,$ (146)

with $C_{\mathrm{Bj}}(Q^{2})$ defined in Eq.(119).

###  VI.4 Born contribution

The Born contribution exhibits itself as a delta function in the structure
functions at $x=1$. Plugging $X=\phi_{f}$ into Eq.(140), one obtains its
contribution to the structure functions in terms of the form factors in
Eq.(126):

$\displaystyle F_{3,B}^{(0)}(x,Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[f_{1}^{S}(Q^{2})+f_{2}^{S}(Q^{2})\right]G_{A}(Q^{2})\delta(x-1)$
$\displaystyle g_{1,B}^{(0)}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[f_{1}^{S}(Q^{2})\left(2G_{V}(Q^{2})+G_{M}(Q^{2})\right)+f_{2}^{S}(Q^{2})G_{V}(Q^{2})\right]\delta(x-1)$
$\displaystyle g_{2,B}^{(0)}(x,Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{Q^{2}}{32m^{2}}\left[f_{1}^{S}(Q^{2})G_{M}(Q^{2})+f_{2}^{S}(Q^{2})\left(G_{V}(Q^{2})+2G_{M}(Q^{2})\right)\right]\delta(x-1)\leavevmode\nobreak\
,$

where $f_{1,2}^{S}=f_{1,2}^{\phi_{i}}+f_{1,2}^{\phi_{f}}$ are the isoscalar
electromagnetic form factors. With the above we obtain:

$\displaystyle\left(\Box_{\gamma W}^{V}\right)_{B}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{2\pi\mathring{g}_{V}}\int_{0}^{\infty}\frac{dQ^{2}}{Q^{2}}\frac{1+2r_{B}}{\left(1+r_{B}\right)^{2}}\left[f_{1}^{S}(Q^{2})+f_{2}^{S}(Q^{2})\right]G_{A}(Q^{2})$
$\displaystyle\left(\Box_{\gamma W}^{A}\right)_{B}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi\mathring{g}_{A}}\int_{0}^{\infty}\frac{dQ^{2}}{Q^{2}}\frac{1}{(1+r_{B})^{2}}\Bigg{\\{}\frac{5+4r_{B}}{3}\left[f_{1}^{S}(Q^{2})\left(2G_{V}(Q^{2})+G_{M}(Q^{2})\right)\right.$
$\displaystyle\left.+f_{2}^{S}(Q^{2})G_{V}(Q^{2})\right]+f_{1}^{S}(Q^{2})G_{M}(Q^{2})+f_{2}^{S}(Q^{2})\left(G_{V}(Q^{2})+2G_{M}(Q^{2})\right)\Bigg{\\}}\leavevmode\nobreak\
,$

where $r_{B}\equiv r|_{x=1}$. Notice that we have set
$M_{W}^{2}/(M_{W}^{2}+Q^{2})\rightarrow 1$ as the form factors only survive at
$Q^{2}\ll M_{W}^{2}$.

###  VI.5 Exact isospin relations

The structure functions $F_{3}^{(0)}$ and $S_{1,2}^{(0)}$ are not directly
measurable, because (1) they come from the hadronic tensor $W_{\mu\nu}$ that
involves different initial and final states, and (2) the isosinglet
electromagnetic current $J_{\mathrm{em}}^{(0)}$ does not exist in nature.
Fortunately, we can relate them to physically measurable structure functions
through isospin symmetry. For $F_{3}^{(0)}$, we have the following isospin
relation:

$2F_{3}^{(0)}=F_{3,\phi_{f}}^{\gamma Z}-F_{3,\phi_{i}}^{\gamma
Z}\leavevmode\nobreak\ ,$ (149)

where

$\frac{1}{4\pi}\int d^{4}xe^{iq\cdot
x}\langle\phi(p,s)|[J_{\mathrm{em}}^{\mu}(x),J_{Z}^{\nu}(0)]+[J_{Z}^{\mu}(x),J_{\mathrm{em}}^{\nu}(0)]|\phi(p,s)\rangle=-i\epsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}p_{\beta}}{2p\cdot
q}F_{3,\phi}^{\gamma Z}+...$ (150)

The structure function $F_{3,\phi}^{\gamma Z}$ is in principle measurable
through the parity-odd observables in the inclusive $e\phi$ scattering
experiments.

Meanwhile, for $g_{j}^{(0)}$ ($j=1,2$), we have the following isospin
relation:

$2g_{j}^{(0)}=g_{j,\phi_{f}}^{\gamma}-g_{j,\phi_{i}}^{\gamma}\leavevmode\nobreak\
,$ (151)

where

$\frac{1}{4\pi}\int d^{4}xe^{iq\cdot
x}\langle\phi(p,s)|[J_{\mathrm{em}}^{\mu}(x),J_{\mathrm{em}}^{\nu}(0)]|\phi(p,s)\rangle=i\epsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}}{p\cdot
q}\left[s_{\beta}g_{1,\phi}^{\gamma}+\left(s_{\beta}-\frac{s\cdot q}{p\cdot
q}p_{\beta}\right)g_{2,\phi}^{\gamma}\right]+...$ (152)

The polarized structure functions $g_{1,\phi}^{\gamma}$ and
$g_{2,\phi}^{\gamma}$ are well-measured quantities in ordinary DIS experiments
with nucleon and light nuclei.

To summarize, we presented in this section the dispersive representation of
the inner RC to the beta decay of a generic $I=J=1/2$ particle, and derived
analytic formulas and relations that hold in general. The remaining unanalyzed
pieces are process-dependent and have to be studied case-by-case. In the next
section we will use a particularly important example, namely the free neutron,
to demonstrate how these process-dependent contributions can be pinned down
with appropriate experimental inputs.

##  VII Free neutron

The beta decay of free neutron currently provides the second best measurement
of $|V_{ud}|$ after superallowed beta decays. The master formula reads
Zyla:2020zbs :

$|V_{ud}|^{2}=\frac{5024.7\leavevmode\nobreak\
\mathrm{s}}{\tau_{n}(1+3\lambda^{2})(1+\Delta_{R}^{V})}$ (153)

where $\tau_{n}$ is the neutron lifetime and $\lambda$ is the renormalized
axial-to-vector coupling ratio. The numerator at the right hand side includes
the effects from recoil corrections and the outer RCs, while $\Delta_{R}^{V}$
is the so-called “nucleus-dependent RC” to the Fermi matrix element which
takes the following form in the Sirlin’s representation:

$\Delta_{R}^{V}=\frac{\alpha}{2\pi}\left[3\ln\frac{M_{Z}}{m_{p}}+\ln\frac{M_{Z}}{M_{W}}+\tilde{a}_{g}\right]+\delta_{\mathrm{HO}}^{\mathrm{QED}}+2\Box_{\gamma
W}^{V}\leavevmode\nobreak\ .$ (154)

It is nothing but the inner correction to $g_{V}^{2}$ as we discussed in
Eq.(132). The current limiting factor to the extraction of $|V_{ud}|$ from the
neutron beta decay comes from experiment rather than theory. The experimental
determination of the neutron lifetime $\tau_{n}$ suffers from the well-known
beam-bottle discrepancy Bowman:2014nsk , while the measurements of $\lambda$
before Bopp:1986rt ; Erozolimsky:1997wi ; Liaud:1997vu ; Mostovoi:2001ye  and
after 2002 Schumann:2007hz ; Mund:2012fq ; Darius:2017arh ; Brown:2017mhw ;
Markisch:2018ndu  also show a large systematic disagreement (see
Ref.Czarnecki:2018okw  for more discussions). Several experiments are under
construction to measure $\tau_{n}$ with a precision better than 0.4 s and
resolve the beam-bottle discrepancy Pattie:2019brb ; Wietfeldt:2014gia , as
well as to reach an accuracy level of $10^{-4}$ in the $\lambda$ measurement
Fry:2018kvq ; Dubbers:2007st ; Wang:2019pts .

Despite being limited by the experimental precision, the EWRC in free neutron
has long been a main focus in precision physics. In particular, the quantity
$\Delta_{R}^{V}$ is of great interest because it not only appears in the
neutron, but also in nuclear beta decays as the single-nucleon contribution to
the RCs. For instance, it has long been the major source of uncertainty in the
most precise extraction of $|V_{ud}|$ from superallowed nuclear beta decays.
In this section, we will briefly review the past attempts to pin down the
EWRCs in free neutron, and discuss the recent progress based on the dispersive
representation.

###  VII.1 Earlier attempts

Similar to Eq.(109) for pion, a famous parameterization of $\Box_{\gamma
W}^{V}$ for neutron in the early days reads Sirlin:1977sv :

$\Box_{\gamma
W}^{V}=\frac{\alpha}{2\pi}\left[\frac{1}{2}\ln\frac{M_{W}}{M_{A}}+C_{\mathrm{Born}}+\frac{1}{2}A_{g}\right]\leavevmode\nobreak\
,$ (155)

where the first term at the right hand side represents the large electroweak
logarithm with $M_{A}$ an effective IR cutoff scale, $C_{\mathrm{Born}}$ is
the Born contribution, and $A_{g}$ is the pQCD correction. While
$C_{\mathrm{Born}}$ and $A_{g}$ are easily calculable (see Sec.VI), there was
no unique method to determine the scale $M_{A}$. Based on a simple vector-
meson-dominance (VMD) picture, Ref.Marciano:1985pd  set $M_{A}\approx 1.2$
GeV, i.e. the mass of the $A_{1}$ resonance, and estimated the error by
varying $M_{A}$ up or down by a factor 2. Such a simple estimation returned
$\Delta_{R}^{V}=0.0240(8)$ Hardy:2004id .

Ref.Marciano:2005ec  improved upon the determination above by adopting an
interpolation function approach. First, they wrote:

$\Box_{\gamma
W}^{V}=\frac{\alpha}{8\pi}\int_{0}^{\infty}dQ^{2}\frac{M_{W}^{2}}{M_{W}^{2}+Q^{2}}F(Q^{2})\leavevmode\nobreak\
,$ (156)

and assumed the following dominant physics that contribute to $F(Q^{2})$ at
different regions of $Q^{2}$:

  * •

$0<Q^{2}<Q_{1}^{2}$ (long distances): Pure Born contribution, which is
completely fixed by nucleon form factors, dominates.

  * •

$Q_{1}^{2}<Q^{2}<(1.5\leavevmode\nobreak\ \mathrm{GeV})^{2}$ (intermediate
distances): A VMD-inspired interpolating function is constructed:

$F(Q^{2})=\frac{c_{\rho}}{Q^{2}+M_{\rho}^{2}}+\frac{c_{A}}{Q^{2}+M_{A}^{2}}+\frac{c_{\rho^{\prime}}}{Q^{2}+M_{\rho^{\prime}}^{2}}\leavevmode\nobreak\
,$ (157)

with $M_{\rho}=0.776$ GeV, $M_{A}=1.230$ GeV, $M_{\rho^{\prime}}$=1.465 GeV.

  * •

$Q^{2}>(1.5\leavevmode\nobreak\ \mathrm{GeV})^{2}$ (short distances):
$F(Q^{2})$ is given by the leading-twist OPE + pQCD correction.

There are four free parameters in this parameterization, namely the
coefficients $\\{c_{\rho},c_{A},c_{\rho^{\prime}}\\}$ in the interpolating
function, and the matching scale $Q_{1}^{2}$ between the long and intermediate
distances. Ref.Marciano:2005ec  fixed these four parameters with the four
criteria as follows (we will see later than some of them are invalidated by
more recent studies):

  1. 1.

The result of the integral (156) at $Q^{2}>(1.5\leavevmode\nobreak\
\mathrm{GeV})^{2}$ is required to be the same using the VMD parameterization
and the asymptotic expression.

  2. 2.

In the large-$Q^{2}$ limit, the coefficient of the $1/Q^{4}$ term in Eq.(157)
is required to vanish by chiral symmetry.

  3. 3.

Eq.(157) is required to vanish at $Q^{2}=0$ by ChPT.

  4. 4.

Finally, the connection scale $Q_{1}^{2}$ is chosen through the matching of
$F_{\mathrm{Born}}$ and $F_{\mathrm{Interpolation}}$ at $Q^{2}=Q_{1}^{2}$.

With the above, Ref.Marciano:2005ec  obtained $c_{\rho}=-1.490$,
$c_{A}=6.855$, $c_{\rho^{\prime}}=-4.414$ and
$Q_{1}^{2}=(0.823\leavevmode\nobreak\ \mathrm{GeV})^{2}$, which give
$\Delta_{R}^{V}=0.02361(38)$ in total. This value was taken as the state-of-
the-art determination of $\Delta_{R}^{V}$ until 2018.

In the meantime, neutron beta decay had also been studied within the HBChPT
framework we described in Sec.IV.5. The outcome to one-loop agreed naturally
with the Sirlin’s representation in terms of the outer corrections, and a
comparison of the remaining terms between the two formalisms yields the
following matching relation between the renormalized LECs in the EFT and the
inner correction Ando:2004rk ,

$e_{V}^{R}\approx-\frac{5}{4}+\frac{2\pi}{\alpha}\Delta_{R}^{V}\leavevmode\nobreak\
.$ (158)

Of course, since chiral symmetry does not impose any constraint on
$e_{V}^{R}$, the EFT framework itself does not improve our understanding of
the inner RCs.

###  VII.2 $\Box_{\gamma W}^{V}$: DR analysis

Figure 10: The $W^{2}-Q^{2}$ phase diagram for free neutron, taken from
Ref.Seng:2018yzq .

A next important breakthrough occurred in 2018 when the dispersive
representation of $\Box_{\gamma W}^{V}$ was first introduced in
Ref.Seng:2018yzq . Within this formalism, the function $F(Q^{2})$ defined in
Ref.Marciano:2005ec  is expressed in terms of an integral with respect to the
P-odd structure function $F_{3}^{(0)}(x,Q^{2})$ (notice that
$\mathring{g}_{V}=1$ for free neutron):

$F(Q^{2})=\frac{8}{Q^{2}}\int_{0}^{1}dx\frac{1+2r}{(1+r)^{2}}F_{3}^{(0)}(x,Q^{2})=\frac{6}{Q^{2}}M_{3}^{(0)}(1,Q^{2})\leavevmode\nobreak\
,$ (159)

where we have defined the so-called “first Nachtmann moment” of $F_{3}^{(0)}$
Nachtmann:1973mr ; Nachtmann:1974aj :

$M_{3}^{(0)}(1,Q^{2})\equiv\frac{4}{3}\int_{0}^{1}dx\frac{1+2r}{(1+r)^{2}}F_{3}^{(0)}(x,Q^{2})\leavevmode\nobreak\
,$ (160)

which reduces to the simple Mellin moment at large $Q^{2}$.

The main challenge now is to identify all the dominant on-shell intermediate
states $|X\rangle$ in Eq.(140) that contribute to $F_{3}^{(0)}$ at different
regions of $\\{x,Q^{2}\\}$. They are most conveniently represented by the
phase diagram in Fig.10 (where $W^{2}=(p+q)^{2}$): First, at $W^{2}=m_{N}^{2}$
there is an isolated Born contribution. The continuum contribution starts at
the pion production threshold $W^{2}=(m_{N}+M_{\pi})^{2}$, which should be
further separated to regions of high and low $Q^{2}$. At $Q^{2}>2$ GeV2 (which
we will justify later), the parton + pQCD description is valid so the
contribution from this region can be computed to high accuracy. The situation
is more complicated at low $Q^{2}$: When $W^{2}$ is small, what we mainly
observe are baryon resonances (res) on top of a smooth background; approaching
large $W^{2}$, the contributions from multi-hadron intermediate states start
to dominate, which can be described economically by a Regge exchange picture
($\mathbb{R}$).

None of the discussions above would really be necessary if we had precise
experimental data of $F_{3,N}^{\gamma Z}$ at all values of $\\{x,Q^{2}\\}$
because they could then be related to $F_{3}^{(0)}$ using the isospin
relations in Sec.VI.5. Unfortunately, no such data exists to our knowledge
(there are some measurement of $F_{3}^{\gamma Z}$ at very large $Q^{2}$
Abramowicz:2015mha ; Argento:1983dj ), although they are in principle
measurable in parity-violating electron-nucleus scattering experiments, e.g.
in Jefferson Lab. Therefore, we are forced to look at other possible
experimental inputs. We consider the inclusive $\nu p(\bar{\nu}p)$ scattering,
of which differential cross section is given by:

$\frac{d^{2}\sigma^{\nu
p(\bar{\nu}p)}}{dxdy}=\frac{G_{F}^{2}m_{N}E}{\pi(1+Q^{2}/M_{W}^{2})^{2}}\left[xy^{2}F_{1}^{\nu
p(\bar{\nu}p)}+\left(1-y-\frac{m_{N}xy}{2E}\right)F_{2}^{\nu p(\bar{\nu}p)}\pm
x\left(y-\frac{y^{2}}{2}F_{3}^{\nu p(\bar{\nu}p)}\right)\right]$ (161)

where $y=\nu/E$ with $E$ the initial neutrino energy. The parity-odd, spin-
independent structure functions probed in these processes are Onengut:2005kv :

$\displaystyle\frac{|V_{ud}|^{-2}}{4\pi}\sum_{X}(2\pi)^{4}\delta^{(4)}(p+q-p_{X})\langle
p|(J_{W}^{\mu})^{\dagger}|X\rangle\langle X|J_{W}^{\nu}|p\rangle$
$\displaystyle=$
$\displaystyle-\frac{i\epsilon^{\mu\nu\alpha\beta}q_{\alpha}p_{\beta}}{2p\cdot
q}F_{3}^{\nu p}+...,$
$\displaystyle\frac{|V_{ud}|^{-2}}{4\pi}\sum_{X}(2\pi)^{4}\delta^{(4)}(p+q-p_{X})\langle
p|J_{W}^{\mu}|X\rangle\langle X|(J_{W}^{\nu})^{\dagger}|p\rangle$
$\displaystyle=$
$\displaystyle-\frac{i\epsilon^{\mu\nu\alpha\beta}q_{\alpha}p_{\beta}}{2p\cdot
q}F_{3}^{\bar{\nu}p}+...,$

and we can define their average, $F_{3}^{\nu p+\bar{\nu}p}=(F_{3}^{\nu
p}+F_{3}^{\bar{\nu}p})/2$, which is obtained from the difference between the
neutrino and antineutrino cross sections.

$F_{3}^{\nu p+\bar{\nu}p}$ resides in a different isospin channel and is not
directly related to $F_{3}^{(0)}$ through a simple isospin rotation. However,
we could parameterize both of them based on the physical picture in Fig.10 as
follows:

$F_{3}=F_{3,B}+\begin{cases}F_{3,\text{pQCD}}&Q^{2}\gtrsim 2\text{
GeV}^{2}\\\\[5.69054pt] F_{3,\pi
N}+F_{3,\text{res}}+F_{3,\mathbb{R}}&Q^{2}\lesssim 2\text{
GeV}^{2}\leavevmode\nobreak\ .\end{cases}$ (163)

We may separate all the terms at the RHS into two classes:

  * •

The “non-asymptotic” pieces (Born, low-energy continuum, resonances): They are
clearly different for different $F_{3}$, and need to be calculated case-by-
case.

  * •

The “asymptotic” pieces ($F_{3,\text{pQCD}}$ at large $Q^{2}$ and
$F_{3,\mathbb{R}}$ at large $W^{2}$): They are largely universal for different
$F_{3}$ (up to multiplicative factors), so we can either calculate them
explicitly ($F_{3,\text{pQCD}}$), or infer one from the other.

Let us investigate the non-asymptotic pieces in more detail. First, the Born
contribution to $F_{3}^{(0)}$ is given in Eq.(LABEL:eq:Born), while

$F_{3,B}^{\nu
p+\bar{\nu}p}(x,Q^{2})=-\left[f_{1}^{V}(Q^{2})+f_{2}^{V}(Q^{2})\right]G_{A}(Q^{2})\delta(x-1)\leavevmode\nobreak\
,$ (164)

where $f_{i}^{V}=f_{i}^{p}-f_{i}^{n}$ are the isovector EM form factors. Both
$F_{3,B}^{(0)}$ and $F_{3,B}^{\nu p+\bar{\nu}p}$ can be fixed to high
precision using the experimental measurements of the nucleon electromagnetic
Ye:2017gyb ; Lorenz:2012tm ; Lorenz:2014yda ; Lin:2021umk ; Lin:2021umz  and
axial form factors Bernard:2001rs ; Bhattacharya:2011ah . Next, the $N\pi$
contribution can be calculated using ChPT, and is found to be small
Seng:2018qru . In doing so, we find that the simplest inelastic (i.e. $N\pi$)
contribution to $F(Q^{2})$ does not vanish at $Q^{2}=0$, which invalidates one
of the four criteria imposed in Ref.Marciano:2005ec  as we advertised in the
previous subsection. Finally, $F_{3,\text{res}}^{\nu p+\bar{\nu}p}$ is
dominated by the $\Delta$-resonance, which can be calculated using the
parameterizations in Ref.Lalakulich:2005cs . Meanwhile, since $F_{3}^{(0)}$
only probes $I=1/2$ intermediate states, the $\Delta$-resonance cannot
contribute and thus $F_{3,\text{res}}^{(0)}$ is very small.

Next we turn to the asymptotic pieces. First, at large $Q^{2}$, $F_{3}^{(0)}$
obeys the pQCD-corrected polarized Bjorken sum rule as we have already
discussed:

$M_{3}^{(0)}(1,Q^{2})=3\left[1-\sum_{n=1}^{\infty}c_{n}\left(\frac{\alpha_{S}}{\pi}\right)^{n}\right]\leavevmode\nobreak\
.$ (165)

Meanwhile, $F_{3}^{\nu p+\bar{\nu}p}$ follows the pQCD-corrected GLS sum rule
Gross:1969jf :

$M_{3}^{\nu
p+\bar{\nu}p}(1,Q^{2})=3\left[1-\sum_{n=1}^{\infty}\tilde{c}_{n}\left(\frac{\alpha_{S}}{\pi}\right)^{n}\right]\leavevmode\nobreak\
,$ (166)

where the coefficients $\tilde{c}_{n}$ are again known to $n=4$ Baikov:2010iw
; Baikov:2010je ; Baikov:2012zn :

$\displaystyle\tilde{c}_{1}$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle\tilde{c}_{2}$ $\displaystyle=$ $\displaystyle 4.583-0.333n_{f}$
$\displaystyle\tilde{c}_{3}$ $\displaystyle=$ $\displaystyle
41.44-8.020n_{f}+0.1775n_{f}^{2}$ $\displaystyle\tilde{c}_{4}$
$\displaystyle=$ $\displaystyle
479.4-129.2n_{f}+7.930n_{f}^{2}-0.1037n_{f}^{3}\leavevmode\nobreak\ .$ (167)

They are very similar to the pQCD corrections to the polarized Bjorken sum
rule, except for very small differences starting from
$\mathcal{O}((\alpha_{S}/\pi)^{3})$ due to singlet (disconnected) diagrams.
Neglecting these small differences give us the following approximate matching
at large $Q^{2}$:

$M_{3}^{(0)}(1,Q^{2})\approx\frac{1}{18}M_{3}^{\nu
p+\bar{\nu}p}(1,Q^{2})\leavevmode\nobreak\ .\quad\text{Large $Q^{2}$}$ (168)

Figure 11: The Regge exchange picture for (a) $F_{3}^{(0)}$ and (b)
$F_{3}^{\nu p+\bar{\nu}p}$, taken from Ref.Seng:2018yzq .

We next turn to the small-$Q^{2}$, large-$W^{2}$ region. The leading Regge-
exchange contributions to $F_{3}^{(0)}$ and $F_{3}^{\nu p+\bar{\nu}p}$ are
depicted in Fig.11, where the gauge bosons first fluctuate into vector/axial
mesons, which in turn couple to the nucleon through the exchange of a
$t$-channel Regge trajectory. We may adopt the following simple
parameterization of the Regge contribution Seng:2018yzq ; Seng:2018qru :

$\displaystyle F_{3,\mathbb{R}}^{(0)}$ $\displaystyle=$ $\displaystyle
f_{\mathrm{th}}C_{\gamma
W}\frac{M_{\omega}^{2}}{M_{\omega}^{2}+Q^{2}}\frac{M_{a_{1}}^{2}}{M_{a_{1}}^{2}+Q^{2}}\left(\frac{\nu}{\nu_{0}}\right)^{\alpha_{0}^{\rho}}$
$\displaystyle F_{3,\mathbb{R}}^{\nu p+\bar{\nu}p}$ $\displaystyle=$
$\displaystyle
f_{\mathrm{th}}C_{WW}\frac{M_{\rho}^{2}}{M_{\rho}^{2}+Q^{2}}\frac{M_{a_{1}}^{2}}{M_{a_{1}}^{2}+Q^{2}}\left(\frac{\nu}{\nu_{0}}\right)^{\alpha_{0}^{\omega}}\leavevmode\nobreak\
.$ (169)

Let us explain the notations above: $\alpha_{0}^{\phi}$ is the Regge
trajectory intercept, with $\alpha_{0}^{\rho}\approx\alpha_{0}^{\omega}\approx
0.477$ Kashevarov:2017vyl , $\nu_{0}=1$ GeV is a threshold scale,
$f_{\mathrm{th}}$ is a threshold function that approaches 1(0) when $W^{2}$
approaches $\infty(0)$, and $C_{\gamma W}$, $C_{WW}$ are functions of $Q^{2}$
that account for the residual $Q^{2}$-dependence not captured by the VMD
picture.

We assume that the ratio between $F_{3,\mathbb{R}}^{(0)}$ and
$F_{3,\mathbb{R}}^{\nu p+\bar{\nu}p}$ at $Q^{2}=0$ and $Q^{2}=2$ GeV2 follow
the prediction of VMD Lichard:1997ya ; deSwart:1963pdg  and the parton picture
(i.e. Eq.(168)) respectively. Given that the $\rho$ and $\omega$ trajectories
are almost degenerate Kashevarov:2017vyl , it turns out that the two
conditions predict the same ratio:

$\frac{M_{3}^{(0)}(1,0)}{M_{3}^{\nu
p+\bar{\nu}p}(1,0)}\approx\frac{M_{3}^{(0)}(1,2\leavevmode\nobreak\
\mathrm{GeV}^{2})}{M_{3}^{\nu p+\bar{\nu}p}(1,2\leavevmode\nobreak\
\mathrm{GeV}^{2})}\approx\frac{1}{18}\leavevmode\nobreak\ ,$ (170)

which suggests the following simple relation between the coefficient
functions:

$C_{\gamma W}(Q^{2})\approx\frac{1}{18}C_{WW}(Q^{2})\leavevmode\nobreak\ .$
(171)

Figure 12: Left panel: Fitting $M_{3}^{\nu p+\bar{\nu}p}(1,Q^{2})$ to
experimental data. Right panel: Comparison between the theory predictions of
$M_{3}^{(0)}(1,Q^{2})$ using the DR method Seng:2018yzq  (blue) and the method
in Ref.Marciano:2005ec  (red). Figures are taken from Ref.Seng:2018yzq .

Eq.(171) provides a systematic procedure to infer the values of
$M_{3}^{(0)}(1,Q^{2})$ at small-$Q^{2}$. First, the full $M_{3}^{\nu
p+\bar{\nu}p}(1,Q^{2})$ was measured by the CCFR Kataev:1994rj ; Kim:1998kia ,
BEBC and Gargamelle Bolognese:1982zd , and WA25 Allasia:1985hw  collaborations
over 0.15 GeV${}^{2}<Q^{2}<$600 GeV2, which are summarized in the left panel
of Fig.12. In particular, the high-precision measurement from CCFR agrees with
the parton + pQCD prediction at $Q^{2}>2\leavevmode\nobreak\ $GeV2, which
justifies our choice of $Q_{\mathrm{cut}}^{2}=2\leavevmode\nobreak\ $GeV2 as
the separation scale between the perturbative and non-perturbative regions.
One may then subtract the Born, $N\pi$ and resonance contribution from the
experimental result to obtain $M_{3,\mathbb{R}}^{\nu p+\bar{\nu}p}(1,Q^{2})$
at low-$Q^{2}$, which fixes the function $C_{WW}(Q^{2})$. Then, through
Eq.(171) we obtain $M_{3,\mathbb{R}}^{(0)}(1,Q^{2})$. Combining this with
theory calculations of other contributions give us the full
$M_{3}^{(0)}(1,Q^{2})$. Readers should be aware that Eq.(171) is strictly
speaking just a model and is subject to a residual theory uncertainty;
however, given the rather poor quality of the data of $M_{3}^{\nu
p+\bar{\nu}p}(1,Q^{2})$ at low $Q^{2}$, we may assume at this stage that the
final uncertainty in $M_{3}^{(0)}(1,Q^{2})$ is predominantly from experiments.

Refs.Seng:2018yzq ; Seng:2018qru  quoted a new result of
$\Delta_{R}^{V}=0.02467(22)$ based on the analysis above. While reducing the
existing theory uncertainty in Ref.Marciano:2005ec  by a half, the new result
also showed a significant shift in the central value. The origin of such
difference is most clearly explained in the right panel of Fig.12. One
observes an unphysical dip in the red curve around
$Q^{2}=(0.823\leavevmode\nobreak\ \mathrm{GeV})^{2}$ due to the incorrect
small-$Q^{2}$ behavior of the interpolating function in Ref.Marciano:2005ec ,
and the incorrect assumption that only the Born contribution exists at
$Q^{2}\leq(0.823\leavevmode\nobreak\ \mathrm{GeV})^{2}$. Also, there is a
discontinuity in the red curve at the UV-matching point due to the requirement
that the integral of the interpolating function, instead of the interpolating
function itself, matches with the partonic description at large-$Q^{2}$. The
disagreement between the DR and interpolating function approach is thus
apparent because $\Box_{\gamma W}^{V}$ is proportional to the area under the
curve.

The increase of the size of $\Delta_{R}^{V}$ led to a reduction of $|V_{ud}|$
and eventually the appearance of a deficit in the top-row CKM matrix unitarity
as it is well-known by now. Due to this reason, the DR result must be
carefully scrutinized. As a response, Ref.Czarnecki:2019mwq  improved the
original interpolating function method by abandoning the incorrect $Q^{2}=0$
constraint, and using a holographic QCD model to describe the physics at
intermediate $Q^{2}$. This treatment returned a somewhat smaller value
$\Delta_{R}^{V}=0.02426(32)$. However, it was pointed out shortly
Hayen:2020cxh ; Hayen:2021iga  that this treatment missed the target mass
correction at low-$Q^{2}$ which, after adding back, gives
$\Delta_{R}^{V}=0.02473(27)$ which is now consistent with Refs.Seng:2018yzq ;
Seng:2018qru . Several later, DR-based calculations also reported consistent
results Seng:2020wjq ; Shiells:2020fqp .

###  VII.3 $\Box_{\gamma W}^{A}$: DR analysis

$\Box_{\gamma W}^{A}$ is responsible for the renormalization of the axial
coupling constant $g_{A}$. As we explained before, in beta decay experiments
one does not have direct access to the bare coupling constant
$\mathring{g}_{A}$, but rather the renormalized axial-to-vector ratio
$\lambda=g_{A}/g_{V}$. And in fact, since the latter is directly measured
through various vector-vector correlations in the differential decay rate
(e.g. $\vec{p}_{e}\cdot\vec{p}_{\nu}$, $\hat{s}\cdot\vec{p}_{e}$ and
$\hat{s}\cdot\vec{p}_{\nu}$; see Eq.(129)), the quantity $\Box_{\gamma W}^{A}$
plays no role in the determination of $|V_{ud}|$ from the free-neutron beta
decay and hence had received much less attention than $\Box_{\gamma W}^{V}$
from the theory side.

The situation is changed following the increased precision of the first-
principles calculations of the bare coupling constant $\mathring{g}_{A}$ with
lattice QCD Khan:2006de ; Lin:2008uz ; Capitani:2012gj ; Horsley:2013ayv ;
Bali:2014nma ; Abdel-Rehim:2015owa ; Alexandrou:2017hac ; Capitani:2017qpc ;
Edwards:2005ym ; Yamazaki:2008py ; Yamazaki:2009zq ; Bratt:2010jn ;
Green:2012ud ; Yamanaka:2018uud ; Liang:2018pis ; Ishikawa:2018rew ;
Ottnad:2018fri ; Bhattacharya:2016zcn ; Berkowitz:2017gql ; Chang:2018uxx ;
Gupta:2018qil . In particular, the authors in Ref.Chang:2018uxx  reported a
percent-level determination of $\mathring{g}_{A}=-1.271(10)(7)$ and aimed for
sub-percent precision in the near future Walker-Loud:2019cif . Once the
lattice precision reaches $10^{-3}$ or above, it would be possible to search
for signals of BSM physics, in particular the right-handed currents Gonzalez-
Alonso:2016etj ; Alioli:2017ces ; Gonzalez-Alonso:2018omy ; Falkowski:2020pma
, by comparing the experimental measurement of $\lambda$ and the SM prediction
in the right hand side of Eq.(133). Due to this reason, one needs to
understand the difference between $\Box_{\gamma W}^{A}$ and $\Box_{\gamma
W}^{V}$. The first comprehensive analysis of this inner RC was done in
Refs.Hayen:2020cxh ; Hayen:2021iga  based on the holographic QCD model
inspired by Ref.Czarnecki:2019mwq . The model-dependent nature of this method,
however, makes it challenging to have a rigorous estimation of the associated
theoretical uncertainty.

As we detailed in Sec.VI.2, the DR analysis of $\Box_{\gamma W}^{A}$ may
proceed in exactly the same was as $\Box_{\gamma W}^{V}$, and is in fact much
more promising than the latter. This is because its dispersive integral
depends on the polarized structure functions $g_{1,N}^{\gamma}$ and
$g_{2,N}^{\gamma}$ upon an isospin rotation (see Sec.VI.5). These structure
functions are directly measurable in ordinary DIS processes, which means the
low-$Q^{2}$ part in the dispersive integral of $\Box_{\gamma W}^{A}$ can in
principle be fixed completely by experiment, without invoking any extra
modelings. The $g_{1,N}^{\gamma}$ structure function, in particular, is a
well-measured quantity over the past 30 years Anthony:1993uf ; Abe:1994cp ;
Abe:1997cx ; Adams:1994zd ; Alexakhin:2006oza ; Alekseev:2010hc ;
Aghasyan:2017vck ; Ackerstaff:1997ws ; Deur:2004ti ; Wesselmann:2006mw ;
Deur:2008ej ; Guler:2015hsw ; Fersch:2017qrq ; Zheng:2021yrn . The data on
$g_{2,N}^{\gamma}$ are more scarce Anthony:1999py ; Anthony:2002hy ;
Amarian:2003jy ; Wesselmann:2006mw ; Kramer:2005qe ; Fersch:2017qrq , but the
size of its contribution to the inner correction is also generic two orders of
magnitude smaller.

Figure 13:  Data points of $\bar{\Gamma}_{1}^{p-n}$ reconstructed from the
EG1b experiment versus the theory prediction with (red band) and without
(brown curve) the HT correction. Figure taken from Ref.Gorchtein:2021qdf .

The first complete DR analysis of $\Box_{\gamma W}^{A}$ was performed in
Ref.Gorchtein:2021qdf , and we briefly outline the main strategies here.
First, it is separated into three pieces:

$\Box_{\gamma W}^{A}=\frac{\alpha}{2\pi}\left[d_{B}+d_{1}+d_{2}\right]$ (172)

which represent the Born contribution, the inelastic contribution from $g_{1}$
and the inelastic contribution from $g_{2}$, respectively. They are evaluated
as follows:

  * •

The Born contribution is just given by Eq.(LABEL:eq:BoxBorn). Isospin symmetry
requires $G_{V}(Q^{2})=f_{1}^{V}(Q^{2})$ and $G_{M}(Q^{2})=f_{2}^{V}(Q^{2})$,
so the integral is completely fixed by the nucleon electromagnetic form
factors.

  * •

For the inelastic contribution from $g_{1}^{\gamma}$, what we need is the
following quantity:

$\bar{\Gamma}_{1}^{p-n}(Q^{2})\equiv\int_{0}^{x_{\pi}}dx\frac{4(5+4r)}{9(1+r)^{2}}\left\\{g_{1,p}^{\gamma}(x,Q^{2})-g_{1,n}^{\gamma}(x,Q^{2})\right\\}$
(173)

($x_{\pi}=Q^{2}/[(m_{N}+M_{\pi})^{2}-m_{N}^{2}+Q^{2}]$ is the pion production
threshold) as a function of $Q^{2}$. At $Q^{2}>2$ GeV2, one resorts again to
the pQCD-corrected polarized Bjorken sum rule, and include additionally a
small higher-twist (HT) correction which is required to match the theory
prediction with the experimental data Deur:2014vea ; Kotlorz:2017wpu ;
Ayala:2018ulm . At $Q^{2}<2$ GeV2, data are taken from the EG1b experiment at
JLab Fersch:2017qrq ; Guler:2015hsw  that provided the first three moments of
$g_{1,N}^{\gamma}$ from 0.05 GeV2 to 3.5 GeV2, which allow for a precise
reconstruction of $\bar{\Gamma}_{1}^{p-n}(Q^{2})$ at any value of $Q^{2}$
within the range. Fig.13 shows the combination of the experimental and theory
prediction of $\bar{\Gamma}_{1}^{p-n}(Q^{2})$. One then performs the
$Q^{2}$-integral to obtain the full $g_{1}^{\gamma}$ contribution.

  * •

Finally, the $g_{2}^{\gamma}$ contribution is split into two pieces, namely
the twist-two and twist-three (and higher) contributions. The former is
related to $g_{1}^{\gamma}$ through the Wandzura-Wilczek relation
Wandzura:1977qf , while the latter is related to the so-called nucleon color
polarizability Shuryak:1981pi ; Jaffe:1989xx  calculated within the baryon
chiral effective theory Alarcon:2020icz .

Combining everything above, Ref.Gorchtein:2021qdf  reported $\Box_{\gamma
W}^{A}=3.96(6)\times 10^{-3}$. The theory uncertainty is much smaller than
that of $\Box_{\gamma W}^{V}$, due to the much more precise experimental
inputs at low and moderate $Q^{2}$.

###  VII.4 Future prospects with lattice QCD

Figure 14:  The comparison between the leading Regge contribution to the pion
and nucleon $\gamma W$-box diagram. Figure taken from Ref.Seng:2020wjq .

The major limiting factor in the DR analysis is the precision of the
experimental inputs. This is more of an issue for $\Box_{\gamma W}^{V}$
because the existing $\nu(\bar{\nu})p$ scattering data are obsolete and
contain large uncertainties in the non-perturbative region. The Deep
Underground Neutrino Experiment (DUNE) conducted by Fermilab may start to
provide new data with higher precision in the next decade Acciarri:2016crz ;
Alvarez-Ruso:2017oui ; but even with that, one still needs to deal with the
residual model-dependence between the matching of $M_{3}^{(0)}(1,Q^{2})$ and
$M_{3}^{\nu p+\bar{\nu}p}(1,Q^{2})$.

Lattice QCD is so far the only promising approach to solve the two issues
above simultaneously. The lattice calculation of $\Box_{\gamma
W}^{<}(\pi^{+},\pi^{0},M_{\pi})$ we described in Sec.V.5 serves as a first
prototype and has shown great success. In fact, apart from pinning down the
inner RC in $\pi_{e3}$, this calculation also provides an indirect lattice
input to the dispersion integral of $\Box_{\gamma W}^{V}$ for neutron and
serves as a useful cross-check of the experimental data-based DR analysis
Seng:2020wjq ; Seng:2021qdx . The idea is most easily visualized in Fig.14:
There is a one-to-one correspondence between the leading Regge contribution to
the pion and nucleon $\gamma W$-box diagram, which allows us to obtain
$M_{3,\mathbb{R}}^{(0)}(1,Q^{2})$ through a simple rescaling of the lattice
result of $\mathbb{M}_{\pi}(Q^{2})$ (after subtracting the small resonance
contribution to the latter). With the strategy above, Ref.Seng:2020wjq
reported $\Delta_{R}^{V}=0.02477(24)$ which is consistent with the original DR
treatment.

So far the direct lattice calculation of $M_{3}^{(0)}(1,Q^{2})$ is not yet
available but is under intense investigation. A way to proceed is to calculate
the four-point correlation functions in analogy to Ref.Feng:2020zdc . This is,
however, more challenging than that in the pion sector because the quark
contraction diagrams are more complicated, the lattice observables are much
noisier, and the handling of excited-state contaminations are more
challenging. Preliminary results are expected within the next couple of years
BhattacharyaSnowmass . There is also an alternative proposal to calculate
$M_{3}^{(0)}(1,Q^{2})$ using the Feynman-Hellmann theorem (FHT)
Feynman:1939zza ; Hellmann  on lattice Seng:2019plg , which states that the
parity-odd structure function $F_{3}(x,Q^{2})$ can be obtained from the
second-order variation of the nucleon ground state energy upon the
introduction of a periodic source term into the Hamiltonian:

$H_{\mathrm{scr}}(t)=2\lambda_{1}\int
d^{3}x\cos(\vec{q}\cdot\vec{x})J_{\mathrm{em}}^{2}(\vec{x},t)-2\lambda_{2}\int
d^{3}x\sin(\vec{q}\cdot x)J_{A}^{3}(\vec{x},t)\leavevmode\nobreak\ .$ (174)

The proposal was inspired by an earlier study of the parity-even structure
functions Chambers:2017dov . Some other applications of the FHT on lattice
include the calculation of hadronic charges and form factors Chambers:2017tuf
; Chang:2018uxx , Compton scattering amplitude Agadjanov:2016cjc ;
Agadjanov:2018yxh  and hadron resonances RuizdeElvira:2017aet .

Finally, it is worth pointing out that a future lattice calculation of
$\Box_{\gamma W}^{A}$ is equally important, despite that it does not directly
affect the $|V_{ud}|$ extraction. Given that $\Box_{\gamma W}^{A}$ is very
precisely determined through the DR analysis, an extra lattice calculation of
the same quantity serves as an excellent avenue to cross-check the validity of
the lattice methodologies which are also applied to $\Box_{\gamma W}^{V}$. In
particular, through this procedure one could check if there are any large
systematic effects that are missed in the first-principles calculations. This
is a highly non-trivial issue, as we have recently seen cases with significant
disagreement between phenomenological and lattice determination of certain
quantities, some of which have profound impacts on the interpretation of the
results of precision experiments. A well-known example is the hadronic vacuum
polarization contribution to $g_{\mu}-2$ Aoyama:2020ynm ; Borsanyi:2020mff .

##  VIII Superallowed beta decays

Our next example is the beta decay of $J^{P}=0^{+}$ nuclei which is referred
to as “superallowed beta decay”. It is of particular interest as it provides
by far the most precise measurement of $|V_{ud}|$. This is because:

  * •

At tree level only the vector charged weak current is involved, whose matrix
element is exactly known assuming isospin symmetry.

  * •

Experimental data of 23 superallowed transitions had been accumulated over
five decades Towner:1973yrc ; Hardy:1975eq ; Hardy:1990sz ; Hardy:2004id ;
Hardy:2008gy ; Towner:2010zz ; Hardy:2014qxa ; Hardy:2020qwl , with 15 among
them whose $ft$-value precision is 0.23% or better; the large sample size
leads to a huge gain in statistics Hardy:2020qwl . In fact, this makes it the
only avenue where the experimental uncertainty in $|V_{ud}|$ is smaller than
the theory uncertainty.

However, the price to pay is that one needs to carefully account for the
nucleus-dependent corrections that enter each transition. A detailed
description of all the nuclear theory calculations is beyond the scope of this
review, and here we will just briefly outline the current situation.

The matrix element $|V_{ud}|$ is determined from superallowed decays through
the following master formula:

$|V_{ud}|^{2}=\frac{2984.432(3)\>\mathrm{s}}{\mathcal{F}t(1+\Delta_{R}^{V})}\leavevmode\nobreak\
,$ (175)

where $\mathcal{F}t$ is a nucleus-independent quantity which comes from the
nucleus-dependent $ft$-value after applying various nucleus-dependent
corrections:

$\mathcal{F}t=ft(1+\delta_{\mathrm{R}}^{\prime})(1+\delta_{\mathrm{NS}}-\delta_{\mathrm{C}})\leavevmode\nobreak\
.$ (176)

In the equation above, $\delta_{\mathrm{R}}^{\prime}$ represents the nucleus-
dependent outer corrections, $\delta_{\mathrm{NS}}$ entails the nuclear
modifications of the single-nucleon RC, and $\delta_{\mathrm{C}}$ is the
isospin breaking corrections. These three corrections together turn different
nucleus-dependent $ft$-values into a nucleus-independent $\mathcal{F}t$-value.
The only correction among them that is known to satisfactory precision is the
nucleus-dependent outer correction $\delta_{\mathrm{R}}^{\prime}$ which is a
function of the charge $Z_{f}$ of the daughter nucleus and the electron end-
point energy $E_{m}$, and is calculated to the order $Z_{f}^{2}\alpha^{3}$
Sirlin:1967zza ; Sirlin:1987sy ; Sirlin:1986cc ; Towner:2007np . On the other
hand, existing calculations of the remaining two corrections are each plagued
by their respective ambiguities and will be discussed as follows.

###  VIII.1 The nuclear structure correction

To rigorously define the nuclear structure correction term
$\delta_{\mathrm{NS}}$, we start from the nuclear $\gamma W$-box diagram in
the inner correction to the Fermi matrix element, which is the only piece of
the EWRCs that depends on details of the non-perturbative QCD. It can be
written in a dispersive representation, in full analogy to
Eq.(LABEL:eq:DRformula): 333There is a mistake in Ref.Seng:2018qru :
$\Box_{\gamma W}^{\mathrm{nucl}}$ should be defined as the full nuclear
$\gamma W$-box, instead of “per nucleon”.

$\Box_{\gamma
W}^{\mathrm{nucl}}=\frac{\alpha}{\pi\mathring{g}_{V}^{\mathrm{nucl}}}\int_{0}^{\infty}\frac{dQ^{2}}{Q^{2}}\frac{M_{W}^{2}}{M_{W}^{2}+Q^{2}}\int_{0}^{1}dx\frac{1+2r}{\left(1+r\right)^{2}}F_{3,\mathrm{nucl}}^{(0)}(x,Q^{2})\leavevmode\nobreak\
,$ (177)

with $\mathring{g}_{V}^{\mathrm{nucl}}=G_{V}^{\mathrm{nucl}}(0)$. One then
performs an artificial splitting of the nuclear axial $\gamma W$-box diagram
into two pieces:

$\Box_{\gamma W}^{\mathrm{nucl}}=\Box_{\gamma W}^{V}+\left[\Box_{\gamma
W}^{\mathrm{nucl}}-\Box_{\gamma W}^{V}\right]\leavevmode\nobreak\ ,$ (178)

where $\Box_{\gamma W}^{V}$ is the single-nucleon $\gamma W$-box we studied in
Sec.VII. With the splitting above, $\Box_{\gamma W}^{V}$ then enters
$\Delta_{R}^{V}$ in Eq.(175), whereas

$\delta_{\mathrm{NS}}\equiv 2\left[\Box_{\gamma
W}^{\mathrm{nucl}}-\Box_{\gamma W}^{V}\right]$ (179)

describes the nuclear modifications to the single-nucleon $\gamma W$-box
diagram. The benefit of such a separation is apparent: The contributions to
$\Box_{\gamma W}^{\mathrm{nucl}}$ and $\Box_{\gamma W}^{V}$ at high energies
are the same because in this region the gauge bosons mainly probe a single,
free nucleon; for instance, the large-$Q^{2}$ contribution to both
$\Box_{\gamma W}^{\mathrm{nucl}}$ and $\Box_{\gamma W}^{V}$ take the same form
of $(\alpha/(8\pi))\ln(M_{W}^{2}/\Lambda^{2})$. On the other hand, at low
energies the two start to deviate because (a) the single-nucleon absorption
spectrum is distorted in the nuclear medium, and (b) the two gauge bosons may
probe two different nucleons in a nucleus.

####  VIII.1.1 Earlier treatments

Early treatments of $\delta_{\mathrm{NS}}$ mainly consist of calculating
effects (a) and (b) mentioned above with nuclear models, which add up to give:

$\delta_{\mathrm{NS}}=\frac{\alpha}{\pi}\left[C_{a}+C_{b}\right]\leavevmode\nobreak\
.$ (180)

Refs.Towner:1994mw ; Towner:2002rg ; Towner:2007np  estimated the size of
$C_{a}$ by considering the “quenching effect” of the free-nucleon nucleon
axial and EM coupling constants for spin-flip processes in the nuclear medium
Brown:1983zzc ; Brown:1987obh ; Towner:1987zz . That means, one starts from
the Born contribution to the free-nucleon box diagram,

$\Box_{\gamma
W}^{V}=\frac{\alpha}{2\pi}C_{B}^{\mathrm{free}}+...\leavevmode\nobreak\ ,$
(181)

(where $C_{B}^{\mathrm{free}}$ is given in Eq.(LABEL:eq:BoxBorn), with the
free-nucleon form factors) and apply a “quenching factor” $q$ which is
different for different nucleus, to obtain:

$\Box_{\gamma
W}^{\mathrm{nucl}}=\frac{\alpha}{2\pi}qC_{B}^{\mathrm{free}}+...\leavevmode\nobreak\
,$ (182)

and so,

$C_{a}\approx(q-1)C_{B}^{\mathrm{free}}\leavevmode\nobreak\ .$ (183)

Meanwhile, $C_{b}$ (denoted in earlier literature as “$C_{\mathrm{NS}}$”) were
calculated with nuclear shell models in Refs.Jaus:1989dh ; Towner:1992xm ;
Barker:1991tw , which was further improved in Ref.Towner:2007np  by
incorporating the same quenching effect, i.e. $C_{\mathrm{NS}}\rightarrow
C_{\mathrm{NS}}^{\mathrm{quenched}}$. The final numerical results of

$\delta_{\mathrm{NS}}=\frac{\alpha}{\pi}\left[(q-1)C_{B}^{\mathrm{free}}+C_{\mathrm{NS}}^{\mathrm{quenched}}\right]$
(184)

for different superallowed transitions were summarized in Table VI of
Ref.Towner:2007np  and appeared in several subsequent reviews by Hardy and
Towner Hardy:2008gy ; Hardy:2014qxa . In particular, with the theory inputs
above, Ref.Hardy:2014qxa  quoted an average value of

$\overline{\mathcal{F}t}=3072.27\pm 0.72\leavevmode\nobreak\ \mathrm{s}$ (185)

based on the lifetimes of fourteen best-measured superallowed transitions from
10C to 74Rb.

####  VIII.1.2 Recent developments

Figure 15:  Idealized structure of the nuclear absorption spectrum at low
energy.

Recent theory developments of the EWRC in beta decays, in particular the
establishment of the DR formalism, cast doubts on the treatment above. This
can be most clearly seen through the artificial splitting of the full nuclear
$\gamma W$-box in Eq.(178). There is no approximation made in this splitting,
however in practice the two terms at the right hand side are calculated with
completely different fashions. The free nucleon term is evaluated under a
fully relativistic framework of DR, making using inputs from intermediate- and
high-energy data (or lattice inputs), whereas the $\delta_{\mathrm{NS}}$ term
is calculated within non-relativistic shell models. The artificial distinction
in the treatment of these two terms may lead to possible large systematic
uncertainties.

Therefore, a more reasonable starting point is to treat $\Box_{\gamma
W}^{\mathrm{nucl}}$ in the same footing with $\Box_{\gamma W}^{V}$, namely
based on the fully relativistic dispersive representation in Eq.(177). The
dispersive integral depends on the the parity-odd nuclear structure function
$F_{3,\mathrm{nucl}}^{(0)}(x,Q^{2})$ which is defined through:

$\frac{V_{ud}^{*-1}}{4\pi}\sum_{X}(2\pi)^{4}\delta^{(4)}(p+q-p_{X})\langle\phi_{f}(p)|J_{\mathrm{em}}^{(0)\mu}|X\rangle\langle
X|J_{A}^{\nu}|\phi_{i}(p)\rangle=-\frac{i\epsilon^{\mu\nu\alpha\beta}q_{\alpha}p_{\beta}}{2p\cdot
q}F_{3,\mathrm{nucl}}^{(0)}(x,Q^{2})\leavevmode\nobreak\ ,$ (186)

where $\phi_{i,f}$ are the initial and final nuclear states in a given
superallowed transition, and $\\{X\\}$ exhausts all on-shell intermediate
nuclear states. Therefore, the study of $\delta_{\mathrm{NS}}$ boils down to
the calculation of single-current nuclear matrix elements involving
$\phi_{i,f}$ and the leading contributors within $\\{X\\}$.

A rigorous, model-independent study of the many-body effects in Eq.(186) may
involve sophisticated nuclear theory calculations, e.g. with chiral EFT
Weinberg:1990rz ; Weinberg:1991um ; Weinberg:1992yk ; vanKolck:1999mw ;
Epelbaum:2005pn ; Machleidt:2011zz ; Bernard:2007sp ; Bernard:2011zr ;
Girlanda:2011fh ; Krebs:2012yv ; Krebs:2013kha ; Epelbaum:2014sza ;
Entem:2015xwa ; Reinert:2017usi ; Entem:2017gor ; Hammer:2019poc  and its
lattice implementations Lee:2004si ; Borasoy:2006qn ; Lee:2008fa ;
Lahde:2019npb . This is not yet attempted and is an important task that needs
to be carried out in the future. However, here we are still able to get a
flavor of some most important nuclear corrections by simply analyzing the
idealized structure of the virtual photoabsorption spectrum on a nucleus as
depicted in Fig.15. At low energies, a narrow peak is contributed by the
elastic intermediate state and several smaller ones are from discrete energy
levels and the giant dipole resonance (GDR). After that, a broad absorption
peak around $\omega=Q^{2}/(2m_{N})$ is observed; this is the so-called quasi-
elastic (QE) peak that was supposed to be a delta function for free nucleon,
but broadened by nuclear interactions. Above the pion production threshold
$\omega_{\pi}=(Q^{2}+2m_{N}M_{\pi}+M_{\pi}^{2})/(2m_{N})$, the absorption
spectrum starts to resemble that of a single nucleon, and so the contribution
to $\delta_{\mathrm{NS}}$ vanishes.

Figure 16:  Comparison between the contribution to $\Box_{\gamma
W}^{\mathrm{nucl}}$ (a) effectively accounted for in the “quenching mechanism”
and (b) from a QE nucleon in the nucleus. Figure taken from Ref.Seng:2018qru .

With this picture, it is almost immediately apparent that the treatment in
Sec.VIII.1.1 did not account for the most important nuclear structure
corrections. Since the quenching factors $q$ are obtained from spin-flip
processes in Gamow-Teller (GT) transitions, the gauge bosons are effectively
probing deeply off-shell nucleons in the discrete energy levels at the lower
end of the nuclear absorption spectrum. On the other hand, the much more
important contribution from the QE nucleon is missing in this treatment (see
Fig.16). Ref.Seng:2018qru  estimated, for the first time, the contribution
from the latter using a simple Fermi gas model.

Apart from the above, there is yet another nuclear correction which was never
pointed out until recently Gorchtein:2018fxl . In evaluating the axial $\gamma
W$-box diagrams for free neutron and nucleus, one always take
$m_{e},E_{e}\rightarrow 0$ and $p^{\prime}\rightarrow p$. In free neutron,
this approximation is well-justified because the energy splitting between the
ground state and the on-shell intermediate state $X$ is at least $M_{\pi}$, so
the energy-dependent effect is expected to scale as
$(\alpha/\pi)E_{e}/M_{\pi}\sim 10^{-5}$ which is negligible at the present
stage. However, in nucleus, there exists excited states with typical energy
scale $\Lambda_{\mathrm{nucl}}$ of the order MeV, so
$(\alpha/\pi)E_{e}/\Lambda_{\mathrm{nucl}}$ is no longer suppressed. In this
case, a supposedly energy-independent nuclear “inner” correction
$\delta_{\mathrm{NS}}$ generates an energy-dependent effect which is more
commonly classified as an “outer” correction represented by
$\delta_{\mathrm{R}}^{\prime}$.

The two novel effects above are now known as the new nuclear corrections
(NNCs) to superallowed beta decays. Although there is currently no rigorous
theory analysis of their sizes, simple estimations using a Fermi gas model
suggest that they are individually sizable, but carry opposite signs and
partially cancel each other. As a net effect, this leads to an increase of the
theory uncertainty in $\mathcal{F}t$ Hardy:2020qwl :

$\displaystyle\overline{\mathcal{F}t}$ $\displaystyle:$ $\displaystyle
3072.27\pm 0.72\>\mathrm{s}\rightarrow 3072.24\pm
1.85\>\mathrm{s}\leavevmode\nobreak\ .$ (187)

This large theory uncertainty is now the major limiting factor in the precise
determination of $|V_{ud}|$.

###  VIII.2 The isospin-breaking correction

Finally, we briefly discuss the isospin-breaking correction
$\delta_{\mathrm{C}}$ although it is not a part of the EWRC. This correction
plays a central role in obtaining a universal $\mathcal{F}t$-value from
different superallowed transitions, which is required by the conserved vector
current (CVC) hypothesis. However, it turns out that among all existing
nuclear theory calculations of $\delta_{\mathrm{C}}$, including shell-model
with Woods-Saxon (WS) potential Towner:2002rg ; Towner:2007np ; Hardy:2008gy ,
Hartree-Fock wavefunctions Ormand:1989hm ; Ormand:1995df , density functional
theory Satula:2011br , random-phase approximation Liang:2009pf , isovector
monopole resonance sum rule Auerbach:2008ut  and the Damgaard model
Damgaard:1969yyx , only the WS calculation is able to achieve such an
alignment and is taken as the standard input for $\delta_{\mathrm{C}}$.
Whether this indicates the success of this particular simplified nuclear
picture, or rather the existence of some unexpected new physics in Nature
(e.g. second-class currents Weinberg:1958ut ), is not yet clear.

It was recently pointed out Koshchii:2020qkr  that it is possible to relate
$\delta_{\mathrm{C}}$ to experimental observables generated by the same
isospin-breaking effect, such as neutron skins and differences in nuclear
charge radii within a given isomultiplet. Charge radii of stable daughter
nuclei are known to few parts in $10^{4}$ Angeli:2013epw , whereas those for
unstable parent nuclei can be measured at rare-isotope facilities, e.g. the
Isotope Mass Separator On-Line Device facility (ISOLDE) and the Facility for
Rare Isotopes Beams (FRIB). Meanwhile, the size of the parity-violating
asymmetry in electron-nucleus (e.g. 12C) scattering at backward angles is
largely affected by the neutron skin of the nucleus Koshchii:2020qkr , so in
principle future parity-violating electron scattering experiments at Jefferson
Lab Souder:2016xcn  and at the Mainz Energy-Recovering Superconducting
Accelerator (MESA) Becker:2018ggl  will allow one to extract neutron skins of
selected stable superallowed daughter nuclei to few per mille and therefore
better constrain $\delta_{\mathrm{C}}$ in a fully model-independent manner,
provided that the theoretical relations between these quantities are well-
established.

##  IX Kaon semileptonic decays

So far we only discussed semileptonic decay processes that involve a
$u\leftrightarrow d$ transition and are attached to $V_{ud}$. In this section
we switch to the simplest semileptonic decay process that involves a
$u\leftrightarrow s$ transition, namely the $K\rightarrow\pi\ell\nu$ decay. It
is commonly denoted as $K_{\ell 3}$, which represents collectively six types
of decay processes: $K_{e3}^{L}$, $K_{e3}^{S}$, $K_{e3}^{+}$, $K_{\mu 3}^{L}$,
$K_{\mu 3}^{S}$ and $K_{\mu 3}^{+}$. They are of great interest to both
theorists and experimentalists as they provide one of the most precise
measurements of $|V_{us}|$. This is done via the following master formula:

$\Gamma_{K_{\ell
3}}=\frac{G_{F}^{2}|V_{us}|^{2}M_{K}^{5}C_{K}^{2}}{192\pi^{3}}S_{\mathrm{EW}}|f_{+}^{K^{0}\pi^{-}}(0)|^{2}I_{K\ell}^{(0)}\left(1+\delta_{\mathrm{EM}}^{K\ell}+\delta_{\mathrm{SU(2)}}^{K\pi}\right)\leavevmode\nobreak\
.$ (188)

The focus of this review is on the EWRCs to the $K_{\ell 3}$ decay rate, which
are contained in the short-distance electroweak factor $S_{\mathrm{EW}}$
(discussed in Sec.III.4) and long-distance EM correction
$\delta_{\mathrm{EM}}^{K\ell}$. Before doing so, we shall briefly discuss all
the other inputs that are needed to extract $|V_{us}|$ from Eq.(188):

  * •

$\Gamma_{K_{\l 3}}$ is the $K_{\ell 3}$ decay width. The kaon lifetimes,
branching ratios and differential decay widths are measured by BNL E865
Sher:2003fb , KTeV Alexopoulos:2004sw ; Alexopoulos:2004sx ;
Alexopoulos:2004sy ; Abouzaid:2006ir ; KTeV:2010sng , KLOE Ambrosino:2005vx ;
Ambrosino:2006up ; Ambrosino:2005ec ; Ambrosino:2006si ; Sciascia:2005zz ;
Ambrosino:2006gn ; Ambrosino:2007ac ; KLOE:2007jte ; KLOE:2010yit . KLOE-2
KLOE-2:2019rev , NA48 Lai:2004bt ; Lai:2006cf ; Batley:2006cj ; Lai:2004kb
and ISTRA+ Romanovsky:2007qb ; Yushchenko:2004zs  over the past two decades.

  * •

$f_{+}^{K^{0}\pi^{-}}(0)$ is the $K^{0}\pi^{-}$ charged weak form factor in
the (unphysical) $t\rightarrow 0$ limit, with the general definition of the
charged weak form factors in spinless systems given in Eq.(95). High-precision
lattice calculations of this quantity were performed over the past decade by
the FNAL/MILC Gamiz:2013xxa ; Bazavov:2013maa ; Bazavov:2018kjg  and the ETM
Carrasco:2016kpy  collaborations and showed perfect mutual consistencies and a
steady improvement in precision. The most recent Flavor Lattice Averaging
Group (FLAG) online review (updated from the 2019 version
FlavourLatticeAveragingGroup:2019iem ) quoted the following average:

$|f_{+}^{K^{0}\pi^{-}}(0)|=0.9698(17)\leavevmode\nobreak\ .\quad
n_{f}=2+1+1\quad\text{Refs.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Carrasco:2016kpy,Bazavov:2018kjg}{\@@citephrase{(}}{\@@citephrase{)}}}}$
(189)

However, a recent calculation from the PACS collaboration based on a single
lattice spacing returned a somewhat smaller value Kakazu:2019ltq .

  * •

$C_{K}$ is a simple isospin factor defined through the value of
$f_{+}^{K\pi}(0)$ in the SU(3) limit:

$|C_{K}|=|f_{+}^{K\pi}(0)|_{m_{u}=m_{d}=m_{s}}=\left\\{\begin{array}[]{cc}1\leavevmode\nobreak\
,&K^{0}\\\ 1/\sqrt{2}\leavevmode\nobreak\
,&K^{+}\end{array}\right.\leavevmode\nobreak\ .$ (190)

  * •

$I_{K\ell}^{(0)}$ is the tree-level phase space factor defined as: 444Notice
that a number of important review papers, e.g. Refs.Antonelli:2010yf ;
Cirigliano:2011ny , contain a typo in their formula for $I_{K\ell}^{(0)}$.

$I_{K\ell}^{(0)}=\int_{m_{\ell}^{2}}^{(M_{K}-M_{\pi})^{2}}\frac{dt}{M_{K}^{8}}\bar{\lambda}^{3/2}\left(1+\frac{m_{\ell}^{2}}{2t}\right)\left(1-\frac{m_{\ell}^{2}}{t}\right)^{2}\left[\bar{f}_{+}^{2}(t)+\frac{3m_{\ell}^{2}\Delta_{K\pi}^{2}}{\left(2t+m_{\ell}^{2}\right)\bar{\lambda}}\bar{f}_{0}^{2}(t)\right]\leavevmode\nobreak\
,$ (191)

with
$\bar{\lambda}=\left[t-(M_{K}+M_{\pi})^{2}\right]\left[t-(M_{K}-M_{\pi})^{2}\right]$
and $\Delta_{K\pi}=M_{K}^{2}-M_{\pi}^{2}$. Here we have defined the “scalar”
form factor

$f_{0}^{K\pi}(t)\equiv
f_{+}^{K\pi}(t)+\frac{t}{M_{K}^{2}-M_{\pi}^{2}}f_{-}^{K\pi}(t)$ (192)

which coincides with the “vector” form factor $f_{+}^{K\pi}$ at $t=0$. The
normalized form factors $\bar{f}_{+,0}$ are defined by factoring out their
values at $t=0$: $\bar{f}_{+,0}(t)\equiv f_{+,0}^{K\pi}(t)/f_{+}^{K\pi}(0)$.
These form factors are obtained by fitting the $K_{\ell 3}$ differential decay
rate to a specific parameterization. These parameterizations generally fall
into two classes Abouzaid:2009ry : The Class II parameterization that is based
on a systematic mathematical expansion (e.g. the Taylor expansion
Antonelli:2010yf  and the $z$-parameterization Hill:2006bq ) and the Class I
parameterization that imposes additional physical constraints to reduce the
number of independent parameters (e.g. the pole Lichard:1997ya  and dispersive
parameterization Bernard:2006gy ; Bernard:2009zm ; Abouzaid:2009ry ). A
detailed summary of the results from different parameterizations are given in
Ref.Antonelli:2010yf , and a recent update is provided by the NA48/2
collaboration NA482:2018rgv .

  * •

Finally, $\delta_{\mathrm{SU(2)}}^{K\pi}$ is defined as the isospin breaking
correction to the $f_{+}^{K\pi}(0)$ form factor at $t=0$: 555The appearance of
the factor $C_{K^{0}}/C_{K}$ is simply due to our choice of normalization of
the form factors in Eq.(95).

$\delta_{\mathrm{SU(2)}}^{K\pi}\equiv\left(\frac{C_{K^{0}}}{C_{K}}\frac{f_{+}^{K\pi}(0)}{f_{+}^{K^{0}\pi^{-}}(0)}\right)^{2}-1\leavevmode\nobreak\
,$ (193)

which only resides in $K_{\ell 3}^{+}$ by construction. Upon neglecting small
EM contributions, it is completely fixed by the QCD quark mass parameters
$m_{s}/\hat{m}$ and
$\mathcal{Q}\equiv(m_{s}^{2}-\hat{m}^{2})/(m_{d}^{2}-m_{u}^{2})$. The value
frequently quoted in reviews in the early 2010s Antonelli:2010yf ;
Cirigliano:2011ny  is $\delta_{\mathrm{SU(2)}}^{K^{+}\pi^{0}}=0.058(8)$; with
the latest lattice inputs one obtains
$\delta_{\mathrm{SU(2)}}^{K^{+}\pi^{0}}=0.0457(20)$ Seng:2021nar . At the same
time, one also observes a slight discrepancy between the determinations from
lattice QCD and phenomenology Colangelo:2018jxw .

###  IX.1 Kinematics

The main difference between $K_{\ell 3}$ and $\pi_{e3}$ or superallowed decays
is that the initial and final hadron are not nearly-degenerate. This means
many terms that are suppressed in the $\pi_{e3}$ and superallowed decays may
contribute significantly in $K_{\ell 3}$, at both tree level and loops. Thus,
to facilitate the discussion of the EWRC, we first briefly review the
kinematics in $K_{\ell 3}$.

We are interested in the decay process
$K(p)\rightarrow\pi(p^{\prime})+\ell^{+}(p_{\ell})+\nu_{\ell}(p_{\nu})+n\gamma$,
where $\ell=e,\mu$ and $n\geq 0$. The inclusion of processes with emissions of
real photons is necessary to ensure the cancellation of IR divergences. When
all the massless particles (neutrino and photons) as well as the lepton spins
are not undetected, the differential decay rate depends only on three
variables:

$x\equiv\frac{P^{2}}{M_{K}^{2}}\leavevmode\nobreak\ ,\quad
y\equiv\frac{2p\cdot p_{\ell}}{M_{K}^{2}}\leavevmode\nobreak\ ,\quad
z\equiv\frac{2p\cdot p^{\prime}}{M_{K}^{2}}\leavevmode\nobreak\ ,$ (194)

with $P\equiv p-p^{\prime}-p_{\ell}$. Notice that for $n=0$ one must have
$x=0$ due to momentum conservation. Another possible set of independent
variables are the usual Mandelstam variables:
$s\equiv(p^{\prime}+p_{\ell})^{2}$, $t\equiv(p-p^{\prime})^{2}$,
$u\equiv(p-p_{\ell})^{2}$. We also define $r_{\pi}\equiv
M_{\pi}^{2}/M_{K}^{2}$ and $r_{\ell}\equiv m_{\ell}^{2}/M_{K}^{2}$ for future
convenience, and denote the decay amplitude with $n$ real photons in the final
state as $\mathfrak{M}^{(n)}$.

To $\mathcal{O}(G_{F}\alpha)$ precision, one needs only to include the $n=0$
and $n=1$ processes; the former is calculated to one-loop and the latter at
tree level. The total decay rate is then given by:

$\displaystyle\Gamma_{K_{\ell 3}}$ $\displaystyle=$
$\displaystyle\frac{M_{K}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz\left\\{|\mathfrak{M}^{(0)}|^{2}+\frac{M_{K}^{2}}{32\pi^{3}}\int_{0}^{\alpha_{+}}dx\int\frac{d^{3}k}{E_{k}}\frac{d^{3}p_{\nu}}{E_{\nu}}\delta^{(4)}(P-k-
p_{\nu})|\mathfrak{M}^{(1)}|^{2}\right\\}$ (195)
$\displaystyle+\frac{M_{K}^{3}}{8192\pi^{6}}\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}}^{\alpha_{+}}dx\int\frac{d^{3}k}{E_{k}}\frac{d^{3}p_{\nu}}{E_{\nu}}\delta^{(4)}(P-k-
p_{\nu})|\mathfrak{M}^{(1)}|^{2}\leavevmode\nobreak\ ,$

with $k$ the momentum of the real photon in $\mathfrak{M}^{(1)}$, and

$\alpha_{\pm}\equiv
1-y-z+r_{\pi}+r_{\ell}+\frac{yz}{2}\pm\frac{1}{2}\sqrt{y^{2}-4r_{\ell}}\sqrt{z^{2}-4r_{\pi}}\leavevmode\nobreak\
.$ (196)

We have also defined two different regions in the $y-z$ plane, namely:

$\displaystyle\mathcal{D}_{3}$ $\displaystyle:$ $\displaystyle
a(y)-b(y)<z<a(y)+b(y)\leavevmode\nobreak\ ,\quad
2\sqrt{r_{\ell}}<y<1+r_{\ell}-r_{\pi}$ $\displaystyle\mathcal{D}_{4-3}$
$\displaystyle:$ $\displaystyle
2\sqrt{r_{\pi}}<z<a(y)-b(y)\leavevmode\nobreak\ ,\quad
2\sqrt{r_{\ell}}<y<1-\sqrt{r_{\pi}}+\frac{r_{\ell}}{1-\sqrt{r_{\pi}}}\leavevmode\nobreak\
,$ (197)

where

$a(y)=\frac{(2-y)(1+r_{\pi}+r_{\ell}-y)}{2(1+r_{\ell}-y)}\leavevmode\nobreak\
,\quad
b(y)=\frac{\sqrt{y^{2}-4r_{\ell}}(1+r_{\ell}-r_{\pi}-y)}{2(1+r_{\ell}-y)}\leavevmode\nobreak\
.$ (198)

We see that $|\mathfrak{M}^{(0)}|^{2}$ only resides within $\mathcal{D}_{3}$,
while $|\mathfrak{M}^{(1)}|^{2}$ resides within a larger region
$\mathcal{D}_{4}=\mathcal{D}_{3}+\mathcal{D}_{4-3}$. Both squared amplitudes
contain IR-divergences in $\mathcal{D}_{3}$, which cancel each other in the
first line of Eq.(195); meanwhile $|\mathfrak{M}^{(1)}|^{2}$ is IR-finite in
$\mathcal{D}_{4-3}$.

A lot can be learned from the squared amplitude of the $n=0$ process at tree
level:

$\displaystyle|\mathfrak{M}_{0}^{(0)}|^{2}$ $\displaystyle=$ $\displaystyle
2G_{F}^{2}|V_{us}|^{2}M_{K}^{4}\left\\{A_{1}^{(0)}(y,z)|f_{+}^{K\pi}(t)|^{2}+A_{2}^{(0)}(y,z)f_{+}^{K\pi}(t)f_{-}^{K\pi}(t)\right.$
(199) $\displaystyle\left.+A_{3}^{(0)}(y,z)|f_{-}^{K\pi}(t)|^{2}\right\\}$

where

$\displaystyle A_{1}^{(0)}(y,z)$ $\displaystyle=$ $\displaystyle
4(1-y)(y+z-1)+r_{\ell}(4y+3z-3)-4r_{\pi}+r_{\ell}(r_{\pi}-r_{\ell})$
$\displaystyle A_{2}^{(0)}(y,z)$ $\displaystyle=$ $\displaystyle
2r_{\ell}(3-2y-z+r_{\ell}-r_{\pi})$ $\displaystyle A_{3}^{(0)}(y,z)$
$\displaystyle=$ $\displaystyle
r_{\ell}(1-z+r_{\pi}-r_{\ell})\leavevmode\nobreak\ .$ (200)

We find that $A_{2}^{(0)}$ and $A_{3}^{(0)}$ are explicitly suppressed by a
factor of $r_{\ell}$. Recall that $r_{e}\approx 1\times 10^{-6}$ and
$r_{\mu}\approx 0.05$, we see immediately that the contribution from
$f_{-}^{K\pi}$ to $|\mathfrak{M}_{0}^{(0)}|^{2}$ is negligible in the case of
$K_{e3}$ at tree level. The same is for the EWRC: One can show that the one-
loop virtual corrections can always be recast into a shift of the form
factors:

$f_{\pm}^{K\pi}(t)\rightarrow f_{\pm}^{K\pi}(t)+\delta
f_{\pm}^{K\pi}(y,z)\leavevmode\nobreak\ ,$ (201)

so in $K_{e3}$ the only relevant quantity is $\delta f_{+}^{K\pi}(y,z)$. This
is an important observation when we discuss later the recent progress of the
EWRC based on Sirlin’s representation.

###  IX.2 ChPT treatment of the EWRC

ChPT provided the first analysis of the EWRC in $K_{\ell 3}$ with a rigorous
quantification of theory uncertainties Cirigliano:2001mk ; Cirigliano:2004pv ;
Cirigliano:2008wn . Here we will briefly discuss the main results without
diving into any detail.

Figure 17: Meson loop corrections to $K_{\ell 3}$ in ChPT.

The starting point of the ChPT calculation is the effective chiral Lagrangian
we described in Sec.IV.4. The active DOFs are the pNGBs, leptons and photons,
while the contributions from the heavy gauge bosons are contained implicitly
in the LECs. Applying the framework to $\mathcal{O}(e^{2}p^{2})$, one
encounters two kinds of loop diagrams:

  * •

The mesonic loop diagrams depicted in Fig.17. They correct the form factors
$f_{\pm}^{K\pi}(t)$ that enter the phase space factor $I_{K\ell}^{(0)}$.

  * •

The photonic loop diagrams similar to those in $\pi_{e3}$ (see Fig.6). These
diagrams are IR-divergent and must be combined with the bremsstrahlung
contributions in Fig.4.

The loops above contain UV-divergences that are regulated using dimensional
regularization and reabsorbed into the $\mathcal{O}(p^{4})$ LECs
$\\{L_{i}^{r}\\}$ as well as the $\mathcal{O}(e^{2}p^{2})$ LECs
$\\{K_{i}^{r},X_{i}^{r}\\}$. There is a technical detail here which is not
fully intuitive and is worth pointing out, namely: The meson loops not only
describe pure QCD corrections, but also include short-distance QED corrections
from the $\mathcal{O}(e^{2})$ chiral Lagrangian (i.e. the term proportional to
$Z$ in Eq.(80)), which effects enter through the meson mass splitting and the
four-meson vertex. Similarly, one would naïvely think that all
$\\{K_{i}^{r}\\}$ are EM corrections and should all enter
$\delta_{\mathrm{EM}}^{K\ell}$, but in practice, a subset of them which
describes the EM-induced $\pi^{0}-\eta$ mixing is categorized
$\delta_{\mathrm{SU(2)}}^{K^{+}\pi^{0}}$ instead. Interested readers may refer
to Refs.Cirigliano:2001mk ; Cirigliano:2011ny  for more details.

The EWRCs are contained in two quantities: $S_{\mathrm{EW}}$ and
$\delta_{\mathrm{EM}}^{K\ell}$. The former, $S_{\mathrm{EW}}=1.0232(3)$
encodes the short-distance EWRC that is process-independent, whereas
$\delta_{\mathrm{EM}}^{K\ell}$ describes the process-dependent, long-range
EMRC. The full analytic expression of the latter to $\mathcal{O}(e^{2}p^{2})$
can be found in Ref.Cirigliano:2001mk  and will not be repeated here. There
is, however, a slight difference in its numerical implementation in different
literature: Refs.Cirigliano:2001mk ; Cirigliano:2004pv  adopted a partial
resummation scheme where the structure-independent corrections from soft
photons are explicitly factored out. As a consequence, the resulting amplitude
includes an incomplete resummation of $\mathcal{O}(e^{2}p^{2n})$ EM
corrections with $n\geq 2$. On the other hand, Ref.Cirigliano:2008wn
presented the EM corrections at fixed order, $\mathcal{O}(e^{2}p^{2})$. The
two methods agree within error bars, but the fixed-order result is more
frequently quoted in recent reviews.

The two sources of errors in the calculation above are (1) the neglected terms
of the order $\mathcal{O}(e^{2}p^{4})$, and (2) the poorly-constrained LECs.
To estimate the theory uncertainties and the correlations from the first
source, Ref.Cirigliano:2008wn  decomposed $\delta_{\mathrm{EM}}^{K\ell}$ in
the four independent (in terms of EMRCs) channels as:

$\displaystyle\delta_{\mathrm{EM}}^{K^{0}e}$ $\displaystyle=$
$\displaystyle\delta_{1}+\delta_{2}+\delta_{3}+\delta_{4}$
$\displaystyle\delta_{\mathrm{EM}}^{K^{0}\mu}$ $\displaystyle=$
$\displaystyle\delta_{1}+\delta_{2}-\delta_{3}-\delta_{4}$
$\displaystyle\delta_{\mathrm{EM}}^{K^{+}e}$ $\displaystyle=$
$\displaystyle\delta_{1}-\delta_{2}+\delta_{3}-\delta_{4}$
$\displaystyle\delta_{\mathrm{EM}}^{K^{+}\mu}$ $\displaystyle=$
$\displaystyle\delta_{1}-\delta_{2}-\delta_{3}+\delta_{4}\leavevmode\nobreak\
,$ (202)

where $\delta_{1}$ is common to all channels, $\delta_{2}$ is lepton-flavor
independent but anti-correlated in kaon isospin, $\delta_{3}$ is isospin-
independent but anti-correlated in lepton flavor, and $\delta_{4}$ are anti-
correlated to both the isospin and the lepton flavor. Theory uncertainties
from higher-order chiral corrections to different $\delta_{i}$ are assumed to
be uncorrelated, and are obtained by multiplying the respective central values
with $M_{K}^{2}/(4\pi F_{\pi})^{2}\sim 0.2$.

In the meantime, the $\mathcal{O}(e^{2}p^{2})$ LECs that enter
$\delta_{\mathrm{EM}}^{K\ell}$ are $X_{1-3}^{r}$,
$\tilde{X}_{6}^{\mathrm{phys}}$ and $K_{3-6}^{r}$ (see their definitions in
Sec.IV.4 and Eq.(112)), where only $X_{1}$ and $\tilde{X}_{6}^{\mathrm{phys}}$
contribute to $\delta f_{+}^{K\pi}$. These LECs are not constrained by chiral
symmetry, and earlier estimations of their sizes were based on large-$N_{c}$
expansion Bijnens:1996kk  or resonance models Moussallam:1997xx ;
Ananthanarayan:2004qk ; DescotesGenon:2005pw . Recent reviews of mesonic LECs
(e.g. Ref.Bijnens:2014lea ) often quote the results from
Refs.Ananthanarayan:2004qk ; DescotesGenon:2005pw :

$K_{3}^{r}=2.7,\leavevmode\nobreak\ K_{4}^{r}=1.4,\leavevmode\nobreak\
K_{5}^{r}=11.6,\leavevmode\nobreak\ K_{6}^{r}=2.8,\leavevmode\nobreak\
X_{1}=-3.7,\leavevmode\nobreak\ X_{2}^{r}=3.6,\leavevmode\nobreak\
X_{3}^{r}=5.0,\leavevmode\nobreak\ \tilde{X}_{6}^{\mathrm{phys}}=10.4,$ (203)

in units of $10^{-3}$ at $\mu=0.77\leavevmode\nobreak\ $GeV. None of the above
is accompanied by a rigorously-estimated error, and in practice these values
are used by assuming a 100% uncertainty. Also, different LECs are assumed to
be uncorrelated.

| $\delta_{\mathrm{EM}}^{K\ell}(\%)$
---|---
$K_{e3}^{0}$ | $0.99\pm 0.19_{e^{2}p^{4}}\pm 0.11_{\mathrm{LEC}}$
$K_{e3}^{\pm}$ | $0.10\pm 0.19_{e^{2}p^{4}}\pm 0.16_{\mathrm{LEC}}$
$K_{\mu 3}^{0}$ | $1.40\pm 0.19_{e^{2}p^{4}}\pm 0.11_{\mathrm{LEC}}$
$K_{\mu 3}^{\pm}$ | $0.016\pm 0.19_{e^{2}p^{4}}\pm 0.16_{\mathrm{LEC}}$
Table 1: Results of $\delta_{\mathrm{EM}}^{K\ell}$ based on fixed-order ChPT
calculations in Ref.Cirigliano:2008wn .

With the procedures above, Ref.Cirigliano:2008wn  presented the most updated
ChPT determination of $\delta_{\mathrm{EM}}^{K\ell}$, as summarized in Table
1. We observe that both the LECs and the higher-order chiral corrections
contribute to uncertainties of order $10^{-3}$, which sets a natural
limitation to the theory precision within the traditional EFT framework. There
are recently some proposals to compute the full $K_{\ell 3}$ RC, including
both the virtual and real corrections, directly from lattice QCD
Giusti:2018guw ; Sachrajda:2019uhh ; Cirigliano:2019jig  using the similar
approach that was proved successful in the kaon leptonic decay
$K\rightarrow\mu\nu$ ($K_{\mu 2}$) Carrasco:2015xwa ; Lubicz:2016xro ;
Giusti:2017dwk ; DiCarlo:2019thl . However, the generalization from $K_{\mu
2}$ to $K_{\ell 3}$ that involves one more hadron at the final state greatly
complicates the problem. A complete calculation with an overall per mille
level precision is expected to take ten years or more BoyleSnowmass .

###  IX.3 $K_{e3}$ EWRC in Sirlin’s representation

With the recent progress of the EWRC theory in pion, neutron and nuclei, it
becomes apparent that the Sirlin’s representation we described in Sec.III is
also applicable to $K_{\ell 3}$ which enables one to overcome the natural
limitations of the traditional ChPT framework. The underlying principle is
rather straightforward: The ChPT treatment contains a large
$\mathcal{O}(e^{2}p^{4})$ uncertainty because the entire RC, including both
the numerically large and small pieces, is subject to a chiral expansion. On
the other hand, in the Sirlin’s representation the most important pieces in
the one-loop EMRC, namely
$\left(\delta\mathfrak{M}_{2}+\delta\mathfrak{M}_{\gamma
W}^{a}\right)_{\mathrm{int}}$ and $\delta\mathfrak{M}_{\gamma W}^{b}$, are
expressed in terms well-defined hadronic matrix elements involving a two-
current product. This allows us to evaluate them non-perturbatively without
resorting to a chiral expansion, and thus remove a large portion of the chiral
uncertainty. Furthermore, the closed form of $\delta\mathfrak{M}_{\gamma
W}^{b}$ permits a straightforward first-principles study using lattice QCD.
This removes substantially the theory uncertainty from the unknown hadron
physics at $q\sim\Lambda_{\chi}$, which translates into the LEC uncertainties
in the ChPT description.

The idea above was first pioneered in Refs.Seng:2019lxf ; Seng:2020jtz , and
subsequently applied to the study of the EWRC in $K_{e3}$ Seng:2021boy ;
Seng:2021wcf . In what follows, we briefly outline the procedures and discuss
the main results.

####  IX.3.1 $\left(\delta\mathfrak{M}_{2}+\delta\mathfrak{M}_{\gamma
W}^{a}\right)_{\mathrm{int}}$ and $\delta\mathfrak{M}_{\gamma W}^{b,V}$

First, a generic one-loop correction to the $K_{e3}$ decay amplitude reads:

$\delta\mathfrak{M}_{\mathrm{vir}}=-\frac{G_{F}}{\sqrt{2}}L_{\lambda}I^{\lambda}\leavevmode\nobreak\
.$ (204)

The loop integrals are contained in the quantity $I^{\lambda}$. Upon
contracting with the lepton piece and using the on-shell conditions, it simply
translates into corrections to the form factors:

$I^{\lambda}\rightarrow\delta f_{+}^{K\pi}(y,z)(p+p^{\prime})^{\lambda}+\delta
f_{-}^{K\pi}(y,z)(p-p^{\prime})^{\lambda}\leavevmode\nobreak\ .$ (205)

As we advertised before, in $K_{e3}$ only the correction $\delta f_{+}^{K\pi}$
is relevant.

The important hadronic quantities in $I^{\lambda}$ are $T^{\mu\nu}$ and
$\Gamma^{\mu}$ defined in Eq.(7), with $b=\gamma$, $\phi_{i}=K$ and
$\phi_{f}=\pi$. Recall that in Eq.(102) we split the charged weak current into
the vector and axial piece. Accordingly, we may split:

$T^{\mu\nu}=(T^{\mu\nu})_{V}+(T^{\mu\nu})_{A}\leavevmode\nobreak\
,\quad\delta\mathfrak{M}_{\gamma W}^{b}=\delta\mathfrak{M}_{\gamma
W}^{b,V}+\delta\mathfrak{M}_{\gamma W}^{b,A}\leavevmode\nobreak\ .$ (206)

From parity, $(T^{\mu\nu})_{A}$ must contain an antisymmetric
$\epsilon$-tensor, while $(T^{\mu\nu})_{V}$ does not have such a tensor. It is
easy to see that, in near-degenerate decays (e.g. $\pi_{e3}$ and superallowed
beta decays), $\delta\mathfrak{M}_{\gamma W}^{b,V}$ is negligible because when
$p^{\prime}\approx p$ and $p_{e}\approx 0$, there is no structure that
survives after contracting with the $\epsilon$-tensor in the integrand. This
is obviously not the case in $K_{e3}$ so this term must be retained.

Figure 18: The seagull diagram in $T^{\mu\nu}$ and $\Gamma^{\mu}$. The black
dot includes all possible diagrammatic structures except for a single $K$ or
$\pi$ propagator.

We start by discussing the loop integrals that only probe the physics at
$q^{\prime}\sim p-p^{\prime}\sim p_{e}$, namely
$\left(\delta\mathfrak{M}_{2}+\delta\mathfrak{M}_{\gamma
W}^{a}\right)_{\mathrm{int}}+\delta\mathfrak{M}_{\gamma W}^{b,V}$. Their
combined effect reads:

$\displaystyle I_{\mathfrak{A}}^{\lambda}$ $\displaystyle=$
$\displaystyle-e^{2}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{1}{[(p_{e}-q^{\prime})^{2}-m_{e}^{2}][q^{\prime
2}-M_{\gamma}^{2}]}\biggl{\\{}\frac{2p_{e}\cdot
q^{\prime}q^{\prime\lambda}}{q^{\prime
2}-M_{\gamma}^{2}}T^{\mu}_{\>\>\mu}+2p_{e\mu}T^{\mu\lambda}$ (207)
$\displaystyle-(p-p^{\prime})_{\mu}T^{\lambda\mu}+i\Gamma^{\lambda}-i\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}(T_{\mu\nu})_{V}\biggr{\\}}\leavevmode\nobreak\
.$

We may now ask: What are the hadron physics inputs needed to evaluate the
integral above to a satisfactory precision? In the most general terms,
$T^{\mu\nu}$ and $\Gamma^{\mu}$ are fully described by two types of Feynman
diagrams, namely (1) the pole diagrams Fig.5 and (2) the seagull diagram in
Fig.18. The former are completely fixed by the $K$ and $\pi$ electromagnetic
and charged weak form factors (remember $q=p^{\prime}-p+q^{\prime}$): 666We
neglect the contribution from the electromagnetic form factor of $K^{0}$ which
is numerically insignificant.

$\displaystyle\left(T_{K^{0}\pi^{-}}^{\mu\nu}\right)_{\mathrm{pole}}$
$\displaystyle=$ $\displaystyle iV_{us}^{*}F_{\mathrm{em}}^{\pi^{-}}(q^{\prime
2})\frac{(2p^{\prime}+q^{\prime})^{\mu}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}\left[f_{+}^{K^{0}\pi^{-}}(q^{2})(2p+q)^{\nu}-f_{-}^{K^{0}\pi^{-}}(q^{2})q^{\nu}\right]$
$\displaystyle\left(T_{K^{+}\pi^{0}}^{\mu\nu}\right)_{\mathrm{pole}}$
$\displaystyle=$ $\displaystyle iV_{us}^{*}F_{\mathrm{em}}^{K^{+}}(q^{\prime
2})\frac{(2p-q^{\prime})^{\mu}}{(p-q^{\prime})^{2}-M_{K}^{2}}\left[f_{+}^{K^{+}\pi^{0}}(q^{2})(2p^{\prime}-q)^{\nu}-f_{-}^{K^{+}\pi^{0}}(q^{2})q^{\nu}\right]$
$\displaystyle\left(\Gamma_{K^{0}\pi^{-}}^{\mu}\right)_{\mathrm{pole}}$
$\displaystyle=$ $\displaystyle
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p^{\prime}+q^{\prime})^{2}-M_{\pi}^{2}}(2p^{\prime}+q^{\prime})^{\mu}F_{\mathrm{em}}^{\pi^{-}}(q^{\prime
2})f_{0}^{K^{0}\pi^{-}}(q^{2})$
$\displaystyle\left(\Gamma_{K^{+}\pi^{0}}^{\mu}\right)_{\mathrm{pole}}$
$\displaystyle=$ $\displaystyle
V_{us}^{*}\frac{M_{K}^{2}-M_{\pi}^{2}}{(p-q^{\prime})^{2}-M_{K}^{2}}(2p-q^{\prime})^{\mu}F_{\mathrm{em}}^{K^{+}}(q^{\prime
2})f_{0}^{K^{+}\pi^{0}}(q^{2})\leavevmode\nobreak\ ,$ (208)

where

$\langle\phi(p^{\prime})|J_{\mathrm{em}}^{\mu}|\phi(p)\rangle=F_{\mathrm{em}}^{\phi}(t)(p+p^{\prime})^{\mu}$
(209)

defines the electromagnetic form factor of spinless particles, with
$F_{\mathrm{em}}^{K^{+}}(0)=1$ and $F_{\mathrm{em}}^{\pi^{-}}(0)=-1$. With the
above, the pole contribution to $I_{\mathfrak{A}}^{\lambda}$ can be
straightforwardly calculated using the experimental parameterizations of the
form factors Amendolia:1986wj ; Amendolia:1986ui ; Ananthanarayan:2017efc ;
Colangelo:2018mtw ; Lazzeroni:2018glh ; Zyla:2020zbs . In particular, a
monopole parameterization of the form factors allows one to perform the loop
integral analytically using the standard Passarino-Veltman reduction
Passarino:1978jh . Notice that $(\delta
f_{+}^{K\pi})_{\mathfrak{A},\mathrm{pole}}$ is UV-finite even without the form
factors, so the inclusion of the full form factors only give rise to power-
suppressed (instead of logarithmic) corrections.

The seagull diagram is required to ensure the gauge invariance, and it
represents physically the collective effect from all non-elastic intermediate
states. The precise evaluation of its effect seems to be highly non-trivial,
but fortunately, since $I_{\mathfrak{A}}^{\lambda}$ only probes the
small-$q^{\prime}$ region, one needs to only consider some of the lowest
intermediate states. For instance, one may estimate the contribution from the
$1^{--}$ and $1^{++}$ resonances using the resonance chiral theory
Ecker:1988te ; Ecker:1989yg ; Cirigliano:2006hb , and it effect is found to be
less than $10^{-4}$.

To conclude, through the procedures above one can evaluate $(\delta
f_{+}^{K\pi})_{\mathfrak{A}}$ to a high precision. The main theory
uncertainties come from the form factors and a conservative estimation of the
size of the seagull term contribution, which are both of the order $10^{-4}$.

####  IX.3.2 $\delta\mathfrak{M}_{\gamma W}^{b,A}$

The second non-trivial integral is:

$I_{\mathfrak{B}}^{\lambda}=ie^{2}\int\frac{d^{4}q^{\prime}}{(2\pi)^{4}}\frac{M_{W}^{2}}{M_{W}^{2}-q^{\prime
2}}\frac{\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}(T_{\mu\nu})_{A}}{[(p_{e}-q^{\prime})^{2}-m_{e}^{2}]q^{\prime
2}}\leavevmode\nobreak\ ,$ (210)

which comes from $\delta\mathfrak{M}_{\gamma W}^{b,A}$. It is IR-finite, but
probes the physics from $q^{\prime}=0$ all the way up to $q^{\prime}\sim
M_{W}$. In fact, this is nothing but the generalization of the forward axial
$\gamma W$-box in Eq.(105) to include non-forward effects due to the large
$K-\pi$ mass splitting. Therefore, an efficient way to proceed is to first
calculate the forward $K\pi$ $\gamma W$-box on lattice; this single step
captures all the dominant physics in $I_{\mathfrak{B}}^{\lambda}$. After that,
the uncertainty due to the non-forward effect may be estimated through
standard power counting arguments. Following the same procedure in
Ref.Feng:2020zdc , Ref.Ma:2021azh  performed the first calculation of the
forward $K\pi$ $\gamma W$-box diagram (recall the notations in Sec.V.5):

$\Box_{\gamma W}^{<}(K^{0},\pi^{-},M_{\pi})=0.28(4)_{\mathrm{lat}}\times
10^{-3}\leavevmode\nobreak\ ,$ (211)

from which $\Box_{\gamma
W}^{<}(K^{+},\pi^{0},M_{\pi})=1.06(7)_{\mathrm{lat}}\times 10^{-3}$ can also
be obtained through a simple chiral matching Seng:2020jtz .

To connect the result above with $I_{\mathfrak{B}}^{\lambda}$, we first split
the latter as well as $(f_{+}^{K\pi})_{\mathfrak{B}}$ into the “$>$” and “$<$”
components, which correspond to the integral at $Q^{2}>Q_{\mathrm{cut}}^{2}$
and $Q^{2}<Q_{\mathrm{cut}}^{2}$ respectively. The matching in the “$>$”
region is rather trivial:

$(\delta f_{+}^{K\pi})_{\mathfrak{B}}^{>}=\Box_{\gamma
W}^{>}f_{+}^{K\pi}(t)\leavevmode\nobreak\ .$ (212)

Meanwhile, in the “$<$” region, we parameterize the integral as:

$\displaystyle(I_{\mathfrak{B}}^{\lambda})^{<}$ $\displaystyle=$
$\displaystyle
V_{us}^{*}[g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)(p+p^{\prime})^{\lambda}+g_{-}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)(p-p^{\prime})^{\lambda}$
(213)
$\displaystyle+g_{e}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)p_{e}^{\lambda}]\leavevmode\nobreak\
,$

and we are only interested in $(\delta
f_{+}^{K\pi})_{\mathfrak{B}}^{<}=g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)$.
We observe that, $g_{+}$ is simply related to the lattice calculation upon
taking the forward limit:

$g_{+}(M_{\pi}^{2},M_{\pi}^{2},0,M_{\pi}^{2},M_{\pi}^{2})=\Box_{\gamma
W}^{<}(K,\pi,M_{\pi})f_{+}^{K\pi}(0)\leavevmode\nobreak\ .$ (214)

Incorporating the non-forward corrections, we may write:

$g_{+}(M_{K}^{2},M_{\pi}^{2},m_{e}^{2},s,u)=g_{+}(M_{\pi}^{2},M_{\pi}^{2},0,M_{\pi}^{2},M_{\pi}^{2})+\mathcal{O}\left(\frac{E^{2}}{\Lambda_{\chi}^{2}}\right)$
(215)

with $E$ an energy scale that characterizes the non-forward kinematics; we may
take $E=M_{K}^{2}$ as an upper limit. That leads to:

$(\delta f_{+}^{K\pi})_{\mathfrak{B}}^{<}=\left[\Box_{\gamma
W}^{<}(K,\pi,M_{\pi})+\mathcal{O}\left(\frac{M_{K}^{2}}{\Lambda_{\chi}^{2}}\right)\right]f_{+}^{K\pi}(t)\leavevmode\nobreak\
,$ (216)

where the uncertainty from non-forward corrections is estimated by multiplying
the central value with $M_{K}^{2}/\Lambda_{\chi}^{2}$. Utilizing the
procedures above, one is able to fix $(\delta f_{+}^{K\pi})_{\mathfrak{B}}$ to
high precision, with the theory uncertainty from both the lattice inputs and
the non-forward effects controlled at the $10^{-4}$ level.

Finally, in analogy to the discussion at the end of Sec.V.5, the lattice
calculation of the $K\pi$-box diagram also allows us to determine a specific
combination of the $\mathcal{O}(e^{2}p^{2})$ LECs to high precision Ma:2021azh
:

$\displaystyle-\frac{8}{3}X_{1}+\bar{X}_{6}^{\mathrm{phys}}(M_{\rho})$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi\alpha}\left(\Box_{\gamma
W}(K^{0},\pi^{-},M_{\pi})-\frac{\alpha}{8\pi}\ln\frac{M_{W}^{2}}{M_{\rho}^{2}}\right)+\frac{1}{8\pi^{2}}\left(\frac{5}{4}-\tilde{a}_{g}\right)$
(217) $\displaystyle=$ $\displaystyle 22.6(1.0)_{\mathrm{lat}}\times
10^{-3}\leavevmode\nobreak\ .$

####  IX.3.3 $\delta F_{3}^{\lambda}(p^{\prime},p)$

The next required input is $\delta F_{3}^{\lambda}(p^{\prime},p)$ defined in
Eq.(45). To evaluate this quantity we need to know the hadronic matrix
elements $\bar{T}^{\mu}(\delta\bar{p};p^{\prime},p)$ and
$D(\delta\bar{p};p^{\prime},p)$ which, according to their quark-level
definitions in Eqs.(38) and (42), involve a product of three currents which
makes them very challenging for first-principles studies. Fortunately, its
contribution to $\delta f_{+}^{K\pi}$ is also smaller (apart from the IR-
divergent pieces), so one may resort to an EFT calculation without suffering
from a huge loss in precision.

Figure 19: Diagrammatic representation of
$i\bar{T}^{\mu}(\delta\bar{p};p^{\prime},p)$ and
$D(\delta\bar{p};p^{\prime},p)$. An external wiggle line represents an
insertion of $J^{\mu}$ or $\partial\cdot J$, an external dashed line ended
with a black dot represents an insertion of $i\delta\mathcal{L}$, and internal
dashed line with gray blob represents a full single-particle propagator.
Figures taken from Ref.Seng:2019lxf .

Ref.Seng:2019lxf  outlined the full procedure to calculate
$i\bar{T}^{\mu}(\delta\bar{p};p^{\prime},p)$ and
$D(\delta\bar{p};p^{\prime},p)$ in ChPT. The starting point is the following
matrix elements:

$\displaystyle i\bar{T}^{\mu}(\delta\bar{p};p^{\prime},p)$ $\displaystyle=$
$\displaystyle\int d^{4}xe^{i\delta\bar{p}\cdot
x}\bigl{\langle}\pi(p^{\prime})\bigr{|}T\left\\{J^{\mu}(x)i\delta\mathcal{L}(0)\right\\}\bigl{|}K(p)\bigr{\rangle}$
$\displaystyle D(\delta\bar{p};p^{\prime},p)$ $\displaystyle=$
$\displaystyle\int d^{4}xe^{i\delta\bar{p}\cdot
x}\bigl{\langle}\pi(p^{\prime})\bigr{|}T\left\\{\partial\cdot
J(x)i\delta\mathcal{L}(0)\right\\}\bigl{|}K(p)\bigr{\rangle}\leavevmode\nobreak\
,$ (218)

where the QED correction is represented schematically by
$i\delta\mathcal{L}(0)$ instead of the explicit product of two EM currents.
The diagrammatic representation of these matrix elements are given in Fig.19.
To calculate them, we need the ChPT representation of the charged weak current
and its divergence, which can be derived from the mesonic chiral Lagrangian in
Sec.IV.4. The result reads:

$\displaystyle J^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{iF_{0}^{2}}{8\sqrt{2}G_{F}}\left\langle
Q_{\mathrm{L}}^{\mathrm{w}\dagger}\left([D^{\mu}U^{\dagger},U]+[D^{\mu}U,U^{\dagger}]\right)\right\rangle+...$
$\displaystyle\partial\cdot J$ $\displaystyle=$
$\displaystyle-\frac{iF_{0}^{2}}{8\sqrt{2}G_{F}}\left\langle
Q_{\mathrm{L}}^{\mathrm{w}\dagger}\left([\chi^{\dagger},U]+[\chi,U^{\dagger}]\right)\right\rangle+...\leavevmode\nobreak\
,$ (219)

where the ellipses denote terms of higher chiral order. Here we have rewritten
$\partial\cdot J$ using the equations of motion (EOMs), so that it satisfies
the current conservation in the SU(3) limit at the operator level. This is a
necessary step to avoid inconsistencies in the implementation of the Feynman
rules (see discussions in Ref.Seng:2019lxf ). Meanwhile, the effect of
$i\delta\mathcal{L}(0)$ is obtained by applying the Feynman rules derived from
the chiral Lagrangian with dynamical photons. Notice that it does not
necessarily involve a photon loop, but can as well be a local interaction that
originates from short-distance QED effects (i.e. $\mathcal{L}^{e^{2}}$ and
$\mathcal{L}^{e^{2}p^{2}}_{\\{K\\}}$).

As usual, $\delta F_{3}^{\lambda}(p^{\prime},p)$ is expressed in terms of
corrections to the two form factors:

$\delta F_{3}^{\lambda}(p^{\prime},p)=V_{us}^{*}\left[\delta
f_{+,3}^{K\pi}(t)(p+p^{\prime})^{\lambda}+\delta
f_{-,3}^{K\pi}(t)(p-p^{\prime})^{\lambda}\right]\leavevmode\nobreak\ ,$ (220)

and in $K_{e3}$ one only needs $\delta f_{+,3}^{K\pi}$. It is useful to split
it into the IR-divergent and IR-finite terms:

$\delta f_{+,3}^{K\pi}=\left(\delta
f_{+,3}^{K\pi}\right)^{\mathrm{IR}}+\left(\delta
f_{+,3}^{K\pi}\right)^{\mathrm{fin}}\leavevmode\nobreak\ .$ (221)

Both terms were calculated to $\mathcal{O}(e^{2}p^{2})$ in Ref.Seng:2019lxf .
The results read:

$\displaystyle\left(\delta
f_{+,3}^{K^{0}\pi^{-}}\right)^{\mathrm{IR}}_{e^{2}p^{2}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\frac{p^{\prime}\cdot(p-p^{\prime})}{2M_{\pi}^{2}}\left[\ln\frac{M_{\pi}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]$
$\displaystyle\left(\delta
f_{+,3}^{K^{+}\pi^{0}}\right)^{\mathrm{IR}}_{e^{2}p^{2}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\sqrt{2}\pi}\frac{p\cdot(p-p^{\prime})}{2M_{K}^{2}}\left[\ln\frac{M_{K}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]\leavevmode\nobreak\
,$ (222)

and:

$\displaystyle\left(\delta
f_{+,3}^{K^{0}\pi^{-}}\right)_{e^{2}p^{2}}^{\mathrm{fin}}$ $\displaystyle=$
$\displaystyle-8\pi
Z\alpha\left[\frac{1}{2}\bar{h}_{K^{+}\pi^{0}}(t)+\bar{h}_{K^{0}\pi^{-}}(t)+\frac{3}{2}\bar{h}_{K^{+}\eta}(t)\right]$
$\displaystyle\left(\delta
f_{+,3}^{K^{+}\pi^{0}}\right)_{e^{2}p^{2}}^{\mathrm{fin}}$ $\displaystyle=$
$\displaystyle-\frac{8\pi
Z\alpha}{\sqrt{2}}\left[\frac{1}{2}\bar{h}_{K^{+}\pi^{0}}(t)+\bar{h}_{K^{0}\pi^{-}}(t)+\frac{3}{2}\bar{h}_{K^{+}\eta}(t)\right]$
(223)
$\displaystyle+\frac{Z\alpha}{2\sqrt{2}\pi}\frac{M_{K}^{2}}{M_{\eta}^{2}-M_{\pi}^{2}}\left[1+\ln\frac{M_{K}^{2}}{\mu^{2}}\right]-\frac{4\pi\alpha}{\sqrt{2}}\left[-2K_{3}^{r}+K_{4}^{r}+\frac{2}{3}K_{5}^{r}+\frac{2}{3}K_{6}^{r}\right]$
$\displaystyle-\frac{8\pi\alpha}{\sqrt{2}}\frac{M_{\pi}^{2}}{M_{\eta}^{2}-M_{\pi}^{2}}\left[-2K_{3}^{r}+K_{4}^{r}+\frac{2}{3}K_{5}^{r}+\frac{2}{3}K_{6}^{r}-\frac{2}{3}K_{9}^{r}-\frac{2}{3}K_{10}^{r}\right]\leavevmode\nobreak\
.$

The definition of the loop function $\bar{h}_{PQ}(t)$ can be found in Appendix
D.

A direct numerical application of the results above is not possible due to the
incomplete cancellation of IR-divergences. This is because all other IR-
divergence terms (i.e. terms attached to $\ln M_{\gamma}^{2}$) in both virtual
and real corrections (as we will discuss later) are calculated exactly, except
that in $(\delta f_{+,3}^{K\pi})^{\mathrm{IR}}$ which is expanded to
$\mathcal{O}(e^{2}p^{2})$. Fortunately, it is easy to resum this term to all
orders in chiral expansion by simply noticing that it originates from the
convection term contribution Seng:2021wcf . The resummed version reads:

$\displaystyle\left(\delta f_{+}^{K^{0}\pi^{-}}\right)^{\mathrm{IR}}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\frac{p^{\prime}\cdot(p-p^{\prime})}{2M_{\pi}^{2}}\left[f_{+}^{K^{0}\pi^{-}}(t)+f_{-}^{K^{0}\pi^{-}}(t)\right]\left[\ln\frac{M_{\pi}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]$
$\displaystyle\left(\delta f_{+}^{K^{+}\pi^{0}}\right)^{\mathrm{IR}}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\frac{p\cdot(p-p^{\prime})}{2M_{K}^{2}}\left[f_{+}^{K^{+}\pi^{0}}(t)-f_{-}^{K^{+}\pi^{0}}(t)\right]\left[\ln\frac{M_{K}^{2}}{M_{\gamma}^{2}}-\frac{5}{2}\right]\leavevmode\nobreak\
.$ (224)

To summarize, we obtain:

$\delta f_{+,3}^{K\pi}=\left(\delta
f_{+,3}^{K\pi}\right)^{\mathrm{IR}}+\left\\{\left(\delta
f_{+,3}^{K\pi}\right)^{\mathrm{fin}}_{e^{2}p^{2}}+\mathcal{O}(e^{2}p^{4})\right\\}\leavevmode\nobreak\
,$ (225)

where the first term at the right hand side is exact, and only the second term
is associated to a chiral expansion uncertainty. Furthermore, it was realized
that $(\delta f_{+,3}^{K\pi})^{\mathrm{fin}}_{e^{2}p^{2}}$ consists of terms
proportional to $Z$ which are assigned to $I_{K\ell}^{(0)}$, as well as
counterterms for the electromagnetically-induced $\pi^{0}-\eta$ mixing which
are assigned to $\delta_{\mathrm{SU(2)}}^{K^{+}\pi^{0}}$, in the standard ChPT
convention (recall our discussion in Sec.IX.2). This means that only $(\delta
f_{+,3}^{K\pi})^{\mathrm{IR}}$ contributes to $\delta_{\mathrm{EM}}^{Ke}$.

####  IX.3.4 Bremsstrahlung contribution

Finally, we briefly discuss the treatment of the bremsstrahlung contribution
depicted in Fig.4, of which amplitude is given by Eq.(51). In traditional ChPT
treatment, the entire amplitude is evaluated to LO in the chiral expansion.
But here we can do much better. First of all, we retain the full charged weak
matrix element $F_{\mu}$. Next, we split $T^{\mu\nu}$ into two terms, which
separately satisfies the EM Ward identity:

$T^{\mu\nu}=T_{\mathrm{conv}}^{\mu\nu}+\left\\{(T^{\mu\nu}-T^{\mu\nu}_{\mathrm{conv}})_{p^{2}}+\mathcal{O}(p^{4})\right\\}\leavevmode\nobreak\
.$ (226)

The first term at the right hand side is the full convection term (see
Eq.(103)) that contains the full information of the IR-divergence, while the
second is the difference which is evaluated to $\mathcal{O}(p^{2})$:

$\displaystyle(T^{\mu\nu}_{K^{0}\pi^{-}})_{p^{2}}$ $\displaystyle=$
$\displaystyle
iV_{us}^{*}\left[\frac{(2p^{\prime}+k)^{\mu}(p+p^{\prime}+k)^{\nu}}{(p^{\prime}+k)^{2}-M_{\pi}^{2}}-g^{\mu\nu}\right]$
$\displaystyle(T^{\mu\nu}_{K^{+}\pi^{0}})_{p^{2}}$ $\displaystyle=$
$\displaystyle-\frac{iV_{us}^{*}}{\sqrt{2}}\left[\frac{(2p-k)^{\mu}(p+p^{\prime}-k)^{\nu}}{(p-k)^{2}-M_{K}^{2}}-g^{\mu\nu}\right]$
(227)

and

$\displaystyle(T^{\mu\nu}_{K^{0}\pi^{-}})_{p^{2}}^{\mathrm{conv}}$
$\displaystyle=$ $\displaystyle
iV_{us}^{*}\left[\frac{(2p^{\prime}+k)^{\mu}(p+p^{\prime})^{\nu}}{(p^{\prime}+k)^{2}-M_{\pi}^{2}}\right]$
$\displaystyle(T^{\mu\nu}_{K^{+}\pi^{0}})_{p^{2}}^{\mathrm{conv}}$
$\displaystyle=$
$\displaystyle-\frac{iV_{us}^{*}}{\sqrt{2}}\left[\frac{(2p-k)^{\mu}(p+p^{\prime})^{\nu}}{(p-k)^{2}-M_{K}^{2}}\right]\leavevmode\nobreak\
.$ (228)

In this way, we can split the bremsstrahlung amplitude into two pieces:

$\mathfrak{M}_{\mathrm{brems}}=\mathfrak{M}_{A}+\mathfrak{M}_{B}\leavevmode\nobreak\
,$ (229)

where $\mathfrak{M}_{B}$ consists of terms that depend on
$(T^{\mu\nu}-T^{\mu\nu}_{\mathrm{conv}})_{p^{2}}$, and all other terms go to
$\mathfrak{M}_{A}$. This means $\mathfrak{M}_{A}$ is exact, and only
$\mathfrak{M}_{B}$ involves a chiral expansion. Hence, when we calculate the
squared amplitude:

$|\mathfrak{M}_{\mathrm{brems}}|^{2}=|\mathfrak{M}_{A}|^{2}+2\mathfrak{Re}\\{\mathfrak{M}_{A}^{*}\mathfrak{M}_{B}\\}+|\mathfrak{M}_{B}|^{2}\leavevmode\nobreak\
,$ (230)

the only IR-divergent and the numerically-largest contribution
$|\mathfrak{M}_{A}|^{2}$ will not carry a chiral expansion uncertainty. This
single change improves substantially the precision of the bremsstrahlung
contribution. We summarize in Appendix E all the relevant phase-space
integrals for the benefits of interested readers.

####  IX.3.5 Results and discussions

With the methods outlined above, Refs.Seng:2021boy ; Seng:2021wcf  reported an
improved determination of the long-distance EMRC in $K_{e3}$:

$\displaystyle\delta_{\mathrm{EM}}^{K^{+}e}$ $\displaystyle=$ $\displaystyle
0.21(2)_{\mathrm{sg}}(1)_{r_{K}}(1)_{\mathrm{lat}}(4)_{\mathrm{NF}}(1)_{e^{2}p^{4}}\times
10^{-2}$ $\displaystyle\delta_{\mathrm{EM}}^{K^{0}e}$ $\displaystyle=$
$\displaystyle
1.16(2)_{\mathrm{sg}}(1)_{\mathrm{lat}}(1)_{\mathrm{NF}}(2)_{e^{2}p^{4}}\times
10^{-2}\leavevmode\nobreak\ ,$ (231)

The sources of uncertainties are as follows:

  * •

“sg” represents the very conservative estimation of the seagull diagram
contribution in $(\delta\mathfrak{M}_{2}+\delta\mathfrak{M}_{\gamma
W}^{a})_{\mathrm{int}}+\delta\mathfrak{M}_{\gamma W}^{b,V}$.

  * •

The uncertainties due to the form factor parameterization are mostly small.
The only piece that brings a $\sim 10^{-4}$ uncertainty is the uncertainty of
the $K^{+}$ charge radius $r_{K}$ that only enters
$\delta_{\mathrm{EM}}^{K^{+}e}$.

  * •

“lat” represents the lattice uncertainty in the evaluation of the forward
$\gamma W$-box diagrams.

  * •

“NF” is the estimation of the non-forward correction in
$\delta\mathfrak{M}_{\gamma W}^{b,A}$.

  * •

Finally, “$e^{2}p^{4}$” represents the (reduced) chiral uncertainty in the
bremsstrahlung contribution.

These new results are consistent with the ChPT determinations in Table 1, but
show a substantial increase in precision by nearly an order of magnitude. They
indicate the great success of the Sirlin’s representation of EWRC, not only in
the nearly-degenerate decays but also in non-degenerate decay processes.
However, its success in $K_{e3}$ relies critically on the fact that only
$\delta f_{+}^{K\pi}$ is relevant in the discussion. In its future application
to $K_{\mu 3}$ where $\delta f_{-}^{K\pi}$ also starts to be relevant, a more
complicated analysis of the theory uncertainties is expected. A good news is
that, even in $K_{\mu 3}$, the $\delta f_{-}^{K\pi}$ contribution to the
squared amplitude is still suppressed by $r_{\mu}\approx 0.05$, so a possible
way to proceed is to calculate the full $\delta f_{-}^{K\pi}$ within ChPT, and
estimate its associated uncertainty by the usual power-counting argument. This
is an interesting topic for future research.

##  X Summary

In this work we reviewed some recent progress in the calculation of the
$\mathcal{O}(G_{F}\alpha)$ EWRCs to semileptonic beta decays of strongly-
interacting systems, which are important avenues for precision tests of the
SM. Combining the classical Sirlin’s current algebra representation of EWRCs
and non-perturbative techniques such as DR and lattice QCD, a number of
breakthroughs were made in pion, neutron, nuclei and kaon by reaching an
unprecedented level of $10^{-4}$ in precision:

  * •

The first lattice calculation of the forward axial $\gamma W$-box diagram
$\pi_{e3}$ has removed the major theory uncertainty in its ERWCs, making it 3
times as precise as the previous state-of-the-art determination from ChPT.
This makes $\pi_{e3}$ one of the theoretically cleanest avenues to measure
$|V_{ud}|$. The major limitation is from the experimental error of its
branching ratio, which will be improved in next-generation experiments of rare
pion decays.

  * •

The first DR analysis of the single-nucleon axial $\gamma W$-box diagram with
neutrino-nucleus scattering data as inputs improved its precision by a factor
2 and revealed a large systematic effect that was not captured in previous
studies. It led to a reduction of the $|V_{ud}|$ central value.

  * •

A new theory framework for the EWRCs in $K_{\ell 3}$ was constructed to
overcome the natural limitations from the traditional ChPT formalism. Its
first application to $K_{e3}$ has improved the existing theory precision by
almost an order of magnitude.

Some of the progress above revealed new anomalies in beta decays associated to
the top-row CKM matrix elements, and some of them sharpened the existing ones.
They turned beta decays into a promising avenue to search for BSM physics.
There are, however, a number of unresolved problems that must be investigated
in the future in order to make further progress along this direction:

  * •

The current DR treatment of the free neutron inner corrections suffers from
low precision of the experimental inputs and a residual model dependence in
the matching between the data and the nucleon $\gamma W$-box diagram, so one
still cannot claim with full confidence that the outcome is free from
detectable systematics. To resolve these ambiguities, one should perform
direct lattice QCD calculations of the nucleon $\gamma W$-box diagram.

  * •

A similar DR analysis of the nuclear $\gamma W$-box diagram revealed several
potentially large nuclear structure corrections that were failed to be
incorporated in previous nuclear model calculations. Ab-initio studies of
these new effects are urgently needed. Also, the current results of the
nuclear isospin breaking corrections should be scrutinized by model-
independent approaches to make sure no large systematic uncertainties are
hidden.

  * •

The new theory analysis of the $K_{e3}$ EWRC should also be applied to $K_{\mu
3}$ to complete the story. One expects a more complicated error analysis in
the latter due to the non-negligible contributions from $\delta f_{-}^{K\pi}$.

  * •

So far the Sirlin’s representation has only been applied to semileptonic
decays, but in principle it is also valid for leptonic decays. Particularly
interesting examples are $K_{\mu 2}$ and $\pi_{\mu 2}$ which measure the ratio
$|V_{us}/V_{ud}|$. The EWRCs in these channels were previously studied with
ChPT and lattice QCD, and it is interesting to see if the new method is able
to further improve its precision. This also serves as another cross-check of
the $K_{\mu 2}-K_{\ell 3}$ discrepancy in the $|V_{us}|$ measurement.

Future progress in both theories and experiments of these topics could further
sharpen the existing beta decay anomalies and eventually lead to one of the
earliest confirmed signals of BSM physics, of which impact to the fundamental
science is far-reaching.

###### Acknowledgements.

The author thanks Jens Erler, Xu Feng, Daniel Galviz, Mikhail Gorchtein,
Charles J. Horowitz, Lu-Chang Jin, Oleksandr Koshchii, Peng-Xiang Ma, William
J. Marciano, Ulf-G. Meißner, Hiren H. Patel, Jorge Piekarewicz, Michael J.
Ramsey-Musolf, Xavier Roca-Maza and Hubert Spiesberger for collaborations in
related topics. The author is also grateful to the supermarkets in Bonn,
Germany for keep opening during the COVID-19 pandemic. 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), and by the Alexander von Humboldt Foundation through the
Humboldt Research Fellowship.

##  Appendix A  Derivation of the free-field OPE relation

Here we use a simple example to illustrate how the leading-twist, free-field
OPE in Eq.(LABEL:eq:OPE) can be derived in a free-quark picture. Let us
consider a $\beta^{-}$ decay involving only $u$ and $d$ quarks, so the
electroweak currents read:

$J_{\mathrm{em}}^{\mu}=e_{u}\bar{u}\gamma^{\mu}u+e_{d}\bar{d}\gamma^{\mu}d\leavevmode\nobreak\
,\quad J^{\mu}=V_{ud}\bar{u}\gamma^{\mu}(1-\gamma_{5})d\leavevmode\nobreak\ .$
(232)

Now we study the following time-ordered current product at very large
$q^{\prime}$:

$\displaystyle\int d^{4}xe^{iq^{\prime}\cdot
x}T\\{J_{\mathrm{em}}^{\mu}(x)J^{\nu}(0)\\}$ (233) $\displaystyle=$
$\displaystyle\int d^{4}xe^{iq^{\prime}\cdot
x}T\\{(e_{u}\bar{u}(x)\gamma^{\mu}u(x)+e_{d}\bar{d}(x)\gamma^{\mu}d(x))V_{ud}\bar{u}(0)\gamma^{\nu}(1-\gamma_{5})d(0)\\}$
$\displaystyle=$ $\displaystyle V_{ud}\int d^{4}xe^{iq^{\prime}\cdot
x}\left\\{e_{u}\bar{u}(0)\gamma^{\mu}\langle
u(x)\bar{u}(0)\rangle\gamma^{\nu}(1-\gamma_{5})d(0)\right.$
$\displaystyle\left.+e_{d}\bar{u}(0)\gamma^{\nu}(1-\gamma_{5})\langle
d(0)\bar{d}(x)\rangle\gamma^{\mu}d(0)\right\\}+...\leavevmode\nobreak\ ,$

where we have used the Wick’s theorem in the last equality, and display only
terms with one contraction. Also, we have set $x\rightarrow 0$ in the
uncontracted field operators, because $q^{\prime}$ is large so the integral at
large $x$ vanishes due to the oscillating exponential factor. Next, we use the
free propagator of a massless quark:

$\int d^{4}xe^{iq^{\prime}\cdot
x}\langle\psi(x)\bar{\psi}(0)\rangle=\frac{i\not{q}^{\prime}}{q^{\prime
2}}\leavevmode\nobreak\ $ (234)

to get:

$\displaystyle\int d^{4}xe^{iq^{\prime}\cdot
x}T\\{J_{\mathrm{em}}^{\mu}(x)J^{\nu}(0)\\}$ (235) $\displaystyle=$
$\displaystyle\frac{i}{q^{\prime
2}}V_{ud}\bar{u}(0)\left[e_{u}\gamma^{\mu}\not{q}^{\prime}\gamma^{\nu}-e_{d}\gamma^{\nu}\not{q}^{\prime}\gamma^{\mu}\right](1-\gamma_{5})d(0)+...$
$\displaystyle=$ $\displaystyle\frac{i}{q^{\prime
2}}V_{ud}\bar{u}(0)\left[(e_{u}-e_{d})(q^{\prime\mu}\gamma^{\nu}-g^{\mu\nu}\not{q}^{\prime}+q^{\prime\nu}\gamma^{\mu})-i(e_{u}+e_{d})\epsilon^{\mu\nu\alpha\beta}q^{\prime}_{\alpha}\gamma_{\beta}\right](1-\gamma_{5})d(0)$
$\displaystyle+...$ $\displaystyle=$ $\displaystyle\frac{i}{q^{\prime
2}}\left\\{\eta\left[g^{\mu\nu}q^{\prime\lambda}-g^{\nu\lambda}q^{\prime\mu}-g^{\mu\lambda}q^{\prime\nu}\right]-2i\bar{Q}\epsilon^{\mu\nu\alpha\lambda}q^{\prime}_{\alpha}\right\\}J_{\lambda}(0)+...\leavevmode\nobreak\
,$

where in the third line we have used the identity in Eq.(20), and in the last
line we have used $e_{u}-e_{d}=1$, $e_{u}+e_{d}=2\bar{Q}$ and $\eta=-1$ for
$\beta^{-}$ decay. Finally, taking the matrix element at both sides of the
equation above returns the OPE relation of
$T_{(\gamma)}^{\mu\nu}(q^{\prime};p^{\prime},p)$ in Eq.(LABEL:eq:OPE). Other
OPE relations can be derived accordingly.

##  Appendix B pQCD corrections in $\Box_{\gamma W}^{>}$

In this Appendix we prove that the pQCD corrections to $\Box_{\gamma W}^{>}$
is identical to that to the polarized Bjorken sum rule.

First, from the definition of $\Box_{\gamma W}$ it is clear that only the
component of $T^{\mu\nu}$ with an $\epsilon$-tensor is probed. The polarized
Bjorken sum rule originates from the following pQCD-corrected OPE, where we
only display the structure with an $\epsilon$-tensor (See, e.g.
Ref.Larin:1991tj , but take into account the definition of $\gamma_{5}$):

$\int d^{4}xq^{iq\cdot
x}T\\{V_{\mu}^{a}V_{\nu}^{b}(0)\\}=-\epsilon_{\mu\nu\rho\sigma}\frac{q^{\rho}}{q^{2}}C_{\mathrm{Bj}}(Q^{2})d^{abc}A^{c\sigma}(0)+...\leavevmode\nobreak\
.$ (236)

In the above we have defined the vector and axial SU(3) currents:

$V_{\mu}^{a}\equiv\bar{\psi}\gamma_{\mu}\frac{\lambda^{a}}{2}\psi\leavevmode\nobreak\
,\quad
A_{\mu}^{a}\equiv\bar{\psi}\gamma_{\mu}\gamma_{5}\frac{\lambda^{a}}{2}\psi\leavevmode\nobreak\
,$ (237)

from which the electroweak currents can be reconstructed (for notational
simplicity, let us concentrate on $u\leftrightarrow d$ transition, and neglect
the factor $V_{ud}$):

$J_{\mathrm{em}}^{\mu}=V_{3}^{\mu}+\frac{1}{\sqrt{3}}V_{8}^{\mu}\leavevmode\nobreak\
,\quad J_{V}^{\mu}=V_{1}^{\mu}-i\eta V_{2}^{\mu}\leavevmode\nobreak\ ,\quad
J_{A}^{\mu}=-A_{1}^{\mu}+i\eta A_{2}^{\mu}\leavevmode\nobreak\ ,$ (238)

and $C_{\mathrm{Bj}}(Q^{2})$ is the pQCD correction factor. Substituting
Eq.(238) into Eq.(236) gives:

$\int d^{4}xe^{iq\cdot
x}T\\{J^{\mathrm{em}}_{\mu}J^{V}_{\nu}(0)\\}=\frac{1}{3}\epsilon_{\mu\nu\rho\sigma}\frac{q^{\rho}}{q^{2}}C_{\mathrm{Bj}}(Q^{2})J_{A}^{\sigma}(0)+...\leavevmode\nobreak\
.$ (239)

To further proceed, we perform the following chiral rotation to the quark
field:

$\psi_{R}\rightarrow\psi_{R}\leavevmode\nobreak\ ,\quad\psi_{L}\rightarrow
L\psi_{L}\leavevmode\nobreak\ ,$ (240)

where $L=\mathrm{diag}(1,-1,-1)$ is an SU(3)L matrix. This rotation is
equivalent to $d\rightarrow\gamma_{5}d$, $s\rightarrow\gamma_{5}s$, which
transforms the currents as:

$J^{\mathrm{em}}_{\mu}\rightarrow J^{\mathrm{em}}_{\mu}\leavevmode\nobreak\
,\quad J^{V}_{\mu}\rightarrow-J^{A}_{\mu}\leavevmode\nobreak\ ,\quad
J^{A}_{\mu}\rightarrow-J^{V}_{\mu}.$ (241)

Now suppose chiral symmetry is a good symmetry, then Eq.(239) should hold even
after the chiral rotation. That means, we should similarly obtain:

$\int d^{4}xe^{iq\cdot
x}T\\{J^{\mathrm{em}}_{\mu}J^{A}_{\nu}(0)\\}=\frac{1}{3}\epsilon_{\mu\nu\rho\sigma}\frac{q^{\rho}}{q^{2}}C_{\mathrm{Bj}}(Q^{2})J_{V}^{\sigma}(0)+...\leavevmode\nobreak\
.$ (242)

So, adding Eqs.(239) and (242) gives:

$\int d^{4}xe^{iq\cdot
x}T\\{J^{\mathrm{em}}_{\mu}J_{\nu}(0)\\}=\frac{1}{3}\epsilon_{\mu\nu\rho\sigma}\frac{q^{\rho}}{q^{2}}C_{\mathrm{Bj}}(Q^{2})J^{\sigma}(0)+...\leavevmode\nobreak\
,$ (243)

which reproduces the structure with an $\epsilon$-tensor of our previous OPE
relation in Eq.(LABEL:eq:OPE), together with the pQCD correction factor
$C_{\mathrm{Bj}}(Q^{2})$ as a bonus. This completes our proof.

##  Appendix C Crossing symmetry of the invariant amplitudes

In this Appendix we outline a derivation of the crossing behavior of the
invariant amplitudes $T_{3}$, $S_{1}$ and $S_{2}$ under $\nu\rightarrow-\nu$
using time-reversal invariance and isospin symmetry. Notice that a similar
derivation was shown in Appendix A of Ref.Seng:2018qru . Also, for
definiteness we take $\phi_{i}=n$ and $\phi_{f}=p$, but the derivation
obviously applies to general $I=1/2$ external states.

We start with the following identity involving the time-reversal operator
$\mathbb{T}$ acting on a matrix element of a generic current operator:

$\langle\beta|J^{\mu}(\vec{x},t)|\alpha\rangle=\langle\tilde{\alpha}|\mathbb{T}J_{\mu}^{\dagger}(\vec{x},t)\mathbb{T}^{-1}|\tilde{\beta}\rangle=\langle\tilde{\alpha}|J^{\mu\dagger}(\vec{x},-t)|\tilde{\beta}\rangle\leavevmode\nobreak\
,$ (244)

where $|\tilde{\alpha}\rangle$ and $|\tilde{\beta}\rangle$ are the time-
reversed states of $|\alpha\rangle$ and $|\beta\rangle$ respectively. The
momentum and spin in the external states should also transformed as
$p^{\mu}\rightarrow\tilde{p}^{\mu}\equiv p_{\mu}$ and
$s^{\mu}\rightarrow\tilde{s}^{\mu}\equiv s_{\mu}$. Furthermore, we decompose
the electromagnetic current into isosinglet and isotriplet components:
$J_{\mathrm{em}}^{\mu}=J_{\mathrm{em}}^{(0)\mu}+J_{\mathrm{em}}^{(1)\mu}$ and
subsequently $T_{\mu\nu}=T_{\mu\nu}^{(0)}+T_{\mu\nu}^{(1)}$. With all the
above, we obtain the following identity for the forward generalized Compton
tensor at $q\rightarrow-q$:

$\displaystyle T^{(I)\mu\nu}_{ss}(-q;p,p)$ $\displaystyle=$ $\displaystyle\int
d^{4}xe^{-iq\cdot x}\langle
p(p,s)|T[J_{\mathrm{em}}^{(I)\mu}(x)J^{\nu}(0)]|n(p,s)\rangle$ (245)
$\displaystyle=$ $\displaystyle\int d^{4}xe^{i\tilde{q}\cdot x}\langle
n(\tilde{p},\tilde{s})|T[J^{(I)\mathrm{em}}_{\mu}(x)J^{\dagger}_{\nu}(0)]|p(\tilde{p},\tilde{s})\rangle$
$\displaystyle=$ $\displaystyle-\xi^{(I)}\int d^{4}xe^{i\tilde{q}\cdot
x}\langle
p(\tilde{p},\tilde{s})|T[J^{(I)\mathrm{em}}_{\mu}(x)J_{\nu}(0)]|n(\tilde{p},\tilde{s})\rangle\leavevmode\nobreak\
,$

where $\xi^{I}=-1(+1)$ for $I=0(1)$, and the third line is obtained through a
simple isospin rotation of the second line (or by applying the Wigner-Eckart
theorem to the isospin space).

Now we may express both sides of the equation above in terms of invariant
functions. That gives (we suppress the $Q^{2}$-dependence for simplicity):

$\displaystyle-i\epsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}p_{\beta}}{2p\cdot
q}T_{3}^{(I)}(-\nu)+i\epsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}}{p\cdot
q}\left[s_{\beta}S_{1}^{(I)}(-\nu)+\left(s_{\beta}-\frac{s\cdot q}{p\cdot
q}p_{\beta}\right)S_{2}^{(I)}(-\nu)\right]$ $\displaystyle=$
$\displaystyle-\xi^{(I)}\left\\{-i\epsilon_{\mu\nu\alpha\beta}\frac{\tilde{q}^{\alpha}\tilde{p}^{\beta}}{2p\cdot
q}T_{3}^{(I)}(\nu)+i\epsilon_{\mu\nu\alpha\beta}\frac{\tilde{q}^{\alpha}}{p\cdot
q}\left[\tilde{s}^{\beta}S_{1}^{(I)}(\nu)+\left(\tilde{s}^{\beta}-\frac{s\cdot
q}{p\cdot q}\tilde{p}^{\beta}\right)S_{2}^{(I)}(\nu)\right]\right\\}$
$\displaystyle=$
$\displaystyle\xi^{(I)}\left\\{-i\epsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}p_{\beta}}{2p\cdot
q}T_{3}^{(I)}(\nu)+i\epsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}}{p\cdot
q}\left[s_{\beta}S_{1}^{(I)}(\nu)+\left(s_{\beta}-\frac{s\cdot q}{p\cdot
q}p_{\beta}\right)S_{2}^{(I)}(\nu)\right]\right\\}\leavevmode\nobreak\ ,$

where we have used $\tilde{a}\cdot\tilde{b}=a\cdot b$ and
$\epsilon^{\mu\nu\alpha\beta}=-\epsilon_{\mu\nu\alpha\beta}$. The equation
above implies:

$T_{3}^{(I)}(-\nu,Q^{2})=\xi^{(I)}T_{3}(\nu,Q^{2})\leavevmode\nobreak\ ,\quad
S_{1,2}^{(I)}(-\nu,Q^{2})=\xi^{(I)}S_{1,2}^{(I)}(\nu,Q^{2})\leavevmode\nobreak\
,$ (247)

which is our required crossing relation.

##  Appendix D ChPT loop functions

The loop functions $\bar{h}_{PQ}(t)$ that appear in Eq.(223) are defined as:

$\displaystyle\bar{h}_{K^{+}\pi^{0}}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{64\pi^{2}(M_{K}^{2}-M_{\pi}^{2})}\left[M_{\pi}^{2}-M_{K}^{2}+M_{\pi}^{2}\ln\frac{M_{K}^{2}}{M_{\pi}^{2}}\right]+\frac{M_{K}^{2}-M_{\pi}^{2}-t}{4t}\bar{J}_{K\pi}(t)$
$\displaystyle\bar{h}_{K^{0}\pi^{-}}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{64\pi^{2}(M_{K}^{2}-M_{\pi}^{2})}\left[M_{\pi}^{2}-M_{K}^{2}+M_{K}^{2}\ln\frac{M_{K}^{2}}{M_{\pi}^{2}}\right]+\frac{M_{\pi}^{2}-M_{K}^{2}-t}{4t}\bar{J}_{K\pi}(t)$
$\displaystyle\bar{h}_{K^{+}\eta}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{64\pi^{2}(M_{K}^{2}-M_{\eta}^{2})}\left[M_{\eta}^{2}-M_{K}^{2}+M_{\eta}^{2}\ln\frac{M_{K}^{2}}{M_{\eta}^{2}}\right]+\frac{M_{K}^{2}-M_{\eta}^{2}-t}{4t}\bar{J}_{K\eta}(t)\leavevmode\nobreak\
,$

where $\bar{J}_{PQ}$ is the standard mesonic loop function Gasser:1984gg :

$\displaystyle\bar{J}_{PQ}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}\left[2+\frac{\Delta_{PQ}}{t}\ln\frac{M_{Q}^{2}}{M_{P}^{2}}-\frac{\Sigma_{PQ}}{\Delta_{PQ}}\ln\frac{M_{Q}^{2}}{M_{P}^{2}}-\frac{\lambda^{1/2}(t,M_{P}^{2},M_{Q}^{2})}{t}\right.$
(249)
$\displaystyle\left.\times\ln\left(\frac{[t+\lambda^{1/2}(t,M_{P}^{2},M_{Q}^{2})]^{2}-\Delta_{PQ}^{2}}{[t-\lambda^{1/2}(t,M_{P}^{2},M_{Q}^{2})]^{2}-\Delta_{PQ}^{2}}\right)\right]\leavevmode\nobreak\
,$

with $\Sigma_{PQ}=M_{P}^{2}+M_{Q}^{2}$, $\Delta_{PQ}=M_{P}^{2}-M_{Q}^{2}$, and
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+xz)$ the Källén function.

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

In the bremsstrahlung process
$K(p)\rightarrow\pi(p^{\prime})e^{+}(p_{e})\nu(p_{\nu})\gamma(k)$, there is
only one IR-divergent integral in the squared amplitude:

$I_{i}(y,z)\equiv\int_{0}^{\alpha_{+}(y,z)}dx\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(P-k-
p_{\nu})\left(\frac{p_{e}}{p_{e}\cdot k}-\frac{p_{i}}{p_{i}\cdot
k}\right)^{2}\leavevmode\nobreak\ ,$ (250)

where $i=K,\pi$. Regularizing the IR-divergence with a photon mass, one can
write:

$I_{i}(y,z)=I_{i}^{\mathrm{IR}}(y,z)+I_{i}^{\mathrm{fin}}(y,z)\leavevmode\nobreak\
,$ (251)

where the IR-divergent piece reads:

$I_{i}^{\mathrm{IR}}(y,z)=\frac{1}{2\pi
M_{K}^{2}}\left\\{\left(1-\frac{1}{\beta_{i}(0)}\tanh^{-1}\beta_{i}(0)\right)\ln\left[\frac{M_{K}^{2}}{M_{\gamma}^{2}}\right]-\frac{1}{2}\ln\left[\frac{M_{K}^{2}}{m_{e}^{2}}\right]\right\\}\leavevmode\nobreak\
,$ (252)

and the IR-finite piece reads:

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

with

$\beta_{i}(x)\equiv\sqrt{1-\frac{M_{i}^{2}m_{e}^{2}}{(p_{i}\cdot
p_{e})^{2}}}\leavevmode\nobreak\ ,\>\>P_{0}(x)\equiv\frac{p_{i}\cdot
P}{M_{i}}\leavevmode\nobreak\
,\>\>P_{1}(x)\equiv\frac{1}{\beta_{i}(x)}\left(P_{0}(x)-\frac{p_{e}\cdot
P}{p_{i}\cdot p_{e}}M_{i}\right)\leavevmode\nobreak\ .$ (254)

Of course all the quantities in Eq.(254) are functions of $\\{y,z\\}$ as well.
Notice that $I_{i}^{\mathrm{fin}}$ includes a residual integral over $x$ that
exists only for $i=\pi$ and can be easily performed numerically.

There are also IR-finite integrals, which can be cast in the following form:

$I_{m,n}(p_{1},p_{2})\equiv\frac{1}{2\pi}\int\frac{d^{3}k}{E_{k}}\frac{d^{3}p_{\nu}}{E_{\nu}}\frac{\delta^{(4)}(P-k-
p_{\nu})}{(p_{1}\cdot k)^{m}(p_{2}\cdot k)^{n}}\leavevmode\nobreak\ ,$ (255)

where $m,n$ are integers. The analytic expressions of those appear in $K_{\ell
3}$ were first derived in Ref.Ginsberg:1969jh :

$\displaystyle I_{0,0}(p_{1},p_{2})$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle I_{1,0}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\beta_{1}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)$
$\displaystyle I_{2,0}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{4}{M_{1}^{2}M_{K}^{2}x}$ $\displaystyle
I_{1,1}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{2}{\gamma_{12}M_{K}^{2}x}\ln\left(\frac{p_{1}\cdot
p_{2}+\gamma_{12}}{p_{1}\cdot p_{2}-\gamma_{12}}\right)$ $\displaystyle
I_{1,-1}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{(p_{1}p_{2}:P)}{\beta_{1}^{2}}+\frac{M_{K}^{2}x(p_{2}P:p_{1})}{2\beta_{1}^{3}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)$
$\displaystyle I_{2,-1}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{2(p_{2}P:p_{1})}{M_{1}^{2}\beta_{1}^{2}}+\frac{(p_{1}p_{2}:P)}{\beta_{1}^{3}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)$
$\displaystyle I_{1,-2}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{M_{K}^{4}x^{2}(p_{2}P:p_{1})^{2}}{4\beta_{1}^{5}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)+\frac{M_{K}^{2}x(p_{1}p_{2}:P)(p_{2}P:p_{1})}{\beta_{1}^{4}}$
$\displaystyle+\frac{\alpha_{1}(p_{1}p_{2}:P)^{2}}{2\beta_{1}^{4}}+\left[\frac{\beta_{1}^{2}\beta_{2}^{2}-(p_{1}p_{2}:P)^{2}}{4\beta_{1}^{4}}\right]\left[\alpha_{1}-\frac{M_{1}^{2}M_{K}^{2}x}{2\beta_{1}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)\right]$
$\displaystyle I_{2,-2}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{M_{K}^{2}x(p_{2}P:p_{1})^{2}}{M_{1}^{2}\beta_{1}^{4}}+\frac{M_{K}^{2}x(p_{2}P:p_{1})(p_{1}p_{2}:P)}{\beta_{1}^{5}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)$
$\displaystyle+\frac{(p_{1}p_{2}:P)^{2}}{\beta_{1}^{4}}-\left[\frac{\beta_{1}^{2}\beta_{2}^{2}-(p_{1}p_{2}:P)^{2}}{2\beta_{1}^{4}}\right]\left[2-\frac{\alpha_{1}}{\beta_{1}}\ln\left(\frac{\alpha_{1}+\beta_{1}}{\alpha_{1}-\beta_{1}}\right)\right]$
$\displaystyle I_{-1,0}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}}{2}\leavevmode\nobreak\ ,$ (256)

where

$\displaystyle p_{i}^{2}$ $\displaystyle=$ $\displaystyle M_{i}^{2}$
$\displaystyle\alpha_{i}$ $\displaystyle=$ $\displaystyle p_{i}\cdot P$
$\displaystyle\beta_{i}$ $\displaystyle=$
$\displaystyle(\alpha_{i}^{2}-M_{i}^{2}M_{K}^{2}x)^{1/2}$
$\displaystyle\gamma_{ij}$ $\displaystyle=$ $\displaystyle[(p_{i}\cdot
p_{j})^{2}-M_{i}^{2}M_{j}^{2}]^{1/2}$ $\displaystyle(ab:c)$ $\displaystyle=$
$\displaystyle(a\cdot c)(b\cdot c)-c^{2}(a\cdot b)\leavevmode\nobreak\ .$
(257)

## References

  * (1) L. M. Brown, The idea of the neutrino, Phys. Today 31N9 (1978) 23–28.
  * (2) E. Fermi, An attempt of a theory of beta radiation. 1., Z. Phys. 88 (1934) 161–177.
  * (3) C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Experimental Test of Parity Conservation in $\beta$ Decay, Phys. Rev. 105 (1957) 1413–1414.
  * (4) T. D. Lee and C.-N. Yang, Question of Parity Conservation in Weak Interactions, Phys. Rev. 104 (1956) 254–258.
  * (5) R. P. Feynman and M. Gell-Mann, Theory of Fermi interaction, Phys. Rev. 109 (1958) 193–198.
  * (6) E. C. G. Sudarshan and R. e. Marshak, Chirality invariance and the universal Fermi interaction, Phys. Rev. 109 (1958) 1860–1860.
  * (7) N. Cabibbo, Unitary Symmetry and Leptonic Decays, Phys. Rev. Lett. 10 (1963) 531–533.
  * (8) M. Kobayashi and T. Maskawa, CP Violation in the Renormalizable Theory of Weak Interaction, Prog. Theor. Phys. 49 (1973) 652–657.
  * (9) J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Evidence for the $2\pi$ Decay of the $K_{2}^{0}$ Meson, Phys. Rev. Lett. 13 (1964) 138–140.
  * (10) Planck Collaboration, N. Aghanim et. al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [1807.06209].
  * (11) J. D. Simon, The Faintest Dwarf Galaxies, Ann. Rev. Astron. Astrophys. 57 (2019), no. 1 375–415 [1901.05465].
  * (12) P. Salucci, The distribution of dark matter in galaxies, Astron. Astrophys. Rev. 27 (2019), no. 1 2 [1811.08843].
  * (13) S. W. Allen, A. E. Evrard and A. B. Mantz, Cosmological Parameters from Observations of Galaxy Clusters, Ann. Rev. Astron. Astrophys. 49 (2011) 409–470 [1103.4829].
  * (14) Supernova Search Team Collaboration, A. G. Riess et. al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009–1038 [astro-ph/9805201].
  * (15) Supernova Cosmology Project Collaboration, S. Perlmutter et. al., Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565–586 [astro-ph/9812133].
  * (16) A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Sov. Phys. Usp. 34 (1991), no. 5 392–393.
  * (17) V. Mossa et. al., The baryon density of the Universe from an improved rate of deuterium burning, Nature 587 (2020), no. 7833 210–213.
  * (18) CMS Collaboration, S. Chatrchyan et. al., Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716 (2012) 30–61 [1207.7235].
  * (19) ATLAS Collaboration, G. Aad et. al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1–29 [1207.7214].
  * (20) Central scientific results page of the ATLAS Collaboration: https://twiki.cern.ch/twiki/bin/view/AtlasPublic/.
  * (21) Central scientific results page of the CMS Collaboration: http://cms-results.web.cern.ch/cms-results/public-results/publications/.
  * (22) FCC Collaboration, A. Abada et. al., FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1, Eur. Phys. J. C 79 (2019), no. 6 474.
  * (23) FCC Collaboration, A. Abada et. al., FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2, Eur. Phys. J. ST 228 (2019), no. 2 261–623.
  * (24) FCC Collaboration, A. Abada et. al., FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3, Eur. Phys. J. ST 228 (2019), no. 4 755–1107.
  * (25) CEPC Study Group Collaboration, W. Chou et. al., CEPC Conceptual Design Report: Volume 1 - Accelerator, 1809.00285.
  * (26) CEPC Study Group Collaboration, M. Dong et. al., CEPC Conceptual Design Report: Volume 2 - Physics \& Detector, 1811.10545.
  * (27) Particle Data Group Collaboration, P. Zyla et. al., Review of Particle Physics, PTEP 2020 (2020), no. 8 083C01.
  * (28) D. Bryman and R. Shrock, Improved Constraints on Sterile Neutrinos in the MeV to GeV Mass Range, Phys. Rev. D 100 (2019), no. 5 053006 [1904.06787].
  * (29) D. Bryman and R. Shrock, Constraints on Sterile Neutrinos in the MeV to GeV Mass Range, Phys. Rev. D 100 (2019) 073011 [1909.11198].
  * (30) M. Kirk, Cabibbo anomaly versus electroweak precision tests: An exploration of extensions of the standard model, Phys. Rev. D 103 (2021), no. 3 035004 [2008.03261].
  * (31) Y. Grossman, E. Passemar and S. Schacht, On the Statistical Treatment of the Cabibbo Angle Anomaly, JHEP 07 (2020) 068 [1911.07821].
  * (32) B. Belfatto, R. Beradze and Z. Berezhiani, The CKM unitarity problem: A trace of new physics at the TeV scale?, Eur. Phys. J. C 80 (2020), no. 2 149 [1906.02714].
  * (33) K. Cheung, W.-Y. Keung, C.-T. Lu and P.-Y. Tseng, Vector-like Quark Interpretation for the CKM Unitarity Violation, Excess in Higgs Signal Strength, and Bottom Quark Forward-Backward Asymmetry, JHEP 05 (2020) 117 [2001.02853].
  * (34) Y. Jho, S. M. Lee, S. C. Park, Y. Park and P.-Y. Tseng, Light gauge boson interpretation for (g-2)μ and the $K_{L}\rightarrow\pi^{0}$ \+ (invisible) anomaly at the J-PARC KOTO experiment, JHEP 04 (2020) 086 [2001.06572].
  * (35) C. X. Yue and X. J. Cheng, Constraints on the charged-current non-standard neutrino interactions induced by the gauge boson W’, Nucl. Phys. B 963 (2021) 115280 [2008.10027].
  * (36) M. Endo and S. Mishima, Muon g-2 and CKM unitarity in extra lepton models, JHEP 08 (2020), no. 08 004 [2005.03933].
  * (37) B. Capdevila, A. Crivellin, C. A. Manzari and M. Montull, Explaining $b\to s\ell^{+}\ell^{-}$ and the Cabibbo angle anomaly with a vector triplet, Phys. Rev. D 103 (2021), no. 1 015032 [2005.13542].
  * (38) O. Eberhardt, A. P. n. Martínez and A. Pich, Global fits in the Aligned Two-Higgs-Doublet model, 2012.09200.
  * (39) A. Crivellin and M. Hoferichter, $\beta$ Decays as Sensitive Probes of Lepton Flavor Universality, Phys. Rev. Lett. 125 (2020), no. 11 111801 [2002.07184].
  * (40) A. M. Coutinho, A. Crivellin and C. A. Manzari, Global Fit to Modified Neutrino Couplings and the Cabibbo-Angle Anomaly, Phys. Rev. Lett. 125 (2020), no. 7 071802 [1912.08823].
  * (41) M. Gonzalez-Alonso, O. Naviliat-Cuncic and N. Severijns, New physics searches in nuclear and neutron $\beta$ decay, Prog. Part. Nucl. Phys. 104 (2019) 165–223 [1803.08732].
  * (42) A. Falkowski, M. González-Alonso and Z. Tabrizi, Reactor neutrino oscillations as constraints on Effective Field Theory, JHEP 05 (2019) 173 [1901.04553].
  * (43) V. Cirigliano, A. Garcia, D. Gazit, O. Naviliat-Cuncic, G. Savard and A. Young, Precision Beta Decay as a Probe of New Physics, 1907.02164.
  * (44) A. Falkowski, M. González-Alonso and O. Naviliat-Cuncic, Comprehensive analysis of beta decays within and beyond the Standard Model, JHEP 04 (2021) 126 [2010.13797].
  * (45) D. Bečirević, F. Jaffredo, A. Peñuelas and O. Sumensari, New Physics effects in leptonic and semileptonic decays, 2012.09872.
  * (46) A. Crivellin, M. Hoferichter and C. A. Manzari, The Fermi constant from muon decay versus electroweak fits and CKM unitarity, 2102.02825.
  * (47) W. Tan, Laboratory tests of the ordinary-mirror particle oscillations and the extended CKM matrix, 1906.10262.
  * (48) A. Crivellin, M. Hoferichter, M. Kirk, C. A. Manzari and L. Schnell, First-Generation New Physics in Simplified Models: From Low-Energy Parity Violation to the LHC, 2107.13569.
  * (49) A. Crivellin, F. Kirk, C. A. Manzari and M. Montull, Global Electroweak Fit and Vector-Like Leptons in Light of the Cabibbo Angle Anomaly, JHEP 12 (2020) 166 [2008.01113].
  * (50) A. Crivellin, F. Kirk, C. A. Manzari and L. Panizzi, Searching for lepton flavor universality violation and collider signals from a singly charged scalar singlet, Phys. Rev. D 103 (2021), no. 7 073002 [2012.09845].
  * (51) W. Dekens, L. Andreoli, J. de Vries, E. Mereghetti and F. Oosterhof, A low-energy perspective on the minimal left-right symmetric model, 2107.10852.
  * (52) Muon g-2 Collaboration, B. Abi et. al., Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801.
  * (53) T. Aoyama et. al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1–166 [2006.04822].
  * (54) J. P. Miller, E. de Rafael and B. L. Roberts, Muon (g-2): Experiment and theory, Rept. Prog. Phys. 70 (2007) 795 [hep-ph/0703049].
  * (55) J. P. Miller, E. de Rafael, B. L. Roberts and D. Stöckinger, Muon (g-2): Experiment and Theory, Ann. Rev. Nucl. Part. Sci. 62 (2012) 237–264.
  * (56) F. Jegerlehner and A. Nyffeler, The Muon g-2, Phys. Rept. 477 (2009) 1–110 [0902.3360].
  * (57) LHCb Collaboration, R. Aaij et. al., Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 122 (2019), no. 19 191801 [1903.09252].
  * (58) LHCb Collaboration, R. Aaij et. al., Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 113 (2014) 151601 [1406.6482].
  * (59) LHCb Collaboration, R. Aaij et. al., Measurement of the ratio of branching fractions $\mathcal{B}(\bar{B}^{0}\to D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\to D^{*+}\mu^{-}\bar{\nu}_{\mu})$, Phys. Rev. Lett. 115 (2015), no. 11 111803 [1506.08614]. [Erratum: Phys.Rev.Lett. 115, 159901 (2015)].
  * (60) LHCb Collaboration, R. Aaij et. al., Angular analysis of the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity, JHEP 02 (2016) 104 [1512.04442].
  * (61) R. E. Behrends, R. J. Finkelstein and A. Sirlin, Radiative corrections to decay processes, Phys. Rev. 101 (1956) 866–873.
  * (62) T. Kinoshita and A. Sirlin, Muon Decay with Parity Nonconserving Interactions and Radiative Corrections in the Two-Component Theory, Phys. Rev. 107 (1957) 593–599.
  * (63) T. Kinoshita and A. Sirlin, Radiative corrections to Fermi interactions, Phys. Rev. 113 (1959) 1652–1660.
  * (64) A. Sirlin, General Properties of the Electromagnetic Corrections to the Beta Decay of a Physical Nucleon, Phys. Rev. 164 (1967) 1767–1775.
  * (65) G. Kallen, Radiative Corrections to $\beta$-Decay and Nucleon Form Factors, Nucl. Phys. B 1 (1967), no. 5 225–263.
  * (66) D. Wilkinson and B. Macefield, The numerical evaluation of radiative corrections of order $\alpha$ to allowed nuclear $\beta$-decay, Nucl. Phys. A 158 (1970) 110–116.
  * (67) S. L. Glashow, Partial Symmetries of Weak Interactions, Nucl. Phys. 22 (1961) 579–588.
  * (68) S. Weinberg, A Model of Leptons, Phys. Rev. Lett. 19 (1967) 1264–1266.
  * (69) A. Salam, Weak and Electromagnetic Interactions, Conf. Proc. C 680519 (1968) 367–377.
  * (70) A. Sirlin, Radiative corrections to g(v)/g(mu) in simple extensions of the su(2) x u(1) gauge model, Nucl. Phys. B 71 (1974) 29–51.
  * (71) A. Sirlin, Current Algebra Formulation of Radiative Corrections in Gauge Theories and the Universality of the Weak Interactions, Rev. Mod. Phys. 50 (1978) 573. [Erratum: Rev. Mod. Phys.50,905(1978)].
  * (72) D. J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343–1346.
  * (73) H. D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973) 1346–1349.
  * (74) A. Sirlin, Large m(W), m(Z) Behavior of the O(alpha) Corrections to Semileptonic Processes Mediated by W, Nucl. Phys. B196 (1982) 83–92.
  * (75) E. S. Ginsberg, Radiative Corrections to Kl-3 + /- Decays, Phys. Rev. 142 (1966) 1035–1040.
  * (76) E. S. Ginsberg, Radiative corrections to k-e-3-neutral decays and the delta-i=1/2 rule. (erratum), Phys. Rev. 171 (1968) 1675. [Erratum: Phys.Rev. 174, 2169 (1968)].
  * (77) E. S. Ginsberg, Radiative corrections to the k-l-3 +- dalitz plot, Phys. Rev. 162 (1967) 1570. [Erratum: Phys.Rev. 187, 2280 (1969)].
  * (78) E. Ginsberg, Radiative corrections to k-mu-3 decays, Phys. Rev. D 1 (1970) 229–239.
  * (79) T. Becherrawy, Radiative Correction to K(l3) Decay, Phys. Rev. D 1 (1970) 1452–1468.
  * (80) V. Bytev, E. Kuraev, A. Baratt and J. Thompson, Radiative corrections to the K+-(e3) decay revised, Eur. Phys. J. C 27 (2003) 57–71 [hep-ph/0210049]. [Erratum: Eur.Phys.J.C 34, 523–524 (2004)].
  * (81) T. C. Andre, Radiative corrections in K0(l3) decays, Annals Phys. 322 (2007) 2518–2544 [hep-ph/0406006].
  * (82) A. Garcia and M. Maya, MODEL INDEPENDENT RADIATIVE CORRECTIONS TO M+-(l3) DECAYS, Phys. Rev. D 23 (1981) 2603.
  * (83) C. Juarez-Leon, A. Martinez, M. Neri, J. Torres and R. Flores-Mendieta, Radiative corrections to the Dalitz plot of $K_{l3}^{\pm}$ decays, Phys. Rev. D 83 (2011) 054004 [1010.5547]. [Erratum: Phys.Rev.D 86, 059901 (2012)].
  * (84) J. Torres, A. Martinez, M. Neri, C. Juarez-Leon and R. Flores-Mendieta, Radiative corrections to the Dalitz plot of $K_{l3}^{\pm}$ decays: Contribution of the four-body region, Phys. Rev. D 86 (2012) 077501 [1209.5759].
  * (85) M. Neri, A. Martínez, C. Juárez-León, J. Torres and R. Flores-Mendieta, Radiative corrections to the Dalitz plot of $K_{l3}^{0}$ decays, Phys. Rev. D 92 (2015), no. 7 074022 [1510.00401].
  * (86) R. Urech, Virtual photons in chiral perturbation theory, Nucl. Phys. B433 (1995) 234–254 [hep-ph/9405341].