# Constrained Vector-like quark model with perturbative unitarity

Elena Accomando<EMAIL_ADDRESS>School of Physics & Astronomy,
University of Southampton, Highfield, Southampton SO17 1BJ, UK  Particle
Physics Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11
0QX, UK  Jake Brannigan<EMAIL_ADDRESS>School of Physics & Astronomy,
University of Southampton, Highfield, Southampton SO17 1BJ, UK  Jacob Gunn
<EMAIL_ADDRESS>School of Physics & Astronomy, University of Southampton,
Highfield, Southampton SO17 1BJ, UK  Yuheng Huyan<EMAIL_ADDRESS>School
of Physics & Astronomy, University of Southampton, Highfield, Southampton SO17
1BJ, UK  Sam Mulligan<EMAIL_ADDRESS>School of Physics & Astronomy,
University of Southampton, Highfield, Southampton SO17 1BJ, UK

###### Abstract

We propose a Vector-like quark (VLQ) model constrained by the requirement of
perturbative unitarity. In this scenario, the neutral and charged couplings of
the new heavy VLQs to SM quarks and bosons are strictly related. We derive the
corresponding sum rules and display the definite structure of the couplings.
We analyse, qualitatively, the phenomenological consequences of this model in
terms of expected width, production mechanism and decay channels. We show that
the upcoming LHC run 3 could have sensitivity to such particles but the search
strategy should be adapted to cover wide resonances decaying mainly in the
charged channel $T(B)\rightarrow Wq$.

##  I Introduction

Though the Standard Model (SM) has proved itself amazingly successful, well
established tensions of the SM with precision measurements indicate the need
for new physics. The CKM matrix encodes couplings for each of the three
generation quarks. By construction, within the SM the CKM matrix is unitarity.
For the first row, unitarity relations predict that
$|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1=C^{1}_{3}$. Yet, the current best
experimental determinations of $V_{ud}$ and $V_{us}$ find
$C^{1}_{3}=0.99798\pm 0.0003$ that is a disagreement greater than 4$\sigma$
Belfatto  _et al._ (2020). The recent dataset collected after 2018 Aoki  _et
al._ (2020) disfavours the CKM unitarity for three generations of quarks at
99.998% C.L. Such a deviation from the CKM unitarity seems to suggest the
presence of new physics. The hypothetical VLQs could explain this misalignment
between the SM predictions and observed data; this note will address this
Beyond Standard Model (BSM) scenario. This class of models can indeed
naturally lead to deviations from the CKM unitarity capable of accommodating
the recent measurements of $|V_{us}|$ and $|V_{ud}|$ Belfatto  _et al._
(2020); Branco  _et al._ (2021).

Vector-like quarks (VLQs) are spin 1/2 particles characterised by having the
left- and right-handed components defined by the same colour and electroweak
quantum numbers. Under a given gauge group, their left and right-handed
projections belong to the same representation. They naturally emerge in many
BSM scenarios. VLQs with a TeV-scale mass are indeed strongly motivated by two
main theoretical frameworks: models where the Higgs is a pseudo-Goldstone
boson Perelstein  _et al._ (2004); Contino  _et al._ (2007a); Matsedonskyi
_et al._ (2013) and partial compositeness flavour theories Contino  _et al._
(2007b). In the first case, vector-like quarks are required to induce the
electroweak symmetry breaking and explain the lightness of the observed Higgs
boson. In the latter, they are part of the predicted extra heavy fermion
resonances. Explicit realisations of these two basic mechanisms are little
Higgs models, composite Higgs models, and their holographic versions. VLQs
also appear in extra dimension theories with quarks propagating in the bulk,
and in grand unified or string $E_{6}$-models. There is a vast literature on
the development of general VLQ models realised by extending the SM with the
sole inclusion of VLQs coming in three possible representations i.e. singlet,
doublet and triplet. For a review see Refs. Cacciapaglia  _et al._ (2010,
2012, 2018); Botella  _et al._ (2012).

From the experimental point of view, vector-like quarks are the only coloured
fermions still allowed by the most recent data. The forth generation of quarks
with chiral couplings is indeed highly disfavoured by the electroweak
precision tests combined with Higgs data Eberhardt  _et al._ (2012); Djouadi
and Lenz (2012). Vector-like quarks can evade such exclusion bounds as they
are not chiral, a priori, and do not have to acquire their mass via the Higgs
mechanism. Therefore, VLQs are receiving a lot of attention at the LHC. The
CERN CMS collaboration has set mass exclusion bounds up to 1650 ${\rm GeV}$,
depending on the vector-like quark type, coupling and decay width Sirunyan
_et al._ (2019). The CERN ATLAS collaboration has set analogous exclusion
limits at 1030 (1210) ${\rm GeV}$ for singlet (doublet) up-type VLQ and 1010
(1140) ${\rm GeV}$ for singlet (doublet) down-type VLQ Vale (2018).

Direct experimental searches for such heavy states assume that a VLQ resonance
can be described by a Breit-Wigner (BW) line-shape, standing over the SM
background, when looking at the cross section distribution in the invariant
mass of the VLQ decay products. Hence, the new physics signal is expected to
have a peaking structure, concentrated in an interval centred around its mass.
The VLQs existence could also be inferred from indirect searches. These new
particles could in fact give rise to potentially striking effects in low
energy physics. New tree-level flavour-changing neutral currents (FCNC),
mediated by the weak $Z$-boson and/or the Higgs boson, could appear. VLQs
could also introduce new sources of CP violation. The landscape of signatures
is quite varied and interesting, a priori.

There is a strong interplay between direct and indirect searches for VLQs. The
mixing between the SM quarks and VLQs, which generates the FCNC, has an impact
both on the production process and the open decay channels of these latter
particles. This result, in turn, affects the way the new heavy particles are
searched for at the LHC. Direct experimental searches are typically optimised
for narrow resonances and, generally, assume a pure BW line-shape over the SM
background, when looking at the invariant mass of the VLQs decay products. On
the basis of this assumption, in absence of evidence, the 95% Confidence Level
(CL) upper bound on the cross section is derived and limits on the mass and
couplings of the resonance are extracted within some benchmark model. In this
approach, theoretical cross sections are usually calculated in Narrow Width
Approximation (NWA); more rarely, they might include Finite Width (FW) and
interference effects. The main production channel of VLQs at the LHC is the
pair production, mediated by the gluon. The extra heavy quarks subsequently
decay into SM particles, namely ordinary quarks plus a Higgs or a gauge boson,
with branching ratios that are mostly determined by their gauge quantum
numbers. These decays occur through the mixing of the new quarks with the SM
ones. The very same mixing opens up the possibility to singly produce the new
quarks. This becomes the dominant production mechanism for high enough masses.

The purpose of this paper is building a VLQ model that preserves the
perturbative unitarity of the theory, explicitly. The inclusion of this
property constrains the VLQs couplings to the SM gauge bosons and matter,
enhancing the predictivity of the model. The advantage of this approach is
twofold. Firstly, one can find relations between the charged current and
neutral current couplings. Secondly, these relations can be used to extract
bounds on such couplings from different experiments. We consider two possible
group representations for the VLQs: singlet, $T$ or $B$, and doublet $(T,B)$.
To model the VLQs couplings to SM particles, by imposing perturbative
unitarity, we then use candle processes involving SM vector bosons and quarks
along with at least one vector-like quark.

The format of this note is as follows. In Sect. II, we define the VLQs
representations considered in this paper and the corresponding Lagrangian for
the charged and neutral currents. In Sect. III, we derive the relations
between charged and neutral VLQs couplings, by requiring perturbative
unitarity, and define our constrained model. In Sect. IV, we illustrate some
phenomenological aspects of the vector-like quarks with constrained couplings,
which could have significant consequences for their search at the LHC.
Finally, we conclude in Sect. V.

##  II VLQs representations

New heavy vector-like quarks, with renormalisable couplings to the SM gauge
bosons and quarks, can only appear in seven possible multiplets characterised
by well defined $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ quantum numbers del
Aguila  _et al._ (2000a, b): $T$\- and $B$-singlets, $(X,T)$-, $(T,B)$\- and
$(B,Y)$-doublets, $(X,T,B)$\- and $(T,B,Y)$-triplets. The new heavy partners
of the SM top and bottom quarks, $T$ and $B$, have electric charge $2/3$ and
$-1/3$, respectively. The exotic vector-like quarks, $X$ and $Y$, have instead
a charge different from the SM up-type and down-type quarks. They are indeed
characterised by a fractional charge of $5/3$ and $-4/3$, respectively.

The Lagrangian for the charged current induced by the $W$-boson can be written
as:

${\mathcal{L}}_{W}=-{g_{W}\over\sqrt{2}}\bar{U}\gamma^{\mu}(V^{L}_{UD}P_{L}+V^{R}_{UD}P_{R})DW^{+}_{\mu}+H.c.$
(II.1)

where $P_{L}$ and $P_{R}$ are the left- and right-handed projectors,
respectively. The symbol $U(D)$ stands for any up(down)-type quarks, including
the extra vector-like quark. The couplings $V^{L,R}_{UD}$ are the entries of
the left-handed and right-handed CKM matrix, extended by the addition of the
hypothetical vector-like quark(s). In this scenario, the SM CKM matrix appears
as the $3\times 3$ block of the left-handed matrix (within the SM the right-
handed entries are null).

The Lagrangian for the neutral current induced by the weak $Z$-boson is:

${\mathcal{L}}_{Z}=-{g_{W}\over{c_{W}}}\bar{q}\gamma^{\mu}\left((\pm
c^{ZL}_{qq^{\prime}}P_{L}\pm
c^{ZR}_{qq^{\prime}}P_{R})-2Q_{q}s^{2}_{W}\delta_{qq^{\prime}}\right)q^{\prime}Z_{\mu}+H.c.$
(II.2)

where $q$ and $q^{\prime}$ describe any quarks, including the heavy vector-
like quark(s). Here, $s_{W}(c_{W})$ is the Weinberg angle sine(cosine) and
$Q_{q}$ the electric charge of the quark $q$. The plus sign is for up-type
quarks, the minus for down-type quarks. The first two terms in the Lagrangian
can give rise to flavour changing neutral currents, in case $q$ and
$q^{\prime}$ were different. In that event, the complex parameters $c^{ZL}$
and $c^{ZR}$ represent the strength of the FCNC of the weak neutral $Z$-boson.

Finally, the Lagrangian for the neutral current induced by the Higgs boson is:

${\mathcal{L}}_{h}=-{g_{W}\over{2M_{W}}}\bar{q}\gamma^{\mu}(\pm
M_{L}c^{hL}_{qq^{\prime}}P_{L}\pm
M_{R}c^{hR}_{qq^{\prime}}P_{R})q^{\prime}H+H.c.$ (II.3)

where $M_{L,R}$ are generic masses, which will later be related to the masses
of the $q$ and $q^{\prime}$ quarks. One can notice that, as for the $Z$-boson
mediated neutral current, also in the case of the Higgs boson one can have the
appearance of FCNC whose strength is represented by the complex parameters
$c^{hL}$ and $c^{hR}$. Moreover, oppositely to the SM predictions, in these
BSM scenarios the left- and right-handed couplings of the Higgs to fermions
are not identical. This implies that the Higgs might have a pseudo-scalar
component whose magnitude depends on the difference
$(c^{hL}_{qq^{\prime}}-c^{hR}_{qq^{\prime}})/(c^{hL}_{qq^{\prime}}+c^{hR}_{qq^{\prime}})$.

In this paper, we shall restrict ourselves to extensions of the SM with only
one extra multiplet. In particular, as anticipated in the introduction, we
analyse three types of multiplet: $T$-singlet, $B$-singlet and
$(T,B)$-doublet.

##  III Perturbative unitarity

In this section, we analyse the consequences of imposing the perturbative
unitarity on the Standard Model extended by the inclusion of vector-like
quarks. Our approach in this investigation is intended to be generic and model
independent; therefore, we will not specify beforehand a representation with
which we will derive our results. For the same reason, we do not make any
assumptions on the couplings of the VLQs with SM particles.

We select as candle process the vector-boson-fermion scattering (VBFS)

$W^{+}_{L}(k_{1})d(p_{1})\rightarrow W^{+}_{L}(k_{2})b(p_{2})$ (III.1)

where $d$ and $b$ are the down- and bottom-quark, respectively, and $W^{+}$
the weak $W$-bosons. The symbol $L$ denotes the longitudinal polarisation of
the vector bosons while $k_{1},p_{1},k_{2},p_{2}$ are the 4-momenta of the
particles. We generalise the result in the next section. The choice of this
channel is well motivated. When considering weak interactions, the
preservation of perturbative unitarity is intimately tied to the Higgs
mechanism and the Higgs boson. The Higgs mechanism is indeed responsible for
the longitudinal polarisations of the weak vector bosons, whose charged
version is present in our chosen interaction. As a thumb rule, for each weak
gauge boson appearing as an external particle in a two-by-two scattering
process, the matrix element acquires an energy growth proportional to the
total Centre-of-Mass (CoM) energy, $\sqrt{s}=E^{CM}_{TOT}$. The Higgs boson is
then required to cancel these energy dependent terms in the matrix element, if
all the couplings involved have the correct structure. Our candle process
(including its generalisations) is therefore excellent for extracting
informations on the relations between the different fermion couplings to gauge
bosons by using the requirement of perturbative unitarity. The above channel
has in fact two $W$-bosons, giving rise to an increase with energy of the
matrix element proportional to $s$, and two fermions allowing for the presence
of extra vector-like quarks either as external particles in generalised
processes or as a mediator.

As already mentioned, we consider three group representations of the VLQs:
$T$-singlet, $B$-singlet and $(T,B)$ doublet. From the requirement of
perturbative unitarity, sum rules for the VLQs couplings with SM gauge bosons
and quarks will be derived. The sum rules have implications on the
phenomenology of the VLQs, which will be discussed in the next section.

At the tree level, there are three Feynman diagrams potentially contributing
to this process. They are shown in Fig. 1. The first of these diagrams is the
only interaction of the three to be ordinarily permitted by the standard
model. The remaining two diagrams represent flavour-changing neutral currents
(FCNC), which can only happen at loop level in the SM and are highly
suppressed by the GIM mechanism. As such, modification of SM observables by
contributions from FCNCs could provide evidence for the presence of new
vector-like quarks.

Figure 1: Complete set of Feynman diagrams, at the tree level, contributing to
the process $W^{+}d\rightarrow W^{+}b$. From left to right, the first graph is
mediated by up-type quarks, the second and the third by the Z- and Higgs
boson, respectively.

In order to analyse the behaviour of the matrix element at high energies and
see what conditions on couplings and masses are required to guarantee
perturbative unitarity, we consider longitudinally polarised $W$-bosons. We
adopt the representation of the longitudinal polarisation vectors,
$\epsilon^{\mu}_{L}$, given by the Mathematica package Feyncalc Shtabovenko
_et al._ (2020, 2016):

$\epsilon^{\mu}_{L}(p_{1})={{p_{1}^{\mu}*(k_{1}\cdot p_{1}+k_{2}\cdot
p_{1})-(k_{1}+k_{2})^{\mu}p_{1}^{2}}\over{\sqrt{p_{1}^{2}[(k_{1}\cdot
p_{1}+k_{2}\cdot p_{1})^{2}-p_{1}^{2}(k_{1}+k_{2})^{2}}}}$ (III.2)

with the 4-momentum assignment given above. Performing a Taylor expansion in
the mass over energy ratio, we obtain the analytical expression for the matrix
elements corresponding to the Feynman diagrams in Fig. 1. The amplitude of the
first graph mediated by the up-type quarks is given by:

$\displaystyle M_{U}$
$\displaystyle=\sum_{u}F_{u}\bar{u}_{b}(k_{2})[A_{u}P_{L}+B_{u}P_{R}+C_{u}\not{p_{1}}P_{R}+D_{u}\not{p_{1}}P_{L}]u_{d}(p_{2})$

where the sum goes over all the possible up-type quarks: up, charm, top and
the heavy $T$ vector like quark. The constants $A_{u},B_{u},C_{u},D_{u}$ are
functions of the CoM energy $s$ and masses:

$\displaystyle A_{u}$
$\displaystyle=m_{b}(m_{b}^{2}-M_{W}^{2}-s)\left[V_{bu}^{L*}V_{ud}^{L}\left(s(s-M_{W}^{2})+m_{d}^{2}(m_{d}^{2}-M_{W}^{2}-2s)\right)+V_{bu}^{L*}V_{ud}^{R}(-2m_{u}m_{d}M_{W}^{2})\right]+$
$\displaystyle(s-M_{W}^{2}-m_{b}^{2})\left[V_{bu}^{R*}V_{ud}^{L}m_{u}(s^{2}-M_{W}^{2}s-2m_{d}^{2}s-M_{W}^{2}m_{d}^{2}+m_{d}^{4})-2m_{d}M_{W}^{2}sV_{bu}^{R*}V_{ud}^{R}\right]$

$\displaystyle B_{u}$
$\displaystyle=m_{b}(m_{b}^{2}-M_{W}^{2}-s)\left[V_{bu}^{R*}V_{ud}^{R}\left(s(s-M_{W}^{2})+m_{d}^{2}(m_{d}^{2}-M_{W}^{2}-2s)\right)+V_{bu}^{R*}V_{ud}^{L}(-2m_{u}m_{d}M_{W}^{2})\right]+$
$\displaystyle(s-M_{W}^{2}-m_{b}^{2})\left[V_{bu}^{L*}V_{ud}^{R}m_{u}(s^{2}-M_{W}^{2}s-2m_{d}^{2}s-M_{W}^{2}m_{d}^{2}+m_{d}^{4})-2m_{d}M_{W}^{2}sV_{bu}^{L*}V_{ud}^{L}\right]$

$\displaystyle C_{u}$
$\displaystyle=(s-M_{W}^{2}-m_{b}^{2})\left[V_{bu}^{R*}V_{ud}^{R}s(s-M_{W}^{2}-m_{d}^{2})-m_{u}m_{d}V_{bu}^{R*}V_{ud}^{L}(s+M_{W}^{2}-m_{d}^{2})\right]+$
$\displaystyle
m_{b}(m_{b}^{2}-M_{W}^{2}-s)\left[-m_{d}V_{bu}^{L*}V_{ud}^{L}(s+M_{W}^{2}-m_{d}^{2})+m_{u}V_{bu}^{L*}V_{ud}^{R}(s-M_{W}^{2}-m_{d}^{2})\right]$

$\displaystyle D_{u}$
$\displaystyle=(s-M_{W}^{2}-m_{b}^{2})\left[V_{bu}^{L*}V_{ud}^{L}s(s-M_{W}^{2}-m_{d}^{2})-m_{u}m_{d}V_{bu}^{L*}V_{ud}^{R}(s+M_{W}^{2}-m_{d}^{2})\right]+$
$\displaystyle
m_{b}(m_{b}^{2}-M_{W}^{2}-s)\left[-m_{d}V_{bu}^{R*}V_{ud}^{R}(s+M_{W}^{2}-m_{d}^{2})+m_{u}V_{bu}^{R*}V_{ud}^{L}(s-M_{W}^{2}-m_{d}^{2})\right]$

$F_{u}=\left({{-ig_{W}^{2}}\over{2M_{W}^{2}}}\right){1\over{s^{3}}}\left[1+{{(m_{u}^{2}+2M_{W}^{2}+m_{b}^{2}+m_{d}^{2})}\over
s}\right].$ (III.3)

In these formulae, $m_{d}$ and $m_{b}$ are the masses of the down- and bottom-
quark, respectively. The mass of the up-type quark, mediator of the
interaction, is indicated by $m_{u}$.

The amplitude of the second graph in Fig. 1, mediated by the $Z$-boson, is
given by:

$\displaystyle M_{Z}$
$\displaystyle=F_{z}{\bar{u}}_{b}(k_{2})[A_{z}P_{L}+B_{z}P_{R}+C_{z}\not{p_{1}}P_{R}+D_{z}\not{p_{1}}P_{L}]u_{d}(p_{2})$

where

$A_{z}=m_{b}c^{ZL}_{bd}A+m_{d}c^{ZR}_{bd}B,~{}~{}~{}~{}B_{z}=m_{b}c^{ZR}_{bd}A+m_{d}c^{ZL}_{bd}B,~{}~{}~{}~{}C_{z}=c^{ZR}_{bd}C,~{}~{}~{}~{}D_{z}=c^{ZL}_{bd}C.$
(III.4)

The functions $A,B,C$ and $F_{z}$ are defined as follows:

$\displaystyle A$ $\displaystyle=M_{W}^{2}\left[{t\over
2}(M_{W}^{2}-m_{b}^{2})-(s-M_{W}^{2})^{2}+m_{b}^{2}(m_{b}^{2}-M_{W}^{2}-s)\right]+{{ts}\over
2}(2M_{W}^{2}-m_{b}^{2}+s)+$ $\displaystyle m_{d}^{2}\left[M_{W}^{2}(-{t\over
2}+3s+3M_{W}^{2}-m_{b}^{2}-m_{d}^{2})+{t\over 2}(m_{b}^{2}-s)\right]$

$\displaystyle B$ $\displaystyle={t\over
2}M_{W}^{2}(7M_{W}^{2}-3m_{b}^{2})+{{ts}\over
2}(6M_{W}^{2}+m_{b}^{2}-s)+M_{W}^{2}\left[m_{b}^{2}(s+M_{W}^{2})+(s-M_{W}^{2})^{2}-m_{b}^{4}\right]+$
$\displaystyle m_{d}^{2}\left[{t\over
2}(s-3M_{W}^{2}-m_{b}^{2})+M_{W}^{2}(m_{d}^{2}-3s-3M_{W}^{2}+m_{b}^{2})\right]$

$\displaystyle C$
$\displaystyle=2M_{W}^{2}(s-M_{W}^{2})^{2}-M_{W}^{2}m_{b}^{2}(t+2s+2M_{W}^{2})+ts(m_{b}^{2}+2M_{W}^{2})+t(3M_{W}^{4}-s^{2})+$
$\displaystyle
m_{d}^{2}\left[2M_{W}^{2}(m_{b}^{2}-s-M_{W}^{2})+t(s-m_{b}^{2}-M_{W}^{2})\right]$

and

$F_{z}=\left({{ig_{W}^{2}}\over{M_{W}^{2}}}\right){1\over{s^{2}t}}\left[1+{{(2M_{W}^{2}+m_{b}^{2}+m_{d}^{2})}\over
s}+{M_{z}^{2}\over t}\right]$ (III.5)

Finally, the diagram mediated by the Higgs boson is given by:

$M_{H}=F_{h}A_{h}{\bar{u}}_{b}(k_{2})[M_{L}c^{hL}_{bd}P_{L}+M_{R}c^{hR}_{bd}P_{R}]u_{d}(p_{2})$
(III.6)

with

$\displaystyle A_{h}$
$\displaystyle=(s-M_{W}^{2}-m_{d}^{2})[(s-M_{W}^{2}-m_{b}^{2})(M_{W}^{2}-{t\over
2})-M_{W}^{2}(M_{W}^{2}+m_{b}^{2}-u)]$ $\displaystyle-
M_{W}^{2}(s-M_{W}^{2}-m_{b}^{2})(M_{W}^{2}+m_{d}^{2}-u)+2M_{W}^{4}(m_{b}^{2}+m_{d}^{2}-t)$

and

$F_{h}={{ig_{W}^{2}}\over{2M_{W}^{2}}}{1\over{ts^{2}}}\left[1+{{2M_{W}^{2}+m_{d}^{2}+m_{b}^{2}}\over
s}+{{M_{h}^{2}}\over t}\right]$ (III.7)

where $M_{h}$ is the Higgs mass. In the above formulae, $s,t,u$ are the three
Mandelstam variables defined as:

$s=(p_{1}+p_{2})^{2},~{}~{}~{}~{}t=(p_{1}-k_{1})^{2},~{}~{}~{}~{}u=(p_{1}-k_{2})^{2}.$
(III.8)

The modulus squared of the sum of the three amplitudes represented by the
Feynman diagrams in Fig. 1, $|M|^{2}=|M_{U}+M_{Z}+M_{H}|^{2}$, has been
computed using the UNIX-based program FORM. Once expanded in terms of the
Mandelstam variables, it displays a term proportional to the product $su$,
which grows with the fourth power of the CoM energy $E_{CM}^{4}$, terms
proportional to $t$ and $u$ increasing as $E_{CM}^{2}$, and terms constant or
decreasing with increasing the energy. The highest order $su$-term comes from
the modulus square of the sum of the two matrix elements mediated by the up-
type quarks and the $Z$-boson, specifically, and it can be written as

$I_{us}={g_{W}^{4}\over M_{W}^{4}}(-us)\left[\left|{1\over
2}\sum_{u}V_{bu}^{R*}V_{ud}^{R}+c^{ZR}_{bd}\right|^{2}+\left|{1\over
2}\sum_{u}V_{bu}^{L*}V_{ud}^{L}+c^{ZL}_{bd}\right|^{2}\right].$ (III.9)

In order to restore perturbative unitarity at high energies, this term must be
cancelled out. By imposing this requirement, a natural relation between the
weak charged and neutral couplings can be extracted:

$-{1\over 2}\sum_{u}V_{bu}^{R*}V_{ud}^{R}=c^{ZR}_{bd};~{}~{}~{}~{}-{1\over
2}\sum_{u}V_{bu}^{L*}V_{ud}^{L}=c^{ZL}_{bd}.$ (III.10)

The above equations are satisfied by the left-handed and right-handed
components of the couplings, separately. The immediate consequence of these
equalities is that the FCNC induced by the $Z$-boson are null if the extended
CKM matrix is unitary. We shall analyse the phenomenological implications of
this result in more detail in the next section.

The second highest order terms increasing with $t$ are more cumbersome as they
are proportional to the different masses of the external fermions involved in
the scattering process. These terms are generated by all the three amplitudes
contributing to the scattering, therefore they include all the fermion
couplings to the $W$, $Z$ and Higgs bosons. After applying the relation
between the weak couplings given in Eq. III.10, this contribution reads:

$\displaystyle I_{t}$ $\displaystyle={g_{W}^{4}\over
M_{W}^{4}}(-t)\Big{[}(m_{b}^{2}+m_{d}^{2}){{(|c^{ZL}_{bd}|^{2}+|c^{ZR}_{bd}|^{2})}\over
4}+m_{b}m_{d}{{(c^{ZL}_{bd}c^{ZR*}_{bd}+h.c.)}\over
2}+{{(m_{L}^{2}|c^{hL}_{bd}|^{2}+m_{R}^{2}|c^{hR}_{bd}|^{2})}\over{16}}$
$\displaystyle+m_{b}{{(m_{L}c^{ZL}_{bd}c^{hL*}_{bd}+m_{R}c^{ZR}_{bd}c^{hR*}_{bd}+h.c.)}\over
8}+m_{d}{{(m_{L}c^{ZR}_{bd}c^{hL*}_{bd}+m_{R}c^{ZL}_{bd}c^{hR*}_{bd}+h.c.)}\over
8}\Big{]}$ (III.11)

where couplings and masses have been defined in Sec. II. The first two terms
in the above equation clearly show that the Higgs contribution is needed to
restore perturbative unitarity. The two terms purely dependent on the
$Z$-boson couplings cannot in fact be cancelled out by themselves.
Furthermore, from the quadratic dependence of the first term on the $m_{b}$
and $m_{d}$ masses it is clear that, in order to suppress the $I_{t}$ term,
the generic masses $m_{L,R}$ entering the fermion-Higgs coupling must be equal
to either $m_{b}$ or $m_{d}$. This is expected as the Higgs couplings to
fermions are proportional to their masses. The novelty in this case, as
compared to the Standard Model, is that there are potential FCNC induced by
the Higgs connecting two different fermions characterised by two different
masses. A priori, one cannot know whether $m_{L}$ is equal to $m_{d}$ or
$m_{b}$ (and consequently $m_{R}$ equal to $m_{b}$ or $m_{d}$). The
perturbative unitarity requirement allows one to attribute $m_{L,R}$ the
correct fermion mass.

There are two possible scenarios where the $I_{t}$ term is nullified:

$m_{L}=m_{b},~{}~{}~{}m_{R}=m_{d},~{}~{}~{}c^{hL}_{bd}=c^{h}_{bd}=-2c^{ZL}_{bd},~{}~{}~{}c^{hR}_{bd}=c^{h}_{bd}=-2c^{ZL}_{bd},~{}~{}~{}c^{zR}_{bd}=0$
(III.12)
$m_{L}=m_{d},~{}~{}~{}m_{R}=m_{b},~{}~{}~{}c^{hL}_{bd}=c^{h}_{bd}=-2c^{ZR}_{bd},~{}~{}~{}c^{hR}_{bd}=c^{h}_{bd}=-2c^{ZR}_{bd},~{}~{}~{}c^{ZL}_{bd}=0.$
(III.13)

In principle, the above two scenarios could be equally realised. However, if
we take as limit case the process $W^{+}b\rightarrow W^{+}b$, the latter
scenario will be excluded. In fact, for that setup
$c^{ZL}_{bb}=-0.5\sum_{u}|V_{bu}^{L}|^{2}\leq-0.5$ taking into account the
experimental limits on the SM-like positive definite terms and considering the
additional contribution of the hypothetical $|V_{bT}^{L}|^{2}$ coming from the
new heavy $T$-VLQ. This result is clearly not compatible with the zero value
of the left-handed $Z$-boson coupling to the bottom-quarks provided within the
scenario in Eq. III.13. One is therefore left with the mass and coupling
configuration given in Eq. III.12. There, the condition $c^{ZR}_{bd}=0$
applied to the process with one incoming and one outgoing bottom-quark brings
a further constraint. From the explicit expression
$c^{ZR}_{bb}=-0.5\sum_{u^{\prime}}|V_{bu^{\prime}}^{R}|^{2}$ with
$u^{\prime}=u,c,t,T$, one can evince that all the entries
$V_{bu^{\prime}}^{R}$ of the right-handed CKM matrix, extended by the addition
of one or more VLQs, are null.

An important advantage of the method we have used in this investigation is
that, due to its generality, the amplitudes that have been calculated for the
process $W^{+}d\rightarrow W^{+}b$ can be interpreted through a number of
different frameworks, in each case deriving unique relations between the
couplings. Just for illustrative purposes, so far we have considered
interactions involving an initial down quark state and a final bottom quark
state. If we so wished, however, the calculated matrix elements could easily
describe interactions between any other two types of down-type quark,
including the new heavy VLQs, by simply relabelling the relevant mass terms
and the elements of the extended CKM matrix. We could even use it to describe
transitions between up-type quarks and/or corresponding VLQs, by replacing any
term containing a sum over the up-type quarks, with a sum over down-type
quarks instead. Taking into account all possible processes, the net outcome is
that all the entries of the right-handed CKM matrix, extended by the addition
of one or more VLQs, are null: $V_{qq^{\prime}}^{R}=0$ with $q=d,s,b,B$ and
$q^{\prime}=u,c,t,T$.

Within this scenario, the last term of the full amplitude square growing with
$E_{CM}^{2}$ via the Mandelstam variables $u$ and $t$ is cancelled out as
well. This term depends on the mass of the up-type quarks appearing in the
first graph of Fig. 1; it can be written as:

$\displaystyle I_{mu}$ $\displaystyle={g_{W}^{4}\over
M_{W}^{4}}\Big{\\{}\Big{(}-{t\over
4}\Big{)}\Big{[}|\sum_{u}m_{u}V_{bu}^{R*}V_{ud}^{L}|^{2}+|\sum_{u}m_{u}V_{bu}^{L*}V_{ud}^{R}|^{2}\Big{]}+{u\over
2}m_{d}\sum_{u}m_{u}[V_{bu}^{R*}V_{ud}^{L}c^{ZR*}+V_{bu}^{L*}V_{ud}^{R}c^{ZL*}+H.c.]$
$\displaystyle+{u\over
2}m_{b}\sum_{u}m_{u}[V_{bu}^{L*}V_{ud}^{R}c^{ZR*}+V_{bu}^{R*}V_{ud}^{L}c^{ZL*}+H.c.].$
(III.14)

Since this contribution is proportional to the entries of the right-handed
CKM, it automatically vanishes in the above setup. Only the terms constant or
decreasing with energy are left, thus recovering the good behaviour of the
scattering amplitude at high energies.

###  III.1 Unitary VLQ model

In this section, we summarise the VLQ model preserving perturbative unitarity.
The masses and couplings must satisfy the following relations:

$m_{L}=m_{q},~{}~{}~{}m_{R}=m_{q^{\prime}},~{}~{}~{}c^{hL}_{qq^{\prime}}=c^{hR}_{qq^{\prime}}=-2c^{ZL}_{qq^{\prime}}=\sum_{q^{\prime\prime}}V_{qq^{\prime\prime}}^{L*}V_{q^{\prime\prime}q^{\prime}}^{L},~{}~{}~{}V_{qq^{\prime\prime}}^{R}=0,~{}~{}~{}c^{ZR}_{qq^{\prime}}=0$
(III.15)

where $q$ is the incoming quark, $q^{\prime}$ is the outgoing quark and
$q^{\prime\prime}$ is the quark mediator of the weak interaction.
Independently on their representation, vector-like quarks thus behave in the
same manner as the SM-like quarks. Only the left-handed VLQs interact via the
weak force, the Lagrangian being

${\mathcal{L}}_{W}=-{g_{W}\over\sqrt{2}}\bar{q}\gamma^{\mu}V^{L}_{qq^{\prime}}P_{L}q^{\prime}W^{+}_{\mu}+H.c.$
(III.16)

Moreover, if the extended CKM matrix is unitary like in the Standard Model,
FCNC are not allowed. Alternatively, the FCNC induced by the $Z$-boson is
purely left-handed as the weak charged current is. The weak neutral current
Lagrangian is expressed as:

${\mathcal{L}}_{Z}=-{g_{W}\over{c_{W}}}\bar{q}\gamma^{\mu}\left(\pm{1\over
2}V^{L*}_{qq^{\prime\prime}}V^{L}_{q^{\prime\prime}q^{\prime}}P_{L}-2Q_{q}s^{2}_{W}\delta_{qq^{\prime}}\right)q^{\prime}Z_{\mu}+H.c.$
(III.17)

where the plus(minus) sign is for down-type(up-type) quarks. On the contrary,
the FCNC generated by the Higgs has both a left-handed and a right-handed part
proportional to the outgoing and incoming quark mass, respectively. The
corresponding Lagrangian is given by:

$\displaystyle{\mathcal{L}}_{h}$
$\displaystyle={g_{W}\over{2M_{W}}}\bar{q}\gamma^{\mu}V^{L*}_{qq^{\prime\prime}}V^{L}_{q^{\prime\prime}q^{\prime}}(m_{b}P_{L}+m_{d}P_{R})q^{\prime}H+H.c.$
$\displaystyle={g_{W}\over{4M_{W}}}\bar{q}\gamma^{\mu}V^{L*}_{qq^{\prime\prime}}V^{L}_{q^{\prime\prime}q^{\prime}}((m_{b}+m_{d})+(m_{d}-m_{b})\gamma_{5})q^{\prime}H+H.c.$
(III.18)

Owing to the fermion mass splitting, the Higgs boson acquires a pseudo-scalar
component.

As noted earlier, the general nature of our approach means that the VLQ model
in Eq. III.15 can incorporate the new heavy VLQs in singlet or doublet
representations, and no assumption has been made about the level of mixing
with the SM quarks. This model agrees with the one proposed in Ref. Aguilar-
Saavedra (2009); Aguilar-Saavedra  _et al._ (2013), which can be considered as
a sub-case of our setup. There, in fact, the $B$-singlet and $T$-singlet
representations only are considered under the assumption that the extra VLQs
would mix with the third generation SM quarks uniquely. Our result reinforces
this model, explicitly showing that the structure of its couplings is dictated
by the fundamental requirement of perturbative unitarity. In addition, our
findings generalise such a structure to the standard doublet representation,
without any conditions on the mixing.

Figure 2: Matrix element square for the processes $W^{-}t\rightarrow W^{-}u$
and $W^{-}t\rightarrow W^{-}c$ versus the CoM energy. We assume
$V_{qq^{\prime}}^{R}=0$ with $q(q^{\prime})$ any up-type (down-type) quarks,
including the VLQs, and $c^{ZR}=0$. The blue (cien) contour refers to the
scattering amplitude squared of the channel with an outgoing up-quark (charm-
quark) when no relations between couplings are imposed. The orange solid line
represents the matrix element squared of the two overlapping processes when
the relation in Eq. III.10 between the $W$\- and $Z$-boson couplings to quarks
is imposed. The green solid line shows the amplitude square for the two
processes when all the relations in Eq. III.15 between the $W$-, $Z$\- and
Higgs-boson couplings to quarks are implemented.

Before discussing the phenomenological implications of our outcomes, we
conclude this section with a graphic representation of the behaviour of the
amplitude squared corresponding to two illustrative scattering processes,
$W^{-}t\rightarrow W^{-}u$ and $W^{-}t\rightarrow W^{-}c$, as a function of
the CoM energy. The choice of these processes, specifically, is motivated by
the presence of experimental bounds on the strength of the FCNC inducing both
the Higgs-up-top and Higgs-charm-top interactions. In line with our findings
and for ease of calculation, we assume $V_{qq^{\prime}}^{R}=0$ with
$q(q^{\prime})$ any up-type (down-type) quarks, including the VLQs, and
$c^{ZR}=0$. The blue (cien) contour refers to the scattering amplitude squared
of the channel with an outgoing up-quark (charm-quark) when no relations
between couplings are imposed. The growth with the CoM energy to the power
four, coming from the dominant $us$-term, is clearly displayed. The orange
solid line represents the matrix element squared of the two processes, which
overlap, when the relation in Eq. III.10 between the $W$\- and $Z$-boson
couplings to quarks is imposed. In this case, the $us$-term is cancelled out
and the sub-dominant terms, proportional to the $u$ and $t$ Mandelstam
variables, lead the growth of the matrix element squared with the C.o.M.
energy to the power two. The green solid line shows the amplitude square for
the two processes when all the relations in Eq. III.15 between the $W$-, $Z$\-
and Higgs-boson couplings to quarks are implemented. This is when both the
dominant and sub-dominant terms in the CoM energy are washed out so that the
amplitude square is given by a constant plus terms decreasing with energy.

We will now turn our attention to the phenomenological consequences of
imposing the fulfilment of perturbative unitarity within VLQs models.

##  IV Phenomenology of the unitary VLQ model

Flavour violating couplings between quarks and neutral $Z$ and Higgs bosons
can be generated in presence of either extra neutral gauge bosons or new heavy
vector-like quarks with $+2/3$ and $-1/3$ electric charges. Recently,
extensive analyses of flavour violation in the presence of new heavy VLQs have
been presented Bobeth  _et al._ (2017) Up to now, the presence of such VLQs
has been considered essential for generating flavour violating couplings to SM
up- and down-quarks, respectively. Our results show that FCNC are instead
correlated to the unitarity of the extended CKM and therefore depend on the
type of group representation the VLQs are described by. In the following, we
analyse the various representations individually.

###  IV.1 T-singlet

We consider the SM with the minimal extension of one up-type iso-singlet
vector-like quark, $T_{L,R}$, with fractional electric charge $Q=+2/3$. In the
flavour basis, the mass terms in the Lagrangian read as:

${\mathcal{L}}_{m}=(\bar{u}_{L}~{}\bar{T}_{L})M_{u}\Big{(}\begin{matrix}u_{R}\\\
T_{R}\end{matrix}\Big{)}+\bar{d}_{L}M_{d}d_{R}+H.c.$ (IV.1)

where $u_{L},d_{L}$ are represented by the SM quark doublets and $u_{R},d_{R}$
by the SM quark singlets. The two mass matrices $M_{u}$ and $M_{d}$ can be
diagonalised via $4\times 4$ and $3\times 3$ bi-unitary transformations,
respectively, as

$D_{u}=V_{L}^{u{\dagger}}M_{u}V_{R}^{u}~{}~{}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}~{}D_{d}=V_{L}^{d{\dagger}}M_{d}V_{R}^{d}$ (IV.2)

with $D_{u}=diag(m_{u},m_{c},m_{t},m_{T})$, where $m_{T}$ is the mass of the
new heavy up-type VLQ, and $D_{d}=diag(m_{d},m_{s},m_{b})$. Like in the SM,
going from the flavour basis to the mass eigenstates basis, the charged
current Lagrangian becomes:

${\mathcal{L}}_{W}=-{g_{w}\over\sqrt{2}}(\bar{u}_{L}~{}\bar{T}_{L})V_{CKM}^{ext}\gamma^{\mu}d_{L}W^{+}_{\mu}+H.c.$
(IV.3)

where the mixing matrix, $V_{CKM}^{ext}=V^{u{\dagger}}_{L}V^{d}_{L}$, is now
an extended $4\times 3$ matrix

$V_{CKM}^{ext}=\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\\ V_{cd}&V_{cs}&V_{cb}\\\
V_{td}&V_{ts}&V_{tb}\\\ V_{Td}&V_{Ts}&V_{Tb}\end{pmatrix}$

whose first three rows play the role of the $3\times 3$ CKM mixing matrix.
This latter sub-matrix is not unitary anymore, unless there is no mixing of
the SM quarks with the new heavy VLQ. Neither the enlarged $V_{CKM}^{ext}$ is
unitary, a priori. Therefore, the appearance of tree-level FCNC mediated by
the $Z$ and Higgs bosons is a natural consequence of the $T_{L,R}$ existence.
In the physical (or mass) basis, the vectorial neutral current Lagrangian
becomes:

${\mathcal{L}}_{Z}=-{g_{W}\over c_{w}}\big{[}{1\over
2}(\bar{u}_{L}~{}\bar{T}_{L})V_{CKM}^{ext}V_{CKM}^{ext{\dagger}}\gamma^{\mu}\Big{(}\begin{matrix}u_{L}\\\
T_{L}\end{matrix}\Big{)}-{1\over
2}\bar{d}_{L}V_{CKM}^{ext{\dagger}}V_{CKM}^{ext}\gamma^{\mu}d_{L}-{2\over
3}s^{2}_{w}(\bar{u}_{i}\gamma^{\mu}u_{i}+\bar{T}\gamma^{\mu}T)+{1\over
3}(\bar{d}_{i}\gamma^{\mu}d_{i})\big{]}Z_{\mu}+H.c.$ (IV.4)

where the $4\times 4$ $V_{CKM}^{ext}V_{CKM}^{ext{\dagger}}$ and the $3\times
3$ $V_{CKM}^{ext{\dagger}}V_{CKM}^{ext}$ matrices are not diagonal and can
induce FCNC in the quark sector. An analogous result arises for the Higgs-
mediated FCNC. In this scenario, the FCNC connecting two down-type quarks are
suppressed compared to the FCNC giving rise to two up-type quarks. This can be
better visualised by choosing a weak basis where the down-type quark mass
matrix is diagonal and therefore $V_{L,R}^{d}=I_{3}$. The phenomenological
consequence is that in the down-type quark sector the couplings inducing the
FCNC mediated by the $Z$-boson are null, $c^{ZL}_{bd}=c^{ZL}_{bs}=0$, as well
as those generating the FCNC mediated by the Higgs,
$c^{hL}_{bd}=c^{hL}_{bs}=0$. Therefore, the $b$-quark physics cannot provide
useful observables for this type of VLQ model. On the contrary, the rare top
decays into a lighter up-type quark plus the associate production of a neutral
boson, $Z$ or Higgs, represent a valuable signature. The recent $95\%$ C.L.
exclusion bounds coming from the analyses of the top decays $t\rightarrow Hu$
and $t\rightarrow Hc$ could be used to constrain the model and extract limits
not only on the couplings $c^{hL,R}_{tu},c^{hL,R}_{tc}$ but also on
$c^{ZL}_{tu},c^{ZL}_{tc}$, using Eq. III.15. In order to estimate the size of
such couplings as predicted by the $T$-singlet model, one can use the present
measurements of the SM-like entries of the CKM matrix and apply Eq. III.15.
One obtains

$c^{ZL}_{tu}=-{1\over 2}\sum_{D=d,s,b}V^{L}_{tD}V^{*L}_{uD}=(1.05\pm
0.05)\cdot 10^{-4}+{\rm i}(3.2\pm 0.2)\cdot
10^{-3};~{}~{}~{}~{}~{}~{}~{}~{}|c^{ZL}_{tu}|=(3.2\pm 0.2)\cdot 10^{-3}$
(IV.5) $c^{ZL}_{tc}=-{1\over 2}\sum_{D=d,s,b}V^{L}_{tD}V^{*L}_{cD}=(3.90\pm
0.40)\cdot 10^{-4}-{\rm i}(3.7\pm 0.2)\cdot
10^{-4};~{}~{}~{}~{}~{}~{}~{}~{}|c^{ZL}_{tc}|=(5.4\pm 0.4)\cdot 10^{-4}$
(IV.6) $c^{hL,R}_{tu}=\sum_{D=d,s,b}V^{L}_{tD}V^{*L}_{uD}=(-2.1\pm 0.1)\cdot
10^{-4}-{\rm i}(6.4\pm 0.3)\cdot
10^{-3};~{}~{}~{}~{}~{}~{}~{}~{}|c^{hL,R}_{tu}|=(6.4\pm 0.4)\cdot 10^{-3}$
(IV.7) $c^{hL,R}_{tc}=\sum_{D=d,s,b}V^{L}_{tD}V^{*L}_{cD}=(-7.7\pm 0.7)\cdot
10^{-4}+{\rm i}(7.4\pm 0.4)\cdot
10^{-4};~{}~{}~{}~{}~{}~{}~{}~{}|c^{hL,R}_{tc}|=(1.1\pm 0.08)\cdot 10^{-3}$
(IV.8)

Figure 3: Upper limits on Higgs FCNC couplings from CMS experiment (see Ref.
Sirunyan  _et al._ (2018)).

These values are more than two orders of magnitude smaller than the couplings
probed at the present LHC run. Fig. 3 shows the 95$\%$ upper limits obtained
by CMS on the neutral couplings inducing the flavour-changing decay of the
top-quark into the lighter up-type quarks $u$ and $c$ plus the Higgs. In the
CMS analysis of Ref. Sirunyan  _et al._ (2018), the Lagrangian is given by

${\mathcal{L}}_{h}=\sum_{q=u,c}{g_{W}\over\sqrt{2}}\bar{t}X_{hqt}(f^{L}_{hq}P_{L}+f^{R}_{hq}P_{R})qh+H.c.$
(IV.9)

where $X_{hqt}$ is the flavour violating coupling between the top-quark and a
lighter up-type quark $q=u,c$, while $f^{L,R}_{hq}$ are complex parameters and
$P_{L,R}$ are the projection operators as usual. It is immediate to see that
in our notation the FCNC coupling $c_{tq}^{hL}$ can be re-expressed as

$c^{hL,R}_{tq}=\sqrt{2}{M_{W}\over M_{t}}X_{hqt}.$ (IV.10)

Therefore, considering the scenario given in Eq. III.15, we can recast the
parameter that is used to interpret the experimental data and visualise the
exclusion bounds as

$X_{hqt}={M_{t}\over{\sqrt{2}M_{W}}}\sum_{D=d,s,b}V^{L}_{tD}V^{*L}_{qD}~{}~{}~{}~{}~{}{\rm
with}~{}~{}~{}~{}~{}q=u,c.$ (IV.11)

Inserting the measured values of the CKM matrix entries, as in Eq. IV.8, we
obtain values of the order of permill, which are well below the current
sensitivity at the LHC. We can however estimate the branching ratios of the
rare decays of the top-quark, via the neutral FCNC mediated by the $Z$-boson
and the Higgs, to evaluate the potential of future accelerators in this
respect. The partial width of the top decay into a lighter up-type quark and a
$Z$-boson is given by

$\Gamma_{Z}(T\rightarrow
Zq)={{G_{F}m_{t}^{3}}\over{4\sqrt{2}\pi\cos^{2}\theta_{W}}}|c^{ZL}_{tq}|^{2}\Big{(}1-{M_{Z}^{2}\over
m_{t}^{2}}\Big{)}^{2}\Big{(}1+2{M_{Z}^{2}\over m_{t}^{2}}\Big{)}$ (IV.12)

with $q=u,c$ and neglecting their mass. Thus we obtain,
$\Gamma_{Z}(t\rightarrow Zu)=3.6\cdot 10^{-6}$ GeV and
$\Gamma_{Z}(t\rightarrow Zc)=10^{-6}$. Analogously, the partial width of the
top decay into a lighter up-type quark and a Higgs boson is given by

$\Gamma_{h}(t\rightarrow
hq)={{G_{F}m_{t}^{3}}\over{16\sqrt{2}\pi}}|c^{hL}_{Tt}|^{2}\Big{(}1-{M_{h}^{2}\over
m_{t}^{2}}\Big{)}^{2}.$ (IV.13)

In this case, we get $\Gamma_{h}(t\rightarrow hu)=7.6\cdot 10^{-6}$ GeV and
$\Gamma_{h}(t\rightarrow hc)=2.2\cdot 10^{-7}$. Taking the top-quark width to
be $\Gamma_{top}=1.35$ GeV, the individual branching ratios in the four
different channels, assuming that all other processes are zero, are:
$Br(t\rightarrow Zu)=2.7\cdot 10^{-6}$, $Br(t\rightarrow Zc)=0.7\cdot
10^{-6}$, $Br(t\rightarrow hu)=5.6\cdot 10^{-6}$ and $Br(t\rightarrow
hc)=1.6\cdot 10^{-7}$. From Fig. 4, which shows a summary of the 95$\%$ C.L.
limits in the search for FCNC in top-quark production and decay for various
future colliders, it is clear that FCNC in the top sector generated by the
$T$-singlet (see purple lines) could only be observed at the FCC-hh or FCC-ee
in the channel $t\rightarrow Zu$.

Figure 4: Summary of 95$\%$ C.L. limits in the search for FCNC in top
production or decays for various future collider options, compared to current
LHC limits Abada  _et al._ (2019). The thick purple lines represent the
Branching fractions within the $T$-singlet.

From the point of view of the direct search for these new heavy particles, the
mixing between $T$ and the SM quarks is a priori different from zero. This
means that the charged couplings $V_{Td^{\prime}}^{L}$ ($d^{\prime}=d,s,b$)
and the neutral couplings $c^{zL}_{Tu^{\prime}}$ and $c^{h}_{Tu^{\prime}}$
with ($u^{\prime}=u,c,t$) could be sizeable. This would allow the new VLQ to
be singly produced and afterwards to decay into SM quarks with the associate
production of either a vector or a scalar boson: $T\rightarrow
W^{+}d^{\prime}$, $T\rightarrow Zu^{\prime}$ and $T\rightarrow Hu^{\prime}$
with $d^{\prime}(u^{\prime})$ any down-type (up-type) SM quark.

We can estimate the value of such couplings by combining the unitarity
condition on the multiplication of any two columns pertaining to the extended
CKM matrix with the measurements of the SM-like CKM elements. From the first
condition, we get three relations:

$\displaystyle 0$
$\displaystyle=\sum_{U=u,c,t}V^{L}_{dU}V^{L*}_{Ub}+V^{L}_{dT}V^{L*}_{Tb}=(-3.2\pm
0.75)\cdot 10^{-5}-{\rm i}(6.4\pm 0.2)\cdot 10^{-3}+V^{L}_{dT}V^{L*}_{Tb}$
$\displaystyle 0$
$\displaystyle=\sum_{U=u,c,t}V^{L}_{dU}V^{L*}_{Us}+V^{L}_{dT}V^{L*}_{Ts}=(-3.4\pm
0.2)\cdot 10^{-4}-{\rm i}(1.4\pm 0.1)\cdot 10^{-4}+V^{L}_{dT}V^{L*}_{Ts}$
$\displaystyle 0$
$\displaystyle=\sum_{U=u,c,t}V^{L}_{sU}V^{L*}_{Ub}+V^{L}_{sT}V^{L*}_{Tb}=(-7.7\pm
0.7)\cdot 10^{-4}-{\rm i}(7.4\pm 0.4)\cdot 10^{-4}+V^{L}_{sT}V^{L*}_{Tb}$
(IV.14)

where the numerical values are taken from the PDG. Solving the system, the
absolute value of the three independent elements of the extended CKM matrix,
which represent the mixing between the new $T$ VLQ and the SM down-type
quarks, can be estimated to be:

$V^{L}_{dT}=0.046;~{}~{}~{}~{}~{}V^{L}_{sT}=0.008;~{}~{}~{}~{}~{}V^{L}_{bT}=0.14\pm
0.04.$ (IV.15)

By making use of these estimates, we can now extract values for the neutral
FCNC couplings between the $T$ VLQ, an up-type SM quark and either the $Z$ or
the Higgs boson. We have in fact

$\displaystyle c^{hL,R}_{Tu}$
$\displaystyle=\sum_{D=d,s,b}V^{L}_{TD}V^{L*}_{Du}\simeq
V^{L}_{Td}V^{L*}_{du}=0.046$ $\displaystyle c^{hL,R}_{Tc}$
$\displaystyle=\sum_{D=d,s,b}V^{L}_{TD}V^{L*}_{Dc}\simeq
V^{L}_{Ts}V^{L*}_{sc}=0.008$ $\displaystyle c^{hL,R}_{Tt}$
$\displaystyle=\sum_{D=d,s,b}V^{L}_{TD}V^{L*}_{Dt}\simeq
V^{L}_{Tb}V^{L*}_{bt}=0.14$

We are now ready to compute decay widths and branching ratios of the new heavy
$T$ VLQ. The decay rate into a $W$-boson and bottom-quark pair increases with
the third power of the $T$ VLQ mass

$\displaystyle\Gamma_{W}(T\rightarrow W^{+}b)$
$\displaystyle={{G_{F}M_{T}^{3}}\over{8\sqrt{2}\pi}}|V_{bT}^{L}|^{2}\sqrt{\Big{(}1-{M_{W}^{2}\over
M_{T}^{2}}-{m_{b}^{2}\over M_{T}^{2}}\Big{)}^{2}-4{{M_{W}^{2}m_{b}^{2}}\over
M_{T}^{4}}}\Big{[}\Big{(}1-{M_{W}^{2}\over M_{T}^{2}}+{m_{b}^{2}\over
M_{T}^{2}}\Big{)}\Big{(}1+2{M_{W}^{2}\over M_{T}^{2}}+{m_{b}^{2}\over
M_{T}^{2}}\Big{)}-2{m_{b}^{2}\over M_{T}^{2}}\Big{]}$ $\displaystyle\simeq
175{\rm MeV}\Big{(}{M_{T}\over M_{W}}\Big{)}^{3}|V_{bT}^{L}|^{2}.$

If we consider as a range of hypothetical masses the interval $1{\rm TeV}\leq
M_{T}\leq 5{\rm TeV}$, the partial width in the charged channel could vary
between $7{\rm GeV}\leq\Gamma_{W}\leq 837{\rm GeV}$. As to the neutral
channel, the decay rate into a $Z$-boson and top-quark pair is given by

$\displaystyle\Gamma_{Z}(T\rightarrow Zt)$
$\displaystyle={{G_{F}M_{T}^{3}}\over{4\sqrt{2}\pi\cos^{2}\theta_{W}}}|c^{ZL}_{Tt}|^{2}\sqrt{\Big{(}1-{M_{Z}^{2}\over
M_{T}^{2}}-{m_{t}^{2}\over M_{T}^{2}}\Big{)}^{2}-4{{M_{Z}^{2}m_{t}^{2}}\over
M_{T}^{4}}}\Big{[}\Big{(}1-{M_{Z}^{2}\over M_{T}^{2}}+{m_{t}^{2}\over
M_{T}^{2}}\Big{)}\Big{(}1+2{M_{Z}^{2}\over M_{T}^{2}}+{m_{t}^{2}\over
M_{T}^{2}}\Big{)}-2{m_{t}^{2}\over M_{T}^{2}}\Big{]}$ $\displaystyle\simeq
443{\rm MeV}\Big{(}{M_{T}\over M_{W}}\Big{)}^{3}|c^{ZL}_{Tt}|^{2}.$

For the same mass range as above, we get a partial width which is three orders
of magnitude bigger than the previous one: $4.2{\rm GeV}\leq\Gamma_{Z}\leq
530{\rm GeV}$. Analogously, for the decay rate into a Higgs boson and top-
quark pair we have

$\displaystyle\Gamma_{h}(T\rightarrow ht)$
$\displaystyle={{G_{F}M_{T}^{3}}\over{16\sqrt{2}\pi}}|c^{hL}_{Tt}|^{2}\sqrt{\Big{(}1-{M_{h}^{2}\over
M_{T}^{2}}-{m_{t}^{2}\over M_{T}^{2}}\Big{)}^{2}-4{{M_{h}^{2}m_{t}^{2}}\over
M_{T}^{4}}}\Big{[}\Big{(}1-{M_{h}^{2}\over M_{T}^{2}}+{m_{t}^{2}\over
M_{T}^{2}}\Big{)}\Big{(}1+{m_{t}^{2}\over M_{T}^{2}}\Big{)}+{m_{t}^{2}\over
M_{T}^{2}}\Big{]}$ $\displaystyle\simeq 85{\rm MeV}\Big{(}{M_{T}\over
M_{W}}\Big{)}^{3}|c^{hL}_{Tt}|^{2}.$

For the same mass range as above, we get a partial width of the same order as
the previous one: $3.3{\rm GeV}\leq\Gamma_{h}\leq 407{\rm GeV}$. The total
width can be easily computed for the same range of masses: $15{\rm
GeV}\leq\Gamma_{TOT}\leq 1774{\rm GeV}$. This means that the $T$-singlet VLQ
would appear as a narrow-mid resonance. For masses $M_{T}\leq 2650$ GeV, the
$T$-singlet would have ${\Gamma_{TOT}\over M_{T}}\leq 10\%$ and could be
analysed in narrow width approximation. For higher masses, $2.7\leq M_{T}\leq
5$ TeV, the new $T$ VLQ would be broader, having a width over mass ratio
$10\%\leq{\Gamma_{TOT}\over M_{T}}\leq 35\%$. The signal could manifest itself
as a large shoulder spread over the SM background. Novel strategies,
specifically tailored for wide resonances, should be applied. Moreover, given
the very large width of the predicted $T$-singlet, heavy interferences with
the SM contribution to the same final state could take place, a priori.
Therefore, finite width and interference effects should be both taken into
consideration when searching for these signals in the mid-large mass regime.

For sake of completeness, at fixed mass, we can also calculate the expected
branching ratio of the three individual channels:

$Br(T\rightarrow Wb)=47\%;~{}~{}~{}Br(T\rightarrow
Zt)=30\%;~{}~{}~{}Br(T\rightarrow ht)=23\%.$ (IV.16)

###  IV.2 B-singlet

Analogously to what done in the previous section, we now consider the SM with
the minimal extension of one down-type iso-singlet vector-like quark,
$B_{L,R}$, with fractional electric charge $Q=-1/3$. In the flavour basis, the
mass terms in the Lagrangian read as:

${\mathcal{L}}_{m}=(\bar{d}_{L}~{}\bar{B}_{L})M_{d}\Big{(}\begin{matrix}d_{R}\\\
B_{R}\end{matrix}\Big{)}+\bar{u}_{L}M_{u}u_{R}+H.c.$ (IV.17)

where $u_{L},d_{L}$ are represented by the SM quark doublets and $u_{R},d_{R}$
by the SM quark singlets. The two mass matrices $M_{u}$ and $M_{d}$ can be
diagonalised via $3\times 3$ and $4\times 4$ bi-unitary transformations,
respectively, as

$D_{u}=V_{L}^{u{\dagger}}M_{u}V_{R}^{u}~{}~{}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}~{}D_{d}=V_{L}^{d{\dagger}}M_{d}V_{R}^{d}$ (IV.18)

with $D_{d}=diag(m_{d},m_{s},m_{b},m_{B})$, where $m_{B}$ is the mass of the
new heavy down-type VLQ, and $D_{u}=diag(m_{u},m_{c},m_{t})$. Like in the SM,
going from the flavour basis to the mass eigenstates basis, the charged
current Lagrangian becomes:

${\mathcal{L}}_{W}=-{g_{w}\over\sqrt{2}}(\bar{d}_{L}~{}\bar{B}_{L})V_{CKM}^{ext}\gamma^{\mu}u_{L}W^{-}_{\mu}+H.c.$
(IV.19)

where the mixing matrix, $V_{CKM}^{ext}=V^{d{\dagger}}_{L}V^{u}_{L}$, is now
an extended $3\times 4$ matrix

$V_{CKM}^{ext}=\begin{pmatrix}V_{ud}&V_{us}&V_{ub}&V_{uB}\\\
V_{cd}&V_{cs}&V_{cb}&V_{cB}\\\ V_{td}&V_{ts}&V_{tb}&V_{tB}\end{pmatrix}$

whose first three columns play the role of the $3\times 3$ CKM mixing matrix.
This latter sub-matrix is not unitary anymore, unless there is no mixing of
the SM quarks with the new heavy VLQ. Neither the enlarged $V_{CKM}^{ext}$ is
unitary, a priori. Therefore, as for the T-singlet above, the appearance of
tree-level FCNC mediated by the $Z$ and Higgs bosons is a natural consequence
of the $B_{L,R}$ existence as well. In the physical (or mass) basis, the
vectorial neutral current Lagrangian becomes:

${\mathcal{L}}_{Z}=-{g_{w}\over c_{w}}\big{[}{1\over
2}(\bar{d}_{L}~{}\bar{B}_{L})V_{CKM}^{ext}V_{CKM}^{ext{\dagger}}\gamma^{\mu}\Big{(}\begin{matrix}d_{L}\\\
B_{L}\end{matrix}\Big{)}-{1\over
2}\bar{u}_{L}V_{CKM}^{ext{\dagger}}V_{CKM}^{ext}\gamma^{\mu}u_{L}-{2\over
3}s^{2}_{w}(\bar{u}_{i}\gamma^{\mu}u_{i}+\bar{T}\gamma^{\mu}T)+{1\over
3}(\bar{d}_{i}\gamma^{\mu}d_{i})\big{]}Z_{\mu}+H.c.$ (IV.20)

where the $4\times 4$ $V_{CKM}^{ext}V_{CKM}^{ext{\dagger}}$ and the $3\times
3$ $V_{CKM}^{ext{\dagger}}V_{CKM}^{ext}$ matrices are not diagonal and can
induce FCNC in the quark sector. An analogous result arises for the Higgs-
mediated FCNC. In this scenario, the FCNC connecting two up-type quarks are
suppressed compared to the FCNC giving rise to two down-type quarks. This can
be better visualised by choosing a weak basis where the up-type quark mass
matrix is diagonal and therefore $V_{L,R}^{u}=I_{3}$. The phenomenological
consequence is that the couplings inducing the FCNC mediated by the $Z$-boson
are null, $c^{ZL}_{tu}=c^{ZL}_{tc}=0$, as well as those generating the FCNC
mediated by the Higgs, $c^{hL}_{tu}=c^{hL}_{tc}=0$. The rare top decays into a
lighter up-type quark plus the associate production of a neutral boson, $Z$ or
Higgs, are not a signature for this type of VLQ model. Therefore, the recent
exclusion bounds coming from these neutral FCNC couplings cannot be used to
constrain the model. One would need experimental analyses on the down-type
quark sector to claim evidence or extract bounds on the couplings
$c^{ZL}_{bd},c^{ZL}_{bs},c^{hL,R}_{bd},c^{hL,R}_{bs}$.

From the point of view of the direct search of these new heavy particles, the
mixing between $B$ and the SM quarks is a priori different from zero. This
means that the charged couplings $V_{Bu^{\prime}}^{L}$ ($u^{\prime}=u,c,t$)
and the neutral couplings $c^{ZL}_{Bd^{\prime}}$ and $c^{hL,R}_{Bd^{\prime}}$
with ($d^{\prime}=d,s,b$) could be sizeable. This would allow the new VLQ to
be singly produced and afterwards to decay into SM quarks with the associate
production of either a vector or a scalar boson: $B\rightarrow
W^{-}u^{\prime}$, $B\rightarrow Zd^{\prime}$ and $B\rightarrow Hd^{\prime}$
with $d^{\prime}(u^{\prime})$ any down-type (up-type) SM quark.

We can estimate the value of such couplings by combining the unitarity
condition on the multiplication of any two rows pertaining to the extended CKM
matrix with the measurements of the SM-like CKM elements. From the first
condition, we get three relations:

$\displaystyle 0$
$\displaystyle=\sum_{D=d,s,b}V^{L}_{uD}V^{L*}_{Dc}+V^{L}_{uB}V^{L*}_{Bc}=6.4\cdot
10^{-5}-{\rm i}1.7\cdot 10^{-4}+V^{L}_{uB}V^{L*}_{Bc}$ $\displaystyle 0$
$\displaystyle=\sum_{D=d,s,b}V^{L}_{uD}V^{L*}_{Dt}+V^{L}_{uB}V^{L*}_{Bt}=-2.1\cdot
10^{-4}-{\rm i}6.4\cdot 10^{-3}+V^{L}_{uB}V^{L*}_{Bt}$ $\displaystyle 0$
$\displaystyle=\sum_{D=d,s,b}V^{L}_{cD}V^{L*}_{Dt}+V^{L}_{cB}V^{L*}_{Bt}=-7.7\cdot
10^{-4}+{\rm i}7.4\cdot 10^{-4}+V^{L}_{cB}V^{L*}_{Bt}$ (IV.21)

where the numerical values are taken from the PDG. Solving the system, the
value of the three independent elements of the extended CKM matrix, which
represent the mixing between the new $B$ VLQ and the SM up-type quarks, can be
estimated to be:

$V^{L}_{uB}=0.03;~{}~{}~{}~{}~{}V^{L}_{cB}=0.06;~{}~{}~{}~{}~{}V^{L}_{tB}=0.2.$
(IV.22)

By making use of these estimates, we can now extract values for the neutral
FCNC couplings between the $B$ VLQ, a down-type SM quark and either the $Z$ or
the Higgs boson. We have in fact

$\displaystyle c^{hL,R}_{Bd}$
$\displaystyle=\sum_{U=u,c,t}V^{L}_{BU}V^{L*}_{Ud}\simeq
V^{L}_{Bu}V^{L*}_{ud}=0.03$ $\displaystyle c^{hL,R}_{Bs}$
$\displaystyle=\sum_{U=u,c,t}V^{L}_{BU}V^{L*}_{Us}\simeq
V^{L}_{Bc}V^{L*}_{cs}=0.06$ $\displaystyle c^{hL,R}_{Bb}$
$\displaystyle=\sum_{U=u,c,t}V^{L}_{BU}V^{L*}_{Ub}\simeq
V^{L}_{Bt}V^{L*}_{tb}=0.2.$

We are now ready to compute decay widths and branching ratios of the new heavy
$T$ VLQ. The decay rate into a $W$-boson and bottom-quark pair increases with
the third power of the $T$ VLQ mass

$\displaystyle\Gamma_{W}(B\rightarrow W^{+}t)$
$\displaystyle={{G_{F}M_{B}^{3}}\over{8\sqrt{2}\pi}}|V_{tB}^{L}|^{2}\sqrt{\Big{(}1-{M_{W}^{2}\over
M_{B}^{2}}-{m_{t}^{2}\over M_{B}^{2}}\Big{)}^{2}-4{{M_{W}^{2}m_{t}^{2}}\over
M_{B}^{4}}}\Big{[}\Big{(}1-{M_{W}^{2}\over M_{B}^{2}}+{m_{t}^{2}\over
M_{B}^{2}}\Big{)}\Big{(}1+2{M_{W}^{2}\over M_{B}^{2}}+{m_{t}^{2}\over
M_{B}^{2}}\Big{)}-2{m_{t}^{2}\over M_{B}^{2}}\Big{]}$ $\displaystyle\simeq
175{\rm MeV}\Big{(}{M_{B}\over M_{W}}\Big{)}^{3}|V_{tB}^{L}|^{2}.$

If we consider as a range of hypothetical masses the interval $1{\rm TeV}\leq
M_{B}\leq 5{\rm TeV}$, the partial width in the charged channel could vary
between $14{\rm GeV}\leq\Gamma_{W}\leq 1.71{\rm TeV}$. As to the neutral
channel, the decay rate into a $Z$-boson and top-quark pair is given by

$\displaystyle\Gamma_{Z}(B\rightarrow Zb)$
$\displaystyle={{G_{F}M_{B}^{3}}\over{4\sqrt{2}\pi\cos^{2}\theta_{W}}}|c^{ZL}_{Bb}|^{2}\sqrt{\Big{(}1-{M_{Z}^{2}\over
M_{B}^{2}}-{m_{b}^{2}\over M_{B}^{2}}\Big{)}^{2}-4{{M_{Z}^{2}m_{b}^{2}}\over
M_{B}^{4}}}\Big{[}\Big{(}1-{M_{Z}^{2}\over M_{B}^{2}}+{m_{b}^{2}\over
M_{B}^{2}}\Big{)}\Big{(}1+2{M_{Z}^{2}\over M_{B}^{2}}+{m_{b}^{2}\over
M_{B}^{2}}\Big{)}-2{m_{b}^{2}\over M_{B}^{2}}\Big{]}$ $\displaystyle\simeq
443{\rm MeV}\Big{(}{M_{B}\over M_{W}}\Big{)}^{3}|c^{ZL}_{Bb}|^{2}.$

For the same mass range as above, we get a partial width which is three orders
of magnitude bigger than the previous one: $9{\rm GeV}\leq\Gamma_{Z}\leq
1.08{\rm TeV}$. Analogously, for the decay rate into a Higgs boson and top-
quark pair we have

$\displaystyle\Gamma_{h}(B\rightarrow hb)$
$\displaystyle={{G_{F}M_{B}^{3}}\over{16\sqrt{2}\pi}}|c^{hL}_{Bb}|^{2}\sqrt{\Big{(}1-{M_{h}^{2}\over
M_{B}^{2}}-{m_{b}^{2}\over M_{B}^{2}}\Big{)}^{2}-4{{M_{h}^{2}m_{b}^{2}}\over
M_{B}^{4}}}\Big{[}\Big{(}1-{M_{h}^{2}\over M_{B}^{2}}+{m_{b}^{2}\over
M_{B}^{2}}\Big{)}\Big{(}1+{m_{b}^{2}\over M_{B}^{2}}\Big{)}+{m_{b}^{2}\over
M_{B}^{2}}\Big{]}$ $\displaystyle\simeq 85{\rm MeV}\Big{(}{M_{B}\over
M_{W}}\Big{)}^{3}|c^{hL}_{Bb}|^{2}.$

For the same mass range as above, we get a partial width of the same order as
the previous one: $7{\rm GeV}\leq\Gamma_{h}\leq 0.8{\rm TeV}$. The total width
can be easily computed for the same range of masses: $30{\rm
GeV}\leq\Gamma_{TOT}\leq 3.6{\rm TeV}$. This means that the $B$-singlet VLQ
would appear either as a narrow or a wide resonance, depending on its mass.
For masses $M_{B}\leq 1900$ GeV, the $B$-singlet would have
${\Gamma_{TOT}\over M_{B}}\leq 10\%$ and could be analysed in narrow width
approximation. For higher masses, $1.9\leq M_{T}\leq 5$ TeV, the new $B$ VLQ
would be much broader, having a width over mass ratio
$10\%\leq{\Gamma_{TOT}\over M_{B}}\leq 70\%$. The signal could manifest itself
as a large shoulder or even worse as an excess of events spread over the SM
background. Present searches, performed under the narrow width approximation,
might have missed these new particles. Novel strategies, specifically tailored
for wide resonances, should be applied. Moreover, given the very large width
of the predicted $B$-singlet, heavy interferences with the SM contribution to
the same final state could take place, a priori. Therefore, finite width and
interference effects should be both taken into consideration when searching
for these signals in the mid-large mass regime.

For sake of completeness, at fixed mass, we can also calculate the expected
branching ratio of the three individual channels:

$Br(B\rightarrow Wt)=48\%;~{}~{}~{}Br(B\rightarrow
Zb)=30\%;~{}~{}~{}Br(B\rightarrow hb)=22\%.$ (IV.23)

###  IV.3 (T, B)-doublet

We consider the SM with the minimal extension of one doublet containing one
up-type and one down-type vector-like quark, $T_{L,R}$ and $B_{L,R}$, with
fractional electric charge $Q=+2/3$ and $Q=-1/3$, respectively. This doublet
can be seen as an heavier copy of the SM one. In the flavour basis, the mass
terms in the Lagrangian read as:

${\mathcal{L}}_{m}=(\bar{u}_{L}~{}\bar{T}_{L})M_{u}\Big{(}\begin{matrix}u_{R}\\\
T_{R}\end{matrix}\Big{)}+(\bar{d}_{L}~{}\bar{B}_{L})M_{d}\Big{(}\begin{matrix}d_{R}\\\
D_{R}\end{matrix}\Big{)}+H.c.$ (IV.24)

where $u_{L},d_{L}$ are represented by the SM quark doublets and $u_{R},d_{R}$
by the SM quark singlets. The two mass matrices $M_{u}$ and $M_{d}$ can be
diagonalised via $4\times 4$ bi-unitary transformations as

$D_{u}=V_{L}^{u{\dagger}}M_{u}V_{R}^{u}~{}~{}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}~{}D_{d}=V_{L}^{d{\dagger}}M_{d}V_{R}^{d}$ (IV.25)

with $D_{u}=diag(m_{u},m_{c},m_{t},m_{T})$ and
$D_{d}=diag(m_{d},m_{s},m_{b},m_{B})$, where $m_{T}$ and $m_{B}$ are the
masses of the two new heavy up-type and down-type VLQs. Like in the SM, going
from the flavour basis to the mass eigenstates basis, the charged current
Lagrangian becomes:

${\mathcal{L}}_{W}=-{g_{w}\over\sqrt{2}}(\bar{u}_{L}~{}\bar{T}_{L})V_{CKM}^{ext}\gamma^{\mu}\Big{(}\begin{matrix}d_{L}\\\
B_{L}\end{matrix}\Big{)}W^{+}_{\mu}+H.c.$ (IV.26)

where the mixing matrix, $V_{CKM}^{ext}=V^{u{\dagger}}_{L}V^{d}_{L}$, is now
an extended $4\times 4$ matrix

$V_{CKM}^{ext}=\begin{pmatrix}V_{ud}&V_{us}&V_{ub}&V_{uB}\\\
V_{cd}&V_{cs}&V_{cb}&V_{cB}\\\ V_{td}&V_{ts}&V_{tb}&V_{tB}\\\
V_{Td}&V_{Ts}&V_{Tb}&V_{TB}\end{pmatrix}$

including the $3\times 3$ CKM mixing matrix in the upper left corner. This
latter sub-matrix is not unitary anymore, unless there is no mixing of the SM
quarks with the new heavy VLQ. Oppositely to the T-singlet and B-singlet
cases, now the enlarged $V_{CKM}^{ext}$ is unitary by construction (assuming
there is no new physics beyond the VLQs).

We now proceed by examining a few specific assumptions that may affect the
predictions which are made by our unitary VLQ model. As noted earlier, the
general nature of our approach means that this model can incorporate vector-
like quarks in singlet or doublet representations, and no assumption has been
made about the level of mixing with the standard model quarks. Therefore, we
are free to analyse each case individually.

If we decide to permit mixing with all generations of quarks, all the entries
of the extended CKM matrix are non zero, a priori. From the point of view of
the new heavy particles search, the mixing between the new heavy $T$ and $B$
VLQs with the SM quarks may in principle be different from zero as well. This
means that the charged couplings $V_{Td^{\prime}}^{L}$ ($d^{\prime}=d,s,b$)
and $V_{Bu^{\prime}}^{L}$ ($u^{\prime}=u,c,t$) could be sizeable. On the
contrary, as the extended $4\times 4$ CKM matrix is unitary, the tree-level
FCNC mediated by the $Z$ and Higgs bosons are absent, owing to the relation
between the charged and neutral couplings shown in Eq. III.15.

In the physical (or mass) basis, the vectorial neutral current Lagrangian
becomes:

${\mathcal{L}}_{Z}=-{g_{w}\over c_{w}}\big{[}{1\over
2}(\bar{u}_{L}~{}\bar{T}_{L})\gamma^{\mu}\Big{(}\begin{matrix}u_{L}\\\
T_{L}\end{matrix}\Big{)}-{1\over 2}\bar{d}_{L}\gamma^{\mu}d_{L}-{2\over
3}s^{2}_{w}(\bar{u}_{i}\gamma^{\mu}u_{i}+\bar{T}\gamma^{\mu}T)+{1\over
3}(\bar{d}_{i}\gamma^{\mu}d_{i})\big{]}Z_{\mu}+H.c.$ (IV.27)

where only same flavour quarks can couple. An analogous result arises for the
Higgs-mediated neutral current. This implies that the neutral couplings
$c^{zL}_{Tu^{\prime}}$, $c^{h}_{Tu^{\prime}}$ with ($u^{\prime}=u,c,t$) and
$c^{zL}_{Bd^{\prime}}$, $c^{h}_{Bd^{\prime}}$ with ($d^{\prime}=d,s,b$) are
zero. This would allow the new VLQ to be singly produced and afterwards to
decay into SM quarks with the associate production of a $W$-boson only:
$T\rightarrow W^{+}d^{\prime}$ with $d^{\prime}$ any down-type SM quark and
$B\rightarrow W^{-}u^{\prime}$ with $u^{\prime}$ any up-type SM quark. The
neutral current single production and decay would not be allowed instead, for
the above-mentioned reasons. Therefore, the search for these new particles
should be different from what performed at the moment. Only the charged
channel is indeed open for the single production and the sub-sequent decay of
these VLQs.

By exploiting the six unitarity conditions on the product of any two of the
first three columns and any two of the first three rows, one can estimate the
magnitude of the following matrix elements:

$V^{L}_{dT}=0.046;~{}~{}~{}~{}~{}V^{L}_{sT}=0.008;~{}~{}~{}~{}~{}V^{L}_{bT}=0.14\pm
0.04$ (IV.28)
$V^{L}_{uB}=0.03;~{}~{}~{}~{}~{}V^{L}_{cB}=0.06;~{}~{}~{}~{}~{}V^{L}_{tB}=0.2\pm
0.04.$ (IV.29)

With these values, the width of the $T$ VLQ

$\displaystyle\Gamma(T\rightarrow W^{+}b)$
$\displaystyle={{G_{F}M_{T}^{3}}\over{8\sqrt{2}\pi}}|V_{bT}^{L}|^{2}\sqrt{\Big{(}1-{M_{W}^{2}\over
M_{T}^{2}}-{m_{b}^{2}\over M_{T}^{2}}\Big{)}^{2}-4{{M_{W}^{2}m_{b}^{2}}\over
M_{T}^{4}}}\Big{[}\Big{(}1-{M_{W}^{2}\over M_{T}^{2}}+{m_{b}^{2}\over
M_{T}^{2}}\Big{)}\Big{(}1+2{M_{W}^{2}\over M_{T}^{2}}+{m_{b}^{2}\over
M_{T}^{2}}\Big{)}-2{m_{b}^{2}\over M_{T}^{2}}\Big{]}$ $\displaystyle\simeq
175{\rm MeV}\Big{(}{M_{T}\over M_{W}}\Big{)}^{3}|V_{bT}^{L}|^{2}$

can vary between $7{\rm GeV}\leq\Gamma_{W}\leq 837{\rm GeV}$, if we consider
as a range of hypothetical masses the interval $1{\rm TeV}\leq M_{T}\leq 5{\rm
TeV}$. The width of the $B$ VLQ would be instead

$\displaystyle\Gamma(B\rightarrow W^{+}t)$
$\displaystyle={{G_{F}M_{B}^{3}}\over{8\sqrt{2}\pi}}|V_{tB}^{L}|^{2}\sqrt{\Big{(}1-{M_{W}^{2}\over
M_{B}^{2}}-{m_{t}^{2}\over M_{B}^{2}}\Big{)}^{2}-4{{M_{W}^{2}m_{t}^{2}}\over
M_{B}^{4}}}\Big{[}\Big{(}1-{M_{W}^{2}\over M_{B}^{2}}+{m_{t}^{2}\over
M_{B}^{2}}\Big{)}\Big{(}1+2{M_{W}^{2}\over M_{B}^{2}}+{m_{t}^{2}\over
M_{B}^{2}}\Big{)}-2{m_{t}^{2}\over M_{B}^{2}}\Big{]}$ $\displaystyle\simeq
175{\rm MeV}\Big{(}{M_{B}\over M_{W}}\Big{)}^{3}|V_{tB}^{L}|^{2}$

and it could vary between $14{\rm GeV}\leq\Gamma_{W}\leq 1.71{\rm TeV}$, if we
consider as a range of hypothetical masses the interval $1{\rm TeV}\leq
M_{B}\leq 5{\rm TeV}$. These values show that the two new VLQs in
$(T,B)$-doublet would typically appear as narrow-mid resonances. The $T$ VLQ
would have ${\Gamma_{TOT}\over M_{T}}\leq 10\%$ for masses $M_{T}\leq 3850$
GeV and could be analysed in narrow width approximation. Only in the large
mass domain, $3.9\leq M_{T}\leq 5$ TeV, the new $T$ VLQ would be slightly
broader, having a width over mass ratio $10\%\leq{\Gamma_{TOT}\over M_{B}}\leq
17\%$. The $B$ VLQ would have ${\Gamma_{TOT}\over M_{T}}\leq 10\%$ for masses
$M_{B}\leq 2700$ GeV and there it could be analysed in narrow width
approximation. In the mid-large mass domain, $2.7\leq M_{T}\leq 5$ TeV, the
new $B$ VLQ would be broader, having a width over mass ratio
$10\%\leq{\Gamma_{TOT}\over M_{B}}\leq 34\%$.

If we assume the existence of new physics beyond the VLQs, the unitarity
condition of the $4\times 4$ CKM matrix could be released. This matrix could
in fact be considered as a sub-matrix belonging to a larger unitary matrix. In
this event, both charged currents and FCNCs could be present across all
generations of up-type and down-type quarks. The new heavy VLQs could then be
singly-produced and subsequently decay into SM quarks plus the associate
production of a $W$, $Z$ or Higgs boson.

A different assumption, largely present in the literature, is that VLQs would
couple only to the third generation of SM quarks. If we stipulate that the
$4\times 4$ extended matrix $V=V_{CKM}^{ext}$ is unitary, then it must satisfy

$\sum_{k}V_{ik}V^{*}_{kj}=0$ (IV.30)

for any different $i$ and $j$. If we set $i=T$ (or $i=B$ indifferently), the
above-mentioned unitarity condition becomes just

$\sum_{k}V_{Tk}V^{*}_{kj}=V_{Tb}V^{*}_{bj}=0;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{k}V_{Bk}V^{*}_{kj}=V_{Bt}V^{*}_{tj}=0$
(IV.31)

as the $T(B)$-quark couples only to the $b(t)$-quark. However, In general, the
$V^{*}_{bj}$ and $V^{*}_{tj}$ entries take demonstrably non-zero values for
all other up(down)-type quarks $j$. Thus, for the above condition to hold, it
must be the case that

$V_{Tb}=0~{}~{}~{}~{}~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}~{}~{}~{}~{}V_{Bt}=0.$
(IV.32)

In this case, no mixing is present neither between the $T$-VLQ and the SM
$b$-quark nor between the $B$-VLQ and the SM $t$-quark. This realisation of
the VLQ model has troubling phenomenological implications to say the least, as
it means that VLQs cannot decay into the lighter SM quarks plus the associated
emission of a $W$-boson. Neither they can decay into SM quarks via FCNCs
mediated by the $Z$ or the Higgs boson, in virtue of the relation in Eq.
III.15 and the unitarity of the extended CKM matrix. Along the same reasoning,
they cannot be singly produced. Barring some other form of radiative decay,
the $T$ and $B$ quarks must therefore be either long-lived or stable
particles, calling in to question the predictive capabilities of this
particular implementation. Cosmological constraints would indeed forbid the
existence of electrically charged stable particles, the VLQs in this case, as
source of Dark Matter.

If we do not require that the extended CKM matrix is unitary, then this
problem does not arise, and we should observe new signals as the $T(B)$-quark
couplings with SM ordinary matter are no longer necessarily zero and mixing
effects may be observed in interactions involving the SM third-generation
quarks. Not only the new heavy VLQs could interact with the third generation
SM quarks via the charged weak current, but also the possibility of FCNCs
mediated by the $Z$ and the Higgs boson could open up, in principle. The
$(T,B)$-doublet scenario, with mixing in the third-generation only, requires
therefore to assume the existence of new physics beyond the VLQs to be
realised. This is consistent with findings by other groups in more model
specific approaches.

##  V Conclusions

In this paper, we have presented a new Vector-like quark model constrained by
the requirement of preserving the perturbative unitarity of the theory. We
have considered a general framework, where the SM is extended by the sole
presence of VLQs. We have chosen to address, specifically, three group
representations: $T$-singlet, $B$-singlet and $(T,B)$-doublet. The first
outcome is that perturbative unitarity imposes three main relations between
charged and neutral current couplings between quarks (including VLQs) and
bosons. These sum rules highly constrain the structure of such couplings and
their magnitude. They moreover relate such couplings to the properties of the
extended CKM matrix, whether being unitary or not. From this point of view,
two are the main consequences. First, the extended CKM matrix is purely left-
handed. Second, the SM-like Higgs displays a pseudo- scalar component in its
FCNC couplings, which is proportional to the difference in mass of the
involved quarks.

This result gives rise to novel phenomenological aspects relevant to VLQs,
which might have a significant impact on the search strategies at the LHC. For
$T$-singlet VLQs, only the up-type quark sector can have FCNCs mediated by the
$Z$ and Higgs bosons, therefore $b$-quark physics cannot provide useful
observables. On the contrary, the rare top decays into a lighter up-type quark
and a neutral boson, $Z$ or Higgs, represent a valuable signature a priori.
The branching ratios of the rare top decays within this model could be within
the reach of the FCC-ee and FCC-hh. As to the direct search, the $T$-quarks
could be either doubly- or singly-produced. They would then consequently decay
in all three possible channels, charged and neutral, with branching ratios
$Br(T\rightarrow Wb)=47\%,Br(T\rightarrow Zt)=30\%$ and $Br(T\rightarrow
Ht)=23\%$. The results of the data analyses obtained at the LHC, assuming 100%
decay into a single channel, should be then appropriately rescaled to extract
the exclusion bounds on the mass of the VLQs. Moreover, the search strategy
should be tailored according to the $T$-quark mass and width. In the mid-mass
range, $M_{T}\leq 2650{\rm GeV}$, the VLQs would in fact appear as narrow
resonances, while for higher masses their width-over-mass ratio could be
$35\%$ or more.

An analog result holds for the $B$-singlet VLQ. In this case, only the down-
type quark sector can have FCNCs mediated by the $Z$ and Higgs bosons,
therefore the rare top decays into a lighter up-type quark and a neutral
boson, $Z$ or Higgs, cannot provide any useful measurement. The $B$-quarks
could instead contribute to $b$-quark physics. As the $T$-quark, also the
$B$-quark could be doubly or singly produced and then decay into SM quarks and
bosons with similar branching ratios as its $T$ counterpart. A noticeable
difference compared to the $T$-quark consists in the width of the $B$-quark.
This time, while for $M_{T}\leq 1900{\rm GeV}$ they would appear as narrow
resonances, with increasing the mass their width-over-mass value could reach
$70\%$ and more. The signal would manifest itself as a large shoulder or even
worse as an excess of events spread over the SM background. Present searches,
performed under the narrow width approximation, might have missed these new
particles.

The $(T,B)$-doublet case is instead different in more than one aspect. If the
extended CKM is assumed to be unitary, all FCNCs are absent independently on
the type of quarks. This would allow the new VLQs to be doubly- or singly-
produced and afterwards to decay into SM quarks with the associate production
of a $W$-boson, only. Therefore, the search for these new particles should be
different from what performed at the moment; only the charged channel should
be considered. If in the doublet representation, the $T$ and $B$ quarks are
expected to appear as rather narrow resonances, with a width-over-mass ratio
not exceeding $35\%$. In a scenario where the extended CKM were not unitary,
all decay channels would open up again. Finally, a comment on an assumption
largely present in the literature, that is VLQs only couple to the third
generation of SM quarks. In this scenario, if we stipulate that the extended
CKM is unitary, no mixing would be present between VLQs and SM quarks
rendering the VLQs (nearly) stable particles. Cosmological constraints would
indeed forbid the existence of electrically charged stable particles as source
of Dark Matter, so the model would be ruled out. Relaxing the unitarity
condition of the CKM would of course make the scenario viable again. The
$(T,B)$-doublet scenario, with mixing in the third-generation only, requires
therefore to assume the existence of new physics beyond the VLQs to be
realised.

To summarise, the unitary VLQ model constrained by the requirement of
perturbative unitarity is more predictable than the more general VLQ models
and could be tested at the upcoming LHC run 3.

## Acknowledgements

This work is supported by the Science and Technology Facilities Council, grant
number ST/L000296/1. EA acknowledges partial financial support through the
NExT Institute.

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