# MeV Sterile Neutrino in light of the Cabibbo-Angle Anomaly

Teppei Kitahara<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  Kobayashi-Maskawa Institute for the Origin of Particles and the
Universe, Nagoya University, Nagoya 464–8602, Japan  Kohsaku Tobioka
<EMAIL_ADDRESS>Department of Physics, Florida State University, Tallahassee,
FL 32306, USA  KEK Theory Center, IPNS, KEK, Tsukuba 305–0801, Japan

###### Abstract

A modified neutrino sector could imprint a signature on precision measurements
of the quark sector because many such measurements rely on the semi-leptonic
decays of the charged currents. Currently, global fits of the determinations
of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements point to a
$3\sigma$-level deficit in the first-row CKM unitarity test, commonly referred
to as the Cabibbo-angle anomaly. We find that a MeV sterile neutrino that
mixes with the electron-type neutrino increases the extracted $|V_{ud}|$,
accommodating the Cabibbo-angle anomaly. This MeV sterile neutrino affects the
superallowed nuclear $\beta$ decays and neutron decay, but it barely modifies
the other measurements of the CKM elements. While various constraints may
apply to such a sterile neutrino, we present viable scenarios within the
inverse seesaw models.

††preprint: KEK–TH–2544

##  I Introduction

While precision measurements have largely confirmed the predictions of the
standard model (SM), there are now unignorable deviations. Notably, certain
observables, such as the lepton flavor universality in meson decays [1], have
begun to diverge from SM predictions. Against this backdrop, the neutrino
sector, often considered a relatively unexplored domain in direct
determinations, offers intriguing possibilities. Many of these anomalies
seemingly in the quark sector could be actually due to new physics (NP) of the
neutrino sector, given that the relevant measurements heavily rely on the
semi-leptonic decays involving neutrinos.

In this paper, we consider, as one such example, the unitarity relation of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix $VV^{\dagger}={\boldsymbol{I}}_{3}$ [2,
3]. A deficit has been observed in the first-row CKM unitarity test [4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. According to a recent review in Ref.
[16], we obtain

$\displaystyle(VV^{\dagger})_{11}\equiv|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1+\Delta_{\rm
CKM}^{\rm global}\,,$ (1)

with

$\displaystyle\\!\\!\Delta_{\rm CKM}^{\rm
global}=\left\\{\begin{array}[]{lll}\\!\\!-1.51(53)\times
10^{-3}\leavevmode\nobreak\ \text{(w/\,bottle\,UCN\,best)}\,,\\\
\\!\\!-2.34(62)\times 10^{-3}\leavevmode\nobreak\ \text{(w/\,in-
beam\,best)}\,,\end{array}\right.\\!\\!$ (4)

which deviate from the unitarity relation ($\Delta_{\rm CKM}^{\rm global}=0$)
at a significance of $-2.8\sigma$ and $-3.8\sigma$, respectively. Here, all
available data from the kaon, pion, tau lepton, hyperon decays, and various
types of $\beta$ decays are used in the global fit [16], and all the decays
involve neutrinos. The difference in Eq. (4) comes from the two different
input data of the neutron decays:

$\displaystyle|V_{ud,n}|=0.974\,13(43)\quad\text{(bottle\,UCN\,best)}\,,$ (5)

based on the single most precise bottle ultracold neutron (UCN) lifetime data
$\tau_{n}^{\rm bottle}=877.75(36)$ sec [18], while

$\displaystyle|V_{ud,n}|=0.968\,66(131)\quad\text{(in-beam\,best)}\,,$ (6)

based on the in-beam one $\tau_{n}^{\rm beam}=887.7(2.2)$ sec [19]. Here, the
single most precise data of the nucleon isovector axial charge $g_{A}/g_{V}$
[20] and an updated radiative correction $\Delta_{R}$ [15] are used in both
cases. It is found that the beam neutron data provides a little bit larger
significance in the deficit. Note that as discussed in the following,
$|V_{ud}|$ is mostly determined by the data of the superallowed nuclear
$\beta$ decays.

Since the size of $|V_{ub}|^{2}$ is significantly smaller than uncertainties
of the other components [21, 22], the deficit is essentially inherent in the
two-generation Cabibbo angle. Hence, it is referred to as the Cabibbo angle
anomaly (CAA) [6, 7]. On the other hand, the ordinary CKM unitarity triangle
(in $B$-meson decays) corresponds to $(V^{\dagger}V)_{31}=0$, which is
currently consistent with the SM prediction within the experimental errors
[22]. The representative measurements specific to the $(VV^{\dagger})_{11}$
unitarity test are: the superallowed nuclear $\beta$ decay and the neutron
decay for $|V_{ud}|$; the kaon decays such as $K\to\pi\ell\nu$, $K\to\mu\nu$,
and tau hadronic decays for $|V_{us}|$ and their ratio $|V_{us}/V_{ud}|$. The
recent global fit of them and assessment of the deficit of
$(VV^{\dagger})_{11}$ have been performed in Refs. [15, 16].

There have been several attempts to resolve this anomaly by TeV-scale new
physics models [5, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35]. In this article, we develop deeper the
extensions of neutrino sector concerning the anomaly. Our findings uniquely
suggest that the CAA might be pointing to an intermediate mass scale for new
particles, specifically, a sterile neutrino at the MeV scale that mixes with
the electron-type neutrino [36, 37]. This scenario could arise because all the
relevant precision measurements use (semi-)leptonic decays and involve
neutrinos, considering that theoretical and experimental uncertainties of
these decays are well managed. As we will show, the MeV sterile neutrino could
be the underlying cause of the anomaly seemingly in the quark sector
quantities. It’s worth noting that several studies have indicated that the CAA
is not easily resolved by heavy or massless sterile neutrinos [7, 35, 26].

In the next section, we delve into the contributions of sterile neutrinos to
the observables for the CAA. In section III, we summarize and discuss the
current constraints on the favored parameter region, and conclude in section
IV.

##  II Sterile Neutrino in light of CAA

This section outlines how a sterile neutrino might resolve the CAA. We
initially examine two simple limits: very heavy and massless sterile neutrinos
and show that they cannot accommodate the CAA. On the other hand, a MeV-scale
sterile neutrino presents a potential solution by influencing the $|V_{ud}|$
determinations. To ascertain this solution, we conduct global fits in search
of the optimal parameter space. (Discussions on experimental constraints and
underlying models will be presented in Sec. III.)

Typically, a sterile neutrino’s characteristics in most models can be
effectively captured by its mass, $m_{\nu 4}$, and its mixing angles with the
SM neutrinos, $U_{\ell 4}$. Below the electroweak scale, the SM neutrinos from
the weak doublets, $\nu_{\ell}$, split into two pieces in the mass eigenbasis,

$\displaystyle\nu_{\ell}\simeq\cos U_{\ell 4}\nu_{\ell}^{\prime}+\sin U_{\ell
4}\nu_{4}\quad\text{for\leavevmode\nobreak\ }\ell=e,\mu,\tau\,,$ (7)

where $\nu_{\ell}^{\prime}$ are the SM-like neutrinos, often called active
neutrinos. For the active neutrinos, the mixing ($\cos U_{\ell 4}\lesssim 1$)
results in the coupling reduction of the weak interaction, and the deficit
gives the sterile neutrino a feeble coupling ($\sin U_{\ell 4}\ll 1$) to the
SM. These couplings alter observables primarily governed by the weak
interaction, i.e., all the measurements relevant to the CAA are potentially
affected, see Fig. 1. The sign of the modification depends on the mass scale
of the sterile neutrino, pinpointing a specific scale.

Figure 1:  Neutrino energy $E_{\nu}$ of each channel relevant to the first-row
CKM unitarity test is summarized. The sterile neutrino channel is open when
$m_{\nu 4}<E_{\nu}$.

###  II.1 Sterile neutrino above GeV

If the sterile neutrino mass exceeds $\mathcal{O}$(1) GeV, it would not be
kinematically permissible in the relevant measurements. In this case, an
important modification occurs in the Fermi constant measurement via the muon
decay. The observed value, $G_{F}^{\rm obs}$, deviates from the true value,
$G_{F}$, as described by

$\displaystyle G_{F}^{\rm obs}=G_{F}\cos U_{e4}\cos U_{\mu 4}\,.$ (8)

This relationship is pivotal in all the measurements employed in the CKM
determinations. The CKM element $V_{ud}$ is obtained by the superallowed
nuclear $\beta$ decay and neutron decay associated with $\nu^{\prime}_{e}$,
and we have $V_{ud}^{\rm obs}G_{F}^{\rm obs}=V_{ud}G_{F}\cos U_{e4}$ which
leads to

$\displaystyle V_{ud}^{\rm obs}$ $\displaystyle=V_{ud}/\cos U_{\mu 4}\,.$ (9)

Other elements $V_{us}$ and its ratio $V_{us}/V_{ud}$ are derived from the
measurements involving $\nu^{\prime}_{\mu}$.111$V_{us}$ is also determined
from the electron mode, $K\to\pi e\nu_{e}$ (with better precision than the
muon mode), which leads to $V_{us}^{\rm obs}=V_{us}/\cos U_{\mu 4}$. But,
combining Eq. (9), $\Delta_{\rm CKM}>0$ is predicted in either way. Analogous
to Eq. (9), the relationships between the observed and true values are given
by

$\displaystyle V_{us}^{\rm obs}$ $\displaystyle=V_{us}/\cos U_{e4}\,,$ (10)
$\displaystyle(V_{us}/V_{ud})^{\rm obs}$ $\displaystyle=V_{us}/V_{ud}\,.$ (11)

Now one can check the unitarity relation, for example, combining Eqs. (9) with
(10),

$\displaystyle 1+\Delta_{\rm CKM}=|V_{ud}^{\rm obs}|^{2}+|V_{us}^{\rm
obs}|^{2}=\frac{|V_{ud}|^{2}}{\cos^{2}U_{\mu
4}}+\frac{|V_{us}|^{2}}{\cos^{2}U_{e4}}\,.$ (12)

Given the inherent CKM unitarity, represented by
$|V_{ud}|^{2}+|V_{us}|^{2}\simeq 1$, it follows $\Delta_{\rm CKM}>0$, which is
in contrast to the experimental fits presented in Eq. (4).222The $m_{W}$
anomaly seen at the CDF experiment [38] can be resolved in this framework,
because the modification goes in the right direction. Modification of $G_{F}$
is essential in this case. See also Ref. [35]. Similarly, combining Eqs. (9)
and (11) draws the same conclusion.

###  II.2 Light sterile neutrino

Given that a heavy sterile neutrino cannot address the anomaly, it is
instructive to examine the opposite scenario where the sterile neutrino mass
is significantly smaller than $E_{\nu}$, the maximum energy of neutrino in the
relevant processes. Intriguingly, under these conditions, experimental
observations align with the SM predictions. This is because the processes
involving $\nu_{4}$ are kinematically allowed and must be summed incoherently.
To illustrate, consider the muon decay where we account for four processes,

$\displaystyle\sum_{\nu_{i}=\nu^{\prime}_{\mu},\nu_{4},\leavevmode\nobreak\
\bar{\nu}_{j}=\bar{\nu}^{\prime}_{e},\bar{\nu}_{4}}\Gamma(\mu\to
e\nu_{i}\bar{\nu}_{j})$
$\displaystyle\qquad\simeq(\cos^{2}U_{e4}+\sin^{2}U_{e4})(\cos^{2}U_{\mu
4}+\sin^{2}U_{\mu 4})\Gamma_{\rm SM}$ $\displaystyle\qquad=\Gamma_{\rm SM}\,.$
(13)

Similarly, other processes reproduce the SM predictions. Hence, $\Delta_{\rm
CKM}$ is expected to be zero, which is again incompatible with the current
data.

###  II.3 MeV sterile neutrino interacting with electron

The above discussion tells the CAA may suggest an intermediate mass scale of
the sterile neutrino. If the mass is in the MeV scale, the coupling reduction
of the weak interaction remains in the neutron/nuclear decays, while other
observables, especially the Fermi constant, stay almost the same as in the SM
as seen in the Eq. (13). Consequently, only $V_{ud}$ is modified,

$\displaystyle V_{ud}^{\rm obs}=V_{ud}\cos U_{e4}\,.$ (14)

The other quantities, $G_{F}^{\rm obs},V_{us}^{\rm obs}$, and
$(V_{us}/V_{ud})^{\rm obs}$, are the same as the SM ones because the
corresponding $E_{\nu}$ is much larger than MeV. This realizes the
experimentally favored value,

$\displaystyle 1+\Delta_{\rm
CKM}=|V_{ud}|^{2}\cos^{2}U_{e4}+|V_{us}|^{2}<1\,.$ (15)

From the size of the anomaly $\Delta_{\rm CKM}\approx 10^{-3}$, we can infer
that the favored mixing-angle-squared is $U^{2}_{e4}\approx 10^{-3}$, and
$U^{2}_{\mu 4}$ is not necessary.333In the interest of the $m_{W}$ anomaly,
this parameter space is not necessarily favored, but it can be easily fixed if
another heavy sterile neutrino ($\nu_{5}$) affects $G_{F}$ by $U_{\mu 5}$.
Then, $V_{ud}$ is modified as $V_{ud}^{\rm obs}\simeq
V_{ud}(1-U_{e4}^{2}/2+U_{\mu 5}^{2}/2)$.

The sterile neutrino mass can be in the same order of $E_{\nu}$ of the
neutron/nuclear decays relevant to $V_{ud}$ measurements. In the following, we
evaluate the sterile mass dependence in the neutron and nuclear decays, and we
focus on the mixing with the electron neutrino.

####  II.3.1 Superallowed nuclear $\beta$ decay

The measurements of $V_{ud}$ in nuclear physics have been conducted through
so-called superallowed $0^{+}\to 0^{+}$ nuclear $\beta$ decay, and the latest
survey was given by Ref. [39]. A heavy nucleus of $J^{P}=0^{+}$ decay to
another nucleus of $0^{+}$ with a significant wave function overlap
(superallowed), emitting $e^{+}\nu_{e}$ ($\beta^{+}$ decay). Also, only the
vector current of the quark weak interaction can contribute (Fermi decay).
There are 15 transitions utilized to determine $|V_{ud}|$ [39], and the
released energy in these transitions varies according to the specific process.
Note that the determination of $V_{ud}$ is significantly dominated by a single
transition, ${}^{26m}\text{Al}\to^{26}\text{Mg}$ [40, 41]. While a sterile
neutrino might be absent in some processes, it could emerge in others with
larger released energy, and the effect is suppressed by the phase space as
well as the small mixing angle $\sin^{2}U_{e4}$.

Let us explore how $V_{ud}$ is measured and the impact of the massive sterile
neutrino. The decay width for both the active and sterile neutrinos is given
by

$\displaystyle\\!\\!\\!\\!\frac{d\Gamma_{0^{+}}}{dE_{e}}=\left(\frac{\sqrt{E_{e}^{2}-m_{e}^{2}}\sqrt{E_{\nu}^{2}-m_{\nu}^{2}}E_{e}E_{\nu}}{16\pi^{3}}\right)\frac{|{\cal
M}|^{2}}{E_{e}E_{\nu}MM^{\prime}}\,,$ (16)

where the parenthesis comes from the phase space calculation. The outgoing
neutrino (positron) energy is $E_{\nu(e)}$, and $M(M^{\prime})$ is the mass of
the parent (daughter) nucleus. $\delta M$ denotes the nucleus mass difference,
which is $\delta M\equiv M-M^{\prime}=Q_{\rm EC}-m_{e}$ neglecting the
electron binding energy of $\mathcal{O}(10)\,\text{eV}$ [42]. The electron-
capture (EC) $Q$-value $Q_{\rm EC}$ is the experimentally measured quantity
summarized in Table I of Ref. [39]. For example, $Q_{\rm EC}({{}^{26m}{\rm
Al}\to^{26}\\!{\rm Mg}})=4.23$ MeV. The nucleus recoil energy $E_{\rm recoil}$
can be ignored because it is much smaller than the typical electron and
nuclear energy. Thus the energy conservation is approximately held as $\delta
M\simeq E_{\nu}+E_{e}$, leading to $m_{e}\leq E_{e}\leq\delta M-m_{\nu}$. This
approximation greatly simplifies the calculation, and is valid until the
precision becomes as good as $\delta M/M\sim 10^{-4}$. With this accuracy, the
matrix element is given by

$\displaystyle|{\cal M}|^{2}\simeq CE_{e}E_{\nu}M^{2}G_{F}^{2}|V_{ud}|^{2}\,,$
(17)

where $C$ is a numerical coefficient. Normally, one can extract the
combination of $G_{F}^{2}|V_{ud}|^{2}$ based on the ${\cal F}t$ values of each
nuclear $\beta$ decay, which will be described later, under the assumption
that the neutrinos are massless.

We present a simple formalism to account for the modification due to the
massive sterile neutrino. Given the mass of neutrino barely modifies the
matrix element of Eq. (17), the massive neutrino effect appears through the
phase space. We define the modified phase space integral as

$\displaystyle\\!\\!\\!I\left(m_{\nu},\delta
M\right)\equiv\int_{m_{e}}^{\delta
M-m_{\nu}}\\!\\!\\!\\!\\!\\!dE_{e}\sqrt{E_{e}^{2}-m_{e}^{2}}\sqrt{E_{\nu}^{2}-m_{\nu}^{2}}E_{e}E_{\nu}\,,$
(18)

where $I(0,\delta M)$ corresponds to the SM case (maximizes for a given
$\delta M$), and the kinematically forbidden neutrino yields $I(m_{\nu}>\delta
M-m_{e},\delta M)=0$. A single decay mode is given by the sum of the active
and sterile neutrino channels. To extract the modification factor depending on
the sterile neutrino mass and mixing angle, we normalize the decay width by
the one with $m_{\nu_{4}}=0$,

$\displaystyle\frac{\Gamma_{0^{+}}}{\Gamma_{{0^{+}},m_{\nu_{4}}=0}}$
$\displaystyle=\cos^{2}U_{e4}+\frac{I(m_{\nu_{4}},\delta M)}{I(0,\delta
M)}\sin^{2}U_{e4}$ (19) $\displaystyle=1-\epsilon(m_{\nu_{4}},\delta
M)\sin^{2}U_{e4}\,,$ (20)

with

$\displaystyle\epsilon\left(m_{\nu_{4}},\delta M\right)\equiv
1-\frac{I(m_{\nu_{4}},\delta M)}{I(0,\delta M)}\,,$ (21)

where $m_{\nu_{4}}$ represents the sterile neutrino mass. We found that Eq.
(19) is consistent with a result of Ref. [36]. In Fig. 2, the phase space
modification factor $\epsilon(m_{\nu_{4}},\delta M)$ is plotted for each
superallowed channel. There, the charge repulsion or attraction effect is
included in accordance with Ref. [43], which has little impact.

Figure 2:  The phase space modification factors $\epsilon(m_{\nu_{4}},\delta
M)$ are shown for the ten most accurate superallowed decays (solid lines and a
dotted line) and the neutron decay (black dashed line).

The measured $|V_{ud,0^{+}}^{\rm obs}|$ through the nuclear $\beta$ decay is
different from the value in the presence of the MeV sterile neutrino, and,
therefore, we obtain

$\displaystyle\Gamma_{0^{+}}^{\rm obs}\propto|V_{ud,0^{+}}^{\rm
obs}|^{2}=\left[1-\epsilon(m_{\nu_{4}},\delta
M)\sin^{2}U_{e4}\right]|V_{ud,0^{+}}|^{2}$ (22)

by using Eq. (20). This reproduces the concise expression of Eq. (14) when
$m_{\nu 4}>\delta M-m_{e}$, corresponding to $\epsilon=1$. Thus, we expect
$U_{e4}^{2}\approx\Delta_{\rm CKM}^{\rm global}\approx 10^{-3}$ to accommodate
the CAA.

####  II.3.2 Neutron decay

The neutron decay is currently the second-best probe of $V_{ud}$. The neutron
lifetime ($\tau_{n}$) measurements and the theoretical calculations are put
together to obtain $V_{ud,n}$. It is known that there are two different
measurements for the neutron lifetime, so-called neutron decays in the bottle
or in the beam, and these data indicate $\approx 4\sigma$ discrepancy [44].
Although they are controversial, both results imply the violation of CKM
unitarity, see details in the Introduction. In our fit, we treat two methods
separately and combine it with other measurements. The sensitivity for
$V_{ud}$ is still dominated by the superallowed nuclear decay, but the favored
sterile neutrino mass changes a little depending on the method of neutron
lifetime measurements.

The effect of massive sterile neutrino is very similar to the case of
superallowed nuclear decay. The neutron decay width is given by

$\displaystyle\frac{\Gamma_{n}}{\Gamma_{{n},m_{\nu_{4}}=0}}$
$\displaystyle=1-\epsilon\left(m_{\nu 4},\delta
M_{np}\right)\sin^{2}U_{e4}\,,$ (23)

where $\delta M_{np}$ is the mass difference between a neutron and a proton.
Hence, the observed $V_{ud,n}^{\rm obs}$ can be smaller than the true value,

$\displaystyle|V^{\rm obs}_{ud,n}|^{2}=\left[1-\epsilon\left(m_{\nu 4},\delta
M_{np}\right)\sin^{2}U_{e4}\right]|V_{ud,n}|^{2}\,.$ (24)

The phase space modification factor $\epsilon\left(m_{\nu 4},\delta
M_{np}\right)$ is shown in Fig. 2.

As we already addressed, the neutron lifetime measurements are mutually
inconsistent between the bottle UCN and the beam methods. In this article, we
treat them separately rather than combining.

Figure 3: Favored parameter regions are shown. The red points represent the
best-fit point with the pulls from the SM hypothesis. For the neutron decay
data, the most precise neutron lifetime results from the bottle UCN and the
beam measurements are used in the left and right panels, respectively. The
blue and purple shaded regions are excluded by the nuclear $\beta$-decay kink
searches [45, 46, 47, 48] and the EC-decay search [49], respectively. The
regions above green dashed lines can be constrained by $\pi^{+}\to e^{+}\nu$
measurements [50, 51, 36], and the regions between dotted or dash-dotted green
lines are allowed with the dimension-six operator. See more details in the
text.

####  II.3.3 Statistical combinations

Basically we construct $\chi^{2}$ functions of $m_{\nu 4},\leavevmode\nobreak\
\sin^{2}U_{e4}\simeq U_{e4}^{2},\leavevmode\nobreak\ |V_{ud}|$, and
$|V_{us}|$, and impose the unitarity at the truth level,
$|V_{ud}|^{2}+|V_{us}|^{2}=1$, safely dropping $|V_{ub}|^{2}\simeq 1.4\times
10^{-5}$ for the uncertainty of $\Delta_{\rm CKM}$ [21, 22]. In this
subsection, we omit the absolute value notation for simplicity.

We combine the $V_{ud}$ measurements which depend on the sterile neutrino mass
less than $10\,$MeV and the mixing, with $V_{us}$ and $V_{us}/V_{ud}$
measurements which have negligible shifts from the sterile neutrino, since
$U_{e4}^{2}(m_{\nu 4}/E_{\nu})\lesssim 10^{-4}$. We adopt the following fitted
values: $V_{us}$ from the semi-leptonic kaon decays ($K_{\ell 3}$), inclusive-
hadronic $\tau$ decays, and hyperon semi-leptonic decays, and $V_{us}/V_{ud}$
from the leptonic kaon-decay rate over the pion one ($K_{\mu 2}/\pi_{\mu 2}$)
[16]

$\displaystyle V_{us,{\rm fit}}$ $\displaystyle=0.223\,14(51)\,,$ (25)
$\displaystyle\left(V_{us}/V_{ud}\right)_{\rm fit}$
$\displaystyle=0.231\,08(51)\,.$ (26)

Then, we construct two $\chi^{2}$ functions,

$\displaystyle\chi^{2}_{us}\left(V_{ud}\right)=\frac{\left(\sqrt{1-V_{ud}^{2}}-V_{us,\rm
fit}\right)^{2}}{\left(\delta V_{us}\right)^{2}}\,,$ (27)
$\displaystyle\chi^{2}_{us/ud}\left(V_{ud}\right)=\frac{\left[\sqrt{1-V_{ud}^{2}}/V_{ud}-(V_{us}/V_{ud})_{\rm
fit}\right]^{2}}{\left[\delta(V_{us}/V_{ud})\right]^{2}}\,,$ (28)

where $\delta V_{us}$ represents the uncertainty of $V_{us,\rm fit}$, and so
does $\delta(V_{us}/V_{ud})$.

For the $V_{ud}$ determination through the neutron lifetime, we have to
consider the effect of massive sterile neutrino, and then we have

$\displaystyle\\!\\!\\!\\!\\!\\!\chi^{2}_{n}(m_{\nu
4},U_{e4}^{2},V_{ud})=\frac{\left[V_{ud}\sqrt{1-\epsilon(m_{\nu_{4}},\delta
M_{np})U_{e4}^{2}}-V_{ud,\rm n}\right]^{2}}{(\delta V_{ud,n})^{2}}\,.$ (29)

We have two different $\chi^{2}_{n}$ functions for the $V_{ud,n}=0.97413(43)$
at the bottle UCN experiment (5) and $V_{ud,n}=0.96866(131)$ in the beam (6).
We do not combine them due to their disagreement.

Next, we construct $\chi^{2}$ function for the superallowed $0^{+}\to 0^{+}$
nuclear $\beta$ decay, where the ten most accurate nuclei data are used. Each
superallowed transition is characterized by its own $ft$ value, where $f$ is a
dimensionless constant that comes from an integral over phase space and $t$ is
the half-lifetime of the $0^{+}\to 0^{+}$ transition [52, 53, 54]. By
subtracting the nuclear structure-dependent corrections from each $ft$ value,
one can obtain each corrected $\mathcal{F}t$ value, which is a nucleus-
independent quantity according to the conserved vector current hypothesis, and
sensitive to $(G_{F}^{\rm obs}V_{ud}^{\rm obs})^{-2}$. In the fit of ten
superallowed nuclear decays, one needs a common nuisance parameter $\alpha$ to
account for the total systematic uncertainty. With this consideration,
$\chi^{2}$ functions from the superallowed transitions are given by

$\displaystyle\chi^{2}_{0^{+}}(m_{\nu 4},U_{e4}^{2},V_{ud},\alpha)$
$\displaystyle=\sum_{i}\left({\cal
F}t_{i}-\frac{K(1+\Delta_{R}^{V})^{-1}(1+\alpha)}{2G_{F}^{2}V_{ud}^{2}\left[1-\epsilon(m_{\nu
4},\delta M_{i})U_{e4}^{2}\right]}\right)^{2}/(\delta{\cal F}t_{i})^{2}\,,$
(30) $\displaystyle\chi^{2}_{\rm
sys}(\alpha)=\frac{\alpha^{2}}{\sigma_{\Delta_{R}^{V}}^{2}+\sigma_{\delta^{\prime}_{R}}^{2}+\sigma_{\delta_{\rm
NS}}^{2}}\,,$ (31)

where $G_{F}=1.1663788(6)\times 10^{-5}\,\text{GeV}^{-2}$,
$K=8120.27648(26)\times 10^{-10}\,\text{GeV}^{-4}\text{s}$, and the index $i$
runs 10C, 14O, 26mAl, 34Cl, 34Ar, 38mK, 42Sc, 46V, 50Mn, and 54Co, shown in
Fig. 2. We use the transition-independent radiative correction
$\Delta_{R}^{V}=2.467(27)\times 10^{-2}$ from Eq. (A.8) of Ref. [15], giving
$\sigma_{\Delta_{R}^{V}}=\delta\Delta_{R}^{V}/(1+\Delta_{R}^{V})=2.63\times
10^{-4}$. $\mathcal{F}t_{i}$ values are obtained from Table XVI of Ref. [39].
We extract
$\sigma_{\delta^{\prime}_{R}}=0.36\text{\,s}/3072\text{\,s}=1.17\times
10^{-4}$ and $\sigma_{\delta_{\rm
NS}}=1.73\text{\,s}/3072\text{\,s}=5.63\times 10^{-4}$ from Eq. (22) of Ref.
[39].

For a crosscheck of our statistical scheme of the superallowed nuclear decays,
when a sterile neutrino decoupling limit ($U_{e4}\to 0$ or $m_{\nu 4}\to 0$)
is examined with $\Delta_{R}^{V}=2.454(19)\times 10^{-2}$ given in Ref. [39],
we obtained $V_{ud,0^{+}}=0.97369(32)$, which well agrees with
$V_{ud,0^{+}}=0.97373(31)$ of Ref. [39].

Finally, we combine all the $\chi^{2}$ functions using only one neutron
$\chi^{2}$ at a time,

$\displaystyle\chi^{2}(m_{\nu
4},U_{e4}^{2},V_{ud},\alpha)=\chi_{us}^{2}+\chi_{us/ud}^{2}+\chi_{n}^{2}+\chi_{0^{+}}^{2}+\chi^{2}_{\rm
sys}\,.$ (32)

We find the global minimum $\chi^{2}_{\rm min}$ and search for the favored
parameter space of the sterile neutrino by evaluating

$\displaystyle\Delta\chi^{2}\equiv\chi^{2}(m_{\nu
4},U_{e4}^{2},V_{ud},\alpha)-\chi^{2}_{\rm min}\,,$ (33)

where $V_{ud}$, as well as $\alpha$, is treated as the nuisance parameters and
thus minimized for a given set of $(m_{\nu 4},U_{e4}^{2})$.

The results are shown in Fig. 3. As expected, the favored mixing angle squared
is $U_{e4}^{2}\approx 10^{-3}$, and there is a plateau once the sterile
neutrino mass $m_{\nu 4}$ is heavier than $\delta M-m_{e}\approx 3.2\,$MeV,
which is the maximum neutrino energy at the superallowed nuclear decay
${}^{26m}\text{Al}\to^{26}\text{Mg}$, see also Fig. 2.

We show two panels depending on the neutron lifetime measurements. In the left
panel, we use the bottle measurement to extract $V_{ud}$, which is consistent
with the one from the superallowed nuclear decays. The original tension of the
CAA is at 2.8$\sigma$, and the pull in the presence of sterile neutrino is
2.1$\sigma$ at the best-fit point. On the other hand, since the beam
measurement prefers smaller $V_{ud}$ than the superallowed nuclear decays, a
significance of the CAA is enhanced to be 3.8$\sigma$, and the sterile
neutrino can relax it by 3.2$\sigma$ at the best-fit point.

##  III Constraints of MeV sterile neutrino

In the previous section, we identified the favored ranges of the mass and
mixing of the sterile neutrino for the given anomaly. However, these parameter
regions are subject to constraints from the laboratory to the cosmology. In
this section, we list all the relevant bounds and show consistent resolutions
if exist.

###  III.1 Direct bounds

There have been kink searches in the Kurie plot, the emitted electron energy
spectrum measurements in the nuclear $\beta$ decays, to test the mass of
sterile neutrino [45, 46, 47, 48]. One relevant measurement is the kink search
in ${}^{20}{\rm F}$ decay which covers $m_{\nu 4}\lesssim 3$ MeV [46]. The
blue-shaded regions in Fig. 3 are excluded by the kink searches. This is a
robust bound on the sterile neutrino.

Furthermore, recently the BeEST experiment has set a constraint on the sterile
neutrino through the EC-$\beta$ decay using the superconducting quantum sensor
[49]. This bound is shown by the purple-shaded regions in Fig. 3, which
significantly improved the bound on the sterile neutrino mixing $U_{e4}$ with
$m_{\nu 4}=$0.1 MeV – 0.8 MeV. This bound is also robust in our scenario.

###  III.2 $0\nu\beta\beta$ bound

Searches for the neutrinoless double beta decay ($0\nu\beta\beta$) are
sensitive to Majorana neutrino mass and mixing. See the recent review [48,
55]. If a $3+1$ sterile neutrino scenario (three active neutrinos plus one
Majorana sterile neutrino) is considered, the $0\nu\beta\beta$ rate is
proportional to $U_{e4}^{2}m_{\nu 4}|^{2}$. In this simplest case, the current
measurements [56, 57] set the upper limit on $U_{e4}^{2}<10^{-7}$ at $m_{\nu
4}=\mathcal{O}(1)\,$MeV [48], which excludes the whole parameter region of
Fig. 3.

However, it is known that the $0\nu\beta\beta$ bound can be totally suppressed
in the inverse seesaw model [58, 59, 60]. In this scenario, the left-handed
singlet fermion $S$ as well as the right-handed neutrino $N$ are introduced to
the SM Lagrangian [61],

$\displaystyle\\!\\!\\!\\!-\mathcal{L}$
$\displaystyle=\frac{1}{2}\overline{N}^{c}\mu_{N}N+y\overline{L}\widetilde{H}N+\frac{1}{2}\overline{S}^{c}\mu_{S}S+\overline{S}M_{N}N+\textrm{h.c.}\,,$
(34)

where $\mu_{N}$ and $\mu_{S}$ are Majorana masses, which are small lepton
number violations and are technically natural. After taking the mass
eigenbasis, the active neutrino mass is

$\displaystyle
m_{\nu^{\prime}_{\ell}}\simeq\frac{M_{D}^{2}}{M_{N}^{2}}\mu_{S}\simeq
U_{e4}^{2}\mu_{S}\,,$ (35)

where the mixing is $U_{c}^{2}\simeq(M_{D}/M_{N})^{2}$, while the sterile
neutrino becomes a pseudo-Dirac fermion with mass as

$\displaystyle m_{\nu 4}\simeq M_{N}\pm\frac{\mu_{S}}{2}\,.$ (36)

Here, $M_{N}\gg M_{D}=vy\leavevmode\nobreak\ (v\simeq 174\,$GeV) is assumed.
Note that $\mu_{N}$ is an irrelevant parameter at the leading order.

The smallness of the active neutrino masses requires $\mu_{S}$ to be
suppressed in the inverse seesaw model, namely
$\mu_{s}\approx\mathcal{O}(1)\,$eV in the parameter of our interest.
Consequently, the $0\nu\beta\beta$ rate, which has to pick up the Majorana
mass $\mu_{s}$, is significantly smaller than in the 3+1 scenario and
contained well below the observed limit for $m_{\nu
4}\lesssim\mathcal{O}(100)\,$MeV [48].

###  III.3 Meson decay bounds

The sterile neutrino bounds from meson leptonic decays have been discussed in
Ref. [62]. Within the SM, $\pi^{+}\to e^{+}\nu$ decay is helicity-suppressed
by the electron mass ($m_{e}^{2}/m_{\pi^{+}}^{2}$). On the other hand, when
the sterile neutrino is heavier than the electron, the branching ratio is
significantly modified. Strong constraints come from the $e$-$\mu$
universality measurements in the two-body leptonic decays of $\pi^{+}$ and
$K^{+}$ [62, 63, 64, 65],

$\displaystyle R_{e/\mu}^{M}=\frac{\text{BR}(M^{+}\to
e^{+}\nu(\gamma))}{\text{BR}(M^{+}\to\mu^{+}\nu(\gamma))}\,,\quad\text{for\leavevmode\nobreak\
}M=\pi,\,K.$ (37)

In Fig. 3, the areas above the green dashed lines are excluded by the latest
analysis of $R_{e/\mu}^{\pi}$ at the PiENu experiment [50, 51, 36] (the bound
from $R_{e/\mu}^{K}$ is weaker [66]). In the minimal scenario of the sterile
neutrino with the mass and mixing to the weak interaction, this bound excludes
the significant part of the favored parameter space for the CAA.

However, if there are some higher dimensional operators that contain the
sterile neutrino and induce $\pi^{+}\to e^{+}\nu_{4}$, they can reduce
$\pi^{+}\to e^{+}\nu_{4}$ while keeping the nuclear and neutron $\beta$ decay
unaffected. This is because the SM-like amplitude of the pion decay is
actually suppressed by the lepton mass, so its electron mode is sensitive to
new physics contributions. Even with new physics of $\mathcal{O}(10)$ TeV
scale, the $R_{e/\mu}^{\pi}$ bound can be easily compensated by small
modification due to the higher dimensional operators.

For example, $R_{e/\mu}^{M}$ is sensitive to a dimension-six scalar operator
$(\overline{u}_{R}V_{ui}d_{L}^{i})(\overline{e}_{L}N)$, which can
destructively interfere with the mixing contribution of Eq. (7), and we
introduce

$\displaystyle\mathcal{L}_{\rm
eff}=\frac{1}{\Lambda^{2}}\left[\overline{u}_{R}(V_{ud}d_{L}+V_{us}s_{L})\right](\overline{e}_{L}N)+\textrm{h.c.}\,.$
(38)

In Fig. 3, we show that modified allowed regions of $R_{e/\mu}^{M}$ for the
cases of $\Lambda=$20 TeV and 30 TeV by the green dash-dotted and dotted
lines, respectively. It is found that the $R_{e/\mu}^{M}$ bound is
significantly sensitive to the operator of Eq. (38), and the CAA favors the
dimension-six operator with $\Lambda=$20 – 30 TeV.

###  III.4 Long-lived sterile neutrino and cosmology

When the sterile neutrino lifetime is determined by the weak decay, the
reactor [67, 68] and Borexino experiments [69] use $\nu_{4}\to
e^{+}e^{-}\nu^{\prime}_{e}$ decay mode to probe the range of 1 MeV
$<m_{\nu_{4}}<$ 14 MeV [48]. These bounds potentially exclude the favored
regions of Fig. 3. However, the bounds are not applied if the lifetime is
shorter by a factor of $\mathcal{O}(10-100)$ due to the additional decay
modes.

The decay of sterile neutrino to three active neutrinos becomes significant in
the presence of a real scalar mediator $\phi$ described by

$\displaystyle-\mathcal{L}=\frac{m_{\phi}^{2}}{2}\phi^{2}+\left(\frac{\lambda}{2}\phi\overline{S}^{c}S+\text{h.c.}\right)\,.$
(39)

If the effective interaction scale satisfies $\lambda^{2}/m_{\phi}^{2}\gtrsim
10\,U_{e4}^{-2}G_{F}\sim 0.1/{\rm GeV}^{2}$, the decay of the sterile neutrino
is short enough. We find, when $m_{\phi}\simeq 0.5$ GeV – 30 GeV, this
condition is compatible with the bounds, such as the meson decays, studied in
Ref. [70] since the mediator interactions with the active-neutrino are
suppressed by the mixing angle as $\frac{1}{2}\lambda
U_{e4}^{2}\phi\bar{\nu}_{e}^{c}\nu_{e},\lambda
U_{e4}\phi\overline{S}^{c}\nu_{e}$.

The cosmological observations typically constrain the MeV scale sterile
neutrino. However, with the required mediator-interaction, the sterile
neutrino decoupling from the SM thermal bath occurs together with the active
neutrinos decoupling at $T\sim 2$ MeV. As a result, the remaining bound is
from the effective number of neutrinos, $N_{\rm eff}$. As the sterile
neutrinos are not completely non-relativistic at the decoupling temperature,
they would increase the effective number of neutrinos by $\Delta N_{\rm
eff}\simeq 1.0\text{\,--\,}0.25$ for $m_{\nu 4}=5$ – 10 MeV. Although $\Delta
N_{\rm eff}\gtrsim 0.3$ is constrained by the CMB observations [71],
additional well-motivated particles, such as heavy axions with a lifetime of
about 0.1 – 1 sec, can consistently compensate $\Delta N_{\rm eff}$ [72, 73].

In the absence of additional long-lived particles affecting $N_{\rm eff}$, a
different set of mediator interactions,

$\displaystyle-\mathcal{L}_{\phi}=\frac{m_{\phi}^{2}}{2}\phi^{2}+\frac{\phi}{4\Lambda_{\gamma}}F_{\mu\nu}F^{\mu\nu}+\left(\lambda^{\prime}\phi\overline{S}_{L}N+\text{h.c.}\right)\,,$
(40)

can lead to another consistent scenario. In this case, the sterile neutrino
decay is predominantly
$\nu_{4}\to\phi^{(\ast)}\nu^{\prime}_{e}\to\gamma\gamma\nu^{\prime}_{e}$ which
dismisses the reactor and Borexino bounds. The thermal history is modified
such that the sterile neutrino interaction with the photons determines the
decoupling from the thermal bath. The temperature when the entire neutrino
sector decouples can be lower to $1(0.75)$ MeV, which is about the temperature
of neutron decoupling. The sterile neutrino at this temperature is non-
relativistic enough such that $\Delta N_{\rm eff}\simeq
0.3\text{\,--\,}9\times 10^{-3}(0.1\text{\,--\,}6\times 10^{-4})$ for $m_{\nu
4}=5$ – 10 MeV is allowed by the CMB. This scenario requires
$\lambda^{\prime}/(m_{\phi}^{2})\Lambda\gtrsim 0.03/\text{GeV}^{3}$.
Considering the bounds involving neutrino [70] and photon [74, 75], we find
that the mediator is allowed when $m_{\phi}\simeq 0.5$ GeV – 4 GeV and
$\lambda\gtrsim 1$.

###  III.5 Summary of the constraints

Given the potential constraints for the parameter space favored by the CAA,
the inverse seesaw models with a mediator $\phi$ and a higher dimensional
operator emerge as viable scenarios. Even in the presence of the additional
decay modes of the sterile neutrino via the mediator, it remains effectively
stable in most laboratory experiments, in particular, measurements of neutron
and nuclear $\beta$ decays.

##  IV Conclusions

Currently, it is reported that the first-row CKM unitarity test is violated at
the $2.8\sigma$ level, referred to as the Cabibbo angle anomaly (CAA). This
violation is worse when the neutron lifetime data of the in-beam experiment is
used in the global fit. In this article, we point out that the MeV sterile
neutrino that mixes the electron-type neutrino with $U_{e4}^{2}\sim 10^{-3}$
can decrease the value of extracted $|V_{ud}^{\rm obs}|$ from the superallowed
nuclear $\beta$ decays and the neutron decay relative to its true value,
without modifying other observables relevant to the CKM determinations. As a
result, it is found that the MeV sterile neutrino can alleviate the CAA. The
parameter space of the sterile neutrino favored as the solution of the CAA is
typically subject to various constraints. Although the nuclear $\beta$ decay
bounds are robust, we show that the sterile neutrino within the inverse seesaw
models can evade the other constraints.

While we highlighted the sterile neutrino resolving the anomaly of the
unitarity test, this type of scenario where new physics primarily in the
neutrino sector gives apparent deviations in the precision measurements on the
quark sector could be interesting in a broader context.

###### Acknowledgements.

We would like to thank Daniel Egana-Ugrinovic, Matheus Hostert, Takemichi
Okui, Robert Shrock, Mark-Christoph Spieker, Vandana Tripathi, and Nodoka
Yamanaka for useful discussions. T. K. was supported by the Japan Society for
the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (Grant
No. 21K03572) and the JSPS Core-to-Core Program (Grant No. JPJSCCA20200002).
K. T. is supported by in part the US Department of Energy grant DE-SC0010102
and JSPS Grant-in-Aid for Scientific Research (Grant No. 21H01086).

## References

  * [1] Heavy Flavor Averaging Group, HFLAV Collaboration, “Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021,” Phys. Rev. D 107 (2023) 052008 [arXiv:2206.07501].
  * [2] N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Phys. Rev. Lett. 10 (1963) 531–533.
  * [3] M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory of Weak Interaction,” Prog. Theor. Phys. 49 (1973) 652–657.
  * [4] C.-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, “Reduced Hadronic Uncertainty in the Determination of $V_{ud}$,” Phys. Rev. Lett. 121 (2018) 241804 [arXiv:1807.10197].
  * [5] 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) 149 [arXiv:1906.02714].
  * [6] Y. Grossman, E. Passemar, and S. Schacht, “On the Statistical Treatment of the Cabibbo Angle Anomaly,” JHEP 07 (2020) 068 [arXiv:1911.07821].
  * [7] 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) 071802 [arXiv:1912.08823].
  * [8] 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 [arXiv:2001.02853].
  * [9] C.-Y. Seng, X. Feng, M. Gorchtein, and L.-C. Jin, “Joint lattice QCD–dispersion theory analysis confirms the quark-mixing top-row unitarity deficit,” Phys. Rev. D 101 (2020) 111301 [arXiv:2003.11264].
  * [10] 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 [arXiv:2008.01113].
  * [11] M. Kirk, “Cabibbo anomaly versus electroweak precision tests: An exploration of extensions of the Standard Model,” Phys. Rev. D 103 (2021) 035004 [arXiv:2008.03261].
  * [12] A. Crivellin, M. Hoferichter, and C. A. Manzari, “Fermi Constant from Muon Decay Versus Electroweak Fits and Cabibbo-Kobayashi-Maskawa Unitarity,” Phys. Rev. Lett. 127 (2021) 071801 [arXiv:2102.02825].
  * [13] B. Belfatto and Z. Berezhiani, “Are the CKM anomalies induced by vector-like quarks? Limits from flavor changing and Standard Model precision tests,” JHEP 10 (2021) 079 [arXiv:2103.05549].
  * [14] D. Bryman, V. Cirigliano, A. Crivellin, and G. Inguglia, “Testing Lepton Flavor Universality with Pion, Kaon, Tau, and Beta Decays,” Ann. Rev. Nucl. Part. Sci. 72 (2022) 69–91 [arXiv:2111.05338].
  * [15] V. Cirigliano, A. Crivellin, M. Hoferichter, and M. Moulson, “Scrutinizing CKM unitarity with a new measurement of the $K_{\mu 3}/K_{\mu 2}$ branching fraction,” Phys. Lett. B 838 (2023) 137748 [arXiv:2208.11707].
  * [16] A. Crivellin, M. Kirk, T. Kitahara, and F. Mescia, “Global fit of modified quark couplings to EW gauge bosons and vector-like quarks in light of the Cabibbo angle anomaly,” JHEP 03 (2023) 234 [arXiv:2212.06862].
  * [17] B. Belfatto and S. Trifinopoulos, “The remarkable role of the vector-like quark doublet in the Cabibbo angle and $W$-mass anomalies.” arXiv:2302.14097.
  * [18] UCN$\tau$ Collaboration, “Improved neutron lifetime measurement with UCN$\tau$,” Phys. Rev. Lett. 127 (2021) 162501 [arXiv:2106.10375].
  * [19] A. T. Yue, et al., “Improved Determination of the Neutron Lifetime,” Phys. Rev. Lett. 111 (2013) 222501 [arXiv:1309.2623].
  * [20] B. Märkisch et al., “Measurement of the Weak Axial-Vector Coupling Constant in the Decay of Free Neutrons Using a Pulsed Cold Neutron Beam,” Phys. Rev. Lett. 122 (2019) 242501 [arXiv:1812.04666].
  * [21] CKMfitter Group Collaboration, “CP violation and the CKM matrix: Assessing the impact of the asymmetric $B$ factories,” Eur. Phys. J. C 41 (2005) 1–131 [hep-ph/0406184]. updated results and plots available at: http://ckmfitter.in2p3.fr.
  * [22] UTfit Collaboration, “New UTfit Analysis of the Unitarity Triangle in the Cabibbo-Kobayashi-Maskawa scheme,” Rend. Lincei Sci. Fis. Nat. 34 (2023) 37–57 [arXiv:2212.03894].
  * [23] A. Crivellin and M. Hoferichter, “$\beta$ Decays as Sensitive Probes of Lepton Flavor Universality,” Phys. Rev. Lett. 125 (2020) 111801 [arXiv:2002.07184].
  * [24] M. Endo and S. Mishima, “Muon $g-2$ and CKM unitarity in extra lepton models,” JHEP 08 (2020) 004 [arXiv:2005.03933].
  * [25] 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) 015032 [arXiv:2005.13542].
  * [26] T. Li, X.-D. Ma, and M. A. Schmidt, “Constraints on the charged currents in general neutrino interactions with sterile neutrinos,” JHEP 10 (2020) 115 [arXiv:2007.15408].
  * [27] A. Crivellin, C. A. Manzari, M. Alguero, and J. Matias, “Combined Explanation of the Z→bb¯ Forward-Backward Asymmetry, the Cabibbo Angle Anomaly, and $\tau$→$\mu$$\nu$$\nu$ and b→s$\ell$+$\ell$\- Data,” Phys. Rev. Lett. 127 (2021) 011801 [arXiv:2010.14504].
  * [28] 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) 073002 [arXiv:2012.09845].
  * [29] T. Felkl, J. Herrero-Garcia, and M. A. Schmidt, “The Singly-Charged Scalar Singlet as the Origin of Neutrino Masses,” JHEP 05 (2021) 122 [arXiv:2102.09898]. [Erratum: JHEP 05, 073 (2022)].
  * [30] G. C. Branco, J. T. Penedo, P. M. F. Pereira, M. N. Rebelo, and J. I. Silva-Marcos, “Addressing the CKM unitarity problem with a vector-like up quark,” JHEP 07 (2021) 099 [arXiv:2103.13409].
  * [31] D. Marzocca and S. Trifinopoulos, “Minimal Explanation of Flavor Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabibbo Angle,” Phys. Rev. Lett. 127 (2021) 061803 [arXiv:2104.05730].
  * [32] A. J. Buras, A. Crivellin, F. Kirk, C. A. Manzari, and M. Montull, “Global analysis of leptophilic Z’ bosons,” JHEP 06 (2021) 068 [arXiv:2104.07680].
  * [33] 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,” JHEP 10 (2021) 221 [arXiv:2107.13569].
  * [34] V. Cirigliano, D. Díaz-Calderón, A. Falkowski, M. González-Alonso, and A. Rodríguez-Sánchez, “Semileptonic tau decays beyond the Standard Model,” JHEP 04 (2022) 152 [arXiv:2112.02087].
  * [35] M. Blennow, P. Coloma, E. Fernández-Martínez, and M. González-López, “Right-handed neutrinos and the CDF II anomaly,” Phys. Rev. D 106 (2022) 073005 [arXiv:2204.04559].
  * [36] D. A. Bryman and R. Shrock, “Constraints on Sterile Neutrinos in the MeV to GeV Mass Range,” Phys. Rev. D 100 (2019) 073011 [arXiv:1909.11198].
  * [37] T. Kitahara and K. Tobioka, “Sterile neutrinos in light of the Cabibbo-angle anomaly,” J. Phys. Conf. Ser. 2446 (2023) 012009.
  * [38] CDF Collaboration, “High-precision measurement of the W boson mass with the CDF II detector,” Science 376 (2022) 170–176.
  * [39] J. C. Hardy and I. S. Towner, “Superallowed $0^{+}\to 0^{+}$ nuclear $\beta$ decays: 2020 critical survey, with implications for Vud and CKM unitarity,” Phys. Rev. C 102 (2020) 045501.
  * [40] S. A. Brindhaban and P. H. Barker, “The Q value for the Alm26 superallowed beta decay,” Phys. Rev. C 49 (1994) 2401–2406.
  * [41] T. Eronen et al., “Q-values of the Superallowed beta-Emitters Al-26-m, Sc-42 and V-46 and their impact on V(ud) and the Unitarity of the CKM Matrix,” Phys. Rev. Lett. 97 (2006) 232501 [nucl-ex/0606035].
  * [42] A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team. “NIST Atomic Spectra Database (ver. 5.10),” Available: https://www.nist.gov/pml/atomic-spectra-database. National Institute of Standards and Technology, Gaithersburg, MD., 2022.  doi:10.18434/T4W30F.
  * [43] M. Kleesiek et al., “$\beta$-Decay Spectrum, Response Function and Statistical Model for Neutrino Mass Measurements with the KATRIN Experiment,” Eur. Phys. J. C 79 (2019) 204 [arXiv:1806.00369].
  * [44] A. Czarnecki, W. J. Marciano, and A. Sirlin, “Neutron Lifetime and Axial Coupling Connection,” Phys. Rev. Lett. 120 (2018) 202002 [arXiv:1802.01804].
  * [45] K. Schreckenbach, G. Colvin, and F. Von Feilitzsch, “Search for mixing of heavy neutrinos in the $\beta^{+}$ and $\beta^{-}$ spectra of the 64Cu decay,” Phys. Lett. B 129 (1983) 265–268.
  * [46] J. Deutsch, M. Lebrun, and R. Prieels, “Searches for admixture of massive neutrinos into the electron flavor,” Nucl. Phys. A 518 (1990) 149–155.
  * [47] A. V. Derbin, et al., “Search for a Neutrino with a Mass of 0.01–1.0 MeV in Beta Decays of 144Ce–144Pr Nuclei,” JETP Lett. 108 (2018) 499–503.
  * [48] P. D. Bolton, F. F. Deppisch, and P. S. Bhupal Dev, “Neutrinoless double beta decay versus other probes of heavy sterile neutrinos,” JHEP 03 (2020) 170 [arXiv:1912.03058].
  * [49] S. Friedrich et al., “Limits on the Existence of sub-MeV Sterile Neutrinos from the Decay of 7Be in Superconducting Quantum Sensors,” Phys. Rev. Lett. 126 (2021) 021803 [arXiv:2010.09603].
  * [50] PiENu Collaboration, “Improved Measurement of the $\pi\to\textrm{e}\nu$ Branching Ratio,” Phys. Rev. Lett. 115 (2015) 071801 [arXiv:1506.05845].
  * [51] D. A. Bryman and R. Shrock, “Improved Constraints on Sterile Neutrinos in the MeV to GeV Mass Range,” Phys. Rev. D 100 (2019) 053006 [arXiv:1904.06787].
  * [52] J. C. Hardy and I. S. Towner, “New limit on fundamental weak-interaction parameters from superallowed beta decay,” Phys. Rev. Lett. 94 (2005) 092502 [nucl-th/0412050].
  * [53] J. C. Hardy and I. S. Towner, “Superallowed $0^{+}\to 0^{+}$ nuclear beta decays: A Critical survey with tests of CVC and the standard model,” Phys. Rev. C 71 (2005) 055501 [nucl-th/0412056].
  * [54] J. C. Hardy and I. S. Towner, “Superallowed $0^{+}\to 0^{+}$ nuclear beta decays: A New survey with precision tests of the conserved vector current hypothesis and the standard model,” Phys. Rev. C 79 (2009) 055502 [arXiv:0812.1202].
  * [55] W. Dekens, et al., “Neutrinoless double-beta decay in the neutrino-extended Standard Model.” arXiv:2303.04168.
  * [56] KamLAND-Zen Collaboration, “Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen,” Phys. Rev. Lett. 117 (2016) 082503 [arXiv:1605.02889]. [Addendum: Phys.Rev.Lett. 117, 109903 (2016)].
  * [57] GERDA Collaboration, “Improved Limit on Neutrinoless Double-$\beta$ Decay of 76Ge from GERDA Phase II,” Phys. Rev. Lett. 120 (2018) 132503 [arXiv:1803.11100].
  * [58] R. N. Mohapatra and J. W. F. Valle, “Neutrino Mass and Baryon Number Nonconservation in Superstring Models,” Phys. Rev. D 34 (1986) 1642.
  * [59] R. N. Mohapatra, “Mechanism for Understanding Small Neutrino Mass in Superstring Theories,” Phys. Rev. Lett. 56 (1986) 561–563.
  * [60] S. Nandi and U. Sarkar, “A Solution to the Neutrino Mass Problem in Superstring E6 Theory,” Phys. Rev. Lett. 56 (1986) 564.
  * [61] P. S. Bhupal Dev and A. Pilaftsis, “Light and Superlight Sterile Neutrinos in the Minimal Radiative Inverse Seesaw Model,” Phys. Rev. D 87 (2013) 053007 [arXiv:1212.3808].
  * [62] R. E. Shrock, “General Theory of Weak Leptonic and Semileptonic Decays. 1. Leptonic Pseudoscalar Meson Decays, with Associated Tests For, and Bounds on, Neutrino Masses and Lepton Mixing,” Phys. Rev. D 24 (1981) 1232.
  * [63] J. Tran Thanh Van, ed., “Search for Massive Neutrinos in $\pi\to\nu_{e}$ Decay,” Phys. Rev. Lett. 50 (1983) 1546.
  * [64] D. I. Britton et al., “Measurement of the $\pi^{+}\to e^{+}\nu$ branching ratio,” Phys. Rev. Lett. 68 (1992) 3000–3003.
  * [65] D. I. Britton et al., “Improved search for massive neutrinos in $\pi^{+}\to e^{+}\nu$ decay,” Phys. Rev. D 46 (1992) R885–R887.
  * [66] NA62 Collaboration, “Precision Measurement of the Ratio of the Charged Kaon Leptonic Decay Rates,” Phys. Lett. B 719 (2013) 326–336 [arXiv:1212.4012].
  * [67] A. I. Derbin, et al., “Experiment on anti-neutrino scattering by electrons at a reactor of the Rovno nuclear power plant,” JETP Lett. 57 (1993) 768–772. http://jetpletters.ru/ps/1202/index.shtml.
  * [68] C. Hagner, et al., “Experimental search for the neutrino decay $\nu_{3}\to\nu_{j}+e^{+}+e^{-}$ and limits on neutrino mixing,” Phys. Rev. D 52 (1995) 1343–1352.
  * [69] Borexino Collaboration, “New limits on heavy sterile neutrino mixing in B8 decay obtained with the Borexino detector,” Phys. Rev. D 88 (2013) 072010 [arXiv:1311.5347].
  * [70] A. de Gouvêa, et al., “Leptonic Scalars at the LHC,” JHEP 07 (2020) 142 [arXiv:1910.01132].
  * [71] Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6 [arXiv:1807.06209]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [72] D. Cadamuro, S. Hannestad, G. Raffelt, and J. Redondo, “Cosmological bounds on sub-MeV mass axions,” JCAP 02 (2011) 003 [arXiv:1011.3694].
  * [73] D. I. Dunsky, L. J. Hall, and K. Harigaya, “Dark Radiation Constraints on Heavy QCD Axions.” arXiv:2205.11540.
  * [74] M. J. Dolan, T. Ferber, C. Hearty, F. Kahlhoefer, and K. Schmidt-Hoberg, “Revised constraints and Belle II sensitivity for visible and invisible axion-like particles,” JHEP 12 (2017) 094 [arXiv:1709.00009]. [Erratum: JHEP 03, 190 (2021)].
  * [75] S. Gori, G. Perez, and K. Tobioka, “KOTO vs. NA62 Dark Scalar Searches,” JHEP 08 (2020) 110 [arXiv:2005.05170].