# Electron Mass Singularities in Semileptonic Kaon Decays

Chien-Yeah Seng1 William J. Marciano2 Ulf-G. Meißner1,3,4 1Helmholtz-Institut
für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,
Universität Bonn, 53115 Bonn, Germany  2Department of Physics, Brookhaven
National Laboratory, Upton, New York 11973, USA  3Institute for Advanced
Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics,
Forschungszentrum Jülich, 52425 Jülich, Germany  4Tbilisi State University,
0186 Tbilisi, Georgia

(August 27, 2024)

###### Abstract

We show that recent improvements in the $O(\alpha)$ long-distance quantum
electrodynamics (QED) corrections to the radiative inclusive $K_{e3}$ decay
rate using the Sirlin representation are free from infrared divergences and
collinear electron mass singularities in the limit $m_{e}\rightarrow 0$, as
predicted by the Kinoshita-Lee-Nauenberg theorem. We also verify that in
massless QED with the simultaneous dimensional regularization of QED photon
infrared divergences and electron mass singularities leads to the same result
for the inclusive rate in the limit of four space-time dimensions. The
equivalence of the two approaches results in part from an interesting
interplay between a small chirality-breaking effect in the massless electron
limit and the generalization of space-time algebra and phase space integrals
to $d>4$ dimensions. Our finding supports the small theoretical uncertainty
claimed for $K_{e3}$ radiative inclusive rates and reaffirms its utility in
precision unitarity tests of the quark mixing matrix.

##  I Introduction

Radiative inclusive semileptonic kaon decays into electrons or muons along
with soft or hard bremsstrahlung ($K\rightarrow\pi\ell\nu(\gamma)$, denoted as
$K_{\ell 3}$) provide one of the most precise determinations of the first-row
Cabibbo-Kobayashi-Maskawa Cabibbo (1963); Kobayashi and Maskawa (1973) quark
mixing matrix element $|V_{us}|$. The existence of 6 distinct charged and
neutral such $K$ decay modes allows consistency checks and averaging that
currently result in a $\pm 0.2\%$ determination of $|V_{us}|$. That quantity
can be identified with sine of the Cabibbo angle in the effective (two
generations) four quark limit. It plays an essential role in precision
unitarity tests of the Standard Model (SM).

In recent years, a significant roughly 1% difference has been observed between
the values of $|V_{us}|$ obtained from $K_{\ell 3}$ in comparison with the
radiative inclusive leptonic kaon decay mode ($K\rightarrow\mu\nu(\gamma)$,
denoted as $K_{\mu 2}$) Seng et al. (2022a):

$|V_{us}|=\left\\{\begin{array}[]{ccc}0.22308(55)&,&K_{\ell 3}\\\
0.22520(50)&,&K_{\mu 2}\end{array}\right.\leavevmode\nobreak\ .$ (1)

The discrepancy is about 2.9$\sigma$. Similarly, improvements in the
calculation of the inner radiative corrections (RC) to free neutron and
nuclear beta decays Seng et al. (2018, 2019); Czarnecki et al. (2019);
Gorchtein (2019); Seng et al. (2020a); Shiells et al. (2021); Hayen (2021)
reduce the central value of $|V_{ud}|$ Hardy and Towner (2020) and its
theoretical uncertainty, resulting in an apparent violation of the first-row
CKM unitarity relation $|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1$ (with
negligible $|V_{ub}|^{2}$) up to about 3.2$\sigma$ Seng et al. (2022b). These
various inconsistencies between different precision experimental extractions
of the Cabibbo angle $\theta_{C}=\sin^{-1}|V_{us}|$ and the apparent violation
of first row CKM unitarity are commonly referred to as the “Cabibbo angle
anomaly”. Its possible implications for “new physics” beyond the Standard
Model (BSM) expectations have been extensively discussed in the literature
Gonzalez-Alonso et al. (2019); Bryman and Shrock (2019a, b); Belfatto et al.
(2020); Tan (2019); Grossman et al. (2020); Coutinho et al. (2020); Falkowski
et al. (2019); Cirigliano et al. (2019); Jho et al. (2020); Yue and Cheng
(2021); Endo and Mishima (2020); Capdevila et al. (2021); Eberhardt et al.
(2021); Cheung et al. (2020); Crivellin et al. (2020); Crivellin et al.
(2021a); Crivellin and Hoferichter (2020); Kirk (2021); Crivellin et al.
(2021b); Falkowski et al. (2021); Bečirević et al. (2021); Crivellin et al.
(2021c).

To confirm or negate the current disagreements between experiments and SM
theory predictions, highly precise theory inputs are needed. In $K_{\ell 3}$
decays, they include the $K^{0}\pi^{-}$ transition form factor
$f_{+}^{K^{0}\pi^{-}}(0)$ at zero momentum transfer Lubicz et al. (2009);
Bazavov et al. (2013); Boyle et al. (2015); Carrasco et al. (2016); Bazavov et
al. (2019) (determined to about 0.2% using lattice QCD and currently
representing the dominant theory uncertainty), the phase space factors Lichard
(1997); Antonelli et al. (2010); Hill (2006); Bernard et al. (2006, 2009);
Abouzaid et al. (2010); Moulson (2021), quark mass parameters used to estimate
the isospin-breaking correction for $f_{+}^{K^{+}\pi^{0}}$ (mainly due to
$\pi^{0}\eta$-mixing) Blum et al. (2016); Durr et al. (2011a, b); Bazavov et
al. (2009); Fodor et al. (2016); Bazavov et al. (2018); Carrasco et al.
(2014); Bazavov et al. (2014); Giusti et al. (2017); Colangelo et al. (2018),
and the long-distance electromagnetic radiative corrections (EMRC). In
particular, a recent re-analysis of the $K_{\ell 3}$ EMRC Seng et al. (2021a,
b, 2022a) based on a new theory framework Seng et al. (2020b, c) that
hybridizes the classic Sirlin representation Sirlin (1978); Seng (2021), the
modern language of Chiral Perturbation Theory (ChPT) and the latest lattice
calculations of mesonic $\gamma W$-box diagrams Feng et al. (2020); Ma et al.
(2021) achieves a precision level of $O(10^{-4})$, an order of magnitude
better than the existing pure ChPT analyses Cirigliano et al. (2002, 2004,
2008). These new results sharpen the $K_{\ell 3}$–$K_{\mu 2}$ discrepancy in
the $|V_{us}|$ extractions.

At the $O(10^{-4})$ level, every approximation and uncertainty estimate made
in the theory analysis must be thoroughly scrutinized. In particular, the
procedure of separating the EMRC into perturbative and non-perturbative pieces
must be compatible with general properties of quantum field theory. Measurable
(infrared safe) quantities must be properly regularized such that QED photonic
“infrared (IR)-divergences” due to (virtual and real) emission of low-energy
photons cancel. For decays such as $K_{e3}$ with a light electron in the final
state, “mass singularities” (i.e. terms enhanced by $\ln m_{e}$) due to the
emission of a photon collinear to the positron (or electron) are generally
present in the differential decay rate. However, such terms which are
divergent in the massless limit ($m_{e}=0$) cancel in the total integrated
radiative inclusive decay rate. Since the experimental values of the $K_{e3}$
partial widths are derived from kaon lifetime measurements which are by
definition radiative inclusive and experimental branching ratios, which are
assumed to be fully radiative inclusive, theory calculations used for
comparison must, therefore, also be radiative inclusive.

The cancellation of infrared photonic divergences between real and virtual
radiative corrections was realized as early as the 1930s by Bloch and
Nordsieck Bloch and Nordsieck (1937), and later generalized by Yennie,
Frautschi and Suura Yennie et al. (1961). The cancellation of electron mass
singularities in radiative inclusive decays was first observed by Kinoshita
and Sirlin in their calculation of QED corrections to the muon total decay
rate in the Fermi V-A theory Kinoshita and Sirlin (1959). Later generalization
of that feature is known as the Kinoshita-Lee-Nauenberg (KLN) theorem
Kinoshita (1962); Lee and Nauenberg (1964). The theorem is valid to all orders
in perturbation theory for fully inclusive rate expansions in terms of bare
couplings and masses. However, renormalization can induce mass singularities.
For example, electric charge renormalization at zero momentum transfer used to
define $\alpha$, the fine structure constant, if used as the expansion
parameter will induce two loop $\ln m_{e}$ dependent corrections. Based on
that connection, Roos and Sirlin derived the leading $\ln m_{e}$ logarithmic
QED radiative corrections to the muon lifetime that start to appear at two
loop order Roos and Sirlin (1971); Marciano (1975).

In Refs. Seng et al. (2021a, b, 2022a), the cancellation of the photonic IR
divergences is explicitly demonstrated for $K_{e3}$ by computing the so-called
“convection term” contribution Yennie et al. (1961); Meister and Yennie (1963)
to the virtual and real corrections analytically. In Sirlin’s representation,
that term represents the complete IR divergence, but not the full electron
mass singularities contribution. The missing part was computed numerically
using the physical electron mass as input. Therefore, the expected
cancellation of electron mass singularities in the total numerical result for
the inclusive decay rate was not directly tested. In principal, some $\ln
m_{e}$-enhanced contributions could have been missed or lost in the
approximations made. If so, it could invalidate the final error analysis and
might shift the theoretical result somewhat. Indeed, the explicit cancellation
of mass singularities in radiative inclusive processes is a useful tool for
checking difficult calculations.

The first part of this paper serves to fill in this missing check. By
extending the analytically-calculable terms in both virtual and real
corrections and expanding them with respect to $m_{e}$ (which we will call the
“mass-expansion” method), we demonstrate explicitly that the approximations
made in Refs. Seng et al. (2021a, b, 2022a) within Sirlin’s representation
indeed result in a singularity-free total $K_{e3}$ decay rate. A similar study
was done previously in Ref. Bytev et al. (2003) but only in the $K_{e3}^{+}$
channel. Our analysis constitutes a more general proof. Since cancellation is
achieved in the total decay rate, this calculation provides a nice check not
just to the validity of the approximations made on the interaction dynamics at
the level of the squared amplitude, but also the correctness of the kinematic
settings of the three- and four-body phase space integrations.

The second part of this paper is more theoretically oriented. It examines the
possibility of employing $M_{\gamma}=m_{e}=0$, i.e. massless QED, from the
start of a radiative inclusive calculation. Because the kinematics are simpler
than the massive electron calculation, it represents a simplified method of
checking the $K_{e3}$ decay rate in the limit of zero electron mass.

For such studies, a regulator that preserves the properties of QED must be
used. A powerful technique for this purpose is the continuous dimensional
regularization (DR) approach ’t Hooft and Veltman (1972); Bollini and
Giambiagi (1972); Ashmore (1972); Gastmans and Meuldermans (1973); Marciano
and Sirlin (1975); Marques and Papanicolaou (1975); Marciano et al. (1975);
Marciano (1975) which generalizes the space-time dimension to $d=4-\epsilon$.
IR and mass singularities manifest themselves as poles of order
$(2/\epsilon)^{2}$ and $(2/\epsilon)$. Note that $\epsilon$ is negative for
${d>4}$, the space-time domain where infrared effects are finite. This method
is used extensively in the study of perturbative Quantum Chromodynamics (pQCD)
with massless quarks (see, e.g., Collins (2013); Becher et al. (2015) and
references therein). An interesting comparison is the relationship between the
prediction of singularity-free quantities for the $\epsilon\rightarrow 0$
limit in the DR method and the $m_{e}\rightarrow 0$ limit in the mass-
expansion method. Ref. Marciano (1975) demonstrated the equivalence between
these two methods in the total radiatively inclusive $W\rightarrow
e\nu(\gamma)$ decay rate, but similar comparisons in processes with more
complicated kinematics like $K_{e3}$ have, to the best of our knowledge, not
been studied before. They are of interest both in terms of testing the
robustness of the regularization techniques and achieving a better
understanding of the general properties of the underlying quantum field
theory.

In this paper we demonstrate that the mass-expansion and DR methods both
satisfy the KLN theorem and give the same result for the total radiatively
inclusive $K_{e3}$ decay rate in the massless electron limit. That finding is
highly non-trivial, particularly for the final finite decay rate prediction.
Two essential steps are needed: In the case of DR, a generalization of the
three- and four-body phase space integrals along with algebraic manipulations
lead to an ${\epsilon}$ dependence which produces a finite contribution in the
limit ${\epsilon=0}$. In the mass expansion approach, terms in the
bremsstrahlung process that are proportional to $(p_{e}\cdot k)^{-2}$, with
$k$ the photon momentum must be kept even though they are suppressed by
$m_{e}^{2}$ in the squared amplitude. They acquire a $1/m_{e}^{2}$-enhancement
upon performing the phase space integration which lifts the suppression and
results in a small finite contribution in the $m_{e}\rightarrow 0$ limit. When
all such effects are properly taken into account, the finite parts in the two
methods are in perfect agreement.

The contents of this paper are arranged as follows. In Sec. II and III we
briefly review the kinematics of $K_{e3}$ and Sirlin’s representation of the
virtual EMRC. In Sec. IV we study the integrals with IR divergences and mass
singularities in both the virtual corrections and bremsstrahlung process using
the mass-expansion method, and prove the exact cancellation of both in the
total decay rate. Next, in Sec. V we discuss the aforementioned contribution
from the bremsstrahlung which remains finite in the $m_{e}\rightarrow 0$ limit
but is not explicitly present in massless QED; this subtle chirality-breaking
effect is crucial in reconciling the mass-expansion method and DR. In Sec. VI
we repeat a similar analysis using DR and demonstrate the equivalence between
the two methods. Final discussions are given in Sec.VII.

##  II Brief review of the $\boldsymbol{K_{e3}}$ kinematics

Despite being available in many papers (e.g. Appendix A in Ref. Seng et al.
(2021b)), we still start with a brief review of the essential kinematics of
the $K_{e3}$ decay to keep the discussion self-contained. We are interested in
the following inclusive decay process:
$K(p)\rightarrow\pi(p^{\prime})+e^{+}(p_{e})+\nu_{e}(p_{\nu})+n\gamma$ ($n\geq
0$) with realistic physical masses, i.e. $m_{e}\neq 0$. If all the massless
particles in the final state are left unobserved, then the decay process is
fully described by three scalar kinematic variables $\\{x,y,z\\}$ defined as:

$P^{2}\equiv(p-p^{\prime}-p_{e})^{2}=M_{K}^{2}x\leavevmode\nobreak\
,\leavevmode\nobreak\ p\cdot p_{e}=\frac{1}{2}M_{K}^{2}y\leavevmode\nobreak\
,\leavevmode\nobreak\ p\cdot
p^{\prime}=\frac{1}{2}M_{K}^{2}z\leavevmode\nobreak\ ,$ (2)

notice that when $n=0$ we must have $x=0$ given that $P^{2}=p_{\nu}^{2}=0$
assuming massless neutrinos. We also define $r_{\pi}\equiv
M_{\pi}^{2}/M_{K}^{2}$ and $r_{e}\equiv m_{e}^{2}/M_{K}^{2}$ for notational
simplicity. The $K$–$\pi$ squared momentum transfer is given by
$t=(p-p^{\prime})^{2}=M_{K}^{2}(1-z+r_{\pi})$. Finally, in the $K^{0}$
($K^{+}$) decay channel it is also customary to define the respective
Mandelstam variables $s=(p^{\prime}+p_{e})^{2}$ and $u=(p-p_{e})^{2}$.

Up to $\mathcal{O}(G_{F}^{2}\alpha)$ in the decay rate, only the $n=0$ (three-
body) and $n=1$ (four-body) decay processes need to be included. The
corresponding decay rate formula are:

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

and

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

respectively, where:

$\alpha_{\pm}(y,z)\equiv
1-y-z+r_{\pi}+r_{e}+\frac{yz}{2}\pm\frac{1}{2}\sqrt{y^{2}-4r_{e}}\sqrt{z^{2}-4r_{\pi}}\leavevmode\nobreak\
,$ (5)

and we have also defined the following shorthand for the integral measure:

$d\Gamma_{k}\equiv\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\leavevmode\nobreak\ .$ (6)

The regions $\mathcal{D}_{3}$ and $\mathcal{D}_{4-3}$ are defined as:

$\displaystyle\mathcal{D}_{3}$ $\displaystyle:$ $\displaystyle
c(z)-d(z)<y<c(z)+d(z)\leavevmode\nobreak\
,\>\>2\sqrt{r_{\pi}}<z<1+r_{\pi}-r_{e}$ $\displaystyle\mathcal{D}_{4-3}$
$\displaystyle:$ $\displaystyle 2\sqrt{r_{e}}<y<c(z)-d(z)\leavevmode\nobreak\
,\>\>2\sqrt{r_{\pi}}<z<1-\sqrt{r_{e}}+\frac{r_{\pi}}{1-\sqrt{r_{e}}}\leavevmode\nobreak\
,$ (7)

where

$c(z)=\frac{(2-z)(1+r_{e}+r_{\pi}-z)}{2(1+r_{\pi}-z)}\leavevmode\nobreak\
,\leavevmode\nobreak\
d(z)=\frac{\sqrt{z^{2}-4r_{\pi}}(1+r_{\pi}-r_{e}-z)}{2(1+r_{\pi}-z)}\leavevmode\nobreak\
.$ (8)

The tree-level $n=0$ decay amplitude can be written as:

$M_{0}=-\frac{G_{F}}{\sqrt{2}}L_{\lambda}F^{\lambda}(p^{\prime},p)\leavevmode\nobreak\
,$ (9)

where $L_{\lambda}=\bar{u}_{\nu}\gamma_{\lambda}(1-\gamma_{5})v_{e}$ is the
lepton current, and

$F^{\lambda}(p^{\prime},p)=\langle\pi(p^{\prime})|J_{W}^{\lambda\dagger}(0)|K(p)\rangle=V_{us}^{*}\left[f_{+}^{K\pi}(t)(p+p^{\prime})^{\lambda}+f_{-}^{K\pi}(t)(p-p^{\prime})^{\lambda}\right]\leavevmode\nobreak\
$ (10)

with $J_{W}^{\lambda}$ the charged weak current, defines the two form factors
$f_{\pm}^{K\pi}(t)$. For simplicity we will omit the superscript $K\pi$,
knowing that they refer to $K^{0}\pi^{-}$ ($K^{+}\pi^{0}$) in the $K^{0}$
($K^{+}$)-channel. The squared amplitude (summing over lepton spins) is given
by:

$|M_{0}|^{2}(x,y,z)\equiv
G_{F}^{2}F_{\mu}(p^{\prime},p)F_{\nu}^{*}(p^{\prime},p)\mathrm{Tr}[\not{P}\gamma^{\mu}(\not{p}_{e}-m_{e})\gamma^{\nu}(1-\gamma_{5})]\leavevmode\nobreak\
.$ (11)

Here, we purposely retain the $x$-dependence so that the same structure can be
reused in the bremsstrahlung process. By evaluating the trace, we find that
terms involving $f_{-}(t)$ are suppressed by $r_{e}$, which makes their
contribution to the decay rate negligible. Given that the virtual corrections
can always be expressed as corrections to the form factors:

$f_{\pm}(t)\rightarrow f_{\pm}(t)+\delta f_{\pm}(y,z)\leavevmode\nobreak\ ,$
(12)

it is therefore only $\delta f_{+}(y,z)$ which is of relevance in practise.

##  III Sirlin’s representation of the long-distance EMRC

Figure 1: One-loop EMRC in Sirlin’s representation. $\gamma_{<}$ denotes a
photon which propagator is attached to a Pauli-Viilars-regulator
$M_{W}^{2}/(M_{W}^{2}-q^{\prime 2})$.

In Sirlin’s representation (Ref. Sirlin (1978), see also Refs. Seng et al.
(2020b, c); Seng (2021) for comprehensive reviews), the full
$\mathcal{O}(\alpha)$ virtual electroweak RC splits into two parts. First is
the “weak” RC that depend only on physics at the scale $M_{W}$; they come from
one-loop diagrams that involve only heavy gauge bosons, or with photon but
only pick up the first term in the following splitting of the photon
propagator:

$\frac{-ig_{\mu\nu}}{q^{\prime
2}-M_{\gamma}^{2}}=\frac{-ig_{\mu\nu}}{q^{\prime
2}-M_{W}^{2}}+\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{-ig_{\mu\nu}}{q^{\prime
2}-M_{\gamma}^{2}}+\mathcal{O}(M_{\gamma}^{2})\leavevmode\nobreak\ .$ (13)

The weak RC are either reabsorbed into the definition of the Fermi constant
$G_{F}$ or provides a regular correction that is proportional to the tree-
level amplitude which is irrelevant for our discussion. What remains are the
long-distance EMRC represented by the three Feynman diagrams in Fig.1:

  1. 1.

Contribution from the electron wavefunction renormalization: $\delta
M=(1/2)\delta Z_{e}M_{0}$,

  2. 2.

The long-distance EMRC to the $K\pi$ form factor, which can be divided into
“two-point function” and “three-point function”: $\delta M=\delta M_{2}+\delta
M_{3}$,

  3. 3.

The $\gamma W$-box diagram, which splits into the piece $\delta M_{\gamma
W}=\delta M_{\gamma W}^{a}+\delta M_{\gamma W}^{b}$ without (a) and with (b) a
totally-antisymmetric tensor.

Furthermore, using Ward identities, one may combine $\delta M_{2}$ and $\delta
M_{\gamma W}^{a}$ to get an “analytic” piece and a “residual integral” piece:

$\delta M_{2}+\delta M_{\gamma W}^{a}=(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{ana}}+(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{int}}\leavevmode\nobreak\ .$ (14)

Throughout this work we ignore small corrections from perturbative QCD that
are relevant to the actual numerical calculation at the level of $10^{-4}$
Sirlin (1978) but do not affect the structure of the infrared and mass
singularities. This gives:

$\displaystyle(\delta M_{2}+\delta M_{\gamma W}^{a})_{\mathrm{ana}}$
$\displaystyle=$
$\displaystyle-\left(\frac{\alpha}{8\pi}+2ie^{2}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{[(p_{e}-q^{\prime})^{2}-m_{e}^{2}]q^{\prime 2}}\right)M_{0}$
$\displaystyle(\delta M_{2}+\delta M_{\gamma W}^{a})_{\mathrm{int}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}e^{2}}{\sqrt{2}}L_{\lambda}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{1}{(p_{e}-q^{\prime})^{2}-m_{e}^{2}}\biggl{\\{}\frac{2p_{e}\cdot
q^{\prime}q^{\prime\lambda}}{(q^{\prime
2}-M_{\gamma}^{2})^{2}}T^{\mu}_{\>\>\mu}$ (15)
$\displaystyle+\frac{2p_{e\mu}}{q^{\prime
2}-M_{\gamma}^{2}}T^{\mu\lambda}-\frac{(p-p^{\prime})_{\mu}}{q^{\prime
2}-M_{\gamma}^{2}}T^{\lambda\mu}+\frac{i}{q^{\prime
2}-M_{\gamma}^{2}}\Gamma^{\lambda}\biggr{\\}}\leavevmode\nobreak\ .$

The antisymmetric piece of the $\gamma W$-box diagram reads:

$\delta M_{\gamma
W}^{b}=-i\frac{G_{F}e^{2}}{\sqrt{2}}L_{\lambda}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{1}{(p_{e}-q^{\prime})^{2}-m_{e}^{2}}\frac{1}{q^{\prime
2}}\epsilon^{\mu\nu\alpha\lambda}q_{\alpha}^{\prime}T_{\mu\nu}\leavevmode\nobreak\
.$ (16)

The non-trivial integrals above are expressed in terms of the following
quantities:

$\displaystyle T^{\mu\nu}(q^{\prime};p^{\prime},p)$ $\displaystyle\equiv$
$\displaystyle\int d^{4}xe^{iq^{\prime}\cdot
x}\langle\pi(p^{\prime})|T\\{J_{\mathrm{em}}^{\mu}(x)J_{W}^{\nu\dagger}(0)\\}|K(p)\rangle\leavevmode\nobreak\
,$ $\displaystyle\Gamma^{\mu}(q^{\prime};p^{\prime},p)$ $\displaystyle\equiv$
$\displaystyle\int d^{4}xe^{iq^{\prime}\cdot
x}\langle\pi(p^{\prime})|T\\{J_{\mathrm{em}}^{\mu}(x)\partial\cdot
J_{W}^{\dagger}(0)\\}|K(p)\rangle\leavevmode\nobreak\ .$ (17)

##  IV Cancellation of IR and mass singularities

To check the cancellation of IR and mass singularities we only need the
divergent pieces from the virtual and real corrections; but to prove the
equivalence between the mass-expansion method and DR we need the finite pieces
as well. Fortunately, not all the finite pieces are relevant: there are parts
of the loop/bremsstrahlung integrals that are intrinsically finite even at
$M_{\gamma}=m_{e}=0$, so these parts must be the same in both methods and can
be discarded in our discussion. What we are interested in are the
loop/bremsstrahlung integrals with IR and mass singularities, which we always
split into divergent and finite pieces:

$F=F^{\mathrm{div,m}}+F^{\mathrm{fin}}+\mathcal{O}(M_{\gamma}^{2},m_{e}^{2})\leavevmode\nobreak\
,$ (18)

where $F^{\mathrm{div,m}}$ (“m” stands for “mass-expansion”) contains only the
terms proportional to $\ln(m_{e}^{2}/\mu^{2})$ or
$\ln(M_{\gamma}^{2}/\mu^{2})$ (or both), with $\mu$ an arbitrarily chosen
scale, while $F^{\mathrm{fin}}$ represents all the remaining finite terms
independent of $M_{\gamma}$ and $m_{e}$. The sum of the two pieces is of
course $\mu$-independent. Analytic formulas for these integrals are in given
in Ref. Seng et al. (2021b) at non-zero $m_{e}$; here, we just need to expand
those expression with respect to $m_{e}$. Notice that this procedure
implicitly assumes the hierarchy $m_{e}\gg M_{\gamma}$ in the mass-expansion
method.

Throughout this study, we work in the Feynman gauge. We start from the
electron wavefunction renormalization $\delta Z_{e}=\delta
Z_{e}^{\mathrm{div,m}}+\delta Z_{e}^{\mathrm{fin}}$, where

$\delta
Z_{e}^{\mathrm{div,m}}=-\frac{\alpha}{4\pi}\left[-3\ln\frac{m_{e}^{2}}{\mu^{2}}+2\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\right]\leavevmode\nobreak\
,\leavevmode\nobreak\ \delta
Z_{e}^{\mathrm{fin}}=-\frac{\alpha}{4\pi}\left[\ln\frac{M_{W}^{2}}{\mu^{2}}+\frac{9}{2}\right]\leavevmode\nobreak\
.$ (19)

Next, the “analytic” piece in $\delta M_{2}+\delta M_{\gamma W}^{a}$ gives:

$(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{ana}}^{\mathrm{div,m}}=-\frac{\alpha}{2\pi}\ln\frac{m_{e}^{2}}{\mu^{2}}M_{0}\leavevmode\nobreak\
,\leavevmode\nobreak\ (\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{ana}}^{\mathrm{fin}}=\frac{\alpha}{2\pi}\left(\ln\frac{M_{W}^{2}}{\mu^{2}}+\frac{3}{4}\right)M_{0}\leavevmode\nobreak\
.$ (20)

These are all exactly-known contributions independent of hadronic structure.

###  IV.1 Born contribution to the remaining loop integrals

Evaluating the remaining loop integrals $(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{int}}$ and $\delta M_{\gamma W}^{b}$ in the virtual
correction requires the knowledge of the hadronic tensor $T^{\mu\nu}$ and the
vertex function $\Gamma^{\mu}$. A particularly important set of constraints
comes from the EM and charged weak Ward identities:

EM $\displaystyle:$ $\displaystyle
q_{\mu}^{\prime}T^{\mu\nu}(q^{\prime};p^{\prime},p)=-iF^{\nu}(p^{\prime},p)\leavevmode\nobreak\
,$ Charged weak $\displaystyle:$ $\displaystyle
q_{\nu}T^{\mu\nu}(q^{\prime};p^{\prime},p)=-iF^{\mu}(p^{\prime},p)-i\Gamma^{\mu}(q^{\prime};p^{\prime},p)\leavevmode\nobreak\
,$ (21)

where $q=p^{\prime}-p+q^{\prime}$. We have used both in the derivation of
Eqs.(15), (16).

In Refs. Seng et al. (2021a, b, 2022a), the so-called “convection term
contribution” Meister and Yennie (1963) was analytically calculated. It
involves the following substitution for the hadronic tensor and the vertex
function:

$T_{K^{0}\pi^{-}}^{\mu\nu,\mathrm{conv}}=-\frac{i(2p^{\prime}+q^{\prime})^{\mu}F^{\nu}(p^{\prime},p)}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\
T_{K^{+}\pi^{0}}^{\mu\nu,\mathrm{conv}}=\frac{i(2p-q^{\prime})^{\mu}F^{\nu}(p^{\prime},p)}{(p-q^{\prime})^{2}-M_{K}^{2}}\leavevmode\nobreak\
,$ (22)

and:

$\Gamma^{\mu,\mathrm{conv}}_{K^{0}\pi^{-}}=\frac{(2p^{\prime}+q^{\prime})^{\mu}(p^{\prime}-p)_{\lambda}F^{\lambda}(p^{\prime},p)}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\
\Gamma^{\mu,\mathrm{conv}}_{K^{+}\pi^{0}}=-\frac{(2p-q^{\prime})^{\mu}(p^{\prime}-p)_{\lambda}F^{\lambda}(p^{\prime},p)}{(p-q^{\prime})^{2}-M_{K}^{2}}\leavevmode\nobreak\
.$ (23)

Those relationships represent the simplest structures that satisfy the exact
EM Ward identity and are able to reproduce the full IR divergence structure.
However, they fail to satisfy the charged weak Ward identity even in the
SU(3)f limit, and therefore do not reproduce the full $\ln m_{e}$ structure.

Fortunately, in the actual non-perturbative numerical calculation of those
works, what enters is not just the convection term, but the full “Born”
contribution which includes more complete pole and seagull structures. Its
original implementation in the numerical calculation can be found in
Eqs.(4.3)-(4.5) of Ref. Seng et al. (2021b), but there the $K\pi$ form factors
depend on the loop momentum $q^{\prime}$, which complicates its analytic
study. In this paper we make a further simplification, namely to discard the
$q^{\prime}$-dependence in the electromagnetic and the $f_{\pm}$ form factors,
which is again relevant to the actual numerical calculation but do not affect
the divergent structure:

$\displaystyle T_{K^{0}\pi^{-}}^{\mu\nu,\mathrm{B}}$ $\displaystyle=$
$\displaystyle-
iV_{us}^{*}\left[\frac{(2p^{\prime}+q^{\prime})^{\mu}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}\left(f_{+}(t)(2p+q)^{\nu}-f_{-}(t)q^{\nu}\right)-g^{\mu\nu}\left(f_{+}(t)-f_{-}(t)\right)\right],$
$\displaystyle T_{K^{+}\pi^{0}}^{\mu\nu,\mathrm{B}}$ $\displaystyle=$
$\displaystyle
iV_{us}^{*}\left[\frac{(2p-q^{\prime})^{\mu}}{(p-q^{\prime})^{2}-M_{K}^{2}}\left(f_{+}(t)(2p^{\prime}-q)^{\nu}-f_{-}(t)q^{\nu}\right)-g^{\mu\nu}\left(f_{+}(t)+f_{-}(t)\right)\right],$
$\displaystyle\Gamma^{\mu,\mathrm{B}}_{K^{0}\pi^{-}}$ $\displaystyle=$
$\displaystyle-
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}(2p^{\prime}+q^{\prime})^{\mu}\left(f_{+}(t)+\frac{q^{2}}{M_{K}^{2}-M_{\pi}^{2}}f_{-}(t)\right),$
$\displaystyle\Gamma^{\mu,\mathrm{B}}_{K^{+}\pi^{0}}$ $\displaystyle=$
$\displaystyle
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p-q^{\prime})^{2}-M_{K}^{2}}(2p-q^{\prime})^{\mu}\left(f_{+}(t)+\frac{q^{2}}{M_{K}^{2}-M_{\pi}^{2}}f_{-}(t)\right)\leavevmode\nobreak\
.$ (24)

They satisfy the following relations:

$\displaystyle
q_{\mu}^{\prime}T^{\mu\nu,\mathrm{B}}(q^{\prime};p^{\prime},p)+iF^{\nu}(p^{\prime},p)$
$\displaystyle=$ $\displaystyle 0\leavevmode\nobreak\ ,$ $\displaystyle
q_{\nu}T^{\mu\nu,\mathrm{B}}(q^{\prime};p^{\prime},p)+iF^{\mu}(p^{\prime},p)+i\Gamma^{\mu,\mathrm{B}}(q^{\prime};p^{\prime},p)$
$\displaystyle=$ $\displaystyle
iV_{us}^{*}f_{-}(t)\left(2(p-p^{\prime})^{\mu}-q^{\prime\mu}\right)\leavevmode\nobreak\
,$ (25)

so the EM Ward identity is obeyed. Meanwhile, the charged weak Ward identity
is not exactly satisfied. There is an extra pole-free term on the right hand
side of the second line. Fortunately, this term has only
$(p-p^{\prime})^{\mu}$ and $q^{\prime\mu}$ structures, which means: upon
plugging into the loop integrals, it can only give rise to $\delta f_{-}$ and
not $\delta f_{+}$. So, as far as its contribution to $\delta f_{+}$ is
concerned, the Born term satisfies both Ward identities. Therefore, it is able
to describe the complete mass singularities in the virtual correction within
the framework of Sirlin’s representation.

Now, we can discuss the divergence structures in the remaining integrals.
First, as explained in Ref. Seng et al. (2021b), the IR divergences from
$\delta M_{2}$ (contained in the first term in $(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{int}}$) combined with $\delta M_{3}$ gives:

$(\delta
f_{+})_{\mathrm{2pt+3pt}}^{\mathrm{div,m}}=-\frac{\alpha}{4\pi}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}f_{+}\leavevmode\nobreak\
.$ (26)

Its finite contribution is irrelevant for our discussion and need not be
retained. Next, the second term in $(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{int}}$ contains both IR and mass singularities, which comes
from the following scalar function 111Notice that the definition of
$C_{0}^{\mathrm{fin}}$ here and $I_{i}^{\mathrm{fin}}$ in Eq.(33) are
different from those in Ref. Seng et al. (2021b), because the full results
there were not expanded with respect to $m_{e}$.:

$C_{0}\equiv\frac{16\pi^{2}}{i}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{[(p_{1}-k)^{2}-M_{1}^{2}][(p_{e}-k)^{2}-m_{e}^{2}][k^{2}-M_{\gamma}^{2}]}=C_{0}^{\mathrm{div,m}}+C_{0}^{\mathrm{fin}}$
(27)

where

$\displaystyle C_{0}^{\mathrm{div,m}}$ $\displaystyle=$
$\displaystyle\frac{1}{M_{1}^{2}-v}\left\\{-\frac{1}{2}\ln\frac{m_{e}^{2}}{\mu^{2}}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}+\frac{1}{4}\ln^{2}\frac{m_{e}^{2}}{\mu^{2}}+\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\left[\frac{1}{2}\ln\frac{M_{1}^{2}}{\mu^{2}}-\ln\frac{M_{1}^{2}}{M_{1}^{2}-v}\right]\right\\}$
$\displaystyle C_{0}^{\mathrm{fin}}$ $\displaystyle=$
$\displaystyle\frac{1}{M_{1}^{2}-v}\left\\{-\frac{1}{4}\ln^{2}\frac{M_{1}^{2}}{\mu^{2}}+\ln\frac{M_{1}^{2}}{\mu^{2}}\ln\frac{M_{1}^{2}}{M_{1}^{2}-v}-\frac{1}{2}\ln^{2}\frac{M_{1}^{2}}{M_{1}^{2}-v}+\mathrm{Li}_{2}\left(\frac{v}{v-M_{1}^{2}}\right)\right\\}\leavevmode\nobreak\
.$

Here we should take $M_{1}=M_{\pi}$, $v=s$ in the $K^{0}$-decay, and
$M_{1}=M_{K}$, $v=u$ in the $K^{+}$-decay. Finally, the mass singularities
from the third and fourth terms in $(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{int}}$ cancel out with that from $\delta M_{\gamma W}^{b}$,
resulting in a finite sum. (If only the convection term contribution is
considered, the cancellation does not occur and one obtains an extra $\ln
m_{e}$-divergent piece from $\delta M_{\gamma W}^{b}$). Finally, the sum of
the divergent contributions to the virtual corrections reads:

$\displaystyle(\delta f_{+})^{\mathrm{div,m}}$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{2}\delta
Z_{e}^{\mathrm{div,m}}-\frac{\alpha}{2\pi}(v-M_{i}^{2})C_{0}^{\mathrm{div,m}}-\frac{\alpha}{4\pi}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\right\\}f_{+}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\left\\{\ln\frac{m_{e}^{2}}{\mu^{2}}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}-\frac{1}{2}\ln^{2}\frac{m_{e}^{2}}{\mu^{2}}+\frac{1}{2}\ln\frac{m_{e}^{2}}{\mu^{2}}+\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\left(2-\ln\frac{M_{i}^{2}}{\mu^{2}}+2\ln\frac{M_{i}^{2}}{M_{i}^{2}-v}\right)\right\\}f_{+}$

Throughout this work we use “$i$” to label the charged meson in the decay
process, i.e. $\pi^{-}$ ($K^{+}$) in the $K^{0}$ ($K^{+}$)-decay.

###  IV.2 Divergent integrals in bremsstrahlung

Figure 2: Tree-level bremsstrahlung diagrams contributing to the $K_{e3}$
decay rate.

Next we study the IR and mass singularities from the bremsstrahlung process
depicted in Fig.2. Following the treatment in Refs. Seng et al. (2021a, b,
2022a), we split the bremsstrahlung amplitude as:

$M_{K\rightarrow\pi e\nu\gamma}=M_{A}+M_{B}\leavevmode\nobreak\ ,$ (30)

where $M_{A}$ contains the full convection term contribution:

$M_{A}=-\frac{eG_{F}}{\sqrt{2}}F_{\mu}(p^{\prime},p)\varepsilon_{\nu}^{*}(k)\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})\left\\{\left(\frac{p_{e}}{p_{e}\cdot
k}-\frac{p_{i}}{p_{i}\cdot k}\right)^{\nu}+\frac{1}{2p_{e}\cdot
k}\not{k}\gamma^{\nu}\right\\}v_{e}\leavevmode\nobreak\ ,$ (31)

and $M_{B}$ is the remaining regular term that admits a ChPT expansion. It is
easy to check that all the singularities exist only in the phase space
integral over $|M_{A}|^{2}$. In particular, the IR divergence exists only in
the $\mathcal{D}_{3}$ region, which can be isolated through the following
separation:

$|M_{A}|^{2}=-e^{2}\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p_{i}}{p_{i}\cdot
k}\right)^{2}|M_{0}|^{2}(0,y,z)+|M_{A}|_{\mathrm{res}}^{2}\leavevmode\nobreak\
.$ (32)

The integral of the first term gives the full IR divergence (and a part of the
$\ln m_{e}$ divergences), while the second term gives only $\ln m_{e}$ but not
IR divergences. Meanwhile, the integral of the whole $|M_{A}|^{2}$ in the
$\mathcal{D}_{4-3}$ region gives the last piece with mass singularities.
Therefore, for the purpose of this paper only these three contributions need
to be studied.

First, the IR divergence is fully contained in the following integral:

$I_{i}\equiv\int_{0}^{\alpha_{+}}dx\int
d\Gamma_{k}d\Gamma_{p_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})\left(\frac{p_{i}}{p_{i}\cdot k}-\frac{p_{e}}{p_{e}\cdot
k}\right)^{2}=I_{i}^{\mathrm{div,m}}+I_{i}^{\mathrm{fin}}\leavevmode\nobreak\
.$ (33)

Its analytic expression is given in Appendix D of Ref. Seng et al. (2021b),
and here we further expand it with respect to $m_{e}$. This gives:

$\displaystyle I_{i}^{\mathrm{div,m}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi
M_{K}^{2}}\biggl{\\{}-\ln\frac{m_{e}^{2}}{\mu^{2}}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}+\frac{1}{2}\ln^{2}\frac{m_{e}^{2}}{\mu^{2}}+\left[1+2\ln\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)\right]\ln\frac{m_{e}^{2}}{\mu^{2}}$
$\displaystyle-\left[2+\ln\left(\frac{\mu^{2}P_{0}^{2}(0)(1-\cos\chi)^{2}}{M_{K}^{4}(1-z+r_{\pi})^{2}}\right)\right]\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\biggr{\\}}$
$\displaystyle I_{i}^{\mathrm{fin}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi
M_{K}^{2}}\biggl{\\{}-\frac{1}{2}\ln^{2}\left(\frac{\mu^{2}P_{0}^{2}(0)(1-\cos\chi)^{2}}{M_{K}^{4}\alpha_{+}^{2}}\right)+4\ln^{2}\left(\sin\frac{\chi}{2}\right)+\ln\left(\frac{M_{K}^{4}\alpha_{+}^{2}}{4P_{0}^{2}(0)\mu^{2}}\right)$
(34)
$\displaystyle+2\mathrm{Li}_{2}\left(\cos^{2}\frac{\chi}{2}\right)+2\ln^{2}\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)+2\ln\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)+4\mathrm{Li}_{2}\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)\biggr{\\}}$
$\displaystyle+\frac{1}{\pi
M_{K}^{2}}\int_{0}^{\alpha_{+}}dx\frac{1}{x}\ln\left(\frac{P_{0}(x)-P_{1}(x)}{P_{0}(0)-P_{1}(0)}\right)\leavevmode\nobreak\
.$

where we have taken the $m_{e}=0$ kinematics:

$P_{0}(x)=\frac{p_{i}\cdot P}{M_{i}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
P_{0}(x)-P_{1}(x)=\frac{p_{e}\cdot P}{p_{i}\cdot
p_{e}}M_{i}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\chi\equiv\cos^{-1}\frac{P_{1}(0)}{P_{0}(0)}\leavevmode\nobreak\ .$ (35)

The remaining bremsstrahlung integral with mass singularities is:

$I_{m,1}(p_{i})\equiv 8\pi\int
d\Gamma_{k}d\Gamma_{p_{\nu}}\frac{(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})}{(p_{i}\cdot k)^{m}p_{e}\cdot
k}=I_{m,1}^{\mathrm{div,m}}(p_{i})+I_{m,1}^{\mathrm{fin}}(p_{i})\leavevmode\nobreak\
,$ (36)

with $m=1,0,-1,-2$. The divergent piece of these integrals is:

$I_{m,1}^{\mathrm{div,m}}(p_{i})=-\frac{2(2p_{e}\cdot
P)^{m-1}}{(M_{K}^{2}x)^{m}(p_{i}\cdot
p_{e})^{m}}\ln\frac{m_{e}^{2}}{\mu^{2}}\leavevmode\nobreak\ ,$ (37)

while the finite pieces are:

$\displaystyle I_{1,1}^{\mathrm{fin}}(p_{i})$ $\displaystyle=$
$\displaystyle-\frac{2}{p_{e}\cdot
p_{i}M_{K}^{2}x}\ln\left(\frac{\mu^{2}M_{i}^{2}}{4(p_{e}\cdot
p_{i})^{2}}\right)\leavevmode\nobreak\ ,$ $\displaystyle
I_{0,1}^{\mathrm{fin}}(p_{i})$ $\displaystyle=$
$\displaystyle-\frac{1}{p_{e}\cdot
P}\ln\left(\frac{\mu^{2}M_{K}^{2}x}{4(p_{e}\cdot
P)^{2}}\right)\leavevmode\nobreak\ ,$ $\displaystyle
I_{-1,1}^{\mathrm{fin}}(p_{i})$ $\displaystyle=$
$\displaystyle-\frac{p_{i}\cdot p_{e}}{2(p_{e}\cdot
P)^{2}}M_{K}^{2}x\left[2+\ln\left(\frac{\mu^{2}M_{K}^{2}x}{4(p_{e}\cdot
P)^{2}}\right)\right]+\frac{p_{i}\cdot P}{p_{e}\cdot P}\leavevmode\nobreak\ ,$
$\displaystyle I_{-2,1}^{\mathrm{fin}}(p_{i})$ $\displaystyle=$
$\displaystyle-\frac{(p_{e}\cdot p_{i})^{2}}{4(p_{e}\cdot
P)^{3}}M_{K}^{4}x^{2}\left[3+\ln\left(\frac{\mu^{2}M_{K}^{2}x}{4(p_{e}\cdot
P)^{2}}\right)\right]$ (38) $\displaystyle+\frac{1}{4p_{e}\cdot
P}\left[\frac{2p_{i}\cdot Pp_{i}\cdot p_{e}}{p_{e}\cdot
P}M_{K}^{2}x-M_{i}^{2}M_{K}^{2}x+2(p_{i}\cdot
P)^{2}\right]\leavevmode\nobreak\ .$

These are all the divergent integrals that appear in the bremsstrahlung.

###  IV.3 Full cancellation of the IR and mass singularities

Now we are ready to demonstrate the complete cancellation of IR divergences
and electron mass singularities in the radiative inclusive total $K_{e3}$
decay rate. Since in the previous subsections we explicitly isolated all the
terms that are logarithmically divergent with respect to $M_{\gamma}$ and
$m_{e}$, in what follows we will take $M_{\gamma}=m_{e}=0$ everywhere except
in those logs. We start with the contribution from $(\delta
f_{+})^{\mathrm{div,m}}$:

$\displaystyle(\delta\Gamma)_{(\delta f_{+})^{\mathrm{div,m}}}$
$\displaystyle=$
$\displaystyle\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz\left\\{\frac{2\mathfrak{Re}(\delta
f_{+})^{\mathrm{div,m}}}{f_{+}}|M_{0}|^{2}(0,y,z)\right\\}$ (39)
$\displaystyle=$ $\displaystyle\frac{\alpha
G_{F}^{2}|V_{us}|^{2}M_{K}^{5}}{64\pi^{4}}\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}}dzf_{+}^{2}(t)\int_{c(z)-d(z)}^{c(z)+d(z)}dy\biggl{[}\ln\frac{m_{e}^{2}}{\mu^{2}}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}-\frac{1}{2}\ln^{2}\frac{m_{e}^{2}}{\mu^{2}}+\frac{1}{2}\ln\frac{m_{e}^{2}}{\mu^{2}}$
$\displaystyle+\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\left(2-\ln\frac{M_{i}^{2}}{\mu^{2}}+\ln\frac{M_{i}^{4}}{(M_{i}^{2}-v)^{2}}\right)\biggr{]}[r_{\pi}+(y-1)(y+z-1)]\leavevmode\nobreak\
.$

Next, the divergent part of the bremsstrahlung integral $I_{i}$ contributes:

$\displaystyle(\delta\Gamma)_{I_{i}^{\mathrm{div,m}}}$ $\displaystyle=$
$\displaystyle\frac{M_{K}^{3}}{512\pi^{4}}\int_{\mathcal{D}_{3}}dydz(-e^{2})I_{i}^{\mathrm{div,m}}|M_{0}|^{2}(0,y,z)$
(40) $\displaystyle=$ $\displaystyle\frac{\alpha
G_{F}^{2}|V_{us}|^{2}M_{K}^{5}}{64\pi^{4}}\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}}dzf_{+}^{2}(t)\int_{c(z)-d(z)}^{c(z)+d(z)}dy\biggl{[}-\ln\frac{m_{e}^{2}}{\mu^{2}}\ln\frac{M_{\gamma}^{2}}{\mu^{2}}+\frac{1}{2}\ln^{2}\frac{m_{e}^{2}}{\mu^{2}}$
$\displaystyle+\ln\frac{m_{e}^{2}}{\mu^{2}}\left(1+\ln\frac{\alpha_{+}^{2}}{(1-z+r_{\pi})^{2}}\right)-\ln\frac{M_{\gamma}^{2}}{\mu^{2}}\left(2-\ln\frac{M_{i}^{2}}{\mu^{2}}+\ln\frac{M_{i}^{4}}{(M_{i}^{2}-v)^{2}}\right)\biggr{]}\leavevmode\nobreak\
.$ $\displaystyle\times[r_{\pi}+(y-1)(y+z-1)]\leavevmode\nobreak\ .$

Summing up the above two terms, the IR divergences completely cancel, as
originally demonstrated in Ref. Seng et al. (2021b). However, the
$\ln(m_{e}^{2}/\mu^{2})$ terms do not fully cancel; they must be combined with
those from $|M_{A}|_{\mathrm{res}}^{2}$ in the $\mathcal{D}_{3}$ region and
from $|M_{A}|^{2}$ in the $\mathcal{D}_{4-3}$ region. To include these
contributions, we first write:

$|M_{A}|^{2}_{\mathrm{res}}=\sum_{m,n}\frac{\mathbb{C}_{m,n}(x,y,z)}{(p_{i}\cdot
k)^{m}(p_{e}\cdot k)^{n}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
|M_{A}|^{2}=\sum_{m,n}\frac{\mathbb{C}^{\prime}_{m,n}(x,y,z)}{(p_{i}\cdot
k)^{m}(p_{e}\cdot k)^{n}}\leavevmode\nobreak\ ,$ (41)

where the coefficients $\mathbb{C}_{m,n}^{(\prime)}(x,y,z)$ are known
quantities. We are only interested in the coefficients with $n=1$ as they are
attached to the $\ln m_{e}$-divergent integrals $I_{m,1}$. To that end, it is
useful to define a new set of coefficients

$\sum_{m}\mathbb{C}_{m1}^{(\prime)}(x,y,z)\left\\{-\frac{2(2p_{e}\cdot
P)^{m-1}}{(M_{K}^{2}x)^{m}(p_{i}\cdot p_{e})^{m}}\right\\}\equiv 64\pi\alpha
M_{K}^{2}G_{F}^{2}|V_{us}|^{2}f_{+}^{2}(t)\bar{\mathbb{C}}^{(\prime)}(x,y,z)\leavevmode\nobreak\
,$ (42)

which turns out to be channel-independent:

$\displaystyle\bar{\mathbb{C}}(x,y,z)$ $\displaystyle=$
$\displaystyle-\frac{x^{2}y^{2}}{(1-x-z+r_{\pi})^{3}}+\frac{xy(r_{\pi}-x+3y-1)}{(1-x-z+r_{\pi})^{2}}+\frac{2y(r_{\pi}+2y-1)-3xy+x}{1-x-z+r_{\pi}}-2y,$
$\displaystyle\bar{\mathbb{C}}^{\prime}(x,y,z)$ $\displaystyle=$
$\displaystyle\bar{\mathbb{C}}(x,y,z)+\frac{2(r_{\pi}+(y-1)(y+z-1))}{x}\leavevmode\nobreak\
.$ (43)

Based on the above, the final $\ln m_{e}$ singularities contribution from
bremsstrahlung becomes:

$\displaystyle(\delta\Gamma)_{I_{m,1}^{\mathrm{div,m}}}$ $\displaystyle=$
$\displaystyle\frac{M_{K}^{3}}{4096\pi^{5}}\sum_{m}\left\\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}}dx\mathbb{C}_{m,1}+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}}^{\alpha_{+}}dx\mathbb{C}_{m,1}^{\prime}\right\\}I_{m,1}^{\mathrm{div,m}}$
(44) $\displaystyle=$ $\displaystyle\frac{\alpha
G_{F}^{2}|V_{us}|^{2}M_{K}^{5}}{64\pi^{4}}\ln\frac{m_{e}^{2}}{\mu^{2}}\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}}dzf_{+}^{2}(t)\left\\{\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\alpha_{+}}dx\bar{\mathbb{C}}(x,y,z)\right.$
$\displaystyle\left.+\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\alpha_{+}}dx\bar{\mathbb{C}}^{\prime}(x,y,z)\right\\}\leavevmode\nobreak\
.$

and the sum of divergent contributions to the total $K_{e3}$ decay rate
becomes:

$\displaystyle(\delta\Gamma)^{\mathrm{div,m}}$ $\displaystyle=$
$\displaystyle(\delta\Gamma)_{(\delta
f_{+})^{\mathrm{div,m}}}+(\delta\Gamma)_{I_{i}^{\mathrm{div,m}}}+(\delta\Gamma)_{I_{m,1}^{\mathrm{div,m}}}$
(45) $\displaystyle=$ $\displaystyle\frac{\alpha
G_{F}^{2}|V_{us}|^{2}M_{K}^{5}}{64\pi^{4}}\ln\frac{m_{e}^{2}}{\mu^{2}}\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}}dzf_{+}^{2}(t)\biggl{\\{}\int_{c(z)-d(z)}^{c(z)+d(z)}dy\left[\frac{3}{2}+\ln\frac{\alpha_{+}^{2}}{(1-z+r_{\pi})^{2}}\right]$
$\displaystyle\times[r_{\pi}+(y-1)(y+z-1)]+\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\alpha_{+}}dx\bar{\mathbb{C}}(x,y,z)$
$\displaystyle+\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\alpha_{+}}dx\bar{\mathbb{C}}^{\prime}(x,y,z)\biggr{\\}}\leavevmode\nobreak\
.$

The sum would vanish if the terms in the curly brackets add up to zero for all
values of $z$. Indeed that happens, by direct integration one can show that:

$\displaystyle\int_{c(z)-d(z)}^{c(z)+d(z)}dy\left[\frac{3}{2}+\ln\frac{\alpha_{+}^{2}}{(1-z+r_{\pi})^{2}}\right][r_{\pi}+(y-1)(y+z-1)]$
$\displaystyle=$
$\displaystyle\frac{1}{36}(z^{2}-4r_{\pi})^{3/2}\left[12\ln\left(\frac{2-z+\sqrt{z^{2}-4r_{\pi}}}{2\sqrt{z^{2}-4r_{\pi}}}\right)+1\right]\equiv
g_{1}(z)\leavevmode\nobreak\ ,$
$\displaystyle\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\mathrm{\alpha_{+}}}dx\bar{\mathbb{C}}(x,y,z)$
$\displaystyle=$
$\displaystyle\frac{1}{36}\biggl{\\{}-4r_{\pi}\left[2\sqrt{z^{2}-4r_{\pi}}+9z-18\right]+8(z^{2}-3z+3)\sqrt{z^{2}-4r_{\pi}}+9z^{2}(z-2)$
$\displaystyle+6\left[(z^{2}-6z+2r_{\pi}+6)\sqrt{z^{2}-4r_{\pi}}+(z-2)^{3}\right]\ln\left(\frac{2\sqrt{1-z+r_{\pi}}}{2-z-\sqrt{z^{2}-4r_{\pi}}}\right)\biggr{\\}}\equiv
g_{2}(z)\leavevmode\nobreak\ ,$

and

$\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\mathrm{\alpha_{+}}}dx\bar{\mathbb{C}}^{\prime}(x,y,z)=-g_{1}(z)-g_{2}(z)\leavevmode\nobreak\
.$ (47)

So, we have $(\delta\Gamma)^{\mathrm{div,m}}=0$. This calculation shows that
the full numerical result of the $K_{e3}$ decay rate in Refs. Seng et al.
(2021a, b, 2022a) is free from photonic IR divergences and electron mass
singularities. It supports the small error estimate given there for the 1 loop
RC by demonstrating that no potentially numerically-large contributions
enhanced by $\ln m_{e}$ were missed.

##  V Extra finite terms when $\boldsymbol{m_{e}}$ is infinitesimal but non-
zero

In the previous section the emphasis was on terms with $\ln M_{\gamma}$
divergences and/or $\ln m_{e}$ singularities, assuming that all the remaining
terms without such divergences would not distinguish between the mass-
expansion and DR in the zero mass limit. However, the way equality comes about
is quite novel and particularly interesting. We will show in this section that
there exists a contribution from the bremsstrahlung that remains finite when
$m_{e}\rightarrow 0$, i.e. exists in the mass-expansion method, but naively
appears to be absent when we set $m_{e}=0$ from the beginning. Its presence in
DR will be subsequently discussed.

So far, in the bremsstrahlung process, we only examined the integrals
$I_{m,1}$ that give a $\ln m_{e}$-divergence. However, there exists another
set of integrals $I_{m,2}$ that gives a potentially more severe,
$1/m_{e}^{2}$-power divergence:

$\displaystyle I_{m,2}(p_{i},p_{e})$ $\displaystyle=$ $\displaystyle 8\pi\int
d\Gamma_{k}d\Gamma_{p_{\nu}}\frac{(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})}{(p_{i}\cdot k)^{m}(p_{e}\cdot k)^{2}}$ (48) $\displaystyle=$
$\displaystyle\frac{4(2p_{e}\cdot
P)^{m}}{m_{e}^{2}(M_{K}^{2}x)^{m+1}(p_{i}\cdot
p_{e})^{m}}+\mathcal{O}(m_{e}^{0})\leavevmode\nobreak\ .$

Upon inspection, one finds that their coefficients
$\mathbb{C}^{(\prime)}_{m,2}(x,y,z)$ always contain a factor $m_{e}^{2}$, i.e.
they are chirally-suppressed in the squared-amplitude level. Therefore, in the
$m_{e}\rightarrow 0$ limit such terms make a finite contribution to the total
decay rate. In other words, they represent a novel non-vanishing chirality-
breaking effect in the $m_{e}\rightarrow 0$ limit. We emphasize that such
contributions exists only in the mass-expansion method and not in DR, because
in the latter the coefficients $\mathbb{C}^{(\prime)}_{m,2}$ are identically
zero as we have set $m_{e}=0$ from the beginning. The existence of such
phenomena were recognized in the literature, e.g. Refs. Gorsky et al. (1989);
Smilga (1991), where it was argued that the $m_{e}=0$ is not smooth for
certain observables in quantum electrodynamics because of such terms. Our
interpretation will be somewhat different. However, we will first evaluate the
magnitude of those contributions to the $K_{e3}$ radiative inclusive decay
rate.

To account for the effects of such terms, we may define a new set of
coefficients $\bar{\mathbb{D}}^{(\prime)}(x,y,z)$ by:

$\lim_{m_{e}\rightarrow
0}\sum_{m}\mathbb{C}_{m,2}^{(\prime)}(x,y,z)\left\\{\frac{4(2p_{e}\cdot
P)^{m}}{m_{e}^{2}(M_{K}^{2}x)^{m+1}(p_{i}\cdot
p_{e})^{m}}\right\\}=64\pi\alpha
M_{K}^{2}G_{F}^{2}|V_{us}|^{2}f_{+}^{2}(t)\bar{\mathbb{D}}^{(\prime)}(x,y,z)\leavevmode\nobreak\
.$ (49)

Again, these coefficients turn out to be channel-independent:

$\displaystyle\bar{\mathbb{D}}(x,y,z)$ $\displaystyle=$
$\displaystyle\frac{2y[r_{\pi}(2y+z-2)-x(y+z-2)-(z-1)(2y+z-2)]}{(1-x-z+r_{\pi})^{2}},$
$\displaystyle\bar{\mathbb{D}}^{(\prime)}(x,y,z)$ $\displaystyle=$
$\displaystyle\bar{\mathbb{D}}(x,y,z)+\frac{2[r_{\pi}+(y-1)(y+z-1)]}{x}\leavevmode\nobreak\
.$ (50)

With them, the extra finite contribution to the decay rate in the mass-
expansion method is given in the $m_{e}\rightarrow 0$ limit as:

$\displaystyle(\delta\Gamma)_{\mathrm{ext,m}}$ $\displaystyle=$
$\displaystyle\frac{M_{K}^{3}}{4096\pi^{5}}\lim_{m_{e}\rightarrow
0}\sum_{m}\left\\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}}dx\mathbb{C}_{m,2}+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}}^{\alpha_{+}}dx\mathbb{C}_{m,2}^{\prime}\right\\}I_{m,2}$
(51) $\displaystyle=$ $\displaystyle\frac{\alpha
G_{F}^{2}|V_{us}|^{2}M_{K}^{5}}{64\pi^{4}}\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}}dzf_{+}^{2}(t)\left\\{\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\alpha_{+}}dx\bar{\mathbb{D}}(x,y,z)\right.$
$\displaystyle\left.+\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\alpha_{+}}dx\bar{\mathbb{D}}^{\prime}(x,y,z)\right\\}\leavevmode\nobreak\
.$

Direct integration of the terms in the curly brackets returns a rather elegant
expression:

$\displaystyle\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\alpha_{+}}dx\bar{\mathbb{D}}(x,y,z)+\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\alpha_{+}}dx\bar{\mathbb{D}}^{\prime}(x,y,z)$
(52) $\displaystyle=$
$\displaystyle\frac{1}{18}(z^{2}-4r_{\pi})^{3/2}\left\\{3\ln\left[\frac{(z^{2}-4r_{\pi})(2-z-\sqrt{z^{2}-4r_{\pi}})}{(1-z+r_{\pi})(2-z+\sqrt{z^{2}-4r_{\pi}})}\right]+1\right\\}\equiv
H(z)\leavevmode\nobreak\ .$

Numerically, utilizing a simple monopole parameterization of $f_{+}(t)$ Batley
et al. (2018) one obtains
$(\delta\Gamma)_{\mathrm{ext,m}}/\Gamma_{K_{e3}}\approx-0.01$%, making up a
negligibly small part in the full long-distance EM correction
$\delta_{\mathrm{EM}}^{K\ell}$, which is 1.16(3)% in the $K^{0}e$ channel and
0.21(5)% in the $K^{+}e$ channel respectively. In Refs. Seng et al. (2021a, b,
2022a), this contribution was automatically included in the numerical results
because no $m_{e}=0$ simplifications were made in the analytic expressions.

##  VI The same calculation with dimensional regularization

Now we proceed to the second goal of this paper, namely to demonstrate the
full equivalence between DR and the mass-expansion in reproducing the finite
terms of the total radiatively inclusive $K_{e3}$ decay rate in the zero
electron mass limit.

We begin by setting $M_{\gamma}=m_{e}=0$ from the beginning, and generalize
the space-time dimension to $d=4-\epsilon$. Some obvious generalizations of
the basic formula in the previous section are:

$\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\rightarrow\mu^{\epsilon}\frac{d^{d}q^{\prime}}{(2\pi)^{d}}$
(53)

in the loop integrals, and

$(2\pi)^{4}\delta^{(4)}(k)\rightarrow\mu^{-\epsilon}(2\pi)^{d}\delta^{(d)}(k)\leavevmode\nobreak\
,\leavevmode\nobreak\
\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\rightarrow\mu^{\epsilon}\frac{d^{d-1}k}{(2\pi)^{d-1}2E_{k}}\equiv\mu^{\epsilon}d\Gamma_{k}$
(54)

in the phase space integrals. This is, however, not the full story. We will
now demonstrate that extra modifications of the phase space formula are
required to reproduce the correct finite pieces.

###  VI.1 Three- and four-body phase space in $\boldsymbol{d}$-dimension

We start by considering the following Lorentz-invariant integral:

$I=\frac{1}{2M_{K}}\mu^{\epsilon}\int d\Gamma_{p^{\prime}}\mu^{\epsilon}\int
d\Gamma_{p_{e}}A(x,y,z)$ (55)

where $A(x,y,z)$ is an arbitrary scalar function of $x,y,z$. This is a
straightforward generalization of the integral defined in Eq.(A.1) of Ref.
Seng et al. (2021b). In general, the integral measure of the $d-1$ spatial
components of a momentum $\vec{k}$ can be written as Marciano and Sirlin
(1975):

$\int d^{d-1}k=\int
d|\vec{k}||\vec{k}|^{d-2}\left(\prod_{a=1}^{d-3}\int_{0}^{\pi}d\theta_{a}\sin^{d-a-2}\theta_{a}\right)\int_{0}^{2\pi}d\theta_{d-2}\leavevmode\nobreak\
.$ (56)

Now, since $A(x,y,z)$ is Lorentz-invariant, we may evaluate it in the kaon
rest-frame, namely:

$\vec{p}=\vec{0},\>\>\vec{p}^{\prime}=|\vec{p}^{\prime}|(1,0,0,...)\,\>\>\vec{p}_{e}=E_{e}(\cos\theta_{e},\sin\theta_{e},0,...)\leavevmode\nobreak\
.$ (57)

In this way, we can integrate all angles in $\vec{p}^{\prime}$ and all except
the first angle in $\vec{p}_{e}$ using the formula:

$\int_{0}^{\pi}d\theta\sin^{m}\theta=\sqrt{\pi}\frac{\Gamma\left(\frac{1}{2}(m+1)\right)}{\Gamma\left(\frac{1}{2}(m+2)\right)}\leavevmode\nobreak\
,$ (58)

which gives:

$I=\frac{1}{2M_{K}}\frac{\sqrt{\pi}}{64\pi^{4}\Gamma\left(\frac{3-\epsilon}{2}\right)\Gamma\left(\frac{2-\epsilon}{2}\right)}\int
dE^{\prime}dE_{e}\int_{-1}^{+1}d\cos\theta_{e}\left(\frac{16\pi^{2}\mu^{4}}{|\vec{p}^{\prime}|^{2}E_{e}^{2}(1-\cos^{2}\theta_{e})}\right)^{\epsilon/2}E_{e}|\vec{p}^{\prime}|A(x,y,z)\leavevmode\nobreak\
.$ (59)

To further proceed, we notice that Eq. (57) allows us to express $E^{\prime}$,
$E_{e}$ and $\cos\theta_{e}$ using $x,y,z$. In particular, we find:

$|\vec{p}^{\prime}|^{2}E_{e}^{2}(1-\cos^{2}\theta_{e})=M_{K}^{4}f(x,y,z)\leavevmode\nobreak\
,$ (60)

where

$f(x,y,z)\equiv\frac{1}{4}\left[-r_{\pi}^{2}+r_{\pi}(2x-y(y+z-2)+2(z-1))-(x+y+z-1)(x-yz+y+z-1)\right]\leavevmode\nobreak\
.$ (61)

With this definition, we obtain:

$I=K\frac{M_{K}^{3}}{512\pi^{4}}\int_{2\sqrt{r_{\pi}}}^{\infty}dz\int_{0}^{\infty}dy\int_{\alpha_{-}}^{\alpha_{+}}dx[f(x,y,z)]^{-\epsilon/2}A(x,y,z)\leavevmode\nobreak\
,$ (62)

where

$K\equiv\frac{\sqrt{\pi}}{2\Gamma\left(\frac{3-\epsilon}{2}\right)\Gamma\left(\frac{2-\epsilon}{2}\right)}\left(\frac{16\pi^{2}\mu^{4}}{M_{K}^{4}}\right)^{\epsilon/2}$
(63)

is an overall constant multiplicative factor, which equals 1 when
$\epsilon=0$. Although the equation above is defined by choosing a specific
frame, since the final form is explicitly Lorentz-invariant, it obviously
holds in every frame. This expression is analogous to the one in Eq. (A.1) of
Ref. Seng et al. (2021b), so we may follow exactly the same logic in Appendix
A of that paper to derive the three- and four-body phase space formula in
$d$-dimension:

$\displaystyle\Gamma_{3-\mathrm{body}}$ $\displaystyle=$ $\displaystyle
K\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz(f(0,y,z))^{-\epsilon/2}|M|_{K\rightarrow\pi
e\nu}^{2}$ $\displaystyle\Gamma_{4-\mathrm{body}}$ $\displaystyle=$
$\displaystyle
K\frac{M_{K}^{3}}{512\pi^{4}}\left\\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}}dx+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}}^{\alpha_{+}}dx\right\\}\mu^{\epsilon}\int
d\Gamma_{k}d\Gamma_{p_{\nu}}$ (64)
$\displaystyle\times(2\pi)^{d}\delta^{(d)}(P-k-
p_{\nu})(f(x,y,z))^{-\epsilon/2}|M|_{K\rightarrow\pi
e\nu\gamma}^{2}\leavevmode\nobreak\ .$

There are two new ingredients: (1) the overall multiplicative factor $K$, and
(2) the factor $(f(x,y,z))^{-\epsilon/2}$ in the integrand. The former is
irrelevant as long as the final decay rate is singularity-free, but the latter
is important in order to get the correct finite contributions.

It turns out that if we keep $m_{e}$ finite (i.e. do not pay attention to mass
singularities), then the two new ingredients above are irrelevant (which is
implicitly assumed in Ref. Seng et al. (2021b)). The reasons are of twofold:
(1) when there are only IR singularities, the poles in both virtual and real
corrections are only at the order $(2/\epsilon)$, and (2) these poles already
cancel each other upon the $x$-integration, without touching $y$ and $z$. To
explain this idea more clearly, let us write:

$\displaystyle\Gamma_{3-\mathrm{body}}+\Gamma_{4-\mathrm{body}}$
$\displaystyle=$ $\displaystyle
K\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz(f(0,y,z))^{-\epsilon/2}\Bigl{\\{}|M|^{2}_{K\rightarrow\pi
e\nu}+\frac{M_{K}^{2}}{2\pi}\int_{0}^{\alpha_{+}}dx\mu^{\epsilon}\int
d\Gamma_{k}d\Gamma_{p_{\nu}}$ $\displaystyle\times(2\pi)^{d}\delta^{(d)}(P-k-
p_{\nu})|M|^{2}_{K\rightarrow\pi
e\nu\gamma}\Bigr{\\}}+K\frac{M_{K}^{3}}{512\pi^{4}}\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}}dx\mu^{\epsilon}\int
d\Gamma_{k}d\Gamma_{p_{\nu}}$ $\displaystyle\times(2\pi)^{d}\delta^{(d)}(P-k-
p_{\nu})\left\\{(f(x,y,z))^{-\epsilon/2}-(f(0,y,z))^{-\epsilon/2}\right\\}|M|_{K\rightarrow\pi
e\nu\gamma}^{2}$
$\displaystyle+K\frac{M_{K}^{3}}{512\pi^{4}}\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}}^{\alpha_{+}}dx\mu^{\epsilon}\int
d\Gamma_{k}d\Gamma_{p_{\nu}}(2\pi)^{d}\delta^{(d)}(P-k-
p_{\nu})(f(x,y,z))^{-\epsilon/2}|M|_{K\rightarrow\pi
e\nu\gamma}^{2}\leavevmode\nobreak\ .$

There are three terms on the right-hand side, and the IR divergence occurs
only within the curly brackets of the first term; in particular, in
bremsstrahlung it appears as a log-divergence in the $x$-integral at
$x\rightarrow 0$. The claim is that the IR divergences within the two terms in
the curly bracket cancel out each other, so we can set $\epsilon=0$ in the
remaining parts of the first integral, which brings $K$,
$(f(0,y,z))^{-\epsilon/2}$ both to 1. In the second term, the factor
$(f(x,y,z))^{-\epsilon/2}-(f(0,y,z))^{-\epsilon/2}$ renders the
$x$-integration finite, so we can set $\epsilon=0$ from the beginning, which
simply kills the entire term. Finally, the third term is by itself IR-finite;
so, we can again take $\epsilon=0$ which brings $K$,
$(f(x,y,z))^{-\epsilon/2}$ to 1. This proves our assertion.

Things are more complicated when mass singularities are also regularized using
DR, because now one obtains poles of the order $(2/\epsilon)^{2}$ and
$(2/\epsilon)$; the former cancels within the $x$-integration, but the latter
only cancels upon the $y$-integration, therefore $(f(x,y,z))^{-\epsilon/2}$
gives a non-zero finite contribution which is the analog of the small chiral
breaking contribution found in the massive electron calculation, as we shall
show.

###  VI.2 Divergent quantities in virtual corrections

We may study the total $K_{e3}$ decay rate at $m_{e}=M_{\gamma}=0$ in Sirlin’s
representation using DR, starting from the virtual corrections. As an
effective bookkeeping method for possible finite differences with the mass-
expansion method, we apply the following strategy: for every quantity $F$ with
singularities, we always make the following separation:

$F=F^{\mathrm{div,DR}}+F^{\mathrm{fin}}+\mathcal{O}(\epsilon)\leavevmode\nobreak\
,$ (66)

where $F^{\mathrm{fin}}$ is defined to be exactly the same as that in the
mass-expansion method (see Eq. (18)), which means $F^{\mathrm{div,DR}}$ may
contain finite terms that are not proportional to $(2/\epsilon)^{2}$ or
$(2/\epsilon)$. In this way, the comparison between the finite terms in DR and
mass-expansion method can simply be done by comparing the effect of
$F^{\mathrm{div,DR}}$ and $F^{\mathrm{div,m}}$ in the total decay rate.

One more technical detail is in order. In Sirlin’s representation, one
utilizes the following identity:

$\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}=g^{\mu\nu}\gamma^{\alpha}-g^{\mu\alpha}\gamma^{\nu}+g^{\nu\alpha}\gamma^{\mu}-i\epsilon^{\mu\nu\alpha\beta}\gamma_{\beta}\gamma_{5}$
(67)

to split the full $\gamma W$-box diagram into $\delta M_{\gamma W}^{a}$ and
$\delta M_{\gamma W}^{b}$; the former is combined with $\delta M_{2}$ such
that the residual integrals are ultraviolet-insensitive. A conceptual problem
then arises when standard dimensional regularization (which we shall
abbreviate as “DReg” to be specific) is applied to the formalism, because in
DReg the generalization into $d$-dimensional vectors applies both to momenta
and to the Dirac matrices $\gamma^{\mu}$, and the latter renders the totally-
antisymmetric tensor $\epsilon^{\mu\nu\alpha\beta}$ ill-defined. This is more
of a problem since there is a mass singularity in $\delta M_{\gamma W}^{b}$
(which contains an antisymmetric tensor) that has to be canceled with that in
$(\delta M_{2}+\delta M_{\gamma W}^{a})_{\mathrm{int}}$, as we discussed in
Sec. IV.1. Although in transforming the $\delta M_{\gamma W}^{b}$ contribution
into $\delta f_{+}$ one again makes use of Eq. (67) to get rid of the
antisymmetric tensor, so it is possible that the final result is unambiguous
since we have made “even number of mistakes”, but it is still important to
keep this in mind in order to track down possible extra finite terms in the
final result.

An approach to bypass the ambiguity above is to adopt the “dimensional
reduction” (which we abbreviate as “DRed” to be specific) formalism Siegel
(1979), where only momenta are treated in $d$-dimension whereas the Dirac
matrices remain 4-vectors. In this way the $\epsilon$-tensor can be rigorously
defined, but the formalism itself has its own problems, e.g. mathematical
inconsistencies Siegel (1980). To stay on the safe side, in this work we apply
both methods in the analysis; the finite difference between the two methods
comes mainly from the fact that $\gamma^{\mu}\gamma_{\alpha}\gamma_{\mu}$
equals $(2-d)\gamma_{\alpha}$ in DReg but $-2\gamma_{\alpha}$ in DRed. In what
follows we label both methods collectively as “DR”, but will specify “DReg” or
“DRed” whenever they make a difference.

Now we present the essential results of our DR analysis to the virtual
corrections, where the replacement in Eq.(53) applies to all loop integrals.
First, the divergent part of the electron wavefunction renormalization reads:

$\delta
Z_{e}^{\mathrm{div,DR}}=-\frac{\alpha}{4\pi}R\left[-\frac{2}{\epsilon}-5+\delta\right]\leavevmode\nobreak\
,$ (68)

with $\delta$ equals 1 in DReg and 0 in DRed. Here we have defined:

$R\equiv\left(\frac{e^{\gamma_{E}}}{4\pi}\right)^{-\epsilon/2}$ (69)

as a convenient multiplicative constant to absorb the effect of $\gamma_{E}$
and $\ln 4\pi$ which always appear in DR. Next, for the analytic piece in
$\delta M_{2}+\delta M_{\gamma W}^{a}$ we have:

$(\delta M_{2}+\delta M_{\gamma
W}^{a})_{\mathrm{ana}}^{\mathrm{div,DR}}=-\frac{\alpha}{2\pi}R\left(\frac{2}{\epsilon}+2\right)M_{0}\leavevmode\nobreak\
.$ (70)

Next we study the sum of the two-point and three-point function; since it
involves only IR divergence and no mass singularity, there is a universal
matching relation between these two schemes Marciano and Sirlin (1975):
$\ln(M_{\gamma}^{2}/\mu^{2})\leftrightarrow R(2/\epsilon)$. Applying this to
Eq. (26) gives:

$(\delta
f_{+})_{\mathrm{2pt+3pt}}^{\mathrm{div,DR}}=-\frac{\alpha}{4\pi}R\left(\frac{2}{\epsilon}\right)f_{+}\leavevmode\nobreak\
.$ (71)

And finally, the divergent part of the scalar function $C_{0}$ reads:

$C_{0}^{\mathrm{div,DR}}=\frac{R}{M_{1}^{2}-v}\left\\{-\frac{1}{2}\left(\frac{2}{\epsilon}\right)^{2}+\left(\frac{2}{\epsilon}\right)\left[\frac{1}{2}\ln\frac{M_{1}^{2}}{\mu^{2}}-\ln\frac{M_{1}^{2}}{M_{1}^{2}-v}\right]-\frac{\pi^{2}}{24}\right\\}\leavevmode\nobreak\
.$ (72)

With all the above, the total divergent contribution to $\delta f_{+}$ is
given in DR by:

$\displaystyle(\delta f_{+})^{\mathrm{div,DR}}$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{2}\delta
Z_{e}^{\mathrm{div,DR}}-\frac{\alpha}{2\pi}(v-M_{i}^{2})C_{0}^{\mathrm{div,DR}}-\frac{\alpha}{4\pi}R\left(\frac{2}{\epsilon}\right)\right\\}f_{+}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}R\left\\{\left(\frac{2}{\epsilon}\right)^{2}+\left(\frac{2}{\epsilon}\right)\left(\frac{5}{2}-\ln\frac{M_{i}^{2}}{\mu^{2}}+2\ln\frac{M_{i}^{2}}{M_{i}^{2}-v}\right)+\frac{3+\delta}{2}+\frac{\pi^{2}}{12}\right\\}f_{+}\leavevmode\nobreak\
,$

which is to be compared with the same quantity in the mass-expansion method,
Eq.(LABEL:eq:deltafpmass). One observes that, unlike the case of finite
$m_{e}$, in the presence of collinear mass singularity there is no simple
matching that connects $\ln(m_{e}^{2}/\mu^{2})$ and
$\ln(M_{\gamma}^{2}/\mu^{2})$ to $(2/\epsilon)$.

###  VI.3 Divergent integrals in bremsstrahlung

Next we study the divergent integrals in the bremsstrahlung process, starting
from the integral $I_{i}$. Since the original definition in Eq. (33) involves
the integration over $x$, in DR it must include the factor
$(f(x,y,z))^{-\epsilon/2}$ due to the generalized phase space formula
discussed in Sec. VI.1. It is defined as:

$\displaystyle I_{i}$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\alpha_{+}}dx(f(x,y,z))^{-\epsilon/2}\mu^{\epsilon}\int
d\Gamma_{k}d\Gamma_{p_{\nu}}(2\pi)^{d}\delta^{(d)}(P-k-
p_{\nu})\left(\frac{p_{i}}{p_{i}\cdot k}-\frac{p_{e}}{p_{e}\cdot
k}\right)^{2}$ (74)

where the divergent piece reads:

$\displaystyle I_{i}^{\mathrm{div,DR}}$ $\displaystyle=$
$\displaystyle\frac{R}{4\pi
M_{K}^{2}}(f(0,y,z))^{-\epsilon/2}\biggl{\\{}-\left(\frac{2}{\epsilon}\right)^{2}-\left(\frac{2}{\epsilon}\right)\left[1+\ln\left(\frac{\mu^{2}P_{0}^{2}(0)(1-\cos\chi)^{2}}{M_{K}^{4}\alpha_{+}^{2}}\right)\right]$
(75)
$\displaystyle+2\mathrm{Li}_{2}\left(\frac{\alpha_{+}}{\alpha_{-}}\right)+\frac{\pi^{2}}{4}-2\ln^{2}\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)-2\ln\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)$
$\displaystyle-4\mathrm{Li}_{2}\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)\biggr{\\}}\leavevmode\nobreak\
.$

The remaining bremsstrahlung integrals with mass singularities are those in
Eq. (36), which are defined in DR as:

$I_{m,1}(p_{i})\equiv 8\pi\mu^{\epsilon}\int
d\Gamma_{k}d\Gamma_{p_{\nu}}\frac{(2\pi)^{d}\delta^{(d)}(P-k-
p_{\nu})}{(p_{i}\cdot k)^{m}p_{e}\cdot k}\leavevmode\nobreak\ .$ (76)

Their divergent piece is given by:

$I_{m,1}^{\mathrm{div,DR}}(p_{i})=-\frac{2(2p_{e}\cdot
P)^{m-1}}{(M_{K}^{2}x)^{m}(p_{i}\cdot
p_{e})^{m}}R\left(\frac{2}{\epsilon}+\ln\frac{(1-x-z+r_{\pi})^{2}}{x^{2}}\right)\leavevmode\nobreak\
.$ (77)

###  VI.4 Reconciling DR and the mass-expansion

Now we may collect all the aforementioned divergent contributions in DR to the
total decay rate. They are:

$\displaystyle(\delta\Gamma)_{(\delta f_{+})^{\mathrm{div,DR}}}$
$\displaystyle=$ $\displaystyle
K\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz(f(0,y,z))^{-\epsilon/2}\left\\{\frac{2\mathfrak{Re}(\delta
f_{+})^{\mathrm{div,DR}}}{f_{+}}|M_{0}|^{2}(0,y,z)\right\\},$
$\displaystyle(\delta\Gamma)_{I_{i}^{\mathrm{div,DR}}}$ $\displaystyle=$
$\displaystyle
K\frac{M_{K}^{3}}{512\pi^{4}}\int_{\mathcal{D}_{3}}dydz(-e^{2})I_{i}^{\mathrm{div,DR}}|M_{0}|^{2}(0,y,z)\leavevmode\nobreak\
,$ $\displaystyle(\delta\Gamma)_{I_{m,1}^{\mathrm{div,DR}}}$ $\displaystyle=$
$\displaystyle
K\frac{M_{K}^{3}}{4096\pi^{5}}\sum_{m}\left\\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}}dx\mathbb{C}_{m,1}+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}}^{\alpha_{+}}dx\mathbb{C}_{m,1}^{\prime}\right\\}$
(78)
$\displaystyle\times(f(x,y,z))^{-\epsilon/2}I_{m,1}^{\mathrm{div,DR}}\leavevmode\nobreak\
.$

Based on the analysis in Sec. IV.3, it is easy to check that all the
divergences of order $(2/\epsilon)^{2}$ and $(2/\epsilon)$ cancel upon summing
the above three terms. and what left over is the extra finite terms in the DR
method relative to the mass-expansion method. It is given in the
$\epsilon\rightarrow 0$ limit by:

$(\delta\Gamma)_{\mathrm{ext,DR}}=(\delta\Gamma)_{(\delta
f_{+})^{\mathrm{div,DR}}}+(\delta\Gamma)_{I_{i}^{\mathrm{div,DR}}}+(\delta\Gamma)_{I_{m,1}^{\mathrm{div,DR}}}=\frac{\alpha
G_{F}^{2}|V_{us}|^{2}M_{K}^{5}}{64\pi^{4}}\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}}dzf_{+}^{2}(t)h(z)\leavevmode\nobreak\
,$ (79)

where

$\displaystyle h(z)$ $\displaystyle\equiv$
$\displaystyle\int_{c(z)-d(z)}^{c(z)+d(z)}dy[r_{\pi}+(y-1)(y+z-1)]\Bigl{[}-\ln
f(0,y,z)\left(\frac{3}{2}+\ln\frac{\alpha_{+}^{2}}{(1-z+r_{\pi})^{2}}\right)$
(80)
$\displaystyle+\frac{3+\delta}{2}+2\mathrm{Li}_{2}\left(\frac{\alpha_{+}}{\alpha_{-}}\right)+\frac{\pi^{2}}{3}-2\ln^{2}\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)-2\ln\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)$
$\displaystyle-4\mathrm{Li}_{2}\left(\frac{\alpha_{+}}{1-z+r_{\pi}}\right)\Bigr{]}+\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\alpha_{+}}dx\Bigl{[}\bar{\mathbb{C}}(x,y,z)\Bigl{(}-\ln
f(x,y,z)$
$\displaystyle+\ln\frac{(1-x-z+r_{\pi})^{2}}{x^{2}}\Bigr{)}+\delta\frac{4xf(x,y,z)}{(1-x-z+r_{\pi})^{3}}\Bigr{]}+\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\alpha_{+}}dx$
$\displaystyle\times\Bigl{[}\bar{\mathbb{C}}^{\prime}(x,y,z)\Bigl{(}-\ln
f(x,y,z)+\ln\frac{(1-x-z+r_{\pi})^{2}}{x^{2}}\Bigr{)}+\delta\frac{4xf(x,y,z)}{(1-x-z+r_{\pi})^{3}}\Bigr{]}\leavevmode\nobreak\
.$

Let us study the complicated expression above. First we observe that, apart
from the electron wavefunction renormalization, there are also contributions
from bremsstrahlung in the $\mathcal{D}_{3}$ and $\mathcal{D}_{4-3}$ regions
that are proportional to $\delta$, and thus differentiate DReg from DRed. This
comes from a part of $|M_{A}|^{2}$:

$|M_{A}|^{2}=-\frac{e^{2}G_{F}^{2}}{4}\left(\frac{1}{p_{e}\cdot
k}\right)^{2}\mathrm{Tr}\left[(\not{P}-\not{k})\gamma^{\mu}\not{k}\gamma^{\alpha}\not{p}_{e}\gamma_{\alpha}\not{k}\gamma^{\nu}(1-\gamma_{5})\right]F_{\mu}F_{\nu}^{*}+....$
(81)

The two schemes give a difference in the value of
$\gamma^{\alpha}\not{p}_{e}\gamma_{\alpha}$ at the order $(\epsilon/2)$, which
then combines with the $(2/\epsilon)$-divergence in the $I_{m,1}$ integrals to
give a finite difference. However, it is easy to check that all the terms
proportional to $\delta$ add up to zero:

$\displaystyle-\frac{1}{12}(z^{2}-4r_{\pi})^{3/2}$ $\displaystyle=$
$\displaystyle\int_{c(z)-d(z)}^{c(z)+d(z)}dy\frac{1}{2}[r_{\pi}+(y-1)(y+z-1)]$
$\displaystyle=$
$\displaystyle-\left(\int_{c(z)-d(z)}^{c(z)+d(z)}dy\int_{0}^{\alpha_{+}}dx+\int_{0}^{c(z)-d(z)}dy\int_{\alpha_{-}}^{\alpha_{+}}dx\right)\frac{4xf(x,y,z)}{(1-x-z+r_{\pi})^{3}}\leavevmode\nobreak\
,$

which means DReg and DRed give the same total decay rate. This is an important
confirmation of the formalism-independence of our result.

Finally, all the integrals in $h(z)$ can in fact be explicitly performed. The
outcome turns out to be exactly equal to $H(z)$ in Eq. (52):

$h(z)=\frac{1}{18}(z^{2}-4r_{\pi})^{3/2}\left\\{3\ln\left[\frac{(z^{2}-4r_{\pi})(2-z-\sqrt{z^{2}-4r_{\pi}})}{(1-z+r_{\pi})(2-z+\sqrt{z^{2}-4r_{\pi}})}\right]+1\right\\}=H(z)\leavevmode\nobreak\
,$ (83)

That equality can also be checked numerically. This implies
$(\delta\Gamma)_{\mathrm{ext,m}}=(\delta\Gamma)_{\mathrm{ext,DR}}$, i.e. the
DR method at $\epsilon\rightarrow 0$ agrees perfectly with the mass-expansion
method at $m_{e}\rightarrow 0$ in the total, IR-finite $K_{e3}$ decay rate.
The effect of small chiral breaking terms proportional to ${m_{e}^{2}}$ in the
bremsstrahlung squared amplitude integral from collinear electron-photon
configurations are reproduced by the space-time DR dependence.

##  VII Final discussions

In our discussion of the QED RC to ${K_{e3}}$ decays, we have tried to further
clarify some aspects of the relatively recent improvements in the theory Seng
et al. (2021a, b, 2022a) and to provide cross-checks for the very small
uncertainty, ${O(10^{-4})}$, they found. By introducing an electron mass
expansion, we were able to demonstrate the cancellation of electron mass
singularities in the radiative inclusive decay rates and verify agreement with
the KLN theorem. As a further check on the results, a study of the inclusive
rates for the case $m_{e}=0$, that is massless QED, was carried out using
dimensional regularization of infrared photonic divergences and electron mass
singularities. In the $d\rightarrow 4$ limit, it confirmed the
$m_{e}\rightarrow 0$ electron mass expansion results. Particularly interesting
is a very small contribution of about $-0.01\%$ from a set of
$m_{e}^{2}$-suppressed terms in the bremsstrahlung squared amplitude that was
subsequently power-enhanced by 1/${m_{e}^{2}}$ from the phase-space
integration. That contribution was found to be needed for full agreement with
the extra finite terms in the DR massless QED calculation that originate from
the loop/bremsstrahlung integrals and from the non-trivial modification of the
phase space formula in $d$ dimension.

The major sources of theory uncertainty in the $K_{e3}$ RC are: (1) lattice
QCD uncertainties in the mesonic $\gamma W$-box diagram that are quantifiable
through standard procedures, and (2) residual hadronic structure-dependent
uncertainties incalculable within Sirlin’s representation (and not constrained
by lattice). In particular, (2) is kept under control through a careful
separation of the non-perturbative pieces from the perturbative ones. Our
results in this work support the error analysis in Refs. Seng et al. (2021a,
b, 2022a), that all the $\ln m_{e}$-enhanced terms are fully contained in the
non-perturbative pieces that were precisely calculated either analytically or
numerically. The remaining, perturbative terms are free from $\ln
m_{e}$-enhancement, hence the standard chiral power counting argument adopted
in those papers provides a reliable estimation of its theory error.

Given the smallness of the RC QED theory uncertainty, the emphasis should now
be placed on other SM theory inputs. In particular, we strongly encourage more
precise lattice QCD calculations of the form factor $f_{+}(t)$, not only its
$t=0$ value but also its $t$-dependence. The latter can provide useful
independent constraints on the $K\pi$ phase space factors. In addition, a
lattice QCD calculation of $f_{+}(t)$ that is free of QED effects avoids any
issue of double counting which is potentially present in experimental
measurements which presumably use an approximate Monte Carlo program to
subtract out QED effects. Of course improved experimental measurements of all
the kaon lifetimes and measurements of the radiative inclusive $K_{\ell 3}$
branching ratios, including the least-precise $K_{S}\mu$ channel, are also
very desirable.

###### Acknowledgements.

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 Chinese Academy of Sciences (CAS)
through a President’s International Fellowship Initiative (PIFI) (Grant No.
2018DM0034), by the VolkswagenStiftung (Grant No. 93562) (U-G.M), and by the
U.S. Department of Energy under Grant DE-SC0012704 (WJM).

## References

  * Cabibbo (1963) N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
  * Kobayashi and Maskawa (1973) M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
  * Seng et al. (2022a) C.-Y. Seng, D. Galviz, M. Gorchtein, and U.-G. Meißner (2022a), eprint 2203.05217.
  * Seng et al. (2018) C.-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, Phys. Rev. Lett. 121, 241804 (2018), eprint 1807.10197.
  * Seng et al. (2019) C. Y. Seng, M. Gorchtein, and M. J. Ramsey-Musolf, Phys. Rev. D100, 013001 (2019), eprint 1812.03352.
  * Czarnecki et al. (2019) A. Czarnecki, W. J. Marciano, and A. Sirlin, Phys. Rev. D 100, 073008 (2019), eprint 1907.06737.
  * Gorchtein (2019) M. Gorchtein, Phys. Rev. Lett. 123, 042503 (2019), eprint 1812.04229.
  * Seng et al. (2020a) C.-Y. Seng, X. Feng, M. Gorchtein, and L.-C. Jin, Phys. Rev. D 101, 111301 (2020a), eprint 2003.11264.
  * Shiells et al. (2021) K. Shiells, P. G. Blunden, and W. Melnitchouk, Phys. Rev. D 104, 033003 (2021), eprint 2012.01580.
  * Hayen (2021) L. Hayen, Phys. Rev. D 103, 113001 (2021), eprint 2010.07262.
  * Hardy and Towner (2020) J. C. Hardy and I. S. Towner, Phys. Rev. C 102, 045501 (2020).
  * Seng et al. (2022b) C.-Y. Seng, D. Galviz, W. J. Marciano, and U.-G. Meißner, Phys. Rev. D 105, 013005 (2022b), eprint 2107.14708.
  * Gonzalez-Alonso et al. (2019) M. Gonzalez-Alonso, O. Naviliat-Cuncic, and N. Severijns, Prog. Part. Nucl. Phys. 104, 165 (2019), eprint 1803.08732.
  * Bryman and Shrock (2019a) D. Bryman and R. Shrock, Phys. Rev. D 100, 053006 (2019a), eprint 1904.06787.
  * Bryman and Shrock (2019b) D. Bryman and R. Shrock, Phys. Rev. D 100, 073011 (2019b), eprint 1909.11198.
  * Belfatto et al. (2020) B. Belfatto, R. Beradze, and Z. Berezhiani, Eur. Phys. J. C 80, 149 (2020), eprint 1906.02714.
  * Tan (2019) W. Tan (2019), eprint 1906.10262.
  * Grossman et al. (2020) Y. Grossman, E. Passemar, and S. Schacht, JHEP 07, 068 (2020), eprint 1911.07821.
  * Coutinho et al. (2020) A. M. Coutinho, A. Crivellin, and C. A. Manzari, Phys. Rev. Lett. 125, 071802 (2020), eprint 1912.08823.
  * Falkowski et al. (2019) A. Falkowski, M. González-Alonso, and Z. Tabrizi, JHEP 05, 173 (2019), eprint 1901.04553.
  * Cirigliano et al. (2019) V. Cirigliano, A. Garcia, D. Gazit, O. Naviliat-Cuncic, G. Savard, and A. Young (2019), eprint 1907.02164.
  * Jho et al. (2020) Y. Jho, S. M. Lee, S. C. Park, Y. Park, and P.-Y. Tseng, JHEP 04, 086 (2020), eprint 2001.06572.
  * Yue and Cheng (2021) C. X. Yue and X. J. Cheng, Nucl. Phys. B 963, 115280 (2021), eprint 2008.10027.
  * Endo and Mishima (2020) M. Endo and S. Mishima, JHEP 08, 004 (2020), eprint 2005.03933.
  * Capdevila et al. (2021) B. Capdevila, A. Crivellin, C. A. Manzari, and M. Montull, Phys. Rev. D 103, 015032 (2021), eprint 2005.13542.
  * Eberhardt et al. (2021) O. Eberhardt, A. P. n. Martínez, and A. Pich, JHEP 05, 005 (2021), eprint 2012.09200.
  * Cheung et al. (2020) K. Cheung, W.-Y. Keung, C.-T. Lu, and P.-Y. Tseng, JHEP 05, 117 (2020), eprint 2001.02853.
  * Crivellin et al. (2020) A. Crivellin, F. Kirk, C. A. Manzari, and M. Montull, JHEP 12, 166 (2020), eprint 2008.01113.
  * Crivellin et al. (2021a) A. Crivellin, F. Kirk, C. A. Manzari, and L. Panizzi, Phys. Rev. D 103, 073002 (2021a), eprint 2012.09845.
  * Crivellin and Hoferichter (2020) A. Crivellin and M. Hoferichter, Phys. Rev. Lett. 125, 111801 (2020), eprint 2002.07184.
  * Kirk (2021) M. Kirk, Phys. Rev. D 103, 035004 (2021), eprint 2008.03261.
  * Crivellin et al. (2021b) A. Crivellin, C. A. Manzari, M. Alguero, and J. Matias, Phys. Rev. Lett. 127, 011801 (2021b), eprint 2010.14504.
  * Falkowski et al. (2021) A. Falkowski, M. González-Alonso, and O. Naviliat-Cuncic, JHEP 04, 126 (2021), eprint 2010.13797.
  * Bečirević et al. (2021) D. Bečirević, F. Jaffredo, A. Peñuelas, and O. Sumensari, JHEP 05, 175 (2021), eprint 2012.09872.
  * Crivellin et al. (2021c) A. Crivellin, M. Hoferichter, and C. A. Manzari, Phys. Rev. Lett. 127, 071801 (2021c), eprint 2102.02825.
  * Lubicz et al. (2009) V. Lubicz, F. Mescia, S. Simula, and C. Tarantino (ETM), Phys. Rev. D 80, 111502 (2009), eprint 0906.4728.
  * Bazavov et al. (2013) A. Bazavov et al., Phys. Rev. D 87, 073012 (2013), eprint 1212.4993.
  * Boyle et al. (2015) P. A. Boyle et al. (RBC/UKQCD), JHEP 06, 164 (2015), eprint 1504.01692.
  * Carrasco et al. (2016) N. Carrasco, P. Lami, V. Lubicz, L. Riggio, S. Simula, and C. Tarantino, Phys. Rev. D 93, 114512 (2016), eprint 1602.04113.
  * Bazavov et al. (2019) A. Bazavov et al. (Fermilab Lattice, MILC), Phys. Rev. D99, 114509 (2019), eprint 1809.02827.
  * Lichard (1997) P. Lichard, Phys. Rev. D 55, 5385 (1997), eprint hep-ph/9702345.
  * Antonelli et al. (2010) M. Antonelli et al. (FlaviaNet Working Group on Kaon Decays), Eur. Phys. J. C 69, 399 (2010), eprint 1005.2323.
  * Hill (2006) R. J. Hill, Phys. Rev. D 74, 096006 (2006), eprint hep-ph/0607108.
  * Bernard et al. (2006) V. Bernard, M. Oertel, E. Passemar, and J. Stern, Phys. Lett. B638, 480 (2006), eprint hep-ph/0603202.
  * Bernard et al. (2009) V. Bernard, M. Oertel, E. Passemar, and J. Stern, Phys. Rev. D80, 034034 (2009), eprint 0903.1654.
  * Abouzaid et al. (2010) E. Abouzaid et al. (KTeV), Phys. Rev. D 81, 052001 (2010), eprint 0912.1291.
  * Moulson (2021) M. Moulson (2021), $V_{us}$ from kaon decays, 11th International Workshop on the CKM Unitarity Triangle (CKM 2021), https://indico.cern.ch/event/891123/contributions/4601856/attachments/2351074/4011941/CKM202021.pdf.
  * Blum et al. (2016) T. Blum et al. (RBC, UKQCD), Phys. Rev. D 93, 074505 (2016), eprint 1411.7017.
  * Durr et al. (2011a) S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert, Phys. Lett. B 701, 265 (2011a), eprint 1011.2403.
  * Durr et al. (2011b) S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert, JHEP 08, 148 (2011b), eprint 1011.2711.
  * Bazavov et al. (2009) A. Bazavov et al. (MILC), PoS CD09, 007 (2009), eprint 0910.2966.
  * Fodor et al. (2016) Z. Fodor, C. Hoelbling, S. Krieg, L. Lellouch, T. Lippert, A. Portelli, A. Sastre, K. K. Szabo, and L. Varnhorst, Phys. Rev. Lett. 117, 082001 (2016), eprint 1604.07112.
  * Bazavov et al. (2018) A. Bazavov et al., Phys. Rev. D 98, 074512 (2018), eprint 1712.09262.
  * Carrasco et al. (2014) N. Carrasco et al. (European Twisted Mass), Nucl. Phys. B 887, 19 (2014), eprint 1403.4504.
  * Bazavov et al. (2014) A. Bazavov et al. (Fermilab Lattice, MILC), Phys. Rev. D 90, 074509 (2014), eprint 1407.3772.
  * Giusti et al. (2017) D. Giusti, V. Lubicz, C. Tarantino, G. Martinelli, F. Sanfilippo, S. Simula, and N. Tantalo, Phys. Rev. D 95, 114504 (2017), eprint 1704.06561.
  * Colangelo et al. (2018) G. Colangelo, S. Lanz, H. Leutwyler, and E. Passemar, Eur. Phys. J. C 78, 947 (2018), eprint 1807.11937.
  * Seng et al. (2021a) C.-Y. Seng, D. Galviz, M. Gorchtein, and U. G. Meißner, Phys. Lett. B 820, 136522 (2021a), eprint 2103.00975.
  * Seng et al. (2021b) C.-Y. Seng, D. Galviz, M. Gorchtein, and U.-G. Meißner, JHEP 11, 172 (2021b), eprint 2103.04843.
  * Seng et al. (2020b) C.-Y. Seng, D. Galviz, and U.-G. Meißner, JHEP 02, 069 (2020b), eprint 1910.13208.
  * Seng et al. (2020c) C.-Y. Seng, X. Feng, M. Gorchtein, L.-C. Jin, and U.-G. Meißner, JHEP 10, 179 (2020c), eprint 2009.00459.
  * Sirlin (1978) A. Sirlin, Rev. Mod. Phys. 50, 573 (1978), [Erratum: Rev. Mod. Phys.50,905(1978)].
  * Seng (2021) C.-Y. Seng, Particles 4, 397 (2021), eprint 2108.03279.
  * Feng et al. (2020) X. Feng, M. Gorchtein, L.-C. Jin, P.-X. Ma, and C.-Y. Seng, Phys. Rev. Lett. 124, 192002 (2020), eprint 2003.09798.
  * Ma et al. (2021) P.-X. Ma, X. Feng, M. Gorchtein, L.-C. Jin, and C.-Y. Seng, Phys. Rev. D 103, 114503 (2021), eprint 2102.12048.
  * Cirigliano et al. (2002) V. Cirigliano, M. Knecht, H. Neufeld, H. Rupertsberger, and P. Talavera, Eur. Phys. J. C23, 121 (2002), eprint hep-ph/0110153.
  * Cirigliano et al. (2004) V. Cirigliano, H. Neufeld, and H. Pichl, Eur. Phys. J. C 35, 53 (2004), eprint hep-ph/0401173.
  * Cirigliano et al. (2008) V. Cirigliano, M. Giannotti, and H. Neufeld, JHEP 11, 006 (2008), eprint 0807.4507.
  * Bloch and Nordsieck (1937) F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).
  * Yennie et al. (1961) D. R. Yennie, S. C. Frautschi, and H. Suura, Annals Phys. 13, 379 (1961).
  * Kinoshita and Sirlin (1959) T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959).
  * Kinoshita (1962) T. Kinoshita, J. Math. Phys. 3, 650 (1962).
  * Lee and Nauenberg (1964) T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964).
  * Roos and Sirlin (1971) M. Roos and A. Sirlin, Nucl. Phys. B 29, 296 (1971).
  * Marciano (1975) W. J. Marciano, Phys. Rev. D 12, 3861 (1975).
  * Meister and Yennie (1963) N. Meister and D. Yennie, Phys. Rev. 130, 1210 (1963).
  * Bytev et al. (2003) V. Bytev, E. Kuraev, A. Baratt, and J. Thompson, Eur. Phys. J. C 27, 57 (2003), [Erratum: Eur.Phys.J.C 34, 523–524 (2004)], eprint hep-ph/0210049.
  * ’t Hooft and Veltman (1972) G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44, 189 (1972).
  * Bollini and Giambiagi (1972) C. G. Bollini and J. J. Giambiagi, Nuovo Cim. B 12, 20 (1972).
  * Ashmore (1972) J. F. Ashmore, Lett. Nuovo Cim. 4, 289 (1972).
  * Gastmans and Meuldermans (1973) R. Gastmans and R. Meuldermans, Nucl. Phys. B 63, 277 (1973).
  * Marciano and Sirlin (1975) W. Marciano and A. Sirlin, Nucl. Phys. B 88, 86 (1975).
  * Marques and Papanicolaou (1975) G. C. Marques and N. Papanicolaou, Phys. Rev. D 12, 1052 (1975).
  * Marciano et al. (1975) W. J. Marciano, G. C. Marques, and N. Papanicolaou, Nucl. Phys. B 96, 237 (1975).
  * Collins (2013) J. Collins, _Foundations of perturbative QCD_ , vol. 32 (Cambridge University Press, 2013), ISBN 978-1-107-64525-7, 978-1-107-64525-7, 978-0-521-85533-4, 978-1-139-09782-6.
  * Becher et al. (2015) T. Becher, A. Broggio, and A. Ferroglia, _Introduction to Soft-Collinear Effective Theory_ , vol. 896 (Springer, 2015), eprint 1410.1892.
  * Gorsky et al. (1989) A. S. Gorsky, B. L. Ioffe, and A. Y. Khodjamirian, Phys. Lett. B 227, 474 (1989).
  * Smilga (1991) A. V. Smilga, Comments Nucl. Part. Phys. 20, 69 (1991).
  * Batley et al. (2018) J. R. Batley et al. (NA48/2), JHEP 10, 150 (2018), eprint 1808.09041.
  * Siegel (1979) W. Siegel, Phys. Lett. B 84, 193 (1979).
  * Siegel (1980) W. Siegel, Phys. Lett. B 94, 37 (1980).