$\mathcal{V}_{L}^{d}=\begin{pmatrix}V\\\ B_{L}^{d}\end{pmatrix}\,,$ (3.78)

where eq. (3.19) was used in combination with

$A_{L}^{u\dagger}=\mathds{1}_{3}\leavevmode\nobreak\ ,$ (3.79)

which is valid for the WB in use. $\mathcal{V}_{L}^{d}$ is a
$(3+n_{d})\times(3+n_{d})$ unitary matrix. Again, the physical
$3\times(3+n_{d})$ mixing matrix, $V$, can be read off directly from it - _its
first three rows_.

####  3.2.1 Model with $n_{u}=1$, $n_{d}=0$

As an example, consider the case $n_{u}=1$, which is the case of ref. [34].
This reference intended to provide a simple and minimal answer to the CKM
unitarity problem. Consider the following parameterisation for the matrix in
eq. (3.76) in terms of 6 mixing angles and 3 phases, first used in ref. [158],

$\mathcal{V}_{L}^{u\dagger}=\mkern-4.0mu\begin{pmatrix}1\\\ &1\\\
&&c_{34}&s_{34}\\\
&&-s_{34}&c_{34}\end{pmatrix}\mkern-8.0mu\begin{pmatrix}1\\\
&c_{24}&&s_{24}e^{-i\delta_{24}}\\\ &&1\\\
&-s_{24}e^{i\delta_{24}}&&c_{24}\end{pmatrix}\mkern-8.0mu\begin{pmatrix}c_{14}&&&s_{14}e^{-i\delta_{14}}\\\
&1\\\ &&1\\\
-s_{14}e^{i\delta_{14}}&&&c_{14}\end{pmatrix}\mkern-8.0mu\begin{pmatrix}&&&0\\\
&V_{\text{PDG}}&&0\\\ &&&0\\\ 0&0&0&1\end{pmatrix}\,,$ (3.80)

where $V_{\text{PDG}}$ is the standard PDG parameterization of a $3\times 3$
unitary matrix [169, 3],

$V_{\text{PDG}}=\mkern-4.0mu\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{pmatrix}\,,$
(3.81)

and $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$, with
$\theta_{ij}\in[0,\pi/2]$, $\delta_{ij}\in[0,2\pi]$. Note that
$V_{\text{CKM}}$ is described by the traditional PDG picture when all
$\theta_{i4}\rightarrow 0$. With this parameterization, the matrices which
control the tree-level FCNC of these models are $F^{d}=\mathds{1}_{3}$ and

$F^{u}=\mkern-4.0mu\begin{pmatrix}1-s^{2}_{14}&-s_{14}c_{14}s_{24}e^{-i(\delta_{14}-\delta_{24})}&-s_{14}c_{14}c_{24}s_{34}e^{-i\delta_{14}}&-s_{14}c_{14}c_{24}c_{34}e^{-i\delta_{14}}\\\
-s_{14}c_{14}s_{24}e^{i(\delta_{14}-\delta_{24})}&1-c^{2}_{14}s^{2}_{24}&-c^{2}_{14}s_{24}c_{24}s_{34}e^{-i\delta_{24}}&-c^{2}_{14}s_{24}c_{24}c_{34}e^{-i\delta_{24}}\\\
-s_{14}c_{14}c_{24}s_{34}e^{i\delta_{14}}&-c^{2}_{14}s_{24}c_{24}s_{34}e^{i\delta_{24}}&1-c^{2}_{14}c^{2}_{24}s^{2}_{34}&-c^{2}_{14}c^{2}_{24}s_{34}c_{34}\\\
-s_{14}c_{14}c_{24}c_{34}e^{i\delta_{14}}&-c^{2}_{14}s_{24}c_{24}c_{34}e^{i\delta_{24}}&-c^{2}_{14}c^{2}_{24}s_{34}c_{34}&1-c^{2}_{14}c^{2}_{24}c^{2}_{34}\end{pmatrix}\,.$
(3.82)

For small angles and neglecting the $\mathcal{O}(\theta^{3}_{i4})$ terms, one
can express the latter as

$F^{u}\simeq\begin{pmatrix}1-\theta^{2}_{14}&-\theta_{14}\theta_{24}e^{-i(\delta_{14}-\delta_{24})}&-\theta_{14}\theta_{34}e^{-i\delta_{14}}&-\theta_{14}e^{-i\delta_{14}}\\\
-\theta_{14}\theta_{24}e^{i(\delta_{14}-\delta_{24})}&1-\theta^{2}_{24}&-\theta_{24}\theta_{34}e^{-i\delta_{24}}&-\theta_{24}e^{-i\delta_{24}}\\\
-\theta_{14}\theta_{34}e^{i\delta_{14}}&-\theta_{24}\theta_{34}e^{i\delta_{24}}&1-\theta^{2}_{34}&-\theta_{34}\\\
-\theta_{14}e^{i\delta_{14}}&-\theta_{24}e^{i\delta_{24}}&-\theta_{34}&\theta^{2}_{14}+\theta^{2}_{24}+\theta^{2}_{34}\end{pmatrix}\,.$
(3.83)

Comparing eq. (3.49) and eq. (3.83), one concludes that

$\Theta_{u}=-\theta_{14}e^{-i\delta_{14}}\,,\quad\Theta_{c}=-\theta_{24}e^{-i\delta_{24}}\,,\quad\Theta_{t}=-\theta_{34}\,,$
(3.84)

where the angles and phases now refer to the parameters introduced in 3.80.
Like before, the vector $\Theta_{u}$ is implicitly defined as
$(\Theta_{u},\Theta_{c},\Theta_{t})^{T}$.

Using the definitions in eq. (3.54) one can derive for this case

$\displaystyle\delta\,\equiv\,\delta_{1}^{u}$
$\displaystyle=\,1-|V_{ud}|^{2}-|V_{us}|^{2}-|V_{ub}|^{2}\,$
$\displaystyle\\!\\!\\!\\!=\,\left|\mathcal{V}_{L_{41}}^{*}\right|^{2}$
$\displaystyle=\,s_{14}^{2}\,,$ (3.85) $\displaystyle\delta_{2}^{u}$
$\displaystyle=\,1-|V_{cd}|^{2}-|V_{cs}|^{2}-|V_{cb}|^{2}\,$
$\displaystyle\\!\\!\\!\\!=\,\left|\mathcal{V}_{L_{42}}^{*}\right|^{2}$
$\displaystyle=\,c_{14}^{2}\,s_{24}^{2}\,,$ $\displaystyle\delta_{3}^{u}$
$\displaystyle=\,1-|V_{td}|^{2}-|V_{ts}|^{2}-|V_{tb}|^{2}\,$
$\displaystyle\\!\\!\\!\\!=\,\left|\mathcal{V}_{L_{43}}^{*}\right|^{2}$
$\displaystyle=\,c_{14}^{2}\,c_{24}^{2}\,s_{34}^{2}\,.$

hence obtaining

$\displaystyle
F^{u}\simeq\,\small\begin{pmatrix}1-\delta_{1}^{u}&-\sqrt{\delta_{1}^{u}}\sqrt{\delta_{2}^{u}}e^{-i(\delta_{14}-\delta_{24})}&-\sqrt{\delta_{1}^{u}}\sqrt{\delta_{3}^{u}}e^{-i\delta_{14}}&-\sqrt{\delta_{1}^{u}}e^{-i\delta_{14}}\\\
-\sqrt{\delta_{1}^{u}}\sqrt{\delta_{2}^{u}}e^{i(\delta_{14}-\delta_{24})}&1-\delta_{2}^{u}&-\sqrt{\delta_{2}^{u}}\sqrt{\delta_{3}^{u}}e^{-i\delta_{24}}&-\sqrt{\delta_{2}^{u}}e^{-i\delta_{24}}\\\
-\sqrt{\delta_{1}^{u}}\sqrt{\delta_{3}^{u}}e^{i\delta_{14}}&-\sqrt{\delta_{2}^{u}}\sqrt{\delta_{3}^{u}}e^{i\delta_{24}}&1-\delta_{3}^{u}&-\sqrt{\delta_{3}^{u}}\\\
-\sqrt{\delta_{1}^{u}}e^{i\delta_{14}}&-\sqrt{\delta_{2}^{u}}e^{i\delta_{24}}&-\sqrt{\delta_{3}^{u}}&\delta_{1}^{u}+\delta_{2}^{u}+\delta_{3}^{u}\end{pmatrix}\,.$
(3.86)

Finally, one can conclude that, in approximation,

$\delta_{1}^{u}=|\Theta_{u}|^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\delta_{2}^{u}=|\Theta_{c}|^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\delta_{3}^{u}=|\Theta_{t}|^{2}\leavevmode\nobreak\ .$ (3.87)

This result shows that there is a simple connection between the deviations
from unitarity of the rows of the CKM matrix and the standard quark FCNC. This
is a consequence of what was discussed in general in section 3.1.2.

Since in the context of the CKM unitarity problem one has
$\delta\equiv\delta_{1}^{u}\gg\delta_{2}^{u}\sim\delta_{3}^{u}$, using eq.
(3.87) in the perturbativity discussion in section 3.1.2 gives

$m_{T}\lesssim\frac{m_{t}}{\sqrt{\delta_{1}^{u}+\delta_{2}^{u}+\delta_{3}^{u}}}\sim\frac{m_{t}}{\sqrt{\delta}}\,.$
(3.88)

Which, within the framework of trying to address the CKM unitarity puzzle, see
eq. (3.74), implies $m_{T}\lesssim 4.4$ TeV for $\sqrt{\delta}=0.04$. Of
course that introducing a VLQ does not come without consequences. The rest of
this section follows [34]. The phenomenological impacts of a up-type VLQ will
be discussed, and will be used to restrict the parameter space of this model.
With that, a global fit will be performed, with the goal of obtaining a region
of the parameter space that is allowed by experiment and that can explain the
CKM unitarity problem, while predicting a up-type VLQ, $T$, with a mass that
can be probed in the current and next round of experiments.

#####  3.2.1.1 New Physics and Global Fit

A variety of measurable quantities can undergo modifications either at tree
level or through loop processes when up-type vector-like quarks (VLQs) are
present [170, 171, 172, 173]. The effects of new physics contributions on
neutral meson mixing, specifically in the $D^{0}$, $K^{0}$, and $B^{0}_{d,s}$
neutral-meson systems, as well as the enhanced rare decay of the top quark,
$t\to qZ$, are particularly significant. These phenomenological quantities
play a crucial role in constraining the parameter space of this model.
Furthermore, the existence of a new heavy quark, $T$, could potentially allow
its production at current colliders if its mass is sufficiently low.

This quark can be generated either through single production or pair
production processes [174, 175, 176, 177]. The dominant contributions to pair
production cross sections are dependent solely on the mass of $T$, $m_{T}$. On
the other hand, single production mechanisms are controlled by functions
related to the elements of the mixing matrix, namely $V_{Td}$, $V_{Ts}$, and
$V_{Tb}$, hence more model-dependent.

In scenarios where significant couplings to light quark generations are
present, single production processes can dominate pair production by several
orders of magnitude within the few TeV energy range. This is primarily due to
a larger available phase space, an enhancement from longitudinal gauge bosons,
and the involvement of the proton’s valence quarks ($u$ and $d$) in the
initial state [178, 179]. Our current scenario aligns with this description,
as deviations from unitarity in the first row of the CKM matrix necessitate
relatively large values for the quantities $|V_{Td}|\sim|F^{u}_{41}|\sim 0.04$
(note the approximation
$|\mathcal{V}_{L{41}}^{u}|\sim|\mathcal{V}_{L{14}}^{u}|$ under the small
mixing angle assumption). These quantities govern the primary contributions to
the single VLQ production cross-sections via $W$ and $Z$ boson exchanges.

Regarding the decay of the heavy quark, the decay widths can be determined by
generalizing the expressions provided in Appendix B of ref. [177] (see e.g.
[180, 33]) and they are also influenced by $|V_{Td}|$ and $|F^{u}_{41}|$. In
the parameter space of this model, as will be discussed later, decays to the
lightest generation can be approximately an order of magnitude more probable
compared to decays to the second and third generations.

At the time of writing of ref. [34], the most stringent lower limits on the
VLQ mass were $m_{T}>1.3$ TeV at 95% confidence level (CL) by the ATLAS
collaboration [181], and $m_{T}\gtrsim 1.0$ TeV at 95% CL by the CMS
collaboration [182], based on searches for pair-produced $T$-quarks.

For more details regarding new physics, effects on physical observables, the
best experimental constraints and the most stringent bounds for models with
VLQs see section 4.2.

In order to explore the viability of this model, and to check if it can indeed
solve the CKM unitarity process, a numerical scan of the parameter space of
the model was performed. Initially, the model is restricted by the magnitudes
of the elements in the CKM matrix. The current best-fit values of these
elements, without enforcing unitarity, are obtained from [3].

$|V_{\text{CKM}}|=\begin{pmatrix}0.97370\pm 0.00014&0.2245\pm 0.0008&(3.82\pm
0.24)\times 10^{-3}\\\ 0.221\pm 0.004&0.987\pm 0.011&(41.0\pm 1.4)\times
10^{-3}\\\ (8.0\pm 0.3)\times 10^{-3}&(38.8\pm 1.1)\times 10^{-3}&1.013\pm
0.030\end{pmatrix}\,.$ (3.89)

In addition, it is assumed that the presence of an up-type VLQ does not affect
the value of the phase $\gamma=\arg(-V_{ud}V_{cb}V_{ub}^{*}V_{cd}^{*})$ [183,
165], which is obtained from SM tree-level dominated $B$ decays,
$\gamma=(72.1^{+4.1}_{-4.5})^{\circ}\,$ [3]. To assess the goodness of fit,
the measure $N\sigma=\sqrt{\chi^{2}}$ is employed, where $\chi^{2}$ is
approximated as the sum of priors,

$\chi^{2}=\sum_{ij}\left(\frac{V_{ij}-V_{ij}^{c}}{\sigma(V_{ij})}\right)^{2}+\left(\frac{\gamma-\gamma^{c}}{\sigma(\gamma)}\right)^{2}\,,$
(3.90)

with the superscript $c$ denoting central values and
$\sigma(\gamma)=4.5^{\circ}\,$. Furthermore, it is assumed
$m_{c}(M_{Z})=0.619\pm 0.084$ GeV, $m_{t}(M_{Z})=171.7\pm 3.0$ GeV [184] and
require $m_{T}>1$ TeV, in line with collider bounds.

The values of the new angles and phases compatible with the above criteria are
shown as the dashed regions in the correlation plot of Figure
LABEL:fig:correlations. These regions contract to the solid green $2\sigma$
and $3\sigma$ contours after all the phenomenological constraints described
before are taken into account. These constraints comprise the bounds on
$\Delta m_{N}$ ($N=D^{0},K^{0},B^{0},B^{0}_{s}$), and $|\epsilon_{K}|$
discussed in section 4.2.2 and the perturbativity bound of eq. (3.88). The
data suggests that relatively large values for both
$\theta_{14}\simeq\sqrt{\delta}$ and $\theta_{34}$ are favored, indicating a
disfavoring of $\theta_{34}=0$ at a significance exceeding $2\sigma$. On the
other hand, $\theta_{24}$ is consistent with zero, and the inclination towards
smaller values of this angle is influenced by the constraint imposed by
$D^{0}$-$\overline{D}^{0}$ mixing (refer to section 4.2.2.2).

The constraint of perturbativity also limits the permissible values of
$m_{T}$, which are depicted in Figure 3.1 against the deviations from
unitarity in the first row. As shown by eq. (3.88), the maximum value of
$m_{T}$ relies on the magnitude of these deviations. Assuming
$\sqrt{\delta}=0.04$, it follows that $m_{T}\lesssim 5$ TeV. Considering the
entire $3\sigma$ region of the fit, this bound extends to $m_{T}\lesssim 7$
TeV.

Figure 3.1:  Masses $m_{T}$ of the new heavy quark and first-row deviations
from unitarity ($\sqrt{\delta}\simeq\theta_{14}$) compatible with the bounds
on $|V_{ij}|$, $\gamma$, meson mixing, $|\epsilon_{K}|$ and with the
requirement of perturbativity. The latter constraint imposes an upper limit on
$m_{T}$, while lower bounds are set by ATLAS [181] and CMS [182] (95% CL).

Finally, in Figure 3.2 and Figure 3.3 the predicted values for the branching
ratios of rare top decays $t\to uZ$ and $t\to cZ$, respectively, as a function
of $\sqrt{\delta}$. These may considerably exceed the SM predictions. One
finds $2.0\times 10^{-8}<\text{Br}(t\to uZ)<3.5\times 10^{-7}$ at the
$2\sigma$ level, which is still 3 orders of magnitude below present bounds
(see section 4.2.2.1). The decay $t\to cZ$ may instead be arbitrarily
suppressed, even at the $2\sigma$ level, by the allowed smallness of
$\theta_{24}$.

Figure 3.2: Branching ratios for the rare top decays $t\rightarrow uZ$ as a
function of first-row deviations from unitarity $\delta$
($\sqrt{\delta}\simeq\theta_{14}$). These may considerably exceed the SM
expectations Br$(t\to uZ)_{\text{SM}}\sim 10^{-16}$ [185] (not shown). Dashed
contours refer to different values for $\theta_{34}$, in the small angle
approximation. Figure 3.3: Branching ratios for the rare top decays
$t\rightarrow cZ$ as a function of first-row deviations from unitarity
$\delta$ ($\sqrt{\delta}\simeq\theta_{14}$). These may considerably exceed the
SM expectations Br$(t\to cZ)_{\text{SM}}\sim 10^{-14}$ [185] (not shown).
Dashed contours refer to different values for the product
$\theta_{24}\,\theta_{34}$, in the small angle approximation.

Considering the correlation plot in Figure LABEL:fig:correlations, the
$2\sigma$ region of the fit, it is observed that $|V_{41}|>|V_{42}|,|V_{43}|$,
indicating that the up-type vector-like quark $T$ exhibits stronger coupling
to the up quark compared to the charm and top quarks. This challenges the
conventional assumption that the new quark primarily couples to the third
generation of Standard Model quarks. Consequently, the VLQ is more inclined to
decay into quarks of the first generation, as there are no suppressions
associated with the masses of these quarks. The decay widths primarily depend
on $m_{T}$ and $|V_{4j}|^{2}$ or $|F_{4j}^{u}|^{2}$ [177, 33]. A discussion of
a low-$\chi^{2}$ benchmark belonging to this $2\sigma$ region now follows. The
selected point in parameter space corresponds to a mass $m_{T}=1.5$ TeV and is
described by

$\displaystyle\theta_{12}=0.2265\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \theta_{13}=0.003818\leavevmode\nobreak\
,\leavevmode\nobreak\ \theta_{23}=0.03998\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ $ (3.91)
$\displaystyle\theta_{14}\simeq\sqrt{\Delta_{(1)}}=0.03951\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\theta_{24}\simeq\sqrt{\Delta_{2}}=0.002078\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\theta_{34}\simeq\sqrt{\Delta_{3}}=0.01271\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\delta_{13}=0.396\,\pi\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \delta_{14}=1.818\,\pi\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \delta_{24}=0.728\,\pi\,,$

corresponding to $\chi^{2}\simeq 3.2$,
$m_{T}\approx\frac{0.36}{\sqrt{\delta}}m_{t}$ and to the following values for
observables:

$\begin{split}&x_{D}\simeq 0.18\%\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \Delta m_{K}\simeq 6.3\times 10^{-13}\leavevmode\nobreak\
\text{MeV}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta m_{B}\simeq 3.0\times 10^{-10}\leavevmode\nobreak\
\text{MeV}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\\\\[2.84526pt] &\Delta m_{B_{s}}\simeq 2.7\times 10^{-10}\leavevmode\nobreak\
\text{MeV}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
|\epsilon_{K}|\simeq 1.9\times 10^{-4}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \\\\[2.84526pt] &\text{Br}(t\to
uZ)\simeq 1.2\times 10^{-7}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \text{Br}(t\to cZ)\simeq 3.2\times
10^{-10}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\\\\[2.84526pt] &\alpha\simeq 84.92^{\circ}\,\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \beta\simeq
23.85^{\circ}\,\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma\simeq 71.23^{\circ}\,\,,\end{split}$ (3.92)

where $\alpha=\arg(-V_{td}V_{ub}V_{tb}^{*}V_{ud}^{*})$ and
$\beta=\arg(-V_{cd}V_{tb}V_{cb}^{*}V_{td}^{*})$. The absolute values of the
diagonalizing matrix in eq. (3.80) read

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{}{}{}}}{{ {}{}{}}}}}$\left.\vbox{\hrule
height=29.75389pt,depth=29.75389pt,width=0.0pt}\right)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}}}}{{{{}{}{}}}}{{{ {}{}{}}}}{{{ {}{}{}}}}{ {}{}{}}{{{
{}{}{}}}}{{{{}{}{}}}}{{{ {}{}{}}}}{{{ {}{}{}}}}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}}
{{}\pgfsys@rect{4.33302pt}{-15.64345pt}{106.7315pt}{37.73132pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.33302pt}{3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{108.7315pt}\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{{{
{}{}{}}}}{{{{}{}{}}}}{{{ {}{}{}}}}{{{ {}{}{}}}}{ {}{}{}}{{{
{}{}{}}}}{{{{}{}{}}}}{{{ {}{}{}}}}{{{ {}{}{}}}}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@rect{2.33301pt}{-30.75389pt}{110.7315pt}{54.84177pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.33301pt}{-3.33301pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{108.7315pt}\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (3.93)

where $|V|$ is given by the first 3 columns of $|\mathcal{V}_{L}^{\dagger}|$
(dashed block) and $|V_{\text{CKM}}|$ is the $3\times 3$ upper-left block of
$|\mathcal{V}_{L}^{\dagger}|$ (unbroken line). The absolute values of the
entries of $F^{u}$ are

$|F^{u}|=\begin{pmatrix}0.99843969&0.00008203&0.00050179&0.03946672\\\
0.00008203&0.99999569&0.00002638&0.00207496\\\
0.00050179&0.00002638&0.99983863&0.01269224\\\
0.03946672&0.00207496&0.01269224&0.001726\end{pmatrix}\,,$ (3.94)

As mentioned in section 3.1.2, this matrix encodes the structure of FCNC and
deviations from unitarity of $V_{CKM}$.

Lastly, the mass matrix $\mathcal{M}_{u}$ for this particular scenario can be
explicitly expressed in the weak basis, where the down-type quark mass matrix
is diagonal, denoted as
$\mathcal{M}_{d}=\mathcal{D}_{d}=\text{diag}(m_{d},m_{s},m_{b})$. In this
basis, the upper-left $2\times 2$ block of $\mathcal{M}_{u}$ undergoes a
rotation resulting in a zero matrix,

$|\mathcal{M}_{u}|=\left(\begin{array}[]{cccc}0&0&1.39&58.57\\\
0&0&6.776&9.775\\\ 0.001658&15.38&170.9&18.92\\\ 0.03206&1.486&2.166&1499\\\
\end{array}\right)\leavevmode\nobreak\ \text{GeV}\,.$ (3.95)

The latter rotation can be achieved solely via transformations from the right.
In this WB, one can directly identify $(\mathcal{M}_{u})_{33}\simeq m_{t}$,
$(\mathcal{M}_{u})_{44}\simeq m_{T}$, and
${(\mathcal{M}_{u})_{14}}/{(\mathcal{M}_{u})_{44}}\simeq\sqrt{\Delta}$. It is
evident in this specific instance that
$(\mathcal{M}_{u})_{14}<(\mathcal{M}_{u})_{33}$.

It is worth noting that if in the scenario where a down-type VLQ is added to
the SM, the down-type quark mass matrix $\mathcal{M}_{d}$ would exhibit
similar qualitative characteristics to the previously described
$\mathcal{M}_{u}$. Specifically, in the WB where the up-type quark mass matrix
$\mathcal{M}_{u}$ is diagonal and the upper-left $2\times 2$ block of the
down-type quark mass matrix $\mathcal{M}_{d}$ is transformed into a zero
matrix, the structure of $\mathcal{M}_{d}$ would need to resemble the form
presented in eq. (3.95) to achieve the necessary deviation from unitarity in
the first row of the CKM matrix. In such a basis, one expects
$(\mathcal{M}_{d})_{33}\sim m_{b}$, with $m_{b}(M_{Z})\simeq 2.89$ GeV [186].
On the other hand, experimental bounds on the mass $m_{B}$ of the new heavy
quark in this scenario are close to those on $m_{T}$, i.e. $m_{B}\gtrsim 1$
TeV [181, 182], implying a large $(\mathcal{M}_{d})_{44}\sim m_{B}$. The
requirement of reproducing the observed deviation from unitarity
$\sqrt{\delta}\simeq 0.04$ would force $(\mathcal{M}_{d})_{14}\gtrsim
50\leavevmode\nobreak\ \text{GeV}\gg(\mathcal{M}_{d})_{33}$. Since the terms
in the first three rows of the mass matrix share a common origin in
electroweak symmetry breaking, this hierarchy between $(\mathcal{M}_{d})_{14}$
and $(\mathcal{M}_{d})_{33}$ may not be appealing or plausible in a theory of
flavour addressing the gap $m_{b}\ll m_{t}$ in third-generation quark masses.
In conclusion, the addition of one up-type VLQ may explain the CKM unitarity
problem, in, possibly, a more appealing way - in terms of flavour -, than
adding one down-type VLQ.

####  3.2.2 Model $n_{d}=1$, $n_{u}=0$

For completeness and comparison to the case of section 3.2.1, consider the
case where $n_{d}=1$, which is briefly discussed in ref. [33]. Consider the
following parameterisation for the matrix in eq. (3.78) in terms of 6 mixing
angles and 3 phases

$\mathcal{V}_{L}^{d}=\mkern-4.0mu\begin{pmatrix}&&&0\\\ &V_{\text{PDG}}&&0\\\
&&&0\\\
0&0&0&1\end{pmatrix}\mkern-8.0mu\begin{pmatrix}c_{14}&&&-s_{14}e^{-i\delta_{14}}\\\
&1\\\ &&1\\\
s_{14}e^{i\delta_{14}}&&&c_{14}\end{pmatrix}\mkern-8.0mu\begin{pmatrix}1\\\
&c_{24}&&-s_{24}e^{-i\delta_{24}}\\\ &&1\\\
&s_{24}e^{i\delta_{24}}&&c_{24}\end{pmatrix}\mkern-8.0mu\begin{pmatrix}1\\\
&1\\\ &&c_{34}&-s_{34}\\\ &&s_{34}&c_{34}\end{pmatrix}\,.$ (3.96)

With this choice, the traditional picture of $V_{\text{CKM}}$ is recovered
when all $\theta_{i4}\to 0$. Meanwhile, in these models the matrices which
control all tree-level FCNC are given by

$F^{d}=\mkern-4.0mu\begin{pmatrix}1-s^{2}_{14}&-s_{14}c_{14}s_{24}e^{-i(\delta_{14}-\delta_{24})}&-s_{14}c_{14}c_{24}s_{34}e^{-i\delta_{14}}&-s_{14}c_{14}c_{24}c_{34}e^{-i\delta_{14}}\\\
-s_{14}c_{14}s_{24}e^{i(\delta_{14}-\delta_{24})}&1-c^{2}_{14}s^{2}_{24}&-c^{2}_{14}s_{24}c_{24}s_{34}e^{-i\delta_{24}}&-c^{2}_{14}s_{24}c_{24}c_{34}e^{-i\delta_{24}}\\\
-s_{14}c_{14}c_{24}s_{34}e^{i\delta_{14}}&-c^{2}_{14}s_{24}c_{24}s_{34}e^{i\delta_{24}}&1-c^{2}_{14}c^{2}_{24}s^{2}_{34}&-c^{2}_{14}c^{2}_{24}s_{34}c_{34}\\\
-s_{14}c_{14}c_{24}c_{34}e^{i\delta_{14}}&-c^{2}_{14}s_{24}c_{24}c_{34}e^{i\delta_{24}}&-c^{2}_{14}c^{2}_{24}s_{34}c_{34}&1-c^{2}_{14}c^{2}_{24}c^{2}_{34}\end{pmatrix}$
(3.97)

and $F^{u}=\mathds{1}_{3}$. For small angles, $F^{d}$ becomes

$F^{d}\simeq\mkern-4.0mu\begin{pmatrix}1-\theta^{2}_{14}&-\theta_{14}\theta_{24}e^{-i(\delta_{14}-\delta_{24})}&-\theta_{14}\theta_{34}e^{-i\delta_{14}}&-\theta_{14}e^{-i\delta_{14}}\\\
-\theta_{14}\theta_{24}e^{i(\delta_{14}-\delta_{24})}&1-\theta^{2}_{24}&-\theta_{24}\theta_{34}e^{-i\delta_{24}}&-\theta_{24}e^{-i\delta_{24}}\\\
-\theta_{14}\theta_{34}e^{i\delta_{14}}&-\theta_{24}\theta_{34}e^{i\delta_{24}}&1-\theta^{2}_{34}&-\theta_{34}\\\
-\theta_{14}e^{i\delta_{14}}&-\theta_{24}e^{i\delta_{24}}&-\theta_{34}&\theta^{2}_{14}+\theta^{2}_{24}+\theta^{2}_{34}\end{pmatrix}\mkern-4.0mu$
(3.98)

plus terms of order $\mathcal{O}(\theta^{3}_{i4})$. Notice that, due to
similar choices, eq. (3.97) and eq. (3.82) are identical. However, this is
just a mere coincidence, and fits should prefer different values of the
$\theta_{i4}$ in each case, since they represent different things.

Again, comparing eq. (3.49) and eq. (3.98), one concludes that

$\Theta_{d}=-\theta_{14}e^{-i\delta_{14}}\,,\quad\Theta_{s}=-\theta_{24}e^{-i\delta_{24}}\,,\quad\Theta_{b}=-\theta_{34}\,,$
(3.99)

where the angles and phases now refer to the parameters introduced in 3.96.
Like before, the vector $\Theta_{d}$ is implicitly defined as
$(\Theta_{d},\Theta_{s},\Theta_{b})^{T}$. Fore more details regarding this
model and how it can solve the CKM unitarity problem, see section 4 of ref.
[23].

##  Chapter 4 New Physics and Experimental Signals

The smoking gun signals that can reveal the presence of ’heavy’111Here ’heavy’
should be understood as having a larger mass than the active ’light’ neutrinos
which due to the PLANCK bound [110] are expected to have a sub eV mass - cf.
section 2.3.2 of this thesis. neutrinos and VLQs primarily hinge on two
crucial quantities: their mass and their mixing with their respective SM
counterparts. Notably, the mass of the hypothetical particle is a critical
factor, as it determines whether the particle can be produced in experimental
setups. In models featuring ’heavy’ neutrinos and VLQs, a certain seesaw
between their mass and mixing is expected in order to have electroweak
perturbativity. This connection is elaborated upon in the final section of 2.2
and section 2.3.2 for neutrinos, and section 3.1.2 for VLQs. Consequently,
these models share a common characteristic: as the mass of the hypothetical
particle increases, its mixing with the active counterparts must decrease.
This leads to the existence of a fruitful region in the parameter space of
these models, where the particle’s mass is not excessively large and its
mixing is sufficiently significant to produce measurable effects in observable
quantities.
The phenomenological analysis presented in refs. [33, 34, 31, 30], which form
the basis of this thesis, precisely focuses on this particular region in the
parameter space, yielding results that can be probed in ongoing or forthcoming
experimental endeavors.
In this chapter, these signals will be reviewed, starting in the neutrino
sector and finishing in the quark sector. A comprehensive analysis of each
signal will be done, along with an assessment of the existing experimental
bounds associated with them. Future prospects and current bounds will be
discussed according to the heavy particle’s mass. For neutrinos, the effects
of ’heavy’ neutrinos with a mass of some GeV up to a few TeV and with a mass
between the MeV and the eV scale will be analysed, as these were discussed in
section 2.1 and in ref. [30] and in section 2.2 and in ref. [31]. Finally, for
VLQs, only those with a mass of a few TeV - discussed in section 3.2 and in
refs. [34, 33], will be considered.

###  4.1 Neutrinos

As implied earlier in this thesis, in terms of experimental viability, only
scenarios with $M\lesssim$ TeV hold promise for being tested in the
foreseeable future [89], as neutrinos within this mass range can potentially
leave discernible signatures in ongoing experiments at the energy, cosmic, and
intensity frontiers. The next paragraph will list the main signals ’heavy’
Majorana neutrinos may leave in current and future experiments. The rest of
the section will contain a brief description of each one and their current
experimental bounds.
Neutrinos, if Majorana particles, can in principle be tested via Lepton number
violating processes such as the neutrinoless double beta decay [187, 188,
189]. If their mixing with the active neutrinos is sizeable, they can be
produced with enough statistics to declare a discovery in the LHC [190]. This
mixing with their active counterparts can be inferred from the non-unitarity
of $3\times 3$ leptonic mixing matrix [54, 191, 56, 57, 192] and have impact
on low-energy observables related to lepton flavour violation (LFV), lepton
flavour non-universality (LFNU) and electroweak precision tests [52, 56].
Special emphasis will be given to the LFV process $\mu\rightarrow e\gamma$
[95], as it was one of the most restrictive to the parameter space of the
models studied in ref. [31]. Sizeable deviations from $3\times 3$ unitary may
also leave a clear trace in short baseline and long baseline oscillation
experiments [192, 52, 78]. Beta decay experiments, such as KATRIN, may also
play an important role since the existence of a eV almost-sterile ’heavy
neutrino’ [111] or a keV one [112] changes the shape of the beta decay
spectrum. Neutrinos in the discussed mass and mixing ranges may also leave
traces of CP Violation at low-energy [55, 31] detectable in future oscillation
experiments and, if long-lived, leave displaced vertex [125, 127, 128] and
resolvable oscillation [136, 131] signals at colliders.
To end section 4.1, a discussion on the prospects of future experiments
regarding neutrino physics is presented.

####  4.1.1 Neutrinoless double beta decay

Neutrinoless double $\beta$ decay ($0\nu\beta\beta$) is a a Lepton number
violating process [187, 122] that consists of

$(A,Z)\rightarrow(A,Z+2)+e^{-}+e^{-}\leavevmode\nobreak\ ,$ (4.1)

where $Z$ is the atomic number of a given atom, or, put simply, its number of
protons, and $A$ is its atomic mass number, or its number of protons and
neutrons, $e^{-}$ represents an electron. The decay rate for this process is
given by:

$\Gamma_{0\nu\beta\beta}=G(Q,Z)|M^{0}_{nuc}|^{2}|=G(Q,Z)|M_{nuc}|^{2}\cdot|m_{\beta\beta}|^{2}\leavevmode\nobreak\
,$ (4.2)

where $G(Q,Z)$ is a phase-space factor that depends on the nucleus and
$M^{0}_{nuc}$ is the total nuclear matrix element. $M^{0}_{nuc}$ contains the
leptonic amplitude
$(W^{-}_{\mu})^{*}(p_{3})+(W^{-}_{\nu})^{*}(p_{4})\rightarrow
e^{-}(p_{1})+e^{-}(p_{2})$, which has two diagrams (channel t and u) because
of the anti-symmetrization of the final state:

$M^{0}_{nuc}\propto(M_{t}-M_{u})\leavevmode\nobreak\ .$ (4.3)

It is only possible to write the amplitude for this process using a lepton
number violating vertex, thus making the process only possible if neutrinos
are Majorana [188]. For illustrative purposes, consider the t-channel
amplitude, specifically the part that is sandwiched between the spinors,

$iM_{t}=\sum_{j}\left[\overline{u(p_{1})}\left(\frac{-ig}{\sqrt{2}}\gamma_{\mu}P_{L}(K)_{1j}\right)\frac{i(\gamma^{\delta}p_{\delta}+m_{j})}{p^{2}-m_{j}^{2}}\left(-\frac{-ig}{\sqrt{2}}\gamma_{\nu}P_{L}(K)_{1j}\right)v(p_{2})\right]\leavevmode\nobreak\
,$ (4.4)

where $p=p_{1}-p_{3}=p_{4}-p_{2}$ and
$e^{-}(p_{2})=\left(e^{+}(p_{2})\right)^{c}$ represents the Conjugated Dirac
fermion, the vertex connecting momenta $p_{2}$ and $p_{4}$ exhibits lepton
number violation.

In the computation of $|M^{0}_{nuc}|^{2}$, upon evaluating the traces of the
leptonic part, it becomes evident that terms proportional to
$\gamma^{\delta}p_{\delta}$ vanish due to the presence of an odd number of
gamma matrices. Consequently, the only surviving terms are those proportional
to $m_{j}$. Consequently, by neglecting the neutrino mass in the denominator,
it becomes possible to factor out a meaningful quantity,

$|M^{0}_{nuc}|^{2}=|M_{nuc}|^{2}\cdot|\sum_{j}(K_{1j})^{2}m_{j}|^{2}\leavevmode\nobreak\
,$ (4.5)

where $K$ is the $3\times 3$ non-unitary leptonic mixing matrix, defined in
eq. (2.21). Combining eq. (4.5) with eq. (4.2) yields the definition

$m_{\beta\beta}=\sum_{j}(K_{1j})^{2}m_{j}\leavevmode\nobreak\ ,$ (4.6)

Note that, in general, this quantity is sensitive to all the CP Violating
Phases in the leptonic sector of the $n_{R}\nu$SM. The unitary phases, the
Dirac phase and the Majorana phases, and the hermitian phases, the phases of
the deviations from unitarity, as described in section 2.3.1. The quantity
$m_{\beta\beta}$ changes if $p$ of the $n_{R}$ "heavy" states have a mass of
the order of the light neutrinos, $\sim eV,keV$, such that it is kinematically
allowed, since this is a nuclear process [105, 72, 193]. In that case,

$m_{\beta\beta}=\sum_{i}^{3}(K_{1i})^{2}m_{i}+\sum_{j}^{p}((KX^{\dagger})_{1j})^{2}M_{j}\leavevmode\nobreak\
.$ (4.7)

The current experimental bounds on $m_{\beta\beta}$, considering 3
light/active neutrinos and very heavy almost-sterile neutrinos, are depicted
graphically in fig. 4.1:

Figure 4.1: $m_{\beta\beta}$ as given in eq. (4.6) as a function of the
lightest neutrino mass. The dark shaded regions are predictions based on best-
fit values of neutrino oscillation parameters for the normal ordering (NO) and
the inverted ordering (IO), and the light shaded regions indicate the
$3\sigma$ ranges calculated from oscillation parameter uncertainties [194,
195]. The regions below the horizontal lines are allowed at 90% C.L. with
$\textrm{Xe}^{136}$ from KamLAND-Zen [196] considering an improved phase space
factor calculation [197, 198] and commonly used nuclear matrix element
estimates: energy density functional (EDF) theory [199, 200, 201] (solid
lines), interacting boson model (IBM)[202, 203] (dashed lines), shell model
(SM) [204, 205, 206] (dot-dashed lines), and quasi-particle random phase
approximation (QRPA) [207, 208, 209, 210, 211] (dotted lines). The side panel
shows the corresponding limits for $\textrm{Xe}^{136}$, $\textrm{Ge}^{76}$
[212], and $\textrm{Te}^{130}$ [213], and theoretical model predictions on
$m_{\beta\beta}$, (a) Ref. [214], (b) Ref. [215], and (c) Ref. [216] (shaded
boxes), in the IO region. Image taken from ref. [196].

#####  4.1.1.1 $M\gtrsim$ MeV

Due to the inverted-ordering (IO) hypothesis of light neutrino masses being
disfavoured in global fits of leptonic mixing angles [58, 103, 59], the
prospects of measuring $m_{\beta\beta}$ have been affected, since for the
normal-ordering (NO) scenario $m_{\beta\beta}$ can be significantly smaller.
If the ’heavy’ neutrinos all have masses much beyond $\sim 100$ MeV, the
nuclear scale of the process, then they can’t be produced and contribute to
$m_{\beta\beta}$. In that case, the leading experiments in neutrinoless double
beta decay, namely KamLAND-Zen [196], GERDA [212], and CUORE [213], currently
lack immediate prospects of observing significant signals, especially with IO
being effectively excluded. Figure 4.1 presents the current most stringent
constraints on $m_{\beta\beta}$ under this framework.

#####  4.1.1.2 $M\lesssim$ MeV $\sim$ eV, keV

The layout obviously changes if there are ’heavy’ neutrinos with masses closer
to the MeV scale [217], since in this case $m_{\beta\beta}$ has new
contributions and its value might be substantially larger [105, 72, 193]. For
instance, the upper bound of the values of $m_{\beta\beta}$, calculated with
eq. (4.7), for the cases of section 2.2 ranged from $4.6$ meV to $27$ meV,
below the current experimental upper bounds. Nonetheless, if heavy neutrinos
with these masses and mixings exist, a discovery may be imminent in the next
round of experiments [189].

####  4.1.2 Prompt Decay Searches at LHC: ATLAS, CMS and LHCb

The LHC is the most powerful proton-proton collider built by mankind and is
host to nine approved experiments. The primary experiments involved in the
pursuit of new physics are ATLAS, CMS and LHCb. To date, these collisions have
taken place at center-of-mass energies of 7, 8, and 13 TeV.

Given the limitations of the LHC, ATLAS and CMS searches mainly focused on
heavy neutrinos with masses in the GeV/TeV scale [217], as these had the
better discovery prospects. The LHC’s array of detector experiments offers
comprehensive coverage across a wide range of potential scenarios related to
heavy neutrinos, with multiple searches [89, 218, 219, 220, 221, 222]
addressing the matter.

Another experimental configuration within the LHC is the LHCb experiment,
which primarily focuses on searching for new physics through the decay of
heavy-flavour hadrons. Its unique setup has yielded intriguing indications of
potential LFNU, as reported by the LHCb experiment [223, 224]. However, these
findings require confirmation and establishment with greater statistical
significance and preferably through independent measurements.

To summarize, ATLAS, CMS, and LHCb experiments have conducted searches for
heavy neutrinos in different regions of mass, active-sterile neutrino mixing,
and lifetimes, encompassing both minimal and non-minimal scenarios. This
thesis will only cover the minimal scenario, $n_{R}\nu$SM, as discussed in
chapter 2.

The LHC has the capability to generate heavy neutrinos in the mass range of
GeV/TeV by means of decays involving heavy mesons, $\tau$ leptons, $W$ bosons,
Higgs bosons, and even top quarks. If there are novel interactions or gauge
symmetries present in nature, or if the masses of heavy neutrinos exceed the
electro-weak scale, then additional production mechanisms for heavy neutrinos
become feasible. In the minimal model, the production of high-mass heavy
neutrinos involves processes such as the Drell-Yan process [225, 226], gluon
fusion [227, 228, 229], $W\gamma$ fusion [230, 231, 232, 233] and same-sign WW
fusion [234, 235]. When neutrinos are Majorana particles, they can decay into
both lepton number conserving and lepton number violating channels. However,
if they are Dirac particles, only lepton number conserving processes are
allowed. However, both Majorana and Dirac neutrinos can have lepton flavour
violating processes. Furthermore, depending on the mass and mixing angle of
the heavy neutrino, its decays can either be prompt (short lifetime) or
displaced (long lifetime) from the production vertex. This section will only
cover prompt decays, for displaced vertex decay searches see section 4.1.7.

It is important to note that in the experimental searches for heavy neutrinos
performed at the LHC, it is common to assume the presence of a single heavy
neutrino mixing with a single neutrino flavour. Naturally, the exclusion
limits obtained from these searches may not directly apply to the mixing
angles in realistic neutrino models that can explain neutrino oscillations. It
is evident that many of these searches require reinterpretation[236].

#####  4.1.2.1 $M\sim$ GeV/TeV

A CMS search from 2018 [93] excluded at 95% confidence level heavy neutrinos
with masses from $1$ GeV up to $1.2$ TeV for mixing $|V_{lN}|^{2}$ from
$1.2\times 10^{-5}$ up to 1.8, with $l=e,\mu$, at a center-of-mass energy of
13 TeV, with an integrated luminosity of $35.9\textrm{fb}^{-1}$. The results
are displayed in Figure 4.2.

Figure 4.2: The CMS experiment has established exclusion regions at a 95%
confidence level (CL) for heavy neutrinos in the $|V_{eN}|^{2}$ vs. $m_{N}$
(left) and $|V_{\mu N}|^{2}$ vs. $m_{N}$ (right) planes. In these plots, the
dashed black curve represents the expected upper limit, accompanied by dark
green and light yellow bands denoting one and two standard-deviation
uncertainties, respectively. The solid black curve represents the observed
upper limit, while the dotted black curve represents the observed limit
assuming prompt decays of heavy neutrinos [93]. Additionally, the plots
include the best upper limits at a 95% CL obtained from other collider
searches conducted by L3, DELPHI, ATLAS (8 TeV), and CMS (8 TeV). Image taken
from ref. [93].

This search looked for a signal comprised of a heavy neutrino produced via W
boson decay $W\rightarrow Nl$ with subsequent decay $N\rightarrow Wl$ and the
$W$ proceed to decay to $\nu l$, hence the mixing element involved is
$V_{lN}$, in the notation of eq. (2.9) and eq. (2.16).

Another CMS search from 2018 [94] excluded, at 95% C.L., heavy neutrinos with
masses from $20$ GeV to $1.6$ TeV for mixing $V_{lN}$ from $2.3\times 10^{-5}$
up to $1$, with $l=e,\mu$, at a center-of-mass energy of 13 TeV, with an
integrated luminosity of $35.9\textrm{fb}^{-1}$. The results are displayed in
Figure 4.3.

Figure 4.3: The CMS experiment has established exclusion regions at a 95%
confidence level (CL) for heavy neutrinos in the $|V_{eN}|^{2}$ vs. $m_{N}$
(left) and $|V_{\mu N}|^{2}$ vs. $m_{N}$ (right) planes. In these plots, the
dashed black curve represents the expected upper limit, accompanied by dark
green and light yellow bands denoting one and two standard-deviation
uncertainties, respectively. The solid black curve represents the observed
upper limit and the dashed cyan line shows constraints from EWPD. Also shown
are the upper limits from other direct searches: DELPHI, L3, ATLAS, and the
upper limits from the CMS $\sqrt{s}=8$ TeV 2012 data and the trilepton
analysis based on the same 2016 data set as used in this analysis. Image taken
from ref. [94].

The search in ref. [94] probes the decay of a W boson, in which an SM neutrino
oscillates into a new heavy state $N$.

An ATLAS search from 2023 [190] excluded, at 95% C.L., heavy neutrinos with
masses from $50$ GeV to $20$ TeV for mixing $V_{\mu N}$ from $5\times 10^{-2}$
up to $1$, at a center-of-mass energy of 13 TeV, with an integrated luminosity
of $140\textrm{fb}^{-1}$. The results are displayed in Figure 4.4.

Figure 4.4: Observed and expected 95% CL upper limits on the heavy Majorana
neutrino mixing element $|V_{\mu N}|^{2}$ function of $m_{N}$ in the
Phenomenological Type-I Seesaw model. The one and two standard deviation bands
of the expected limit are indicated in green and yellow, respectively. Image
taken from ref. [190].

The search in ref. [190] probes same-sign $\mu^{\pm}\mu^{\pm}$ production in
$W^{\pm}W^{\pm}$ scattering mediated by a Majorana neutrino $N$.

All these experiments exclude neutrinos with a mass between $100$ GeV and $1$
TeV for a $|V_{lN}|^{2}\gg 10^{-3}$. The models in section 2.1 and section 2.2
have neutrinos in these mass ranges with $|V_{lN}|^{2}\in[10^{-6},10^{-4}]$,
hence these models are not yet ruled out and the next round of experiments may
start probing them [123, 88, 237, 217, 90, 91, 92]. For low GeV masses, LHCb
provides more stringent bounds [238, 239, 240, 217]. A search for a heavy
Majorana neutrino involved in the $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$
decay mode was performed using $3\leavevmode\nobreak\ \textrm{fb}^{-1}$ of
integrated luminosity, at a center-of-mass energies of $7$ TeV and $8$ TeV.
Note that the search in ref. [238] was revised in ref. [239], its results are
summarized in fig. 4.5.

Figure 4.5: Upper limit on $|V_{\mu 4}|^{2}$ at 95% confidence level from the
LHCb experiment. The dashed line shows the limit from [238]. The solid line
shows the limit that would be extracted using the decay width formulae in this
paper. For comparison, the lower dotted line shows the recently revised limit
from Belle [241]. All three limit curves are constructed with the assumption
$V_{e4}=V_{\tau 4}=0$. Image taken from ref. [239].

Recently, a search for heavy neutrinos in the decay of a W boson into two
muons and a jet was performed at the LHCb [240]. The data set was collected in
the same conditions as the aforementioned search in refs. [238, 239]. Its
results are depicted in fig. 4.6

Figure 4.6: Observed upper limit on the mixing parameter $|V_{\mu N}|^{2}$
between a heavy neutrino and a muon neutrino in the mass range $5-50$ GeV for
same-sign and opposite-sign muons in the final states with and without
lifetime correction. Image taken from ref. [240].

#####  4.1.2.2 $M\lesssim$ GeV

Bounds for heavy neutrinos with masses below the GeV scale have better future
prospects and are more stringent in other experiments that do not work in the
energy frontier [242, 87, 243, 72, 81]. For keV neutrinos refer to section
4.1.5, for eV neutrinos see section 4.1.4 and section 4.1.5.

####  4.1.3 LFV, LFNU and Electroweak Precision Observables

The presence of heavy almost-sterile neutrinos produces effects on electroweak
precision observables, specifically lepton number and flavour violating decays
[244, 245, 246]. Constraints on these typically do not rely on the mass of the
heavy neutrinos, but it is important to consider the underlying assumptions
made when deriving these constraints [57]. In Section 2.2.1, it is explained
that if all heavy neutrinos have masses above the electroweak scale
(approximately $M\gtrsim m_{W}$, where $m_{W}$ represents the mass of the W
boson and $M$ the mass scale of all heavy neutrinos), one can solely constrain
the hermitian part of the leptonic mixing matrix, as defined in eq. (2.84).

According to the polar decomposition of a square invertible matrix, it is
possible to express such a matrix as the product of a unitary matrix with a
hermitian matrix. This hermitian matrix can be positioned either on the left
or on the right of the unitary matrix [55]. In terms of notation, the
hermitian part on the left is commonly written in terms of a matrix $\eta$,
cf. eq. (2.31). The most stringent constraints on the $\eta$ matrix are
provided in equation (2.35), originating from references [52, 56, 192]. In
particular, this discussion pertains to scenarios where the mass of the heavy
neutrinos is around or above the GeV/TeV scale ($M\gtrsim$ GeV/TeV), while
more stringent bounds for lighter heavy neutrinos can be found in sections
4.1.1, 4.1.4 and 4.1.5.

In the $n_{R}\nu$SM, when all $M\gtrsim$ GeV/TeV, the charged and neutral
electroweak current are modified [56]

$j_{\mu}^{\pm}=l_{\alpha}\gamma_{\mu}K_{\alpha j}\nu_{j}\leavevmode\nobreak\
,\leavevmode\nobreak\
j_{\mu}^{0}=\overline{\nu}_{i}(K^{\dagger}K)_{ij}\gamma_{\mu}\nu_{j}\leavevmode\nobreak\
,$ (4.8)

where $l$ are charged leptons, $\gamma$ are the Gamma matrices, $\nu$ are
light active neutrinos and $K$ is the $3\times 3$ non-unitary leptonic mixing
matrix, defined in eqs. (2.9) and (2.16). Using eq. (2.16) and comparing eq.
(4.8) with eqs. (2.8), (2.9) and eqs. (2.10), (2.12) one can see indeed that
eq. (4.8) is an approximation of the exact result, on which it is assumed that
only the active light neutrinos enter in the low-energy electroweak processes
one is studying. This is consistent with a effective description approach,
where the heavy neutrinos are integrated out, compatible with what is known in
the literature as the Minimum Unitarity Violation (MUV) scheme[56]. The
$n_{R}\nu$SM along with all $M\gtrsim$ GeV/TeV is equivalent to such approach
and the decay rates and cross sections are proportional to
$(KK^{\dagger})_{\alpha\beta}$ and $\sum_{\beta}(KK^{\dagger})_{\alpha\beta}$
for processes involving charged currents and
$\sum_{ij}|(K^{\dagger}K)_{ij}|^{2}$ for processes involving neutral currents.

The $\eta$ matrix is commonly used in the literature because, using eq.
(2.84), there is a straight forward connection between $KK^{\dagger}$, what
appears in the charged currents, and $\eta$

$\displaystyle
KK^{\dagger}\,=\,\mathds{1}-2\eta+\eta^{2}\approx\mathds{1}-2\eta\leavevmode\nobreak\
,$ (4.9)

where the unitarity relation eq. (2.17) and eq. (2.29) were used.

Similarly, there is a connection between $K^{\dagger}K$, what appears in
neutral currents, and $\eta$

$\displaystyle
K^{\dagger}K\,=U_{K}^{\dagger}(\mathds{1}-2\eta+\eta^{2})U_{K}\approx\mathds{1}-2U_{K}^{\dagger}\eta
U_{K}\leavevmode\nobreak\ .$ (4.10)

Neglecting the off-diagonal entries of $\eta$ with respect to the diagonal
entries, motivated by the Schwarz inequality for square hermitian matrices

$|\eta_{\alpha\beta}|\leq\sqrt{\eta_{\alpha\alpha}\eta_{\beta\beta}}\leavevmode\nobreak\
,$ (4.11)

and keeping only terms linear in $\eta_{ij}$, one can easily obtain simple
expression for the factors that appear in charged currents

$\sum_{\beta}(KK^{\dagger})_{\alpha\beta}\approx
1-2\eta_{\alpha\alpha}\approx(KK^{\dagger})_{\alpha\alpha}\leavevmode\nobreak\
,$ (4.12)
$(KK^{\dagger})_{\alpha\beta}=-2\eta_{\alpha\beta}\equiv\epsilon_{\alpha\beta}\leavevmode\nobreak\
,\leavevmode\nobreak\ \alpha\neq\beta\leavevmode\nobreak\ ,$ (4.13)

and neutral currents

$\sum_{ij}|(K^{\dagger}K)_{ij}|^{2}\approx
3-4\eta_{11}-4\eta_{22}-4\eta_{33}\leavevmode\nobreak\ ,$ (4.14)

where the unitarity of $U_{K}$ was used to obtain the last equation.

When $s$ light heavy neutrinos are kinematically accessible, the usual bounds
cannot be directly applied because the observables are no longer only
proportional to $(KK^{\dagger})_{\alpha\beta}$,
$\sum_{\beta=1}^{3}(KK^{\dagger})_{\alpha\beta}$ and
$\sum_{i,j=1}^{3}|(K^{\dagger}K)_{ij}|^{2}$ but to

$(KK^{\dagger})_{\alpha\beta}+(KX^{\dagger})_{\alpha
s}(XK^{\dagger})_{s\beta}\leavevmode\nobreak\ ,$
$\sum_{\beta=1}^{3}\Big{(}(KK^{\dagger})_{\alpha\beta}+(KX^{\dagger})_{\alpha
s}(XK^{\dagger})_{s\beta}\Big{)}$

and

$\sum_{i,j=1}^{3+s}\Bigg{|}\begin{pmatrix}K^{\dagger}K&K^{\dagger}KX^{\dagger}\\\
XK^{\dagger}K&XK^{\dagger}KX^{\dagger}\\\
\end{pmatrix}_{ij}\Bigg{|}^{2}\leavevmode\nobreak\ ,$

respectively [57, 31]. An updated version of the study described in refs. [52,
56, 192] that takes into account such cases would be pertinent but maybe
unfeasible in the present due to the expected smallness of the deviations from
unitarity of the leptonic mixing matrix, which are proportional to the
couplings among light active neutrinos and heavy almost-sterile neutrinos, cf.
chapter 2 and ref. [31].

A discussion on the most relevant observables that are sensitive to these
effects follows.

#####  4.1.3.1 Muon Decay: $G_{F}$

The muon lifetime is one of the best measured quantities in particle physics
[247]. The best measurement of $G_{F}$, which is one of most important
parameters in the SM, is obtained from it. The decay rate of the muon in the
$3n_{R}\nu$SM with all $M\gtrsim$ GeV/TeV is approximately given by [52, 56,
192]

$\Gamma_{\mu}=\frac{m_{\mu}^{5}G_{F}^{2}}{192\pi^{3}}\sum_{i}(KK^{\dagger})_{2i}\sum_{j}(KK^{\dagger})_{1j}\approx\frac{m_{\mu}^{5}G_{F}^{2}}{192\pi^{3}}(1-2\eta_{11}-2\eta_{22})\equiv\frac{m_{\mu}^{5}G_{\mu}^{2}}{192\pi^{3}}\leavevmode\nobreak\
,$ (4.15)

where eq. (4.12) was used and only terms linear in $\eta$ were kept. In eq.
(4.15) it was also implicitly defined

$G_{\mu}=G_{F}\sqrt{(1-2\eta_{11}-2\eta_{22})}\approx
G_{F}(1-\eta_{11}-\eta_{22})\leavevmode\nobreak\ ,$ (4.16)

which is $G_{F}$ as determined through muon decay. Using the smallness of
$\eta_{ii}$ one can also obtain

$G_{F}=G_{\mu}(1+\eta_{11}+\eta_{22})\leavevmode\nobreak\ .$ (4.17)

The present value of $G_{\mu}$ is [3]

$G_{\mu}=1.1663788(6)\times 10^{-5}\leavevmode\nobreak\
\textrm{GeV}^{-2}\leavevmode\nobreak\ .$ (4.18)

Hence, in this model, it acquires a non-unitary correction that spreads to
other observables [52, 56, 192]. Thus, it is crucial to compare the
measurement of $G_{F}$ via the muon decay with other independent
determinations of it [247], like via super-allowed $\beta$ decays, where
similarly to eq. (4.16), one can define $G_{F}$ determined through it as

$G_{\beta}=G_{F}\sqrt{\sum_{j}(KK^{\dagger})_{1j}}\approx\sqrt{(1-2\eta_{11})}\approx
G_{F}(1-\eta_{11})\leavevmode\nobreak\ ,$ (4.19)

where eq. (4.12) was used. It is important to keep in mind that one may be
propagating non-unitary corrections to other measurements when using $G_{F}$
determined through these weak processes.

#####  4.1.3.2 Invisible $Z$ width: $N_{\nu}$

An important test to the number of active neutrino species with masses below
the Z boson mass, $N_{\nu}$, was the measurement of the Z invisible width at
LEP [248], obtaining $N_{\nu}=2.984\pm 0.008$. In the $3n_{R}\nu$SM with all
$M\gtrsim$ GeV/TeV the Z invisible width is approximately given by [52, 56,
192]

$\Gamma_{\textrm{inv}}=\frac{G_{F}M_{Z}^{3}}{12\sqrt{2}\pi}\sum_{ij}|(K^{\dagger}K)_{ij}|^{2}\approx\frac{G_{\mu}M_{Z}^{3}}{12\sqrt{2}\pi}(3-(4\eta_{33}+\eta_{11}+\eta_{22}))\equiv\frac{G_{\mu}M_{Z}^{3}}{12\sqrt{2}\pi}N_{\nu}\leavevmode\nobreak\
,$ (4.20)

where eqs. (4.14), (4.16) and (4.17) were used. In eq. (4.20) it was also
implicitly defined

$N_{\nu}=3-(4\eta_{33}+\eta_{11}+\eta_{22})\leavevmode\nobreak\ ,$ (4.21)

The present value of $N_{\nu}$, averaged from several measurements [3] is

$N_{\nu}=2.92\pm 0.05\leavevmode\nobreak\ .$ (4.22)

It is remarkable that the $n_{R}\nu$SM with all $M\gtrsim$ GeV/TeV predicts
this deviation from $3$, caused by the deviations from unitarity expressed in
$\eta_{ii}$, cf. eq. (4.21).

#####  4.1.3.3 $W$ Decay to leptons: $W\rightarrow l_{\alpha}\nu_{\alpha}$

The decay width of the W boson into a charged lepton-neutrino pair in the
$3n_{R}\nu$SM with all $M\gtrsim$ GeV/TeV is given by [52, 56, 192]

$\Gamma_{W,\alpha}=\sum_{i}\Gamma(W\rightarrow
l_{\alpha}\nu_{i})=\frac{G_{F}m_{W}^{3}}{6\sqrt{2}\pi}(KK^{\dagger})_{\alpha\alpha}F_{W}(m_{l_{\alpha}})\approx\frac{G_{\mu}m_{W}^{3}}{6\sqrt{2}\pi}F_{W}(m_{l_{\alpha}})(1+\eta_{22}+\eta_{11}-2\eta_{\alpha\alpha})\leavevmode\nobreak\
,$ (4.23)

where eqs. (4.12) and (4.17) were used and with

$F_{W}(m_{l_{\alpha}})=\Big{(}1-\frac{m^{2}_{l_{\alpha}}}{m^{2}_{W}}\Big{)}\Big{(}1+\frac{m^{2}_{l_{\alpha}}}{m^{2}_{W}}\Big{)}\leavevmode\nobreak\
.$ (4.24)

The flavour structure of this decay is common to many weak processes, hence
its importance. Nevertheless, the $W$ boson branching ratios to leptons is a
well measured quantity [3], making it a great test for non-unitarity.

#####  4.1.3.4 LFNU: Lepton Flavour Universality Tests

From eq. (4.23) one can define the following lepton-universality observables

$R_{\alpha\beta}^{W}=\sqrt{\frac{\Gamma_{W,\alpha}F_{W}(m_{l_{\beta}})}{\Gamma_{W,\beta}F_{W}(m_{l_{\alpha}}}}=\sqrt{\frac{(KK^{\dagger})_{\alpha\alpha}}{(KK^{\dagger})_{\beta\beta}}}\approx(1-\eta_{\alpha\alpha}+\eta_{\beta\beta})\leavevmode\nobreak\
.$ (4.25)

These observables allow to directly constrain the ratios of the diagonal
elements of $\eta_{ii}$. The flavour structure of $\Gamma_{W,\alpha}$, defined
in eq. (4.23), is the same in the width for other weak processes such as
$\pi/K\rightarrow l_{\alpha}\nu_{\alpha}$ for instance, thus one can define
[52, 56, 192]

$R_{\alpha\beta}=\sqrt{\frac{(KK^{\dagger})_{\alpha\alpha}}{(KK^{\dagger})_{\beta\beta}}}\approx(1-\eta_{\alpha\alpha}+\eta_{\beta\beta})\leavevmode\nobreak\
,$ (4.26)

for ratios of decay widths of $W\rightarrow l_{\alpha}\nu_{\alpha}$ processes,
as in eq. (4.23), for ratios of decay widths of $l_{\alpha}\rightarrow
l_{\beta}\nu_{\alpha}\overline{\nu}_{\beta}$ processes or for ratios of decay
widths of $K/\pi\rightarrow l_{\alpha}\overline{\nu}_{\alpha}$ processes to
$l_{\alpha}\rightarrow K/\pi\nu_{\alpha}$ processes. Note that the following
ratios of decay widths

$R_{\mu
e}=\sqrt{\frac{\Gamma(\tau\rightarrow\nu_{\tau}\mu\overline{\nu}_{\mu})}{\Gamma(\tau\rightarrow\nu_{\tau}e\overline{\nu}_{e})}}\leavevmode\nobreak\
,\leavevmode\nobreak\
R_{\tau\mu}=\sqrt{\frac{\Gamma(\tau\rightarrow\nu_{\tau}e\overline{\nu}_{e})}{\Gamma(\mu\rightarrow\nu_{\mu}e\overline{\nu}_{e})}}\leavevmode\nobreak\
,\leavevmode\nobreak\ R_{\mu
e}=\sqrt{\frac{\Gamma(\pi\rightarrow\mu\overline{\nu}_{\mu})}{\Gamma(\pi\rightarrow
e\overline{\nu}_{e})}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
R_{\tau\mu}=\sqrt{\frac{\Gamma(\tau\rightarrow
K\nu_{\tau})}{\Gamma(K\rightarrow\mu\overline{\nu}_{\mu})}}\leavevmode\nobreak\
,$ (4.27)

have the same flavour dependence, expressed in eq. (4.26). It is important to
emphasize that in the SM $R_{\alpha\beta}$ is given by a function of the
masses of the involved charged leptons, incorporating phase space and
chirality flip factors, along with various loop corrections. Since a flavour
dependence is induced in the presence of non-unitary mixing, weak interaction
lepton flavour universality constraints become powerful probes of light-heavy
neutrino mixing.

#####  4.1.3.5 Unitarity of the CKM Matrix

Both within the Standard Model (SM) and in the $n_{R}\nu$SM, the CKM matrix is
unitary. Denoting the theoretical values of the CKM matrix elements with a
superscript "th," the unitary condition for the first row can be expressed as
follows

$|V_{ud}^{\textrm{th}}|^{2}+|V_{us}^{\textrm{th}}|^{2}+|V_{ub}^{\textrm{th}}|^{2}=1\leavevmode\nobreak\
.$ (4.28)

As discussed in section 3.2, presently a tension exists regarding the
unitarity of the first row of the CKM matrix. This is known in the literature
as the CKM unitarity problem or Cabibbo angle anomaly [24, 25, 18, 19, 20,
21].

The current experimental values for the entries of the first row of the CKM
matrix are given in eq. (3.72). From those values, one can obtain the
experimental version of eq. (4.28) [22],

$|V_{ud}^{\textrm{exp}}|^{2}+|V_{us}^{\textrm{exp}}|^{2}+|V_{ub}^{\textrm{exp}}|^{2}\approx
0.9984\pm 0.0001\leavevmode\nobreak\ .$ (4.29)

It is remarkable that in the MUV scheme, equivalent to the $3n_{R}\nu$SM with
all $M\gtrsim$ GeV/TeV, the weak processes involving leptons from which the
$V_{ij}^{\textrm{exp}}$ are measured get modified such that [52, 56, 192]

$|V_{ij}^{\textrm{exp}}|^{2}=|V_{ij}^{\textrm{th}}|^{2}(1+f^{\textrm{process}}(\eta_{\alpha\alpha}))\leavevmode\nobreak\
,$ (4.30)

where $f^{\textrm{process}}$ refers to a function of $\eta_{\alpha}$ with
$\alpha$ the leptonic flavour involved in the process in which the CKM matrix
element is measured. For instance, in the determination of $|V_{ud}|$ via
super-allowed $\beta$ decays, one has

$|V_{ud}^{\textrm{exp},\beta}|^{2}\approx|V_{ud}^{\textrm{th}}|^{2}(1-2\eta_{11})(1+2\eta_{11}+2\eta_{22})\approx|V_{ud}^{\textrm{exp},\beta}|(1+2\eta_{22})\leavevmode\nobreak\
,$ (4.31)

where eq. (4.17) and the smallness of $\eta_{ii}$ was used, since the process
is proportional to $G_{F}^{2}$, and the factor $(1-2\eta_{11})$ comes from the
electrons in the final state. A similar argument can be made regarding
$V_{us}$ and its measurement via kaon or tau decays [52, 56, 192]

$|V_{us}^{\textrm{exp},K\rightarrow\pi
e\overline{\nu}_{e}}|^{2}\approx|V_{us}^{\textrm{th}}|^{2}(1+2\eta_{22})\leavevmode\nobreak\
,$ (4.32)
$|V_{us}^{\textrm{exp},K\rightarrow\pi\mu\overline{\nu}_{\mu}}|^{2}\approx|V_{us}^{\textrm{th}}|^{2}(1+2\eta_{11})\leavevmode\nobreak\
,$ (4.33)

and

$|V_{us}^{\textrm{exp},\tau\rightarrow
K\nu_{\tau}}|^{2}\approx|V_{us}^{\textrm{th}}|^{2}(1+2\eta_{11}+2\eta_{22}-2\eta_{33})\leavevmode\nobreak\
.$ (4.34)

As explained in section 3.2, the present accuracy and smallness of $V_{ub}$
dictates that it can be safely neglected in eq. (4.28) and in this discussion,
hence, when doing these types of studies, one can consider
$|V_{ub}^{\textrm{exp}}|^{2}\equiv|V_{ub}^{\textrm{th}}|^{2}$.

The corollary of this section is that the measurement of $V_{\textrm{CKM}}$
elements via weak processes involving leptons may propagate leptonic non-
unitarity effects to the quark sector, which may alleviate the tension
regarding the unitarity of the first row of the CKM matrix [249, 250].

#####  4.1.3.6 LFV: Rare Charged Lepton Decays, $\mu\rightarrow e\gamma$

Although they are absent in the SM due to the masslessness of neutrinos,
charged lepton decays of the type $l_{\sigma}\rightarrow l_{\beta}\gamma$
occur at one loop in the MUV scheme. The decay width for this LFV process is
given by

$\Gamma_{l_{\sigma}\rightarrow l_{\beta}\gamma}=\frac{\sigma
G_{\mu}^{2}m_{\sigma}^{5}}{2048\pi}\Big{|}\sum_{j}K_{\sigma
j}K_{j\beta}^{\dagger}F(x_{j})\Big{|}^{2}\leavevmode\nobreak\ ,$ (4.35)

where $\alpha$ is the fine structure constant, $m_{\sigma}$ is the mass of the
charged lepton with flavour $\sigma$ and the terms of order
$\mathcal{O}(m_{l_{\sigma}}/m_{l_{\beta}})^{2})$ were neglected. $F(x_{j})$ is
a loop function where $x_{j}=m_{j}/m_{W}\approx 0$, where $m_{i}$ are the
masses of the light neutrinos. The smallness but non-zero nature of $F(x_{j})$
allows the excellent approximation

$\sum_{j}K_{\sigma
j}K_{j\beta}^{\dagger}F(x_{j})\approx-2F(0)\eta_{\sigma\beta}\leavevmode\nobreak\
,\leavevmode\nobreak\ F(0)=\frac{10}{3}\leavevmode\nobreak\ .$ (4.36)

Using this approximation one can write simple formulas for the branching
ratios of the LFV processes $l_{\sigma}\rightarrow l_{\beta}\gamma$ [52, 56,
192]

$\textrm{BR}(l_{\sigma}\rightarrow
l_{\beta}\gamma)=G_{\sigma\beta}\frac{100\alpha}{96\pi}4|\eta_{\sigma\beta}|^{2}\leavevmode\nobreak\
,$ (4.37)

where $G_{\sigma\beta}$ is a numerical factor that contains phase space
factors

$G_{\mu e}=1\leavevmode\nobreak\ ,\leavevmode\nobreak\ G_{\tau
e}=\frac{1}{5.6}\leavevmode\nobreak\ ,\leavevmode\nobreak\
G_{\tau\mu}=\frac{1}{5.9}\leavevmode\nobreak\ .$ (4.38)

As pointed out in section 2.2.1, the experimental bound obtained from the
search for the LFV decay $\mu\rightarrow e\gamma$ is one of the most stringent
constraints on the parameter space of $n_{R}\nu$SM models with sizeable
deviations from unitarity. Hence, the search by the MEG collaboration [95] and
also for the other rare tau decays $\tau\rightarrow e\gamma$,
$\tau\rightarrow\mu\gamma$ [251] provide a great probe for the non-unitarity
of the leptonic mixing matrix and one of the best ways to constrain the off-
diagonal terms of $\eta$.

####  4.1.4 Signals in Neutrino Oscillation Experiments

Neutrino oscillation experiments, such as SNO [7] and Super-Kamiokande [6],
played a crucial role in the discovery that neutrinos have mass. These
experiments are known as long-baseline (LBL) experiments and are characterized
by having a large source-detector distance, often denominated as $L$, allowing
for oscillations due to extremely small mass differences to develop, leading
to a sufficient phase difference to cause flavour transformations [252]. T2K
[253] and No$\nu$A [254] are prominent examples.
In contrast, short-baseline (SBL) experiments, as implied by their name,
involve significantly shorter source-detector distances, typically on the
order of a few meters [255], as is the case of MiniBooNE [256].

Considering a unitary leptonic mixing matrix, $U\equiv U_{\textrm{PMNS}}$,
defined in eq. (2.27), the oscillation probability between a (anti-)neutrino
of flavour $\alpha$ to a (anti-)neutrino of flavour $\beta$, with
$\alpha,\beta=e,\mu,\tau$, can be shown to take the form

$\displaystyle
P_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\alpha}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\beta}}(L,E)\,=\,\delta_{\alpha\beta}$
$\displaystyle-4\sum_{i>j}^{3+n}\,\operatorname{Re}\left(U_{\alpha
i}^{*}\,U_{\beta i}\,U_{\alpha j}\,U_{\beta j}^{*}\right)\sin^{2}\Delta_{ij}$
(4.39) $\displaystyle\pm 2\sum_{i>j}^{3+n}\,\operatorname{Im}\left(U_{\alpha
i}^{*}\,U_{\beta i}\,U_{\alpha j}\,U_{\beta j}^{*}\right)\sin 2\Delta_{ij}\,,$

where the plus or minus sign in the second line refers to neutrinos or anti-
neutrinos, respectively. Here, again $L$ denotes the source-detector distance,
$E$ is the (anti-)neutrino energy, and one has defined

$\Delta_{ij}\,\equiv\,\frac{\Delta
m^{2}_{ik}\,L}{4E}\,\simeq\,1.27\,\frac{\Delta
m_{ij}^{2}[\text{eV}^{2}]\,L[\text{km}]}{E[\text{GeV}]}\,,$ (4.40)

with mass-squared differences $\Delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$.
In order for experiments to be sensitive to neutrino oscillations, it is
necessary for the neutrino energy and the distance to the detector to satisfy
the condition $\Delta_{ij}\,\equiv\,\frac{\Delta m^{2}_{ik}\,L}{4E}\gtrsim 1$.
However, it is often impractical to physically move either the detector or the
source to cover a wide range of $L/E$ values. Therefore, having a broad energy
spectrum proves advantageous as it allows for the detection of oscillations
through the distortion of the expected neutrino energy distribution, as
depicted in figure 4.7, which is a reconstructed $\nu_{\mu}$ energy spectrum
from the MINOS experiment [257].

Figure 4.7: Distribution of muon neutrino events plotted as a function of
reconstructed energy in the MINOS far detector at a distance of $735$ km. The
solid grey line shows the expected spectrum under no oscillation hypothesis.
The depletion of events at $\gtrsim 1.5$ GeV and the shape of the depleted
spectrum is consistent with dominant $\nu_{\mu}\rightarrow\nu_{\tau}$
oscillations with maximal mixing. Image taken from ref. [252].

The sensitivity ranges for different types of neutrino experiments are as
follows:

  * •

For solar neutrino experiments ($L\approx 10^{11}$ m, $E\approx 15$ MeV), the
sensitivity range is $\Delta m_{ij}\approx 10^{-10}\leavevmode\nobreak\
\textrm{eV}^{2}$.

  * •

For atmospheric neutrino experiments ($20$ km $<L<10^{3}$ km, $E\approx 1$
GeV), the sensitivity range is $\Delta m_{ij}\approx
10^{-4}\leavevmode\nobreak\ \textrm{eV}^{2}$.

  * •

For reactor neutrino experiments ($L\approx[10,10^{4}]$ m, $E\approx 3$ MeV),
the sensitivity range is $\Delta m_{ij}\approx 10^{-5}\leavevmode\nobreak\
\textrm{eV}^{2}$.

  * •

For accelerator neutrino experiments ($L\leq 500$ km, $E\approx 1$ GeV), the
sensitivity range is $\Delta m_{ij}\approx 10^{-3}\leavevmode\nobreak\
\textrm{eV}^{2}$.

  * •

For a typical SBL experiment ( $L\sim\mathcal{O}(100)$ m,
$E\approx\mathcal{O}(100)$ MeV), the sensitivity range is $\Delta
m_{ij}\approx\mathcal{O}(1)\leavevmode\nobreak\ \textrm{eV}^{2}$.

One cannot end this summary without mentioning IceCube [258], an experiment
that is sensitive to typical LBL $\Delta m_{ij}$ [259] but also to the usual
SBL $\Delta m_{ij}$ [260], as it can detect both cosmic and atmospheric
neutrinos, hence being able to acquire data for various values of $L/E$.

Comprehensive summaries of current results based on global fits from LBL
experiments, considering a unitary leptonic mixing matrix, can be found in
references [261, 59]. The results from ref. [59] can be found in Table 2.2.
These results provide information about the leptonic mixing angles and the
Dirac phase $\delta$. However, the models studied in this thesis do not have
unitary leptonic mixing matrices, as discussed in chapter 2. Consequently, the
equations governing neutrino oscillation probabilities, such as eq. (4.39),
are modified due to non-unitarity effects, cf. section 2.2 and eq. (2.47).
This modified equation can be simplified for specific cases, such as SBL and
LBL experimental setups with the consideration of matter effects in LBL
experiments being mandatory in order to obtain realistic results.

Interestingly, experiments with sensitivities of $\Delta
m_{ij}\sim\mathcal{O}(1)\leavevmode\nobreak\ \textrm{eV}^{2}$ are particularly
susceptible to non-unitarity effects [31], as discussed in section 2.2.
Notably, SBL experiments, which fall within this sensitivity range [262],
observed an anomaly now known as SBL anomaly, initially referred to as the
LSND anomaly [74, 75], based on the experiment that detected it. This anomaly
involves an excess of $\overline{\nu}_{e}$ at the detector from a
$\overline{\nu}_{\mu}$ source. The $L/E$ settings of the detector points
towards a large $\Delta m^{2}$ if interpreted as a two-neutrino oscillation
$\overline{\nu}_{\mu}\xrightarrow{}\overline{\nu}_{e}$. Reactor and
radioactive source experiments also reported excess or deficit of $\nu_{e}$ or
$\overline{\nu}_{e}$ in the detector [76, 77, 78, 263]. Later, KARMEN reported
results that disagreed with the results from LSND [62], but the MiniBOONE
experiment [256] confirmed the results obtained by LSND [79, 80]. MiniBOONE,
despite having similar $L/E$ as LSND, differs in terms of energy, beam,
detector systematics, event signatures and backgrounds, which makes it an
excellent tester of the LSND results.

A global fit combining all these results and other constraints [264, 66]
suggests the existence of a heavy sterile neutrino with the data exhibiting a
preference for the $(3+1)$ scenario, where there are three light neutrinos and
one neutrino with a mass in the eV range and a sizeable coupling with at least
one light neutrino[72, 65, 265]. Hence, these anomalies may be explained if
the spectrum of heavy neutrinos includes at least one neutrino with an eV-
scale mass [78, 64], and if there are significant deviations from the
unitarity of the $3\times 3$ mixing matrix (DU) [71, 31].

However, it is crucial to note that a closer examination of the dataset
reveals tension between the appearance and disappearance datasets, as well as
the neutrino and anti-neutrino datasets [81]. Furthermore, recently, the
IceCube experiment also reported results that conflict with the ones from LSND
and MiniBOONE [260]. In conclusion, all of these anomalies and hypotheses are
still subject to ongoing debate and discussion.

#####  4.1.4.1 Effects in LBL Experiments

Obtaining direct evidence of heavy neutrinos in LBL oscillation experiments is
extremely challenging because these experiments are sensitive to very small
mass differences, $\Delta m_{ij}$. One possibility for detection would arise
if there exists a pair of quasi-degenerate heavy neutrinos, hence with a small
enough mass splitting, with significant mixing and adequate mass in order for
it to be produced and/or detectable in the experiment. For the spectra
considered in this thesis, which contain a quasi-degenerate pair of heavy
neutrinos with TeV masses, cf. case II of 2.2 , an experiment like
IceCube[266, 258] could potentially detect them, although the likelihood of
this occurring is extremely low.

LBL experiments have a higher probability of detecting non-unitarity effects
associated with large mass splittings. This is because terms proportional to
$\sin^{2}(\Delta m\leavevmode\nobreak\ L/4E)$ can be effectively averaged out
to approximately 0.5 [86], thus the oscillation probabilities in these cases
only depend on the usual mixing among light neutrinos and the mixing between
light and heavy neutrinos. The modifications of neutrino oscillation
probabilities in vacuum in LBL experiments for the cases studied in ref. [31]
is discussed in section 2.2.2. Using the definitions of eq. (2.55) and eq.
(2.65), the transition probability of muon to electron (anti-)neutrinos for
the Ia case can be approximated by eq. (2.64), for the Ib case by eq. (2.68)
and for the II case by eq. (2.79).

Hence, heavy almost-sterile neutrinos may leave traces in LBL experiments.
This should come as no surprise since, as stated earlier in this section, in
the presence of a non-unitary mixing matrix, and possibly a sizeable mixing
with almost-sterile neutrinos, the neutrino oscillation probabilities are
modified. This can have noticeable effects in the conversion and survival
probabilities of a neutrino species in a LBL experiment[192], mostly due to
the effect of the hermitian phases, the phases of the deviations from
unitarity, first discussed in this thesis in section 2.3.1. A summary of the
effect of these phases in CP Violation measurements can also be found in
section 4.1.6. However, the limited sensitivity of active LBL experiments to
these effects, see ref. [86] and the relatively weak limits obtained, means
that perhaps only experiments like DUNE [267, 107] have the potential to truly
test them.

Nevertheless, is interesting to point out that a global-fit from a combined
analysis of 2019 and 2020 data from T2K and NO$\nu$A prefers non-unitarity at
2 $\sigma$ C.L. [17]. The tension between NO$\nu$A and T2K 2019 data is also
reduced when both the experiments are analysed with the non-unitary
hypothesis. If in the future the two experiments consistently show improved
agreement with the non-unitary hypothesis, it could serve as a compelling
indication of the existence of new physics in the neutrino sector.

#####  4.1.4.2 Effects in SBL Experiments

SBL experiments and LBL experiments have distinct discovery prospects. SBL
experiments are capable of detecting oscillations only when the magnitude of
$\Delta m_{ij}$ falls within their typical sensitivity range, which is
approximately $\Delta m_{ij}\approx\mathcal{O}(1)\textrm{eV}^{2}$.
Consequently, SBL experiments can test the existence of almost-sterile
neutrinos with a mass of around $1$ eV, as in case Ia of ref. [31], or the
presence of a pair of nearly-sterile neutrinos with $\Delta
m_{ij}\approx\mathcal{O}(1)\textrm{eV}^{2}$, as outlined in case Ib of ref.
[31]. The observability of these oscillations is enhanced when there is a
substantial mixing between the heavy and light neutrinos, as demonstrated by
the figures and benchmarks presented in section 2.2.2 and reference [31]. In
fact, the resolution of the aforementioned SBL anomalies, within the 3+1
scheme, which closely resembles the Ia case, requires $\Delta m^{2}=0.209$ eV2
and $\sin^{2}2\vartheta^{(4)}_{\mu e}=0.0316$ and $|V_{e4}|^{2}=0.016$, a 4.3
$\sigma$ preference comparing to the no-oscillation scenario, per the most
recent work of the MiniBooNE and MicroBooNE collaborations[268], cf. figure
4.8.

Figure 4.8:  The results of the MiniBooNE-only and combined fits with
MicroBooNE’s CCQE sample [269] (top) and MicroBooNE’s Inclusive sample [270]
(bottom). The likelihood is obtained by profiling over all parameters $\Delta
m^{2}$ and $\sin^{2}(2\theta_{\mu e})\equiv\sin^{2}2\vartheta^{(4)}_{\mu e}$.
The two best-fit points are shown as appropriately colored stars, and the
contours are obtained by comparing the profile-likelihood-ratio test-statistic
to the asymptotic distribution provided by Wilks’ theorem, and assuming a
difference of two degrees of freedom. Image taken from ref. [268].

MicroBooNE [108] is another SBL experiment built in the same apparatus as
MiniBooNE, located in the Booster Neutrino Beam Beamline at Fermilab.
MicroBooNE’s two main physics goals are to investigate the MiniBooNE low-
energy excess and neutrino-argon cross sections.

The modifications of neutrino oscillation probabilities in vacuum for the
cases examined in ref. [31] are discussed in section 2.2.2. By utilizing the
definitions in equation (2.55) and equation (2.65), the transition probability
from muon to electron (anti-)neutrinos can be approximated as equation (2.67)
for the Ia case, equation (2.70) for the Ib case, and equation (2.81) for the
II case. Hence, SBL experiments provide the most stringent constraints on
light heavy neutrinos with eV-scale masses [57]. In these cases, the
observables directly constrain the entries of the mixing matrix $V_{\alpha
k}=(KX^{\dagger})_{\alpha k}$, where $\alpha=1,2,3$ and $k=1,..,n_{R}$,
defined in eq. (2.9) and eq. (2.16). To incorporate these constraints from the
experiments, the relevant exclusion curves in the
$\sin^{2}2\vartheta{\alpha\beta}$–$\Delta m^{2}$ planes are considered and
translated into constraints on the elements of the mixing matrix $V_{\alpha
k}$.

For values $\Delta m^{2}\in[10^{-2},1]\textrm{eV}^{2}$ the combined result of
MINOS [83], MINOS+ [271], Daya Bay [272] and Bugey-3 [82] gives the best
result, shown in fig. 4.9.

Figure 4.9: The combined limits at 99% C.L.(CLs) on $\sin^{2}(2\theta_{\mu
e})\equiv\sin^{2}2\vartheta{\alpha\beta}$ using the CLs method (red). The
exclusion contours are compared to the LSND and MiniBooNE allowed regions.
Also shown are the allowed regions from global fits by Gariazzo et al. [105]
and Dentler et al. [78]. Note that $\sin^{2}(2\theta_{\mu
e})\equiv\sin^{2}2\vartheta^{(4)}_{\mu e}$. Image taken from ref. [273].

Thus, for a heavy neutrino with a $1$ eV mass and $\Delta m^{2}\sim
1\textrm{eV}^{2}$, one has the bound

$\sin^{2}2\vartheta^{(4)}_{\mu e}=4|V_{e4}|^{2}|V_{\mu 4}|^{2}\lesssim
10^{-3}\leavevmode\nobreak\ .$ (4.41)

Note that the benchmarks presented in tables 6a, 6b and 6c satisfy this bound.

For larger values of $\Delta m^{2}$, KARMEN [274], MiniBooNE [80] and
MicroBooNE [275] give the most stringent limit, depicted in fig. 4.10.

Figure 4.10: MicroBooNE constraints on sterile neutrino parameter space at
3$\sigma$ C.L. (blue, Inclusive and orange, CCQE). For reference, the
MiniBooNE 1-, 2-, and 3-$\sigma$ preferred regions is shown in shades of grey
[79], the future sensitivity of the three SBN detectors (pink) [276], and
existing constraints from KARMEN (green) [274] and OPERA (gold) [277]. Note
that $\sin^{2}(2\theta_{\mu e})\equiv\sin^{2}2\vartheta^{(4)}_{\mu e}$. Image
taken from ref.[275].

Note that a large portion of the LSND and MiniBooNE allowed region [275] and
the 3+1 scheme best-fit zone [268] are excluded by the combined analysis of
MINOS, MINOS+, Daya Bay and Bugey-3 [273]. The tension between the indications
of electron (anti)neutrino appearance and the absence of corresponding signals
in disappearance channels is heightened, further contributing to the ongoing
debate surrounding SBL neutrino oscillations.

####  4.1.5 Beta Decay

Beta ($\beta$) decay is a nuclear process described by

$n\xrightarrow{}p+e^{-}+\overline{\nu_{e}}\leavevmode\nobreak\ ,$ (4.42)

where $n$ represents a neutron in a given atomic nucleus, $p$ a proton,
$e^{-}$ an electron and $\overline{\nu_{e}}$ an electron-flavoured anti-
neutrino. The spectra of the emitted electron depends on it energy, $E_{e}$,
and is given by the Kurie function [105]

$\begin{split}K^{2}(E_{e})&=(Q-E_{e})\sum_{i}|K_{1i}|^{2}\sqrt{(Q-E_{e})^{2}-m^{2}_{i}}\times\Theta(Q-T_{e}-m_{i})\leavevmode\nobreak\
.\\\ \end{split}$ (4.43)

$Q$ is the amount of energy released by the reaction, $\Theta$ is the
Heaviside step function and $m_{i}$ is the mass of the emitted anti-neutrino.
Since there are no prospects of achieving a resolution that may distinguish
the light neutrino states, it is customary to perform an approximation that is
valid for an experiment in which the energy resolution is such that
$m_{i}<<Q-E_{e}$,

$K^{2}(E_{e})\approx(Q-E_{e})\sqrt{(Q-E_{e})^{2}-m^{2}_{\beta}}\times\Theta(Q-E_{e}-m_{\beta})\leavevmode\nobreak\
,$ (4.44)

where $m_{\beta}$ is the “electron neutrino mass” and the Kurie function
depends on it thanks to the phase space factor. It is determined as a real
average over the three light neutrino mass eigenstates contributing to the
electron neutrino,

$m^{2}_{\beta}=\sum_{i}^{3}|K_{1i}|^{2}m^{2}_{i}\leavevmode\nobreak\ ,$ (4.45)

where $K$ is defined in eq. (2.21) and eq. (2.15). Note that, in general, this
quantity is sensitive to all the CP Violating Phases in the leptonic sector of
the $n_{R}\nu$SM. The unitary phases, the Dirac phase and the Majorana phases,
and the hermitian phases, the phases of the deviations from unitarity, as
described in section 2.3.1. Using the full equation for $K$, eq. (2.21) and
eq. (2.84), with eq. (2.27) to parameterise the unitary part of $K$ and eq.
(2.86) to parameterise the hermitian part of $K$, one can calculate
$|K_{11}|^{2}$, to prove the previous statement,

$\begin{split}|K_{11}|^{2}=&\Bigl{(}c_{12}c_{13}h_{11}+h_{12}s_{12}c_{13}\cos(\alpha_{21}-p_{12})+h_{13}s_{13}\cos(\alpha_{31}-\delta-
p_{13})\Bigr{)}^{2}\\\
&+\Bigl{(}h_{12}s_{12}c_{13}\sin(\alpha_{21}-p_{12})+h_{13}s_{13}\sin(\alpha_{31}-\delta-
p_{13})\Bigr{)}^{2}\leavevmode\nobreak\ .\end{split}$ (4.46)

However, in practical terms, the effect of the phases is only measurable if
the deviations from unitarity are sizeable, otherwise $h_{11}\sim 1$ and
$h_{12}\sim h_{13}\sim 0$, and $m^{2}_{\beta}$ only depends on mixing angles.
The measurement of $m_{\beta}$ consists of looking for a decrease in the
observed maximum electron energy (the end-point of the spectrum) and a subtle
distortion in the spectral shape, relative to the expected theoretical
prediction. Figure 4.11 illustrates this. Note how a $m_{\beta}\neq 0$
($m_{\beta}=m_{\nu}$ in the figure) affects the shape and the end-point of the
spectrum.

Figure 4.11: Illustration of the shape distortion induced by a non-zero
$m_{\beta}$ on the tritium $\beta$-decay spectrum. Image taken from ref.
[278].

As discussed in section 2.2, the deviations from unitarity, and as a
consequence the effect of the phases on $m^{2}_{\beta}$, might be sizeable
precisely in scenarios where $k$ of the $n_{R}$ "heavy" states have a mass of
the order of the eV or keV. These neutrinos would be kinematically allowed to
be produced in a beta decay, hence the electron energy spectrum would be a
superposition of the light neutrino spectrum and the "heavy" neutrino spectrum
[105, 72]. In this case, eq. (4.43) becomes

$\begin{split}K^{2}(E_{e})&\approx(Q-E_{e})\sqrt{(Q-E_{e})^{2}-m^{2}_{\beta}}\times\Theta(Q-E_{e}-m_{\beta})\\\
&+(Q-E_{e})\sum_{k=4}^{n_{R}}|V_{ek}|^{2}\sqrt{(Q-E_{e})^{2}-m^{2}_{k}}\times\Theta(Q-E_{e}-m_{k})\leavevmode\nobreak\
,\end{split}$ (4.47)

The above expression shows that a "heavy" neutrino mass, $m_{k}$ can be
measured by observing a kink of the kinetic energy spectrum at
$E_{e}=Q-m_{k}$, the point where the "heavy" neutrino spectrum ends [105].
Figure 4.12 illustrates how the existence of a light heavy neutrino that is
almost-sterile affects the spectrum with a kink-like signature.

Figure 4.12: Illustration of the almost-sterile neutrino signature in a
differential tritium $\beta$-decay spectrum. For the sake of clarity, a
almost-sterile neutrino with a mass of $10$ keV and an unrealistic large
mixing angle is considered. Image taken from ref. [278].

KATRIN [279, 280, 281, 282, 283] is an experiment that aims to measure
$m_{\beta}$, from the beta decay of Tritium ($H^{3}$) with endpoint $Q=18.6$
keV. The experiment is sensitive to $m_{\beta}>0.2\leavevmode\nobreak\ eV$
[284]. With the data acquired during the first two measurement campaigns of
2019, the collaboration obtained the most stringent limit on $m_{\beta}$ with
$m_{\beta}<0.8$ eV at 90% CL. [285, 278]. TRISTAN (TRitium Investigation on
STerile to Active Neutrino mixing) is a subsequent experiment to KATRIN,
building upon its advancements by utilizing the complete KATRIN source. The
primary objective of TRISTAN is to explore the existence of heavy neutrinos
within the mass range of 1 keV to 18 keV. By mitigating theoretical and
experimental systematic uncertainties, TRISTAN has the potential to
investigate $|V_{14}|^{2}$ values as low as $10^{-7}$ [278, 217]. The planned
upgrade for TRISTAN is scheduled to occur following the completion of the
KATRIN experiment’s $m_{\beta}$ measurement, which is expected to take place
around 2025. At the conclusion of the data collection, it is anticipated that
a total of approximately $70\times 10^{6}$ electrons will be obtained, aiming
to achieve a target sensitivity $m_{\beta}<0.2-0.3$ eV (90% confidence level).
Of particular interest is the observation that the upper bound of the values
of $m_{\beta}$, for the cases of section 2.2 ranged from $0.0093$ eV to
$0.015$ eV, considerably lower than this target sensitivity.

#####  4.1.5.1 $M\sim$ eV

Before KATRIN started acquiring data it was already speculated that it could
probe heavy neutrinos with a eV mass [111, 243, 72]. By utilizing the same
datasets employed for determining the value of $m_{\beta}$, the KATRIN
experiment conducted an investigation into the possible existence of a fourth
neutrino mass eigenstate [285, 278]. This analysis possesses sensitivity to
the range of $m_{4}^{2}\lesssim 1600\leavevmode\nobreak\ \textrm{eV}^{2}$ for
the fourth neutrino mass eigenstate, as well as active-to-sterile mixing
$|V_{14}|^{2}\gtrsim 6\times 10^{-3}$ within the $3\nu+1$ framework [72]. No
significant signal indicating the presence of an eV-scale sterile neutrino was
observed, and an improved exclusion limit was established. The outcomes of the
analysis are summarized in Figure 4.13, which also includes the constraints
derived from previous experiments.

Figure 4.13: eV-scale sterile neutrino exclusion contours at 95% C.L. for the
first and second campaign of KATRIN [285]. Note that $\sin^{2}(\theta_{ee})$
in the figure refers to $|V_{14}|^{2}$ in the text. The combined results of
the two campaigns as well the projected final sensitivity are also shown.
These exclusions are compared to the most recent results from reactor,
gallium, beta decay and double beta decay experiments. References for the
different experimental results used in this plot can be found in ref. [285].
Image taken from ref. [278].

The eV neutrinos in the models from section 2.2 have $|V_{14}|^{2}\sim
10^{-4}$, hence they are not yet ruled out by KATRIN, and may start being
probed in the next round of results.

#####  4.1.5.2 $M\sim$ keV

Prior to the commencement of the operations at KATRIN, preliminary studies
highlighted its potential to investigate almost-sterile neutrinos at the keV
scale[112, 286, 193, 287, 87, 288]. Hence, in 2022, KATRIN also presented the
outcomes of its initial search for keV-scale almost-sterile neutrinos using
tritium data obtained during the commissioning run [289]. No significant
signal indicating the presence of keV-scale sterile neutrinos was observed,
leading to the establishment of exclusion limits. These results significantly
enhance the existing laboratory constraints on the amplitude of active-to-
sterile mixing within the mass range of $0.1$ keV $<m_{4}<1.0$ keV by up to an
order of magnitude. Consequently, an active-sterile mixing amplitude of
$\sin^{2}\theta<5\times 10^{-4}$ is excluded for an almost-sterile neutrino
mass of $m_{4}=300$ eV [289, 278]. These results are summarised in figure
4.14.

Figure 4.14: 95 % C.L. exclusion limit obtained based on the first tritium
data set of KATRIN with respect to the best fit (BF) (blue). The exclusion
limit obtained by comparing the $\chi^{2}$ values to the null hypothesis (NH)
(orange) is in reasonable agreement with the sensitivity (dashed black). The
current laboratory limits were improved (colored shaded areas) on the active-
to-sterile mixing amplitude in a mass range of $0.1$ keV $<m_{4}<1.0$ keV by
up to an order of magnitude. The main result is given by the exclusion limit
with respect to the best fit (blue line). Image taken from ref. [289].

The keV neutrinos in the models from section 2.2 have
$\sin^{2}\theta\in[10^{-6},10^{-5}]$, note that in case Ia of section 2.2
$\sin^{2}\theta\equiv|V_{15}|^{2}$, hence they are not yet ruled out by
KATRIN, and may start being probed in the next round of results.

#####  4.1.5.3 $M>$ keV

Masses beyond the keV scale are not kinematically allowed since the process is
nuclear and its scale is the MeV. For tests for these mass scales see section
4.1.2 and section 4.1.3.

####  4.1.6 CP Violation Effects at low-energy

The $n_{R}\nu$SM contains several CP Violating phases. The unitary phases, the
Dirac phase and the Majorana phases, and the hermitian phases, the phases of
the deviations from unitarity, as described in section 2.3.1. All these phases
play a role in neutrino oscillations, since in the presence of deviations from
unitarity, neutrino oscillation probabilities are modified [54, 57]. This
subject was originally discussed in ref. [31], as an extension of ref. [55].
In order to analyse the CP Violation effects of these phase, it is instructive
to define CP asymmetries $A_{\nu\overline{\nu}}^{\alpha\beta}$ at the level of
oscillation probabilities [290]:

$\displaystyle
A_{\nu\overline{\nu}}^{\alpha\beta}\,\equiv\,\frac{P_{\nu_{\alpha}\to\nu_{\beta}}-P_{\overline{\nu}_{\alpha}\to\overline{\nu}_{\beta}}}{P_{\nu_{\alpha}\to\nu_{\beta}}+P_{\overline{\nu}_{\alpha}\to\overline{\nu}_{\beta}}}\,\equiv\,\frac{\Delta
P_{\alpha\beta}}{P_{\nu_{\alpha}\to\nu_{\beta}}+P_{\overline{\nu}_{\alpha}\to\overline{\nu}_{\beta}}}\,.$
(4.48)

This analysis focus exclusively on the vacuum scenario, acknowledging that in
a more realistic context, the violation of CP and CPT due to the matter-
induced asymmetry encountered by neutrinos needs to be considered. The
principle of CPT invariance gives rise to the relations $\Delta
P_{\alpha\beta}=-\Delta P_{\beta\alpha}$ and $\Delta P_{\alpha\alpha}=0$.
Additionally, the unitarity of the complete mixing matrix imposes the
constraints

$\displaystyle\sum_{\beta}\,\Delta P_{\alpha\beta}=0\,,$ (4.49)

for all indices $\alpha$ and $\beta$ encompassing the complete set, which
includes $e$, $\mu$, $\tau$, $s_{1}$, $s_{2}$, …, $s_{q}$. In a $3\times 3$
unitary framework, these relations indicate the existence of a single
independent difference, which can be selected as $\Delta P_{e\mu}$. On the
other hand, within a $4\times 4$ unitary framework, as demonstrated in ref.
[290], these relations imply the presence of three independent differences,
denoted as $\Delta P_{e\mu}$, $\Delta P_{\mu\tau}$, and $\Delta P_{\tau e}$.

In the case of a $6\times 6$ unitary context, it is determined that there are
10 independent differences $\Delta P_{\alpha\beta}$ (as also mentioned in ref.
[291]), but only the three involving solely active neutrinos hold experimental
significance. Consequently, it is generally anticipated that different values
for $\Delta P_{e\mu}$, $\Delta P_{\mu\tau}$, and $\Delta P_{\tau e}$ will
manifest in a given minimal seesaw-type model. Using Eq. (2.47), with $n$
mostly-sterile neutrinos kinematically accessible at an oscillation
experiment, one finds:

$\displaystyle\Delta
P_{\alpha\beta}\,=\,\frac{4}{(\Theta\Theta^{\dagger})_{\alpha\alpha}(\Theta\Theta^{\dagger})_{\beta\beta}}\,\sum_{i>j}^{3+n}\,\operatorname{Im}\left(\Theta_{\alpha
i}^{*}\,\Theta_{\beta i}\,\Theta_{\alpha j}\,\Theta_{\beta j}^{*}\right)\sin
2\Delta_{ij}\,.$ (4.50)

Even in scenarios where none of the new sterile states are accessible, meaning
$n=0$, it is still anticipated that $\Delta P_{e\mu}$, $\Delta P_{\mu\tau}$,
and $\Delta P_{\tau e}$ will remain independent. This is due to the fact that
the relevant $3\times 3$ mixing submatrix $\Theta$, defined in eq. (2.46),
(which in that case is equivalent to $K$) is not unitary. This means that it
is possible for CP invariance to hold in one oscillation channel, such as
$\nu_{\mu}\rightarrow\nu_{e}$ and yet be violated in another, such as
$\nu_{\mu}\rightarrow\nu_{\tau}$ Indeed, one has:

$\displaystyle\Delta P_{\mu\tau}\,$ $\displaystyle=\,\Delta
P_{e\mu}\,+\,\frac{4}{\prod_{\alpha=e,\mu,\tau}(\Theta\Theta^{\dagger})_{\alpha\alpha}}\,\sum_{i>j}^{3}\,\operatorname{Im}\Big{[}\Theta_{\mu
i}^{*}\,\Theta_{\mu
j}\Big{(}\Theta_{ei}\,\Theta_{ej}^{*}\,(\Theta\Theta^{\dagger})_{\tau\tau}+\Theta_{\tau
i}\,\Theta_{\tau j}^{*}\,(\Theta\Theta^{\dagger})_{ee}\Big{)}\Big{]}\sin
2\Delta_{ij}\,,$ (4.51) $\displaystyle\Delta P_{\tau e}\,$
$\displaystyle=\,\Delta
P_{e\mu}\,-\,\frac{4}{\prod_{\alpha=e,\mu,\tau}(\Theta\Theta^{\dagger})_{\alpha\alpha}}\,\sum_{i>j}^{3}\,\operatorname{Im}\Big{[}\Theta_{ei}^{*}\,\Theta_{ej}\Big{(}\Theta_{\mu
i}\,\Theta_{\mu j}^{*}\,(\Theta\Theta^{\dagger})_{\tau\tau}+\Theta_{\tau
i}\,\Theta_{\tau j}^{*}\,(\Theta\Theta^{\dagger})_{\mu\mu}\Big{)}\Big{]}\sin
2\Delta_{ij}\,.$ (4.52)

In such cases, it is possible for $\Delta P_{e\mu}$ to be zero while $\Delta
P_{\mu\tau}$ and/or $\Delta P_{\tau e}$ are non-zero. It should be noted that
if the mixing submatrix $\Theta$ were unitary, one would have $\Delta
P_{e\mu}=\Delta P_{\mu\tau}=\Delta P_{\tau e}$. Therefore, deviations from
unitarity can serve as a potential source of CP violation. This observation
aligns with the fact that the hermitian part of $K$, defined in eq. (2.84) and
eq. (2.85), contains complex hermitian elements, which can encompass CP-
violating physical phases, eq. (2.86).

#####  4.1.6.1 $M\sim$ eV, keV

In the cases examined in section 2.2, where $n=1,2$, we can derive explicit
expressions for the CP asymmetries that are relevant in the context of short-
baseline oscillations. These expressions can be obtained from the approximate
relations given by equations (2.70) and (2.81) for cases Ib and II,
respectively. For case Ia, it can be observed from equation (2.67) that the CP
asymmetries in that case are negligible in the short-baseline regime. One has,
for case Ib:

$\displaystyle\Delta
P_{e\mu}^{\text{SBL,\>Ib}}\,\simeq\,4\,\operatorname{Im}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\Theta_{\mu 5}\,\Theta_{e5}^{*}\right)\sin
2\Delta_{54}\,,$ (4.53)

while for case II:

$\displaystyle\Delta
P_{e\mu}^{\text{SBL,\>II}}\,\simeq\,8\,\operatorname{Im}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,(KX^{\dagger})_{\mu
2}\,(KX^{\dagger})_{e2}^{*}\right)\sin 2\Delta_{41}\,.$ (4.54)

These asymmetries might be measurable in future short-baseline oscillation
experiments under two conditions:

  * •

There is a mass-squared difference, $\Delta_{ij}$, among any neutrino (light
and heavy) pair of the order of the sensitivity of the experiment. For
instance, active SBL experiment MicroBooNE [292] is sensitive to
$\Delta_{ij}\sim\mathcal{O}(1)\leavevmode\nobreak\ \textrm{eV}^{2}$.

  * •

The mixing between the neutrinos that comprise this pair and the active light
neutrinos, $\Theta_{in}$ with $i=e,\mu,\tau$ and $n=4$ or $n=4,5$, is sizeable
enough. That implies sizeable deviations from unitarity, only possible if
there are heavy neutrinos with a mass in the eV or keV scale, cf. beginning of
chapter 4.

####  4.1.7 Displaced Vertex Signatures

In recent years it has been pointed out how in many realistic models these
heavy neutrinos may have a large lifetime [125, 133, 127, 135]. This means
they can leave a noticeable signature in detectors: a displaced vertex decay
[132, 129, 134]. This can be searched for using recently developed machine
learning techniques [293]. The signal is especially traceable if these
neutrinos have a mass between $1$ and $50$ GeV [130, 127], as in realistic
models with $\mathcal{O}(1)$ Yukawa couplings, these naturally have smaller
mixing with active neutrinos, cf. beginning of chapter 4, hence having larger
lifetimes. Figure 4.15 illustrates this.

Figure 4.15: The heavy neutrino life time as a function of its mass and
coupling calculated with MadWidth. Image taken from ref. [135].

Recently, the possibility of a new signature - a displaced shower-, was
suggested [294]. It is claimed heavy neutrinos masses between $1-6$ GeV can
can be accessed for mixings as low as $|V_{\tau 4}|^{2}\sim 10^{-7}$, enabling
the probing of unique areas of the parameter space in the $\tau$ sector. The
concern mentioned in section 4.1.2 regarding the need to interpret many of
these searches [236] also applies to these kinds of searches, as it also
customary to to assume the presence of a single heavy neutrino mixing with a
single neutrino flavour.

#####  4.1.7.1 $M\in[1,50]$ GeV

A 2020 ATLAS search [295] published results obtained at a center-of-mass
energy of 13 TeV, which places constraints on the mixing of heavy neutrinos
with muon-neutrinos at the level of $|V_{\mu 4}|^{2}\approx 2\times 10^{-6}$
for heavy neutrino masses ranging from $4$ to $10$ GeV, based on a data sample
of $33-36\textrm{fb}^{-1}$ . These results are illustrated in Figure 5.

Figure 4.16: Observed 95% confidence-level exclusion in $|V_{\mu 4}|^{2}$
versus the heavy neutrino mass for the prompt signature (the region above the
black line is excluded) and the displaced signature (the region enclosed by
the red line is excluded). The solid lines show limits assuming lepton-number
violation (LNV) for 50% of the decays and the long-dashed line shows the limit
in the case of lepton-number conservation (LNC). The dotted lines show
expected limits and the bands indicate the ranges of expected limits obtained
within $1\sigma$ and $2\sigma$ of the median limit, reflecting uncertainties
in signal and background yields. Image taken from ref. [295].

In a recent study by CMS [296], the focus was on searching for low-mass heavy
neutrinos. The analysis made use of the complete dataset from Run 2,
corresponding to an integrated luminosity of $138\leavevmode\nobreak\
\textrm{fb}^{-1}$. The investigation specifically targeted the decay of heavy
neutrinos into oppositely charged leptons and a neutrino, resulting in a final
state characterized by three charged leptons, two of which exhibit
displacement. The results of this study are presented in Figure 4.17. The
obtained results impose bounds on the mixing of heavy neutrinos with muon-
neutrinos, with a value of approximately $|V_{\mu 4}|^{2}\approx 4\times
10^{-7}$ for neutrino masses ranging from 8 to 14 GeV. In the case of
electron-neutrino channels, the sensitivity is approximately
$|V_{e4}|^{2}\approx 4\times 10^{-6}$ within the same mass range.

Figure 4.17: The 95% CL limits on $|V_{4e}|^{2}$ (left) and $|V_{4\mu}|^{2}$
(right) as functions of $m_{N}$ for a Majorana heavy neutrino. The area inside
the solid (dashed) black curve indicates the observed (expected) exclusion
region. Older results from the DELPHI and the CMS Collaborations are shown for
reference. Image taken from ref. [296].

The most recent displaced vertex search was done by the ATLAS collaboration
[297], and it addressed the concerns explained in ref. [236] regarding the
need to reinterpret searches considering multiflavour scenarios. Hence, it is
the first study that provides limits for both single-flavour and multiflavour
mixing scenarios, which are motivated by the findings of neutrino flavour
oscillation studies [236]. The search utilizes a dataset of
$139\leavevmode\nobreak\ \textrm{fb}^{-1}$ of proton-proton collision data at
a center-of-mass energy of $13$ TeV. The search is centered around the
production of heavy neutrinos through the decay process of $W$ bosons,
specifically $W\rightarrow Nl_{\alpha}$, where $\alpha=e,\mu$ represents the
flavour of the prompt lepton. The heavy neutrinos subsequently decays into two
oppositely charged leptons and a neutrino: $N\rightarrow
l_{\beta}l_{\alpha}\nu_{\gamma}$ via an intermediate $W^{*}$ boson, or
$N\rightarrow l_{\gamma}l_{\gamma}\nu_{\beta}$ via a $Z^{*}$ boson, where
$\beta$ and $\gamma$ can be either $e$ or $\mu$. The search specifically
focuses on the mixing and mass range (up to $20$ GeV) in which the heavy
neutrino exhibits long lifetimes.

No excess was observed and limits were set at 95% CL on the squared mixing
coefficient $|V_{N\alpha}|^{2}$, where $\alpha=e,\mu,\tau$ in different
scenarios for neutrino masses masses in the approximate range $M\in[3,15]$
GeV. The results of the search are summarised in fig. 4.18.

Figure 4.18: The observed and expected 95% CL limits on $|V_{N\alpha}|^{2}$
vs. $m_{N}$ in the Majorana-limit case, with green and yellow bands showing
the one and two standard deviation spreads for the expected limits. (b,c) The
observed limits in the 2QDH scenario with inverted (IH) and normal (NH) mass
hierarchy, and in 1SFH scenarios. 1SFH refers to the model where the heavy
neutrino only couples with one flavour of active neutrinos. 2QDH refers to the
more realistic case of two quasi-degenerate neutrinos with masses in the range
$m_{N}\in[1,20]$ GeV that couple to all active neutrino flavours. Image taken
from ref. [297].

####  4.1.8 Signals at Future Experiments: FCC, DUNE, HyperK

Several experiments are scheduled for the upcoming years, including the High-
Luminosity Large Hadron Collider (HL-LHC), which will collect data at a
significantly higher luminosity than the LHC. This increase in luminosity will
greatly enhance the statistical significance of future analyses conducted with
the acquired data. It is anticipated that these analyses significantly improve
the constraints on the direct detection of heavy neutrinos [90, 91, 92].

The next step in the energy frontier is the the Future Circular Collider (FCC)
[298, 299]. The initial phase of the FCC consists of an $e^{+}e^{-}$ collider
known as FCC-ee. This collider is designed to serve as a Higgs factory, an
electroweak and top factory, operating at the highest luminosities. It will
operate at four distinct center-of-mass energies, specifically targeting the
$Z$ pole, the $WW$ threshold, the $ZH$ production peak, and the $tt$
threshold. Subsequently, in the second phase, the FCC-ee would be followed by
the FCC-hh, a proton collider with a center-of-mass energy of $100$ TeV.

During the electron-positron stage of the Future Circular Collider (FCC),
there will be a focus on conducting high-precision measurements. However, the
FCC-ee also presents significant opportunities for the exploration and
discovery of new phenomena. By combining precise measurements and direct
searches, both indirect and direct evidence for new physics can be obtained.
In particular, the high-luminosity Z pole run at FCC-ee holds exceptional
potential for the detection of heavy neutrinos [88, 123, 128]. The primary
production mode for heavy neutrinos at this stage will be through the process
$Z\rightarrow\nu N$, where $N$ is the heavy neutrino and $\nu$ a light active
neutrino, followed by the $N$ decaying into off-shell $W$ or $Z$ bosons. The
FCC-ee is expected to exhibit excellent sensitivity above the charm mass and
extending up to the mass of the W boson, for various mixing angles with the
known neutrinos. In refs. [88, 123, 128, 300, 301, 302], sensitivity down to
heavy-light mixing of $|V_{\nu N}|^{2}\sim 10^{-8}-10^{-12}$ was achieved,
covering a broad parameter space for heavy neutrino masses in the GeV scale.

As for the FCC-hh, it will have a much higher sensitivity than the FCC-ee to
higher masses, both for prompt and displaced searches [90, 91, 92, 217, 303].
For instance, refs. [221, 217] suggest that it could be sensitive to masses up
to $10$ TeV ($100$ GeV) to the level of $|V_{\nu N}|^{2}\sim 10^{-3}$ (
$|V_{\nu N}|^{2}\sim 10^{-4}$) and much smaller mixings can potentially be
probed at lower masses [221].

The FCC and other projected colliders such as the International Linear
Collider (ILC) or the Compact Linear Collider (CLIC) also hold great promise
in detecting heavy neutrino-antineutrino oscillations [136, 131, 304], a
phenomenon that occurs when there is a pair of quasi-degenerate heavy
neutrinos, and also in probing some specific models where leptogenesis
explains the BAU [305, 306].

Regarding future neutrino oscillation experiments, the most notable ones are
JUNO [307], DUNE [107, 267] and HyperKamiokande (HyperK) [308]. JUNO is a
reactor anti-neutrino experiment in China that intends on performing a
$>3\sigma$ mass ordering determination, and to measure $\sin^{2}\theta_{12}$,
$\Delta m^{2}_{21}$ and $|\Delta m^{2}_{21}|$ with a precision better than
$0.6\%$ [309]. DUNE, a long-baseline experiment in the USA, and HyperK, an
improved iteration of SuperKamiokande in Japan, have similar goals - measuring
$\sin^{2}2\theta_{13}$, $\sin^{2}\theta_{23}$ and $\Delta m^{2}_{32}$ with
great precision with a special emphasis in the determination of the octant of
$\theta_{23}$ and the discovery (or not) of CP violation in the leptonic
sector via the precise determination of $\delta_{CP}$. Furthermore, both DUNE
and HyperK will have a nearby detector complex situated close to the neutrino
source. These detectors serve multiple purposes, including observing and
tracking the neutrino flux generated by the facility. Moreover, the high-
intensity beam interactions with the target offer the possibility of
generating heavy neutrinos and other novel particles, particularly through the
decay of light mesons. Given the energy and intensity of the beams, the near
detectors of these experiments provide an intriguing and unexplored search
area for heavy neutrinos undergoing decay while in motion, with masses of up
to $2$ GeV[217]. Despite having similar goals and operational principles, DUNE
and HyperK have largely uncorrelated systematic uncertainties. This is due to
their differences in neutrino energies, far detector target nuclei, and the
effects of calibration, reconstruction, and event selection. As a result,
these two experiments complement each other greatly.

###  4.2 Vector-like Quarks

In recent decades, extensive searches for heavy quarks have been conducted at
the Tevatron and the LHC. Despite the absence of concrete evidence for the
existence of heavy quarks, experimental limits have been progressively
tightened.

The presence of VLQs has a significant impact on various electroweak precision
observables. In the SM, the absence of FCNC at tree level in the quark sector
results in the suppression of several quark-related processes, such as
$D^{0}$-$\overline{D}^{0}$ mixing, as they only occur at the one-loop through
a version of the GIM mechanism. However, VLQs introduce FCNC at tree level,
and the amplitude of these processes is proportional to $|F^{q}_{ij}|^{2}$,
which are defined in equations (3.22) and (3.35). The matrix $F$ controls FCNC
and is proportional to the deviations from unitarity of the quark mixing
matrix $V$, as defined in equations (3.19) and (3.34), cf. eq. (3.23). A
detailed explanation of the connection among $F$, $V$, FCNC, and the
deviations from unitarity of $V_{\textrm{CKM}}$ (the upper-left $3\times 3$
block of $V$) can be found in section 3.1.2. The absence of significant
signals in the searches for these rare quark processes can be interpreted in
one of two ways: either there are no additional quarks, or the elements of
$F^{q}$ are highly suppressed. The latter scenario could indicate the
existence of VLQs with large masses, as elaborated in section 3.1.2, as large
masses naturally suppress FCNC in VLQ models [33]. Consequently, the precision
frontier is imposing increasingly stringent constraints on the matrix elements
of $F$ and $V$ and, consequently, the deviations from unitarity of
$V_{\textrm{CKM}}$. A discussion on the effects of VLQ on these observables
and the most recent experimental bounds can be found in section 4.2.2.

In terms of direct searches, the bounds on the matrix elements of $V$,
particularly those connecting the SM quarks with the new quark states, are
becoming increasingly restrictive. Otherwise, these new quark states would
have already been detected at the LHC, considering the current center of mass
energy it operates at. Moreover, the lower mass bounds on VLQs have been
steadily increasing over the last decade and VLQs that decay predominantly to
third generation quarks are essentially ruled out for masses below the TeV
scale [33, 310, 311, 312, 313, 314, 315, 316].

The majority of model-independent constraints are derived from searches
focusing on pair production, as the production mechanism of heavy quarks is
attributed to Quantum Chromodynamics (QCD). Typically, the outcome of a pair
production search is an exclusion, which establishes, at a specific confidence
level (e.g. 95% C.L.), a lower bound for VLQ masses, indicating that masses
below this threshold are ruled out by the search.

The single production of VLQs is highly dependent on the specific theoretical
model, as the production mechanism involves electroweak interactions.
Consequently, the outcomes of single production VLQ searches are typically
presented in terms of constraints on cross sections and couplings of the heavy
quarks to SM quarks, for different heavy quark masses. In the context of a
particular model, it becomes feasible to establish bounds on VLQ masses at
specific confidence levels. It is worth noting that if the coupling between
light quarks and VLQs is substantial, the single production of VLQs is
anticipated to exhibit a higher cross section compared to pair production,
particularly in proton colliders like the LHC, as outlined in refs. [178, 179,
34]. The results of the most recent direct searches for VLQ are discussed in
section 4.2.1.

Before proceeding, it is crucial to address the customary conventions,
assumptions, and terminology found in the relevant literature. The majority of
searches are conducted with the assumption that VLQs predominantly decay to
third-generation quarks or solely decay to third-generation quarks. However,
it is important to acknowledge that such an assumption may not always be
justified, particularly in scenarios where VLQs are proposed to address the
CKM unitarity problem, cf. ref. [31] and section 3.2.1. Nevertheless, this
hypothesis of only decaying into third-generation quarks, as expressed in eq.
(4.56), has become ingrained in the literature as a form of "conventional
wisdom".

Figure 4.19:  Allowed branching ratios for singlet VLQs $T$ (left) and $B$
(right) assuming they only couple with the third family quarks. VLQs in non-
trivial $SU(2)_{L}$ multiplets are also shown. Red dots correspond to
$M_{T,B}=2\leavevmode\nobreak\ \mathrm{TeV}$ and are representative of the
asymptotic high-mass value. The crosses indicate
$M_{T}=640\leavevmode\nobreak\ \mathrm{GeV}$ or $M_{B}=590\leavevmode\nobreak\
\mathrm{GeV}$. The blue lines refer to intermediate masses. Taken from ref.
[177].

Figure 4.19 depicts the potential BRs of VLQs, based on their gauge group
representation, with the constraint that they exclusively decay into third-
generation quarks. While these BRs are contingent on the masses of VLQs, it is
evident that $SU(2)$ singlet VLQs, limited to decaying into third-generation
quarks, exhibit a distinct range of permissible BRs. These values closely
align with the specific points where

$\displaystyle\textrm{BR}(T\rightarrow Wb)$
$\displaystyle=0.5\,,\,\,\textrm{BR}(T\rightarrow
Zt)=0.25\,,\,\,\textrm{BR}(T\rightarrow ht)=0.25\,,\text{ and}$ (4.55)
$\displaystyle\textrm{BR}(B\rightarrow Wt)$
$\displaystyle=0.5\,,\,\,\textrm{BR}(B\rightarrow
Zb)=0.25\,,\,\,\textrm{BR}(B\rightarrow hb)=0.25\,.$

To maintain consistency, the term "$SU(2)$ singlet (VLQ) BRs" will be employed
to refer to the BRs in eq. (4.55), as it is commonly used in the majority of
literature. Nevertheless, it is important to note that these BRs specifically
pertain to the allowed branching ratios of $SU(2)$ singlet VLQs under the
condition that they exclusively decay into third-generation quarks. Under that
same strong condition but allowing for deviations from the values of eq.
(4.55), one has instead the weaker constraints

$\displaystyle\textrm{BR}(T\rightarrow Wb)+\textrm{BR}(T\rightarrow
Zt)+\textrm{BR}(T\rightarrow ht)$ $\displaystyle=1\,,\text{ and}$ (4.56)
$\displaystyle\textrm{BR}(B\rightarrow Wt)+\textrm{BR}(B\rightarrow
Zb)+\textrm{BR}(B\rightarrow hb)$ $\displaystyle=1\,.$

It is crucial to bear in mind that these assumptions may be violated in more
complex scenarios, where other significant decay channels can arise. Examples
of such extended scenarios can be found in references such as [317, 318, 319,
320, 321, 322, 323, 324, 325, 326, 327]. This motivates searches for VLQs
which decay predominantly to light quarks $u,d,s$ [328, 329]. Because of this,
in section 4.2.1, where the results of the most recent searches at ATLAS and
CMS will be discussed, only mass bounds that stem from a search whose
assumptions roughly correspond to an $SU(2)$ singlet VLQ with realistic BRs
will be covered.

Future VLQ search prospects are discussed in section 4.2.3.

####  4.2.1 Direct Searches

Pair production and single production are the two primary mechanisms by which
heavy VLQs can be generated. The specific production mechanism and subsequent
decay modes of VLQs at high energies are distinctive characteristics of
different VLQ models. In most experimental analyses, it is assumed that VLQs
predominantly decay into or exclusively decay to third-generation quarks.
Under this assumption, previous studies have extensively examined pair
production [330] and single production [177] of VLQs. These studies have
emphasized the relationship between decay widths, electroweak observables, and
the production modes of VLQs. Various types of VLQs, including singlets,
doublets, and triplets of $SU(2)_{L}$, have been considered in these
investigations. The decay widths of isosinglet VLQs, the focus of this thesis,
via the $Z$, $h$, and $W$ bosons can be expressed at the tree-level as
follows:

$\displaystyle\Gamma\big{(}Q_{i}\rightarrow Z\psi^{q}_{j}\big{)}$
$\displaystyle=\frac{G_{F}M^{3}_{i}}{16\pi\sqrt{2}}\lambda[r_{Z},r_{j}]\Big{[}\big{(}1-r^{2}_{j}\big{)}^{2}+r^{2}_{Z}\big{(}1+r^{2}_{j}\big{)}-2r^{4}_{Z}\Big{]}\big{|}F^{q}_{ji}\big{|}^{2}\,,$
(4.57) $\displaystyle\Gamma\big{(}Q_{i}\rightarrow h\psi^{q}_{j}\big{)}$
$\displaystyle=\frac{G_{F}M^{3}_{i}}{16\pi\sqrt{2}}\lambda[r_{h},r_{j}]\Big{[}\big{(}1+r^{2}_{j}\big{)}^{2}+4r^{2}_{j}-r^{2}_{h}\big{(}1+r^{2}_{j}\big{)}\Big{]}\big{|}F^{q}_{ji}\big{|}^{2}\,,$
(4.58) $\displaystyle\Gamma\big{(}U_{i}\rightarrow W^{+}\psi^{d}_{j}\big{)}$
$\displaystyle=\frac{G_{F}M^{3}_{i}}{8\pi\sqrt{2}}\lambda[r_{W},r_{j}]\Big{[}\big{(}1-r^{2}_{j}\big{)}^{2}+r^{2}_{W}\big{(}1+r^{2}_{j}\big{)}-2r^{4}_{W}\Big{]}\big{|}V_{ij}\big{|}^{2}\,,$
(4.59) $\displaystyle\Gamma\big{(}D_{i}\rightarrow W^{-}\psi^{u}_{j}\big{)}$
$\displaystyle=\frac{G_{F}M^{3}_{i}}{8\pi\sqrt{2}}\lambda[r_{W},r_{j}]\Big{[}\big{(}1-r^{2}_{j}\big{)}^{2}+r^{2}_{W}\big{(}1+r^{2}_{j}\big{)}-2r^{4}_{W}\Big{]}\big{|}V_{ji}\big{|}^{2}\,,$
(4.60)

where $\psi^{q}$ ($q=u,d$) contains the SM and heavy quarks, with
$\psi^{u}=(u,c,t,U_{1},\ldots,U_{n_{u}})^{T}$ and
$\psi^{d}=(d,s,b,D_{1},\ldots,D_{n_{d}})^{T}$. In the ratio
$r_{j}=m_{j}/M_{i}$, the $m_{j}$ is the mass of the final-state quark and
$M_{i}$ the mass of the VLQ — the decaying particle. The function
$\lambda[r,r^{\prime}]$ is given by

$\lambda^{2}\big{[}a,b\big{]}=1-2\big{(}a^{2}+b^{2}\big{)}+\big{(}a^{2}-b^{2}\big{)}^{2}\,,$
(4.61)

and for the Fermi constant $G_{F}$ one has, at tree level,
$G_{F}=\frac{\sqrt{2}g^{2}}{8m_{W}^{2}}$. The quark mixing matrix $V$ was
defined in equations (3.19) and (3.34), while the matrices
$F^{u}=VV^{\dagger}$ and $F^{d}=V^{\dagger}V$, controlling the FCNC in the up
sector and down sector, respectively, were defined in eq (3.23).

For sufficiently heavy quarks decaying into SM quarks, with all $r\ll 1$, the
expressions above reduce to

$\displaystyle\Gamma\big{(}Q_{i}\rightarrow
Zq_{j}\big{)}\simeq\frac{G_{F}M^{3}_{i}}{16\pi\sqrt{2}}\big{|}F^{q}_{ji}\big{|}^{2}$
$\displaystyle\,,\quad\Gamma\big{(}U_{i}\rightarrow
W^{+}d_{j}\big{)}\simeq\frac{G_{F}M^{3}_{i}}{8\pi\sqrt{2}}\big{|}V_{ij}\big{|}^{2}\,,$
(4.62) $\displaystyle\Gamma\big{(}Q_{i}\rightarrow
hq_{j}\big{)}\simeq\frac{G_{F}M^{3}_{i}}{16\pi\sqrt{2}}\big{|}F^{q}_{ji}\big{|}^{2}$
$\displaystyle\,,\quad\Gamma\big{(}D_{i}\rightarrow
W^{-}u_{j}\big{)}\simeq\frac{G_{F}M^{3}_{i}}{8\pi\sqrt{2}}\big{|}V_{ji}\big{|}^{2}\,,$
(4.63)

which can be used to derive the following useful relations,

$\displaystyle\frac{\Gamma(U_{i}\rightarrow Zu_{j})}{|F^{u}_{ji}|^{2}}$
$\displaystyle\simeq\frac{\Gamma(U_{i}\rightarrow
hu_{j})}{|F^{u}_{ji}|^{2}}\simeq\frac{\Gamma(U_{i}\rightarrow
W^{+}d_{j})}{2|V_{ij}|^{2}}\,,$ (4.64)
$\displaystyle\frac{\Gamma(D_{i}\rightarrow Zd_{j})}{|F^{d}_{ji}|^{2}}$
$\displaystyle\simeq\frac{\Gamma(D_{i}\rightarrow
hd_{j})}{|F^{d}_{ji}|^{2}}\simeq\frac{\Gamma(D_{i}\rightarrow
W^{-}u_{j})}{2|V_{ji}|^{2}}\,.$ (4.65)

The limiting values in eq. (4.55) can be seen in the relations of eq. (4.64)
and eq. (4.65), taking $|F_{ji}|\simeq|V_{ji}|$ or $|F_{ji}|\simeq|V_{ij}|$,
valid when the index $i$ refers to the heavy quark with high mass and $j$
refers to the third-generation quark, neglecting CKM mixing, cf. (3.47) and
eq. (3.49). It is important to note the relation between the decay and single
production of VLQs. In a single production event, the vertex involving the
heavy quark is of the same form as that of one of its decays. Evidently,
single production also depends on the flavour couplings between VLQ, SM quarks
and $Z$, $W$ and $h$ bosons. These couplings consist of the elements of the
matrices $F^{q}$ and $V$, previously defined.

#####  4.2.1.1 Searches for Pair-Produced VLQs that Decay Predominantly to
$t,b$ Quarks