+}}(0)=1$.
Furthermore, the normalization of the seagull term is fixed as:

$\displaystyle f_{\rm
seagull}^{K^{0}\pi^{-}}=f_{+}^{K^{0}\pi^{-}}(q^{2})-f_{-}^{K^{0}\pi^{-}}(q^{2})\,,\quad\quad
f_{\rm
seagull}^{K^{+}\pi^{0}}=f_{+}^{K^{+}\pi^{0}}(q^{2})+f_{-}^{K^{+}\pi^{0}}(q^{2})\,.$
(28)

One can check that the electromagnetic Ward identity is satisfied upon
neglecting the $q^{\prime}$-dependence of the form factors in
$q_{\mu}^{\prime}T^{\mu\nu,\mathrm{B}}_{K\pi}$. With the same diagrams and
keeping in mind that we must apply the equation of motion to the charged weak
vertex so that it vanishes exactly when $M_{K}=M_{\pi}$ (see, e.g. the
discussion in Sec. 7 of Ref.Seng:2019lxf ), the Born contribution to
$\Gamma^{\mu}_{K\pi}$ reads,

$\displaystyle\Gamma^{\mu,\,\rm B}_{K^{0}\pi^{-}}$ $\displaystyle=$
$\displaystyle
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}(2p^{\prime}+q^{\prime})^{\mu}F_{\mathrm{em}}^{\pi^{-}}(q^{\prime
2})f_{0}^{K^{0}\pi^{-}}(q^{2})$ $\displaystyle\Gamma^{\mu,\,\rm
B}_{K^{+}\pi^{0}}$ $\displaystyle=$ $\displaystyle
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p-q^{\prime})^{2}-M_{K}^{2}}(2p-q^{\prime})^{\mu}F_{\mathrm{em}}^{K^{+}}(q^{\prime
2})f_{0}^{K^{+}\pi^{0}}(q^{2})$ (29)

that depends on the scalar but not the vector charged weak form factor.

While the seagull term (in both the Born and the non-Born part) is generally
model-dependent, its effect on the loop integrals turns out to be very small.
In $\left(\delta M_{2}+\delta M_{\gamma W}^{a}\right)_{\mathrm{int}}$, it only
contributes to $\delta f_{-}^{K\pi}$, whose effect in the decay rate is
further suppressed by $r_{e}\approx 10^{-6}$ (which is yet another reason why
we restrict ourselves to $K_{e3}$ throughout this study). The seagull
contribution to $\delta M_{\gamma W}^{b,V}$ vanishes trivially due to
symmetry. Of course all of the above is due to the fact that only the
$g^{\mu\nu}$ structure is present in the seagull term at $\mathcal{O}(p^{2})$.
Starting from $\mathcal{O}(p^{4})$ one expects new structures such as
$p^{\mu}p^{\nu}/\Lambda^{2}$ to enter. Observe that a new mass scale $\Lambda$
is present for dimensional reasoning, and an obvious choice is the mass of the
lowest resonances. This means we are able to get a handle of the effect of the
seagull term by computing the contributions from the resonances at low energy.
We perform that calculation based on the framework of resonance chiral theory
(this is fine, as we are only dealing with tree graphs, see details in
Appendix B), and find that their contribution to $\delta_{K_{e3}}$ through
$\left(\delta M_{2}+\delta M_{\gamma W}^{a}\right)_{\mathrm{int}}$ and $\delta
M_{\gamma W}^{b,V}$ is smaller than $10^{-4}$, which indicates that the
contribution from the seagull term is negligible. However, to stay on the safe
side, we may still introduce manually a common uncertainty of $2\times
10^{-4}$, which is roughly four times the magnitude of the resonance
contribution estimated in Appendix B, to $\delta_{K_{e3}}$ as a very
conservative estimation of the effects from the neglected seagull term.

Before proceeding directly with the numerical calculations, we prefer to
further isolate a particularly important piece from $T_{K\pi}^{\mu\nu,\,\rm
B}$ and $\Gamma_{K\pi}^{\mu,\,\rm B}$ known as the “convection term”
Meister:1963zz ,

$\displaystyle T^{\mu\nu,\,\rm conv}_{K^{0}\pi^{-}}$ $\displaystyle=$
$\displaystyle-\frac{i(2p^{\prime}+q^{\prime})^{\mu}F_{K^{0}\pi^{-}}^{\nu}(p^{\prime},p)}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}$
$\displaystyle T^{\mu\nu,\,\rm conv}_{K^{+}\pi^{0}}$ $\displaystyle=$
$\displaystyle\frac{i(2p-q^{\prime})^{\mu}F_{K^{+}\pi^{0}}^{\nu}(p^{\prime},p)}{(p-q^{\prime})^{2}-M_{K}^{2}}\leavevmode\nobreak\
,$ (30) $\displaystyle\Gamma^{\mu,\,\rm conv}_{K^{0}\pi^{-}}$ $\displaystyle=$
$\displaystyle\frac{(2p^{\prime}+q^{\prime})^{\mu}(p^{\prime}-p)_{\lambda}F_{K^{0}\pi^{-}}^{\lambda}(p^{\prime},p)}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}$
$\displaystyle\Gamma^{\mu,\,\rm conv}_{K^{+}\pi^{0}}$ $\displaystyle=$
$\displaystyle-\frac{(2p-q^{\prime})^{\mu}(p^{\prime}-p)_{\lambda}F_{K^{+}\pi^{0}}^{\lambda}(p^{\prime},p)}{(p-q^{\prime})^{2}-M_{K}^{2}}\leavevmode\nobreak\
.$ (31)

It corresponds to taking the contribution of the point electric charge in the
Born term. This contribution contains the full IR-divergent structure and is
numerically the largest. Being $q^{\prime}$-independent, it leads to a
contribution to the loop integrals that does not depend on the specific
parameterization of the hadronic form factors. Therefore it gives rise to the
so-called “model-independent” contribution emphasized in Refs.Garcia:1981it ;
JuarezLeon:2010tj ; Torres:2012ge ; Neri:2015eba , which is more commonly
known as the “outer correction” in the case of free neutron and nuclear beta
decays Sirlin:1967zza ; Wilkinson:1970cdv . We thus choose to split the full
Born contribution to $\delta f_{+}^{K\pi}$ into three pieces as follows:

$\left(\delta f_{+}^{K\pi}\right)_{\mathrm{Born}}=\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{II}}+\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{conv}}^{\mathrm{fin}}+\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{Born-conv}}\leavevmode\nobreak\ .$ (32)

The first and the second piece on the right-hand side of the equation above
represent the IR-divergent and IR-finite contributions from the convection
term, respectively. The last piece, $\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{Born-conv}}$, represents the difference between
the full Born contribution and the convection term contribution. In what
follows we provide the analytic results for the first two pieces:

$\displaystyle\left(\delta f_{+}^{K^{0}\pi^{-}}\right)_{\mathrm{II}}$
$\displaystyle=$ $\displaystyle-\frac{\alpha}{4\pi}\left\\{-\frac{4p_{e}\cdot
p^{\prime}x_{s}}{m_{e}M_{\pi}(1-x_{s}^{2})}\ln
x_{s}\ln\left(\frac{M_{\gamma}^{2}}{m_{e}M_{\pi}}\right)f_{+}^{K^{0}\pi^{-}}(t)\right.$
$\displaystyle\left.+\left(\frac{5}{2}-\ln\frac{M_{\pi}^{2}}{M_{\gamma}^{2}}\right)\left(\frac{p^{\prime}\cdot(p+p^{\prime})}{2M_{\pi}^{2}}f_{+}^{K^{0}\pi^{-}}(t)+\frac{p^{\prime}\cdot(p-p^{\prime})}{2M_{\pi}^{2}}f_{-}^{K^{0}\pi^{-}}(t)\right)\right\\}$
$\displaystyle\left(\delta f_{+}^{K^{+}\pi^{0}}\right)_{\mathrm{II}}$
$\displaystyle=$ $\displaystyle-\frac{\alpha}{4\pi}\left\\{\frac{4p_{e}\cdot
px_{u}}{m_{e}M_{K}(1-x_{u}^{2})}\ln
x_{u}\ln\left(\frac{M_{\gamma}^{2}}{m_{e}M_{K}}\right)f_{+}^{K^{+}\pi^{0}}(t)\right.$
$\displaystyle\left.+\left(\frac{5}{2}-\ln\frac{M_{K}^{2}}{M_{\gamma}^{2}}\right)\left(\frac{p\cdot(p+p^{\prime})}{2M_{K}^{2}}f_{+}^{K^{+}\pi^{0}}(t)+\frac{p\cdot(p-p^{\prime})}{2M_{K}^{2}}f_{-}^{K^{+}\pi^{0}}(t)\right)\right\\}\leavevmode\nobreak\
,$

and

$\displaystyle(\delta f_{+}^{K^{0}\pi^{-}})_{\mathrm{conv}}^{\mathrm{fin}}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\biggl{\\{}\left(C_{00}^{\mathrm{fin}}+4p_{e}\cdot
p^{\prime}C_{0}^{\mathrm{fin}}+2p_{e}\cdot
p^{\prime}C_{1}-2m_{e}^{2}C_{2}\right)f_{+}^{K^{0}\pi^{-}}(t)\biggr{.}$
$\displaystyle+\left(p^{\prime}\cdot(p+p^{\prime})f_{+}^{K^{0}\pi^{-}}(t)+p^{\prime}\cdot(p-p^{\prime})f_{-}^{K^{0}\pi^{-}}(t)\right)\left(C_{1}+\frac{1}{2}C_{11}\right)$
$\displaystyle-\frac{1}{2}\left(p_{e}\cdot(p+p^{\prime})f_{+}^{K^{0}\pi^{-}}(t)+p_{e}\cdot(p-p^{\prime})f_{-}^{K^{0}\pi^{-}}(t)\right)C_{12}$
$\displaystyle\biggl{.}+\left(p_{e}\cdot(p^{\prime}-p)+m_{e}^{2}\right)\left(f_{+}^{K^{0}\pi^{-}}(t)+f_{-}^{K^{0}\pi^{-}}(t)\right)C_{2}\biggr{\\}}$
$\displaystyle(\delta f_{+}^{K^{+}\pi^{0}})_{\mathrm{conv}}^{\mathrm{fin}}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\biggl{\\{}\left(C_{00}^{\mathrm{fin}}-4p_{e}\cdot
pC_{0}^{\mathrm{fin}}-2p_{e}\cdot
pC_{1}-2m_{e}^{2}C_{2}\right)f_{+}^{K^{+}\pi^{0}}(t)\biggr{.}$ (34)
$\displaystyle+\left(p\cdot(p+p^{\prime})f_{+}^{K^{+}\pi^{0}}+p\cdot(p-p^{\prime})f_{-}^{K^{+}\pi^{0}}\right)\left(C_{1}+\frac{1}{2}C_{11}\right)$
$\displaystyle+\frac{1}{2}\left(p_{e}\cdot(p+p^{\prime})f_{+}^{K^{+}\pi^{0}}(t)+p_{e}\cdot(p-p^{\prime})f_{-}^{K^{+}\pi^{0}}(t)\right)C_{12}$
$\displaystyle\biggl{.}+\left(p_{e}\cdot(p^{\prime}-p)+m_{e}^{2}\right)\left(f_{+}^{K^{+}\pi^{0}}(t)-f_{-}^{K^{+}\pi^{0}}(t)\right)C_{2}\biggr{\\}}\leavevmode\nobreak\
.$

The variables $x_{s}$, $x_{u}$ and the loop functions are defined in Appendix
C. Notice that one needs to substitute $m_{1}=M_{\pi}$, $m_{2}=m_{e}$, and
$v=s=(p^{\prime}+p_{e})^{2}$ in the $C$-functions for the case of
$K_{e3}^{0}$, and $m_{1}=M_{K}$, $m_{2}=m_{e}$, $v=u=(p-p_{e})^{2}$ for the
case of $K_{e3}^{+}$.

Next, we shall study $\left(\delta f_{+}^{K\pi}\right)_{\mathrm{Born-conv}}$,
which is the only piece that requires a specific parameterization of the
hadronic form factors in order to perform the loop integral. Our first
observation is that the Born contribution to $\delta f_{+}^{K\pi}$ is UV-
finite even without the form factors (it is UV-divergent for $\delta
f_{-}^{K\pi}$ without the form factors, which is however irrelevant for
$K_{e3}$). Therefore, we expect the effect of the form factors to receive a
regular power suppression instead of a logarithmic enhancement.

There are different ways of parameterizing the form factors which are
practically indistinguishable in the region $q^{\prime}\sim p_{e}\sim
p-p^{\prime}\sim M_{K}-M_{\pi}$ relevant to the integrals. However, in
practice a simpler parameterization allows for a more straightforward
evaluation of the loop integrals. Therefore, in this work, we shall adopt the
monopole representation for both the electromagnetic and charged weak form
factors. It is advantageous because the monopole resembles an ordinary
propagator, so the $q^{\prime}$-integral reduces to standard Passarino-Veltman
loop functions which can be integrated numerically with respect to
$\\{y,z\\}$.666Throughout this research we make extensive use of Package-X
Patel:2015tea ; Patel:2016fam . It is a Mathematica package that provides very
efficiently all the analytic expressions of one-loop integrals that can be
directly applied to the numerical phase-space integration. For the
electromagnetic form factors, we have:

$F_{\mathrm{em}}^{\pi^{-}}(q^{\prime 2})=\frac{-1}{1-\frac{1}{6}\left\langle
R_{\pi}^{2}\right\rangle q^{\prime 2}}\leavevmode\nobreak\
,\>\>F_{\mathrm{em}}^{K^{+}}(q^{\prime 2})=\frac{1}{1-\frac{1}{6}\left\langle
R_{K}^{2}\right\rangle q^{\prime 2}}\leavevmode\nobreak\ ,$ (35)

where $\left\langle R_{\pi}^{2}\right\rangle$ and $\left\langle
R_{K}^{2}\right\rangle$ are the mean-square charge radius of $\pi^{-}$ and
$K^{+}$, respectively777A general monopole form factor would read
$F=1/(1-q^{2}/\Lambda^{2})$. Here, we simply express the cut-off $\Lambda$ in
terms of the charge radius, as we are interested in a precise low-energy
representation.. The former was quite precisely measured Amendolia:1986wj :

$\left\langle R_{\pi}^{2}\right\rangle=(0.431\pm 0.010)\leavevmode\nobreak\
\mathrm{fm}^{2}$ (36)

which signifies $\rho$-exchange dominance. The 2% experimental uncertainty is
completely negligible in our analysis. The kaon mean-square charge radius, on
the other hand, was measured with a 15% uncertainty Amendolia:1986ui :

$\left\langle R_{K}^{2}\right\rangle=(0.34\pm 0.05)\leavevmode\nobreak\
\mathrm{fm}^{2}\leavevmode\nobreak\ .$ (37)

We will include this uncertainty later in our error analysis. Finally, for the
vector and scalar charged weak form factor, the monopole parameterization
reads:

$\bar{f}_{+}(q^{2})=\frac{M_{V}^{2}}{M_{V}^{2}-q^{2}}\leavevmode\nobreak\
,\>\>\bar{f}_{0}(q^{2})=\frac{M_{S}^{2}}{M_{S}^{2}-q^{2}}\leavevmode\nobreak\
,$ (38)

where the fitted vector and scalar pole masses are Lazzeroni:2018glh :

$M_{V}=(884.4\pm 7.4)\leavevmode\nobreak\ \mathrm{MeV}\leavevmode\nobreak\
,\>\>M_{S}=(1208.3\pm 52.1)\leavevmode\nobreak\
\mathrm{MeV}\leavevmode\nobreak\ .$ (39)

The uncertainties are less than 5% and can be safely neglected in our
analysis.

To end this section, we summarize the numerical contributions to
$\delta_{K_{e3}}$ from the different pieces in Eq. (32) (except $\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{II}}$ that we need to combine with other terms to
achieve IR-finiteness). For the error analysis, we retain only the
uncertainties of the order $10^{-4}$ or larger which, in this case, only arise
from $\left\langle R_{K}^{2}\right\rangle$. The first column represents the
physical results, but we also consider two other cases for comparison. In the
second column, we retain only the $\mathcal{O}(e^{2}p^{2})$ contributions,
which corresponds to taking $\bar{f}_{+}=\bar{f}_{0}=1$ and
$F_{\mathrm{em}}^{\pi^{-}}(q^{\prime 2})=-1$,
$F_{\mathrm{em}}^{K^{+}}(q^{\prime 2})=1$. Comparing to the numbers in the
first column, we find the inclusion of form factors has a larger impact on the
$\delta_{K_{e3}^{+}}$ than on the $\delta_{K_{e3}^{0}}$ channel. In fact, the
amount of shift in the former exceeds the estimated $\mathcal{O}(e^{2}p^{4})$
uncertainty of $0.19\%$ in the ChPT analysis Cirigliano:2008wn . This is
understandable because the effect of the form factors scales typically as
$M_{i}^{2}/M^{2}$, where $M^{2}$ is the typical mass scale in the monopole
parameterization, and $M_{i}$ is the mass of the charged meson. In
$K_{e3}^{+}$ we have $M_{i}=M_{K}$ so the numerical impact is larger. Finally,
in the third column, we consider an unphysical case where $M_{K}=1.1M_{\pi}$.
We observe in this case that
$\left(\delta_{K_{e3}}\right)_{\mathrm{conv}}^{\mathrm{fin}}\gg\left(\delta_{K_{e3}}\right)_{\mathrm{Born-
conv}}$, which proves our previous assertion that the contribution from the
convection term dominates when the initial and final hadronic states are
nearly degenerate.

| Full | $\mathcal{O}(e^{2}p^{2})$ | $M_{K}=1.1M_{\pi}$
---|---|---|---
$\left(\delta_{K_{e3}^{0}}\right)_{\mathrm{conv}}^{\mathrm{fin}}$ | $-5.0\times 10^{-3}$ | $-5.3\times 10^{-3}$ | $3.08\times 10^{-2}$
$\left(\delta_{K_{e3}^{0}}\right)_{\mathrm{Born-conv}}$ | $4.1\times 10^{-3}$ | $3.6\times 10^{-3}$ | $1\times 10^{-4}$
$\left(\delta_{K_{e3}^{+}}\right)_{\mathrm{conv}}^{\mathrm{fin}}$ | $9.6\times 10^{-3}$ | $9.2\times 10^{-3}$ | $9.9\times 10^{-3}$
$\left(\delta_{K_{e3}^{+}}\right)_{\mathrm{Born-conv}}$ | $1(1)_{\left\langle R_{K}^{2}\right\rangle}\times 10^{-4}$ | $-1.8\times 10^{-3}$ | $-1\times 10^{-4}$
Table 1: The IR-finite Born contribution to $\delta M_{2}+\delta M_{\gamma
W}^{a}+\delta M_{\gamma W}^{b,V}$.

##  V Virtual correction: $\delta M_{3}$

Next, we study $\delta M_{3}$, namely the “three-point function” correction to
the charged weak form factors. It was suggested in Ref.Seng:2019lxf  to
calculate such contributions in fixed-order ChPT, and we obtain the following
results at $\mathcal{O}(e^{2}p^{2})$:

$\displaystyle\left(\delta f_{+,3}^{K^{+}\pi^{0}}\right)_{e^{2}p^{2}}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\sqrt{2}\pi}\frac{p\cdot(p-p^{\prime})}{2M_{K}^{2}}\left[\ln\frac{M_{K}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]+\left(\delta
f_{+,3}^{K^{+}\pi^{0}}\right)_{e^{2}p^{2}}^{\mathrm{fin}}$
$\displaystyle\left(\delta f_{+,3}^{K^{0}\pi^{-}}\right)_{e^{2}p^{2}}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\frac{p^{\prime}\cdot(p-p^{\prime})}{2M_{\pi}^{2}}\left[\ln\frac{M_{\pi}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]+\left(f_{+,3}^{K^{0}\pi^{-}}\right)_{e^{2}p^{2}}^{\mathrm{fin}}\leavevmode\nobreak\
,$ (40)

where the IR-finite pieces read:

$\displaystyle\left(\delta
f_{+,3}^{K^{+}\pi^{0}}\right)_{e^{2}p^{2}}^{\mathrm{fin}}$ $\displaystyle=$
$\displaystyle-\frac{8\pi
Z\alpha}{\sqrt{2}}\left[\frac{1}{2}\bar{h}_{K^{+}\pi^{0}}(t)+\bar{h}_{K^{0}\pi^{-}}(t)+\frac{3}{2}\bar{h}_{K^{+}\eta}(t)\right]$
$\displaystyle+\frac{Z\alpha}{2\sqrt{2}\pi}\frac{M_{K}^{2}}{M_{\eta}^{2}-M_{\pi}^{2}}\left[1+\ln\frac{M_{K}^{2}}{\mu^{2}}\right]-\frac{4\pi\alpha}{\sqrt{2}}\left[-2K_{3}^{r}+K_{4}^{r}+\frac{2}{3}K_{5}^{r}+\frac{2}{3}K_{6}^{r}\right]$
$\displaystyle-\frac{8\pi\alpha}{\sqrt{2}}\frac{M_{\pi}^{2}}{M_{\eta}^{2}-M_{\pi}^{2}}\left[-2K_{3}^{r}+K_{4}^{r}+\frac{2}{3}K_{5}^{r}+\frac{2}{3}K_{6}^{r}-\frac{2}{3}K_{9}^{r}-\frac{2}{3}K_{10}^{r}\right]$
$\displaystyle\left(f_{+,3}^{K^{0}\pi^{-}}\right)_{e^{2}p^{2}}^{\mathrm{fin}}$
$\displaystyle=$ $\displaystyle-8\pi
Z\alpha\left[\frac{1}{2}\bar{h}_{K^{+}\pi^{0}}(t)+\bar{h}_{K^{0}\pi^{-}}(t)+\frac{3}{2}\bar{h}_{K^{+}\eta}(t)\right]\leavevmode\nobreak\
.$ (41)

The parameter $Z\approx 0.8$ represents the short-distance electromagnetic
effects that causes the $M_{\pi^{\pm}}-M_{\pi^{0}}$ mass splitting, while
$\\{K_{i}^{r}\\}$ are the $\mathcal{O}(e^{2}p^{2})$ LECs in the chiral
Lagrangian with dynamical photons Urech:1994hd . Finally, the loop functions
$\bar{h}_{PQ}(t)$ are defined in Appendix A of Ref.Seng:2019lxf .

The strategy above has a caveat, namely: There is an IR-divergent piece in
$\left(\delta f_{+,3}^{K\pi}\right)$ that is numerically large, so its
associated $\mathcal{O}(e^{2}p^{4})$ uncertainty can also be significant.
Fortunately, it is straightforward to resum the IR-divergent piece to all
orders in the chiral power counting by appropriately putting back the charged
weak form factors based on two simple criteria as follows:

  1. 1.

The combination $\ln(M_{i}^{2}/M_{\gamma}^{2})-5/2$ originates from the
convection term contribution and should stay intact after the resummation.
This is apparent by noticing that the same combination appears also in
$\left(\delta f_{+}^{K\pi}\right)_{\mathrm{II}}$.

  2. 2.

As we will show in Section VII, the IR-divergent piece from the bremsstrahlung
contribution takes the following form:

$\delta|M|_{\mathrm{brem}}^{2}=\frac{\alpha}{\pi}\left(\frac{1}{\beta_{i}(0)}\tanh^{-1}\beta_{i}(0)-1\right)\ln\left[\frac{M_{K}^{2}}{M_{\gamma}^{2}}\right]|M_{0}|^{2}(0,y,z)+...\leavevmode\nobreak\
,$ (42)

(the definition of $\delta|M|^{2}_{\mathrm{brem}}$ is given in Eq.(92)) where
$\beta_{i}(0)$ is the speed of the positron in the rest frame of the charged
meson (i.e. $\pi^{-}$ in $K_{e3}^{0}$ and $K^{+}$ in $K_{e3}^{+}$). The
$M_{\gamma}$-dependence above must be canceled exactly by the corresponding
$M_{\gamma}$-dependence in $\left(\delta f_{+}^{K\pi}\right)_{\mathrm{I}}$,
$\left(\delta f_{+}^{K\pi}\right)_{\mathrm{II}}$ and the IR-divergent piece in
$\delta f_{+,3}^{K\pi}$.

The arguments above lead straightforwardly to the following expression for
$\delta f_{+,3}^{K\pi}$:

$\delta f_{+,3}^{K\pi}=\left(\delta
f_{+}^{K\pi}\right)_{\mathrm{III}}+\left\\{\left(\delta
f_{+,3}^{K\pi}\right)_{e^{2}p^{2}}^{\mathrm{fin}}+\mathcal{O}(e^{2}p^{4})\right\\}\leavevmode\nobreak\
,$ (43)

where the fully-resummed IR-divergent terms read:

$\displaystyle\left(\delta f_{+}^{K^{+}\pi^{0}}\right)_{\mathrm{III}}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\frac{p\cdot(p-p^{\prime})}{2M_{K}^{2}}\left[f_{+}^{K^{+}\pi^{0}}(t)-f_{-}^{K^{+}\pi^{0}}(t)\right]\left[\ln\frac{M_{K}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]$
$\displaystyle\left(\delta f_{+}^{K^{0}\pi^{-}}\right)_{\mathrm{III}}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\frac{p^{\prime}\cdot(p-p^{\prime})}{2M_{\pi}^{2}}\left[f_{+}^{K^{0}\pi^{-}}(t)+f_{-}^{K^{0}\pi^{-}}(t)\right]\left[\ln\frac{M_{\pi}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]\leavevmode\nobreak\
,$ (44)

while the IR-finite terms stay unchanged as in Eq. (41). A significant
advantage of Eq. (43) over the $\mathcal{O}(e^{2}p^{2})$ expression in Eq.
(40) is that now only the $\\{...\\}$ term involves a chiral expansion and
must be associated with an $\mathcal{O}(e^{2}p^{4})$ uncertainty.

We end this section by summarizing the numerical contribution from
$\left(\delta f_{+,3}^{K\pi}\right)_{e^{2}p^{2}}^{\mathrm{fin}}$ to the decay
rate in Table 2. The $\mathcal{O}(e^{2}p^{4})$ uncertainty is obtained by
multiplying the central value by $M_{K}^{2}/\Lambda_{\chi}^{2}$. The numerical
values of the LECs $\\{K_{i}^{r}\\}$ at $\mu=M_{\rho}$ are obtained from
Refs.Moussallam:1997xx ; DescotesGenon:2005pw  (also summarized in
Ref.Bijnens:2014lea ), and we assign a 100% uncertainty to the sum of the LEC
contributions.

| $\left(\delta_{K_{e3}}\right)_{\mathrm{3}}^{\mathrm{fin}}$
---|---
$K_{e3}^{0}$ | $0.5(1)_{e^{2}p^{4}}\times 10^{-3}$
$K_{e3}^{+}$ | $1.4(3)_{e^{2}p^{4}}(8)_{\mathrm{LEC}}\times 10^{-3}$
Table 2: The IR-finite contribution from the three-point function.

##  VI Virtual correction: $\delta M_{\gamma W}^{b,A}$

The last piece of the virtual corrections to $f_{+}^{K\pi}(t)$ comes from
$\delta M_{\gamma W}^{b,A}$, which is fundamentally different from those we
studied in Section IV and V in the sense that it probes the strong interaction
physics in $T_{K\pi}^{\mu\nu}$ from $Q^{2}\equiv-q^{\prime 2}=0$ all the way
up to $Q^{2}\sim M_{W}^{2}$. At large $Q$, one could perform a leading-twist,
free-field OPE that gives us the large electroweak logarithm, but this
treatment breaks down at small $Q$. Also, due to parity, there is no Born
contribution in $\delta M_{\gamma W}^{b,A}$ that can be easily accounted for
as in the previous two sections. Instead, one needs to deal with contributions
from inelastic intermediate states residing at $Q\sim\Lambda_{\chi}$ that are
governed by non-perturbative QCD. In the language of ChPT, their corresponding
uncertainties are buried in the poorly-constrained LECs $X_{1}$ and
$\bar{X}_{6}^{\mathrm{phys}}$ Cirigliano:2001mk ; Cirigliano:2004pv ;
Cirigliano:2008wn ; Seng:2020jtz .

As we mentioned in the Introduction, an important breakthrough happened in
early 2020 as lattice QCD started to pick up its role in this subject. A
series of first-principles calculations were performed to study the so-called
“forward axial $\gamma W$-box” defined as follows:

$\Box_{\gamma
W}^{VA}(\phi_{i},\phi_{f},M)\equiv\frac{ie^{2}}{2M^{2}}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{(q^{\prime
2})^{2}}\epsilon^{\mu\nu\alpha\beta}q^{\prime}_{\alpha}p_{\phi\beta}\frac{T_{\mu\nu}^{if}(q^{\prime};p_{\phi},p_{\phi})}{F_{+}^{if}(0)}\leavevmode\nobreak\
,$ (45)

where $\phi_{i}$ and $\phi_{f}$ are two degenerate hadrons with mass $M$, and
carry the same external momentum $p_{\phi}$, and $F_{+}^{if}(0)$ is the form
factor $f_{+}^{if}(0)$ multiplied by the appropriate CKM matrix element. The
first calculation of $\Box_{\gamma W}^{VA}(\pi^{+},\pi^{0},M_{\pi})$ in Ref.
Feng:2020zdc  led to the reduction of the RC uncertainty in the pion
semileptonic decay by a factor of three. Shortly after that, a new calculation
of $\Box_{\gamma W}^{VA}(K^{0},\pi^{-},M_{\pi})$ in the flavor SU(3) limit was
performed Ma:2021azh  following the suggestion in Ref.Seng:2020jtz . These two
calculations together provided an improved determination of the LECs $X_{1}$
and $\bar{X}_{6}^{\mathrm{phys}}$. However, in the pure ChPT representation,
the major source of theory uncertainty in the long-range electromagnetic
corrections to $K_{e3}$ comes from $\mathcal{O}(e^{2}p^{4})$ instead of the
LECs. Therefore, the significance of the calculations above was not fully
revealed within the traditional framework.

In this section, we will demonstrate how the above-mentioned lattice QCD
results play a decisive role within the new theory framework, namely to pin
down $\delta M_{\gamma W}^{b,A}$. We start by splitting the forward axial
$\gamma W$-box into two pieces:

$\Box_{\gamma W}^{VA}(\phi_{i},\phi_{f},M)=\Box_{\gamma W}^{VA>}+\Box_{\gamma
W}^{VA<}(\phi_{i},\phi_{f},M)$ (46)

which come from the integral in Eq.(45) at $Q^{2}>Q_{\mathrm{cut}}^{2}$ and
$Q^{2}<Q_{\mathrm{cut}}^{2}$, respectively, where $Q_{\mathrm{cut}}$ is a
scale above which the leading-twist, free-field OPE is applicable. Throughout
this work we choose $Q^{2}_{\mathrm{cut}}=2$ GeV2, in accordance with the
original lattice QCD paper Feng:2020zdc 888The validity of this choice is
justified by the observation that the difference between the pQCD corrections
to $\mathcal{O}(\alpha_{s}^{3})$ and to $\mathcal{O}(\alpha_{s}^{4})$ is
negligible above $2$ GeV2 Seng:2020wjq , which demonstrates the convergence of
the perturbative series.. The first term, $\Box_{\gamma W}^{VA>}$, contains a
large electroweak logarithm and is independent of the external states
$\\{\phi_{i},\phi_{f}\\}$ as well as the mass $M$. It is given by:

$\Box_{\gamma
W}^{VA>}=\frac{\alpha}{8\pi}\ln\frac{M_{W}^{2}}{Q_{\mathrm{cut}}^{2}}+\ldots\,,$
(47)

where “$+\ldots$” denotes the pQCD corrections, which are at present
calculated to $\mathcal{O}(\alpha_{s}^{4})$ Baikov:2010je , leading to a very
precise determination: $\Box_{\gamma W}^{VA>}=2.16\times 10^{-3}$. Meanwhile,
$\Box_{\gamma W}^{VA<}(\phi_{i},\phi_{f},M)$ depends
$\\{\phi_{i},\phi_{f},M\\}$ and probe the details of the strong interaction at
$Q\sim\Lambda_{\chi}$.

To proceed further, we perform the same splitting to the integral in $\delta
M_{\gamma W}^{b,A}$:

$ie^{2}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{(p_{e}-q^{\prime})^{2}-m_{e}^{2}}\frac{1}{q^{\prime
2}}\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}\left(T_{\mu\nu}^{K\pi}(q^{\prime};p^{\prime},p)\right)_{A}=\left\\{\int_{Q^{2}>Q_{\mathrm{cut}}^{2}}+\int_{Q^{2}<Q_{\mathrm{cut}}^{2}}\right\\}\left(...\right),$
(48)

where $\left(T_{\mu\nu}^{K\pi}\right)_{A}$ represents the component in
$T_{\mu\nu}^{K\pi}$ that involves the axial charged weak current. The
contributions from these two terms to $\delta f_{+}^{K\pi}$ are denoted as
$\left(\delta f_{+}^{K\pi}\right)_{\gamma W}^{b,A>}$ and $\left(\delta
f_{+}^{K\pi}\right)_{\gamma W}^{b,A<}$, respectively, and will now be related
to the different components of the forward axial $\gamma W$-box. First, since
at $Q^{2}>Q^{2}_{\mathrm{cut}}\gg|p_{e}|^{2}$ we can set $p_{e}\to 0$ in the
integrand, one can show using OPE that,

$\left(\delta f_{+}^{K\pi}\right)_{\gamma W}^{b,A>}=\Box_{\gamma
W}^{VA>}f_{+}^{K\pi}(t)\leavevmode\nobreak\ .$ (49)

Adding this piece to $(\delta f_{+}^{K\pi})_{\mathrm{I}}$ in Eq.(24)
reproduces the full electroweak logarithm in the total RC.

Next, we can parameterize the integral at $Q^{2}<Q_{\mathrm{cut}}^{2}$ as:

$\displaystyle
ie^{2}\int_{Q^{2}<Q_{\mathrm{cut}}^{2}}\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{(p_{e}-q^{\prime})^{2}-m_{e}^{2}}\frac{1}{q^{\prime
2}}\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}\left(T_{\mu\nu}^{K\pi}(q^{\prime};p^{\prime},p)\right)_{A}$
(50) $\displaystyle\equiv$ $\displaystyle
V_{us}^{*}\left[g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)(p+p^{\prime})^{\lambda}+g_{-}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)(p-p^{\prime})^{\lambda}\right.$
$\displaystyle\left.+g_{e}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)p_{e}^{\lambda}\right]\leavevmode\nobreak\
,$

so it is obvious that:

$\left(\delta f_{+}^{K\pi}\right)_{\gamma
W}^{b,A<}=g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)\leavevmode\nobreak\ .$
(51)

To relate this quantity to the recent lattice QCD results, we set $p\to
p^{\prime}$ and $p_{e}\to 0$ on both sides of Eq.(50). That gives999In the
last line we made two implicit approximations: (1) we do not distinguish the
value of $f_{+}^{K\pi}(0)$ between the case of $M_{K}>M_{\pi}$ and
$M_{K}=M_{\pi}$, and (2) we add the $t$-dependence to the form factor. Both
approximations only lead to changes of a few percent in $f_{+}^{K\pi}$, which
is completely negligible after multiplying with $\Box_{\gamma
W}^{VA<}(K,\pi,M_{\pi})$.:

$\displaystyle g_{+}(M_{\pi}^{2},M_{\pi}^{2},0,M_{\pi}^{2},M_{\pi}^{2})$
$\displaystyle=$
$\displaystyle\frac{ie^{2}}{2M_{\pi}^{2}}\int_{Q^{2}<Q_{\mathrm{cut}}^{2}}\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{(q^{\prime
2})^{2}}\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}p^{\prime}_{\lambda}\frac{\left(T_{\mu\nu}^{K\pi}(q^{\prime};p^{\prime},p^{\prime})\right)_{A}}{V_{us}^{*}}$
(52) $\displaystyle=$ $\displaystyle\Box_{\gamma
W}^{VA<}(K,\pi,M_{\pi})f_{+}^{K\pi}(t).$

Since the lattice community has computed $\Box_{\gamma
W}^{VA<}(K,\pi,M_{\pi})$, we can obtain
$g_{+}(M_{\pi}^{2},M_{\pi}^{2},0,M_{\pi}^{2},M_{\pi}^{2})$ which is not
exactly the same as $g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)$ that we seek.
However, remember that the integral in Eq. (50) is dominated by the physics at
the scale $q^{\prime}\sim\Lambda_{\chi}$ (e.g. Regge physics Seng:2020wjq ),
it is then possible to simply take
$g_{+}(M_{\pi}^{2},M_{\pi}^{2},0,M_{\pi}^{2},M_{\pi}^{2})$ together with an
appropriately-assigned uncertainty:

$g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)=g_{+}(M_{\pi}^{2},M_{\pi}^{2},0,M_{\pi}^{2},M_{\pi}^{2})+\mathcal{O}\left(\frac{E^{2}}{\Lambda_{\chi}^{2}}\right)\leavevmode\nobreak\
,$ (53)

where $E$ is an energy scale that characterizes the non-forward (NF)
kinematics in Eq.(50), e.g. $M_{K}-M_{\pi}$, $\left(s-M_{\pi}\right)^{1/2}$ or
$\left(u-M_{\pi}\right)^{1/2}$. Since they are all smaller than $M_{K}$, we
can take $E\to M_{K}$ as a conservative estimation of the uncertainty due to
the NF effects. So, combining Eqs.(49), (51) and (53), we obtain:

$\left(\delta f_{+}^{K\pi}\right)_{\gamma W}^{b,A}=\left\\{\Box_{\gamma
W}^{VA>}+\left[\Box_{\gamma
W}^{VA<}(K,\pi,M_{\pi})+\mathcal{O}\left(\frac{M_{K}^{2}}{\Lambda_{\chi}^{2}}\right)\right]\right\\}f_{+}^{K\pi}(t)\leavevmode\nobreak\
.$ (54)

Notice that only the term in the square bracket is associated to an
$\mathcal{O}(M_{K}^{2}/\Lambda_{\chi}^{2})$ uncertainty.

The recent lattice calculations provided the forward axial $\gamma W$-box in
the charged pion and neutral kaon decay:

$\Box_{\gamma W}^{VA<}(\pi^{+},\pi^{0},M_{\pi})=0.671(28)_{\mathrm{lat}}\times
10^{-3},\>\>\>\Box_{\gamma
W}^{VA<}(K^{0},\pi^{-},M_{\pi})=0.278(44)_{\mathrm{lat}}\times
10^{-3}\leavevmode\nobreak\ .$ (55)

The box diagram in charged kaon decay is not yet computed, but can be related
to the first two through a matching to the $\mathcal{O}(e^{2}p^{2})$ ChPT
expression:

$\Box_{\gamma W}^{VA<}(K^{+},\pi^{0},M_{\pi})=2\Box_{\gamma
W}^{VA<}(\pi^{+},\pi^{0},M_{\pi})-\Box_{\gamma
W}^{VA<}(K^{0},\pi^{-},M_{\pi})=1.064(71)_{\mathrm{lat}}\times
10^{-3}\leavevmode\nobreak\ .$ (56)

The higher-order ChPT corrections to the expression above scales as
$\mathcal{O}(M_{\pi}^{2}/\Lambda_{\chi}^{2})$ and can be safely neglected in
our error analysis101010Nevertheless, a direct lattice calculation of
$\Box_{\gamma W}^{VA}(K^{+},\pi^{0},M_{\pi})$ in the future is still very much
desirable as it provides an excellent test of the convergence speed of the
chiral expansion in the SU(3) limit.. With the numbers above, we obtain the
numerical correction to the $K_{e3}$ decay rate from $\delta M_{\gamma
W}^{b,A}$, as summarized in Table 3. Notice that the NF uncertainty is
obtained by simply multiplying $2\Box_{\gamma W}^{VA<}(K,\pi,M_{\pi})$ with
$M_{K}^{2}/\Lambda_{\chi}^{2}$.

$\left(\delta_{K_{e3}}\right)_{\gamma W}^{b,A}$ | $>$ | $<$ | Total
---|---|---|---
$K_{e3}^{0}$ | $4.3\times 10^{-3}$ | $0.6(1)_{\mathrm{lat}}(1)_{\mathrm{NF}}\times 10^{-3}$ | $4.9(1)_{\mathrm{lat}}(1)_{\mathrm{NF}}\times 10^{-3}$
$K_{e3}^{+}$ | $4.3\times 10^{-3}$ | $2.1(1)_{\mathrm{lat}}(4)_{\mathrm{NF}}\times 10^{-3}$ | $6.4(1)_{\mathrm{lat}}(4)_{\mathrm{NF}}\times 10^{-3}$
Table 3: Contribution from $\delta M_{\gamma W}^{b,A}$.

To end this section, we briefly discuss the future role of the lattice QCD.
The estimation of the NF uncertainty in Eq.(53) is physically sound but can be
further improved with an extra lattice calculation. This can be seen by
considering the following relations:

$\displaystyle-\frac{8}{3}X_{1}+\bar{X}_{6}^{\mathrm{phys}}(M_{\rho})$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi\alpha}\left(\Box_{\gamma
W}^{VA}(K^{0},\pi^{-},M_{\pi})-\frac{\alpha}{8\pi}\ln\frac{M_{W}^{2}}{M_{\rho}^{2}}\right)+\frac{1}{8\pi^{2}}\left(\frac{5}{4}-\tilde{a}_{g}\right)+\mathcal{O}\left(\frac{M_{\pi}^{2}}{\Lambda_{\chi}^{2}}\right)$
$\displaystyle-\frac{8}{3}X_{1}+\bar{X}_{6}^{\mathrm{phys}}(M_{\rho})$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi\alpha}\left(\Box_{\gamma
W}^{VA}(K^{+},K^{0},M_{K})-\frac{\alpha}{8\pi}\ln\frac{M_{W}^{2}}{M_{\rho}^{2}}\right)+\frac{1}{8\pi^{2}}\left(\frac{5}{4}-\tilde{a}_{g}\right)+\mathcal{O}\left(\frac{M_{K}^{2}}{\Lambda_{\chi}^{2}}\right)\leavevmode\nobreak\
.$

Both equations are obtained through a matching between the calculation of the
RC based on Sirlin’s approach and ChPT; the first line was given in
Ref.Seng:2020jtz  and the second line can be derived accordingly. We see that
both $\Box_{\gamma W}^{VA}(K^{0},\pi^{-},M_{\pi})$ and $\Box_{\gamma
W}^{VA}(K^{+},K^{0},M_{k})$ are matched to the same combination of LECs,
except that the latter is subject to larger higher-order corrections because
the involved meson mass is $M_{K}$ which is larger. That means, the difference
in the numerical values between $\Box_{\gamma W}^{VA}(K^{0},\pi^{-},M_{\pi})$
and $\Box_{\gamma W}^{VA}(K^{+},K^{0},M_{k})$ provides an estimation of the
size of the NF corrections in Eq.(53). This strategy is very similar to the
standard lattice QCD technique to estimate the size of the chiral power
corrections through the variation of the quark masses.

##  VII Bremsstrahlung contribution

Figure 2: The real photon emission diagrams.

After going through all the virtual corrections, we switch to the contribution
from the $n=1$ process, which is simply known as the “bremsstrahlung
contribution”. According to the discussions in Appendix A, the bremsstrahlung
process contributes to the differential decay width $d\Gamma_{K_{e3}}/dydz$
not only in the $\mathcal{D}_{3}$ region but also in the $\mathcal{D}_{4-3}$
region, the latter has no correspondence in the $n=0$ process. Therefore, it
is eventually up to the experimentalists to decide in which region of
$\\{y,z\\}$ will the data be taken, and whether or not a veto will be applied
to exclude decay events with hard photons. Of course, the simplest choice is
to not apply any veto, and to collect data from all available regions of
$\\{y,z\\}$. This corresponds to a fully-inclusive prescription of the real
photon emission process, or in other words, we should calculate the sum of the
full $n=0$ and $n=1$ decay width. This prescription was adopted in
Ref.Cirigliano:2008wn  and will be followed in this work.

The bremsstrahlung amplitude, depicted by the two diagrams in Fig.2, reads:

$\displaystyle M_{K\rightarrow\pi e^{+}\nu\gamma}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}e}{\sqrt{2}}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})\left\\{\frac{p_{e}\cdot\varepsilon^{*}(k)}{p_{e}\cdot
k}+\frac{\not{k}\not{\varepsilon}^{*}(k)}{2p_{e}\cdot
k}\right\\}v_{e}F_{\mu}^{K\pi}(p^{\prime},p)$ (58)
$\displaystyle+\frac{iG_{F}e}{\sqrt{2}}\bar{u}_{\nu}\gamma^{\nu}(1-\gamma_{5})v_{e}\varepsilon^{\mu*}(k)T_{\mu\nu}^{K\pi}(k;p^{\prime},p)\leavevmode\nobreak\
.$

We observe that the generalized Compton tensor $T_{\mu\nu}^{K\pi}$ appears
again, only that now one deals with a real photon. Unlike in the loop
diagrams, here we only need to know $T_{\mu\nu}^{K\pi}$ for small (due to the
phase-space constraint) and on-shell photon momentum $k$, so instead of
exhausting the contributions from all intermediate states, it is possible to
adopt a low-energy effective expression $T_{\mu\nu}^{K\pi}$. It should,
however, satisfy three basic criteria:

  * •

It must contain the full convection term contribution to ensure an exact
cancellation of the IR-divergence from the virtual corrections.

  * •

It should include the seagull term, as the effect of the latter is not
particularly suppressed in the decay rate, unlike in the loop diagrams.

  * •

It should satisfy exact electromagnetic gauge invariance, so that one could
perform the usual replacement
$\sum_{s}\varepsilon^{\mu}_{s}(k)\varepsilon^{\nu*}_{s}(k)\rightarrow-g^{\mu\nu}$
in the sum of the outgoing photon polarizations.

The simplest effective expression that satisfies all these criteria is:

$\displaystyle T^{\mu\nu}_{K^{0}\pi^{-}}(k;p^{\prime},p)$ $\displaystyle=$
$\displaystyle-\frac{i(2p^{\prime}+k)^{\mu}F^{\nu}_{K^{0}\pi^{-}}(p^{\prime},p)}{(p^{\prime}+k)^{2}-M_{\pi}^{2}}+\left\\{iV_{us}^{*}\left[\frac{(2p^{\prime}+k)^{\mu}k^{\nu}}{(p^{\prime}+k)^{2}-M_{\pi}^{2}}-g^{\mu\nu}\right]+\mathcal{O}(p^{4})\right\\}$
$\displaystyle T^{\mu\nu}_{K^{+}\pi^{0}}(k;p^{\prime},p)$ $\displaystyle=$
$\displaystyle\frac{i(2p-k)^{\mu}F^{\nu}_{K^{+}\pi^{0}}(p^{\prime},p)}{(p-k)^{2}-M_{K}^{2}}+\left\\{\frac{iV_{us}^{*}}{\sqrt{2}}\left[\frac{(2p-k)^{\mu}k^{\nu}}{(p-k)^{2}-M_{K}^{2}}+g^{\mu\nu}\right]+\mathcal{O}(p^{4})\right\\}.$
(59)

The first term on the right-hand side in the expressions above is just the
convection term, whereas the remainders are the seagull term and the extra
pieces from the Born contribution needed to recover gauge invariance. Notice
that the convection term is exact, and only the terms in the curly bracket
undergo a chiral expansion. In fact, if we expand the convection term to
$\mathcal{O}(p^{2})$, the LO ChPT expression in Eq.(25) is recovered. In fact,
the existing ChPT calculation uses exactly Eq.(25) in their calculation of the
bremsstrahlung effect, but now our expression allows a resummation of the most
important terms in $T^{\mu\nu}_{K\pi}$ to all chiral orders.

With the above, the bremsstrahlung amplitude splits into two pieces:
$M_{K\rightarrow\pi e^{+}\nu\gamma}=M_{A}+M_{B}$ that are separately gauge-
invariant (i.e. we can write
$M_{A,B}=\varepsilon_{\mu}^{*}(k)\tilde{M}_{A,B}^{\mu}$, where
$k_{\mu}\tilde{M}_{A,B}^{\mu}=0$). For $K^{0}\to\pi^{-}e^{+}\nu_{e}\gamma$ we
have:

$\displaystyle M_{A}$ $\displaystyle=$
$\displaystyle-\frac{eG_{F}}{\sqrt{2}}F^{K^{0}\pi^{-}}_{\mu}(p^{\prime},p)\varepsilon_{\nu}^{*}(k)\left\\{\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p^{\prime}}{p^{\prime}\cdot
k}\right)^{\nu}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})v_{e}+\frac{1}{2p_{e}\cdot
k}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})\not{k}\gamma^{\nu}v_{e}\right\\}$
$\displaystyle M_{B}$ $\displaystyle=$
$\displaystyle-\frac{eG_{F}}{\sqrt{2}}V_{us}^{*}\varepsilon_{\mu}^{*}(k)\bar{u}_{\nu}\left\\{\frac{p^{\prime\mu}}{p^{\prime}\cdot
k}\not{k}-\gamma^{\mu}\right\\}(1-\gamma_{5})v_{e}\leavevmode\nobreak\ ,$ (60)

and for $K^{+}\to\pi^{0}e^{+}\nu_{e}\gamma$,

$\displaystyle M_{A}$ $\displaystyle=$
$\displaystyle-\frac{eG_{F}}{\sqrt{2}}F^{K^{+}\pi^{0}}_{\mu}(p^{\prime},p)\varepsilon_{\nu}^{*}(k)\left\\{\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p}{p\cdot
k}\right)^{\nu}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})v_{e}+\frac{1}{2p_{e}\cdot
k}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})\not{k}\gamma^{\nu}v_{e}\right\\}$
$\displaystyle M_{B}$ $\displaystyle=$
$\displaystyle\frac{eG_{F}}{2}V_{us}^{*}\varepsilon_{\mu}^{*}(k)\bar{u}_{\nu}\left\\{\frac{p^{\mu}}{p\cdot
k}\not{k}-\gamma^{\mu}\right\\}(1-\gamma_{5})v_{e}\leavevmode\nobreak\ .$ (61)

The significance of such a splitting is that $M_{A}$ is an exact expression
and only $M_{B}$ involves a chiral expansion. Therefore, in the computation of
the decay rate, only the contribution from
$2\mathfrak{Re}\\{M_{B}^{*}M_{A}\\}+|M_{B}|^{2}$ acquires an
$\mathcal{O}(e^{2}p^{4})$ uncertainty, while the contribution from
$|M_{A}|^{2}$ is exact. As we will show later, this brings an advantage over
the existing treatment as the latter is numerically the largest.

Now we proceed to the phase space integration of the bremsstrahlung
contribution. We first discuss the integration in the $\mathcal{D}_{3}$
region. To isolate the IR-singular term, we first split $|M_{A}|^{2}$ into two
pieces:

$|M_{A}|^{2}=-e^{2}\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p_{i}}{p_{i}\cdot
k}\right)^{2}|M_{0}|^{2}(0,y,z)+|M_{A}|^{2}_{\mathrm{res}},$ (62)

where $p_{i}=p$ ($p^{\prime}$) in $K_{e3}^{+}$ ($K_{e3}^{0}$). The integration
of the first term with respect to $\\{\vec{p}_{\nu},\vec{k},x\\}$ produces an
IR-divergence:

$\int_{0}^{\alpha_{+}(y,z)}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p_{i}}{p_{i}\cdot
k}\right)^{2}=I_{i}^{\mathrm{IR}}(y,z)+I_{i}^{\mathrm{fin}}(y,z)\leavevmode\nobreak\
,$ (63)

where the explicit expressions of $I_{i}^{\mathrm{IR}}$ and
$I_{i}^{\mathrm{fin}}$ can be found in Appendix D, and with this, we verify
our previous assertion about the IR-divergent structure of the bremsstrahlung
contribution in Eq.(42). We can now combine the IR-divergent contributions
from the virtual corrections (which we previously labeled as I, II, III) with
the bremsstrahlung contribution in the $\mathcal{D}_{3}$ region to obtain the
following shift of the $K_{e3}$ decay rate:

$\left(\delta\Gamma_{K_{e3}}\right)_{\mathrm{I,II,III+brem(\mathcal{D}_{3})}}=\frac{M_{K}}{256\pi^{3}}\int_{\mathrm{D}_{3}}dydz\delta|M|^{2}_{\mathrm{I,II,III+brem(\mathcal{D}_{3})}}(y,z)\leavevmode\nobreak\
,$ (64)

where

$\displaystyle\delta|M|^{2}_{\mathrm{I,II,III+brem(\mathcal{D}_{3})}}(y,z)$
$\displaystyle=$
$\displaystyle\left\\{\frac{\alpha}{2\pi}\left[2\ln\frac{M_{Z}^{2}}{m_{e}^{2}}-\frac{1}{2}\ln\frac{M_{W}^{2}}{m_{e}^{2}}+\left(1-\frac{2}{\beta_{i}(0)}\tanh^{-1}\beta_{i}(0)\right)\ln\frac{M_{i}^{2}}{M_{K}^{2}}\right.\right.$
(65)
$\displaystyle\left.\left.+\frac{1}{\beta_{i}(0)}\tanh^{-1}\beta_{i}(0)\ln\frac{M_{i}^{2}}{m_{e}^{2}}-\frac{13}{4}+\tilde{a}_{g}\right]+\delta_{\mathrm{HO}}^{\mathrm{QED}}-\frac{e^{2}M_{K}^{2}}{2\pi}I_{i}^{\mathrm{fin}}(y,z)\right\\}$
$\displaystyle\times|M_{0}|^{2}(0,y,z)+\frac{M_{K}^{2}}{2\pi}\int_{0}^{\alpha_{+}}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})\left\\{|M_{A}|^{2}_{\mathrm{res}}+2\mathfrak{Re}\left\\{M_{A}^{*}M_{B}\right\\}+|M_{B}|^{2}\right\\}\leavevmode\nobreak\
,$

which is now explicitly IR-finite. We observe that the expression above still
contains a residual integral with respect to $\\{\vec{p}_{\nu},\vec{k},x\\}$,
but it is IR-finite and therefore can be straightforwardly carried out with
the method outlined in Appendix E. The numerical result is summarized in Table
4. The HO uncertainty comes from $\delta_{\mathrm{HO}}^{\mathrm{QED}}$, while
the $\mathcal{O}(e^{2}p^{4})$ uncertainty is obtained by multiplying the
contribution from $2\mathfrak{Re}\left\\{M_{A}^{*}M_{B}\right\\}+|M_{B}|^{2}$
by $M_{K}^{2}/\Lambda_{\chi}^{2}$. We see that these uncertainties are as
small as $10^{-4}$, which is a clear success of our strategy in the splitting
of $T_{K\pi}^{\mu\nu}(k^{\prime};p^{\prime},p)$ in Eq.(59).

$\left(\delta_{K_{e3}}\right)_{\mathrm{I,II,III+brem(\mathcal{D}_{3})}}$ | From $2\mathfrak{Re}\left\\{M_{A}^{*}M_{B}\right\\}+|M_{B}^{2}|$ | Remainder | Total
---|---|---|---
$K_{e3}^{0}$ | $0.10(2)_{e^{2}p^{4}}\times 10^{-2}$ | $2.41(3)_{\mathrm{HO}}\times 10^{-2}$ | $2.51(3)_{\mathrm{HO}}(2)_{e^{2}p^{4}}\times 10^{-2}$
$K_{e3}^{+}$ | $-0.03(1)_{e^{2}p^{4}}\times 10^{-2}$ | $0.44(3)_{\mathrm{HO}}\times 10^{-2}$ | $0.40(3)_{\mathrm{HO}}(1)_{e^{2}p^{4}}\times 10^{-2}$
Table 4: Sum of the IR-divergent one-loop contribution I, II, III and the
bremsstrahlung contribution in the $\mathcal{D}_{3}$ region.

Finally, we also need to compute the bremsstrahlung contribution in the
$\mathcal{D}_{4-3}$ region:

$\displaystyle\left(\delta\Gamma_{K_{e3}}\right)_{\mathrm{brem(\mathcal{D}_{4-3})}}$
$\displaystyle=$
$\displaystyle\frac{M_{K}^{3}}{512\pi^{4}}\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})$ (66)
$\displaystyle\times\left\\{|M_{A}|^{2}+2\mathfrak{Re}\left\\{M_{A}^{*}M_{B}\right\\}+|M_{B}|^{2}\right\\}\leavevmode\nobreak\
.$

The integrals are IR-finite and can be carried out similarly using the method
in Appendix E. The numerical results are given in Table 5. In principle one
also acquires an $\mathcal{O}(e^{2}p^{4})$ uncertainty by multiplying the
contribution from $2\mathfrak{Re}\left\\{M_{A}^{*}M_{B}\right\\}+|M_{B}|^{2}$
by $M_{K}^{2}/\Lambda_{\chi}^{2}$, but the outcomes are of the order $10^{-5}$
and so are not displayed in the table.

$\left(\delta_{K_{e3}}\right)_{\mathrm{brem(\mathcal{D}_{4-3})}}$ | From $2\mathfrak{Re}\left\\{M_{A}^{*}M_{B}\right\\}+|M_{B}^{2}|$ | From $|M_{A}|^{2}$ | Total
---|---|---|---
$K_{e3}^{0}$ | $0.2\times 10^{-3}$ | $5.6\times 10^{-3}$ | $5.8\times 10^{-3}$
$K_{e3}^{+}$ | $-0.1\times 10^{-3}$ | $5.3\times 10^{-3}$ | $5.2\times 10^{-3}$
Table 5: The bremsstrahlung contribution in the $\mathcal{D}_{4-3}$ region.
Uncertainties are of order $10^{-5}$ and are not displayed.

##  VIII Comparing with the ChPT result

We have now finished calculating all components of the
$\mathcal{O}(G_{F}^{2}\alpha)$ electroweak RC to the $K_{e3}$ decay rate. The
total result is simply given by:

$\left(\delta_{K_{e3}}\right)_{\mathrm{tot}}=\left(\delta_{K_{e3}}\right)_{\mathrm{conv}}^{\mathrm{fin}}+\left(\delta_{K_{e3}}\right)_{\mathrm{Born-
conv}}+\left(\delta_{K_{e3}}\right)_{3}^{\mathrm{fin}}+\left(\delta_{K_{e3}}\right)_{\gamma
W}^{b,A}+\left(\delta_{K_{e3}}\right)_{\mathrm{I,II,III+brem(\mathcal{D}_{3})}}+\left(\delta_{K_{e3}}\right)_{\mathrm{brem(\mathcal{D}_{4-3})}}\leavevmode\nobreak\
,$ (67)

where the numerical values of different components can be found in Tables 1–5.
On the other hand, instead of grouping everything into one single number, in
the existing standard ChPT treatment the full electroweak RC is broken down
into (somewhat arbitrarily-defined) “short-distance” and “long-distance”
pieces, and are allocated to several different quantities, some of which are
rather implicit hidden. This section serves to perform a rigorous matching
between our result and the values quoted in the existing ChPT literature, with
special attention paid to the so-called “long-distance electromagnetic
corrections” $\delta_{\mathrm{EM}}^{Ke}$.

In the standard ChPT framework, the photon-inclusive $K_{e3}$ decay rate is
parameterized as Zyla:2020zbs :

$\Gamma_{K_{e3}}=\frac{G_{F}^{2}|V_{us}|^{2}M_{K}^{5}C_{K}^{2}}{192\pi^{3}}S_{\mathrm{EW}}|f_{+}^{K^{0}\pi^{-}}(0)|^{2}I_{Ke}^{(0)}(\lambda_{i})\left(1+\delta_{\mathrm{EM}}^{Ke}+\delta_{\mathrm{SU(2)}}^{K\pi}\right)\leavevmode\nobreak\
,$ (68)

where $C_{K}$ is a simple isospin factor. Apart from the quantity
$|f_{+}^{K^{0}\pi^{-}}(0)|$ that requires a lattice input, all the small QCD
and electroweak corrections to $\Gamma_{K_{e3}}$ are distributed into the
following four quantities: $S_{\mathrm{EW}}$, $I_{Ke}^{(0)}(\lambda_{i})$,
$\delta_{\mathrm{SU(2)}}^{K\pi}$ and $\delta_{\mathrm{EM}}^{Ke}$. We shall
take a serious look at each of these quantities, and study their relations to
the different components of electroweak RC we calculated in this work.

###  VIII.1 $S_{\mathrm{EW}}$

The quantity $S_{\mathrm{EW}}$ was first introduced by Marciano and Sirlin in
Ref.Marciano:1993sh  as a process-independent factor that accounts for the
large electroweak logarithm in the electroweak RC Sirlin:1977sv ;
Sirlin:1981ie  including the $\mathcal{O}(\alpha_{s})$ pQCD corrections on top
of it, as well as the resummation of the QED logs (i.e.
$\delta_{\mathrm{HO}}^{\mathrm{QED}}$ in our notation). It was often quoted
schematically in the literature as Cirigliano:2008wn ; Cirigliano:2011ny :

$S_{\mathrm{EW}}=1+\frac{2\alpha}{\pi}\left(1-\frac{\alpha_{s}}{4\pi}\right)\ln\frac{M_{Z}}{M_{\rho}}+\mathcal{O}\left(\frac{\alpha\alpha_{s}}{\pi^{2}}\right)\leavevmode\nobreak\
,$ (69)

where the $\rho$-mass appears here simply as an arbitrarily-chosen low-energy
scale. It is impossible to infer its exact value from the expression above
because some of the important components (e.g.
$\delta_{\mathrm{HO}}^{\mathrm{QED}}$) are not shown, and it is also not clear
what scale one should choose for $\alpha_{s}$. Fortunately, as a common
consensus to some extent, the value $S_{\mathrm{EW}}=1.0232(3)_{\mathrm{HO}}$
was always used for all practical purposes in the recent years (see, e.g.
Refs.Cirigliano:2001mk ; Cirigliano:2004pv  and the FLAVIAnet global analysis,
Ref.Antonelli:2010yf ), where the central value comes from Ref.Marciano:1993sh
and the estimated uncertainty of the QED log resummation comes from
Ref.Erler:2002mv . Notice that although Ref.Cirigliano:2011ny  quoted a
slightly different value of $S_{\mathrm{EW}}=1.0223(5)$, but that number was
never used in any subsequent analysis.

Now, the process-independent physics included in our
$\left(\delta_{K_{e3}}\right)_{\mathrm{tot}}$ are not only those described by
$S_{\mathrm{EW}}$ but even more. For example, the most important pQCD
correction contained in $\left(\delta_{K_{e3}}\right)_{\gamma W}^{b,A}$ is
calculated to $\mathcal{O}(\alpha_{s}^{4})$ instead of just
$\mathcal{O}(\alpha_{s})$ in $S_{\mathrm{EW}}$. Therefore, it is not the most
natural choice to remove $S_{\mathrm{EW}}-1$ analytically from
$\left(\delta_{K_{e3}}\right)_{\mathrm{tot}}$ in order to compare our result
with the ChPT result. Instead, it is more convenient to take the above-
mentioned numerical value of $S_{\mathrm{EW}}$ simply as its definition, i.e.,

$S_{\mathrm{EW}}-1\equiv 0.0232(3)_{\mathrm{HO}}\leavevmode\nobreak\ ,$ (70)

and remove this value numerically from
$\left(\delta_{K_{e3}}\right)_{\mathrm{tot}}$ for the comparison. This
prescription keeps us on the same track with all the recent literature
mentioned above.

###  VIII.2 $I_{Ke}^{(0)}(\lambda_{i})$

According to the description in Ref.Cirigliano:2011ny ,
$I_{Ke}^{(0)}(\lambda_{i})$ is a “phase space integral depending on slope and
curvature of the form factors $f_{\pm}^{K\pi}(t)$”, which means it should
contain only pure QCD effects. This statement is, however, not strictly obeyed
in the existing ChPT treatment. This can be seen in, e.g.,
Refs.Cirigliano:2001mk ; Cirigliano:2004pv : The $t$-dependence of
$f_{\pm}^{K\pi}(t)$ at $\mathcal{O}(p^{4})$ is given by the mesonic loop
functions $H_{PQ}(t)$, and we observe that in these functions the masses of
the charged mesons (e.g. $\pi^{\pm}$) and their neutral counterparts (e.g.
$\pi^{0}$) are kept distinct. Since we know that this mass splitting is
partially induced by short-distance electromagnetic effects, or more
specifically, the $\mathcal{O}(e^{2})$ term in the chiral Lagrangian
Knecht:1999ag :

$\mathcal{L}_{e^{2}}=Ze^{2}F_{0}^{4}\left\langle
Q_{\mathrm{em}}UQ_{\mathrm{em}}U^{\dagger}\right\rangle\leavevmode\nobreak\ ,$
(71)

so the observation above implies that a part of the short-distance
electromagnetic effect proportional to $Z$ is actually assigned implicitly to
$I_{Ke}^{(0)}(\lambda_{i})$ through $H_{PQ}(t)$ within the ChPT framework. In
our notation, this residual effect is represented exactly by the
$\bar{h}_{PQ}(t)$ terms in $\left(\delta
f_{+,3}^{K\pi}(t)\right)_{e^{2}p^{2}}^{\mathrm{fin}}$, since the
$\bar{h}_{PQ}(t)$ functions are simply consequences from the Taylor expansion
of $H_{PQ}(t)$ to $\mathcal{O}(Z)$.

###  VIII.3 $\delta_{\mathrm{SU(2)}}^{K\pi}$

The quantity $\delta_{\mathrm{SU(2)}}^{K\pi}$ is called the “isospin-breaking
correction”, a name suggestive of an $m_{d}-m_{u}$ effect. However, it is
rigorously defined as:

$\delta_{\mathrm{SU(2)}}^{K\pi}\equiv\left(\frac{f_{+}^{K\pi}(0)}{f_{+}^{K^{0}\pi^{-}}(0)}\right)^{2}-1\leavevmode\nobreak\
,$ (72)

that is only present in $K_{l3}^{+}$. And according to the definition above,
it contains not only the strong isospin breaking effect resulting from the
$u$–$d$ mass difference, but also the electromagnetically-induced isospin
breaking. Indeed, according to Eq.(4.42) in Ref.Cirigliano:2011ny , one has:

$\delta_{\mathrm{SU(2)}}^{K^{\pm}\pi^{0}}=2\sqrt{3}\left(\varepsilon^{(2)}+\varepsilon_{\mathrm{S}}^{(4)}+\varepsilon_{\mathrm{EM}}^{(4)}+...\right)\leavevmode\nobreak\
,$ (73)

where $\varepsilon_{\mathrm{EM}}^{(4)}$ originates from the
electromagnetically-induced $\pi^{0}$–$\eta$ mixing. In our notation, this
correction simply comes from $\left(\delta
f_{+,3}^{K^{+}\pi^{0}}(t)\right)_{e^{2}p^{2}}^{\mathrm{fin}}$ after removing
the $\bar{h}_{PQ}(t)$ terms.

###  VIII.4 $\delta_{\mathrm{EM}}^{Ke}$

After all the discussions above, it is now apparent that the most convenient
way to discuss $\delta_{\mathrm{EM}}^{Ke}$ is to simply refer it as “the sum
of all electroweak RC that are not already contained in $S_{\mathrm{EW}}$,
$I_{Ke}^{(0)}(\lambda_{i})$ and $\delta_{\mathrm{SU(2)}}^{K\pi}$”. This means

$\delta_{\mathrm{EM}}^{Ke}=\left(\delta_{K_{e3}}\right)_{\mathrm{tot}}-\left(S_{\mathrm{EW}}-1\right)-\left(\delta_{K_{e3}}\right)_{3}^{\mathrm{fin}}$
(74)

in our notation, where $S_{\mathrm{EW}}-1$ is defined by Eq.(70) as we
discussed earlier. Apart from $S_{\mathrm{EW}}-1$, the quantity
$\left(\delta_{K_{e3}}\right)_{3}^{\mathrm{fin}}$ is also subtracted out
because its contribution is redistributed into $I_{Ke}^{(0)}(\lambda_{i})$ and
$\delta_{\mathrm{SU(2)}}^{K\pi}$ according to the ChPT prescription, as we
discussed above. In fact, $\delta_{\mathrm{EM}}^{Ke}$ is also the only
meaningful quantity to be compared between this work and the existing
literature, because we are taking an $\mathcal{O}(e^{2}p^{2})$ approximation
to $\left(\delta_{K_{e3}}\right)_{3}^{\mathrm{fin}}$ and thus have made no new
improvement on this term.

$\delta_{\mathrm{EM}}^{Ke}$ | This work | Ref.Cirigliano:2008wn
---|---|---
$K_{e3}^{0}$ | $1.16(2)_{\mathrm{sg}}(1)_{\mathrm{lat}}(1^{*})_{\mathrm{NF}}(2)_{e^{2}p^{4}}\times 10^{-2}$ | $0.99(19)_{e^{2}p^{4}}(11)_{\mathrm{LEC}}\times 10^{-2}$
$K_{e3}^{+}$ | $0.21(2)_{\mathrm{sg}}(1)_{\left\langle R_{K}^{2}\right\rangle}(1)_{\mathrm{lat}}(4^{*})_{\mathrm{NF}}(1)_{e^{2}p^{4}}\times 10^{-2}$ | $0.10(19)_{e^{2}p^{4}}(16)_{\mathrm{LEC}}\times 10^{-2}$
Table 6: Comparison between the value of $\delta_{\mathrm{EM}}^{Ke}$ obtained
from this work and from the ChPT calculation.

The comparison between our result of $\delta_{\mathrm{EM}}^{Ke}$ and the ChPT
result is given in Table 6. Let us explain all the different types of
uncertainties that appear in our new evaluation:

  * •

sg: This represents our conservative estimation of the effects from the
seagull term in $\left(\delta M_{2}+\delta M_{\gamma
W}^{a}\right)_{\mathrm{int}}$ and $\delta M_{\gamma W}^{b,V}$. See the
discussions after Eq.(26).

  * •

$\left\langle R_{K}^{2}\right\rangle$: This is the uncertainty originated from
the experimental error of the $K^{+}$ charge radius (see Eq.(37)) that enters
$\left(\delta M_{2}+\delta M_{\gamma W}^{a}\right)_{\mathrm{int}}$ and $\delta
M_{\gamma W}^{b,V}$ in $K_{e3}^{+}$.

  * •

lat: This is the total lattice QCD uncertainty in the calculation of
$\Box_{\gamma W}^{VA}$ (see Eq.(55)).

  * •

NF: This represents our estimation of the uncertainty due to the non-forward
kinematics in $\delta M_{\gamma W}^{b,A}$ at small loop momentum $q^{\prime}$.
We include an asterisk to remind the reader that this error estimation can be
made more rigorous with an extra lattice QCD calculation, as we discussed at
the end of Section VI.

  * •

$e^{2}p^{4}$: This is the chiral expansion uncertainty of the non-convection
term contribution (i.e.
$2\mathfrak{Re}\left\\{M_{B}^{*}M_{A}\right\\}+|M_{B}|^{2}$, see the
discussions after Eq.(61)) in the bremsstrahlung process.

From Table 6 we find that our results are consistent with the ChPT estimation
within the error bars, but with a significant reduction of the total
uncertainty by almost an order of magnitude. This improvement is mainly due to
two reasons:

  1. 1.

Our calculation permits a much better control of the $\mathcal{O}(e^{2}p^{4})$
effects, which are the main source of uncertainty in the ChPT treatment. With
the new theory framework introduced in Refs.Seng:2019lxf ; Seng:2020jtz , all
the hadron physics are contained in quantities such as $T_{K\pi}^{\mu\nu}$ and
$\Gamma_{K\pi}^{\mu}$, from which the full convection/Born contribution can be
explicitly isolated. These contributions govern the full IR-divergent
structure of the decay process, are numerically the largest and, most
importantly, do not involve any chiral expansion. The size of the non-
Born/non-convection term contributions are in general an order of magnitude
smaller (see, for example, Table 4 and 5), so the $\mathcal{O}(e^{2}p^{4})$
uncertainties attached to them are even tinier. On the other hand, in the
traditional ChPT treatment one must multiply the full result by
$M_{K}^{2}/\Lambda_{\chi}^{2}$ to obtain the $\mathcal{O}(e^{2}p^{4})$
uncertainty, so it is much larger.

  2. 2.

We used latest lattice QCD results to pin down $\delta M_{\gamma W}^{b,A}$,
which corresponds to the LECs $X_{1}$ and $\bar{X}_{6}^{\mathrm{phys}}$ in
ChPT. In the existing literature, these LECs were calculated within resonance
models and were assigned a 100% uncertainty. On the other hand, the highly-
precise lattice results of $\Box_{\gamma W}^{VA}$ would correspond exactly to
$\delta M_{\gamma W}^{b,A}$ if $K$ and $\pi$ were degenerate. We investigated
the region of integration in $\delta M_{\gamma W}^{b,A}$ where this non-
degeneracy starts to take effect, and assigned a reasonable NF-uncertainty to
the contribution from this region on top of the lattice results. In the ChPT
language, our treatment above simultaneously take into account the
uncertainties of the LECs themselves as well as the $\mathcal{O}(e^{2}p^{4})$
uncertainties on top of the LEC contributions.

##  IX Final discussions

The $3\sigma$ discrepancy in the extraction of $V_{us}$ from $K_{\mu 2}$ and
$K_{l3}$ decays has triggered renewed interest within the particle physics
community about its possible implications on the existence of BSM physics.
However, the current level of significance is not sufficient to claim a
discovery so one needs further reduction of not just the experimental errors
but also the SM theory uncertainties. Our re-analysis of the SM electroweak RC
in $K_{e3}$ therefore, serves as a crucial step along this direction. We
successfully overcome the natural limitations in traditional ChPT by adopting
a new computational framework that allows for a resummation of the numerical
largest components in the RC, and also utilizing the most recent lattice QCD
outcomes to reduce the uncertainties from the non-perturbative QCD at the
chiral symmetry breaking scale. Our work reduces the existing uncertainties in
the $K_{e3}$ RC by almost an order of magnitude, and finds no large shift in
the central values. This suggests that we should remove the electroweak RC
from the “list of culprits” responsible for the $K_{\mu 2}$–$K_{l3}$
discrepancy.

Is it evident now that the $V_{us}$ anomaly cannot be explained by SM effects?
We would say that it is still too early to decide at this stage. Further
investigations must also be made on other SM inputs, just to mention a few:

  * •

Based on the analysis of a newly-constructed ratio
$R_{V}=\Gamma_{K_{e3}}/\Gamma_{\pi_{e3}}$, Ref.Czarnecki:2019iwz  suggested
that a shift of the lattice QCD input of
$|f_{+}^{K^{0}\pi^{-}}(0)/f_{+}^{\pi^{+}\pi^{0}}(0)|$ from its current value
of $0.970(2)$ to a smaller value of $0.961(4)$ would reconcile the $K_{\mu 2}$
and $K_{l3}$ results, and encouraged the lattice community to examine this
possibility. Lattice calculations of $|f_{+}^{K^{0}\pi^{-}}(0)|$ with
$N_{f}=2+1$ Bazavov:2012cd ; Boyle:2015hfa  and $N_{f}=2+1+1$ Bazavov:2013maa
; Carrasco:2016kpy ; Bazavov:2018kjg  in the recent years have so far been
consistent with each other, which led to the FLAG 2019 averages Aoki:2019cca :

$\displaystyle N_{f}=2+1$ $\displaystyle:$
$\displaystyle|f_{+}^{K^{0}\pi^{-}}(0)|=0.9677(27)$ $\displaystyle
N_{f}=2+1+1$ $\displaystyle:$
$\displaystyle|f_{+}^{K^{0}\pi^{-}}(0)|=0.9706(27)\leavevmode\nobreak\ .$ (75)

However, a new calculation by the PACS collaboration with $N_{f}=2+1$ returned
$|f_{+}^{K^{0}\pi^{-}}(0)|=0.9603(16)(^{+14}_{-4})(44)(19)(1)$ that is
significantly lower than the existing average Kakazu:2019ltq . This
calculation utilized only one lattice spacing $a=0.085$ fm and thus should be
carefully reexamined.

  * •

The quantity $I_{Kl}^{(0)}(\lambda_{i})$ probes the $t$-dependence of the form
factors $\bar{f}_{+,0}(t)$. Adopting a Taylor-expansion parameterization:

$\bar{f}_{+,0}(t)=1+\lambda_{+,0}^{\prime}\frac{t}{M_{\pi^{+}}^{2}}+\frac{1}{2}\lambda_{+,0}^{\prime\prime}\left(\frac{t}{M_{\pi^{+}}^{2}}\right)^{2}+...\leavevmode\nobreak\
,$ (76)

the parameters $\lambda_{+,0}^{\prime}$ and $\lambda_{+}^{\prime\prime}$ are
fit to the experimental distributions of the $K_{l3}$ decays to obtain
$\bar{f}_{+,0}(t)$ in the physical region of $t$. The resulting uncertainties
are 0.13% for $I_{Ke}^{(0)}$ and 0.31% for $I_{K\mu}^{(0)}$ (see Table 21 in
Ref.Antonelli:2009ws ), which look well under control; other forms of
parameterization were also investigated Abouzaid:2009ry ; Bernard:2006gy ;
Bernard:2007tk ; Bernard:2009zm ; Hill:2006bq . However, it is known for some
time that some disagreements occur in the extracted values of the slope
parameter $\lambda_{0}^{\prime}$ of the scalar form factor from different
experiments Cirigliano:2011ny . Also, since $\bar{f}_{+,0}(t)$ are pure QCD
quantities, their fitting to the $K_{l3}$ distributions can only be done after
removing the effects of the electroweak RC from the experimental data. Now
since we have updated the RC analysis, the fitting procedure should in
principle also be updated accordingly. Although in this paper we only present
our updates of $\delta_{\mathrm{EM}}^{Ke}$, but the electromagnetic
corrections to the $K_{e3}$ Dalitz plots can also be derived with the same
method.

  * •

Although the SU(2) isospin breaking correction factor
$\delta_{\mathrm{SU(2)}}^{K\pi}$ exists only in the $K^{+}$ channel by
construction, its associated theory uncertainty is the largest. Upon
neglecting the electromagnetic contributions, it is given by:

$\delta_{\mathrm{SU(2)}}^{K\pi}=\frac{3}{2}\frac{1}{\mathcal{Q}^{2}}\left[\frac{M_{K}^{2}}{M_{\pi}^{2}}+\frac{\mathcal{Q}^{2}}{R}\chi_{p^{4}}\right]$
(77)

in ChPT to $\mathcal{O}(p^{4})$, where
$\mathcal{Q}^{2}\equiv(m_{s}^{2}-\hat{m}^{2})/(m_{d}^{2}-m_{u}^{2})\equiv
R(m_{s}/\hat{m}+1)/2$ and $\chi_{p^{4}}=0.219$ Gasser:1984ux . The main
uncertainties therefore come from $\mathcal{Q}$ and $R$. For instance,
disagreements are observed between the values of $\mathcal{Q}$ and $R$
extracted from phenomenology Colangelo:2018jxw

$\displaystyle\eta\rightarrow 3\pi$ $\displaystyle:$
$\displaystyle\mathcal{Q}=22.1(7),\>\>R=34.4(2.1)$ (78)

and from lattice QCD Aoki:2019cca

$\displaystyle N_{f}=2+1$ $\displaystyle:$
$\displaystyle\mathcal{Q}=23.3(0.5),\>\>R=38.1(1.5)$ $\displaystyle
N_{f}=2+1+1$ $\displaystyle:$
$\displaystyle\mathcal{Q}=24.0(0.8),\>\>R=40.7(2.7)$ (79)

which must be sorted out in order to pin down the isospin breaking correction
precisely.

Finally, we want to mention that we present in this work only our updates on
the electroweak RC but not a new value of $V_{us}$. A part of the reason is
that we work exclusively on $K_{e3}$ and not on $K_{\mu 3}$, given that the
latter involves more sources of uncertainty (e.g. from $\delta f_{-}^{K\pi}$)
and will be a subject of future study. But more importantly, we realize that
the physics of kaon decay is a dynamically progressing field from where the
knowledge in both experiment and theory, including our understanding of the
issues above, is being constantly updated. Therefore, rather than quoting a
new value of $V_{us}$ upon every single improvement, it is more preferable to
have a commonly agreed value that results from a collaborative work between
experimentalists and theorists based on the most updated inputs from their
respective fields, similar to the FLAVIAnet evaluation in the past decade
Antonelli:2010yf . We hope that our research may serve as a useful input for a
possible future collaboration of such kind.

## Acknowledgements

We thank Vincenzo Cirigliano for many inspiring discussions. This work is
supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) and the NSFC through the funds provided to the Sino-German
Collaborative Research Center TRR110 “Symmetries and the Emergence of
Structure in QCD” (DFG Project-ID 196253076 - TRR 110, NSFC Grant No.
12070131001) (U-G.M and C.Y.S), by the Alexander von Humboldt Foundation
through the Humboldt Research Fellowship (C.Y.S), by the Chinese Academy of
Sciences (CAS) through a President’s International Fellowship Initiative
(PIFI) (Grant No. 2018DM0034) and by the VolkswagenStiftung (Grant No. 93562)
(U-G.M), by EU Horizon 2020 research and innovation programme, STRONG-2020
project under grant agreement No 824093 and by the German-Mexican research
collaboration Grant No. 278017 (CONACyT) and No. SP 778/4-1 (DFG) (M.G).

##  Appendix A Three- and four-body phase space in $K_{e3}$

In this Appendix we derive the phase space formula for the $K\rightarrow\pi
e^{+}\nu(\gamma)$ process. We start from the following master formula: suppose
$A(x,y,z)$ is an arbitrary Lorentz-invariant function of the three
dimensionless variables $\\{x,y,z\\}$ defined in Eq.(4), then its integration
with respect to $\vec{p}^{\prime}$ and $\vec{p}_{e}$ can be expressed as

$\frac{1}{2M_{K}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E^{\prime}}\frac{d^{3}p_{e}}{(2\pi)^{3}2E_{e}}A(x,y,z)=\frac{M_{K}^{3}}{512\pi^{4}}\int_{2\sqrt{r_{\pi}}}^{\infty}dz\int_{2\sqrt{r_{e}}}^{\infty}dy\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dxA(x,y,z)\leavevmode\nobreak\
,$ (80)

where

$\alpha_{\pm}(y,z)\equiv
1-y-z+r_{\pi}+r_{e}+\frac{yz}{2}\pm\frac{1}{2}\sqrt{y^{2}-4r_{e}}\sqrt{z^{2}-4r_{\pi}}\leavevmode\nobreak\
.$ (81)

We can apply the master formula above to derive the expressions for the
$K\rightarrow\pi e^{+}\nu(\gamma)$ phase space. First, for
$K(p)\rightarrow\pi(p^{\prime})e^{+}(p_{e})\nu(p_{\nu})$, we can identify:

$\displaystyle A(x,y,z)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-p_{\nu})|M|^{2}_{K\rightarrow\pi
e^{+}\nu}$ (82) $\displaystyle=$
$\displaystyle\frac{2\pi}{M_{K}^{2}}\delta(x)|M|^{2}_{K\rightarrow\pi
e^{+}\nu}\leavevmode\nobreak\ .$

When plugging the expression above into Eq.(80), the $x$-integral is non-zero
only when $\alpha_{-}(y,z)<0<\alpha_{+}(y,z)$, which imposes constraints on
the integration region of $\\{y,z\\}$. Solving these inequalities gives the
well-known formula:

$\Gamma_{K\rightarrow\pi
e^{+}\nu}=\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz|M|^{2}_{K\rightarrow\pi
e^{+}\nu}\leavevmode\nobreak\ ,$ (83)

where the integration region $\mathcal{D}_{3}$ can be represented in two
equivalent ways, namely:

$\displaystyle c(z)-d(z)<y<c(z)+d(z)\leavevmode\nobreak\ ,$ $\displaystyle
2\sqrt{r_{\pi}}<z<1+r_{\pi}-r_{e}$ $\displaystyle
c(z)=\frac{(2-z)(1+r_{e}+r_{\pi}-z)}{2(1+r_{\pi}-z)}\leavevmode\nobreak\ ,$
$\displaystyle
d(z)=\frac{\sqrt{z^{2}-4r_{\pi}}(1+r_{\pi}-r_{e}-z)}{2(1+r_{\pi}-z)}\leavevmode\nobreak\
,$ (84)

or

$\displaystyle a(y)-b(y)<z<a(y)+b(y)\leavevmode\nobreak\ ,$ $\displaystyle
2\sqrt{r_{e}}<y<1+r_{e}-r_{\pi}$ $\displaystyle
a(y)=\frac{(2-y)(1+r_{\pi}+r_{e}-y)}{2(1+r_{e}-y)}\leavevmode\nobreak\ ,$
$\displaystyle
b(y)=\frac{\sqrt{y^{2}-4r_{e}}(1+r_{e}-r_{\pi}-y)}{2(1+r_{e}-y)}\leavevmode\nobreak\
.$ (85)

Next, we discuss the phase space of
$K(p)\rightarrow\pi(p^{\prime})e^{+}(p_{e})\nu(p_{\nu})\gamma(k)$. In this
case we can identify:

$A(x,y,z)=\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})|M|^{2}_{K\rightarrow\pi e^{+}\nu\gamma}\leavevmode\nobreak\ .$ (86)

Without performing the integral, one already sees that the $\delta$-function
imposes the constraint $x\geq 0$ because $P^{2}=M_{K}^{2}x=(k+p_{\nu})^{2}$ is
just the invariant squared mass of the $\nu\gamma$ system, which cannot be
negative. With that one splits the $x$-integral into two terms:

$\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\Theta(x)=\Theta\left(\alpha_{+}(y,z)\right)\Theta\left(-\alpha_{-}(y,z)\right)\int_{0}^{\alpha_{+}(y,z)}dx+\Theta\left(\alpha_{-}(y,z)\right)\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\leavevmode\nobreak\
,$ (87)

and the different step functions in front of each term impose different
constraints on the integration region of $\\{y,z\\}$. The first term requires
$\alpha_{-}(y,z)<0<\alpha_{+}(y,z)$, which simply gives the $\mathcal{D}_{3}$
region we discussed above. Meanwhile, the second term requires
$\alpha_{-}(y,z)>0$, and solving this inequality yields a different
integration region which we may call $\mathcal{D}_{4-3}$. It can again be
represented in two equivalent ways:

$\displaystyle 2\sqrt{r_{e}}<y<c(z)-d(z)\leavevmode\nobreak\ ,$ $\displaystyle
2\sqrt{r_{\pi}}<z<1-\sqrt{r_{e}}+\frac{r_{\pi}}{1-\sqrt{r_{e}}}$ (88)

or

$\displaystyle 2\sqrt{r_{\pi}}<z<a(y)-b(y)\leavevmode\nobreak\ ,$
$\displaystyle
2\sqrt{r_{e}}<y<1-\sqrt{r_{\pi}}+\frac{r_{e}}{1-\sqrt{r_{\pi}}}\leavevmode\nobreak\
.$ (89)

There is no overlap between the region $\mathcal{D}_{3}$ and
$\mathcal{D}_{4-3}$ (see Fig.3). With the above, the $K\rightarrow\pi
e^{+}\nu\gamma$ decay rate can be written as:

$\displaystyle\Gamma_{K\rightarrow\pi e^{+}\nu\gamma}$ $\displaystyle=$
$\displaystyle\frac{M_{K}^{3}}{512\pi^{4}}\left\\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}(y,z)}dx+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\right\\}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}$
(90) $\displaystyle\times(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})|M|^{2}_{K\rightarrow\pi e^{+}\nu\gamma}\leavevmode\nobreak\ .$

In the study of a fully-inclusive kaon semileptonic decay rate up to
$\mathcal{O}(G_{F}^{2}\alpha)$, one should add the $K\rightarrow\pi e^{+}\nu$
and $K\rightarrow\pi e^{+}\nu\gamma$ decay rate to give:

$\displaystyle\Gamma_{K\rightarrow\pi e^{+}\nu}+\Gamma_{K\rightarrow\pi
e^{+}\nu\gamma}$ $\displaystyle=$
$\displaystyle\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz\left\\{|M|^{2}_{K\rightarrow\pi
e^{+}\nu}+\delta|M|^{2}_{\mathrm{brem}}\right\\}+\frac{M_{K}^{3}}{512\pi^{4}}\int_{\mathcal{D}_{4-3}}dydz$
$\displaystyle\times\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})|M|^{2}_{K\rightarrow\pi e^{+}\nu\gamma}\leavevmode\nobreak\ ,$

where

$\delta|M|^{2}_{\mathrm{brem}}\equiv\frac{M_{K}^{2}}{2\pi}\int_{0}^{\mathrm{\alpha_{+}(y,z)}}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})|M|^{2}_{K\rightarrow\pi e^{+}\nu\gamma}\leavevmode\nobreak\ .$ (92)

Both $|M|^{2}_{K\rightarrow\pi e^{+}\nu}$ and $\delta|M|^{2}_{\mathrm{brem}}$
possess IR-divergences that eventually cancel other. Meanwhile, the term with
the integration over the $\mathcal{D}_{4-3}$ region is by itself IR-finite.

Figure 3: Illustration of the $\mathcal{D}_{3}$ region (dark blue) and
$\mathcal{D}_{4-3}$ region (light green) in $K_{e3}^{0}$.

##  Appendix B Resonances at low energy

In this Appendix, we briefly review the basics of the resonance chiral theory
that includes the $1^{--}$ and $1^{++}$ resonances as dynamical DOFs in the
chiral Lagrangian Ecker:1988te ; Ecker:1989yg ; Cirigliano:2006hb . Based on
this formalism we calculate the contribution of these resonance to $\delta
M_{2}+\delta M_{\gamma W}^{a}$ and $\delta M_{\gamma W}^{b,V}$.

In most of the literature on resonance chiral theory, the massive spin-1
particles are described by a totally-antisymmetric tensor field instead of a
vector field Gasser:1983yg , so we start by introducing the formalism. First,
the free Lagrangian of a (real) massive spin-1 particle is written as:

$\mathcal{L}=-\frac{1}{2}\left(\partial^{\lambda}R_{\lambda\mu}\right)\left(\partial_{\nu}R^{\nu\mu}\right)+\frac{1}{4}M_{R}^{2}R_{\mu\nu}R^{\mu\nu}\leavevmode\nobreak\
,$ (93)

where $R_{\mu\nu}$ is the antisymmetric tensor field. It satisfies the
following classical equation of motion:

$\partial^{\mu}\left(\partial_{\lambda}R^{\lambda\nu}\right)-\partial^{\nu}\left(\partial_{\lambda}R^{\lambda\mu}\right)+M^{2}_{R}R^{\mu\nu}=0\leavevmode\nobreak\
.$ (94)

The quantized field takes the form:

$\displaystyle R_{\mu\nu}(x)$ $\displaystyle=$
$\displaystyle\sum_{s}\int\frac{d^{3}k}{(2\pi)^{3}2E_{R}(\vec{k})}\frac{i}{M_{R}}\left\\{\left(k_{\nu}\varepsilon_{\mu}^{s}(\vec{k})-k_{\mu}\varepsilon_{\nu}^{s}(\vec{k})\right)\hat{a}_{s}(\vec{k})e^{-ik\cdot
x}\right.$ (95)
$\displaystyle\left.-\left(k_{\nu}\varepsilon_{\mu}^{s*}(\vec{k})-k_{\mu}\varepsilon_{\nu}^{s*}(\vec{k})\right)\hat{a}^{\dagger}_{s}(\vec{k})e^{ik\cdot
x}\right\\}\leavevmode\nobreak\ ,$

where $\varepsilon_{s}(\vec{k})$ is the polarization vector of the spin-1
particle that satisfies the following relations:

$k\cdot\varepsilon_{s}(\vec{k})=0\leavevmode\nobreak\
,\>\>\>\sum_{s}\varepsilon_{\mu}^{s}(\vec{k})\varepsilon_{\nu}^{s*}(\vec{k})=-g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{M^{2}_{R}}\leavevmode\nobreak\
,$ (96)

and $\hat{a}^{+}_{s}(\vec{k}),\hat{a}_{s}(\vec{k})$ are the creation and
annihilation operators. Finally, by inverting the free Lagrangian one obtains
the covariant propagator of the antisymmetric tensor field:

$\displaystyle\Delta_{\mu\nu\alpha\beta}^{R}(p)$ $\displaystyle=$
$\displaystyle-\frac{i}{p^{2}-M^{2}_{R}+i\varepsilon}\frac{1}{M^{2}_{R}}\left(g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\mu\beta}p_{\nu}p_{\alpha}-g_{\nu\alpha}p_{\mu}p_{\beta}+g_{\nu\beta}p_{\mu}p_{\alpha}\right)$
(97)
$\displaystyle+\frac{i}{M^{2}_{R}}\left(g_{\mu\alpha}g_{\nu\beta}-g_{\nu\alpha}g_{\mu\beta}\right)\leavevmode\nobreak\
.$

We can now construct the chiral Lagrangian with dynamical vector and axial
resonances. The $1^{++}$ octet resonances are represented by $V_{\mu\nu}$
which is a traceless, Hermitian matrix in the flavor space. Its chiral
covariant derivative is given by:

$\nabla_{\lambda}V_{\mu\nu}=\partial_{\lambda}V_{\mu\nu}+[\Gamma_{\lambda},V_{\mu\nu}]\leavevmode\nobreak\
,$ (98)

where

$\Gamma_{\mu}\equiv\frac{1}{2}\left\\{u^{\dagger}\left[\partial_{\mu}-i(v_{\mu}+a_{\mu})\right]u+u\left[\partial_{\mu}-i(v_{\mu}-a_{\mu})\right]u^{\dagger}\right\\}$
(99)

is the standard connection vector, with $v_{\mu},a_{\mu}$ the vector and axial
external sources. Similarly, the $1^{--}$ resonances are represented by the
matrix $A_{\mu\nu}$. Other elementary building blocks of the ordinary ChPT.
include the “vielbein”:

$u_{\mu}\equiv
i\left\\{u^{\dagger}\left[\partial_{\mu}-i(v_{\mu}+a_{\mu})\right]u-u\left[\partial_{\mu}-i(v_{\mu}-a_{\mu})\right]u^{\dagger}\right\\}\leavevmode\nobreak\
,$ (100)

and the anti-symmetric tensors $f_{R,L}^{\mu\mu}$ built from the vector and
axial external sources:

$f_{R,L}^{\mu\nu}\equiv\partial^{\mu}(v^{\mu}\pm
a^{\nu})-\partial^{\nu}(v^{\mu}\pm a^{\mu})-i\left[v^{\mu}\pm
a^{\mu},v^{\nu}\pm a^{\nu}\right]\leavevmode\nobreak\ ,$ (101)

and finally, $f_{\pm}^{\mu\nu}\equiv uf_{L}^{\mu\nu}u^{\dagger}\pm
u^{\dagger}f_{R}^{\mu\nu}u$. With the above we can now write down the chiral
Lagrangian with $1^{++}$ and $1^{--}$ resonances. The LO Lagrangian scales as
$\mathcal{O}(p^{4})$:

$\displaystyle\mathcal{L}_{\mathrm{R}}^{(4)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left\langle(\nabla^{\lambda}V_{\lambda\mu})(\nabla_{\nu}V^{\nu\mu})-\frac{1}{2}M_{V}^{2}V_{\mu\nu}V^{\mu\nu}\right\rangle-\frac{1}{2}\left\langle(\nabla^{\lambda}A_{\lambda\mu})(\nabla_{\nu}A^{\nu\mu})-\frac{1}{2}M_{A}^{2}A_{\mu\nu}A^{\mu\nu}\right\rangle$
(102) $\displaystyle+\frac{F_{V}}{2\sqrt{2}}\left\langle
V_{\mu\nu}f_{+}^{\mu\nu}\right\rangle+\frac{iG_{V}}{\sqrt{2}}\left\langle
V_{\mu\nu}u^{\mu}u^{\nu}\right\rangle+\frac{F_{A}}{2\sqrt{2}}\left\langle
A_{\mu\nu}f_{-}^{\mu\nu}\right\rangle\leavevmode\nobreak\ ,$

where $\left\langle...\right\rangle$ represents the trace over the flavor
space, $M_{V}$ and $M_{A}$ are the vector and axial resonance masses in the
chiral limit, while $F_{V}$, $F_{A}$ and $G_{V}$ are real coupling constants.

Figure 4: The resonance contribution to $T_{\mu\nu}^{K\pi}$ at low energy.

The leading resonance contribution to $T_{K\pi}^{\mu\nu}$ scales as
$\mathcal{O}(p^{4})$ and enters through the s- and u-channel diagrams as
depicted in Fig.4. Since all the couplings in Eq.(102) have even intrinsic
parity, it is evident that only the axial resonances can exist in the
intermediate state. They give rise to the following expressions:

$\displaystyle\left(T^{\mu\nu}_{K^{0}\pi^{-}}(q^{\prime};p^{\prime},p)\right)_{\mathrm{R}}$
$\displaystyle=$
$\displaystyle-\frac{F_{A}^{2}}{F_{0}^{2}}V_{us}^{*}q^{\prime}_{\alpha}(p-p^{\prime}-q^{\prime})_{\beta}\Delta_{A}^{\alpha\mu\beta\nu}(p^{\prime}+q^{\prime})$
$\displaystyle\left(T^{\mu\nu}_{K^{+}\pi^{0}}(q^{\prime};p^{\prime},p)\right)_{\mathrm{R}}$
$\displaystyle=$
$\displaystyle\frac{F_{A}^{2}}{\sqrt{2}F_{0}^{2}}V_{us}^{*}q^{\prime}_{\alpha}(p-p^{\prime}-q^{\prime})_{\beta}\Delta_{A}^{\alpha\mu\beta\nu}(p-q^{\prime})\leavevmode\nobreak\
,$ (103)

where $F_{0}$ is the pion decay constant in the chiral limit. For numerical
estimation, we choose $F_{A}=123$ MeV, $M_{A}=968$ MeV following
Ref.Ecker:1988te , and $F_{0}\approx F_{\pi}=92.1$ MeV. Meanwhile, since
$\Gamma^{\mu}_{K\pi}$ vanishes in the flavor SU(3) limit, it cannot be
generated by the resonance Lagrangian in Eq.(102) at tree level because the
latter is SU(3)-symmetric.

We then plug the expressions above into Eq.(22), (LABEL:eq:deltaMWb) and
evaluate the integrals. Of course, upon setting
$M_{W}^{2}/(M_{W}^{2}-q^{\prime 2})\rightarrow 1$ the integrals are UV-
divergent, but this is expected because the expressions above are only
supposed to work at small $q^{\prime}$ so the integral should be cut off at
$q^{\prime}\sim M_{A}$. As our main purpose here is just to have an order-of-
magnitude estimation of the resonance contribution, we shall adopt a simple
prescription as follows: we first regularize the UV-divergence using
dimensional regularization, and discard the usual divergent combination
$2/(4-d)-\gamma_{E}+\ln 4\pi$. The result is then a function of the
renormalization scale $\mu$, which we vary from $M_{A}$ to $2M_{A}$ as a crude
estimation of the uncertainty. With the above, we obtain the following
resonance contribution to $\delta_{K_{e3}}$:

$\displaystyle\left(\delta_{K_{e3}^{0}}\right)_{\mathrm{resonance}}$
$\displaystyle=$ $\displaystyle(-0.6\pm 2.0)\times 10^{-5}$
$\displaystyle\left(\delta_{K_{e3}^{+}}\right)_{\mathrm{resonance}}$
$\displaystyle=$ $\displaystyle(5.9\pm 0.8)\times 10^{-5}\leavevmode\nobreak\
.$ (104)

They are both smaller than $10^{-4}$.

##  Appendix C Loop functions in the convection term contributions

In this Appendix we provide the analytic formula for the loop functions that
enter the convection term contribution $\delta M_{2}+\delta M_{\gamma W}^{a}$
and $\delta M_{\gamma W}^{b,V}$. We start by parameterizing the relevant loop
integrals:

$\displaystyle\frac{(2\pi\mu)^{4-d}}{i\pi^{2}}\int
d^{d}k\frac{k^{\mu}k^{\nu}}{[(p_{1}-k)^{2}-m_{1}^{2}][(p_{2}-k)^{2}-m_{2}^{2}]k^{2}}$
$\displaystyle=$ $\displaystyle
g^{\mu\nu}C_{00}+(p_{1}^{\mu}p_{2}^{\nu}+p_{1}^{\nu}p_{2}^{\mu})C_{12}$ (105)
$\displaystyle+p_{1}^{\mu}p_{1}^{\nu}C_{11}+p_{2}^{\mu}p_{2}^{\nu}C_{22}\leavevmode\nobreak\
,$ $\frac{1}{i\pi^{2}}\int
d^{4}k\frac{k^{\mu}}{[(p_{1}-k)^{2}-m_{1}^{2}][(p_{2}-k)^{2}-m_{2}^{2}]k^{2}}=-C_{1}p_{1}^{\mu}-C_{2}p_{2}^{\mu}\leavevmode\nobreak\
,$ (106)

and

$\frac{1}{i\pi^{2}}\int
d^{4}k\frac{1}{[(p_{1}-k)^{2}-m_{1}^{2}][(p_{2}-k)^{2}-m_{2}^{2}][k^{2}-M_{\gamma}^{2}]}=C_{0}\leavevmode\nobreak\
.$ (107)

The first expression is UV-divergent and is regularized using dimensional
regularization, while the third expression is IR-divergent and is regularized
by a small photon mass $M_{\gamma}$. All the $C$s above are functions of
$m_{1}^{2}=p_{1}^{2}$, $m_{2}^{2}=p_{2}^{2}$ and $v=(p_{1}-p_{2})^{2}$. The
analytic expressions for the $C_{ij}$ functions are as follows:

$\displaystyle C_{00}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{2}{4-d}-\gamma_{E}+\ln
4\pi+\ln\frac{\mu^{2}}{m_{1}^{2}}+3\right)+\frac{\Lambda(v,m_{1},m_{2})}{4}+\frac{-m_{1}^{2}+m_{2}^{2}+v}{8v}\ln\frac{m_{1}^{2}}{m_{2}^{2}}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{4}\left(\frac{2}{4-d}-\gamma_{E}+\ln
4\pi+\ln\frac{\mu^{2}}{m_{1}^{2}}+3\right)+C_{00}^{\mathrm{fin}}$
$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle-\frac{m_{1}^{4}-2m_{1}^{2}m_{2}^{2}+m_{2}^{4}-2m_{1}^{2}v+v^{2}}{2v\lambda(m_{1}^{2},m_{2}^{2},v)}\Lambda(v,m_{1},m_{2})+\frac{m_{1}^{2}-m_{2}^{2}-v}{4v^{2}}\ln\frac{m_{1}^{2}}{m_{2}^{2}}-\frac{1}{2v}$
$\displaystyle C_{22}$ $\displaystyle=$
$\displaystyle-\frac{m_{1}^{4}-2m_{1}^{2}m_{2}^{2}+m_{2}^{4}-2m_{2}^{2}v+v^{2}}{2v\lambda(m_{1}^{2},m_{2}^{2},v)}\Lambda(v,m_{1},m_{2})+\frac{m_{1}^{2}-m_{2}^{2}+v}{4v^{2}}\ln\frac{m_{1}^{2}}{m_{2}^{2}}-\frac{1}{2v}$
$\displaystyle C_{12}$ $\displaystyle=$
$\displaystyle-\frac{-m_{1}^{4}+2m_{1}^{2}m_{2}^{2}-m_{2}^{4}+m_{1}^{2}v+m_{2}^{2}v}{2v\lambda(m_{1}^{2},m_{2}^{2},v)}\Lambda(v,m_{1},m_{2})-\frac{m_{1}^{2}-m_{2}^{2}}{4v^{2}}\ln\frac{m_{1}^{2}}{m_{2}^{2}}+\frac{1}{2v}\leavevmode\nobreak\
,$

where $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$ is the triangle function,
and

$\Lambda(v,m_{1},m_{2})\equiv\frac{\lambda^{1/2}(m_{1}^{2},m_{2}^{2},v)}{v}\ln\left(\frac{\lambda^{1/2}(m_{1}^{2},m_{2}^{2},v)+m_{1}^{2}+m_{2}^{2}-v}{2m_{1}m_{2}}+i\epsilon\right)\leavevmode\nobreak\
.$ (109)

The analytic expressions for the $C_{i}$ functions read:

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2v}\ln\frac{m_{1}^{2}}{m_{2}^{2}}-\frac{m_{1}^{2}-m_{2}^{2}-v}{\lambda(m_{1}^{2},m_{2}^{2},v)}\Lambda(v,m_{1},m_{2})$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2v}\ln\frac{m_{1}^{2}}{m_{2}^{2}}+\frac{m_{1}^{2}-m_{2}^{2}+v}{\lambda(m_{1}^{2},m_{2}^{2},v)}\Lambda(v,m_{1},m_{2})\leavevmode\nobreak\
.$ (110)

And finally,

$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\frac{x_{v}}{m_{1}m_{2}(1-x_{v}^{2})}\left\\{\ln
x_{v}\left[-\ln\left(\frac{M_{\gamma}^{2}}{m_{1}m_{2}}\right)-\frac{1}{2}\ln
x_{v}+2\ln(1-x_{v}^{2})\right]-\frac{\pi^{2}}{6}\right.$ (111)
$\displaystyle\left.+\mathrm{Li}_{2}(x_{v}^{2})+\frac{1}{2}\ln^{2}\left(\frac{m_{1}}{m_{2}}\right)+\mathrm{Li}_{2}\left(1-x_{v}\frac{m_{1}}{m_{2}}\right)+\mathrm{Li}_{2}\left(1-x_{v}\frac{m_{2}}{m_{1}}\right)\right\\}$
$\displaystyle\equiv$ $\displaystyle-\frac{x_{v}}{m_{1}m_{2}(1-x_{v}^{2})}\ln
x_{v}\ln\left(\frac{M_{\gamma}^{2}}{m_{1}m_{2}}\right)+C_{0}^{\mathrm{fin}}\leavevmode\nobreak\
,$

with

$x_{v}\equiv-\frac{1-\sqrt{1-\frac{4m_{1}m_{2}}{v-(m_{1}-m_{2})^{2}}}}{1+\sqrt{1-\frac{4m_{1}m_{2}}{v-(m_{1}-m_{2})^{2}}}}\leavevmode\nobreak\
.$ (112)

##  Appendix D Dimensional regularization of the IR-divergent integral in the
bremsstrahlung contribution

The only IR-divergent integral in bremsstrahlung process
$K(p)\rightarrow\pi(p^{\prime})e^{+}(p_{e})\nu(p_{\nu})\gamma(k)$ reads:

$\displaystyle I_{i}(y,z)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\alpha_{+}(y,z)}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p_{i}}{p_{i}\cdot
k}\right)^{2}$ (113) $\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\alpha_{+}(y,z)}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\delta(M_{K}^{2}x-2k\cdot
P)\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p_{i}}{p_{i}\cdot k}\right)^{2},$

where $i=K$ or $\pi$ (obviously, $p_{K}=p$ and $p_{\pi}=p^{\prime}$). Here, we
can use the single delta function in the second line to integrate out $E_{k}$,
so the IR-divergence comes from the final integration with respect to $x$,
where the integrand behaves as $x^{-1}$ at small $x$. A common prescription to
regularize this IR-divergence is to introduce a non-zero photon mass such that
$k^{2}=M_{\gamma}^{2}$. This sets a lower bound of $M_{\gamma}^{2}/M_{K}^{2}$
for the $x$-integral that regularizes the IR-divergence, but also introduces a
complicated $M_{\gamma}$-dependence in the integrand that needs to be
carefully taken into account in order to correctly reproduce all the IR-finite
terms in the $M_{\gamma}\rightarrow 0$ limit.

A more elegant way to deal with the IR-divergence is to use dimensional-
regularization Gastmans:1973uv ; Marciano:1974tv . With this prescription, we
first generalize the three-dimensional $k$-integral to $d-1$ dimension:

$\displaystyle\frac{d^{3}k}{(2\pi)^{3}2E_{k}}$ $\displaystyle\rightarrow$
$\displaystyle\mu^{4-d}\frac{d^{d-1}k}{(2\pi)^{d-1}2E_{k}}$ (114)
$\displaystyle=$
$\displaystyle\frac{\mu^{4-d}}{2(2\pi)^{d-1}}E_{k}^{d-2}\sin^{d-3}\theta_{1}\sin^{d-4}\theta_{2}...\sin\theta_{d-3}dE_{k}d\theta_{1}d\theta_{2}...d\theta_{d-3}d\theta_{d-2}\leavevmode\nobreak\
,$

where $0\leq\theta_{1},\theta_{2},...,\theta_{d-3}\leq\pi$ and
$0\leq\theta_{d-2}\leq 2\pi$. The spatial components of $k$ are parameterized
as:

$\displaystyle k_{1}$ $\displaystyle=$ $\displaystyle E_{k}\cos\theta_{1}$
$\displaystyle k_{2}$ $\displaystyle=$ $\displaystyle
E_{k}\sin\theta_{1}\cos\theta_{2}$ $\displaystyle\vdots$ $\displaystyle
k_{d-2}$ $\displaystyle=$ $\displaystyle
E_{k}\sin\theta_{1}\sin\theta_{2}...\sin\theta_{d-3}\cos\theta_{d-2}$
$\displaystyle k_{d-1}$ $\displaystyle=$ $\displaystyle
E_{k}\sin\theta_{1}\sin\theta_{2}...\sin\theta_{d-3}\sin\theta_{d-2}\leavevmode\nobreak\
.$ (115)

With the prescription above, the IR-divergent integral over $x$ can now be
simply performed:

$\int_{0}^{\alpha_{+}(y,z)}dxx^{d-5}=\frac{\left(\alpha_{+}(y,z)\right)^{d-4}}{d-4}\leavevmode\nobreak\
,$ (116)

assuming $d>4$. Meanwhile, the angles can be integrated using the formula:

$\int_{0}^{\pi}\sin^{m}\theta
d\theta=\sqrt{\pi}\frac{\Gamma\left(\frac{1}{2}(m+1)\right)}{\Gamma\left(\frac{1}{2}(m+2)\right)}\leavevmode\nobreak\
.$ (117)

And finally, one expands the result to $\mathcal{O}\left((d-4)^{0}\right)$. It
is also customary to switch the result back to the expression with the
$M_{\gamma}$-regularization. For that purpose one simply performs the
following matching:

$\frac{2}{4-d}-\gamma_{E}+\ln
4\pi\rightarrow\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\leavevmode\nobreak\ .$ (118)

Next, we discuss some useful tricks in the evaluation of $I_{i}(y,z)$ with
dimensional regularization. First, the full integral can be split into three
terms, with the integrand proportional to:

$\frac{1}{(p_{e}\cdot k)^{2}}\leavevmode\nobreak\ ,\>\>\frac{1}{(p_{i}\cdot
k)^{2}}\leavevmode\nobreak\ ,\>\>\frac{1}{(p_{e}\cdot k)(p_{i}\cdot k)}$ (119)

respectively. The integration with respect to the first term is most easily
done in the $\vec{p}_{e}$-rest frame, while the next two terms should be done
in the $\vec{p}_{i}$-rest frame. The following identity is also useful in
performing the integration of the third term:

$\displaystyle\int_{0}^{\alpha_{+}(y,z)}dxx^{d-5}f(d,x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\alpha_{+}(y,z)}dxx^{d-5}f(d,0)+\int_{0}^{\alpha_{+}(y,z)}dxx^{d-5}\left(f(d,x)-f(d,0)\right)$
$\displaystyle=$
$\displaystyle\frac{\left(\alpha_{+}(y,z)\right)^{d-4}}{d-4}+\int_{0}^{\alpha_{+}(y,z)}dx\frac{1}{x}\left(f(4,x)-f(4,0)\right)+\mathcal{O}(d-4)\leavevmode\nobreak\
.$

We are now ready to write down the full result of the integral:

$I_{i}(y,z)=I_{i}^{\mathrm{IR}}(y,z)+I_{i}^{\mathrm{fin}}(y,z)\leavevmode\nobreak\
,$ (121)

where

$I_{i}^{\mathrm{IR}}(y,z)=\frac{1}{2\pi
M_{K}^{2}}\left\\{\left(1-\frac{1}{\beta_{i}(0)}\tanh^{-1}\beta_{i}(0)\right)\ln\left[\frac{M_{K}^{2}}{M_{\gamma}^{2}}\right]-\frac{1}{2}\ln\left[\frac{M_{K}^{2}}{m_{e}^{2}}\right]\right\\}$
(122)

is the IR-divergent piece after switching back to the
$M_{\gamma}$-prescription using Eq.(118), and

$\displaystyle I_{i}^{\mathrm{fin}}(y,z)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi
M_{K}^{2}}\left\\{\left(1-\frac{2}{\beta_{i}(0)}\tanh^{-1}\beta_{i}(0)\right)\ln\left[\frac{M_{K}^{2}\alpha_{+}^{2}}{4P_{0}^{2}(0)}\right]+\ln\left[\frac{\alpha_{+}^{2}(y,z)}{(1-z+r_{\pi}-r_{e})^{2}}\right]\right.$
(123)
$\displaystyle-\frac{1}{\beta_{i}(0)}\mathrm{Li}_{2}\left[\frac{2\beta_{i}(0)}{1+\beta_{i}(0)}\right]+\frac{1}{\beta_{i}(0)}\mathrm{Li}_{2}\left[-\frac{2\beta_{i}(0)}{1-\beta_{i}(0)}\right]$
$\displaystyle\left.+\frac{2}{\beta_{i}(0)}\mathrm{Li}_{2}\left[\frac{\beta_{i}(0)}{1+\beta_{i}(0)}\left(\frac{P_{1}(0)}{P_{0}(0)}+1\right)\right]-\frac{2}{\beta_{i}(0)}\mathrm{Li}_{2}\left[\frac{\beta_{i}(0)}{1-\beta_{i}(0)}\left(\frac{P_{1}(0)}{P_{0}(0)}-1\right)\right]\right\\}$
$\displaystyle-\frac{1}{2\pi
M_{K}^{2}}\int_{0}^{\alpha_{+}(y,z)}dx\frac{1}{x}\left\\{\frac{1}{\beta_{i}(x)}\ln\left[\frac{1+\beta_{i}(x)}{1-\beta_{i}(x)}\right]-\frac{1}{\beta_{i}(0)}\ln\left[\frac{1+\beta_{i}(0)}{1-\beta_{i}(0)}\right]\right\\}$

is the IR-finite piece, with

$\beta_{i}(x)\equiv\sqrt{1-\frac{M_{i}^{2}m_{e}^{2}}{(p_{i}\cdot
p_{e})^{2}}}\leavevmode\nobreak\ ,\>\>P_{0}(x)\equiv\frac{p_{i}\cdot
P}{M_{i}}\leavevmode\nobreak\
,\>\>P_{1}(x)\equiv\frac{1}{\beta_{i}(x)}\left(P_{0}(x)-\frac{p_{e}\cdot
P}{p_{i}\cdot p_{e}}M_{i}\right)\leavevmode\nobreak\ .$ (124)

Of course all the quantities in Eq.(124) are functions of $\\{y,z\\}$ as well.
Their physical meanings are apparent: $\beta_{i}(x)$ is the speed of the
positron, $P_{0}(x)$ is the zeroth component of $P^{\mu}$, and $P_{1}(x)$ is
the spatial component of $P^{\mu}$ along the direction of $\vec{p}_{e}$, all
in the $\vec{p}_{i}$-rest frame. Notice that the residual, IR-finite integral
in the last line of Eq.(123) vanishes for $i=K$, because
$\beta_{K}(x)=\beta_{K}(0)$.

The correct analytic expression for $I_{K}(y,z)$ and $I_{\pi}(y,z)$ first
appeared in Ref.Ginsberg:1969jh  and Ref.Cirigliano:2004pv  respectively
(notice that Ref.Ginsberg:1968pz  also attempted to calculate $I_{\pi}(y,z)$,
but the result there is wrong even with the Errata). It is easy to check the
numerical equivalence between Eq.(121) and those expressions, after accounting
for the difference in the overall normalization.

##  Appendix E IR-finite integrals in the bremsstrahlung contribution

In this Appendix, we outline the general strategy to evaluate the IR-finite
numerical integrations from the bremsstrahlung process, in both the
$\mathcal{D}_{3}$ and $\mathcal{D}_{4-3}$ region. We start by providing the
expressions of the relevant integrands. In $K_{e3}^{0}$ we have:

$\displaystyle|M_{A}|_{\mathrm{res}}^{2}$ $\displaystyle=$
$\displaystyle-e^{2}\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p^{\prime}}{p^{\prime}\cdot
k}\right)^{2}\left\\{|M_{0}|^{2}(x,y,z)-|M_{0}|^{2}(0,y,z)\right\\}$
$\displaystyle+e^{2}G_{F}^{2}F_{\mu}F_{\nu}^{*}\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p^{\prime}}{p^{\prime}\cdot
k}\right)^{2}\mathrm{Tr}\left[\not{k}\gamma^{\mu}(\not{p}_{e}-m_{e})\gamma^{\nu}(1-\gamma_{5})\right]$
$\displaystyle-e^{2}G_{F}^{2}F_{\mu}^{*}F_{\nu}\frac{1}{p_{e}\cdot
k}\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p^{\prime}}{p^{\prime}\cdot
k}\right)_{\alpha}\mathfrak{Re}\mathrm{Tr}\left[(\not{p}_{e}-m_{e})\gamma^{\mu}(\not{P}-\not{k})\gamma^{\nu}\not{k}\gamma^{\alpha}(1-\gamma_{5})\right]$
$\displaystyle+e^{2}G_{F}^{2}F_{\mu}F_{\nu}^{*}\frac{1}{p_{e}\cdot
k}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}\not{k}\gamma^{\nu}(1-\gamma_{5})\right]$
$\displaystyle 2\mathfrak{Re}\left\\{M_{A}M_{B}^{*}\right\\}$ $\displaystyle=$
$\displaystyle-2e^{2}G_{F}^{2}V_{us}F_{\mu}\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p^{\prime}}{p^{\prime}\cdot k}\right)_{\alpha}$
$\displaystyle\times\mathfrak{Re}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}(\not{p}_{e}-m_{e})\left\\{\frac{p^{\prime\alpha}}{p^{\prime}\cdot
k}\not{k}-\gamma^{\alpha}\right\\}(1-\gamma_{5})\right]$
$\displaystyle-e^{2}G_{F}^{2}V_{us}F_{\mu}\frac{1}{p_{e}\cdot
k}\mathfrak{Re}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}\not{k}\gamma^{\alpha}(\not{p}_{e}-m_{e})\left\\{\frac{p^{\prime}_{\alpha}}{p^{\prime}\cdot
k}\not{k}-\gamma_{\alpha}\right\\}(1-\gamma_{5})\right]$
$\displaystyle|M_{B}|^{2}$ $\displaystyle=$
$\displaystyle-e^{2}G_{F}^{2}|V_{us}|^{2}\mathrm{Tr}\left[(\not{P}-\not{k})\left\\{\frac{p^{\prime\mu}}{p^{\prime}\cdot
k}\not{k}-\gamma^{\mu}\right\\}(\not{p}_{e}-m_{e})\left\\{\frac{p^{\prime}_{\mu}}{p^{\prime}\cdot
k}\not{k}-\gamma_{\mu}\right\\}(1-\gamma_{5})\right]\leavevmode\nobreak\ ,$

and similarly for $K_{e3}^{+}$,

$\displaystyle|M_{A}|_{\mathrm{res}}^{2}$ $\displaystyle=$
$\displaystyle-e^{2}\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p}{p\cdot
k}\right)^{2}\left\\{|M_{0}|^{2}(x,y,z)-|M_{0}|^{2}(0,y,z)\right\\}$
$\displaystyle+e^{2}G_{F}^{2}F_{\mu}F_{\nu}^{*}\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p}{p\cdot
k}\right)^{2}\mathrm{Tr}\left[\not{k}\gamma^{\mu}(\not{p}_{e}-m_{e})\gamma^{\nu}(1-\gamma_{5})\right]$
$\displaystyle-e^{2}G_{F}^{2}F_{\mu}^{*}F_{\nu}\frac{1}{p_{e}\cdot
k}\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p}{p\cdot
k}\right)_{\alpha}\mathfrak{Re}\mathrm{Tr}\left[(\not{p}_{e}-m_{e})\gamma^{\mu}(\not{P}-\not{k})\gamma^{\nu}\not{k}\gamma^{\alpha}(1-\gamma_{5})\right]$
$\displaystyle+e^{2}G_{F}^{2}F_{\mu}F_{\nu}^{*}\frac{1}{p_{e}\cdot
k}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}\not{k}\gamma^{\nu}(1-\gamma_{5})\right]$
$\displaystyle 2\mathfrak{Re}\left\\{M_{A}M_{B}^{*}\right\\}$ $\displaystyle=$
$\displaystyle\sqrt{2}e^{2}G_{F}^{2}V_{us}F_{\mu}\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p}{p\cdot k}\right)_{\alpha}$
$\displaystyle\times\mathfrak{Re}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}(\not{p}_{e}-m_{e})\left\\{\frac{p^{\alpha}}{p\cdot
k}\not{k}-\gamma^{\alpha}\right\\}(1-\gamma_{5})\right]$
$\displaystyle+\frac{e^{2}G_{F}^{2}}{\sqrt{2}}V_{us}F_{\mu}\frac{1}{p_{e}\cdot
k}\mathfrak{Re}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}\not{k}\gamma^{\alpha}(\not{p}_{e}-m_{e})\left\\{\frac{p_{\alpha}}{p\cdot
k}\not{k}-\gamma_{\alpha}\right\\}(1-\gamma_{5})\right]$
$\displaystyle|M_{B}|^{2}$ $\displaystyle=$
$\displaystyle-\frac{e^{2}G_{F}^{2}}{2}|V_{us}|^{2}\mathrm{Tr}\left[(\not{P}-\not{k})\left\\{\frac{p^{\mu}}{p\cdot
k}\not{k}-\gamma^{\mu}\right\\}(\not{p}_{e}-m_{e})\left\\{\frac{p_{\mu}}{p\cdot
k}\not{k}-\gamma_{\mu}\right\\}(1-\gamma_{5})\right]\leavevmode\nobreak\ .$

In the above, we have used $F_{\mu}$ as a shorthand of
$F_{\mu}^{K\pi}(p^{\prime},p)$. We do not display the explicit results after
taking the spinor trace, as the latter can be done with, e.g., various
packages in Mathematica such as Tracer or Package-X. After taking the trace,
all the expressions above are functions of $\\{x,y,z\\}$ as well as two of the
three following dot products involving $k$: $\\{k\cdot p,k\cdot
p^{\prime},k\cdot p_{e}\\}$ using the identity
$2k\cdot(p-p^{\prime}-p_{e})=M_{K}^{2}x$.

The integration can be performed with the following strategy. Take
$|M_{A}|^{2}_{\mathrm{res}}$ in $K_{e3}^{0}$ as an example: we first express
the squared amplitude as a finite sum:

$|M_{A}|_{\mathrm{res}}^{2}=\sum_{m,n}c_{m,n}(x,y,z)\frac{1}{(k\cdot
p^{\prime})^{m}(k\cdot p_{e})^{n}}\leavevmode\nobreak\ ,$ (127)

where $-2\leq m,n\leq 2$ and $c_{m,n}(x,y,z)$ are known scalar coefficients.
The $p_{\nu}$ and $k$-integrations return the following functions:

$I_{m,n}(p_{1},p_{2})\equiv\frac{1}{2\pi}\int\frac{d^{3}k}{E_{k}}\frac{d^{3}p_{\nu}}{E_{\nu}}\frac{\delta^{(4)}(P-k-
p_{\nu})}{(p_{1}\cdot k)^{m}(p_{2}\cdot k)^{n}}\leavevmode\nobreak\ ,$ (128)

of which analytic expressions are given in the Appendix of Ref.Ginsberg:1969jh
(we have checked their correctness). With this, we obtain:

$\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})|M_{A}|^{2}_{\mathrm{res}}=\frac{1}{8\pi}\sum_{m,n}c_{m,n}(x,y,z)I_{m,n}(p^{\prime},p_{e})\leavevmode\nobreak\
,$ (129)

where the right-hand side is now a function of $\\{x,y,z\\}$, so the remaining
three-fold integration with respect to these variables are completely regular
and can be performed numerically. The same strategy applies to the IR-finite
integrals in $K_{e3}^{+}$, except that one should choose $1/\left\\{(k\cdot
p)^{m}(k\cdot p_{e})^{n}\right\\}$ as the basis.

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