$m$ (GeV) | $\begin{bmatrix}[r](2.11-5.58\,i)\times 10^{-11}&(1.29+1.65\,i)\times 10^{-9}&11.2-10.9\,i\\\ (0.85+2.22\,i)\times 10^{-10}&(-5.29+3.99\,i)\times 10^{-9}&10.4+0.4\,i\\\ (-0.26+1.98\,i)\times 10^{-10}&(-4.51-1.05\,i)\times 10^{-9}&-10.5-34.6\,i\end{bmatrix}$
$M$ (GeV) | $\begin{bmatrix}[r]8.93\times 10^{-10}&4.45\times 10^{-11}&1.28\times 10^{-13}\\\ 4.45\times 10^{-11}&1.00\times 10^{-6}&6.22\times 10^{-11}\\\ 1.28\times 10^{-13}&6.22\times 10^{-11}&5.00\times 10^{14}\end{bmatrix}$
$K$ | $\begin{bmatrix}[r]-0.797+0.071\,i&0.578+0.006\,i&-0.115+0.096\,i\\\ 0.293-0.086\,i&0.575+0.027\,i&0.719+0.010\,i\\\ -0.516-0.004\,i&-0.570+0.020\,i&0.606\end{bmatrix}$
$KX^{\dagger}$ | $\begin{bmatrix}[r]0.024-0.057\,i&(1.29+1.65\,i)\times 10^{-3}&(-2.24+2.18\,i)\times 10^{-14}\\\ 0.093+0.223\,i&(-5.29+3.99\,i)\times 10^{-3}&(-2.08+0.08\,i)\times 10^{-14}\\\ -0.026+0.199\,i&(-4.51-1.05\,i)\times 10^{-3}&(-2.10+6.92\,i)\times 10^{-14}\end{bmatrix}$
$X$ | $\begin{bmatrix}[r]-0.003-0.015\,i&0.102+0.023\,i&0.050-0.317\,i\\\ (-5.12+1.72\,i)\times 10^{-4}&(0.46-4.33\,i)\times 10^{-3}&(-7.30-2.18\,i)\times 10^{-3}\\\ (0.23+5.33\,i)\times 10^{-14}&(-3.44+2.75\,i)\times 10^{-14}&(0.36-4.41\,i)\times 10^{-14}\end{bmatrix}$
$O_{c}$ (tree level)   | $\begin{bmatrix}[r]-0.53+0.12\,i&0.22-1.12\,i&-1.41-0.22\,i\\\ 0.22+0.56\,i&-1.50-0.13\,i&-0.30+1.03\,i\\\ 1.00-0.06\,i&0.23+0.25\,i&-0.14-0.01\,i\end{bmatrix}$
Masses | $\begin{matrix}[l]m_{1}\simeq 1.06\times 10^{-3}\text{ eV}\,,\quad&m_{2}\simeq 8.48\times 10^{-3}\text{ eV}\,,\quad&m_{3}\simeq 5.02\times 10^{-2}\text{ eV}\,,\,\\\ M_{1}\simeq 1.00\text{ eV}\,,\,&M_{2}\simeq 1.00\text{ keV}\,,\,&M_{3}\simeq 5.00\times 10^{14}\text{ GeV}\end{matrix}$
$3\nu$ $\Delta m^{2}$ | $\Delta m^{2}_{\odot}=\Delta m^{2}_{21}\simeq 7.08\times 10^{-5}\text{ eV}^{2}\,,\,\quad\,\,\,\,\Delta m^{2}_{\text{atm}}=\Delta m^{2}_{31}\simeq 2.52\times 10^{-3}\text{ eV}^{2}$
$3\nu$ mixing angles  | $\sin^{2}\theta_{12}\simeq 0.344\,,\,\quad\,\,\,\,\sin^{2}\theta_{23}\simeq 0.585\,,\,\quad\,\,\,\,\sin^{2}\theta_{13}\simeq 0.0236$
$3\nu$ CPV phases  | $\delta\simeq 1.21\pi\,,\,\quad\,\,\,\,\alpha_{21}\simeq 0.06\pi\,,\,\quad\,\,\,\,\alpha_{31}\simeq 0.06\pi$
$\sin^{2}2\vartheta^{(i)}_{\mu e}$ | $\sin^{2}2\vartheta_{\mu e}^{(4)}\simeq 8.8\times 10^{-4}\,,\,\quad\,\,\,\,\sin^{2}2\vartheta_{\mu e}^{(5)}\simeq 7.7\times 10^{-10}$
Table 6a: Numerical benchmark for case Ia. The ordering of light neutrinos is
NO. From the input matrices $m$ and $M$, and taking into account one-loop
corrections, the other quantities here listed follow. It should be noted that
$O_{c}$ of Eq. (2.20) is only defined at tree level. Values for the mixing
angles and CPV phases of the $3\nu$-framework in the standard parametrisation
[3] are extracted by identifying the unitary matrix $V$ with a unitary
$3\times 3$ PMNS mixing matrix.

The numerical results for scenario Ia are presented in Table 6a, including the
one-loop correction from Eq.(2.54). This scenario features three mostly-active
light neutrinos, as well as three mostly-sterile neutrinos with masses of
approximately 1 eV for $M_{1}$, 1 keV for $M_{2}$, and a few orders of
magnitude below the grand unification scale for $M_{3}$, with a value of
approximately $10^{14}$ GeV. The keV-scale neutrino may serve as a potential
dark matter candidate, as suggested in Refs.[87, 106].

For case Ia, there are two kinematically accessible almost-sterile heavy
neutrinos, meaning $n=2$ in Eq.(2.47). In the context of a LBL experiment like
DUNE [107]), the expression of Eq. (2.47) applied to the transition
probability of muon to electron (anti-)neutrinos can, in this case, be
approximated by:

$\displaystyle
P^{\text{LBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq$
$\displaystyle\frac{1}{(\Theta\Theta^{\dagger})_{\mu\mu}(\Theta\Theta^{\dagger})_{ee}}\Bigg{[}\left|(\Theta\Theta^{\dagger})_{\mu
e}\right|^{2}$ (2.63)
$\displaystyle-4\sum_{i>j}^{3}\,\operatorname{Re}\left(\Theta_{\mu
i}^{*}\,\Theta_{ei}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin^{2}\Delta_{ij}\pm
2\sum_{i>j}^{3}\,\operatorname{Im}\left(\Theta_{\mu
i}^{*}\,\Theta_{ei}\,\Theta_{\mu j}\,\Theta_{ej}^{*}\right)\sin 2\Delta_{ij}$
$\displaystyle-4\,\cdot\,\frac{1}{2}\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)-4\,\cdot\,\frac{1}{2}\,\operatorname{Re}\left(\Theta_{\mu
5}^{*}\,\Theta_{e5}\,\sum_{j=1}^{4}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\Bigg{]}\,,$

where terms depending on $\Delta_{4j},\,\Delta_{5j}\gg 1$ have been replaced
by their averaged versions ($\sin^{2}\Delta_{ij}\to 1/2$, $\sin
2\Delta_{ij}\to 0$).

The loss of unitarity and a zero-distance effect are indicated by the
normalization and the first term in this equation, while the effects of the
two lightest mostly-sterile states on oscillations are explicitly represented
by the last two terms.

If one is in a situation similar to that of the numerical benchmark of Table
6a, for which
$|(\Theta\Theta^{\dagger})_{\mu\mu}(\Theta\Theta^{\dagger})_{ee}-1|$ and
$|(\Theta\Theta^{\dagger})_{\mu e}|^{2}$ are negligible, this expression can
be further approximated by:

$\displaystyle
P^{\text{LBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq\,P^{\text{LBL,
}3\nu}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}+\frac{1}{2}\sin^{2}2\vartheta^{(4)}_{\mu
e}\,,$ (2.64)

where it was defined a $3\nu$-framework transition probability which, however,
incorporates the effects of deviations of $K$ from unitarity,

$\displaystyle P^{\text{LBL,
}3\nu}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\equiv\,-4\sum_{i>j}^{3}\,\operatorname{Re}\left(\Theta_{\mu
i}^{*}\,\Theta_{ei}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin^{2}\Delta_{ij}\pm
2\sum_{i>j}^{3}\,\operatorname{Im}\left(\Theta_{\mu
i}^{*}\,\Theta_{ei}\,\Theta_{\mu j}\,\Theta_{ej}^{*}\right)\sin
2\Delta_{ij}\,,$ (2.65)

and have used the definition of Eq. (2.55), the unitarity of the full $6\times
6$ mixing matrix, and the fact that $|\Theta_{\alpha
4}|^{2}(=|(KX^{\dagger})_{\alpha 1}|^{2})\gg|\Theta_{\alpha
5}|^{2}(=|(KX^{\dagger})_{\alpha 2}|^{2})\gg|(KX^{\dagger})_{\alpha 3}|^{2}$.

In an SBL experiment (e.g. MicroBooNE [108]), the relevant form of Eq. (2.47)
for
$\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}$
transitions is:

$\displaystyle
P^{\text{SBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq$
$\displaystyle\frac{1}{(\Theta\Theta^{\dagger})_{\mu\mu}(\Theta\Theta^{\dagger})_{ee}}\Bigg{[}\left|(\Theta\Theta^{\dagger})_{\mu
e}\right|^{2}-4\,\cdot\,\frac{1}{2}\,\operatorname{Re}\left(\Theta_{\mu
5}^{*}\,\Theta_{e5}\,\sum_{j=1}^{4}\,\Theta_{\mu j}\,\Theta_{ej}^{*}\right)$
(2.66) $\displaystyle-4\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin^{2}\Delta_{41}\pm
2\,\operatorname{Im}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin 2\Delta_{41}\Bigg{]}\,,$

with $\Delta_{41}\simeq\Delta_{42}\simeq\Delta_{43}$, and where terms
depending on $\Delta_{5j}\gg 1$ have been replaced by their averaged versions
($\sin^{2}\Delta_{5j}\to 1/2$, $\sin 2\Delta_{5j}\to 0$). In this context, one
is sensitive to oscillations due to the scale of the mass-squared differences
$\Delta m^{2}_{4j}$ with $j=1,2,3$, whereas the oscillations pertaining to
smaller mass-squared differences have not yet had a chance to develop.

Finally, if one is in a condition similar to that of the numerical benchmark,
this expression can be simply approximated by:

$\displaystyle
P^{\text{SBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq\,\sin^{2}2\vartheta^{(4)}_{\mu
e}\,\sin^{2}\Delta_{41}\,,$ (2.67)

where once again one has taken into account the unitarity of the full mixing
matrix and the fact that $|(KX^{\dagger})_{\alpha
1}|^{2}\gg|(KX^{\dagger})_{\alpha 2}|^{2}\gg|(KX^{\dagger})_{\alpha 3}|^{2}$.

To further study the parameter space of case Ia, numerical seesaw structures
were created by setting the values of the unitary part $V$ of the mixing
matrix $K$, the mostly-active and mostly-sterile masses in $d$ and $D$, and by
scanning the complex orthogonal matrix $O_{c}$ which is represented as a
combination of three complex rotations and a sign that corresponds to its
determinant. The goal is to find seesaw structures that are similar to the
benchmark, for which was set (at tree level) $M_{1}=1$ eV and $M_{2}=1$ keV,
while considering three different values for the heaviest neutrino mass,
$M_{3}\in{10^{13},10^{14},5\times 10^{14}}$ GeV. The minimum lightest neutrino
mass, $m_{\text{min}}$, is examined in the range of $[10^{-4},0.1]$ eV, while
the remaining elements of $d$ are set by determining the solar and atmospheric
mass differences. The $3\nu$ mixing angles and the Dirac CPV phase in $V$, as
well as the three-neutrino mass-squared differences, are selected to be the
central values obtained from the global fit in Ref. [104]. It is important to
emphasize that, as with the numerical benchmark in Table 6a, the $3\nu$ mixing
angles and CPV phases for three neutrinos obtained while identifying $V$ with
a unitary $3\times 3$ mixing matrix may differ slightly from the mixing angles
and CPV phases derived from the full $6\times 6$ mixing matrix $\mathcal{V}$,
due to deviations from unitarity. In Figure 2.2, values of
$\sin^{2}2\vartheta_{\mu e}^{(4)}$ from Eq. (2.55) are displayed against the
lightest neutrino mass for the numerical examples of case Ia. Only the points
for which $\textrm{Tr}\left[m,m^{\dagger}\right]\in[0.01,1],v^{2}$ are
displayed.111By selecting suitable values for $M_{3}$ and $O_{c}$, it is
possible to avoid extremely small Yukawa couplings.

Figure 2.2: Active-sterile mixing measure $\sin^{2}2\vartheta_{\mu e}^{(4)}$
versus the lightest neutrino mass $m_{\text{min}}$ from a scan of the case-Ia
parameter space, with NO ($m_{\text{min}}=m_{1}$). The heavy spectrum at tree
level has $M_{1}=1$ eV and $M_{2}=1$ keV, while three values of the heaviest
mass are considered, $M_{3}=10^{13}\,(10^{14})\,[5\times 10^{14}]$ GeV,
corresponding to the black (dark blue) [light blue] points in the scatter
plot. The horizontal green band shows the $99.7\%$ CL interval of Ref. [105],
while the vertical red exclusion band is obtained by combining the most
stringent bound on the sum of light neutrino masses from cosmology,
$\sum_{i}m_{i}<0.12$ eV ($95\%$ CL) [109, 110], with the $3\sigma$ ranges of
mass-squared differences. The dark green contour delimits the region inside
which loop-stable points have been found (see text), while the benchmark of
Table 6a is marked in yellow.

The horizontal green band in the figure marks the range of
$\sin^{2}2\vartheta_{\mu e}^{(4)}$ that is preferred by the global fit from
Ref.[105], which was cited earlier in the section. The dark green contour
outlines the area where relatively loop-stable points can be found. These are
points that, even after the implementation of the one-loop correction from
Eq.(2.54), still have three-neutrino mass-squared differences and mixing
angles (extracted from $V$) within the $3\sigma$ ranges of the fit [104]. The
areas outside of the dark green lines are not necessarily ruled out, as it is
possible to imagine a scenario where the incorrect tree-level values lead to
acceptable values after corrections. As illustrated in the figure, increasing
the magnitude of $M_{3}$ results in a decrease in the magnitude of light
neutrino masses, thus preventing excessively large values of $m_{\text{min}}$.
The approximations used in deriving the oscillation formulae of Eqs. (2.64)
and (2.67) hold for all the plotted points.

Some quantities of potential phenomenological relevance, unrelated to neutrino
oscillations, include the effective electron neutrino mass in $\beta$-decay,
$m_{\beta}$, the absolute value of the effective neutrino Majorana mass
controlling the rate of neutrinoless double beta
($(\beta\beta)_{0\nu}$-)decay, $|m_{\beta\beta}|$, and the $\mu\to e\gamma$
branching ratio, $BR(\mu\rightarrow e\gamma)$. For all stable numerical
examples pertaining to case Ia that stable under loop corrections, the
branching ratio of $\mu$ to electron gamma decay is so small as to be
unobservable ($BR(\mu\rightarrow e\gamma)\ll 10^{-30}$). The effective
electron neutrino mass in beta decay ($m_{\beta}$) and the magnitude of the
effective Majorana mass that governs the rate of neutrinoless double beta
decay ($|m_{\beta\beta}|$) are still too low to be measured by current or
near-future experiments, with bounds of $m_{\beta}<9.4$ meV and
$|m_{\beta\beta}|<6.7$ meV. The calculation of $|m_{\beta\beta}|$ has taken
into account the effects of the eV- and keV-scale neutrinos.

In scenarios where active-sterile mixing is relatively high, such as this one,
experiments like KATRIN in the future may be able to detect the presence of
sterile neutrinos with masses around $\mathcal{O}$(eV) [111]. The sensitivity
of these experiments is determined by
$|(KX^{\dagger}){e1}|^{2}=|\mathcal{V}{e4}|^{2}$, which for the loop-stable
numerical examples of this case, is found to be limited to
$|(KX^{\dagger})_{e4}|^{2}\lesssim 0.02$. Sterile neutrinos with masses of
$\mathcal{O}$(keV) may be detected through kink-like signatures in next-
generation beta decay experiments, even if the mixing between active and
almost-sterile neutrinos is small
($|(KX^{\dagger}){e2}|^{2}=|\mathcal{V}{e5}|^{2}\sim 10^{-6}$) [112]. Case Ib:
$M_{1}\sim M_{2}\ll M_{3}$

|  Case Ib numerical benchmark
---|---
$m$ (GeV) | $\begin{bmatrix}[r](0.46-2.57\,i)\times 10^{-10}&(2.37+0.54\,i)\times 10^{-10}&11.24-2.72\,i\\\ (-5.50-1.04\,i)\times 10^{-10}&(0.68-6.20\,i)\times 10^{-10}&8.90-27.50\,i\\\ (-3.69+1.78\,i)\times 10^{-10}&(-1.60-4.45\,i)\times 10^{-10}&-1.85+0.43\,i\end{bmatrix}$
$M$ (GeV) | $\begin{bmatrix}[r]2.88\times 10^{-9}&8.24\times 10^{-11}&1.41\times 10^{-11}\\\ 8.24\times 10^{-11}&2.87\times 10^{-9}&1.42\times 10^{-11}\\\ 1.41\times 10^{-11}&1.42\times 10^{-11}&1.00\times 10^{14}\end{bmatrix}$
$K$ | $\begin{bmatrix}[r]-0.799+0.137\,i&0.558+0.001\,i&0.116-0.071\,i\\\ 0.272-0.172\,i&0.582-0.036\,i&-0.695+0.014\,i\\\ -0.480+0.099\,i&-0.560+0.141\,i&-0.620-0.019\,i\end{bmatrix}$
$KX^{\dagger}$ | $\begin{bmatrix}[r]0.039+0.077\,i&0.067-0.040\,i&(-1.12+0.27\,i)\times 10^{-13}\\\ 0.156-0.105\,i&-0.097-0.170\,i&(-0.89+2.75\,i)\times 10^{-13}\\\ 0.061-0.140\,i&-0.115-0.071\,i&(1.85-0.43\,i)\times 10^{-14}\end{bmatrix}$
$X$ | $\begin{bmatrix}[r]-0.003+0.009\,i&0.073-0.064\,i&-0.168-0.196\,i\\\ -0.009-0.005\,i&0.049+0.078\,i&0.170-0.185\,i\\\ (1.40-5.37\,i)\times 10^{-14}&(-1.47-1.74\,i)\times 10^{-13}&(0.37+2.24\,i)\times 10^{-13}\end{bmatrix}$
$O_{c}$ (tree level)   | $\begin{bmatrix}[r]-1.06-0.51\,i&-1.10-1.29\,i&-1.51+1.30\,i\\\ 0.75-1.08\,i&1.40-0.83\,i&-1.46-1.35\,i\\\ 0.91+0.31\,i&-0.60+0.44\,i&0.32-0.05\,i\end{bmatrix}$
Masses | $\begin{matrix}[l]m_{1}\simeq 0.24\times 10^{-3}\text{ eV}\,,\quad&m_{2}\simeq 8.76\times 10^{-3}\text{ eV}\,,\quad&m_{3}\simeq 5.00\times 10^{-2}\text{ eV}\,,\,\\\ M_{1}\simeq 3.00\text{ eV}\,,\,&M_{2}\simeq 3.16\text{ eV}\,,\,&M_{3}\simeq 1.00\times 10^{14}\text{ GeV}\end{matrix}$
$3\nu$ $\Delta m^{2}$ | $\Delta m^{2}_{\odot}=\Delta m^{2}_{21}\simeq 7.66\times 10^{-5}\text{ eV}^{2}\,,\,\quad\,\,\,\,\Delta m^{2}_{\text{atm}}=\Delta m^{2}_{31}\simeq 2.50\times 10^{-3}\text{ eV}^{2}$
$3\nu$ mixing angles  | $\sin^{2}\theta_{12}\simeq 0.327\,,\,\quad\,\,\,\,\sin^{2}\theta_{23}\simeq 0.562\,,\,\quad\,\,\,\,\sin^{2}\theta_{13}\simeq 0.0232$
$3\nu$ CPV phases  | $\delta\simeq 1.26\pi\,,\,\quad\,\,\,\,\alpha_{21}\simeq 0.11\pi\,,\,\quad\,\,\,\,\alpha_{31}\simeq 0.22\pi$
$\sin^{2}2\vartheta^{(i)}_{\mu e}$ | $\sin^{2}2\vartheta_{\mu e}^{(4)}\simeq 1.1\times 10^{-3}\,,\,\quad\,\,\,\,\sin^{2}2\vartheta_{\mu e}^{(5)}\simeq 9.2\times 10^{-4}$
Table 6b: The same as Table 6a for case Ib.

The numerical data for the benchmark corresponding to this case is given in
Table 6b. In addition to the three mainly active neutrinos with low levels of
activity, the spectrum consists of three mainly sterile neutrinos, whose
masses are roughly equal to $M_{1}\sim M_{2}\sim 3$ eV, with a difference of
$M_{2}^{2}-M_{1}^{2}$ being approximately 1 eV2. The mass of the third
neutrino, $M_{3}$, is around $10^{14}$ GeV.

In the spectrum of scenario Ib, $n=2$ in equation (2.47). When applied to the
transition probability of muon to electron (anti-)neutrinos in a Long-Baseline
(LBL) context, the expression in equation (2.47) can be approximated using the
same formula (2.63) as in scenario Ia. Once again, the last two terms in that
equation explicitly show the effects of the two lightest mostly-sterile states
in oscillations. In a scenario similar to the benchmark outlined in Table 6b,
where $|(\Theta\Theta^{\dagger}){\mu\mu}(\Theta\Theta^{\dagger}){ee}-1|$ and
$|(\Theta\Theta^{\dagger})_{\mu e}|^{2}$ are considered to be negligible, the
expression can be simplified further to:

$\displaystyle
P^{\text{LBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq\,P^{\text{LBL,
}3\nu}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,+\,\frac{1}{2}\Big{[}\sin^{2}2\vartheta^{(4)}_{\mu
e}+\sin^{2}2\vartheta^{(5)}_{\mu e}+4\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\Theta_{\mu 5}\,\Theta_{e5}^{*}\right)\Big{]}\,,$ (2.68)

where the unitarity of the full $6\times 6$ mixing matrix was used, and the
fact that $|(KX^{\dagger})_{\alpha 1}|^{2}\sim|(KX^{\dagger})_{\alpha
2}|^{2}\gg|(KX^{\dagger})_{\alpha 3}|^{2}$. This means that
$|(KX^{\dagger}){\alpha 2}|^{2}$ (and thus $\sin^{2}2\vartheta^{(5)}{\mu e}$)
cannot be ignored in comparison to $|(KX^{\dagger}){\alpha 1}|^{2}$ (and
$\sin^{2}2\vartheta^{(4)}{\mu e}$), as was done in the previous scenario.

In a SBL context, the relevant form of Eq. (2.47) for
$\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}$
transitions in case Ib is:

$\displaystyle
P^{\text{SBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq$
$\displaystyle\frac{1}{(\Theta\Theta^{\dagger})_{\mu\mu}(\Theta\Theta^{\dagger})_{ee}}\Bigg{[}\left|(\Theta\Theta^{\dagger})_{\mu
e}\right|^{2}$ (2.69)
$\displaystyle-4\,\cdot\,\frac{1}{2}\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)-4\,\cdot\,\frac{1}{2}\,\operatorname{Re}\left(\Theta_{\mu
5}^{*}\,\Theta_{e5}\,\sum_{j=1}^{3}\,\Theta_{\mu j}\,\Theta_{ej}^{*}\right)$
$\displaystyle-4\,\operatorname{Re}\left(\Theta_{\mu
5}^{*}\,\Theta_{e5}\,\Theta_{\mu
4}\,\Theta_{e4}^{*}\right)\sin^{2}\Delta_{54}\pm
2\,\operatorname{Im}\left(\Theta_{\mu 5}^{*}\,\Theta_{e5}\,\Theta_{\mu
4}\,\Theta_{e4}^{*}\right)\sin 2\Delta_{54}\Bigg{]}\,,$

where terms depending on the large $\Delta_{4j}$ and $\Delta_{5j}$ ($j=1,2,3$)
have been replaced by their averaged versions.

It can be seen that this scenario does not conform to the typical$3+2$
scenario, outlined in reference [105], as the mass-squared differences $\Delta
m^{2}_{4j}$ and $\Delta m^{2}_{5j}$ for $j=1,2,3$ are around 10 eV2. As a
result, oscillations caused by the mass-squared difference of $\Delta
m^{2}_{54}\sim 1$ eV2 can be detected, while oscillations arising from larger
differences are averaged out and those driven by smaller mass-squared
differences are underdeveloped. If one is in a condition similar to that of
the numerical benchmark, this expression can be approximated by:

$\displaystyle
P^{\text{SBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq\,\frac{1}{2}\Big{(}\sin^{2}2\vartheta^{(4)}_{\mu
e}+\sin^{2}2\vartheta^{(5)}_{\mu e}\Big{)}$
$\displaystyle+4\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\Theta_{\mu
5}\,\Theta_{e5}^{*}\right)\cos^{2}\Delta_{54}$ (2.70) $\displaystyle\mp
2\,\operatorname{Im}\left(\Theta_{\mu 4}^{*}\,\Theta_{e4}\,\Theta_{\mu
5}\,\Theta_{e5}^{*}\right)\sin 2\Delta_{54}\,,$

where once again it was taken into account the unitarity of the full mixing
matrix and the fact that $|(KX^{\dagger})_{\alpha
1}|^{2}\sim|(KX^{\dagger})_{\alpha 2}|^{2}\gg|(KX^{\dagger})_{\alpha 3}|^{2}$.
It is important to note that in this scenario, unlike the typical$3+2$ case,
oscillations are dependent on the square of the cosine of the corresponding
$\Delta_{ij}$.

Figure 2.3: The average of $\sin^{2}2\vartheta_{\mu e}^{(4)}$ and
$\sin^{2}2\vartheta_{\mu e}^{(5)}$ versus the lightest neutrino mass
$m_{\text{min}}$, from a scan of the case-Ib parameter space, with NO
($m_{\text{min}}=m_{1}$). The heavy spectrum at tree level has $M_{1}=3.00$ eV
and $M_{2}=3.16$ eV, while three values of the heaviest mass are considered,
$M_{3}=10^{13}\,(10^{14})\,[5\times 10^{14}]$ GeV, corresponding to the black
(dark blue) [light blue] points in the scatter plot. The vertical red band
corresponds to the cosmological constraint, as in Figure 2.2. The dark green
contour delimits the region inside which loop-stable points have been found,
while the benchmark of Table 6b is marked in yellow.

To examine the parameter space of case Ib more thoroughly, numerical seesaw
structures similar to the benchmark have been generated using the method
outlined in the discussion of case Ia. The lightest two neutrino masses were
specified at tree level as $M_{1}=3.00$ eV and $M_{2}=3.16$ eV, and three
different values were considered for the heaviest neutrino mass:
$M_{3}\in{10^{13},\leavevmode\nobreak\ 10^{14},\leavevmode\nobreak\ 5\times
10^{14}}$ GeV. In Figure 2.3, the average of $\sin^{2}2\vartheta_{\mu
e}^{(4)}$ and $\sin^{2}2\vartheta_{\mu e}^{(5)}$ is plotted against the
lightest neutrino mass for the numerical examples found in case Ib. The latter
quantity represents the expected order of magnitude of signals in SBL and LBL
experiments. Only the points where $\textrm{Tr}\left[m,m^{\dagger}\right]$
falls within the range $[0.01,1],v^{2}$ are included. As in the previous case,
the dark green contour outlines the area where relatively stable points can be
found with loop corrections. An increase in the value of $M_{3}$ will result
in a decrease of the light neutrino masses. The approximations used in
deriving the oscillation formulae of Eqs. (2.68) and (2.70) are valid for all
the plotted points.

For all numerical examples pertaining to case Ib which are stable under loop
corrections, $BR(\mu\rightarrow e\gamma)\ll 10^{-30}$ is unobservably small,
while one finds $m_{\beta}<9.3$ meV and $|m_{\beta\beta}|<4.6$ meV, still out
of reach of present and near-future experiments. In the computation of
$|m_{\beta\beta}|$, the effects of both eV-scale neutrinos are important and
have been taken into account. One additionally finds the bounds
$|(KX^{\dagger})_{e4}|^{2},\,|(KX^{\dagger})_{e5}|^{2}\lesssim 0.01$ for the
loop-stable numerical examples of this case.

#####  2.2.2.2 Case II: $\lambda_{\nu}=(1,-1,-1)$, $M_{1}\ll M_{2}\sim M_{3}$

For this case, one is instead led to $\lambda_{L}=(1,1,1)$. In the exact
conservation limit, the mass matrices are given by:

$\displaystyle m\,=\,\left(\begin{array}[]{ccc}a&0&0\\\ b&0&0\\\
c&0&0\end{array}\right)\,,\quad M\,=\,\left(\begin{array}[]{ccc}0&A&B\\\
A&0&0\\\ B&0&0\end{array}\right)\,.$ (2.77)

In this limit, there are two degenerate neutrinos with mass
$\sqrt{|A|^{2}+|B|^{2}}$ and opposite CP parities, forming a single heavy
Dirac particle.

Breaking the symmetry will generate light neutrino masses and another massive
sterile state with a mass that can be much smaller than $|A|$ and $|B|$.
Additionally, it will lift the mass degeneracy for the Dirac neutrino,
resulting in a pseudo-Dirac neutrino pair [113, 114].

As noted in [115], a strong mass degeneracy implies a symmetry in the
$KX^{\dagger}$ block of the mixing matrix, specifically $(KX^{\dagger}){\alpha
2}\simeq\pm i(KX^{\dagger}){\alpha 3}$ ($\alpha=e,\mu,\tau$). Such a relation
has a crucial impact in reducing the effect of large masses $M_{2}$ and
$M_{3}$ on the one-loop correction $\delta M_{L}$ as seen in Eq. (2.54). This
relation indicates a proximity to the limit of lepton number conservation,
even if $(KX^{\dagger})_{\alpha 2}$ and $(KX^{\dagger})_{\alpha 3}$ are not
extremely suppressed, which allows for relatively large Yukawa couplings, in
spite of the fact that $M_{2}\simeq M_{3}$ are not as large as the $M_{3}$ in
case I. This is evident from the form of the Dirac mass matrix $m$, defined in
eq. (2.36). The mass of the pseudo-Dirac pair can be in the TeV range [102,
115, 116, 117, 118], as the lightest neutrino masses are safeguarded by
approximate lepton number conservation. The same symmetry and related effects
are also observed in the examples discussed in section 2.1.

|  Case II numerical benchmark
---|---
$m$ (GeV) | $\begin{bmatrix}[r]-4.15+0.47\,i&(4.51-1.49\,i)\times 10^{-9}&(-1.59+0.13\,i)\times 10^{-9}\\\ 3.98+6.17\,i&(-5.04-4.64\,i)\times 10^{-9}&(1.52+2.31\,i)\times 10^{-9}\\\ 1.53+6.58\,i&(-1.90-2.68\,i)\times 10^{-9}&(0.59+2.59\,i)\times 10^{-9}\end{bmatrix}$
$M$ (GeV) | $\begin{bmatrix}[r]2.18\times 10^{-6}&1390&2.96\\\ 1390&-2.19\times 10^{-6}&5.52\times 10^{-7}\\\ 2.96&5.52\times 10^{-7}&3.33\times 10^{-9}\end{bmatrix}$
$K$ | $\begin{bmatrix}[r]0.825+0.061\,i&0.536+0.027\,i&-0.092+0.108\,i\\\ -0.302+0.113\,i&0.581-0.017\,i&0.728-0.052\,i\\\ 0.455+0.054\,i&-0.599+0.075\,i&0.651+0.002\,i\end{bmatrix}$
$KX^{\dagger}$ | $\begin{bmatrix}[r]0.063-0.056\,i&(2.11-0.24\,i)\times 10^{-3}&(-0.24-2.11\,i)\times 10^{-3}\\\ -0.066-0.147\,i&(-2.03-3.13\,i)\times 10^{-3}&(-3.13+2.03\,i)\times 10^{-3}\\\ -0.021-0.036\,i&(-0.79-3.35\,i)\times 10^{-3}&(-3.35+0.79\,i)\times 10^{-3}\end{bmatrix}$
$X$ | $\begin{bmatrix}[r]0.042+0.014\,i&0.007+0.099\,i&-0.069+0.140\,i\\\ (1.48+0.64\,i)\times 10^{-3}&(2.30+0.66\,i)\times 10^{-4}&(-2.11+4.89\,i)\times 10^{-3}\\\ (-0.64+1.48\,i)\times 10^{-3}&(-0.66+2.30\,i)\times 10^{-4}&(-4.89-2.11\,i)\times 10^{-3}\end{bmatrix}$
$O_{c}$ (tree level)   | $\begin{bmatrix}[r]-0.21+0.62\,i&-1.02+0.06\,i&0.62+0.31\,i\\\ (-1.11+2.54\,i)\times 10^{4}&(-1.05+2.42\,i)\times 10^{3}&(2.55+1.12\,i)\times 10^{4}\\\ (-2.54-1.11\,i)\times 10^{4}&(-2.42-1.05\,i)\times 10^{3}&(-1.12+2.55\,i)\times 10^{4}\end{bmatrix}$
Masses | $\begin{matrix}[l]m_{1}\simeq 4.65\times 10^{-3}\text{ eV}\,,\quad&m_{2}\simeq 9.47\times 10^{-3}\text{ eV}\,,\quad&m_{3}\simeq 5.01\times 10^{-2}\text{ eV}\,,\,\\\ M_{1}\simeq 1.00\text{ eV}\,,\,&M_{2}\simeq 1390\text{ GeV}\,,\,&M_{3}\simeq 1390\text{ GeV}\end{matrix}$
$3\nu$ $\Delta m^{2}$ | $\Delta m^{2}_{\odot}=\Delta m^{2}_{21}\simeq 6.80\times 10^{-5}\text{ eV}^{2}\,,\,\quad\,\,\,\,\Delta m^{2}_{\text{atm}}=\Delta m^{2}_{31}\simeq 2.48\times 10^{-3}\text{ eV}^{2}$
$3\nu$ mixing angles  | $\sin^{2}\theta_{12}\simeq 0.298\,,\,\quad\,\,\,\,\sin^{2}\theta_{23}\simeq 0.563\,,\,\quad\,\,\,\,\sin^{2}\theta_{13}\simeq 0.0212$
$3\nu$ CPV phases  | $\delta\simeq 1.32\pi\,,\,\quad\,\,\,\,\alpha_{21}\simeq 1.99\pi\,,\,\quad\,\,\,\,\alpha_{31}\simeq 0.02\pi$
$\sin^{2}2\vartheta^{(i)}_{\mu e}$ | $\sin^{2}2\vartheta_{\mu e}^{(4)}\simeq 7.4\times 10^{-4}$
Table 6c: The same as Table 6a for case II. For this benchmark,
$M_{3}-M_{2}\simeq 7.6$ eV $\ll M_{2,3}$.

The numerical information for the benchmark scenario of case II is provided in
Table 6c. In this case, apart from the three light mostly-active neutrinos,
there is also a mostly-sterile neutrino with a mass around $1$ eV and a pair
of quasi-degenerate neutrinos with masses close to $1$ TeV. The light neutrino
mass spectrum in this scenario is similar to that of the conventional $\nu$MSM
[119, 120]. However, in the $\nu$MSM, the lightest neutrino has a mass of
about 1 keV, making it a warm dark matter candidate, while the two heavier
neutrinos with masses around 1 GeV generate the baryon asymmetry in the
universe. In contrast, in the scenario under consideration, the lightest
neutrino has a mass around 1 eV and the two heavier neutrinos have masses
around 1 TeV.

From Table 6c one sees that the symmetry in the last two columns of
$KX^{\dagger}$ is tied to an analogous symmetry in the last two rows of $X$
and of $O_{c}$. The latter can be understood from Eqs. (2.23) and (2.20).

The symmetry observed in the last two columns of $KX^{\dagger}$ is also
reflected in the last two rows of $X$ and $O_{c}$, as seen in Table 6c. This
is due to the relationship between $X$, $O_{c}$, and $KX^{\dagger}$ as
described in Eqs. (2.23) and (2.20).

For the case II spectrum, one should consider $n=1$ in the expression in
Eq.(2.47). In a long-baseline experiment, the transition probability of muon
to electron (anti-)neutrinos can be approximately expressed as described in
Eq.(2.47)

$\displaystyle
P^{\text{LBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq$
$\displaystyle\frac{1}{(\Theta\Theta^{\dagger})_{\mu\mu}(\Theta\Theta^{\dagger})_{ee}}\Bigg{[}\left|(\Theta\Theta^{\dagger})_{\mu
e}\right|^{2}-4\,\cdot\,\frac{1}{2}\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu j}\,\Theta_{ej}^{*}\right)$
(2.78) $\displaystyle-4\sum_{i>j}^{3}\,\operatorname{Re}\left(\Theta_{\mu
i}^{*}\,\Theta_{ei}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin^{2}\Delta_{ij}\pm
2\sum_{i>j}^{3}\,\operatorname{Im}\left(\Theta_{\mu
i}^{*}\,\Theta_{ei}\,\Theta_{\mu j}\,\Theta_{ej}^{*}\right)\sin
2\Delta_{ij}\Bigg{]}\,,$

where terms depending on $\Delta_{4j}\gg 1$ have been replaced by their
averaged versions. If the conditions are similar to those of the benchmark in
Table 6c where
$|(\Theta\Theta^{\dagger}){\mu\mu}(\Theta\Theta^{\dagger}){ee}-1|$ and
$|(\Theta\Theta^{\dagger})_{\mu e}|^{2}$ are negligible, the expression for
the muon to electron (anti-)neutrino transition probability in a LBL context
can be simplified to:

$\displaystyle
P^{\text{LBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq\,P^{\text{LBL,
}3\nu}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}+\frac{1}{2}\sin^{2}2\vartheta^{(4)}_{\mu
e}+4\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,(KX^{\dagger})_{\mu
2}\,(KX^{\dagger})_{e2}^{*}\right)\,.$ (2.79)

In this case, the unitarity of the full $6\times 6$ mixing matrix has been
utilized, as well as the approximate symmetry $(KX^{\dagger}){\alpha 2}\simeq
i,(KX^{\dagger}){\alpha 3}$. Additionally, if $|\Theta_{\alpha
4}|^{2}=|(KX^{\dagger}){\alpha 1}|^{2}\gg|(KX^{\dagger}){\alpha
2}|^{2}\simeq|(KX^{\dagger})_{\alpha 3}|^{2}$, the last term in the expression
can be ignored, resulting in Eq. (2.64) from case Ia.

In a SBL context, the relevant form of Eq. (2.47) for
$\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}$
transitions in case II is:

$\displaystyle
P^{\text{SBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}\,\simeq\,\frac{1}{(\Theta\Theta^{\dagger})_{\mu\mu}(\Theta\Theta^{\dagger})_{ee}}\Bigg{[}\left|(\Theta\Theta^{\dagger})_{\mu
e}\right|^{2}$ $\displaystyle-4\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin^{2}\Delta_{41}$ (2.80) $\displaystyle\pm
2\,\operatorname{Im}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,\sum_{j=1}^{3}\,\Theta_{\mu
j}\,\Theta_{ej}^{*}\right)\sin 2\Delta_{41}\Bigg{]}\,,$

with $\Delta_{41}\simeq\Delta_{42}\simeq\Delta_{43}$.

One is sensitive to oscillations due to the mass-squared differences $\Delta
m^{2}_{4j}$ with $j=1,2,3$, while the oscillations related to smaller mass-
squared differences have not yet emerged. In a situation resembling the
numerical benchmark, this expression can be estimated as:

$\displaystyle
P^{\text{SBL}}_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}$
$\displaystyle\simeq\,\left[\sin^{2}2\vartheta^{(4)}_{\mu
e}+8\,\operatorname{Re}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,(KX^{\dagger})_{\mu
2}\,(KX^{\dagger})_{e2}^{*}\right)\right]\sin^{2}\Delta_{41}$ (2.81)
$\displaystyle\mp 4\,\operatorname{Im}\left(\Theta_{\mu
4}^{*}\,\Theta_{e4}\,(KX^{\dagger})_{\mu
2}\,(KX^{\dagger})_{e2}^{*}\right)\sin 2\Delta_{41}\,,$

where once again the unitarity of the full mixing matrix has been taken into
account, as well as the relation $(KX^{\dagger})_{\alpha 2}\simeq
i\,(KX^{\dagger})_{\alpha 3}$. If also $|(KX^{\dagger})_{\alpha
1}|^{2}\gg|(KX^{\dagger})_{\alpha 2}|^{2}\simeq|(KX^{\dagger})_{\alpha
3}|^{2}$, then the two terms containing $(KX^{\dagger})_{\alpha 2}$ in this
equation can be neglected and one recovers Eq. (2.67) of case Ia.

Figure 2.4:  Active-sterile mixing measure $\sin^{2}2\vartheta_{\mu e}^{(4)}$
versus the lightest neutrino mass $m_{\text{min}}$ from a scan of the case-II
parameter space, with NO ($m_{\text{min}}=m_{1}$). The heavy spectrum at tree
level has $M_{1}=1$ eV, while three values of the heaviest quasi-degenerate
masses are considered, $M_{2}\simeq M_{3}=8\,v\,(14\,v)\,[20\,v]$,
corresponding to the black (dark blue) [light blue] points in the scatter
plot. Here, $v\simeq 174$ GeV is the Higgs VEV. The horizontal green band
shows the $99.7\%$ CL interval of Ref. [105], and the vertical red band
corresponds to the cosmological constraint, as in Figure 2.2. The dark green
contour delimits the region inside which loop-stable points have been found,
while the benchmark of Table 6c is marked in yellow.

To further explore the parameter space of case II, numerical seesaw structures
were generated resembling the benchmark in Table 6c, by adopting a method
similar to case Ia. It was set $M_{1}=1$ eV at tree level and three different
values for the second heaviest neutrino mass, $M_{2},(\simeq
M_{3})\in{8,14,20}\leavevmode\nobreak\ ,\leavevmode\nobreak\ v$, where
$v\simeq 174$ GeV represents the Higgs VEV. The mass splitting $M_{3}-M_{2}$
was varied within the range $[0.02,200]$ eV. Figure 2.4 displays the values of
$\sin^{2}2\vartheta_{\mu e}^{(4)}$ in Eq. (2.55) plotted against the lightest
neutrino mass for the numerical examples in case II. Only points for which
$\textrm{Tr}\left[m\,m^{\dagger}\right]\in[0.001,\,1]\,v^{2}$ are kept.
Similar to before, the green band across the horizontal axis represents the
range of $\sin^{2}2\vartheta_{\mu e}^{(4)}$ preferred by the global fit
described in Ref.[105]. The dark green contour outlines the area where
relatively loop-stable points are present. The approximations used in deriving
the oscillation formulae of Eqs. (2.79) and (2.81) are valid for all the
plotted points.

For the numerical examples of case II that remain stable under loop
corrections, the values of $BR(\mu\rightarrow e\gamma)$ can approach the upper
bound of $4.2\times 10^{-13}$ set by MEG. Our scan eliminated points with
higher branching ratios. The benchmark in Table 6c yields $BR(\mu\rightarrow
e\gamma)\simeq 2.0\times 10^{-13}$. These effects are detectable with the MEG
II update [121], which is expected to improve the current sensitivity of MEG
by an order of magnitude. For the loop-stable numerical examples in this case,
the following limits are obtained: $m_{\beta}<15$ meV, $|m_{\beta\beta}|<27$
meV, and $|(KX^{\dagger}){e4}|^{2}\lesssim 0.02$. While KATRIN aims to improve
the current limit on $m\beta$ to $0.2$ eV, the next generation of
$(\beta\beta){0\nu}$-decay experiments [122] may be able to probe values of
$|m{\beta\beta}|\gtrsim 10^{-2}$ eV.

Concerning the prospect of detecting the heavy neutrino pair in future
collider searches, the review [123] is recommended. If the heavy neutrino pair
has a mass in the $1-100$ GeV range and is sufficiently long-lived, it could
potentially produce displaced vertex signatures [124, 125, 126, 127, 128, 129,
130, 131, 132, 133, 134, 135] and resolvable neutrino-antineutrino
oscillations at colliders [136]. Finally, the pseudo-Dirac pair of case II
might play a role in explaining the baryon asymmetry of the Universe through
resonant leptogenesis [137, 138].

However, one should take into account the washout from the interactions of the
lighter sterile neutrino species and to reconcile the light sterile neutrino
paradigm with cosmology, it may be necessary to consider non-standard
interactions.

The numerical examples presented are only for illustration purposes, but they
support the claim that models with an approximate lepton number symmetry and
an eV-scale sterile neutrino mass are viable and may contribute to the
explanation of SBL anomalies.

###  2.3 General Remarks

####  2.3.1 On CP Violating Phases

In table 2.5, the physical parameters of the 3$\nu$SM are summarised. It is
important to parametrise all 6 phases. A general parameterisation for a
$3\times 3$ complex orthogonal $O_{c}$ is given by [96, 139]

$O_{c}=\pm\leavevmode\nobreak\
O_{c}^{23}O_{c}^{13}O_{c}^{12}\leavevmode\nobreak\ ,$ (2.82)

where, for instance,

$O_{c}^{23}=\begin{pmatrix}1&0&0\\\
0&\cos{(r_{23}e^{if_{23}})}&\sin{(r_{23}e^{if_{23}})}\\\
0&-\sin{(r_{23}e^{if_{23}})}&\cos{(r_{23}e^{if_{23}})}\end{pmatrix}$ (2.83)

and $O_{c}^{12}$, $O_{c}^{13}$ have an obviously similar definition. $U_{K}$
has a well-known definition [3], as the $3\times 3$ unitary part of the
leptonic mixing matrix, equivalent to the parameterisation of
$U_{\textrm{PMNS}}$ in eq. (2.27). It is important to note that, eq. (2.29)
can be rewritten as

$K=(\mathds{1}_{3}-\eta)U_{K}=U_{K}\leavevmode\nobreak\
H_{R}\leavevmode\nobreak\ ,$ (2.84)

the unitary part on both decompositions is equal and the hermitian parts are
related by the equation

$H_{R}=(\mathds{1}_{3}+X^{\dagger}X)^{-1/2}=U_{K}^{\dagger}\leavevmode\nobreak\
(\mathds{1}_{3}-\eta)U_{K}\leavevmode\nobreak\ ,$ (2.85)

by the polar decomposition theorem. Due to the rephasing of the charged
leptons, 3 phases can be removed from the left of $K$. The phases of $U_{K}$
are named the unitary phases of the mixing matrix: $\delta$ and the Majorana
phases $\alpha_{1}$, $\alpha_{2}$, defined in eq. (2.27). The phases of
$H_{R}$, $p_{ij}$, are named the hermitian phases of the mixing matrix, or the
phases of the deviations from unitarity.

$H_{R}=\begin{pmatrix}h_{11}&h_{12}e^{ip_{12}}&h_{13}e^{ip_{13}}\\\
h_{12}e^{-ip_{12}}&h_{22}&h_{23}e^{ip_{23}}\\\
h_{13}e^{-ip_{13}}&h_{23}e^{-ip_{23}}&h_{33}\\\
\end{pmatrix}\leavevmode\nobreak\ ,$ (2.86)

where $p_{ij}$ are linear functions of the $O_{c}$ phases $f_{ij}$. This means
that if all $f_{ij}$ can be written as $k\pi$ then the same can be done for
$p_{ij}$, with $k\in\mathbb{N}_{0}$. In the literature, the unitary phases are
also known as low-energy phases and the hermitian phases as high-energy phases
[140, 141, 142, 143]. This is because if deviations from $3\times 3$ unitarity
are negligible the hermitian phases are essentially impossible to measure at
low energy, and only have effects at high energy, possibly being the source of
leptogenesis.
Thus, the 6 physical phases in the $3\nu SM$ are controlled by $\delta$,
$\alpha_{1}$, $\alpha_{2}$ \- the phases of $U_{K}$ and $f_{12}$, $f_{13}$ and
$f_{23}$ \- the phases of $O_{c}$ ( $p_{12}$, $p_{13}$ and $p_{23}$ \- the
phases of $H_{R}$).

####  2.3.2 On the Parameter Space

The parameterisation used in this study was developed in Ref. [30] for heavy
neutrinos with TeV-scale masses, in the quasi-decoupled regime. This is
distinct from the decoupled/seesaw regime, which is more commonly studied in
the literature and where heavy neutrinos are entirely decoupled from light
ones, and deviations from $3\times 3$ unitarity are negligible. This work
demonstrates that this parameterisation can also be applied to other spectra
of heavy neutrinos, including non-decoupled regimes with heavy almost-sterile
neutrinos with masses close to the light ones. Unlike other parameterisations
used in the literature, this one can be used for this purpose without
significant numerical deviations from the exact result. In this discussion,
$m_{i}$ refers to the masses of light neutrinos and $M_{j}$ refers to the
masses of heavy neutrinos.

Regarding its flexibility, in section 2.1 it was discussed the quasi-decoupled
case where $m_{i}/M_{j}\ll 1$, and from Eq.(2.20), it is clear that $X$ can
only have significant entries and deviations from unitarity if $O_{c}$ also
has considerable ($\gg 1$) entries. On the other hand, for the Dirac matrix
given in Eq.(2.37), and considering the perturbativity condition
$m_{ij}\lesssim m_{t}$, the size of the product $O_{c}^{\dagger}\sqrt{D}$ is
limited, which can counterbalance the suppression from $\sqrt{d}$.

Using the same logic for the decoupled/seesaw regime, where heavy neutrino
masses are near the GUT scale, one can observe that $X$ can only have
significant entries if $O_{c}$ also has significant entries. However,
achieving an $X$ with substantial entries in this regime would require entries
in $O_{c}$ that are not compatible with perturbative entries in $m$ generated
by electroweak symmetry breaking (EWSB). As a result, $O_{c}$ must have
entries of $\mathcal{O}(1)$, which results in small entries for $X$ and small
deviations from unitarity. The non-decoupled regimes discussed in section 2.2
present similarities to both the aforementioned regimes.

In those cases, when the complex orthogonal matrix $O_{c}$ has entries of
order $\mathcal{O}(1)$ (which is "small" for such a matrix), it allows for $X$
to have significant entries, leading to connections with short-baseline
anomalies. This is because some of the ${m_{i}}/{M_{j}}$ can be around
$\mathcal{O}(10^{-2}\text{--}10^{-3})$ when $M_{1(,2)}\sim\text{eV}$, as shown
in Eq. (2.20). However, embedding this eV neutrino in a type-I seesaw with 3
sterile neutrinos and with reasonable Yukawa couplings $m_{ij}$ of EWSB
origin, $m_{ij}\sim\mathcal{O}(1-0.01)\,m_{t}$, while keeping keeping $X$
sizeable, leads to non-trivial constraints on $O_{c}$. In particular, spectra
where the highest heavy neutrino(s) mass(es) is (are) not extremely large
(e.g. at the GeV – TeV scale) will need an $O_{c}$ with some sizeable entries
like in the quasi-decoupled regime. On the other hand, spectra where one of
the $M_{j}$ is near the GUT scale require an $O_{c}$ with $\mathcal{O}(1)$
entries, like in the decoupled regime. These requirements on the size of some
entries of $O_{c}$ are necessary to achieve reasonable Yukawa couplings.

Other parameterisations in the literature, such as those in [96, 52], also
begin with an exact parameterisation of $\mathcal{V}$. However, these
approaches define the mixing matrix using an infinite series in a matrix that
controls the deviations from unitarity. To perform explicit calculations, a
truncated series is necessary, leading to a good approximation only for
$M_{j}>100$ GeV, as discussed in section 2.1, and resulting in small
deviations from unitarity. This parameterisation is not suitable for scenarios
in which deviations from unitarity are significant, as in the case of a non-
decoupled sterile neutrino sector. The parameterisation used in this work,
however, is appropriate for such scenarios and may provide a direct connection
to symmetries in an underlying model.

Figure 2.5 is an illustration of the whole parameter space of the
$n_{R}\nu$SM. It is a density plot of the determinant of $d$, hence the
darkest the blue the higher the value of

$det(d)=m_{1}m_{2}m_{3}\leavevmode\nobreak\ .$ (2.87)

In the $x$-axis one can find

$det(d_{X}^{2})=(d_{X}^{1}d_{X}^{2}d_{X}^{3})^{2}\leavevmode\nobreak\ ,$
(2.88)

where $(d_{X}^{i})^{2}$ are the eigenvalues of $X^{\dagger}X$ and $d_{X}^{2}$
a diagonal matrix with them in the diagonal in increasing order. Hence,
$det(d_{X}^{2})$ is a good theoretical measure of the size of deviations from
$3\times 3$ unitarity. Note that due to the properties of orthogonal complex
matrices, see. eq. (2.20), one will always have
$(d_{X}^{3})^{2}\gg(d_{X}^{2})^{2}>(d_{X}^{1})^{2}$.

As for the $y$-axis one has

$det(D)=M_{1}M_{2}M_{3}\leavevmode\nobreak\ .$ (2.89)

Figure 2.5: Projection of the parameter space of the $n_{R}\nu$SM in the
deviations from unitarity - heavy neutrino masses space. The pink and orange
lines are just placeholders to depict the log scale used and to show how fast
$m_{1}$ approaches zero for lighter tones of blue.

It is easy to distinguish the vanilla seesaw region - green symbols - at high
$det(D)$ and low $det(d_{X}^{2})$, the quasi-decoupled region, in the middle
range of $det(D)$ and $det(d_{X}^{2})$ and the non-decoupled region - at low
$det(D)$ and high $det(d_{X}^{2})$.

The red line corresponds to the value of $det(d)$ for which the sum of all
light neutrino masses matches the bound from Planck [144], one of the most
robust bounds [3]. This can be deducted because one can write the light
neutrino masses as [3]

$m_{1}=\sqrt{\Delta m_{31}^{2}z}\leavevmode\nobreak\ ,\leavevmode\nobreak\
m_{2}=\sqrt{\Delta m_{31}^{2}(z+a)}\leavevmode\nobreak\ ,\leavevmode\nobreak\
m_{3}=\sqrt{\Delta m_{31}^{2}(z+1)}$ (2.90)

where

$\Delta m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ a=\frac{\Delta m_{21}^{2}}{\Delta
m_{31}^{2}}\leavevmode\nobreak\ ,$ (2.91)

with $z$ being a free parameter. The current best-fit values for normal
ordering are

$\Delta m_{21}^{2}=7.50\times 10^{-5}\leavevmode\nobreak\
\textrm{eV}^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \Delta
m_{31}^{2}=2.55\times 10^{-3}\textrm{eV}^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ a\approx 0.02941\leavevmode\nobreak\ .$ (2.92)

Thus, using these experimental values and the bound from PLANCK [144], one can
solve numerically for $z$ and obtain a value for $det(d)$, which is only a
function of $z$, as the $m_{i}$, when the $\Delta m_{ij}^{2}$ are fixed

$m_{1}(z)+m_{2}(z)+m_{3}(z)=0.12\leavevmode\nobreak\ \textrm{eV}\implies
z\approx 0.355\implies det(d)\approx 5.54\times 10^{-5}\leavevmode\nobreak\
\textrm{eV}^{3}\leavevmode\nobreak\ .$ (2.93)

Hence, values of $z$ larger than $0.355$, which translates into

$m_{1}>0.0301\leavevmode\nobreak\ \textrm{eV}\leavevmode\nobreak\
,\leavevmode\nobreak\ m_{2}>0.0313\leavevmode\nobreak\
\textrm{eV}\leavevmode\nobreak\ ,\leavevmode\nobreak\
m_{3}>0.0588\leavevmode\nobreak\ \textrm{eV}\leavevmode\nobreak\ ,$ (2.94)

are ruled out. It’s interesting to note that most of this ruled out region,
above the red line, would be ruled out due to perturbativity. This is because
most of the area of the forbidden region is for high values of $det(D)$ and
high values of $det(d_{X}^{2})$. From eq. (2.19) it is easy to see that

$det(m)\sim det(X^{\dagger})\leavevmode\nobreak\ det(D)\leavevmode\nobreak\ ,$
(2.95)

and too large values of $det(D)$ and $det(d_{X}^{2})$ can easily yield non-
perturbative Yukawa couplings. Before proceeding to another section, it is
relevant to discuss how models with very light neutrino masses, $z\ll 1$, are
still allowed, although all the points drawn in the plot, taken from published
papers, are for higher values of $m_{1}$. From eq. (2.20) one can understand
that very small light neutrino masses ($m_{1}\rightarrow 0$) introduces an
extra suppression factor on $X$, beyond the usual $D^{-1/2}$. Nonetheless,
although more challenging, it is still possible to have sizeable deviations
from unitarity, by choosing an appropriate $O_{c}$, with large enough
eigenvalues.

##  Chapter 3 The Standard Model + $n_{u}$ up-type vector-like quarks +
$n_{d}$ down type vector-like quarks ($n_{u}$SM$n_{d}$)

Vector-like quarks are quarks where the right-handed and the left-handed
components transform in the same way under the gauge group. As a consequence
of this, all their interactions are vector-like or, in other words, not
“chiral”. That is, wherever the left-handed component of the VLQ can be
introduced in the Lagrangian, the same can be done for the right-handed
component. This does not happen for the SM fields, since left-handed
components are part of a $SU(2)$ doublet and right-handed components are
$SU(2)$ singlets. This particularity of VLQs allows the existence of bare
mass-terms, meaning that the VLQ mass might have an origin beyond the Higgs
mechanism. Hence, their masses can be much larger than the Higgs vev, while
still leaving considerable traces at low-energy.

There are only seven different types of VLQ multiplets which may mix with the
SM quarks through Yukawa couplings with the Higgs doublet [145, 146]. They
consist of isosinglets, doublets or triplets, all coloured in the fundamental
representation. The complete quantum numbers are listed in Table 3.1.

Multiplet | $U$ | $D$ | $\begin{pmatrix}U\cr D\end{pmatrix}$ | $\begin{pmatrix}X\cr U\end{pmatrix}$ | $\begin{pmatrix}D\cr Y\end{pmatrix}$ | $\begin{pmatrix}X\cr U\cr D\end{pmatrix}$ | $\begin{pmatrix}U\cr D\cr Y\end{pmatrix}$
---|---|---|---|---|---|---|---
$SU(2)_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{2}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{3}$
$U(1)_{Y}$ | $2/3$ | $-1/3$ | $1/6$ | $7/6$ | $-5/6$ | $2/3$ | $-1/3$
Table 3.1:  The only VLQ multiplets that couple to the SM quarks through
Yukawa interactions ($Q=I_{3}+Y$).

This thesis will only cover extensions of the quark sector with vector-like
singlet quarks $U$ and $D$, often denoted in the literature as $T$ and $B$,
respectively. Thus, in the context of this thesis, the names isosinglet
vector-like quarks and vector-like quarks will be used interchangeably.

In section 3.1 of this chapter a model where $n_{u}$ isosinglet vector-like
up-type ($Q=+2/3$) quarks and $n_{d}$ isosinglet vector-like down-type
($Q=-1/3$) quarks are added to the SM, without imposing any symmetry, is
discussed. In section 3.2.1 (3.2.2) a model where only $n_{u}$ ($n_{d}$)
isosinglet vector-like up-type (down-type) quarks are added to the SM, without
imposing any symmetry, will be covered. These extension of the SM will be
treated exactly using the parameterisation first devised in [30], in the more
succinct form of [32] and [33].

###  3.1 The Most General Case ($n_{u}$SM$n_{d}$)

To begin, it is important to establish the notation for the most general case
- the scenario involving $n_{u}$ up ($U^{0}_{Lr}$, $U^{0}_{Rr}$) and $n_{d}$
down ($D^{0}_{Ls}$, $D^{0}_{Rs}$) vector-like isosinglet quarks. Here, the
flavour indices $r$ and $s$ range from 1 to $n_{u}$ and $n_{d}$ respectively,
and the superscript "0" denotes the flavour basis. Once these vector-like
quarks are introduced, the Yukawa interactions in this basis are given by

$\begin{split}\mathcal{L}_{\text{Y}}\,=\,-\,\Big{[}&\,\,\overline{Q}_{Li}^{0}\,\tilde{\Phi}\,\left(Y_{u}\right)_{ij}\,u^{0}_{Rj}\,+\,\overline{Q}_{Li}^{0}\,\tilde{\Phi}\,\left(\overline{Y}\\!_{u}\right)_{is}\,U^{0}_{Rs}\\\
+&\,\,\overline{Q}_{Li}^{0}\,\Phi\,\left(Y_{d}\right)_{ij}\,d^{0}_{Rj}\,+\,\overline{Q}_{Li}^{0}\,\Phi\,\left(\overline{Y}\\!_{d}\right)_{ir}\,D^{0}_{Rr}\,\Big{]}+\text{h.c.}\,,\end{split}$
(3.1)

where the indices $i$ and $j$ run from 1 to 3 as in the SM and $Y_{d}$,
$Y_{u}$ are the SM Yukawa couplings. Here, $Q_{L}=(u_{L}\,\,\,d_{L})^{T}$ are
the SM quark doublets, $u_{R}$ and $d_{R}$ are the SM quark singlets, $\Phi$
is the Higgs doublet and $\tilde{\Phi}=i\tau_{2}\Phi^{*}$ is its $SU(2)_{L}$
conjugate. The matrices ${\overline{Y}\\!_{d}}$ and ${\overline{Y}\\!_{u}}$
denote the new Yukawa couplings to the extra right-handed fields.

Since the VLQs are $SU(2)$ singlets, the ensuing mass terms are, in general,
allowed and must therefore be included

$\begin{split}\mathcal{L}_{\text{b.m.}}\,=\,-\,\Big{[}&\,\,\overline{U}^{0}_{Lr}\,\left(\,\overline{\\!M}\mkern-2.0mu_{u}\right)_{ri}\,u^{0}_{Ri}\,+\,\overline{U}^{0}_{Lr}\,\left(M_{u}\right)_{rr^{\prime}}\,U^{0}_{Rr^{\prime}}\\\
+&\,\,\overline{D}^{0}_{Ls}\,\left(\,\overline{\\!M}\mkern-2.0mu_{d}\right)_{si}\,d^{0}_{Ri}\,+\,\overline{D}^{0}_{Ls}\,\left(M_{d}\right)_{ss^{\prime}}\,D^{0}_{Rs^{\prime}}\,\Big{]}+\text{h.c.}\,,\end{split}$
(3.2)

where $\,\overline{\\!M}\mkern-2.0mu_{q}$ and $M_{q}$ ($q=u,d$) are,
respectively, $n_{q}\times 3$ and $n_{q}\times n_{q}$ general complex
matrices. At this stage, these terms are interpreted as bare mass (b.m.)
terms. In certain models, they may arise from the vacuum expectation value
(VEV) of a complex scalar singlet, for instance.

After SSB, one has $\Phi=\big{(}0\,\,\,\frac{v+h}{\sqrt{2}}\big{)}^{T}$ in the
unitary gauge, where $h$ is the Higgs field and $v\simeq 246\;\mathrm{GeV}$.
Additional mass terms are generated from $\mathcal{L}_{\text{Y}}$ of (3.1).
All mass terms can be collected in a compact form,

$\mathcal{L}_{\text{M}}\,=\,-\begin{pmatrix}\overline{d}^{0}_{L}&\overline{D}^{0}_{L}\end{pmatrix}\,\mathcal{M}_{d}\,\begin{pmatrix}d^{0}_{R}\\\\[2.84526pt]
D^{0}_{R}\end{pmatrix}-\begin{pmatrix}\overline{u}^{0}_{L}&\overline{U}^{0}_{L}\end{pmatrix}\,\mathcal{M}_{u}\,\begin{pmatrix}u^{0}_{R}\\\\[2.84526pt]
U^{0}_{R}\end{pmatrix}+\text{h.c.}\,,$ (3.3)

where

$\displaystyle\mathcal{M}_{q}\,=\,\left(\begin{array}[]{c;{2pt/2pt}c}\,m_{q}&\,\overline{m}\hskip
0.86108pt_q\\\\[4.2679pt]
\hdashline[2pt/2pt]\\!{\,\,\,\overline{\\!M}\mkern-2.0mu_{q}\,\,\,}\\!&\\!{\,\,\,M_q\,\,\,}\\!\\\\[2.84526pt]
\end{array}\right)\,,$ (3.6)

with $m_{q}=\frac{v}{\sqrt{2}}Y_{q}$, $\overline{m}\hskip
0.86108pt_{q}=\frac{v}{\sqrt{2}}\overline{Y}\\!_{q}$ and
$\overline{\\!M}\mkern-2.0mu_{q}$ and $M_{q}$ defined in eq. (3.2). It is
important to note that, generally speaking, the matrices $\mathcal{M}_{q}$
aren’t symmetric or Hermitian. Due to their different origin, a hierarchy
$\overline{\\!M}\mkern-2.0mu_{q}\sim M_{q}\gg\overline{m}\hskip
0.86108pt_{q}\sim m_{q}$ is expected. Whenever there’s no potential for
confusion, the indices $q=u,d$ will be omitted. The mass matrices
$\mathcal{M}_{q}$ can be diagonalized by biunitary transformations — their
singular value decompositions — of the form

$\displaystyle\allowdisplaybreaks[0]{\mathcal{V}^{q}_{L}}^{\dagger}\,\,\mathcal{M}_{q}\,\,\mathcal{V}_{R}^{q}\,=\,\mathcal{D}_{q}\,,\quad\text{with
}\mathcal{D}_{q}\,=\,\left(\begin{array}[]{c;{2pt/2pt}c}\,\,\,d_{q}&\,\,\,0\\\\[4.2679pt]
\hdashline[2pt/2pt]\\!\mathrlap{\smash{\underbrace{\phantom{\begin{matrix}\,\,\,\,0\,\,\,\,\,\end{matrix}}}_{\mbox{$\scriptstyle
3$}}}}\,\,\,\,0\,\,\,\,\,\\!&\mathrlap{\smash{\underbrace{\phantom{\begin{matrix}\,\,\,\,D_{q}\,\,\,\,
\end{matrix}}}_{\mbox{$\scriptstyle\,\,\,n_{q}$}}}}\,\,\,\,D_q\,\,\,\,\\!\\\\[2.84526pt]
\end{array}\right)\\!\\!\\!\begin{array}[]{l}\left.\vphantom{\begin{matrix}0\end{matrix}}\right\\}\scriptstyle\,3\\\\[0.0pt]
\left.\vphantom{\begin{matrix}D\end{matrix}}\right\\}\scriptstyle\,n_{q}\end{array}\,,$
(3.11)

where the unitary rotations $\mathcal{V}_{L,R}^{q}$ connect the flavour and
physical bases by using four rotations (two for each sector). The diagonal
matrices $d_{q}$ and $D_{q}$ contain the light $(d_{q})_{i}\geq 0$ ($i=1,2,3$)
and heavy $(D_{q})_{r}>0$ ($r=1,\ldots,n_{q}$) quark masses. It is important
to note that $D_{q}$ is a matrix, not a down-type quark field. The notations
$d_{u}=\operatorname{diag}(m_{u},m_{c},m_{t})$,
$d_{d}=\operatorname{diag}(m_{d},m_{s},m_{b})$,
$D_{u}=\operatorname{diag}(m_{U1},\ldots,m_{Un_{u}})$, and
$D_{d}=\operatorname{diag}(m_{D1},\ldots,m_{Dn_{d}})$ will be used whenever
there is no potential for confusion. Regardless of how one parameterises the
four diagonalization matrices $\mathcal{V}^{q}_{\chi}$ ($q=u,d$; $\chi=L,R$),
it is useful to divide them into two blocks,

$\mathcal{V}^{q}_{\chi}\,=\,\left(\begin{array}[]{c}{}\hfil\\\\[-11.38109pt]
\qquad A^{q}_{\chi}\\\\[5.69054pt] \hdashline[2pt/2pt]{}\\\\[-11.38109pt]
\qquad B^{q}_{\chi}\\\\[5.69054pt]
\end{array}\right)\begin{array}[]{l}\left.\vphantom{\begin{matrix}A\end{matrix}}\right\\}\scriptstyle\,3\\\\[0.0pt]
\left.\vphantom{\begin{matrix}B\end{matrix}}\right\\}\scriptstyle\,n_{q}\end{array}\,,$
(3.12)

where $A^{q}_{\chi}$ is a $3\times(3+n_{q})$ matrix and $B^{q}_{\chi}$ is an
$n_{q}\times(3+n_{q})$ matrix. The flavour basis and the physical basis (no
‘0’ superscript) are thus connected via

$\begin{pmatrix}\,u^{0}_{\chi}\,\\\\[2.84526pt]
U^{0}_{\chi}\end{pmatrix}\,=\,\begin{pmatrix}\,A^{u}_{\chi}\,\\\\[2.84526pt]
B^{u}_{\chi}\end{pmatrix}\begin{pmatrix}\,u_{\chi}\,\\\\[2.84526pt]
U_{\chi}\end{pmatrix}\,,\qquad\begin{pmatrix}\,d^{0}_{\chi}\,\\\\[2.84526pt]
D^{0}_{\chi}\end{pmatrix}\,=\,\begin{pmatrix}\,A^{d}_{\chi}\,\\\\[2.84526pt]
B^{d}_{\chi}\end{pmatrix}\begin{pmatrix}\,d_{\chi}\,\\\\[2.84526pt]
D_{\chi}\end{pmatrix}\,,$ (3.13)

with $\chi=L,R$.

Within each sector, using the diagonalization relation eq. (3.11), one can
write the mass matrix blocks of eq. (3.6) in terms of the physical masses as

$\displaystyle m_{q}$
$\displaystyle=A^{q}_{L}\,\mathcal{D}_{q}\,{A_{R}^{q}}^{\dagger}\,,\quad\overline{m}\hskip
0.86108pt_{q}=A^{q}_{L}\,\mathcal{D}_{q}\,{B_{R}^{q}}^{\dagger}\,,$ (3.14)
$\displaystyle\overline{\\!M}\mkern-2.0mu_{q}$
$\displaystyle=B^{q}_{L}\,\mathcal{D}_{q}\,{A_{R}^{q}}^{\dagger}\,,\quad
M_{q}=B^{q}_{L}\,\mathcal{D}_{q}\,{B_{R}^{q}}^{\dagger}\,.$

Finally, unitarity of the $\mathcal{V}^{q}_{\chi}$ implies

$\displaystyle\mathcal{V}^{q}_{\chi}\leavevmode\nobreak\
\mathcal{V}^{q{\dagger}}_{\chi}=\begin{pmatrix}\,A^{q}_{\chi}\,\\\\[5.69054pt]
B^{q}_{\chi}\end{pmatrix}\begin{pmatrix}\,{A^{q}_{\chi}}^{\dagger}&{B^{q}_{\chi}}^{\dagger}\,\end{pmatrix}\,$
$\displaystyle=\,\begin{pmatrix}\,A^{q}_{\chi}\,{A^{q}_{\chi}}^{\dagger}&A^{q}_{\chi}\,{B^{q}_{\chi}}^{\dagger}\,\\\\[5.69054pt]
\,B^{q}_{\chi}\,{A^{q}_{\chi}}^{\dagger}&B^{q}_{\chi}\,{B^{q}_{\chi}}^{\dagger}\,\end{pmatrix}\,=\,\begin{pmatrix}\,\mathds{1}_{3}\,&\,0\,\\\\[5.69054pt]
\,0\,&\,\mathds{1}_{n_{q}}\,\end{pmatrix}\,,$ (3.15a)
$\displaystyle\mathcal{V}^{q{\dagger}}_{\chi}\leavevmode\nobreak\
\mathcal{V}^{q}_{\chi}=\begin{pmatrix}\,{A^{q}_{\chi}}^{\dagger}&{B^{q}_{\chi}}^{\dagger}\,\end{pmatrix}\begin{pmatrix}\,A^{q}_{\chi}\,\\\\[5.69054pt]
B^{q}_{\chi}\end{pmatrix}\,$
$\displaystyle=\,\,{A^{q}_{\chi}}^{\dagger}\,A^{q}_{\chi}+{B^{q}_{\chi}}^{\dagger}\,B^{q}_{\chi}\,\,=\,\,\mathds{1}_{3+n_{q}}\,,$
(3.15b)

for each $q=u,d$ and $\chi=L,R$.

Electromagnetic interactions of quarks, including the VLQ isosinglets, are
described by the Lagrangian

$\mathcal{L}_{A}=-e\,J^{\mu}_{\text{em}}A_{\mu}\,,$ (3.16)

with

$\displaystyle J^{\mu}_{\text{em}}$
$\displaystyle\,\equiv\,\frac{2}{3}\Big{(}\overline{u_{i}^{0}}\gamma^{\mu}{u_{i}^{0}}+\overline{U_{r}^{0}}\gamma^{\mu}{U_{r}^{0}}\Big{)}-\frac{1}{3}\Big{(}\overline{d_{i}^{0}}\gamma^{\mu}{d_{i}^{0}}+\overline{D_{s}^{0}}\gamma^{\mu}{D_{s}^{0}}\Big{)}$
(3.17)
$\displaystyle\,=\,\frac{2}{3}\Big{(}\overline{u_{i}}\gamma^{\mu}{u_{i}}+\overline{U_{r}}\gamma^{\mu}{U_{r}}\Big{)}-\frac{1}{3}\Big{(}\overline{d_{i}}\gamma^{\mu}{d_{i}}+\overline{D_{s}}\gamma^{\mu}{D_{s}}\Big{)}\,,$

where $\psi\equiv\psi_{L}+\psi_{R}$ for
$\psi\in\\{u_{i}^{(0)},d_{i}^{(0)},U_{r}^{(0)},D_{s}^{(0)}\\}$. As before, the
index $i$ runs from 1 to 3 as in the SM, while the indices $r,s$ run from 1 to
$n_{u,d}$. Note that the structure of the electromagnetic current
$J^{\mu}_{\text{em}}$ is unchanged in going from the flavour to the mass
basis.

The charged current Lagrangian for the quarks in the mass basis is given by

$\mathcal{L}_{W}^{q}=-\frac{g}{\sqrt{2}}\begin{pmatrix}\overline{u}_{L}&\overline{U}_{L}\end{pmatrix}\,V\,\gamma^{\mu}\begin{pmatrix}d_{L}\\\\[5.69054pt]
D_{L}\end{pmatrix}W_{\mu}^{+}\,+\,\text{h.c.}\leavevmode\nobreak\ ,$ (3.18)

where

$V=A_{L}^{u^{\dagger}}A_{L}^{d}\leavevmode\nobreak\ ,$ (3.19)

is the $(3+n_{u})\times(3+n_{d})$ non-unitary mixing matrix. The mixing matrix
$V$, which may have a rectangular shape when $n_{u}\neq n_{d}$, is typically
not unitary. The CKM quark mixing matrix, $V_{\textrm{CKM}}$, is the upper-
left $3\times 3$ block of $V$, also not unitary in general.

The weak neutral current Lagrangian in the mass basis is given by

$\begin{split}\mathcal{L}_{Z}^{q}&=-\frac{g}{2\cos{\theta_{W}}}Z_{\mu}\left[\begin{pmatrix}\overline{u}_{L}&\overline{U}_{L}\end{pmatrix}\,F^{u}\,\gamma^{\mu}\begin{pmatrix}u_{L}\\\\[5.69054pt]
U_{L}\end{pmatrix}-\begin{pmatrix}\overline{d}_{L}&\overline{D}_{L}\end{pmatrix}\,F^{d}\,\gamma^{\mu}\begin{pmatrix}d_{L}\\\\[5.69054pt]
D_{L}\end{pmatrix}-2\sin^{2}{\theta_{W}}J^{\mu\leavevmode\nobreak\
q}_{\text{em}}\right]\,,\end{split}$ (3.20)

while the part of the Lagrangian that contains the Higgs interactions with the
quarks in the mass basis is given by

$\mathcal{L}_{H}^{q}\,=\,-\frac{h}{v}\bigg{[}\begin{pmatrix}\overline{u}_{L}&\overline{U}_{L}\end{pmatrix}\,F^{u}\,\mathcal{D}_{u}\begin{pmatrix}u_{R}\\\\[5.69054pt]
U_{R}\end{pmatrix}+\begin{pmatrix}\overline{d}_{L}&\overline{D}_{L}\end{pmatrix}\,F^{d}\,\mathcal{D}_{d}\begin{pmatrix}d_{R}\\\\[5.69054pt]
D_{R}\end{pmatrix}\bigg{]}+\,\text{h.c.}\,\leavevmode\nobreak\ ,$ (3.21)

where

$F^{q}=A_{L}^{q^{\dagger}}A_{L}^{q}\leavevmode\nobreak\ .$ (3.22)

Note that the $F^{q}$ are present in interactions with the Higgs and the Z
boson and are the matrices that controls Flavour Changing Neutral Couplings
(FCNC). This might become clearer if these are written as

$\displaystyle F^{u}$
$\displaystyle\equiv\,{A_{L}^{u}}^{\dagger}A_{L}^{u}\,=\,\mathds{1}-{B_{L}^{u}}^{\dagger}B_{L}^{u}\,=\,VV^{\dagger}\,,$
(3.23) $\displaystyle F^{d}$
$\displaystyle\equiv\,{A_{L}^{d}}^{\dagger}A_{L}^{d}\,=\,\mathds{1}-{B_{L}^{d}}^{\dagger}B_{L}^{d}\,=\,V^{\dagger}V\,.$

The strength of FCNC is proportional to the off-diagonal elements of
$F^{u,d}$, and eq. (3.23) shows that these come from the deviations from the
identity matrix of $F^{q}$. This also shows that there is a strong connection
between the strength of FCNC and deviations of $V$ from unitarity.
It should be noted that $F^{u,d}$ matrices are in general not equal to the
identity matrix, which leads to the occurrence of FCNC processes mediated by
both the Higgs boson and the $Z$ boson. The strength of Higgs-mediated FCNC
processes is determined by the off-diagonal elements of $F^{u,d}$ matrices as
well as the ratios of the diagonal elements of $\mathcal{D}_{u,d}$ to the
vacuum expectation value $v$. This is in contrast to the case of $Z$-mediated
FCNC processes. Because of this, in the Higgs sector, for transitions
involving only the lighter quarks $u$ and $c$, there is a suppression factor
of $m_{u}/v\sim 10^{-5}$ or $m_{c}/v\sim 10^{-2}$.

The same steps performed in the beginning of chapter 2 can be done for the
quark sector, allowing one to write the quark form of eq. (2.15) as

$\displaystyle\mathcal{V}_{\chi}^{q}\,=\,\left(\begin{array}[]{c;{2pt/2pt}c}\,\,\,K_{\chi}^{q}&\,\,\,K_\chi^q\,{X_\chi^q}^\textdagger\\\\[4.2679pt]
\hdashline[2pt/2pt]{-\overline{\\!K}\,_{\\!\chi}^{\\!q}\,X_{\chi}^{q}\,\,}\\!&{\quad\,\,\overline{\\!K}\,_{\\!\chi}^{\\!q}\quad\,\,}\\!\\\\[2.84526pt]
\end{array}\right)\leavevmode\nobreak\ .$ (3.26)

Eq. (3.26) gives that the $(3+n_{q})\times(3+n_{q})$ unitary matrices
$\mathcal{V}^{q}_{\chi}$ introduced in eq. (3.11) and eq. (3.12) can be
written, without loss of generality, using a non-singular $3\times 3$ general
complex matrix, $K^{q}_{\chi}$, a non-singular $n_{q}\times n_{q}$ general
complex matrix $\overline{\\!K}\,^{q}_{\chi}$ and a $n_{q}\times 3$ matrix,
$X^{q}_{\chi}$. As in chapter 2, from eq. (3.12) one obtains

$A^{q}_{\chi}=(K^{q}_{\chi}\leavevmode\nobreak\ \leavevmode\nobreak\
K^{q}_{\chi}{X^{q}_{\chi}}^{\dagger})\leavevmode\nobreak\
,\leavevmode\nobreak\
B^{q}_{\chi}=(-\overline{\\!K}\,^{q}_{\chi}X^{q}_{\chi}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{\\!K}\,^{q}_{\chi})\leavevmode\nobreak\ ,$ (3.27)

Using eq. (3.27) and the unitarity relations in eqs. (3.15a) and (3.15b), eq.
(3.14) becomes

$\begin{split}&m_{q}\,=\,K_{L}^{q}\left(d_{q}+{X_{L}^{q}}^{\dagger}\,D_{q}\,X_{R}^{q}\right){K_{R}^{q}}^{\dagger},\\\
&\overline{m}\hskip
0.86108pt_{q}\,=\,K_{L}^{q}\left({X_{L}^{q}}^{\dagger}\,D_{q}-d_{q}\,{X_{R}^{q}}^{\dagger}\right){\overline{\\!K}\,_{R}^{q}}^{\dagger},\\\
&\overline{\\!M}\mkern-2.0mu_{q}\,=\,\overline{\\!K}\,_{L}^{q}\left(D_{q}\,X_{R}^{q}-X_{L}^{q}\,d_{q}\right){K_{R}^{q}}^{\dagger},\\\
&M_{q}\,=\,\overline{\\!K}\,_{L}^{q}\left(D_{q}+X_{L}^{q}\,d_{q}\,{X_{R}^{q}}^{\dagger}\right){\overline{\\!K}\,_{R}^{q}}^{\dagger},\end{split}$
(3.28)

where $m_{q}$ is the $3\times 3$ Dirac mass matrix for the quarks and the
entries of the $3\times n_{q}$ matrix, $\overline{m}\hskip 0.86108pt_{q}$, are
proportional to the Higgs vacuum expectation value, as well. The $n_{q}\times
3$ mass matrix, $\overline{\\!M}\mkern-2.0mu_{q}$, and the $n_{q}\times n_{q}$
mass matrix, $M_{q}$, are bare mass terms involving only singlet quark fields.
It is always possible to go to a WB where $\overline{m}\hskip 0.86108pt_{q}$
is 0, also for $\overline{\\!M}\mkern-2.0mu_{q}$, see section 3.2.1 of ref.
[33]. In the vanishing $\overline{m}\hskip 0.86108pt_{q}$ WB, one can proceed
like in the neutrino case to obtain a formula for $X_{\chi}^{q}$,

$\displaystyle\begin{aligned}
&X_{L}^{q}=\sqrt{D_{q}^{-1}}P^{q}\sqrt{d_{q}}\leavevmode\nobreak\ ,\\\
&X_{R}^{q}=\sqrt{D_{q}}P^{q}\sqrt{d_{q}^{-1}}\leavevmode\nobreak\
,\end{aligned}$ (3.29)

where $P^{q}$ is a general complex matrix and $d_{q}$ ($D_{q}$) is a diagonal
matrix with the masses of the light (heavy) quarks in the diagonal. A keen
reader will notice that eq. (LABEL:eq:Xdefquark) amounts to nothing, since
$P^{q}$ are general complex matrices, so will $X_{L}^{q}$ and $X_{R}^{q}$ be.
$X_{L}^{q}$ and $X_{R}^{q}$ are completely general complex matrices which only
need to satisfy

$\displaystyle d_{q}X_{L}^{q\dagger}$
$\displaystyle=X_{R}^{q\dagger}D_{q}\quad\text{in the vanishing
}\overline{\\!M}\mkern-2.0mu_{q}\text{ WB}\,,$ (3.30)
$\displaystyle\text{and}\quad d_{q}X_{R}^{q\dagger}$
$\displaystyle=X_{L}^{q\dagger}D_{q}\quad\text{in the vanishing
}\overline{m}\hskip 0.86108pt_{q}\text{ WB}\,.$ (3.31)

$X_{L}^{q}$ will, as in the neutrino sector, control the size of the
deviations from unitarity of the $(3+n_{u})\times(3+n_{d})$ non-unitary mixing
matrix $V$. Naturally, its biggest constraint will be how large one wants
these deviations to be. A very important constraint will also be the
perturbativity of the theory. More on that in section 3.1.2. The following
definitions can also be obtained from the unitarity relations in eqs. (3.15a)
and (3.15b),

$\begin{split}&K^{q}_{\chi}=U^{q}_{K\chi}(\mathds{1}_{3}+{X_{\chi}^{q}}^{\dagger}X_{\chi}^{q})^{-1/2}\,,\\\
&\overline{\\!K}\,^{q}_{\chi}=U^{q}_{\overline{\\!K}\,\chi}(\mathds{1}_{n_{q}}+X_{\chi}^{q}{X_{\chi}^{q}}^{\dagger})^{-1/2}\leavevmode\nobreak\
,\end{split}$ (3.32)

where $U^{q}_{K\chi}$ and $U^{q}_{\overline{\\!K}\,\chi}$ are unitary
matrices, since the unitarity relations only define $K$ and
$\overline{\\!K}\,$ up to a unitary matrix on the left. The combination
${K_{L}^{u}}^{\dagger}K_{L}^{d}$ will play the role of the CKM mixing matrix
and is only unitary when $X_{L}^{u},X_{L}^{d}\xrightarrow[]{}\textbf{0}$,

$V_{\textrm{CKM}}={K_{L}^{u}}^{\dagger}K_{L}^{d}\leavevmode\nobreak\ .$ (3.33)

The $(3+n_{u})\times(3+n_{d})$ non-unitary mixing matrix $V$, first defined in
eq. (3.19), can now be written as

$V=A^{u\dagger}_{L}A^{d}_{L}=\left(\begin{array}[]{c;{2pt/2pt}c}\,\,\,V_{\text{CKM}}&\,\,\,
V_\text{CKM}\,{X_L^{d \dagger}}\\\\[4.2679pt]
\hdashline[2pt/2pt]\\!\\!\mathrlap{\smash{\underbrace{\phantom{\begin{matrix}\,\,\,\,{X_{L}^{u}}\,V_{\text{CKM}}\,\,\,\,\,\end{matrix}}}_{\mbox{$\scriptstyle
3$}}}}\,\,\,\,{X_{L}^{u}}\,V_{\text{CKM}}\,\,\,\,\,\\!\\!\\!&\\!\mathrlap{\smash{\underbrace{\phantom{\begin{matrix}\,\,\,{X_{L}^{u}}
\,V_{\text{CKM}}\,{X_{L}^{d}}^{\textdagger}\,\,\,\,
\end{matrix}}}_{\mbox{$\scriptstyle\,\,\,n_{d}$}}}}\,\,\,{X_L^u}
\,V_\text{CKM}\,{X_L^d}^\textdagger\,\,\,\,\\!\\!\\!\\\\[2.84526pt]
\end{array}\right)\\!\\!\\!\begin{array}[]{l}\left.\vphantom{\begin{matrix}V_{\text{CKM}}\,{X_{L}^{d}}^{\dagger}\end{matrix}}\right\\}\scriptstyle\,3\\\\[0.0pt]
\left.\vphantom{\begin{matrix}V_{\text{CKM}}\,{X_{L}^{d}}^{\dagger}\end{matrix}}\right\\}\scriptstyle\,n_{u}\end{array}.\\\
$ (3.34)

The matrices relevant for FCNC, first defined in eq. (3.22), can also be
rewritten as

$F^{q}=\begin{pmatrix}(\mathds{1}_{3}+{X_{L}^{q}}^{\dagger}X_{L}^{q})^{-1}&(\mathds{1}_{3}+{X_{L}^{q}}^{\dagger}{X_{L}^{q}})^{-1}{X_{L}^{q}}^{\dagger}\\\
{X_{L}^{q}}(\mathds{1}_{3}+{X_{L}^{q}}^{\dagger}{X_{L}^{q}})^{-1}&{X_{L}^{q}}(\mathds{1}_{3}+{X_{L}^{q}}^{\dagger}{X_{L}^{q}})^{-1}{X_{L}^{q}}^{\dagger}\\\
\end{pmatrix}\leavevmode\nobreak\ ,$ (3.35)

Finally, for completeness, the exact formula for the
$(3+n_{q})\times(3+n_{q})$ unitary matrix $\mathcal{V}^{q}_{\chi}$ assumes the
form

$\mathcal{V}^{q}_{\chi}=\begin{pmatrix}U^{q}_{K\chi}(\mathds{1}_{3}+{X_{\chi}^{q}}^{\dagger}X_{\chi}^{q})^{-1/2}&U^{q}_{K\chi}(\mathds{1}_{3}+{X_{\chi}^{q}}^{\dagger}X_{\chi}^{q})^{-1/2}X_{\chi}^{q{\dagger}}\\\
-U^{q}_{\overline{\\!K}\,\chi}(\mathds{1}_{n_{q}}+X_{\chi}^{q}{X_{\chi}^{q}}^{\dagger})^{-1/2}X^{q}_{\chi}&U^{q}_{\overline{\\!K}\,\chi}(\mathds{1}_{n_{q}}+X_{\chi}^{q}{X_{\chi}^{q}}^{\dagger})^{-1/2}\\\
\end{pmatrix}\leavevmode\nobreak\ .$ (3.36)

####  3.1.1 Weak Bases and Physical Parameters in the presence of VLQs

Given that in the context of this thesis no symmetry is being assumed when
adding VLQs to the SM Lagrangian, many new parameters will emerge. It is
important to count them properly and to disclose which of them are physical.
This can be done in a general treatment using Weak-basis (WB) transformations.
Even when one is working under a model which restricts the parameter space via
symmetries, the procedure and results that follows will be valid. For this
discussion for the quark sector of the SM see section 3.1 of ref. [33].
As established in eqs. (3.1) (3.2), the flavour sector of the $n_{u}$SM$n_{d}$
looks like

$\begin{split}\mathcal{F}\,=\,-\,\Big{[}&\,\,\overline{Q}_{L}^{0}\,\tilde{\Phi}\,Y_{u}\,u^{0}_{R}\,+\,\overline{Q}_{L}^{0}\,\tilde{\Phi}\,\overline{Y}\\!_{u}\,U^{0}_{R}\,+\,\overline{U}^{0}_{L}\,\,\overline{\\!M}\mkern-2.0mu_{u}\,u^{0}_{R}\,+\,\overline{U}^{0}_{L}\,M_{u}\,U^{0}_{R}\\\
+&\,\,\overline{Q}_{L}^{0}\,\Phi\,Y_{d}\,d^{0}_{R}\,+\,\overline{Q}_{L}^{0}\,\Phi\,\overline{Y}\\!_{d}\,D^{0}_{R}\,+\,\overline{D}^{0}_{L}\,\,\overline{\\!M}\mkern-2.0mu_{d}\,d^{0}_{R}\,+\,\overline{D}^{0}_{L}\,M_{d}\,D^{0}_{R}\,\Big{]}+\text{h.c.}\,,\end{split}$
(3.37)

where all matrices are complex. The most general WB transformation in this
context takes the form

$\begin{array}[]{l@{\qquad}l}u^{0}_{L}\,\to\,W_{L}\,u^{0}_{L}\,,&d^{0}_{L}\,\to\,W_{L}\,d^{0}_{L}\,,\\\\[5.69054pt]
U^{0}_{L}\,\to\,W_{L}^{U}\,U^{0}_{L}\,,&D^{0}_{L}\,\to\,W_{L}^{D}\,D^{0}_{L}\,,\\\\[8.53581pt]
\begin{pmatrix}u^{0}_{R}\\\\[2.84526pt]
U^{0}_{R}\end{pmatrix}\,\to\,\mathcal{W}_{R}^{u}\begin{pmatrix}u^{0}_{R}\\\\[2.84526pt]
U^{0}_{R}\end{pmatrix}\,,&\begin{pmatrix}d^{0}_{R}\\\\[2.84526pt]
D^{0}_{R}\end{pmatrix}\,\to\,\mathcal{W}_{R}^{d}\begin{pmatrix}d^{0}_{R}\\\\[2.84526pt]
D^{0}_{R}\end{pmatrix}\,,\end{array}$ (3.38)

where $W_{L}\sim 3\times 3$, $W_{L}^{U,D}\sim n_{u,d}\times n_{u,d}$ and
$\mathcal{W}^{u,d}_{R}\sim(3+n_{u,d})\times(3+n_{u,d})$ are unitary matrices
in flavour space. It should be noted that, in a WB transformation, changes to
$u_{L}^{0}$ and $d_{L}^{0}$ are interdependent, as in the SM case. However,
this interdependence is not necessary for the transformations of the
isosinglet fields $U_{L}^{0}$ and $D_{L}^{0}$. Furthermore, the transformation
matrices for the right-handed isosinglet fields $\mathcal{W}^{u,d}_{R}$ are
more extensive than those in the Standard Model. They enable mixing between
$u_{R}^{0}$ and $U_{R}^{0}$ (and similarly for $d_{R}^{0}$ and $U_{R}^{0}$),
as the $3+n_{q}$ right-handed quarks in each sector ($q=u,d$) carry the same
quantum numbers.

While the statements above are presented for the case where $n_{f}=3$ (the
number of families of SM quarks), they can be readily generalized and applied
to the case where $n_{f}\neq 3$.

#####  3.1.1.1 Useful weak bases

Different WB choices allow for particular forms of the extended quark mass
matrices $\mathcal{M}_{q}$, defined in eq. (3.6). Since the mass matrices
contain contributions from both Yukawa and bare mass terms, see eqs. (3.1)
(3.2), it is easier to work directly with the mass matrices in this context.
For generic $n_{u}$ and $n_{d}$, under the WB change of eq. (3.38), the mass
matrices are transformed as

$\displaystyle\mathcal{M}_{u}\,=\,\begin{pmatrix}m_{u}&\overline{m}\hskip
0.86108pt_{u}\\\\[5.69054pt]
\overline{\\!M}\mkern-2.0mu_{u}&M_{u}\end{pmatrix}$
$\displaystyle\to\quad\mathcal{M}^{\prime}_{u}\,=\,\begin{pmatrix}W_{L}^{\dagger}\,m_{u}&W_{L}^{\dagger}\,\overline{m}\hskip
0.86108pt_{u}\\\\[5.69054pt]
{W_{L}^{U}}^{\dagger}\,\overline{\\!M}\mkern-2.0mu_{u}&{W_{L}^{U}}^{\dagger}\,M_{u}\end{pmatrix}\mathcal{W}_{R}^{u}\,,$
(3.39)
$\displaystyle\mathcal{M}_{d}\,=\,\begin{pmatrix}m_{d}&\overline{m}\hskip
0.86108pt_{d}\\\\[5.69054pt]
\overline{\\!M}\mkern-2.0mu_{d}&M_{d}\end{pmatrix}$
$\displaystyle\to\quad\mathcal{M}^{\prime}_{d}\,=\,\begin{pmatrix}W_{L}^{\dagger}\,m_{d}&W_{L}^{\dagger}\,\overline{m}\hskip
0.86108pt_{d}\\\\[5.69054pt]
{W_{L}^{D}}^{\dagger}\,\overline{\\!M}\mkern-2.0mu_{d}&{W_{L}^{D}}^{\dagger}\,M_{d}\end{pmatrix}\mathcal{W}_{R}^{d}\,.$

The unitary rotations $\mathcal{W}^{q}_{R}$ that appear on the right-hand side
of these transformations are useful in shaping the mass matrices
$\mathcal{M}_{q}$. Under specific conditions, they can be employed to
eliminate some portion of the matrix and produce a triangular matrix. In
particular, it is worth noting that any complex square matrix $\mathcal{M}$
can be expressed as the product of a lower-triangular matrix $L$ and a unitary
matrix $W$ to its right, $\mathcal{M}=LW$. Alternatively, it can be written as
the product of an upper-triangular matrix $R$ and a different unitary matrix
$W^{\prime}$ to its right, i.e., $\mathcal{M}=RW^{\prime}$. This follows from
the well-known QR decomposition, which states that any complex square matrix
$A$ can be written as $A=QR$, with $Q$ being a unitary matrix and $R$ being an
upper (or right) triangular matrix. By QR-decomposing $\mathcal{M}^{T}$, one
finds the decomposition $\mathcal{M}=LQ^{\prime}$, where $Q^{\prime}$ is
unitary and $L$ is lower (or left) triangular. Similarly, by QR-decomposing
$\mathcal{M}^{T}\mathcal{P}$, where $\mathcal{P}$ is the anti-diagonal
permutation matrix, one can demonstrate that it is possible to write
$\mathcal{M}=R^{\prime}Q^{\prime\prime}$, where $Q^{\prime\prime}$ is unitary
and $R^{\prime}$ is upper triangular.

The matrices $\mathcal{W}^{u}_{R}$ and $\mathcal{W}^{d}_{R}$ can then be used
to eliminate the unitary factors $W^{(\prime)}$ and render the mass matrices
$\mathcal{M}_{u}$ and $\mathcal{M}_{d}$ lower or upper triangular, as desired.
Additionally, they can be utilized to permute the columns of these matrices.
As a result:

  * •

A WB where $\overline{m}\hskip 0.86108pt_{q}=0$ exists for all $n_{q}$
(vanishing $\,\overline{m}\hskip 0.86108pt$ WB).

  * •

A WB where $\overline{\\!M}\mkern-2.0mu_{q}=0$ exists for all $n_{q}$
(vanishing $\,\overline{\\!M}\mkern-2.0mu$ WB).

  * •

A WB where $m_{q}=0$ is guaranteed to exist only if $n_{q}\geq n_{f}=3$.

  * •

A WB where $M_{q}=0$ is guaranteed to exist only if $n_{q}\leq n_{f}=3$.

Typically, only one of the sub-matrices $m$, $\overline{m}\hskip 0.86108pt$,
$\overline{\\!M}\mkern-2.0mu$, or $M$ can be made to vanish at a time through
a WB transformation. This means that one can impose at most one of the above
conditions at a time. However, the vanishing conditions can be enforced
separately in the up and down sectors. Therefore, in each sector, it is always
possible to choose a weak basis where the mass matrix has the following form:

$\mathcal{M}\,=\,\begin{pmatrix}m&\overline{m}\hskip 0.86108pt\\\\[5.69054pt]
0&M\end{pmatrix}\\!\\!\qquad\text{or}\qquad\mathcal{M}\,=\,\begin{pmatrix}m&0\\\\[5.69054pt]
\overline{\\!M}\mkern-2.0mu&M\end{pmatrix}\,.$ (3.40)

Furthermore, in each sector, the matrix blocks located in the lower-right
position, denoted as $M$, can always be transformed into a diagonal form,
denoted as $D^{\prime}$, with real and non-negative entries using a secondary
WB transformation. This transformation involves rotating the left and right
unitary matrices in the singular value decomposition of $M$ by employing
suitable choices of $W^{U,D}_{L}$ and $\mathcal{W}^{u,d}_{R}$ (which are now
block diagonal). This process may impact the matrix
$\overline{\\!M}\mkern-2.0mu$ if it is present, which is currently a general
complex matrix.

On the other hand, the matrix blocks $m$ located in the upper-left position in
each sector cannot be simultaneously converted into a diagonal form using the
same procedure. This limitation arises due to the use of a single $W_{L}$
matrix, which is common to both the up and down sectors.

Consequently, if one begins with upper-triangular mass matrices
$\mathcal{M}_{u}$ and $\mathcal{M}_{d}$, the most that can be achieved through
the WB transformation is a configuration where a unitary rotation remains in
one of the sectors (for example, the down sector).

$\mathcal{M}_{u}\,=\,\begin{pmatrix}d_{u}^{\prime}&\overline{m}\hskip
0.86108pt_{u}\\\\[5.69054pt]
0&D_{u}^{\prime}\end{pmatrix}\,,\qquad\mathcal{M}_{d}\,=\,\begin{pmatrix}V^{\prime}d_{d}^{\prime}&\overline{m}\hskip
0.86108pt_{d}\\\\[5.69054pt] 0&D_{d}^{\prime}\end{pmatrix}\,.$ (3.41)

Here (minimal vanishing $\,\overline{\\!M}\mkern-2.0mu$ WB), $d_{q}^{\prime}$
and $D_{q}^{\prime}$ are diagonal matrices with real non-negative entries,
$V^{\prime}$ is the anticipated residual unitary rotation, and
$\overline{m}\hskip 0.86108pt_{q}$ are general complex matrices.

A similar choice of the WB (Wolfenstein parametrization) has been made in
reference [147], where the matrix $m_{d}$ is allowed to be a general complex
matrix. It is important to note that the primed diagonal matrices, denoted as
$d_{q}^{\prime}$ and $D_{q}^{\prime}$, differ from the matrices containing the
physical masses as defined in eq. (3.11). However, it can be verified that the
determinant of $d_{q}^{\prime}$ multiplied by the determinant of
$D_{q}^{\prime}$ is equal to the absolute value of the determinant of $d_{q}$
multiplied by the determinant of $D_{q}$, which represents the product of all
quark masses of type $q$. Similarly, it is possible to choose a different WB
parametrization where

$\mathcal{M}_{u}\,=\,\begin{pmatrix}d_{u}^{\prime}&0\\\\[5.69054pt]
\overline{\\!M}\mkern-2.0mu_{u}&D_{u}^{\prime}\end{pmatrix}\,,\qquad\mathcal{M}_{d}\,=\,\begin{pmatrix}V^{\prime}d_{d}^{\prime}&0\\\\[5.69054pt]
\overline{\\!M}\mkern-2.0mu_{d}&D_{d}^{\prime}\end{pmatrix}$ (3.42)

(minimal vanishing $\,\overline{m}\hskip 0.86108pt$ WB), with $d_{q}^{\prime}$
and $D_{q}^{\prime}$ diagonal, $V^{\prime}$ unitary and now
$\overline{\\!M}\mkern-2.0mu_{q}$ general complex. Finally, by the above
procedure, it further follows that one may easily arrange for weak bases like

$\mathcal{M}_{u}\,=\,\begin{pmatrix}d_{u}^{\prime}&0\\\\[5.69054pt]
\overline{\\!M}\mkern-2.0mu_{u}&D_{u}^{\prime}\end{pmatrix}\,,\qquad\mathcal{M}_{d}\,=\,\begin{pmatrix}V^{\prime}d_{d}^{\prime}&\overline{m}\hskip
0.86108pt_{d}\\\\[5.69054pt] 0&D_{d}^{\prime}\end{pmatrix}$ (3.43)

(mixed minimal WB), where both $\overline{m}\hskip 0.86108pt_{u}$ and
$\overline{\\!M}\mkern-2.0mu_{d}$ are made to vanish. A closely related WB
choice has been made in Ref. [148], where again $m_{d}$ has been kept general
complex.

By examining eq. (3.41), it becomes apparent that for arbitrary values of
$n_{u}$ and $n_{d}$, it is generally not possible to transform either of the
matrices $\mathcal{M}_{u}$ or $\mathcal{M}_{d}$ into a diagonal form through a
WB transformation.

In other words, a diagonal mass matrix in the flavour basis does not typically
correspond to a straightforward choice in the WB parametrization.

However, in the presence of vector-like quarks (VLQs), exceptions occur in
certain limit cases where either $n_{u}$ or $n_{d}$ equals zero. In these
cases, it is indeed possible to diagonalize one of the mass matrices, either
$\mathcal{M}_{u}$ or $\mathcal{M}_{d}$, respectively. As an illustration,
reference [149] considers the specific scenario with $n_{u}=0$ and $n_{d}=1$,
and a WB parametrization is chosen such that $\mathcal{M}_{u}$ becomes a
diagonal matrix represented by $d_{u}$, while $\mathcal{M}_{d}$ takes the form
described in eq. (3.42).

#####  3.1.1.2 Parameter counting

In the WB choices of eqs. (3.41) (3.42), (3.43), the physical parameter count
becomes apparent. Looking at e.g. eq. (3.41), one finds

  * •

$n_{f}+n_{q}=3+n_{q}$ real parameters (moduli) in the matrices
$d_{q}^{\prime}$ and $D_{q}^{\prime}$ of each sector,

  * •

$n_{f}n_{q}=3n_{q}$ complex parameters in each $\overline{m}\hskip
0.86108pt_{q}$ matrix, i.e. $3n_{q}$ moduli and $3n_{q}$ phases in each
sector, and

  * •

$n_{f}^{2}=9$ parameters in the unitary matrix $V^{\prime}$, corresponding to
$\frac{1}{2}n_{f}(n_{f}-1)=3$ moduli and $\frac{1}{2}n_{f}(n_{f}+1)=6$ phases.

However, it is important to note that, similar to the Standard Model case,
there is still some rephasing freedom in the WB parametrization, and not all
of the phases mentioned above are physically meaningful.

In order to obtain a WB representation with the minimum number of parameters,
one can consider rephasings of the WB where the unitary matrices $W_{L}$,
$W_{L}^{U,D}$, and $\mathcal{W}_{R}^{u,d}$ are diagonal matrices consisting of
phases. By doing so, one can

  * •

remove $2n_{f}-1=5$ phases from $V^{\prime}$ by use of the first $n_{f}=3$
phases of $\mathcal{W}_{R}^{d}$ (commuting with $d_{d}^{\prime}$) and of
$W_{L}$, whose effect on $d_{u}^{\prime}$ may be cancelled by
$\mathcal{W}_{R}^{u}$,

  * •

remove $n_{q}$ phases from $\overline{m}\hskip 0.86108pt_{q}$ in each sector
via the last $n_{q}$ phases of $\mathcal{W}_{R}^{q}$ (see also [147]), whose
effects on $D_{q}^{\prime}$ may be cancelled by the corresponding
$W_{L}^{U,D}$.

Therefore, in this minimal WB the mass matrices display a total of
$N_{\text{phys}}=10+6(n_{u}+n_{d})$ parameters, while keeping the form of eq.
(3.41). Their breakdown into moduli and phases is summarized in table 3.2.

| Moduli | Phases | Total
---|---|---|---
$\mathcal{M}_{u}$ (minimal WB) | $3+4n_{u}$ | $2n_{u}$ | $3+6n_{u}$
$\mathcal{M}_{d}$ (minimal WB) | $6+4n_{d}$ | $1+2n_{d}$ | $7+6n_{d}$
Total | $9+4(n_{u}+n_{d})$ | $1+2(n_{u}+n_{d})$ | $10+6(n_{u}+n_{d})$
Table 3.2: Physical parameter count in the presence of isosinglet VLQs
($n_{f}=3$).

Not only does one recover the SM result when $n_{d}=n_{u}=0$, but one also
expects each singlet vector-like quark to bring about 6 new physical
parameters: 4 moduli (1 mass and 3 mixing angles) and 2 physical phases.

####  3.1.2 Deviations from Unitarity, FCNC and Perturbativity

Unlike the neutrino case, very light VLQs are very difficult (if not
impossible) to accommodate with current phenomenological data. Hence,
deviations from unitary in the quark sector can never be too large. This
statement holds true because, similar to the example of neutrinos - see
chapter 2 \- there is a seesaw between deviations from unitarity and the new
particles’ masses, in order to have sensible Yukawa couplings. This is why one
should worry about perturbativity, since for large masses deviations from
unitarity are very strongly upper bounded. The definitions in section 3.1 are
useful in theory, but due to the aforementioned fact, more useful formulas can
be derived via an approximation valid up to $\mathcal{O}(X^{3})$.

To derive these approximations, it is convenient to simplify eqs. (3.26)
(LABEL:eq:uparam-exactK) and consider a specific WB. The following WB
transformation

$\begin{array}[]{l@{\qquad}l}u^{0}_{L}\,\to\,V_{K^{u}_{L}}\,u^{0}_{L}\,,&d^{0}_{L}\,\to\,V_{K^{u}_{L}}\,d^{0}_{L}\,,\\\\[5.69054pt]
U^{0}_{L}\,\to\,V_{\overline{\\!K}\,\\!^{u}_{L}}\,U^{0}_{L}\,,&D^{0}_{L}\,\to\,V_{\overline{\\!K}\,\\!^{d}_{L}}\,D^{0}_{L}\,,\end{array}$
(3.44)

removes the matrices $V_{K^{u}_{L}}$, $V_{\overline{\\!K}\,\\!^{d}_{L}}$ and
$V_{\overline{\\!K}\,\\!^{u}_{L}}$ from eq. (LABEL:eq:uparam-exactK) without
modifying the charged currents. Now, doing an approximation valid for small
$X$,

$\big{(}\mathds{1}_{3}+X^{\dagger}X\big{)}^{-1/2}\simeq\mathds{1}_{3}-\frac{1}{2}X^{\dagger}X\,,\quad\big{(}\mathds{1}_{n}+XX^{\dagger}\big{)}^{-1/2}\simeq\mathds{1}_{n}-\frac{1}{2}XX^{\dagger}\,,$
(3.45)

valid up to $\mathcal{O}(X^{3})$, which is the same as saying that terms of
higher order are negligible. It is now possible to write eq. (3.26) in the
following way

$\begin{split}\mathcal{V}^{d}_{L}&\simeq\begin{pmatrix}V^{\prime}&0\\\
0&\mathds{1}_{n}\end{pmatrix}\begin{pmatrix}\mathds{1}_{3}-\frac{1}{2}\Theta^{\dagger}_{d}\Theta_{d}&\Theta_{d}^{\dagger}\\\
-\Theta_{d}&\mathds{1}_{n_{d}}-\frac{1}{2}\Theta_{d}\Theta^{\dagger}_{d}\end{pmatrix}\,,\\\\[2.84526pt]
\mathcal{V}^{u}_{L}&\simeq\begin{pmatrix}\mathds{1}_{3}-\frac{1}{2}\Theta^{\dagger}_{u}\Theta_{u}&\Theta_{u}^{\dagger}\\\
-\Theta_{u}&\mathds{1}_{n_{u}}-\frac{1}{2}\Theta_{u}\Theta^{\dagger}_{u}\end{pmatrix}\,,\end{split}$
(3.46)

where $V^{\prime}=V_{K^{u}_{L}}^{\dagger}V_{K^{d}_{L}}$. Furthermore,
$X^{q}_{L}$ was replaced by $\Theta_{q}$, to emphasize that an approximation
has been employed, this convention will be adopted throughout this thesis. In
this notation, the quark mixing matrix is given by

$V\simeq\begin{pmatrix}V^{\prime}-\frac{1}{2}\Theta^{\dagger}_{u}\Theta_{u}V^{\prime}-\frac{1}{2}V^{\prime}\Theta^{\dagger}_{d}\Theta_{d}&V^{\prime}\Theta_{d}^{\dagger}\\\
\Theta_{u}V^{\prime}&\Theta_{u}V^{\prime}\Theta_{d}^{\dagger}\end{pmatrix}\,,$
(3.47)

where the $3\times 3$ unitary matrix $V^{\prime}$ is related to the CKM matrix
$V_{\text{CKM}}$ through

$V_{\text{CKM}}\simeq\bigg{(}\mathds{1}_{3}-\frac{1}{2}\Theta^{\dagger}_{u}\Theta_{u}\bigg{)}\,V^{\prime}\,\bigg{(}\mathds{1}_{3}-\frac{1}{2}\Theta^{\dagger}_{d}\Theta_{d}\bigg{)}\,,$
(3.48)

an expression valid up to terms of order $\Theta^{3}_{q}$, as expected. Note
that the deviation from unitarity is evident for nonzero $\Theta_{u}$ or
$\Theta_{d}$.

It is now appropriate to discuss the standard quark FCNC and their
relationship to the deviations from unitarity of the CKM matrix, taking
advantage of the general and exact parameterization introduced before.

As demonstrated in section 3.1, in models involving singlet VLQs, the FCNC
affecting the standard quarks in each sector can be entirely determined by the
matrices $F^{d}=V^{\dagger}V$ and $F^{u}=VV^{\dagger}$.

The standard quark FCNC are controlled by the upper-left $3\times 3$ block of
each $F_{q}$ matrix, and these are equal to the identity matrix,
$\mathds{1}_{3}$, when the CKM matrix $V_{\textrm{CKM}}$, which represents the
upper-left $3\times 3$ block of the full quark mixing matrix, $V$, is unitary.

Hence, FCNC are closely linked to the aforementioned deviations from
unitarity. This connection becomes explicit when considering the precise
parameterization given by equation (3.26), used to derive eq. (3.35). When the
deviations from unitarity are small, this relation may be approximated by

$F^{q}\simeq\begin{pmatrix}\mathds{1}_{3}-\Theta_{q}^{\dagger}\Theta_{q}&\Theta_{q}^{\dagger}\\\
\Theta_{q}&\Theta_{q}\Theta^{\dagger}_{q}\end{pmatrix}\,,$ (3.49)

with the replacement $X_{L}^{q}\to\Theta_{q}$, an expression valid up to terms
of order $\Theta^{3}_{q}$. Note that in the limit $\Theta_{q}\to 0$, $F^{q}$
approaches the diagonal matrix $\operatorname{diag}(1,1,1,0,\ldots,0)$.

In order to better illustrate the relation between FCNC and deviations from
unitarity, the following useful parameters are introduced

$\Delta^{d}_{n}\equiv
1-\sum^{3+n_{u}}_{j=1}\big{|}V_{jn}\big{|}^{2}\,,\qquad\Delta^{u}_{n}\equiv
1-\sum^{3+n_{d}}_{j=1}\big{|}V_{nj}\big{|}^{2}\,.$ (3.50)

Here, $\Delta^{d}_{n}$ is a test of the unitarity of the $n$-th column of $V$,
with $n=1,...,3+n_{d}$, and $\Delta^{u}_{n}$ is a test of the unitarity of the
$n$-th row of $V$, with $n=1,...,3+n_{u}$. Since $F^{d}=V^{\dagger}V$ and
$F^{u}=VV^{\dagger}$, they are also given by

$\Delta^{q}_{n}=1-F^{q}_{nn}\,.$ (3.51)

Combining this equation with eqs. (3.35) (3.49) gives

$\Delta^{q}_{n}=1-\Big{(}\mathds{1}_{3}+X^{q^{\dagger}}_{L}X^{q}_{L}\Big{)}^{-1}_{nn}\simeq\big{(}\Theta_{q}^{\dagger}\Theta_{q}\big{)}_{nn}=\sum^{n_{q}}_{j=1}\big{|}(\Theta_{q})_{nj}\big{|}^{2}\,.$
(3.52)

Although these definitions are useful in theory, these parameters are not
easily measurable, since one, currently, only has access to the CKM elements
of $V$. Hence, experimentalists test the unitarity of $V_{\text{CKM}}$
directly (as presented in the beginning of section 3.2.1). Taking this into
account, definitions for parameters that test the unitarity of the CKM matrix
follow. To probe the unitarity of the columns,

$\delta^{d}_{n}\equiv
1-\sum^{3}_{j=1}\big{|}V_{jn}\big{|}^{2}=1-|V_{un}|^{2}-|V_{cn}|^{2}-|V_{tn}|^{2}$
(3.53)

and that of its rows,

$\delta^{u}_{n}\equiv
1-\sum^{3}_{j=1}\big{|}V_{nj}\big{|}^{2}=1-|V_{nd}|^{2}-|V_{ns}|^{2}-|V_{nb}|^{2}\,,$
(3.54)

with $n=1,2,3$ here. By comparing these expressions with eq. (3.50), a
connection between the parameters $\Delta^{q}_{n}$ and $\delta^{q}_{n}$
becomes apparent

$\delta^{d}_{n}-\Delta^{d}_{n}=\sum^{n_{u}}_{j=1}\big{|}V_{j+3,n}\big{|}^{2}\,,\qquad\delta^{u}_{n}-\Delta^{u}_{n}=\sum^{n_{d}}_{j=1}\big{|}V_{n,j+3}\big{|}^{2}\,.$
(3.55)

By using eq. (3.34) and eq. (3.46), these relations become

$\begin{split}\delta^{d}_{n}-\Delta^{d}_{n}&=\sum^{n_{u}}_{j=1}\big{|}(X^{u}_{L}V_{\text{CKM}})_{jn}\big{|}^{2}\simeq\sum^{n_{u}}_{j=1}\big{|}(\Theta_{u}V^{\prime})_{jn}\big{|}^{2}\,,\\\
\delta^{u}_{n}-\Delta^{u}_{n}&=\sum^{n_{d}}_{j=1}\big{|}(V_{\text{CKM}}X^{d\dagger}_{L})_{nj}\big{|}^{2}\simeq\sum^{n_{d}}_{j=1}\big{|}(V^{\prime}\Theta^{\dagger}_{d})_{nj}\big{|}^{2}\,.\end{split}$
(3.56)

Finally, it is instructive to consider the following limit scenarios:

  * •

When there are only down-type VLQs in the theory ($n_{u}=0$, $n_{d}\neq 0$),
the rows of $V$ are orthonormal, which gives $F^{u}=\mathds{1}_{3}$ and
$\Delta^{u}_{n}=0$. However, since these rows have more than three entries,
the rows of the $3\times 3$ $V_{\text{CKM}}$ generically deviate from
unitarity, i.e. they are not orthonormal,
$\delta^{u}_{n}=\sum^{n_{d}}_{j=1}|V_{n,j+3}|^{2}\neq 0$. Meanwhile, since in
these models $V$ is a $3\times(3+n_{d})$ matrix,
$\delta^{d}_{n}=\Delta^{d}_{n}$, and the columns of $V_{\text{CKM}}$ are also
not expected to be orthonormal.

  * •

When there are only up-type VLQs in the theory ($n_{u}\neq 0$, $n_{d}=0$), the
columns of $V$ are orthonormal, i.e. $F^{d}=\mathds{1}_{3}$ and
$\Delta^{d}_{n}=0$. Nevertheless,
$\delta^{d}_{n}=\sum^{n_{u}}_{j=1}|V_{j+3,n}|^{2}\neq 0$ generically.
Meanwhile, since in these models $V$ is a $(3+n_{u})\times 3$ matrix,
$\delta^{u}_{n}=\Delta^{u}_{n}$. As in the previous case, neither the rows nor
the columns of the $3\times 3$ $V_{\text{CKM}}$ are expected to be orthonormal
in general.

In the following sections, section 3.2.1 and section 3.2.2, useful
parameterizations for $V$ and $F^{u,d}$ in such limit cases, where only one
VLQ isosinglet is added to the SM, are discussed.

As stated in section 3.1, the perturbativity of the Yukawa couplings of SM
quarks to the Higgs boson is one of the most important constraints on the
mixing matrix, $V$, and the matrices that control its deviations from
unitarity $X_{L}^{u}$ and $X_{L}^{d}$. To make the interplay between
deviations from unitarity, mass scales and perturbativity explicit, consider
the upper-left block on both sides of the relation
$\mathcal{M}_{q}\mathcal{M}^{\dagger}_{q}=\mathcal{V}^{q}_{L}\mathcal{D}^{2}_{q}\mathcal{V}^{q\dagger}_{L}$

$m_{q}m^{\dagger}_{q}+\overline{m}\hskip 0.86108pt_{q}\overline{m}\hskip
0.86108pt^{\dagger}_{q}=K^{q}_{L}\big{(}d^{2}_{q}+X^{q\dagger}_{L}D^{2}_{q}X^{q}_{L}\big{)}K^{q\dagger}_{L}\,,$
(3.57)

combining it with eq. (LABEL:eq:uparam-exactK), this relation implies

$\displaystyle\textrm{Tr}\big{(}m_{q}m^{\dagger}_{q}+\overline{m}\hskip
0.86108pt_{q}\overline{m}\hskip 0.86108pt^{\dagger}_{q}\big{)}$
$\displaystyle=\textrm{Tr}\Big{[}\big{(}d^{2}_{q}+X^{q\dagger}_{L}D^{2}_{q}X^{q}_{L}\big{)}\big{(}1+X^{q\dagger}_{L}X^{q}_{L}\big{)}^{-1}\Big{]}$
(3.58)
$\displaystyle\simeq\sum^{3}_{i=1}m^{2}_{q_{i}}+\sum^{3}_{i=1}\sum^{n_{q}}_{j=1}\big{|}(\Theta_{q})_{ij}\big{|}^{2}M^{2}_{q_{j}}\,,$

where in the second line an approximation valid for small deviations from
unitarity was considered. Here, $m_{q_{i}}$ and $M_{q_{j}}$ refer,
respectively, to the the light and heavy quark masses in a given sector. At
this stage, the link to perturbativity is disclosed via the definitions
$m_{q}=\frac{v}{\sqrt{2}}Y_{q}$ and $\overline{m}\hskip
0.86108pt_{q}=\frac{v}{\sqrt{2}}\overline{Y}\\!_{q}$ and the requirement that

$\textrm{Tr}\big{(}m_{q}m^{\dagger}_{q}+\overline{m}\hskip
0.86108pt_{q}\overline{m}\hskip
0.86108pt^{\dagger}_{q}\big{)}<\rho\,m^{2}_{t}\,,$ (3.59)

for some $\mathcal{O}(1-10)$ numerical factor $\rho$. This requirements
reflects the fact that the individual Yukawa couplings should be at most
$\sim\mathcal{O}(1)$, or the theory may not be perturbative. Combining this
bound with eq. (3.58), one finds

$\sum_{i,j}\big{|}(\Theta_{q})_{ij}\big{|}^{2}M^{2}_{q_{j}}<\rho\,m^{2}_{t}-\sum_{i}m^{2}_{q_{i}}\,,$
(3.60)

showing how the deviations from unitarity are limited by the ratio between the
electroweak and VLQ scales in perturbative models.

For illustrative purposes, a model with only a single up-type VLQ $T$, has

$\sum_{i,j}\big{|}(\Theta_{u})_{ij}\big{|}^{2}M^{2}_{u_{j}}=\left(|\Theta_{u}|^{2}+|\Theta_{c}|^{2}+|\Theta_{t}|^{2}\right)m^{2}_{T}\leavevmode\nobreak\
,$ (3.61)

where the $3\times 1$ matrix, $\Theta_{u}$, here is implicitly defined as
$(\Theta_{u},\Theta_{c},\Theta_{t})^{T}$. In such a model, eq. (3.60) becomes

$\sqrt{|\Theta_{u}|^{2}+|\Theta_{c}|^{2}+|\Theta_{t}|^{2}}\lesssim\frac{m_{t}}{m_{T}}\,,$
(3.62)

which shows that all deviations from unitarity must be smaller than
$\mathcal{O}(m_{t}/m_{T})$, as expected.

####  3.1.3 CP Violation in the Presence of VLQs

CP violation arises from a conflict between the CP transformations demanded by
the gauge interactions and those required by the Yukawa interactions. The most
general CP transformation is defined as the one that preserves the gauge
interactions [150]. Subsequently, one may focus on the restrictions that CP
invariance imposes on the Yukawa couplings. In the framework of the SM, it can
be readily demonstrated that CP invariance implies [151]

$I_{\textrm{SM}}\,\equiv\,\textrm{Tr}\left[Y_{u}Y_{u}^{\dagger},\,Y_{d}Y_{d}^{\dagger}\right]^{3}\,=\,0\,,$
(3.63)

where $Y_{u,d}$ are the Yukawa couplings, i.e. $I_{\textrm{SM}}=0$ is a
necessary condition for CP invariance. The quantity $I_{\textrm{SM}}$ is a
weak-basis invariant, meaning it remains the same in any weak-basis
representation. For a single generation, this condition trivially vanishes.
For two generations, it can be shown that $I_{\textrm{SM}}=0$ follows from the
Hermiticity of $Y_{q}Y_{q}^{\dagger}$ ($q=u,d$). However, with three
generations, equation (3.63) imposes non-trivial constraints on the Yukawa
couplings. As $I_{\textrm{SM}}$ can be evaluated in any weak-basis
representation, one may consider the basis where $Y_{u}$ is diagonal and real.
In this case, the following relationship holds

$\displaystyle I_{\textrm{SM}}$
$\displaystyle\,=\,6i\,(y_{1}^{2}-y_{2}^{2})(y_{2}^{2}-y_{3}^{2})(y_{3}^{2}-y_{1}^{2})\,\operatorname{Im}\left(H_{12}H_{23}H_{31}\right)$
(3.64) $\displaystyle\,\xrightarrow{\textrm{\tiny
EWSB}}\,\frac{6i}{v^{12}}\,(m_{c}^{2}-m_{u}^{2})(m_{t}^{2}-m_{c}^{2})(m_{t}^{2}-m_{u}^{2})$
$\displaystyle\qquad\qquad\quad(m_{b}^{2}-m_{s}^{2})(m_{b}^{2}-m_{d}^{2})(m_{s}^{2}-m_{d}^{2})\,\operatorname{Im}\left(V_{us}V_{cb}V_{ub}^{*}V_{cs}^{*}\right)\,,$

where $Y_{u}=\operatorname{diag}(y_{1},y_{2},y_{3})$ and $H\equiv
Y_{d}Y_{d}^{\dagger}$. Moreover, in this case eq. (3.63) is both a necessary
and sufficient condition to have CP invariance in the SM. It should be noted
that in the case of three generations, the condition for CP invariance is
equivalent to $\det[M_{u}M_{u}^{\dagger},M_{d}M_{d}^{\dagger}]=0$ [152, 153],
where this determinant is proportional to $I_{\textrm{SM}}$. For more than
three generations, equation (3.63) remains a necessary condition for CP
invariance, but it is no longer sufficient (cf. ref. [35]).

In the first line of equation (3.64), the invariant $I_{\textrm{SM}}$ is
expressed solely in terms of Yukawa couplings, and the analysis is conducted
at the Lagrangian level, prior to Electroweak Symmetry Breaking (EWSB).
However, it is also possible to perform this analysis after EWSB, as shown in
the second line of equation (3.64), where the invariant is written in terms of
quark masses and mixing. It is worth emphasizing that the CP conditions can be
investigated at the Lagrangian level without necessarily relying on quark
masses and mixing. The pioneering result by Kobayashi and Maskawa [154] could
have been derived without much effort by considering a general CP
transformation without invoking the matrix $V_{\textrm{CKM}}$.

CP violation is present in the quark sector of the SM if and only if any of
the rephasing-invariant functions of the CKM matrix is not real [35]. Examples
of such functions include _quartets_ like $Q_{uscb}\equiv
V_{us}V_{cb}V_{ub}^{*}V_{cs}^{*}$. In the SM quark sector, there is a single
physical phase parameter, as demonstrated in section 3.1.1.2 with
$n_{u}=n_{d}=0$, which implies the equality of the imaginary parts of all
quartets up to a sign [152, 153, 155]. By defining
$I\equiv\operatorname{Im}Q_{uscb}$, it becomes evident that the condition
$I=0$ is not only consistent with CP invariance but also implies it, as
evident from eq. (3.64). Moreover, the quantity $|I|$ has a geometrical
interpretation, as it corresponds to twice the area of any unitarity triangle.
Taking into consideration the magnitudes of the moduli of $V_{\text{CKM}}$
elements, the standard and most useful unitarity triangle is defined by the
unitarity relation:

$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0\,,$ (3.65)

with each term in the sum corresponding to a side of the triangle in the
complex plane.

The experimental determination of the physical CP-violating phases present in
the quark mixing matrix serves as a rigorous test of the Standard Model (SM).
Without enforcing unitarity, the $3\times 3$ matrix $V_{\text{CKM}}$ contains
four independent rephasing-invariant phases, as five out of the initial nine
phases can be eliminated through the rephasing of the standard quark fields,
as discussed earlier. It is crucial to emphasize that this result is entirely
general and holds true regardless of the presence of VLQs and without relying
on the existence of extra generations These 4 phases can taken to be [156,
157]:

$\displaystyle\beta$
$\displaystyle\,\equiv\,\arg\left(-V_{cd}V_{tb}V_{cb}^{*}V_{td}^{*}\right)\,,$
(3.66) $\displaystyle\gamma$
$\displaystyle\,\equiv\,\arg\left(-V_{ud}V_{cb}V_{ub}^{*}V_{cd}^{*}\right)\,,$
$\displaystyle\chi\text{ or }\beta_{s}$
$\displaystyle\,\equiv\,\arg\left(-V_{cb}V_{ts}V_{cs}^{*}V_{tb}^{*}\right)\,,$
$\displaystyle\chi^{\prime}\text{ or }\beta_{K}$
$\displaystyle\,\equiv\,\arg\left(-V_{us}V_{cd}V_{ud}^{*}V_{cs}^{*}\right)\,.$

Another commonly used definition is
$\alpha\equiv\arg\left(-V_{td}V_{ub}V_{tb}^{*}V_{ud}^{*}\right)$, however, it
is not an independent phase since it is related to the phases mentioned
earlier by $\alpha+\beta+\gamma=\pi$ (mod $2\pi$) by definition.

The matrix $V_{\text{CKM}}$, which describes the mixing among the known
quarks, comprises 13 measurable quantities: 9 moduli and 4 rephasing-invariant
phases. In the limit of the Standard Model where $V_{\text{CKM}}$ is unitary,
there are $n_{f}^{2}=9$ exact and independent relations among these
quantities, such as the one given in equation (3.65), resulting in a total of
4 independent parameters. These 4 parameters are typically chosen as the
standard 3 mixing angles and 1 CP-violation phase. However, they may also be
taken to be i) the phases in eq. (3.66), or ii) four independent moduli. In
fact, ref.[156] demonstrated that assuming $3\times 3$ unitarity of
$V_{\text{CKM}}$ allows the rephasing-invariant phases to be used for
reconstructing the entire CKM matrix. In the SM limit, the phases $\alpha$,
$\beta$, and $\gamma$ also correspond to the inner angles of the standard
unitarity triangle mentioned earlier. Alternatively, the $3\times 3$ unitary
$V_{\text{CKM}}$, up to a two-fold ambiguity and including the strength of CP
violation measured by $|I|$, can be reconstructed by taking 4 independent
moduli $|V_{ij}|$ as input[158, 159]. An additional sign is needed for the
complete reconstruction since the moduli do not determine the sign of CP
violation. Finally, it is important to note that physical processes can only
depend on WB invariant quantities. It has been shown [159] that in the SM, any
four independent moduli of $V_{\text{CKM}}$ can be computed from the four WB
invariants

$\textrm{Tr}(H_{u}H_{d})\,,\quad\textrm{Tr}(H_{u}H^{2}_{d})\,,\quad\textrm{Tr}(H^{2}_{u}H_{d})\,,\quad\textrm{Tr}(H^{2}_{u}H^{2}_{d})\,,$
(3.67)

and the six known quark masses, with $H_{q}\equiv Y_{q}Y_{q}^{\dagger}$
($q=u,d$).

The presence of VLQs introduces new sources of CP violation, as demonstrated
in section 3.1.1.2. To illustrate this, one may consider the case where a
single up-type VLQ is added to the SM ($n_{u}=1$, $n_{d}=0$), while taking the
extreme chiral limit $m_{u}=m_{c}=m_{d}=m_{s}=0$. In this chiral limit,
$I_{\text{SM}}=0$, meaning there is no CP violation in the SM since the masses
of same-charge quarks are degenerate. However, the introduction of VLQ
interactions allows for the presence of CP-violating phases even in this
chiral limit [160]. Specifically, for $n_{u}=1$ and $n_{d}=0$, it is possible
to choose a WB, see section 3.1.1, in which $\mathcal{M}_{d}$ is diagonal
through appropriate choices of $W_{L}$ and $\mathcal{W}^{d}_{R}$, while
$\mathcal{M}_{u}=\mathcal{V}^{u}_{L},\mathcal{D}_{u}$ with
$\mathcal{V}_{R}^{u}$ being cancelable by a suitable $\mathcal{W}^{u}_{R}$.
Consequently, the up-quark mass matrix has two vanishing columns in the chiral
limit. By using the remaining rotation and rephasing freedoms, it is possible
to arrive at a WB where

$\mathcal{M}_{u}\sim\begin{pmatrix}0&0&\times&0\\\ 0&0&0&\times\\\
0&0&\times&\ast\\\
0&0&\times&\times\end{pmatrix}\,,\qquad\mathcal{M}_{d}\sim\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&m_{b}\end{pmatrix}\,.$ (3.68)

In this WB, all entries in the up-quark mass matrix are real except for the
element denoted by “$\ast$”, which is complex and cannot be removed in
general. The presence of this complex element implies the non-vanishing of a
WB invariant [148, 161] that is proportional to both $\operatorname{Im}(\ast)$
and $m_{b}^{2}$, indicating a violation of CP symmetry beyond the SM. The
chiral limit is not only physically natural for studying CP violation beyond
the SM but also phenomenologically relevant in practice. In high-energy
collisions, such as those at the TeV scale, the masses of light fermions are
negligible, and distinguishing light quark jets becomes impractical.
Therefore, significant CP violation at high energy is expected to originate
from beyond the SM physics.

In the presence of VLQs, the extraction of physical phases from data requires
reexamination. The mixing matrix $V$ is now of size $(3+n_{u})\times(3+n_{d})$
and generally non-unitary. As discussed earlier, its upper-left $3\times 3$
block $V_{\text{CKM}}$ contains 4 independent physical phases, as shown in eq.
(3.66). Without loss of generality, one can adopt the following phase
convention [35]

$V=\leavevmode\hbox to168.65pt{\vbox
to76.25pt{\pgfpicture\makeatletter\hbox{\hskip 6.66669pt\lower-36.36504pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{{}}{{}}{{}}{
{}}{{}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-32.83203pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{}|V_{ud}{}|}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}}}}{{{{}{}{}}}}{{{ {}{}{}}}}{{{ {}{}{}}}}{ {}{}{}}{{{
{}{}{}}}}{{{{}{}{}}}}{{{ {}{}{}}}}{{{ {}{}{}}}}{} {{}} {{{}{{}{}}{}}}{{
{}{}{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$V_{\text{CKM}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (3.69)

In the presence of NP effects, the extraction of the CKM phases from
experimental data can be affected. In models with VLQs, where NP contributions
are negligible in weak processes where the SM dominates at tree level, the
determination of $\gamma$ and the moduli from the first two rows of
$V_{\text{CKM}}$ remains unaffected. However, the extraction of $\beta$ and
$\chi$ from $B$ meson decays may be contaminated by NP effects. Experiments
may instead be sensitive to the combinations $\bar{\beta}\equiv\beta-\phi_{d}$
and $\bar{\chi}\equiv\chi+\phi_{s}$, where $\phi_{q}$ ($q=d,s$) are phases
parameterising NP-induced deviations from the SM in the neutral meson systems
$B^{0}_{q}$–$\overline{B^{0}_{q}}$. For example, the decay $B_{d}^{0}\to
J/\psi,K$ is sensitive to $\sin 2(\bar{\beta}-\chi^{\prime})\simeq\sin
2\bar{\beta}$, while the decay $B_{s}^{0}\to J/\psi,\phi$ measures $\sin
2\bar{\chi}$.

In the context of the SM or NP scenarios where $V_{\text{CKM}}$ is unitary,
the remaining phase $\chi^{\prime}$ is expected to be extremely small, with a
value of $\chi^{\prime}\sim 6\times 10^{-4}$[162]. Even if $V_{\text{CKM}}$ is
not strictly unitary but can still be described as the upper-left block of an
auxiliary $(3+n_{u}+n_{d})\times(3+n_{u}+n_{d})$ unitary matrix [147], a bound
can still be placed on $\chi^{\prime}$[163, 35]. Considering the CKM moduli
within their $3\sigma$ ranges [164], it has been found that $|\chi^{\prime}|$
is constrained to be less than or around $0.06$, which is approximately
equivalent to $3^{\circ}$. Consequently, in most cases of interest, this phase
can be safely neglected. Similarly, the phase $\chi$ can be significantly
larger than the SM expectation $\chi\sim 0.02$ in models involving up-type
VLQs, but not in models solely comprising down-type VLQs [162]. Up-type VLQs
have been considered to address tensions in the data by introducing a non-zero
$\phi_{d}$[165].

Furthermore, both up-type and down-type VLQs have been explored to potentially
explain apparent deviations from unitarity in the first row of
$V_{\text{CKM}}$[22, 23, 34]. It is important to note that in the presence of
possible deviations from unitarity, the familiar SM unitary triangle may be
modified. For instance, in extensions involving only one VLQ, the unitary
triangle may be replaced by various quadrangles [160]. As a result, instead of
having only one CP-odd invariant, proportional to
$\operatorname{Im}\left(V_{us}V_{cb}V_{ub}^{*}V_{cs}^{*}\right)$ as in the SM,
one may obtain several rephasing-invariants arising from other quartets. These
new invariants are proportional to
$\operatorname{Im}\left(V_{ij}V_{km}V_{im}^{*}V_{kj}^{*}\right)$ and, in
general, are not equal but may be partly related to each other.

###  3.2 Models with $n_{d}=0$ or $n_{u}=0$

To illustrate some features of VLQ models, this section will discuss models
where only up-like (down-like) VLQs are added to the SM, hence $n_{d}=0$
($n_{u}=0$). These models are relevant in light of the recent CKM unitarity
problem [18, 19, 20, 21, 22, 23], as they are a natural and simple solution
for it. Imposing unitarity of the CKM matrix in the first row gives

$|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1\,.$ (3.70)

However, considering the fact that $|V_{ub}|^{2}$ is much smaller than the
other two entries, with $|V_{ub}|^{2}\simeq 1.6\times 10^{-5}$, the following
approximate relation

$|V_{ud}|^{2}+|V_{us}|^{2}\simeq 1\,,$ (3.71)

is valid up to $\mathcal{O}\left(|V_{ub}|^{2}\right)$. Through precise
experimental data and improved control over theoretical uncertainties, the
values of $|V_{ud}|$, $|V_{us}|$, and $|V_{ud}|/|V_{us}|$ have been determined
with high accuracy. These results, along with the approximate relation stated
in equation (3.71), provide stringent tests for the unitarity of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix.

At energy scales much lower than the mass of the $W$ boson ($M_{W}$), the
matrix elements $|V_{ud}|$ and $|V_{us}|$ play a crucial role in describing
the strength of the effective charged-current interaction responsible for the
leptonic decays of hadrons involving the valence quarks $u$, $d$, and $s$.
Experimental data from neutron decays [166] primarily contribute to the
determination of $|V_{ud}|$, while the entry $|V_{us}|$ is mainly obtained
from kaon decay data. The ratio $\left|V_{us}/V_{ud}\right|$ can be
independently determined by comparing the decay rates of specific radiative
pion and kaon decays.

However, recent advancements in the determination of form factors and
inclusion of radiative corrections for relevant meson and neutron decay
processes have resulted in a downward shift in the central value of
$|V_{ud}|$. This has led to tension with the constraint imposed by the
unitarity of the first row of the CKM matrix [24, 25, 18, 19, 20, 21]. Taking
into account these new theory calculations with reduced hadronic
uncertainties, the authors of Ref. [22] find:

$\begin{array}[]{ccc}|V_{us}|=0.22333(60)\,,&|V_{ud}|=0.97370(14)\,,&\left|V_{us}/V_{ud}\right|=0.23130(50)\,.\end{array}$
(3.72)

These results deviate from the unitarity condition in eq. (3.71) by more than
$4\sigma$, disfavouring the SM CKM unitarity at this level. In fact, the
values in eq. (3.72) are compatible with

$|V_{ud}|^{2}+|V_{us}|^{2}=1-\delta\,,$ (3.73)

where

$\sqrt{\delta}=0.04\pm 0.01\,\quad(95\%\text{ C.L.})\,.$ (3.74)

Note that

$\delta\equiv\delta_{1}^{u}$ (3.75)

was implicitly defined, from the definition of eq. (3.54). Consequently, it is
observed that the parameter $\delta$ is of the order of
$\mathcal{O}(|V_{cb}|^{2})$. Recent lattice calculations performed by the FLAG
collaboration [167] also support these findings. This deviation between the
predictions of the Standard Model and experimental data is commonly known as
the CKM unitarity problem or Cabibbo anomaly.

To address this discrepancy, new physics beyond the Standard Model is required
if it indeed persists. One possible solution involves introducing an
additional mixing channel. One approach to achieve this is by introducing a
singlet vector-like quark (VLQ) into the framework of the Standard Model. The
inclusion of an extra VLQ would result in an enlarged mixing matrix, where the
effective CKM mixing is described by a non-unitary $3\times 3$ sub-matrix
involving the standard quarks. In the literature, models considering both one
additional down-type VLQ and one additional up-type VLQ have been explored
within this context [22, 23, 34, 168].

In theories in which only up-type VLQs are introduced, it is useful to work in
a WB where the down quarks mass matrix $\mathcal{M}_{d}$, in this case a
$3\times 3$ matrix, is diagonal, $d_{d}$. This WB exists when $n_{d}=0$, as
stated in section 3.1.1. In this case, from eq. (3.12) one obtains

$\mathcal{V}_{L}^{u\dagger}=\begin{pmatrix}V&B_{L}^{u\dagger}\end{pmatrix}\,,$
(3.76)

where eq. (3.19) was used in combination with

$A_{L}^{d}=\mathds{1}_{3}\leavevmode\nobreak\ ,$ (3.77)

which is valid for the WB in use. $\mathcal{V}_{L}^{u\dagger}$ is a
$(3+n_{u})\times(3+n_{u})$ unitary matrix, convenient to parameterise because
the parameterisations for unitary matrices are known and the physical
$(3+n_{u})\times 3$ mixing matrix, $V$, can be read off directly from it -
_its first three columns_. Hence, when only one sector has extra mixing
channels, this procedure might be more useful and easier to grasp, as one can
use parameterisations a la PDG, with extra mixing angles and phases.

Again, for models in which only down-type VLQs are introduced, it is useful to
work in a WB where the up quarks mass matrix $\mathcal{M}_{u}$, in this case a
$3\times 3$ matrix, is diagonal, $d_{u}$. This WB exists when $n_{u}=0$, as
stated in section 3.1.1. In this case, from eq. (3.12) one obtains