#  Update on $|V_{us}|$ and $|V_{us}/V_{ud}|$ from semileptonic
kaon and pion decays

Chien-Yeah Seng1 Daniel Galviz1 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

Implications for Cabibbo universality based on progress in the study of
semileptonic kaon and pion decays are discussed. Included are recent updates
of experimental input along with improved radiative corrections, form factors
and isospin breaking effects. As a result, we obtain for the Cabibbo-
Kobayashi-Maskawa (CKM) quark mixing matrix element $|V_{us}|=0.22309(56)$
from semileptonic $K_{\ell 3}$ $(K\rightarrow\pi\ell\nu)$ decays and
$|V_{us}/V_{ud}|=0.22908(87)$ from the ratio between the kaon and pion
($\pi_{e3}$) semileptonic decay rates. In both, a lattice QCD value of the
form factor $|f_{+}^{K}(0)|=0.9698(17)$ is employed. The $V_{us}$ from
$K_{\ell 3}$ decays together with $V_{ud}=0.97373(31)$ found from superallowed
nuclear beta decays implies an apparent $3.2\sigma$ violation of the first-row
CKM unitarity condition. The $|V_{us}|/|V_{ud}|$ obtained from the ratio of
weak vector current induced meson decays is consistent with the observed
unitarity violation but found to differ by $2.2\sigma$ from its extraction
using the ratio of weak axial-vector leptonic decay rates
$\Gamma(K\rightarrow\mu\nu)/\Gamma(\pi\rightarrow\mu\nu)$. The situation
suggests a difference between vector and axial-vector derived CKM matrix
elements or a problem with the lattice QCD form factor input. Prospects for
future improvements in comparative precision tests involving $|V_{ud}|$,
$|V_{us}|$ and their ratios are briefly described.

##  I Introduction

Precision tests of the Standard Model (SM) have become increasingly important
given the null results from high-energy colliders in the search for physics
beyond the Standard Model (BSM) Zyla et al. (2020). Deviation from
expectations in the muon anomalous magnetic moment Abi et al. (2021); Aoyama
et al. (2020); Miller et al. (2007, 2012); Jegerlehner and Nyffeler (2009),
hints of lepton flavor universality violation in B decays Aaij et al. (2019,
2014, 2015, 2016) and tests of unitarity in the Cabibbo-Kobayashi-Maskawa
(CKM) quark mixing matrix are all exhibiting potential BSM effects. In the
last case, improvements in the electroweak radiative corrections (RCs) Seng et
al. (2018, 2019); Czarnecki et al. (2019); Seng et al. (2020a); Hayen (2021a,
b); Shiells et al. (2021); Seng (2021) have revealed tension in the first-row
unitarity requirement $|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1$ Hardy and
Towner (2020). Similarly, the difference in the value of $|V_{us}|$ extracted
from $K_{\ell 3}$ and $K_{\mu 2}$ decays needs to be better understood111All
decay processes are understood to be radiative inclusive; for example,
$K_{\ell 3}$ means $K\rightarrow\pi\ell\nu(\gamma)$.. Possible explanations
based on BSM calculations Bryman and Shrock (2019a, b); Kirk (2021); Grossman
et al. (2020); Belfatto et al. (2020); Cheung et al. (2020); Jho et al.
(2020); Yue and Cheng (2021); Endo and Mishima (2020); Capdevila et al.
(2021); Eberhardt et al. (2021); Crivellin and Hoferichter (2020); Coutinho et
al. (2020); Gonzalez-Alonso et al. (2019); Falkowski et al. (2019); Cirigliano
et al. (2019a); Falkowski et al. (2021); Bečirević et al. (2021); Crivellin et
al. (2021a); Tan (2019); Crivellin et al. (2021b); Crivellin et al. (2020);
Crivellin et al. (2021c); Dekens et al. (2021); Belfatto and Berezhiani (2021)
have been conjectured. The confirmation of such ideas will require
improvements in both experiment and SM theory.

In this work, we focus on the extraction of $|V_{us}|$ from
$K\rightarrow\pi\ell\nu$ semileptonic decay processes known as $K_{\ell 3}$.
Our starting point is the comprehensive 2010 FlaviaNet Working Group Report
Antonelli et al. (2010) which presented a thorough review of all relevant
experimental and theoretical information available at the time. Since then,
significant progress has been made on both the experimental and theoretical
fronts, including new measurements of the $K_{S}$ lifetime Abouzaid et al.
(2011); Ambrosino et al. (2011) and branching ratio (BR) Babusci et al.
(2020), updates of the electroweak RCs Ma et al. (2021); Seng et al. (2021a,
b), phase space factors Cirigliano et al. (2019b), $K\pi$ form factor and
isospin-breaking corrections (Aoki et al. (2020) and references therein). Some
of those advances have been more recently discussed by FlaviaNet Working Group
members in the form of proceedings Moulson (2017) and conference slides
Passemar and Moulson (2018); Cirigliano et al. (2019b) rather than more
detailed research publications, making cross-checking difficult. For that
reason, we present in this paper an updated status report on the value of
$|V_{us}|$ extracted from $K_{\ell 3}$ decay properties. With the recent
theory progress, we find that apart from the lattice input of the $K\pi$ form
factor which is a universal multiplicative constant, the experimental errors
from the kaon lifetimes and BRs are by far the dominant sources of
uncertainties in all six channels of $K_{\ell 3}$, which is quite different
from the situation a few years ago, in particular before the new calculation
of the $K_{e3}$ RC Seng et al. (2021a, b).

The progress described above can also be applied to the determination of the
recently proposed ratio $R_{V}=\Gamma(K_{\ell 3})/\Gamma(\pi_{e3})$ as an
alternative approach to study $|V_{us}/V_{ud}|$ Czarnecki et al. (2020),
complementary to the existing method based on $R_{A}=\Gamma(K_{\mu
2})/\Gamma(\pi_{\mu 2})$. We show, following the recent improvements in the
precision level of the $K_{e3}$ Seng et al. (2021a, b) and $\pi_{e3}$ RC Feng
et al. (2020), that $R_{V}$ is now a theoretically cleaner observable than
$R_{A}$. It provides strong motivation to further improve the experimental
precision of the $\pi_{e3}$ BR and $K_{\ell 3}$ decay properties as much as
possible. An experimental next-generation rare pion decay program proposal:
PIONEER Hertzog (2021); Aguilar-Arevalo et al.  would aim to improve the
experimental $\pi_{e3}$ decay rate by a factor of 3 or better, making the
$K_{\ell 3}$ decay rate the dominant uncertainty in $R_{V}$.

The paper is organized as follows: In Sec. II, we give a detailed update on
$|V_{us}f_{+}^{K}(0)|$ and $|V_{us}|$. Similarly, Sec. III provides updates on
$|V_{us}f_{+}^{K}(0)|/|V_{ud}f_{+}^{\pi}(0)|$ and $|V_{us}/V_{ud}|$. We end
with conclusions in Sec. IV. Some technicalities are relegated to the
Appendix.

##  II Updates on $|V_{us}f_{+}^{K}(0)|$ and $|V_{us}|$

We start from the quantity $|V_{us}f_{+}^{K}(0)|$, with $f_{+}^{K}(0)$ the
vector form factor in the $K^{0}\to\pi^{-}$ transition at zero momentum
transfer. It is extracted from $K_{\ell 3}$ decays through the following
master formula Zyla et al. (2020):

$A_{K\ell}\equiv|V_{us}f_{+}^{K}(0)|_{K\ell}=\sqrt{\frac{192\pi^{3}\mathrm{BR}(K\ell)\Gamma_{K}}{G_{F}^{2}M_{K}^{5}C_{K}^{2}S_{\mathrm{EW}}I_{K\ell}(1+\delta_{\mathrm{EM}}^{K\ell}+\delta_{\mathrm{SU(2)}}^{K\ell})}}\leavevmode\nobreak\
,$ (1)

where $G_{F}=1.1663787(6)\times 10^{-5}$ GeV-2 is the Fermi constant obtained
from muon decay, $\Gamma_{K}$ the total kaon decay width, BR($K\ell$) the
$K_{\ell 3}$ branching ratio, and $C_{K}$ a simple isospin factor which equals
1 ($1/\sqrt{2}$) for $K^{0}$ ($K^{+}$) decay. The SM theory inputs to the
right-hand side are as follows: $S_{\mathrm{EW}}=1.0232(3)_{\mathrm{HO}}$ is a
universal short-distance EW factor Marciano and Sirlin (1993), and the
uncertainty comes from higher-order QED effects Erler (2004) which is common
to all channels and will not take part in the weighted average. Meanwhile,
$I_{K\ell}$, $\delta_{\mathrm{EM}}^{K\ell}$ and
$\delta_{\mathrm{SU(2)}}^{K\ell}$ are the phase space factor, the long-
distance electromagnetic (EM) correction and the isospin-breaking correction,
respectively. These are channel-specific inputs. To facilitate the discussion
of correlation effects, we group the values of $A_{K\ell}$ from six
independent $K_{\ell 3}$ channels into a vector:

$A=(A_{K_{L}e},A_{K_{S}e},A_{K^{+}e},A_{K_{L}\mu},A_{K_{S}\mu},A_{K^{+}\mu})^{T}\leavevmode\nobreak\
.$ (2)

The order of the entries is important, as is seen later.

In what follows, we summarize all the data input needed in this work. All the
experimental data of decay widths and BRs are obtained from the 2021 online
update PDG  of the Particle Data Group (PDG) review Zyla et al. (2020).
Knowing that different choices of inputs of statistical analyses of the same
problem may lead to different quantitative conclusions, throughout the
discussion we will explain the similarities and differences in the data inputs
between our work and existing global analysis (in particular the 2010
FlaviaNet review Antonelli et al. (2010) and its updates Moulson (2017);
Passemar and Moulson (2018); Cirigliano et al. (2019b)), and present all the
essential steps in some detail despite that most of them are familiar to
experts; the basic mathematical tools needed in this work are summarized in
the Appendix. With such, all the intermediate and final results in this paper
are fully transparent and can be easily crossed-checked by interested readers,
or compared to similar analyses with possibly different statistical
approaches.

###  II.1 $K_{L}$ experimental inputs

The $K_{L}$ decay width quoted in PDG is obtained through a combined fit from
Refs. Ambrosino et al. (2006a, 2005); Vosburgh et al. (1972), whereas the
$K_{L}e$, $K_{L}\mu$ BRs are fitted from Refs. Ambrosino et al. (2006a);
Alexopoulos et al. (2004), all which have been included in the FlaviaNet 2010
review. The results read:

$\displaystyle K_{L}^{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\mathrm{BR}(K_{L}e)&\mathrm{BR}(K_{L}\mu)&\Gamma_{K_{L}}\end{array}\right)^{T}$
(4) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0.4055(11)&0.2704(7)&1.2866(53)\times
10^{-14}\>\mathrm{MeV}\end{array}\right)^{T}$ (6)

with the correlation matrix (it is symmetric, so we only show the upper right
components for simplicity)

$\mathrm{Corr}(K_{L}^{\mathrm{exp}})=\left(\begin{array}[]{ccc}1&-0.2149932&-0.2676291\\\
&1&-0.08759115\\\ &&1\end{array}\right)$ (7)

from which we can obtain the covariance matrix
$\mathrm{Cov}(K_{L}^{\mathrm{exp}})$ using Eq.75. The contribution of
$K_{L}^{\mathrm{exp}}$ to the covariance matrix of $A$ is given by:

$\mathrm{Cov}(A)_{K_{L}^{\mathrm{exp}}}=\left(\frac{\partial A}{\partial
K_{L}^{\mathrm{exp}}}\right)\cdot\mathrm{Cov}(K_{L}^{\mathrm{exp}})\cdot\left(\frac{\partial
A}{\partial K_{L}^{\mathrm{exp}}}\right)^{T}$ (8)

where

$\left(\frac{\partial A}{\partial
K_{L}^{\mathrm{exp}}}\right)=\left(\begin{array}[]{ccc}\frac{\partial
A_{K_{L}e}}{\partial\mathrm{BR}(K_{L}e)}&0&\frac{\partial
A_{K_{L}e}}{\partial\Gamma_{K_{L}}}\\\ 0&0&0\\\ 0&0&0\\\ 0&\frac{\partial
A_{K_{L}\mu}}{\partial\mathrm{BR}(K_{L}\mu)}&\frac{\partial
A_{K_{L}\mu}}{\partial\Gamma_{K_{L}}}\\\ 0&0&0\\\
0&0&0\end{array}\right)\leavevmode\nobreak\ .$ (9)

###  II.2 $K_{S}$ experimental inputs

The $K_{S}$ decay width quoted in PDG is fitted from Refs. Abouzaid et al.
(2011); Ambrosino et al. (2011); Lai et al. (2002); Bertanza et al. (1997);
Schwingenheuer et al. (1995); Gibbons et al. (1993), and the $K_{S}e$ BR from
Refs. Batley et al. (2007); Ambrosino et al. (2006b); Aloisio et al. (2002).
Notice that not all of them were utilized in the FlaviaNet review and updates,
but only the more recent results from the KTeV Abouzaid et al. (2011), KLOE
Ambrosino et al. (2006b, 2011) and NA48 Lai et al. (2002); Batley et al.
(2007) collaborations. Also, only five channels in $K_{\ell 3}$ were analyzed
in all the past reviews because the $K_{S}\mu$ BR was not independently
measured. The first direct measurement of this BR appeared in year 2020
Babusci et al. (2020), which allows us to finally include all six channels in
the combined analysis of $V_{us}$ for the first time. The experimental results
read:

$\displaystyle K_{S}^{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\mathrm{BR}(K_{S}e)&\mathrm{BR}(K_{S}\mu)&\Gamma_{K_{S}}\end{array}\right)^{T}$
(11) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}7.04(8)\times
10^{-4}&4.56(20)\times 10^{-4}&7.3510(33)\times
10^{-12}\>\mathrm{MeV}\end{array}\right)^{T}\leavevmode\nobreak\ .$ (13)

The PDG provides the correlation coefficient between the two BRs, but not
between the BRs and the total decay width. The latter is expected to be very
small from Ref. Antonelli et al. (2010), so here we simply set it to zero.
With that, we obtain the following correlation matrix:

$\mathrm{Corr}(K_{S}^{\mathrm{exp}})=\left(\begin{array}[]{ccc}1&-0.00144257&0\\\
&1&0\\\ &&1\end{array}\right)\leavevmode\nobreak\ .$ (14)

The contribution of $K_{S}^{\mathrm{exp}}$ to the covariance matrix of $A$ is
given by:

$\mathrm{Cov}(A)_{K_{S}^{\mathrm{exp}}}=\left(\frac{\partial A}{\partial
K_{S}^{\mathrm{exp}}}\right)\cdot\mathrm{Cov}(K_{S}^{\mathrm{exp}})\cdot\left(\frac{\partial
A}{\partial K_{S}^{\mathrm{exp}}}\right)^{T}$ (15)

where

$\left(\frac{\partial A}{\partial
K_{S}^{\mathrm{exp}}}\right)=\left(\begin{array}[]{ccc}0&0&0\\\ \frac{\partial
A_{K_{S}e}}{\partial\mathrm{BR}(K_{S}e)}&0&\frac{\partial
A_{K_{S}e}}{\partial\Gamma_{K_{S}}}\\\ 0&0&0\\\ 0&0&0\\\ 0&\frac{\partial
A_{K_{S}\mu}}{\partial\mathrm{BR}(K_{S}\mu)}&\frac{\partial
A_{K_{S}\mu}}{\partial\Gamma_{K_{S}}}\\\
0&0&0\end{array}\right)\leavevmode\nobreak\ .$ (16)

###  II.3 $K^{+}$ experimental inputs

The $K^{+}$ decay width quoted in PDG is fitted from Refs. Ambrosino et al.
(2008a); Koptev et al. (1995); Ott and Pritchard (1971); Lobkowicz et al.
(1969); Fitch et al. (1965), and the $K^{+}e$, $K^{+}\mu$ BRs from Refs.
Ambrosino et al. (2008b); Chiang et al. (1972). All of them, except two
earlier experiments Lobkowicz et al. (1969); Chiang et al. (1972), were also
used in the FlaviaNet analysis. The results read:

$\displaystyle K_{+}^{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\mathrm{BR}(K^{+}e)&\mathrm{BR}(K^{+}\mu)&\Gamma_{K^{+}}\end{array}\right)^{T}$
(18) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}5.07(4)\times
10^{-2}&3.352(33)\times 10^{-2}&5.3167(86)\times
10^{-14}\>\mathrm{MeV}\end{array}\right)^{T}$ (20)

with the correlation matrix

$\mathrm{Corr}(K_{+}^{\mathrm{exp}})=\left(\begin{array}[]{ccc}1&0.8959847&0.01425396\\\
&1&0.01376368\\\ &&1\end{array}\right)\leavevmode\nobreak\ .$ (21)

The contribution of $K_{+}^{\mathrm{exp}}$ to the covariance matrix of $A$ is
given by:

$\mathrm{Cov}(A)_{K_{+}^{\mathrm{exp}}}=\left(\frac{\partial A}{\partial
K_{+}^{\mathrm{exp}}}\right)\cdot\mathrm{Cov}(K_{+}^{\mathrm{exp}})\cdot\left(\frac{\partial
A}{\partial K_{+}^{\mathrm{exp}}}\right)^{T}$ (22)

where

$\left(\frac{\partial A}{\partial
K_{+}^{\mathrm{exp}}}\right)=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
\frac{\partial A_{K^{+}e}}{\partial\mathrm{BR}(K^{+}e)}&0&\frac{\partial
A_{K^{+}e}}{\partial\Gamma_{K^{+}}}\\\ 0&0&0\\\ 0&0&0\\\ 0&\frac{\partial
A_{K^{+}\mu}}{\partial\mathrm{BR}(K^{+}\mu)}&\frac{\partial
A_{K^{+}\mu}}{\partial\Gamma_{K^{+}}}\end{array}\right).$ (23)

###  II.4 Phase space factor

The phase space factor is defined as222Notice that there is a typo in the
$I_{K\ell}$ formula in many important references, e.g. Refs. Antonelli et al.
(2010); Cirigliano et al. (2012); Passemar and Moulson (2018); Cirigliano et
al. (2019b).:

$I_{K\ell}=\int_{m_{\ell}^{2}}^{(M_{K}-M_{\pi})^{2}}\frac{dt}{M_{K}^{8}}\bar{\lambda}^{3/2}\left(1+\frac{m_{\ell}^{2}}{2t}\right)\left(1-\frac{m_{\ell}^{2}}{t}\right)^{2}\left[\bar{f}_{+}^{2}(t)+\frac{3m_{\ell}^{2}\Delta_{K\pi}^{2}}{(2t+m_{\ell}^{2})\bar{\lambda}}\bar{f}_{0}^{2}(t)\right]\leavevmode\nobreak\
,$ (24)

where $\bar{\lambda}=[t-(M_{K}+M_{\pi})^{2}][t-(M_{K}-M_{\pi})^{2}]$ and
$\Delta_{K\pi}=M_{K}^{2}-M_{\pi}^{2}$. It probes the $t$-dependence of the
(rescaled) $K\pi$ vector and scalar form factors $\bar{f}_{+,0}(t)$, which are
obtained by fitting to the $K_{\ell 3}$ Dalitz plot. There are different ways
to parameterize the form factors, including the Taylor expansion Antonelli et
al. (2010), the $z$-parameterization Hill (2006), the pole parameterization
Lichard (1997) and the dispersive parameterization Bernard et al. (2006,
2009); Abouzaid et al. (2010). In this work, we take the results of the
dispersive parameterization from the latest FlaviaNet updates which claim the
smallest uncertainty Passemar and Moulson (2018); Cirigliano et al. (2019b);
Moulson (2021):

$\displaystyle I_{K}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}I_{K^{0}e}&I_{K^{+}e}&I_{K^{0}\mu}&I_{K^{+}\mu}\end{array}\right)^{T}$
(26) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0.15470(15)&0.15915(15)&0.10247(15)&0.10553(16)\end{array}\right)^{T}\leavevmode\nobreak\
.$ (28)

with the correlation matrix Moulson (2021):

$\mathrm{Corr}(I_{K})=\left(\begin{array}[]{cccc}1&1&0.530&0.521\\\
&1&0.530&0.521\\\ &&1&1\\\ &&&1\end{array}\right)\leavevmode\nobreak\ .$ (30)

The contribution of $I_{K}$ to the covariance matrix of $A$ is given by:

$\mathrm{Cov}(A)_{I_{K}}=\left(\frac{\partial A}{\partial
I_{K}}\right)\cdot\mathrm{Cov}(I_{K})\cdot\left(\frac{\partial A}{\partial
I_{K}}\right)^{T}$ (31)

where

$\left(\frac{\partial A}{\partial
I_{K}}\right)=\left(\begin{array}[]{cccc}\frac{\partial A_{K_{L}e}}{\partial
I_{K^{0}e}}&0&0&0\\\ \frac{\partial A_{K_{S}e}}{\partial I_{K^{0}e}}&0&0&0\\\
0&\frac{\partial A_{K^{+}e}}{\partial I_{K^{+}e}}&0&0\\\ 0&0&\frac{\partial
A_{K_{L}\mu}}{\partial I_{K^{0}\mu}}&0\\\ 0&0&\frac{\partial
A_{K_{S}\mu}}{\partial I_{K^{0}\mu}}&0\\\ 0&0&0&\frac{\partial
A_{K^{+}\mu}}{\partial I_{K^{+}\mu}}\end{array}\right)\leavevmode\nobreak\ .$
(32)

###  II.5 Isospin-breaking corrections

The isospin-breaking correction $\delta_{\mathrm{SU(2)}}^{K\ell}$ is defined
through the deviation of $f_{+}^{K\pi}(0)$ from $f_{+}^{K^{0}\pi^{-}}(0)$
(after scaling out the isospin factor $C_{K}$):

$\delta_{\mathrm{SU(2)}}^{K\ell}=\left(\frac{f_{+}^{K\pi}(0)}{f_{+}^{K^{0}\pi^{-}}(0)}\right)^{2}-1\leavevmode\nobreak\
,$ (33)

so it resides in the $K^{+}$ channel only by construction. Upon neglecting
small EM contributions, it is given by Antonelli et al. (2010):

$\delta_{\mathrm{SU(2)}}^{K^{+}\ell}=\frac{3}{2}\frac{1}{Q^{2}}\left[\frac{\hat{M}_{K}^{2}}{\hat{M}_{\pi}^{2}}+\frac{\chi_{p^{4}}}{2}\left(1+\frac{m_{s}}{\hat{m}}\right)\right]\leavevmode\nobreak\
,$ (34)

where $\hat{M}_{K,\pi}$ are the meson masses in the isospin limit,
$Q^{2}=(m_{s}^{2}-\hat{m}^{2})/(m_{d}^{2}-m_{u}^{2})$, and $\chi_{p^{4}}\simeq
0.219$ is calculable in chiral perturbation theory (ChPT) Gasser and Leutwyler
(1985). The pure Quantum Chromodynamics (QCD) mass parameters can only be
obtained through lattice simulations. Here we quote the most precise results
of $Q$ and $m_{s}/\hat{m}$ from the latest web-update FLA  of the Flavor
Lattice Averaging Group (FLAG) review Aoki et al. (2020):

$\displaystyle Q=23.3(5)\leavevmode\nobreak\ ,\quad
m_{s}/\hat{m}=27.42(12)\leavevmode\nobreak\ ,\quad
N_{f}=2+1\leavevmode\nobreak\
\qquad\quad\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{RBC:2014ntl,Durr:2010vn,Durr:2010aw,MILC:2009ltw,Fodor:2016bgu}{\@@citephrase{(}}{\@@citephrase{)}}}}$
$\displaystyle Q=24.0(8)\leavevmode\nobreak\ ,\quad
m_{s}/\hat{m}=27.23(10)\leavevmode\nobreak\ ,\quad
N_{f}=2+1+1\leavevmode\nobreak\ .\quad\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bazavov:2017lyh,EuropeanTwistedMass:2014osg,FermilabLattice:2014tsy,Giusti:2017dmp}{\@@citephrase{(}}{\@@citephrase{)}}}}$
(35)

Since $Q$ is by far the main contributor of the uncertainty in
$\delta_{\mathrm{SU(2)}}^{K^{+}\ell}$, we choose the more precise data set
from $N_{f}=2+1$ for numerical applications. Meanwhile, Ref. Aoki et al.
(2020) did not provide the explicit values of $\hat{M}_{K,\pi}$, so we quote
them from the 2017 FLAG review Aoki et al. (2017): $\hat{M}_{\pi}=134.8(3)$
MeV, $\hat{M}_{K}=494.2(3)$ MeV. Putting pieces together, we have:

$\delta_{\mathrm{SU(2)}}^{K^{+}\ell}=0.0457(20)\leavevmode\nobreak\ .$ (36)

The contribution of $\delta_{\mathrm{SU(2)}}$ to the covariance matrix of $A$
is given by:

$\mathrm{Cov}(A)_{\delta_{\mathrm{SU(2)}}}=\left(\frac{\partial
A}{\partial\delta_{\mathrm{SU(2)}}}\right)\cdot\mathrm{Cov}(\delta_{\mathrm{SU(2)}})\cdot\left(\frac{\partial
A}{\partial\delta_{\mathrm{SU(2)}}}\right)^{T}$ (37)

where

$\left(\frac{\partial
A}{\partial\delta_{\mathrm{SU(2)}}}\right)=\left(\begin{array}[]{cccccc}0&0&\frac{\partial
A_{K^{+}e}}{\partial\delta_{\mathrm{SU(2)}}^{K^{+}\ell}}&0&0&\frac{\partial
A_{K^{+}\mu}}{\partial\delta_{\mathrm{SU(2)}}^{K^{+}\ell}}\end{array}\right)^{T}\leavevmode\nobreak\
.$ (38)

It should be pointed out that the parameter $Q$ can also be obtained
phenomenologically. For instance, Ref.Colangelo et al. (2018) obtained
$Q=22.1(7)$ from $\eta\rightarrow 3\pi$ decay, which is marginally discrepant
with the FLAG average based on lattice calculations. We notice that different
versions of FlaviaNet updates in the past few years had adopted different
choices for their quark mass parameters: Ref.Moulson (2017) took the parameter
$Q$ from lattice, while Refs.Passemar and Moulson (2018); Cirigliano et al.
(2019b) adopted the phenomenological value $Q=22.1(7)$ (together with a
slightly different ChPT parameter $\chi_{p^{4}}\simeq 0.252$, of which the
origin was not clearly explained), which returned a somewhat larger isospin
breaking correction $\delta_{\mathrm{SU(2)}}^{K^{+}\ell}=0.0522(34)$.

###  II.6 Long-distance EM corrections

The last theory input is the long-distance EM correction. It was taken in the
FlaviaNet review and its updates from the ChPT calculation at
$\mathcal{O}(e^{2}p^{2})$ Cirigliano et al. (2008), with a theory uncertainty
of the order of $10^{-3}$. However, a novel framework based on Sirlin’s
representation of RC Sirlin (1978) was recently pioneered Seng et al. (2020b,
c). With this framework and new lattice QCD inputs of the meson $\gamma W$-box
diagrams Feng et al. (2020); Ma et al. (2021), $\delta_{\mathrm{EM}}^{Ke}$
were re-evaluated with a significant increase in precision level reaching
$10^{-4}$ Seng et al. (2021a, b). A similar update on
$\delta_{\mathrm{EM}}^{K\mu}$ is not yet available due to the more complicated
error analysis but will be carried out in the near future. Meanwhile, an
important cross-check would be to compute the full $K_{\ell 3}$ RC (both the
virtual and real corrections). This can be based on the existing technique
that was proven successful in the study of the $K_{\mu 2}/\pi_{\mu 2}$ RCs
Giusti et al. (2018), although its generalization to $K_{\ell 3}$ is expected
to be more challenging and could take up to a decade to reach $10^{-3}$
precision Boyle et al. .

In this paper we choose to take the results of $\delta_{\mathrm{EM}}^{K\mu}$
from ChPT, and that of $\delta_{\mathrm{EM}}^{Ke}$ from the new calculation.
Since these two evaluations are based on very different starting points, it is
only natural to assume that they are uncorrelated. In the $K\mu$ channels we
have Cirigliano et al. (2008):

$\displaystyle\delta_{\mathrm{EM}}^{K\mu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\delta_{\mathrm{EM}}^{K^{0}\mu}&\delta_{\mathrm{EM}}^{K^{+}\mu}\end{array}\right)^{T}$
(40) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}1.40(22)\times
10^{-2}&0.016(250)\times 10^{-2}\end{array}\right)^{T}$ (42)

with the correlation matrix

$\mathrm{Corr}(\delta_{\mathrm{EM}}^{K\mu})=\left(\begin{array}[]{cc}1&0.081\\\
&1\end{array}\right)\leavevmode\nobreak\ .$ (43)

Meanwhile, in the $Ke$ channels we have:

$\delta_{\mathrm{EM}}^{Ke}=\left(\begin{array}[]{cc}\delta_{\mathrm{EM}}^{K^{0}e}&\delta_{\mathrm{EM}}^{K^{+}e}\end{array}\right)^{T},$
(44)

where Seng et al. (2021a, b)

$\displaystyle\delta_{\mathrm{EM}}^{K^{0}e}$ $\displaystyle=$ $\displaystyle
1.16(2)_{\mathrm{sg}}(1)_{\mathrm{lat}}(1)_{\mathrm{NF}}(2)_{e^{2}p^{4}}\times
10^{-2}$ $\displaystyle=$ $\displaystyle 1.16(3)\times 10^{-2}$
$\displaystyle\delta_{\mathrm{EM}}^{K^{+}e}$ $\displaystyle=$ $\displaystyle
0.21(2)_{\mathrm{sg}}(1)_{\left\langle
r_{K}^{2}\right\rangle}(1)_{\mathrm{lat}}(4)_{\mathrm{NF}}(1)_{e^{2}p^{4}}\times
10^{-2}$ (45) $\displaystyle=$ $\displaystyle 0.21(5)\times
10^{-2}\leavevmode\nobreak\ .$

The correlation matrix of $\delta_{\mathrm{EM}}^{K\ell}$ was not given in
Refs. Seng et al. (2021a, b) and is derived here for the first time.

First of all, we realize that most of the uncertainties in Eq. (45) were
estimated through simple power-counting arguments on top of the central values
in each respective channel, so the most natural choice is to take them as
independent since assuming any correlation would be equally arbitrary.
However, there is a piece that has well-defined correlations, namely the
lattice calculation of the meson axial $\gamma W$-box diagrams, which enters
$\delta_{\mathrm{EM}}^{Ke}$ effectively through the low-energy constants
(LECs) in ChPT Seng et al. (2020c):

$\displaystyle\delta_{\mathrm{EM}}^{K^{0}e}$ $\displaystyle=$ $\displaystyle
2e^{2}\left[\frac{4}{3}X_{1}-\frac{1}{2}\bar{X}_{6}^{\mathrm{phys}}\right]+...$
$\displaystyle\delta_{\mathrm{EM}}^{K^{+}e}$ $\displaystyle=$ $\displaystyle
2e^{2}\left[2\left(-\frac{2}{3}X_{1}-\frac{1}{2}\bar{X}_{6}^{\mathrm{phys}}\right)-\left(\frac{4}{3}X_{1}-\frac{1}{2}\bar{X}_{6}^{\mathrm{phys}}\right)\right]+...\leavevmode\nobreak\
.$ (46)

These two combinations of LECs were pinned down by two independent lattice
calculations of the axial $\gamma W$-box diagrams Feng et al. (2020); Ma et
al. (2021):

$\displaystyle\Box_{\gamma W}^{\pi}$ $\displaystyle:$
$\displaystyle-\frac{2}{3}X_{1}-\frac{1}{2}\bar{X}_{6}^{\mathrm{phys}}=-7.0(3)\times
10^{-3}$ $\displaystyle\Box_{\gamma W}^{K}$ $\displaystyle:$
$\displaystyle\frac{4}{3}X_{1}-\frac{1}{2}\bar{X}_{6}^{\mathrm{phys}}=-11.3(5)\times
10^{-3}\leavevmode\nobreak\ .$ (47)

Therefore, the variations of $\delta_{\mathrm{EM}}^{Ke}$ due to the lattice
uncertainties are given by:

$\delta\left(\delta_{\mathrm{EM}}^{K^{0}e}\right)=2e^{2}\sigma_{K,\mathrm{lat}}\leavevmode\nobreak\
,\quad\delta\left(\delta_{\mathrm{EM}}^{K^{+}e}\right)=2e^{2}\left[2\sigma_{\pi,\mathrm{lat}}-\sigma_{K,\mathrm{lat}}\right]\leavevmode\nobreak\
,$ (48)

where $\sigma_{K,\mathrm{lat}}=0.5\times 10^{-3}$ and
$\sigma_{\pi,\mathrm{lat}}=0.3\times 10^{-3}$. The two expressions above
depend on a common quantity $\sigma_{K,\mathrm{lat}}$, which gives a non-zero
correlation:

$\left\langle\delta_{\mathrm{EM}}^{K^{0}e}\delta_{\mathrm{EM}}^{K^{+}e}\right\rangle-\left\langle\delta_{\mathrm{EM}}^{K^{0}e}\right\rangle\left\langle\delta_{\mathrm{EM}}^{K^{+}e}\right\rangle=-4e^{4}\sigma_{K,\mathrm{lat}}^{2}\leavevmode\nobreak\
.$ (49)

As a consequence, the correlation matrix reads:

$\mathrm{Corr}(\delta_{\mathrm{EM}}^{Ke})=\left(\begin{array}[]{cc}1&\Delta\\\
&1\end{array}\right)$ (50)

where

$\Delta=-\frac{4e^{4}\sigma_{K,\mathrm{lat}}^{2}}{\sigma_{\delta_{\mathrm{EM}}^{K^{0}e}}\sigma_{\delta_{\mathrm{EM}}^{K^{+}e}}},$
(51)

with $\sigma_{\delta_{\mathrm{EM}}^{K^{0}e}}\simeq 0.03\times 10^{-2}$,
$\sigma_{\delta_{\mathrm{EM}}^{K^{+}e}}\simeq 0.05\times 10^{-2}$ as given
above.

We may now combine all the independent long-distance EM corrections as:

$\delta_{\mathrm{EM}}=\left(\begin{array}[]{cccc}\delta_{\mathrm{EM}}^{K^{0}e}&\delta_{\mathrm{EM}}^{K^{+}e}&\delta_{\mathrm{EM}}^{K^{0}\mu}&\delta_{\mathrm{EM}}^{K^{+}\mu}\end{array}\right)^{T}.$
(52)

Its correlation matrix is given by the following block-diagonal matrix:

$\mathrm{Corr}(\delta_{\mathrm{EM}})=\left(\begin{array}[]{cc}\mathrm{Corr}(\delta_{\mathrm{EM}}^{Ke})&0\\\
0&\mathrm{Corr}(\delta_{\mathrm{EM}}^{K\mu})\end{array}\right).$ (53)

The contribution of $\delta_{\mathrm{EM}}$ to the covariance matrix of $A$ is
given by:

$\mathrm{Cov}(A)_{\delta_{\mathrm{EM}}}=\left(\frac{\partial
A}{\partial\delta_{\mathrm{EM}}}\right)\cdot\mathrm{Cov}(\delta_{\mathrm{EM}})\cdot\left(\frac{\partial
A}{\partial\delta_{\mathrm{EM}}}\right)^{T}$ (54)

where

$\left(\frac{\partial
A}{\partial\delta_{\mathrm{EM}}}\right)=\left(\begin{array}[]{cccc}\frac{\partial
A_{K_{L}e}}{\partial\delta_{\mathrm{EM}}^{K^{0}e}}&0&0&0\\\ \frac{\partial
A_{K_{S}e}}{\partial\delta_{\mathrm{EM}}^{K^{0}e}}&0&0&0\\\ 0&\frac{\partial
A_{K^{+}e}}{\partial\delta_{\mathrm{EM}}^{K^{+}e}}&0&0\\\ 0&0&\frac{\partial
A_{K_{L}\mu}}{\partial\delta_{\mathrm{EM}}^{K^{0}\mu}}&0\\\ 0&0&\frac{\partial
A_{K_{S}\mu}}{\partial\delta_{\mathrm{EM}}^{K^{0}\mu}}&0\\\
0&0&0&\frac{\partial
A_{K^{+}\mu}}{\partial\delta_{\mathrm{EM}}^{K^{+}\mu}}\end{array}\right)\leavevmode\nobreak\
.$ (55)

###  II.7 Final result

| $|V_{us}f_{+}^{K}(0)|$ | Correlation Matrix
---|---|---
$K_{L}e$ | $0.21617(46)_{\mathrm{exp}}(10)_{I_{K}}(3)_{\delta_{\mathrm{EM}}}\,\,\,\,$ | 1 | 0.021 | 0.025 | 0.519 | 0.004 | 0.017
$K_{S}e$ | $0.21530(122)_{\mathrm{exp}}(10)_{I_{K}}(3)_{\delta_{\mathrm{EM}}}\,\,$ |  | 1 | 0.009 | 0.012 | 0.000 | 0.006
$K^{+}e$ | $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.21714(88)_{\mathrm{exp}}(10)_{I_{K}}(21)_{\delta_{\mathrm{SU(2)}}}(5)_{\delta_{\mathrm{EM}}}$ |  |  | 1 | 0.016 | 0.002 | 0.871
$K_{L}\mu$ | $0.21664(50)_{\mathrm{exp}}(16)_{I_{K}}(24)_{\delta_{\mathrm{EM}}}\,\,$ |  |  |  | 1 | 0.029 | 0.047
$K_{S}\mu$ | $\,\,0.21265(466)_{\mathrm{exp}}(16)_{I_{K}}(23)_{\delta_{\mathrm{EM}}}\,$ |  |  |  |  | 1 | 0.006
$K^{+}\mu$ | $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.21703(108)_{\mathrm{exp}}(16)_{I_{K}}(21)_{\delta_{\mathrm{SU(2)}}}(26)_{\delta_{\mathrm{EM}}}$ |  |  |  |  |  | 1
Average: $Ke$ | $0.21626(40)_{K}(3)_{\mathrm{HO}}$ |  |  |  |  |  |
Average: $K\mu$ | $0.21667(52)_{K}(3)_{\mathrm{HO}}$ |  |  |  |  |  |
Average: tot | $0.21635(39)_{K}(3)_{\mathrm{HO}}$ |  |  |  |  |  |
Table 1:  Individual values and weighted average of $|V_{us}f_{+}^{K}(0)|$,
with independent uncertainties displayed separately. Notice that the
uncertainties from the BR and $\Gamma_{K}$ are generally correlated, so we
display only their combined uncertainty as “exp”.

With the above, we may obtain the values of $|V_{us}f_{+}^{K}(0)|$ from each
channel, which are summarized in the left panel of Table 1. Making use of the
total covariance matrix of $A$:

$\mathrm{Cov}(A)=\mathrm{Cov}(A)_{K_{L}^{\mathrm{exp}}}+\mathrm{Cov}(A)_{K_{S}^{\mathrm{exp}}}+\mathrm{Cov}(A)_{K_{+}^{\mathrm{exp}}}+\mathrm{Cov}(A)_{I_{K}}+\mathrm{Cov}(A)_{\delta_{\mathrm{SU(2)}}}+\mathrm{Cov}(A)_{\delta_{\mathrm{EM}}}\leavevmode\nobreak\
,$ (56)

from which the total correlation matrix in the right panel of Table 1 can be
calculated, the weighted average of $|V_{us}f_{+}^{K}(0)|$ can then be
obtained using Eq.(80). Given the different theory statuses of $K_{e3}$ and
$K_{\mu 3}$, we present simultaneously the average values of
$|V_{us}f_{+}^{K}(0)|$ by weighting over the $Ke$ channels, the $K\mu$
channels, and both. Notice that the uncertainty from $S_{\mathrm{EW}}$ is
common to all channels and does not enter the weighting process. Therefore, we
choose to display it only in the weighted averages, i.e. the last three rows
in Table 1. We find that the $Ke$ and $K\mu$ averages agree well with each
other within uncertainties. Finally, the 2020 website update of the FLAG
review quoted FLA :

$\begin{array}[]{ccc}|f_{+}^{K}(0)|=0.9698(17)&N_{f}=2+1+1&\quad\quad\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Carrasco:2016kpy,Bazavov:2018kjg}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
|f_{+}^{K}(0)|=0.9677(27)&N_{f}=2+1&\quad\quad\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bazavov:2012cd,Boyle:2015hfa}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
|f_{+}^{K}(0)|=0.9560(57)(62)&N_{f}=2&\text{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Lubicz:2009ht}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}$
(57)

We choose the most precise value from $N_{f}=2+1+1$ for numerical
applications. With that we obtain:

$|V_{us}|_{K_{\ell
3}}=0.22309(40)_{K}(39)_{\mathrm{lat}}(3)_{\mathrm{HO}}\leavevmode\nobreak\ .$
(58)

Let us discuss the results above. First, both the central value and the total
uncertainty in the weighted average of $|V_{us}f_{+}^{K}(0)|$ experience no
significant change compared to those in previous reviews (e.g. $0.21654(41)$
in Ref. Moulson (2017)), but not the composition of uncertainties in each
channel. In our latest analysis, the combined experimental uncertainty from
the kaon lifetime and BRs are by far the dominant source of uncertainty in all
channels. This is quite different from a few years ago, wherein some channels
(e.g. $K^{+}e$) the theory and experimental uncertainties are comparable. Such
changes are mainly due to the improved theory precision in
$\delta_{\mathrm{SU(2)}}^{K\ell}$ and $\delta_{\mathrm{EM}}^{Ke}$.

We may also review the status of the top-row CKM unitarity. The best
extraction of $|V_{ud}|$ comes from superallowed $0^{+}\to 0^{+}$ beta decays,
but its precise value depends on the theory inputs of the single-nucleon RC
and nuclear structure corrections. In particular, it was recently pointed that
several potentially large new nuclear corrections (NNCs) that reside in the
nuclear $\gamma W$-box diagrams were missed in the existing nuclear structure
calculations Seng et al. (2019); Gorchtein (2019); their true sizes are poorly
understood and are at present only roughly estimated based on a simple Fermi
gas model. After alerting the readers about the possible (small) quantitative
difference due to different choices of theory inputs, let us quote, just for
this work, the result from the latest review by Hardy and Towner Hardy and
Towner (2020):

$|V_{ud}|_{0^{+}}=0.97373(11)_{\mathrm{exp}}(9)_{\mathrm{RC}}(27)_{\mathrm{NS}}\leavevmode\nobreak\
,$ (59)

where “exp” is the combined uncertainty from experiment and the so-called
“outer” correction, “RC” the theory uncertainty from the single-nucleon
(inner) RC, and “NS” the nuclear structure uncertainty that originates
primarily from the NNCs. Combining Eqs.(58) and (59) gives:

$|V_{ud}|_{0^{+}}^{2}+|V_{us}|_{K_{\ell
3}}^{2}-1=-0.0021(2)_{V_{ud},\mathrm{exp}}(2)_{V_{ud},\mathrm{RC}}(5)_{V_{ud},\mathrm{NS}}(2)_{V_{us},K}(2)_{V_{us},\mathrm{lat}}\leavevmode\nobreak\
,$ (60)

while the SM prediction (after neglecting the small $|V_{ub}|^{2}$) is 0. The
above indicates an apparent anomaly in the top-row CKM unitarity with the
significance level of $3.2\sigma$, which could increase to as much as
$5.6\sigma$ if we imagine that the NS uncertainty was significantly reduced
while the central value of $|V_{ud}|$ remained unchanged. This provides a
strong motivation for nuclear theorists to perform ab-initio calculations of
the NS correction in superallowed beta decays to reduce its theory
uncertainty.

##  III  $|V_{us}f_{+}^{K}(0)|/|V_{ud}f_{+}^{\pi}(0)|$ and $|V_{us}/V_{ud}|$
from $K/\pi$ semileptonic decays

In addition to its contribution to an apparent violation of the top-row CKM
unitarity, Eq. (58) also shows a direct disagreement at the level $\sim
2.8\sigma$ with the same quantity extracted from $K_{\mu 2}$ decay:
$|V_{us}|_{K_{\mu 2}}=0.2252(5)$ Zyla et al. (2020). The latter is obtained
from the ratio $R_{A}=\Gamma(K_{\mu 2})/\Gamma(\pi_{\mu 2})$, which gives the
value of $|V_{us}f_{K^{+}}|/|V_{ud}f_{\pi^{+}}|$, with $f_{K^{+}}$ and
$f_{\pi^{+}}$ the decay constants of the charged kaon and pion Marciano
(2004):

$\left|\frac{V_{us}f_{K^{+}}}{V_{ud}f_{\pi^{+}}}\right|=0.23871(20)_{\mathrm{RC}}\times
R_{A}^{1/2}\leavevmode\nobreak\ ,$ (61)

where the theory uncertainty of $0.084\%$ at the right-hand side originates
from residual long-distance RCs that do not cancel in the ratio. This residual
RC has been calculated using both ChPT Cirigliano and Neufeld (2011) and
lattice QCD Giusti et al. (2018) with comparable sizes of theory
uncertainties. The two calculations agree well with each other, showing that
the theory error in this input is under good control. Following PDG, we
utilize the ChPT input in Eq.(61) for illustration (throughout this work we
add nothing new in the $R_{A}$ analysis; everything is the same as in the 2021
online version of the PDG review).

The above discrepancy may indicate the presence of BSM effects or possible
unidentified SM corrections that are not reflected in the existing error
estimation, such as a smaller value for $|f_{+}^{K}(0)|$ outside the range of
the quoted $N_{f}=2+1+1$ lattice QCD result. To further explore these
possibilities, in particular the latter, Ref. Czarnecki et al. (2020)
suggested to study a new ratio $R_{V}=\Gamma(K_{\ell 3})/\Gamma(\pi_{e3})$
(which takes different values in different $K\ell$ channels), where the $V$
denotes the fact that such decays are due to weak vector current interactions.
Like $R_{A}$, it results from a ratio of weak interaction meson decays
(induced by vector rather than axial-vector interactions) for which
theoretical uncertainties partially cancel. A comparison of $R_{V}$ and
$R_{A}$ can, in principle, unveil the influence of BSM physics.

We first recall the SM prediction of the $\pi_{e3}$ decay width:

$\Gamma_{\pi_{e3}}=\frac{G_{F}^{2}|V_{ud}|^{2}M_{\pi^{+}}^{5}|f_{+}^{\pi}(0)|^{2}}{64\pi^{3}}(1+\mathrm{RC}_{\pi})I_{\pi}\leavevmode\nobreak\
.$ (62)

The left-hand side is calculated from the experimental measurement of the
charged pion lifetime and the semileptonic decay BR Zyla et al. (2020):

$\tau_{\pi^{+}}=2.6033(5)\times 10^{-8}\>\mathrm{s}\leavevmode\nobreak\
,\quad\mathrm{BR}(\pi_{e3})=1.038(6)\times 10^{-8}\leavevmode\nobreak\ .$ (63)

Notice the slight modification of the BR from the PDG value that took into
account the effect of the updated $\mathrm{BR}(\pi_{e2})$ normalization
Czarnecki et al. (2020). On the right-hand side, $|f_{+}^{\pi}(0)|$ is the
$\pi^{+}\to\pi^{0}$ form factor at zero momentum transfer which equals 1 in
the isospin limit (isospin-breaking correction is negligible due to the
Behrends-Sirlin-Ademollo-Gatto theorem Behrends and Sirlin (1960); Ademollo
and Gatto (1964)). In this limit, the phase space integral $I_{\pi}$ is
calculable:

$\displaystyle I_{\pi}$ $\displaystyle=$
$\displaystyle\int_{2\sqrt{r_{\pi}}}^{1+r_{\pi}-r_{e}}dz\int_{c(z)-d(z)}^{c(z)+d(z)}dy\left[4(1-y)(y+z-1)+r_{e}(4y+3z-3)-4r_{\pi}+r_{e}(r_{\pi}-r_{e})\right]$
(64) $\displaystyle=$ $\displaystyle 7.3764\times 10^{-8}\leavevmode\nobreak\
,$

where

$r_{\pi}=\frac{M_{\pi^{0}}^{2}}{M_{\pi^{+}}^{2}}\leavevmode\nobreak\
,\>\>r_{e}=\frac{m_{e}^{2}}{M_{\pi^{+}}^{2}}\leavevmode\nobreak\
,\>\>c(z)=\frac{(2-z)(1+r_{e}+r_{\pi}-z)}{2(1+r_{\pi}-z)}\leavevmode\nobreak\
,\>\>d(z)=\frac{\sqrt{z^{2}-4r_{\pi}}(1+r_{\pi}-r_{e}-z)}{2(1+r_{\pi}-z)}\leavevmode\nobreak\
,$ (65)

in analogy to the well-known $K_{\ell 3}$ phase space formula (see, e.g.
Ref.Seng et al. (2021b)). $\mathrm{RC}_{\pi}$ is the electroweak RC in the
pion semileptonic decay which was recently determined to high precision with
lattice QCD: $\mathrm{RC}_{\pi}=0.0332(1)_{\gamma W}(3)_{\mathrm{HO}}$ Feng et
al. (2020). Combining Eqs. (1) and (62) gives:

$\left|\frac{V_{us}f_{+}^{K}(0)}{V_{ud}f_{+}^{\pi}(0)}\right|_{K\ell}=\sqrt{\frac{3}{C_{K}^{2}}\left(\frac{M_{\pi^{+}}}{M_{K}}\right)^{5}\frac{I_{\pi}}{I_{K\ell}}\frac{1+\mathrm{RC_{\pi}}}{S_{\mathrm{EW}}(1+\delta_{\mathrm{EM}}^{K\ell}+\delta_{\mathrm{SU(2)}}^{K\ell})}}\times(R_{V}^{K\ell})^{1/2}\leavevmode\nobreak\
,$ (66)

which provides a measure of $|V_{us}f_{+}^{K}(0)|/|V_{ud}f_{+}^{\pi}(0)|$.

There are several benefits in studying $R_{V}$. First, uncertainties from
short-distance electroweak RCs (contained in $\mathrm{RC}_{\pi}$ and
$S_{\mathrm{EW}}$, although they are numerically smaller than the other SM
theory uncertainties, e.g. those coming from $I_{K}$ and
$\delta_{\mathrm{SU(2)}}$) as well as BSM effects that are common to the
numerator and denominator (e.g. those correcting $G_{F}$ through the muon
lifetime) cancel each other in the $R_{V}$ ratio. This means as follows:
should one observes a significant discrepancy between the values of
$|V_{us}/V_{ud}|$ obtained from $R_{V}$ and $R_{A}$, then its possible BSM
explanations would be more limited than those which could be used to explain
the discrepancy between the values of $|V_{us}|$ from $K_{\mu 3}$ and $R_{A}$.
This makes $R_{V}$ a useful gauge to search for possibly large non-universal
systematic effects, especially those from the SM. Second, the recent
improvements in SM theory precision, in particular the electroweak RCs, makes
$R_{V}$ an extremely clean observable from the theory aspect. Consider, for
instance, the $K_{L}e$ channel. Substituting all the SM theory inputs we
discussed above, one obtains:

$\left|\frac{V_{us}f_{+}^{K}(0)}{V_{ud}f_{+}^{\pi}(0)}\right|_{K_{L}e}=4.9786(24)_{I_{K}}(7)_{\mathrm{RC}_{K}}(2)_{\mathrm{RC}_{\pi}}\cdot
10^{-5}\times(R_{V}^{K_{L}e})^{1/2}\leavevmode\nobreak\ .$ (67)

We see that the total theory uncertainty on the right-hand side is only
0.051%, which is already better than $R_{A}$. Moreover, the above is only for
one channel in $K_{\ell 3}$. A further reduction of uncertainty is achieved
once all six channels are weighted over.

Upon substituting the experimental inputs into Eqs.(61) and (66), we find:

$\displaystyle\left|\frac{V_{us}f_{K^{+}}}{V_{ud}f_{\pi^{+}}}\right|$
$\displaystyle=$ $\displaystyle
0.27600(29)_{\mathrm{exp}}(23)_{\mathrm{RC}}\leavevmode\nobreak\
,\quad\text{0.13\% precision}$
$\displaystyle\left|\frac{V_{us}f_{+}^{K}(0)}{V_{ud}f_{+}^{\pi}(0)}\right|$
$\displaystyle=$ $\displaystyle
0.22216(64)_{\mathrm{BR}(\pi_{e3})}(40)_{K}(2)_{\tau_{\pi^{+}}}(1)_{\mathrm{RC}_{\pi}}\leavevmode\nobreak\
,\quad\text{0.34\% precision}$ (68)

where the first line is from $R_{A}$, and the second line is from $R_{V}$
weighted over all six channels. The precision of the latter is limited
primarily by the uncertainty in $\mathrm{BR}(\pi_{e3})$, and secondarily by
the $K_{\ell 3}$ experiments. Future improvements of the experimental
precision in these areas are therefore urgently needed.

At this point it is interesting to discuss the relevance with respect to
PIONEER, the proposed next-generation experiment for rare pion decays which
may take place in PSI or TRIUMF Aguilar-Arevalo et al. ; Hertzog (2021). It is
originally designed for an improved measurement of the ratio
$R_{e/\mu}=\Gamma(\pi^{+}\to
e^{+}\nu(\gamma))/\Gamma(\pi^{+}\to\mu^{+}\nu(\gamma))$ to test lepton
universality, but the optimized detector is also ideal for a high-precision
measurement of $\mathrm{BR}(\pi_{e3})$. The current best measurement of the
latter is from the PIBETA experiment Pocanic et al. (2004), which leads to the
following extracted value of $|V_{ud}|$ Feng et al. (2020):

$|V_{ud}|_{\pi_{e3}}=0.9740(28)_{\mathrm{exp}}(1)_{\mathrm{th}}\leavevmode\nobreak\
.$ (69)

Despite being theoretically clean, it is 10 times less precise than the
superallowed beta decay extraction (see Eq.(59)). To make
$|V_{ud}|_{\pi_{e3}}$ competitive requires a 10 times reduction of the
experimental uncertainty, which may require 100 times the statistics of the
existing measurement and comparable reduction of systematics and backgrounds,
a very ambitious long-term goal. However, the introduction of $R_{V}$ provides
a new physical significance to $\pi_{e3}$, not just in terms of $|V_{ud}|$ but
also $|V_{us}/V_{ud}|$. With this, it is most beneficial to plan the next-
generation $\pi_{e3}$ experiment for two stages:

  * •

The first stage would primarily aim to improve the precision of the ratio,
$R_{e/\mu}$ (its primary goal) by an order of magnitude. That same phase could
be used to improve the precision of BR$(\pi_{e3})$ by a factor of 3 or better
compared to the existing PIBETA result. That would reduce the uncertainty in
$R_{V}$ to a level comparable to $R_{A}$, making for an interesting
confrontation. If accompanied by future improvement in $K_{\ell 3}$
experiments, $R_{V}$ could eventually surpass $R_{A}$ as the primary means to
constrain $|V_{us}/V_{ud}|$.

  * •

In the second stage an overall improvement of a factor of 10 improvement in
the $\mathrm{BR}(\pi_{e3})$ precision is required to compete with superallowed
beta decays for precision in extracting $V_{ud}$. It is, however, much more
challenging and is not yet at the achievable level in the present technical
design Her .

Figure 1: A combined plot of $|V_{ud}|$ from superallowed beta decays (red
band), $|V_{us}|$ from $K_{\ell 3}$ (blue band), $|V_{us}/V_{ud}|$ from
$R_{A}$ (green band) and $R_{V}$ (brown band), together with the first-row CKM
unitarity requirement (black line). Notice that the blue and brown bands are
highly correlated as they rely on the same set of inputs from $K_{\ell 3}$, so
they should not be taken as independent constraints.

We close this section by reporting the current extracted values of
$|V_{us}/V_{ud}|$ from $R_{A}$ and $R_{V}$ respectively, by supplementing Eq.
(68) with relevant lattice QCD inputs FLA :

$\begin{array}[]{ccc}|f_{K^{+}}/f_{\pi^{+}}|=1.1932(21)&\quad
N_{f}=2+1+1&\quad\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bazavov:2017lyh,Dowdall:2013rya,Carrasco:2014poa,Miller:2020xhy}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
|f_{+}^{K}(0)/f_{+}^{\pi}(0)|\approx|f_{+}^{K}(0)|=0.9698(17)\leavevmode\nobreak\
.&\quad N_{f}=2+1+1&\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Carrasco:2016kpy,Bazavov:2018kjg}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}$
(70)

They give:

$|V_{us}/V_{ud}|=\left\\{\begin{array}[]{ccc}0.23131(41)_{\mathrm{lat}}(24)_{\mathrm{exp}}(19)_{\mathrm{RC}}&&\text{from
$R_{A}$}\\\
0.22908(66)_{\mathrm{BR}(\pi_{e3})}(41)_{K}(40)_{\mathrm{lat}}(2)_{\tau_{\pi^{+}}}(1)_{\mathrm{RC}_{\pi}}&&\text{from
$R_{V}$}\end{array}\right.\leavevmode\nobreak\ .$ (71)

The difference between the two determinations is at the level of $2.2\sigma$.
All the determinations of $|V_{us}|$, $|V_{ud}|$ and their ratio quoted in
this paper are summarized in Fig.1, from which the mutual disagreements
between different determinations and the deviations from the first-row CKM
unitary requirement are clearly shown.

##  IV Conclusions

This work updates the values of $|V_{us}|$ and $|V_{us}/V_{ud}|$ determined
from kaon and pion semileptonic decays using the most recent inputs from
theory and experiment. The uncertainties in these quantities have been
experiment- and lattice-dominated, which is even more the case in the recent
years due to the more precise SM electroweak theory inputs. Their values along
with $V_{ud}$ from superallowed beta decays correspond to 2–3$\sigma$
deviations from CKM unitary and related axial current induced weak decays.
Those differences may provide hints of BSM physics or deficiencies in SM
theory or experiment. Such anomalies provide a strong motivation for future
improvements of the experimental precision of the $\pi_{e3}$ BR as well as the
kaon lifetimes and $K_{\ell 3}$ BRs.

The experiment-dominated uncertainties do not imply that future improvements
from the theory side are not important. It is quite the opposite; the
aforementioned anomalies require us to carefully reexamine all the SM theory
inputs in order to ensure that no unexpected large systematic errors exist.
This was recently done for the long-distance EM corrections to the $K_{e3}$
decay rate and no large corrections were found. Other inputs, such as the
lattice calculations of $f_{+}^{K}(0)$ and $f_{K^{+}}/f_{\pi^{+}}$, should be
cross-checked with the same level of rigor. There are several other theory
works that remain to be done for completeness and internal consistency: For
instance, the reevaluation of the EM corrections should be generalized to the
$K\mu$ channels, and the fitting of the $K\pi$ form factors should, in
principle, also be updated to account for the modified EM corrections to the
$K_{\ell 3}$ Dalitz plot.

###### 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) (UGM and CYS), by the Alexander von Humboldt Foundation through
the Humboldt Research Fellowship (CYS), 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) (UGM), by EU
Horizon 2020 research and innovation programme, STRONG-2020 project under
grant agreement No 824093 (UGM), and by the U.S. Department of Energy under
Grant DE-SC0012704 (WJM).

##  Appendix A Mathematical Tools

In this Appendix, we review all the mathematical tools needed in this work.

###  A.1 Covariance matrix and correlation matrix

For a set of variables $X=(x_{1},x_{2},...,x_{n})^{T}$, we define the
symmetric covariance matrix $\mathrm{Cov}(X)$ as:

$\mathrm{Cov}(X)_{ij}=\left\langle x_{i}x_{j}\right\rangle-\left\langle
x_{i}\right\rangle\left\langle x_{j}\right\rangle\leavevmode\nobreak\ .$ (72)

In particular, the diagonal terms give the variance of $x_{i}$:

$\sigma_{i}^{2}=\mathrm{Cov}(X)_{ii}\leavevmode\nobreak\ .$ (73)

We also define the symmetric correlation matrix $\mathrm{Corr}(X)$ as:

$\mathrm{Corr}(X)_{ij}=\frac{\left\langle x_{i}x_{j}\right\rangle-\left\langle
x_{i}\right\rangle\left\langle
x_{j}\right\rangle}{\sigma_{i}\sigma_{j}}\leavevmode\nobreak\ .$ (74)

Its diagonal elements are always 1, while the off-diagonal elements range
between $-1$ and 1. Its relation to the covariance matrix is given by:

$\mathrm{Cov}(X)=\sigma_{X}\cdot\mathrm{Corr}(X)\cdot\sigma_{X}\leavevmode\nobreak\
,$ (75)

where $\sigma_{X}\equiv\mathrm{diag}(\sigma_{1},\sigma_{2},...,\sigma_{n})$.

###  A.2 Propagation of the covariance matrix

For a set of variables $F=(f_{1},f_{2},...,f_{m})^{T}$ that are functions of
$X$ (i.e. $f_{i}=f_{i}(x_{1},...,x_{n})$), given the covariance matrix of $X$
we can immediately obtain the covariance matrix of $F$ as:

$\mathrm{Cov}(F)=\left(\frac{\partial F}{\partial
X}\right)\cdot\mathrm{Cov}(X)\cdot\left(\frac{\partial F}{\partial
X}\right)^{T}$ (76)

where $\partial F/\partial X$ is a $m\times n$ matrix, with matrix elements:

$\left(\frac{\partial F}{\partial X}\right)_{ij}=\frac{\partial
f_{i}}{\partial x_{j}}\leavevmode\nobreak\ .$ (77)

If $F=(f_{1},f_{2},...,f_{m})^{T}$ depends on two independent sets of
variables $X=(x_{1},x_{2},...,x_{n})^{T}$ and
$Y=(y_{1},y_{2},...,y_{n^{\prime}})^{T}$ with their respective covariance
matrices given, then the covariance matrix of $F$ is simply the sum of the two
contributions:

$\mathrm{Cov}(F)=\left(\frac{\partial F}{\partial
X}\right)\cdot\mathrm{Cov}(X)\cdot\left(\frac{\partial F}{\partial
X}\right)^{T}+\left(\frac{\partial F}{\partial
Y}\right)\cdot\mathrm{Cov}(Y)\cdot\left(\frac{\partial F}{\partial
Y}\right)^{T}\leavevmode\nobreak\ .$ (78)

This is also generalizable if $F$ is a function of more than two independent
sets of variables.

For definiteness, throughout this work, we always calculate partial
derivatives numerically as:

$\frac{\partial f}{\partial a}\approx\frac{f(a+\delta a)-f(a-\delta
a)}{2\delta a}\leavevmode\nobreak\ .$ (79)

###  A.3 Weighted average

If $X=(x_{1},x_{2},...,x_{n})^{T}$ has a covariance matrix $\mathrm{Cov}(X)$,
then the weighted average between $x_{1},...,x_{n}$ is given by:

$\bar{x}=\sigma_{\bar{x}}^{2}\left(J^{T}WX\right)$ (80)

where the variance of $\bar{x}$ is given by:

$\sigma_{\bar{x}}^{2}=\left(J^{T}WJ\right)^{-1}\leavevmode\nobreak\ .$ (81)

Here we have defined $W=\left[\mathrm{Cov}(X)\right]^{-1}$ and
$J=(1,...,1)^{T}$ (length = $n$).

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