11institutetext: Institute for Creative Technologies, University of Southern
California, 90094 USA http://ict.usc.edu/
11email<EMAIL_ADDRESS>Computer Science
Department, University of Southern California, 90007 USA33institutetext:
Information Sciences Institute, University of Southern California, 90292 USA
http://www.isi.edu//
33email<EMAIL_ADDRESS>
# Operational Collective Intelligence of Humans and Machines

Nikolos Gurney 11 0000-0003-3479-2037 Fred Morstatter 33 David V. Pynadath
1122 0000-0003-2452-4733 Adam Russell 33 Gleb Satyukov 33

###### Abstract

We explore the use of aggregative crowdsourced forecasting (ACF) [2, 42] as a
mechanism to help operationalize “collective intelligence” of human-machine
teams for coordinated actions. We adopt the definition for Collective
Intelligence as: “A property of groups that emerges from synergies among data-
information-knowledge, software-hardware, and individuals (those with new
insights as well as recognized authorities) that enables just-in-time
knowledge for better decisions than these three elements acting alone.” [52]
Collective Intelligence emerges from new ways of connecting humans and AI to
enable decision-advantage, in part by creating and leveraging additional
sources of information that might otherwise not be included. Aggregative
crowdsourced forecasting (ACF) is a recent key advancement towards Collective
Intelligence wherein predictions (X% probability that Y will happen) and
rationales (why I believe it is this probability that X will happen) are
elicited independently from a diverse crowd, aggregated, and then used to
inform higher-level decision-making. This research asks whether ACF, as a key
way to enable Operational Collective Intelligence, could be brought to bear on
operational scenarios (i.e., sequences of events with defined agents,
components, and interactions) and decision-making, and considers whether such
a capability could provide novel operational capabilities to enable new forms
of decision-advantage.

###### Keywords:

First keyword Second keyword Another keyword.

##  1 Introduction

Collective Intelligence (CI) emerges from new ways of connecting humans and
machines to enable decision-advantage, at least in part, by creating and
leveraging additional sources of information that decision-makers might
otherwise not include [1, 7, 40]. Operational scenarios present a unique
challenge for typical collective intelligence approaches. The wisdom of the
crowd, the archetypal collective intelligence, works in classic applications,
like predicting the weight of a cow, because the probability distributions of
responses in these instances have median estimates centered near the ground
truth, e.g., the weight of the cow [15, 53]. Typical operational scenarios
necessitate a different approach because the “crowd” is often heavily biased.
For example, in a military setting operators at various levels are privileged
to unique knowledge, some of which may alter their predictions in an
adversarial reasoning scenario. Moreover, operational scenarios increasingly
rely on input from machine intelligence, which decision-makers must integrate
with human judgments. Aggregative crowdsourced forecasting (ACF) is a recent,
pivotal advancement in CI methods. ACF helps decision-makers from commercial
companies, government organizations, and militaries overcome the intractable
problem of individual bias by collecting many predictions and integrating the
best of machine intelligence into the decision process. To date, ACF has been
used for risk management at a strategic level, focusing on tasks like
informing policy, anticipating instability, balancing research portfolios,
identifying emerging technology, and serving as an early warning mechanism for
decision-making.

This document provides a preliminary exploration and evaluation of the
potential use of aggregative crowdsourced forecasting as a mechanism to help
operationalize “collective intelligence” for operational scenarios. After
operationalizing pertinent terminology and laying out the critical research
questions, we identify gaps in the research and propose ways of empirically
testing solutions to the open questions.

##  2 Background

The insight that a group of forecasters may be more reliable than a single
individual, colloquially known as the wisdom of a crowd, is far from new. It
is in Aristotle’s Politics [30] and was famously championed by the British
polymath Sir Francis Galton. (Both use the accuracy achieved by statistical
aggregation of forecasters as motivation for democracy; Aristotle waxed poetic
about art judgments while Galton, more practically, used the statistical
accuracy of a crowd’s guesses of the dressed weight of a cow to argue that
popular judgments are valuable to societies [15].) Modern technology has
undeniably changed the dynamics of such crowd wisdom. Although digital
technology facilitates aggregation of and eases access to crowd wisdom [27],
it can also undermine it because crowd wisdom hinges on natural variation in
the forecasters’ information. Individual forecasters who use digital
technology to change their knowledge of other forecasters’ knowledge may
become biased and reflect their new knowledge in their forecasts, ultimately
undermining the crowd’s wisdom [37]. Studying the process of crowd decision-
making and how the crowd’s wisdom emerges may facilitate mitigation, if not
reversal, of digital technology’s undermining effects on the crowd wisdom [32,
52].

###  2.1 Collective Intelligence

Collective intelligence (CI) is a “form of universally distributed
intelligence, constantly enhanced, coordinated in real-time, and resulting in
the effective mobilization of skills” [32]. We are primarily interested in
collective intelligence that emerges from new ways of connecting humans and AI
to enable decision advantage in instances of adversarial reasoning. Decision
advantages are made possible by connections that allow the collective to
create and leverage information sources that decision-makers might otherwise
not include in a decision process. Adversarial reasoning involves “determining
the states, intents, and actions of one’s adversary, in an environment where
one strives to effectively counter the adversary’s actions.” [28]

Suran, Pattanaik, and Draheim introduce a convenient, generic framework for
conceptualizing CI [52]. They posit that a CI is composed of staff with a goal
they are motivated to achieve through some definable process. The “crowd” in
this instance is the active staff members who contribute to the CI (some staff
in a CI may be passive, e.g., beneficiaries). The CI goal is simply a desired
outcome—for example, anticipating an opponent’s strategy or recognizing
deceptive tactics. Although it is common to think of the goal as being
community-driven, individuals may also participate in a CI to achieve their
own unique goals: A software engineer may commit updates to an open-source
repository to improve its utility for their own project but in so doing
advance the collective goal of having a robust, stable resource for a
community of users. Since the authors excluded AI systems, motivation is
simply the why behind contributions to the collective, be it intrinsic (e.g.,
a passion for the cause) or extrinsic (e.g., money). Lastly, the definable
process of a CI describes the staff’s interactions in pursuit of the
goal—think of the process as the how that describes the way in which a CI
achieves intelligence.

Each aspect of the Suran, Pattanaik, and Draheim framework builds upon
extensive research—they document more than 9,000 publications in the space
[52]. We are interested in the process—the how—of CI. Unsurprisingly, many
publications in this space develop, review, and refine different CI processes.
The project they lean on to develop the process aspect of their generalized
framework looked at argument mapping tools, which diagram evidence for and
against a cause, to improve online collective efforts [24]. Proper deployment
of these decision support systems can improve the process of CI (i.e., lead to
better outcomes) by helping staff understand the interactions that transpire
during collective decision-making. Of course, decision support systems are
continually improving—machine learning and artificial intelligence
technologies are dramatically enhancing their capabilities. Contemporary
technologies facilitate mapping interactions and enhancing the information
generated by those interactions [42]. Such technologies are paving the way for
operational collective intelligence.

###  2.2 Aggregative Crowdsourced Forecasting

We define aggregative crowdsourced forecasting (ACF) as a type of forecasting
wherein predictions (X% probability that Y will happen) and rationales (why I
believe it is this probability that X will happen) are elicited independently
from a diverse crowd and aggregated into a single estimate for informing
higher-level decision-making. Aggregation is carried out via machine learning
models capable of identifying unique data features that may lead to biased
forecasts, accounting for these features, and adjusting the weight of
individual forecasts to produce a more reliable crowdsourced estimate.

Conceptualized in this way, ACF is a type of intelligence that plays on the
relative strengths of each forecaster to achieve otherwise unobtainable
outcomes [2, 11, 47]. Key features of this type of intelligence include humans
and machines working together towards a collective goal, both agent types
continually learning and improving, and the ability to achieve superior
solutions than either type alone could realize [11]. A core strength of ACF is
the machine models and human experts bringing unique skills to bear on their
collective goal. The statistical models underpinning machine intelligence are
able to accurately and efficiently identify patterns in structured data, which
empowers them to make predictions that may avoid human biases. On the other
hand, human experts are not reliant on structured data, which empowers them to
see features in data that may go unnoticed by machine models. Together, the
unique skills of machines and humans allow such an ACF to achieve superior
performance.

##  3 Operational Collective Intelligence

Military, business, and other structured organizations rely on definable
common occurrences, known as operational scenarios, to manage people,
resources, and processes. Operational scenarios describe sequences of events
using a set of defined agents, components, and interactions. Importantly, the
agents, whether human or intelligent machines, that eventually carry out
organizational processes based on operational scenarios are often privy to
unique information that may bias their decision making and behavior. This
feature can undermine the strengths of typical approaches to collective
intelligence. Consider a military scenario in which field operators are
intimately aware of situational factors that command personnel cannot observe
while command personnel are privy to classified mission details that they
cannot share with operators. Applying ACF, which is able to play on the
strengths and weaknesses of different agents, to such operational scenarios
may enable Operational Collective Intelligence (OCI).

###  3.1 Research Questions

Addressing several key, basic research questions will help us assess the
viability of OCI, the main questions being whether and to what degree ACF has
the potential for substantive operational impact. One of the most refined
examples of an ACF is the Synergistic Anticipation of Geopolitical Events
(SAGE) system [2], which was developed as part of the IARPA Hybrid Forecasting
Competition [17]. SAGE is a hybrid forecasting system that gives forecasters
the ability to combine their own judgments with model-based solutions. Its
development emphasized making verifiable probabilistic predictions while being
domain agnostic. Importantly, empirical results show that SAGE can result in
improved aggregate forecasts. One insight is that forecasters needed a base
level of expertise to make the most of the inputs provided by SAGE. This
observation reinforces previous findings related to crowd-based forecasts [1].
A unique feature of SAGE is its ability to facilitate qualitative (e.g., will
a given candidate win a national election) and quantitative questions (e.g.,
what percent of the vote will each candidate receive).

Machine-aided Analytic Triage with Intelligent Crowd Sourcing (MATRICS) is
also a system developed under the IARPA Hybrid Forecasting Competition [23].
Like SAGE, MATRICS combines human and machine predictions to generate a hybrid
forecast. It similarly provides human forecasters guidance from a machine
intelligence intended to improve their judgments before a final aggregation
stage that mixes human and machine forecasts. MATRICS and SAGE differ from
other machine-human hybrids in that they explicitly work to integrate wisdom-
of-the-crowd models in their computation. Although neither system was
explicitly developed as a solution for operational scenarios, the success of
both suggests that they could be developed into relevant operational tools.

An earlier system, CrowdSynth, also sought to leverage the relative strengths
of human and machine intelligence by having an AI decide how to combine
multiple contributions [26]. CrowdSynth was able to optimize engagement with
citizen scientists who contributed to a constellation classification task. It
accomplished this by balancing expected votes from contributors with a model
of what it believed would be the ultimate consensus classification. The
interactive intelligence of MATRICS and SAGE differentiates them from
CrowdSynth—they not only consider the wisdom of the crowd, but work directly
with it to improve the crowd’s performance. Although CrowdSynth was able to
reduce the number of contributors needed for an accurate classification, it
did not have this collective feature.

If we step away from the aggregative, interactive features of SAGE and
MATRICS, there are numerous hybrid intelligence systems [11]. For example,
Dong and coauthors developed a hybrid intelligence algorithm for E-commerce
sales volume forecasting [13]. Their simulations suggest that access to the AI
system is not always helpful in a purely collaborative scenario—particularly
when the humans do not trust the system. Interestingly, more accurate
forecasting occurred when the AI was an opponent in a competitive scenario.
Another example from Wu et al. uses models of human cognition and examples
from human designers, rather than actual humans, to aid an automotive design
AI system [58]. They demonstrate that the system is superior to a baseline
model that lacks the human model plus knowledge integration and argue that it
is capable of exceptional creativity. In yet another example, Russakovsky, Li,
and Fei-Fei developed a model that combined multiple algorithmic and human
annotations of images to improve overall classification [49]. Although these
systems have apparent utility in operational scenarios, including those that
require adversarial reasoning, they are limited in that they do not aggregate
and/or account for the knowledge, experience, and seniority of human staff.
This core feature of ACF is what we believe will set it apart as an
operational collective intelligence. However, even though experimentally
validated models exist in other domains, there are still pertinent questions
to raise about the viability of ACF for operational scenarios.

####  3.1.1 What are the determinants of ACF’s practicality for operational
scenarios?

Aggregative Crowdsourced Forecasting is a data-centric process. The limited
amount of ACF research makes formal estimation of data requirements untenable,
however, related research points to some guidelines. Training the core machine
learning models not only requires more data than is typical for other
statistical models, but the data also need to be easy to access, well-
structured, and relatively errata free [3, 25]. Each element needed to
construct a functional ACF will have these requirements. It follows that
access to high-quality, relevant training data is a key determinant of
successfully deploying ACF in an operational scenario. The very nature of
adversarial scenarios means that troves of valuable data are collected and
maintained. For example, in a military command and control operational
scenario, an effective ACF model will rely on data including features such as
the opponent’s plans for and perceptions of friendly forces; the opponent’s
doctrine, tactics, and training; environmental constraints from terrain to
weather; capital resources such as ammunition, artillery, communications,
personnel, etc.

Access to high-quality data is only half of the modeling task—the modeling
technology needs to match the challenge as well. Modeling technology for
crowd-sourced forecasting has only recently become broadly viable [21]. In
geopolitical modeling, for example, robust forecasting models emerged roughly
when access to big data and high-powered computing were democratized [31].
Additionally, even though combining forecasts from different sources, such as
human and machine intelligence, is not new, the necessary computational power
and algorithm technologies for large-scale modeling are fairly recent
developments [56]. The cost of developing, maintaining, and deploying such
models means that early ACF models will most likely be applied in settings
where the risk of failure and cost of failure are uniquely high.

The need for extant, reliable data and established, state-of-the-art forecast
modeling currently limits the operational scenarios for which ACF is a viable
instrument. In the cases of SAGE [2] and MATRICS [23], the data were
determined and provided by the sponsor of the competition (IARPA [17] plus
they were adapted for a well-studied forecasting domain–geopolitical events.
Pursuing additional work in this area would undoubtedly allow for model
improvement. However, working with new data in other domains may foster
faster, more reliable development [25]. One domain with well-defined, studied,
and documented operational scenarios is military decision-making [54]. And
even though much of military operational data is not publicly available, many
military organizations allow researchers to publish appropriately cleaned data
to gain access to state-of-the-art machine learning [14], meaning it may prove
a viable domain for developing OCI. Other potential domains exist, such as
business intelligence, but access issues commonly fetter these data too [44,
45].

####  3.1.2 What are the key human factors of a successful OCI?

The goal of operational collective intelligence is to provide ultimate
decision-makers with guiding input based on the best of human and machine
intelligence. An important aspect of the how of OCI is the human staff’s
response to the machine intelligence. Historically, people preferred human
over algorithmic judgment (i.e., machine or artificial intelligence) [10, 12].
Whenever and wherever this tendency still exists, it may challenge OCI.
Fortunately, algorithm aversion is not a foregone conclusion of human
behavior—increasingly, researchers are observing instances of the opposite,
that is, people seeking out algorithmic input [36]. Unfortunately, people
occasionally put too much trust in algorithmic decision aids [43, 55], which
may also challenge OCI. Thus, a pivotal human factor of a successful OCI is
well-calibrated trust in its machine intelligence. A well-calibrated decision-
maker will know when to heed and ignore an OCI’s input. From a design
perspective, this reality suggests a need for implementing tools to measure
and predict trust in the OCI [18, 19] along with ways to calibrate that trust.
The trust calibration literature is quite broad, but some obvious, simple
steps include engineering the messages such that they guide users to the
appropriate level of trust [55] and providing interpretable explanations of
the system’s output [46, 59].

Extensive evidence across a variety of domains supports the observation that a
crowd can outperform experts [15, 30, 53]. Nevertheless, expertise still
matters: Clever aggregation that weighs individual contributors’ forecast
relative to their knowledge can yield even wiser crowds [7, 6, 39, 57]. For
example, in an image annotation setting where multiple annotators labeled
images, modeling individual annotators as multidimensional entities with
competence, expertise, and bias variables resulted in better aggregation of
their opinions [57]. A similar result was observed in studies that asked
participants to forecast current events and economic outcomes [6].

Situational factors, however, may undermine the effect of individual
expertise. One study asked participants to estimate corporate earnings and
found that having access to public information resulted in participants under-
weighting their own private information [9]. This behavior resulted in
individual estimates that were more accurate but crowd estimates that were
less accurate—a result that underscores how important aggregation that
accounts for human factors is to an OCI.

####  3.1.3 What are the key machine intelligence factors of a successful OCI?

The machine intelligence (i.e., algorithmic) factors of a successful OCI are
subject to the same limitations as machine learning more generally: they
perform best on well-defined tasks [8], when the environment is simple and
stable [4], and when high-quality training data are available [48].
Operational scenarios are driven by the need for predictable, defined
protocols for common occurrences, meaning that the task is generally well-
defined for the machine intelligence that underpins ACF systems. However, one
factor that could undermine them could also make OCI invaluable: the ability
to handle exceptional task cases. Thus, a critical stage in developing OCI is
engineering pathways into the ACF system for handling exceptions. One solution
is to empower the ACF model with a classification tool for detecting
exceptions [20] and passing exceptional cases off to human staff, who are
typically better at handling them [2].

Using operational scenarios is most common in stable, i.e., predictable,
environments. However, stable and predictable does not necessarily mean an
operational scenario is simple. Many operational scenarios exist as responses
to complicated environments in which simplification can improve a given
operation. Machine intelligence, at least historically, is not considered an
applicable solution for tasks in complex settings, although given sufficient
data and training resources, models are able to generalize their learning
(e.g., [41]). Innovative modeling approaches in the space of neural-symbolic
reasoning, however, may prove a solution for ACF in complex environments [16].

Access to high-quality, operation-relevant data is critical for training an
ACF model. Data quality is simply the degree to which archived training data
reflects the real world [22]. Moreover, a training data set’s relevance is
contingent on a given organization and the operational setting. For example,
network intrusion detection is complicated by the fact that operational data
from one network setting is rarely relevant to another, which creates a lack
of training data [51]. Although network data from other operational settings
could be used to train a model, it likely will not reflect “the real world” in
new network security settings, meaning it is poor-quality data. The impact of
poor data quality on a given model is contingent on the type of model and the
task at hand [5]—if the cost of errors is low, then poor quality data may be a
reasonable tool for getting a model established to facilitate collection of
more relevant data [51]. In the case of ACF, data quality may be the result of
how the crowd was selected [34, 33]. This suggests that an OCI may benefit
from tools that allow it to select which training data are used for a given
operational scenario.

ACF models are subject to the same generalization challenges as other machine
learning models. The models underpinning ACF typically adopt the classic
statistical assumption of independent and identically distributed data
(i.i.d.)—a very strong assumption that is made necessary by the intractable
complexity of its absence. Out-of-Distribution (OOD) data, i.e., real-world
data, can prove a significant challenge for these models. ML models for OOD
data are still in their nascent phase, but improving [35], including in
forecasting applications [50]. Models that are able to handle OOD data that do
not abide by the i.i.d. assumption will also prove a key MI factor of
successful OCI.

###  3.2 Potential Operational Collective Intelligence Impact(s)

Research already documents many impacts of human-machine collective
intelligence. For example, one application in a healthcare setting
demonstrated that significant error reduction is possible by comparing single
medical diagnosticians and a CI system—the diagnosticians achieved 46%
accuracy versus the system’s 76% [29]. A famous, but perhaps underappreciated,
example of collective intelligence is Google. It relies on judgments and
decisions from people worldwide to refine its representations of content on
the web and improve algorithmic recommendations [38]. The general impact of
Google on individual decision-makers, collectives, and society at large is
undeniable (and likely impossible to estimate). Similarly, enumerating the
unique ways in which it has changed the course of human history is probably
impossible.

Military, business, and other organizations define operational scenarios to
improve operational outcomes. Outcome improvements result from using
operational scenarios during training and simulation settings, which aids
staff in learning what decision features to focus on, and using them during
actual operations, which may aid staff in identifying anomalous features of an
operation. We envision OCI improving organizations’ decision accuracy by
improving their ability to listen to the most knowledgeable staff, adapt to
unpredictable scenarios by recognizing situations in which machine
intelligence is underspecified, and increase decision speed by leveraging the
system’s ability to process new data relative to human staff alone. In an
adversarial reasoning scenario, whether in military, business, or political
settings, such an increase in decision time could allow an organization to
gain a competitive advantage.

##  4 Discussion

Research documents impressive outcomes of combining human and machine
intelligence, i.e., Collective Intelligence, to solve complex operational
problems[52]. Significant advances in machine learning have facilitated
aggregative crowdsourced forecasting, a CI capable of taking the best input
from individual staff members of a crowd and intelligently guiding decisions
[2, 42]. Applying this technology to operational scenarios, we believe, is a
way to realize Operational Collective Intelligence.

###  4.1 Advantages

OCI’s main advantage lies in its ability to integrate human and machine
intelligence while applying lessons learned from crowd wisdom quickly and
accurately. Human decision-makers are simply not capable of considering troves
of data or agnostically weighing input from a large number of forecasters in
the same fashion. For example, a robust OCI could consider seasonal variations
in the weather, such variations’ impacts on operators’ well-being, and the
cost of delays without referring to outside resources in minutes if not
seconds. It is not unreasonable for a human decision-maker to rely on a cadre
of workers for each sub-prediction, which might require hours or days, and
then when they integrate the various outputs, allow their own biased judgment
to impact how they weigh the input from different forecasting teams.

###  4.2 Practicality

The key to OCI’s practicality is data. The lock is well-designed machine
intelligence. Without these, OCI may prove more of a hindrance than a benefit.
ACF models are only as good as the data used to train them. Current state-of-
the-art ACF models require more data than is typical for other statistical
models, underscoring the importance of having ready access to well-structured,
errata-free data [3, 25]. Without such data, OCI is not practical.
Additionally, although considerable effort has already gone into designing and
validating ACF models, such as what is reported in [2], much work is still
needed to ensure the practicality of an OCI that hinges on ACF.

The staff interacting with an OCI and their responses to it are of similar
importance. Despite historical evidence of people being averse to algorithmic
input[12], there is increasing evidence that people not only support but seek
out algorithmic aid [36]. With this increased willingness to rely on
algorithmic decision aids like OCI comes an increased need to ensure that the
human staff of an operation is well-calibrated to its use [46, 55].

##  5 Conclusion

Aggregative crowdsourced forecasting helps decision-makers overcome
intractable forecasting problems, such as individual biases, by collecting
many predictions and integrating the best of machine intelligence into the
decision process. Operational scenarios are an obvious application of this
recent advancement in collective intelligence, but translating existing ACF
models for operational scenario application is not straightforward. Data,
modeling, and human factors make this translation a challenge. ACF is a data-
centric process in which each element requires readily available data that are
well-structured and relatively errata-free. Fortunately, there are many
operational applications, such as adversarial scenarios, in which troves of
such data are collected. ACF also hinges on state-of-the-art forecasting
models that require considerable computational resources. Beyond the data and
computational requirements are the human factors of operational scenarios,
from how the human staff responds to machine intelligence to their relative
expertise. Addressing these challenges will enable ACF to unlock operational
collective intelligence, which we believe will prove a pivotal advancement in
the coordinated action of human-machine teams.

####  5.0.1 Acknowledgements

The project or effort depicted was or is sponsored by the U.S. Government
under contract number W911NF-14-D-0005. The content of the information does
not necessarily reflect the position or the policy of the Government, and no
official endorsement should be inferred.

####  5.0.2

The authors have no competing interests to declare that are relevant to the
content of this article.

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[1]$V_{#1}^{*}$

# Theoretical point of view on Cabibbo angle anomaly

Teppei Kitahara  Department of Physics, Graduate School of Science, Chiba
University, Chiba 263-8522, Japan
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
<EMAIL_ADDRESS>
###### Abstract

We present the current situation of the determinations of the first-row CKM
components and show the Cabibbo angle anomaly corresponding to a deficit in
the first-row CKM unitarity condition at the $3\sigma$ level. In this
contribution, we show two new physics interpretations: heavy vector-like quark
models and a MeV scale sterile neutrino models. The super tau-charm facility
will directly probe the other CKM unitarity conditions related to $V_{cd}$.

###### keywords:

CKM matrix; electroweak interactions; sterile neutrino.

Report number: CHIBA-EP-265

##  1 Introduction

00footnotetext: Presented at the 2024 International Workshop on Future Tau
Charm Facilities (FTCF2024), Hefei, China, 14-18 January, 2024. The original
talk title is “Anomalies in the charm-strange sector.”

Precision experiments of flavor physics are becoming increasingly important in
indirectly discovering physics beyond the Standard Model (SM). In particular,
there is the so-called Cabibbo angle anomaly (CAA) [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11] with a significance of currently around the $3\sigma$ level [12, 13,
14, 15]. The CAA consists of two tensions related to the determination of the
Cabibbo angle: First, the different determinations of $|V_{us}|$ from $K_{\mu
2}$, $K_{\ell 3}$, and $\tau$ decays disagree at the $3\sigma$ level. Second,
using the average of these results in combination with $\beta$ decays, a
deficit in first-row Cabibbo-Kobayashi-Maskawa (CKM) unitarity appears with
significance at the $3\sigma$ level. Intriguingly, it is known that both
discrepancies could be explained via a modified $W$ coupling to quarks.

Importantly, due to $SU(2)_{L}$ invariance, such a modified $W$ coupling to
quarks in general leads to modified $Z$-quark-quark couplings as well, that
enter electroweak precision observables (EWPO), affect low-energy parity
violation and can give contributions to the flavor-changing-neutral-current
(FCNC) processes. Therefore, a global fit is required to consistently assess
the agreement of a specific New Physics (NP) scenario with data. The necessity
of such a combined analysis becomes even more obvious when considering a UV
complete model that can generate modified $W$ couplings to quarks.

We will also discuss a light new physics interpretation of the CAA by
considering the sterile neutrino models.

##  2 Current status of Cabibbo angle anomaly

Here, we briefly summarize the current situations of the $|V_{ud}|$ and
$|V_{us}|$ determinations shown in Fig. 1.

Figure 1: (Left) Summary of the determinations of $|V_{ud}|$ from the various
types of $\beta$ decays [15]. The red band represents the world average in Eq.
(1). The details of the extractions from neutron decay (best) and the super-
allowed $\beta$ decays (reduced uncertainty) are given in the main text. Note
that the pion $\beta$ decay, even though it is currently not competitive, is
theoretically clean and will be strikingly improved by the PIONEER experiment
[16]. (Right) Summary of the determinations of $|V_{us}|$ from various
processes [15]. The global fit value of $|V_{us}|$ is obtained in Eq. (5).

First, the CKM element $|V_{ud}|$ can be determined from various types of
$\beta$ decays. The latest determinations are $|V_{ud}|_{0^{+}\to
0^{+}}=0.97367(32)$ from the super-allowed $0^{+}\to 0^{+}$ nuclear $\beta$
decay [17, 13], $|V_{ud}|_{n{\rm(PDG)}}=0.97441(88)$ from the neutron decay
[18], $|V_{ud}|_{\rm mirror}=0.9739(10)$ from $\beta$ transitions of the
mirror nuclei [19], and $|V_{ud}|_{\pi_{e3}}=0.9739(29)$ from the pion $\beta$
decay ($\pi^{+}\to\pi^{0}e^{+}\nu;~{}\pi_{e3}$) [20, 13]. In these
determinations, we use an estimation of Ref. [13] for universal nuclear-
independent radiative corrections from $\gamma W$-box diagrams
$\Delta^{V}_{R}$ [21].***A new lattice calculation derives
$\Delta_{R}^{V}=0.02439(19)$ [22]. If we adopt this result for the global fit
of the super-allowed nuclear $\beta$ decays, significance of the tension is
reduced by about 0.5$\sigma$. For the neutron decay, it is known that the
uncertainty of $|V_{ud}|_{n{\rm(PDG)}}$ is inflated by scale factors that come
from inconsistencies in the data. By using the single most precise result for
the neutron lifetime $\tau_{n}$ [23] and the nucleon isovector axial charge
$g_{A}/g_{V}$ [24], a better determination of
$|V_{ud}|_{n{\rm(best)}}=0.97413(43)$ is possible [13]. Combining
$|V_{ud}|_{0^{+}\to 0^{+}}$, $|V_{ud}|_{n{\rm(best)}}$, $|V_{ud}|_{\rm
mirror}$, and $|V_{ud}|_{\pi_{e3}}$, we obtain a weighted average of

$|V_{ud}|_{\beta}=0.973\,84(25)\,.$ (1)

Note that the $|V_{ud}|$ determination is predominated by super-allowed
$\beta$ decays where the largest uncertainty comes from nuclear-structure (NS)
dependent radiative corrections [25], encoded in $\delta_{{\rm NS},E}$ in Ref.
[17]. Unfortunately, precise estimations of $\delta_{{\rm NS},E}$ are
difficult [26] but the current value is considered to be very conservative
[25]. Recent progress for the NS-dependent radiative corrections has been made
in Refs. [27, 28, 29].

Next, the matrix element $|V_{us}|$ can be determined from semi-leptonic
decays of kaons and hyperons and from inclusive hadronic $\tau$ decays. By
comparing theoretical predictions with data of the semi-leptonic kaon decays
$K_{S,L}\to\pi^{-}\ell^{+}\,\nu$ and $K^{+}\to\pi^{0}\ell^{+}\,\nu$ with
$\ell=e,\mu$ (labeled $K_{\ell 3}$), one can obtain [13], $|V_{us}|_{K_{\ell
3}}=0.223\,30(53),$ where the latest evaluations of the long-distance
electromagnetic (EM) correction [30, 31, 32, 33], the strong isospin-breaking
correction [13], and the recent $K_{S}$ data from the KLOE-2 collaboration
[34, 35] are used. Beyond kaons, one can also use the hyperon semi-leptonic
decays, $(\Lambda\to p,\Sigma\to
n,\Xi\to\Lambda,\Xi\to\Sigma)\,\ell\,\overline{\nu}$, which however lead to a
slightly different yet less precise value $|V_{us}|_{\rm hyperon}=0.2250(27)$
[18]. Inclusive hadronic $\tau$ decays also provide an opportunity to extract
the matrix element $|V_{us}|$ by separating the strange and non-strange
hadronic states. Two representative determinations are reported:
$|V_{us}|_{\rm HFLAV}=0.2184(21)$ [36, 37, 38] and $|V_{us}|_{\rm
OPE+lattice}=0.2212(23)$ [39, 40]. The former is based on the conventional
operator product expansion (OPE) using the vacuum saturation approximation
[41], while the latter is based on improved OPE series by fitting the lattice
result [42]. Although they almost agree, there is no common consensus on which
value, $|V_{us}|_{\rm HFLAV}$ or $|V_{us}|_{\rm lattice}$, to use.
Accordingly, we perform a weighted average of the two values
$|V_{us}|_{\tau_{X_{s}}}=0.2195(21).$ By using these $|V_{us}|$
determinations, we obtain a weighted average of $|V_{us}|_{K_{\ell 3}}$,
$|V_{us}|_{\rm hyperon}$, and $|V_{us}|_{\tau_{X_{s}}}$,

$\displaystyle|V_{us}|_{K,\tau,\Lambda}=0.223\,14(51)\,.$ (2)

We summarize the determinations of $|V_{us}|$ from various observables in the
right panel of Fig. 1. There, the blue band represents the global fit of
$|V_{us}|$ in Eq. (5) in which the CKM unitarity condition is not included. It
is shown that $|V_{us}|_{\tau_{X_{s}}}$ (orange bar) is a little bit smaller
than the other determinations; $3.3\sigma,\,2.6\sigma,\,1.8\sigma$
discrepancies by comparing to $\beta$ decays with unitarity (magenta), $K_{\mu
2}/\pi_{\mu 2}$ with $\beta$ decays (brown), and $K_{\ell 3}$ (green),
respectively.

Third, the ratio $|V_{us}/V_{ud}|$ can be extracted from the several ratios of
leptonic decay rates of kaon, pion and $\tau$ leptons. The leptonic kaon-decay
rate over the pion one, $K_{\mu 2}/\pi_{\mu
2}=\Gamma(K^{+}\to\mu^{+}\nu)/\Gamma(\pi^{+}\to\mu^{+}\nu)$ provides
$\left|V_{us}/V_{ud}\right|_{K_{\mu 2}/\pi_{\mu 2}}=0.231\,08(51)$ [43], where
the latest evaluation of the long-distance EM and strong isospin-breaking
corrections [44, 45] is used. Furthermore, the exclusive $\tau$-decay ratio
$\Gamma(\tau\to K^{-}\nu)/\Gamma(\tau\to\pi^{-}\nu)$ (labelled by
$\tau_{K}/\tau_{\pi}$) provides
$\left|V_{us}/V_{ud}\right|_{\tau_{K}/\tau_{\pi}}=0.2293(15)$ [46]. In
addition, it is recently pointed out in Ref. [47] that the semi-leptonic kaon-
decay rate over the pion $\beta$ decay,
$\Gamma(K\to\pi\ell\nu)/\Gamma(\pi^{+}\to\pi^{0}e^{+}\nu)$ (labelled by
$K_{\ell 3}/\pi_{e3}$), provides $\left|V_{us}/V_{ud}\right|_{K_{\ell
3}/\pi_{e3}}=0.229\,08(88)$ [32]. Again, we obtain a weighted average of
$|V_{us}/V_{ud}|_{K_{\mu 2}/\pi_{\mu 2}}$,
$|V_{us}/V_{ud}|_{\tau_{K}/\tau_{\pi}}$ and $|V_{us}/V_{ud}|_{K_{\ell
3}/\pi_{e3}}$,

$\displaystyle\left|\frac{V_{us}}{V_{ud}}\right|_{\rm
ratios}=0.230\,47(43)\,.$ (3)

Finally, we perform a global analysis within the SM. In Fig 2, the global fit
result including $|V_{ud}|_{\beta}$, $|V_{us}|_{K,\tau,\Lambda}$ and
$|V_{us}/V_{ud}|_{\rm ratios}$ is shown by the blue circles. In the left
panel, only $\beta$ decays, $K_{\ell 3}$, $K_{\mu 2}/\pi_{\mu 2}$ and $K_{\ell
3}/\pi_{e3}$ are displayed (but all data are included in the global fit),
while the right panel shows all the data. The black line stands for the
unitarity condition: $|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1$ with
$|V_{ub}|\approx 0.00377$ [48]. The blue shaded circle corresponds to
$\Delta\chi^{2}\leq 1$, while the dashed circle is $\Delta\chi^{2}=2.3$. In
the $\chi^{2}$ analysis, we included a correlation between $K_{\ell 3}$ and
$K_{\ell 3}/\pi_{e3}$ because they share the same kaon data and common form
factor $f_{+}^{K^{0}\to\pi^{-}}(0)$. We set $100\%$ correlation for these
common uncertainties.

Figure 2: (Left) Global fit of all the available CKM determinations with
$\Delta\chi^{2}=1$ (blue shaded) and $\Delta\chi^{2}=2.3$ (dashed circle)
[15]. Only the $1\sigma$ regions from $\beta$ decays, $K_{\ell 3}$, $K_{\mu
2}/\pi_{\mu 2}$ and $K_{\ell 3}/\pi_{e3}$ observables are shown. The black
line represents the unitarity condition. (Right) The $1\sigma$ regions of
$|V_{us}|_{\rm hyperon}$, $|V_{us}|_{\tau_{X_{s}}}$ and
$|V_{us}/V_{ud}|_{\tau_{K}/\tau_{\pi}}$ observables are also shown [15].

Our global fit results are

$\displaystyle|V_{ud}|_{\text{global}}$ $\displaystyle=0.973\,79(25)\,,$ (4)
$\displaystyle|V_{us}|_{\text{global}}$ $\displaystyle=0.224\,05(35)\,,$ (5)

and

$\displaystyle\Delta_{\text{CKM}}^{\text{global}}$
$\displaystyle\equiv|V_{ud}|^{2}_{\text{global}}+|V_{us}|^{2}_{\text{global}}+|V_{ub}|^{2}-1=-0.001\,51(53)\,,$
(6)

with a $|V_{ud}|_{\text{global}}$–$|V_{us}|_{\text{global}}$ correlation of
$0.09$. This $\Delta_{\text{CKM}}^{\text{global}}$ implies $-2.8\sigma$
deviation from the unitarity condition of the CKM matrix.

One should note that the value of $\Delta_{\text{CKM}}^{\text{global}}$ is
also sensitive to the data of the neutron lifetime. The above result is based
on the bottle ultracold neutron (UCN) lifetime data $\tau_{n}^{\rm
bottle}=877.75(36)$ sec [23]. On the other hand, if instead the in-beam
neutron lifetime data $\tau_{n}^{\rm beam}=887.7(2.2)$ sec [49] is used,
$\Delta_{\text{CKM}}^{\text{global}}$ is slightly changed [50]

$\displaystyle\Delta_{\text{CKM}}^{\text{global}}$
$\displaystyle=-0.002\,34(62)\,,$ (7)

which implies $-3.8\sigma$ deviation from the unitarity condition.

Before closing this section, we give the status of the other unitarity tests:

$\displaystyle\Delta_{\rm CKM}^{1^{\rm st}{\rm column}}$
$\displaystyle\equiv|V_{ud}|^{2}+|V_{cd}|^{2}+|V_{td}|^{2}-1=-0.0028(18)\,,$
(8) $\displaystyle\Delta_{\rm CKM}^{2^{\rm nd}{\rm row}}$
$\displaystyle\equiv|V_{cd}|^{2}+|V_{cs}|^{2}+|V_{cb}|^{2}-1=0.0002^{+15}_{-13}\,,$
(9) $\displaystyle\Delta_{\rm CKM}^{1^{\rm st}\times 2^{\rm nd}{\rm row}}$
$\displaystyle\simeq-|V_{ud}||V_{cd}|+|V_{us}||V_{cs}|=0.003(4)\,.$ (10)

These quantities are still consistent with the CKM unitarity. The above
uncertainties are dominated by $|V_{cd}|$, which is dominated by
$D^{+}\to\mu^{+}\nu$ measurement [51]. The super tau-charm facility (STCF)
will provide $\mathcal{O}(0.1)\%$ statistical accuracy for the $D$-meson
decays, which reduce the uncertainties of $|V_{cd}|$ and $|V_{cs}|$.
Therefore, it is expected that the STCF can probe these CKM unitarity tests.

##  3 SMEFT fittings

Here, we briefly summarize a situation of the Standard Model Effective Field
Theory (SMEFT) interpretation of the CAA in the previous section.

We write the SMEFT Lagrangian as

$\displaystyle\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\rm
SM}+\sum_{i}C_{i}Q_{i}\,,$ (11)

such that the SMEFT coefficients have dimensions of inverse mass squared. We
use the Warsaw basis [52], as well as the corresponding conventions, in which
the operators generating modified gauge-boson couplings to quarks are given by

$\displaystyle\begin{aligned}
Q_{Hq}^{(1)ij}&=(H^{\dagger}i\overset{\leftrightarrow}{D_{\mu}}H)(\bar{q}_{i}\gamma^{\mu}P_{L}q_{j})\,,&Q_{Hq}^{(3)ij}&=(H^{\dagger}i\overset{\leftrightarrow}{D_{\mu}^{I}}H)(\bar{q}_{i}\tau^{I}\gamma^{\mu}P_{L}q_{j})\,,\\\
Q_{Hu}^{ij}&=(H^{\dagger}i\overset{\leftrightarrow}{D_{\mu}}H)(\bar{u}_{i}\gamma^{\mu}P_{R}u_{j})\,,&Q_{Hd}^{ij}&=(H^{\dagger}i\overset{\leftrightarrow}{D_{\mu}}H)(\bar{d}_{i}\gamma^{\mu}P_{R}d_{j})\,,\\\
Q_{Hud}^{ij}&=i(\tilde{H}^{\dagger}D_{\mu}H)(\bar{u}_{i}\gamma^{\mu}P_{R}d_{j})\,.&&\end{aligned}$
(12)

We work in the down-basis such that CKM elements appear in transitions
involving left-handed up-type quarks after the EW symmetry breaking. This
means we write the left-handed quarks doublet as
$q^{T}_{i}=\begin{pmatrix}(V^{\dagger}u_{L})_{i}&d_{L,i}\end{pmatrix}$, where
$V$ is the CKM matrix. With this conventions, the modified $W$ and $Z$
couplings are given by

$\mathcal{L}_{W,Z}=\begin{aligned}
&-\frac{g_{2}}{\sqrt{2}}W^{+}_{\mu}\,\bar{u}_{i}\gamma^{\mu}\left(\left[V\cdot\left(\mathbbm{1}+v^{2}C^{(3)}_{Hq}\right)\right]_{ij}P_{L}+\frac{v^{2}}{2}\left[C_{Hud}\right]_{ij}P_{R}\right)d_{j}+\mathrm{h.c.}\\\
&-\frac{g_{2}}{6c_{W}}Z_{\mu}\,\bar{u}_{i}\gamma^{\mu}\begin{aligned}
\Bigg{(}&\left[(3-4s_{W}^{2})\mathbbm{1}+3v^{2}\,V\cdot\left\\{C^{(3)}_{Hq}-C^{(1)}_{Hq}\right\\}\cdot
V^{\dagger}\right]_{ij}P_{L}\\\
-&\left[4s_{W}^{2}\mathbbm{1}+3v^{2}C_{Hu}\right]_{ij}P_{R}\Bigg{)}u_{j}\end{aligned}\\\
&-\frac{g_{2}}{6c_{W}}Z_{\mu}\,\bar{d}_{i}\gamma^{\mu}\begin{aligned}
\Bigg{(}&\left[(2s_{W}^{2}-3)\mathbbm{1}+3v^{2}\left\\{C^{(3)}_{Hq}+C^{(1)}_{Hq}\right\\}\right]_{ij}P_{L}\\\
+&\left[2s_{W}^{2}\mathbbm{1}+3v^{2}C_{Hd}\right]_{ij}P_{R}\Bigg{)}d_{j}\,,\end{aligned}\end{aligned}$
(13)

with $v\simeq{246}{\,\text{GeV}}$.

Best fit points, $\Delta\chi^{2}$ and pulls w.r.t. the SM hypothesis for the
various EFT scenarios [15]. The best fit points are in units of
$10^{-3}v^{-2}$.  EFT Scenario Best fit point $-\Delta\chi^{2}$ Pull
$\left[C^{(3)}_{Hq}\right]_{11}$ $-0.50$ $3.3$ ${1.8}{\sigma}$
$\left[C^{(3)}_{Hq}\right]_{11}=\left[C^{(3)}_{Hq}\right]_{22}$ $-0.27$ $1.1$
${1.1}{\sigma}$
$\left[C^{(3)}_{Hq}\right]_{11}=\left[C^{(1)}_{Hq}\right]_{11}$ $-0.55$ $3.7$
${1.9}{\sigma}$ $\left[C_{Hud}\right]_{11}$ $-1.0$ $3.1$ ${1.8}{\sigma}$
$\left[C_{Hud}\right]_{12}$ $-2.0$ $7.4$ ${2.7}{\sigma}$
$\left(\left[C_{Hud}\right]_{11},\left[C_{Hud}\right]_{12}\right)$
$(-1.4,-2.1)$ $13$ ${3.2}{\sigma}$
$\left(\left[C^{(3)}_{Hq}\right]_{11},\left[C_{Hud}\right]_{12}\right)$
$(-0.43,-2.0)$ $11$ ${2.8}{\sigma}$
$\left(\left[C^{(3)}_{Hq}\right]_{11},\left[C_{Hud}\right]_{11},\left[C_{Hud}\right]_{12}\right)$
$(0.27,-1.9,-2.4)$ $16$ ${2.9}{\sigma}$
$\left(\left[C^{(3)}_{Hq}\right]_{11},\left[C^{(3)}_{Hq}\right]_{22},\left[C_{Hud}\right]_{11},\left[C_{Hud}\right]_{12}\right)$
$(0.59,0.76,-2.6,-2.5)$ $17$ ${2.9}{\sigma}$
$\left(\left[C^{(3)}_{Hq}\right]_{11},\left[C^{(1)}_{Hq}\right]_{11},\left[C_{Hud}\right]_{11},\left[C_{Hud}\right]_{12}\right)$
$(0.29,0.11,-2.0,-2.4)$ $13$ ${2.6}{\sigma}$

In Table 3, the best fits of the Wilson coefficients are shown for the one,
two, three, and four-dimensional SMEFT operator scenarios. It is found that
the scenarios with modifications of right-handed $W$-$u$-$s$ couplings (by
$C_{Hud}$) provide the best improvement relative to the SM and do not lead to
problems in flavor physics or EWPO since constraints from $SU(2)_{L}$
invariance are not present. The scenarios with both left-handed and right-
handed modifications display a slightly larger $\Delta\chi^{2}$, which can be
understood by the fact left-handed operators change the EW fit by modifying
$Z$-quark couplings.

##  4 A heavy new physics interpretation: vector-like quark

There have been several attempts to resolve the CAA by TeV-scale NP models [1,
4, 53, 8, 6, 7, 11, 10, 12, 13, 15, 54, 55, 9, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66]. Among them, in this contribution, we consider vector-like quarks
(VLQs) as they give rise to the preferred modifications at tree level. VLQs
appear in many extensions of the SM such as grand unified theories [67, 68,
69], composite models or models with extra dimensions [70, 71] and little
Higgs models [72, 73]. For the CAA tension, the effect of modified $Z$
couplings to quarks and loop effects in flavor observables have to be included
in a global analysis.

There are seven possible VLQs mixed with quarks after EW symmetry breaking:

$\displaystyle U$ $\displaystyle:(\mathbf{3},\mathbf{1},\frac{2}{3})\,,$
$\displaystyle D$ $\displaystyle:(\mathbf{3},\mathbf{1},-\frac{1}{3})\,,$
$\displaystyle Q$ $\displaystyle:(\mathbf{3},\mathbf{2},\frac{1}{6})\,,$ (14)
$\displaystyle Q_{5}$ $\displaystyle:(\mathbf{3},\mathbf{2},-\frac{5}{6})\,,$
$\displaystyle Q_{7}$ $\displaystyle:(\mathbf{3},\mathbf{2},\frac{7}{6})\,,$
$\displaystyle T_{1}$ $\displaystyle:(\mathbf{3},\mathbf{3},-\frac{1}{3})\,,$
$\displaystyle T_{2}$ $\displaystyle:(\mathbf{3},\mathbf{3},\frac{2}{3})\,,$

with the representation under the SM gauge group $SU(3)\times SU(2)_{L}\times
U(1)_{Y}$. The Lagrangian describing their interactions with the Higgs and SM
quarks is

$\displaystyle-\mathcal{L}_{\rm VLQ}=$
$\displaystyle\xi_{fi}^{U}\bar{U}_{f}\tilde{H}^{\dagger}q_{i}+\xi_{fi}^{D}\bar{D}_{f}H^{\dagger}q_{i}+\xi_{fi}^{u}\bar{Q}_{f}\tilde{H}u_{i}+\xi_{fi}^{d}\bar{Q}_{f}Hd_{i}$
(15) $\displaystyle+$
$\displaystyle\xi_{fi}^{Q_{5}}\bar{Q}_{5,f}\tilde{H}d_{i}+\xi_{fi}^{Q_{7}}\bar{Q}_{7,f}Hu_{i}+\frac{1}{2}\xi_{fi}^{T_{1}}H^{\dagger}\tau\cdot\bar{T}_{1,f}q_{i}+\frac{1}{2}\xi_{fi}^{T_{2}}\tilde{H}^{\dagger}\tau\cdot\bar{T}_{2,f}q_{i}+\text{h.c.}\,,$

where $q$ is the left-handed quark doublet, $u,d$ are the right-handed quark
singlets, and $i$ and $f$ are flavor indices for the SM quarks and new VLQs,
respectively. Note that $f$ does not necessarily need to run from 1 to 3 as
the number of generations of VLQs is arbitrary. We disregard possible
couplings between two VLQs representations and the SM Higgs as they are not
relevant (at the dimension-six level) for the modification of gauge boson
couplings to quarks.

With these conventions, the matching obtained by integrating out the VLQs at
tree level onto the SMEFT is

$\displaystyle\left[C_{Hu}\right]_{ij}$
$\displaystyle=-\frac{\xi^{u}_{fj}\xi^{u*}_{fi}}{2M_{Q_{f}}^{2}}+\frac{\xi^{Q_{7}}_{fj}\xi^{Q_{7}*}_{fi}}{2M_{Q_{7f}}^{2}}\,,$
(16) $\displaystyle\left[C_{Hd}\right]_{ij}$
$\displaystyle=\frac{\xi^{d}_{fj}\xi^{d*}_{fi}}{2M_{Q_{f}}^{2}}-\frac{\xi^{Q_{5}}_{fj}\xi^{Q_{5}*}_{fi}}{2M_{Q_{5f}}^{2}}\,,$
$\displaystyle\left[C_{Hud}\right]_{ij}$
$\displaystyle=\frac{\xi^{d}_{fj}\xi^{u*}_{fi}}{M_{Q_{f}}^{2}}\,,$
$\displaystyle\left[C^{(1)}_{Hq}\right]_{ij}$
$\displaystyle=\frac{\xi^{U}_{fj}\xi^{U*}_{fi}}{4M_{U_{f}}^{2}}-\frac{\xi^{D}_{fj}\xi^{D*}_{fi}}{4M_{D_{f}}^{2}}-\frac{3\xi^{T_{1}}_{fj}\xi^{T_{1}*}_{fi}}{16M_{T_{1f}}^{2}}+\frac{3\xi^{T_{2}}_{fj}\xi^{T_{2}*}_{fi}}{16M_{T_{2f}}^{2}}\,,$
$\displaystyle\left[C^{(3)}_{Hq}\right]_{ij}$
$\displaystyle=-\frac{\xi^{U}_{fj}\xi^{U*}_{fi}}{4M_{U_{f}}^{2}}-\frac{\xi^{D}_{fj}\xi^{D*}_{fi}}{4M_{D_{f}}^{2}}+\frac{\xi^{T_{1}}_{fj}\xi^{T_{1}*}_{fi}}{16M_{T_{1f}}^{2}}+\frac{\xi^{T_{2}}_{fj}\xi^{T_{2}*}_{fi}}{16M_{T_{2f}}^{2}}\,.$

Figure 3: Global fits for the VLQ $Q$ with either couplings to $u$ and $d$
(left) or $u$ and $s$ scenario (right).

The most important VLQ is the $SU(2)_{L}$ doublet $Q$, which is the only one
that can contribute to the tight-handed $W$ coupling. For the $Q$ VLQ, since
it can have both couplings to right-handed up and down quarks, we instead show
two fits, for either purely $1^{\rm st}$ or purely $2^{\rm nd}$ generation
down quark interactions in Fig. 3. The study of the other VLQs is summarised
in Ref. [15]. It is expected from our previous EFT results that there is a
strong preference towards non-zero couplings to both right-handed $u$ and $d$
(left panel) or $u$ and $s$ (right panel). However, unlike in the simple EFT
scenario, the $Q$ field generates additional correlated effects in $Z$
couplings through $SU(2)_{L}$ invariance, and so parity violation (PO) and
EWPO partially limit the parameter space. In the left panel, the best fit
point is $\xi^{u}_{1}=-0.29$, $\xi^{d}_{1}=0.21$ and has a pull w.r.t. the SM
of ${1.1}{\sigma}$. For the right panel, we find a best fit at
$\xi^{u}_{1}=-0.33$, $\xi^{d}_{2}=0.38$, and a pull of ${2.1}{\sigma}$.
Furthermore, for a single generation of the doublet, NP in the right-handed
$W$-$u$-$d$ and $W$-$u$-$s$ vertices at the same time lead to significant NP
in right-handed $Z$-$d$-$s$ couplings, stringently constrained by
$\epsilon_{K}$ [74, 75]. Updating their result, we find that a single $Q$
doublet coupled to both $d$ and $s$ would have to obey
$M_{Q}/\sqrt{\xi^{d}_{1}\xi^{d}_{2}}>{175}{\,\text{TeV}}$ to be consistent
with experiment, and therefore far too heavy to be relevant to the CAA.

Thus, a full explanation of the tensions in the Cabibbo angle determination
require a modified $W$-$u$-$d$ and $W$-$u$-$s$ coupling and thus multiple
generations of $Q$. Similarly, one can solve the CAA via a modified left-
handed $W$-$u$-$d$ coupling and a right-handed $W$-$u$-$s$ coupling which
again requires at least two VLQs. This means that a full solution of the CAA
demands the presence of at least two VLQs.

##  5 A light new physics interpretation: sterile neutrino

Can the CAA be also explained by the light-scale new physics? The one
possibility is a MeV scale sterile neutrino scenario [50]. In this section, we
briefly summarize this scenario.

Figure 4:  Neutrino energy $E_{\nu}$ of each channel relevant to the first-row
CKM unitarity test is summarized [50]. The sterile neutrino channel is open
when $m_{\nu 4}<E_{\nu}$.

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,

$\displaystyle\nu_{\ell}\simeq\cos U_{\ell 4}\nu_{\ell}^{\prime}+\sin U_{\ell
4}\nu_{4}\quad\text{for~{}}\ell=e,\mu,\tau\,,$ (17)

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. 4. The sign of the modification depends on the mass scale
of the sterile neutrino.

When the sterile neutrino mass exceeds $\mathcal{O}$(1) GeV, it would not be
kinematically permissible in the relevant measurements in Fig. 4. 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

$\displaystyle G_{F}^{\rm obs}=G_{F}\cos U_{e4}\cos U_{\mu 4}\,.$ (18)

This relationship is pivotal in all the measurements employed in the CKM
determinations. Using this relation, one can derive

$\displaystyle\Delta_{\rm CKM}=|V_{ud}^{\rm obs}|^{2}+|V_{us}^{\rm
obs}|^{2}-1=\frac{|V_{ud}|^{2}}{\cos^{2}U_{\mu
4}}+\frac{|V_{us}|^{2}}{\cos^{2}U_{e4}}-1>0\,.$ (19)

This is in contrast to the CAA in Eqs. (6) and (7).

In the opposite limit, it is easily shown that when the sterile neutrinos are
massless, all the measurements reproduce the SM predictions. Hence,
$\Delta_{\rm CKM}$ is expected to be zero, which is again incompatible with
the current data.

Figure 5: Favored parameter regions are shown [50]. 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 [76, 77, 78, 79] and the EC-decay search [80], respectively. The
regions above green dashed lines can be constrained by $\pi^{+}\to e^{+}\nu$
measurements [81, 82, 83], and the regions between dotted or dash-dotted green
lines are allowed with the dimension-six operator. See more details in the
text.

Interestingly, when the sterile neutrino mass is in the MeV scale, the
coupling reduction of the weak interaction remains in the neutron and nuclear
decays, while other observables, especially the Fermi constant, stay almost
the same as in the SM. Consequently, only $V_{ud}$ is modified [50],

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

and

$\displaystyle I\left(m_{\nu},\delta M\right)\equiv\int_{m_{e}}^{\delta
M-m_{\nu}}dE_{e}\sqrt{E_{e}^{2}-m_{e}^{2}}\sqrt{E_{\nu}^{2}-m_{\nu}^{2}}E_{e}E_{\nu}\,,$
(22)

where $\delta M=Q_{\rm EC}-m_{e}$ and the electron-capture (EC) $Q$-value is
$Q_{\rm EC}$. 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, $\Delta_{\rm CKM}<0$. From the size of the
anomaly $\Delta_{\rm CKM}\approx 10^{-3}$, one can infer that the favored
mixing-angle-squared is $U^{2}_{e4}\approx 10^{-3}$, and $U^{2}_{\mu 4}$ is
not necessary.

The global analyses are shown in Fig. 5. 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
super-allowed 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 super-allowed 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.

This mass region of the sterile neutrino receives several constraints from the
laboratory to the cosmology. All constraints are investigated in Ref. [50],
and it is shown that there are viable scenarios within an extension of the
inverse seesaw model.

##  6 Conclusions

Improvements in lattice results for the kaon form factors and also the
radiative corrections have revealed a mild tension in the first-row CKM
unitarity test. The right-handed $W$ couplings are preferred in light of the
tension, and the prime candidate for the UV completion is the vector-like
quark extension. Explanation by a MeV sterile neutrino is also possible,
although the viable model is challenging. The STCF is needed for the first-
column and second-row unitarity tests.

## Acknowledgments

I would like to thank Andreas Crivellin, Matthew Kirk, Federico Mescia, and
Kohsaku Tobioka for fruitful collaborations on the presented works. The
presented works were supported by the Grant-in-Aid for Early-Career Scientists
from the Ministry of Education, Culture, Sports, Science, and Technology
(MEXT), Japan, No. 19K14706 and by the Japan Society for the Promotion of
Science (JSPS) Core-to-Core Program, No. JPJSCCA20200002. I also want to
warmly thank the organizers of the 2024 International Workshop on Future Tau
Charm Facilities for inviting me and giving me the opportunity to present
these results at this great conference.

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