The m331 is built in such a manner that the number of fermionic triplets
equals that of anti-triplets: in the lepton sector, three triplets exist;
within the colored spectrum, we have three triplets and six anti-triplets.
This guarantees the vanishing of this 3-point function amplitude.

The matter of anomaly cancellation in the m331 differs from it in the SM in a
fundamental aspect: it only works when the three generations are taken into
account. The last triangle we evaluated, corresponding to three $SU(3)_{L}$
currents, is enough to arrive at this conclusion. Note that, to achieve a
version of the theory with equal numbers of fundamental and anti-fundamentals,
the balance of two $\bar{\mathbf{3}}$ to one $\mathbf{3}$ must be maintained
within the quark sector. This implies that the theory is only renormalizable
if the number of generations is a multiple of three. Furthermore, to secure
asymptotic freedom at high energies the number of quark flavours must be
smaller than 16. This fixes the number of generations of the m331 to three, as
six families, the next multiple of three available, corresponds to eighteen
quarks and already surpasses the upper limit set by asymptotic freedom. In
this way, the m331 offers a (at least partial) solution to the arbitrariness
of the number of families.

### 4.8 Closing the m331 numerically at the electroweak scale

The Equations (4.18) through (4.26), of masses and rotations between vector
boson eigenstates, are highly dependent on the exotic scalar VEVs. Not only
them, but also vertices and effective couplings between the various particles
are reliant on these quantities. These objects,
$v_{\chi},v_{\eta},v_{\rho},v_{s_{1}},v_{s_{2}}$, are all free parameters of
the theory, with yet unknown values to be fitted through experimental data.
They are not completely free, however, as they parametrize, in the m331,
quantities that are dominated by pure SM phenomena.

In this spirit, Dias et al found that imposing a relation between the VEVs
identically causes known neutral gauge boson masses and neutral current
parameters, at the electroweak scale, to descend into their known values
[128]. The solution to the closing of the 3-3-1 symmetry at the electroweak
scale, as the authors called it, is given by

$v_{\rho}^{2}=\frac{1-4s_{W}^{2}}{2c_{W}^{2}}v_{W}^{2}.$ (4.51)

If the equation above is plugged into the exact expressions of the neutral
gauge boson masses and neutral current (NC) parameters within the m331, their
extremely involved forms descend into the experimentally verified SM
predictions. Because some values differ from those presented in [128], we
write down all the NC parameters, both general and constrained by the
solution, in Appendix A. In particular, an interesting, albeit useless,
consequence of the solution is that it sets a theoretical lower bound on the
absolutely free parameter $v_{\chi}$, being
$v_{\chi}>$54\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ [129].

The RHS of Eq. (4.51) involves only known numerical quantities and gives

$v_{\rho}\approx$54\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ (4.52)

Moreover, $v_{W}$ is actually defined by fitting the Higgs mass and is given
by $v_{W}^{2}=v_{\eta}^{2}+v_{\rho}^{2}+v_{s_{1}}^{2}+v_{s_{2}}^{2}$. Now,
$v_{s_{1}}$ is expected to be small similarly to the VEV of any triplet
extension of the SM [130], and $v_{s_{2}}$ is assumed small simply because it
does not contribute to the quark masses and this is the most immediate way to
fit the lepton ones. Within this most common and reasonable benchmark, one
obtains

$v_{\eta}\approx$240\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.$ (4.53)

In this thesis, this solution will be the preferred choice whenever a
numerical ansatz for the scalar VEVs is needed.

##  5 LHC phenomenology of flavour violating $U^{\pm\pm}$ processes

This section is devoted to the first novel results from the project underlying
this thesis. We perform a preliminary study of the LHC phenomenology of the
vector bilepton, with the mixing between lepton flavours properly implemented.
We start with a short catalogue of previous works and then move on to define a
framework and derive results.

### 5.1 Review of existing phenomenological literature

In order to conduct a useful novel analysis, it is necessary to understand
which aspects of the $U^{\pm\pm}$ phenomenology have already been explored and
which points have not been focused on yet. This is to say that a brief review
of the existing literature is necessary. Wanting to keep this recollection to
a reasonable length, we focus only on LHC results, and for some great early
pure leptonic analysis point the reader to [131, 132, 133]. Furthermore,
because it is impossible to quote the full set of results that could be
extracted from each work, we quote the points of exclusion or discovery of
highest $M_{U}$, with the requirement that the experimental parameters are at
most the current LHC characteristic ones of
$\sqrt{s}\sim$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and
$\mathcal{L}\sim$140\text{\,}\mathrm{f}\mathrm{b}^{-1}$$.

In [134], the authors use the process $ug\to U^{++}j_{1}$ to qualitatively
examine the prospects of discovery of the $U^{++}$ and $j_{1}$. The remaining
exotic quarks are deemed heavy $J,j_{2}\gg U^{++},j_{1}$. The object reaction
occurs through $s$-channel exchange of an up quark or through $t$-channel
exchange of a $j_{1}$. The detectable signal originate from the subsequent
decay chains which, if $m_{j_{1}}>M_{U}$, are given by
$U^{++}\to\ell^{+}\ell^{+}$ and $j_{1}\to U^{--}u\to\ell^{-}\ell^{-}u$. If
$M_{U}>m_{j_{1}}$ , $U^{++}$ acquires the decay mode to $j_{1}u$, in which
case the $j_{1}$ proceeds to decay into charged leptons via off-shell
bileptons. The signal is again composed by four charged leptons, but with a
peak on only one pair. The authors calculate the cross section times branching
ratio of subsequent decay for the $U^{++}$ and $U^{--}$ as a function of mass.
They then perform a qualitative analysis of the LHC background and behaviour
of the signal for four benchmark $m_{j_{1}}$, remarking that the difference in
the position of the intermediate resonance in the invariant mass of pairs of
leptons (like-sign vs. different charge lepton pairs) plus the lower
transverse momentum of the $Z$ mediated background are enough to guarantee
spectacular signal. Finally, they state that, for
$\sqrt{s}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and
$\mathcal{L}=$100\text{\,}\mathrm{f}\mathrm{b}^{-1}$$, masses of up to
$M_{U},m_{j_{1}}<$1.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ could be
explored at the LHC through around 100 events.

In [135], a variety of processes are considered in a model independent
analysis which evokes the m331 when necessary. The subset of their
investigations which concern us look at the hard processes $q\bar{q}\to
U^{++}U^{--}$, $q\bar{q}\to U^{++}e^{-}e^{-}$ and $u\bar{d}\to U^{++}V^{-}$,
where the first two reactions differ in that, although the signal is made up
of four leptons in both cases, in the first both $U$ are real, whereas in the
second one of them is virtual. In order to preserve unitarity, the authors add
an exotic $Z^{\prime}$ to the analysis. They evaluate the full hadronic cross
sections and show the results for four benchmark $M_{Z^{\prime}}$. The authors
define as discovery criteria a minimum cross section of
$2.5\text{\,}\mathrm{f}\mathrm{b}$ corresponding to around 25 events. The most
useful process turns out to be $q\bar{q}\to U^{++}U^{--}$, and the most
powerful findings indicate that bileptons with masses as high as
$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ may be explored at the LHC for
$M_{Z^{\prime}}=1.8\sim$3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$.

[136] carries a study of the specific four leptons signal $pp\to
e^{\mp}e^{\mp}\mu^{\pm}\mu^{\pm}X$ at the LHC. The authors fix the
$Z^{\prime}$ mass, again included by unitarity, to
$M_{Z^{\prime}}=$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, and chose three
benchmark values for all the exotic quark masses, considered the same. In view
of the reconstruction efficiency they set, the authors consider a $5\sigma$
discovery criteria to match the requirement of five events, and use a bayesian
technique to find a maximum 95% exclusion cross section equivalent to the
requirement of zero events of data. The results show that, for an LHC run at
$\sqrt{s}=$14\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and of
$\mathcal{L}=$110\text{\,}\mathrm{f}\mathrm{b}^{-1}$$, with the highest
considered exotic quarks mass
$M_{Q}=$800\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, bileptons of
$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ may be excluded.

The object of Ref. [137] is basically the same. The authors consider pair
production of doubly-charged vector bileptons through $s$-channel exchange of
$\gamma,Z,Z^{\prime}$, i.e., the full hard process given by
$q\bar{q}\to\gamma,Z,Z^{\prime}\to U^{++}U^{--}\to
e^{\mp}e^{\mp}\mu^{\pm}\mu^{\pm}$. The study focus specifically on the m331
Model, scanning over $M_{U}$ which, in this theory, is enough to fix the
masses of the exotic $Z^{\prime}$ as well. As remarked in the earlier works,
the observables of the invariant mass of like-sign leptons and lepton
transverse momentum are verified to allow for distinction between signal and
background. The authors plot the luminosity that is needed to discover the
bilepton at a given mass for
$\sqrt{s}=7,8,$14\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, with the result
that masses of $\sim$700\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ could be
discovered at $\sqrt{s}=$14\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ with
$\mathcal{L}=$1\text{\,}\mathrm{f}\mathrm{b}^{-1}$$ of data.

Reference [138] considers for the first time results of the Run II, using the
limits on the cross section for the process
$pp\to\mu^{\mp}\mu^{\mp}\mu^{\pm}\mu^{\pm}X$ to explore bilepton
phenomenology. The authors first analyse what was learned from Run I,
exploiting the $4.7\text{\,}\mathrm{f}\mathrm{b}^{-1}$ of data provided by the
ATLAS experiment to derive exclusion contours on the $M_{U}\times m_{Q}$
plane, and find that for the highest $m_{Q}$ considered, of
$600\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, bileptons of up to
$520\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ are excluded by the LHC at
$\sqrt{s}=$7\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. The authors then
predicted that, for $m_{Q}=$800\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$M_{Z^{\prime}}=$3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, the Run II at
$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ would improve this bounds up to
way above $1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ for
$100\text{\,}\mathrm{f}\mathrm{b}^{-1}$ of integrated luminosity.

Table 5.1: Summary of the results of the literature we have reviewed. Most of the works considered several benchmarks, in which cases we have chosen the strongest $M_{U}$ bounds or, if the bounds are similar, the highest benchmark masses. Some benchmarks involve extra less important choices that we omit. Moreover, we select the exclusion bounds if presented, otherwise the discovery limits are shown (these are highlighted by an asterisk).  |  Process |  Benchmarks |  Exp. param. |  Excluded
---|---|---|---|---
[134] |  $\\!\\!\\!\\!\\!\\!\begin{aligned} ug&\to U^{++}j_{1}\\\ &\to(4\ell)u\end{aligned}$ | $m_{j_{1}},m_{J}\gg m_{j_{1}}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} \sqrt{s}&=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\\\ \mathcal{L}&=$100\text{\,}\mathrm{f}\mathrm{b}^{-1}$\end{aligned}$ | $M_{U},m_{j_{1}}<$1.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\hskip 2.27621pt*$
[135] |  $\\!\\!\\!\\!\\!\\!\begin{aligned} q\bar{q}&\to U^{++}U^{--}\\\ &\to 4\ell\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} m_{Q}&=$600\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\\\ M_{Z^{\prime}}&=$3.0\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} \sqrt{s}&=$14\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\\\ \mathcal{L}&=$10\text{\,}\mathrm{f}\mathrm{b}^{-1}$\end{aligned}$ | $M_{U}<$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\hskip 2.27621pt*$
[136] |  $\\!\\!\\!\\!\\!\\!\begin{aligned} pp&\to U^{\mp\mp}U^{\pm\pm}X\\\ &\to e^{\mp}e^{\mp}\mu^{\pm}\mu^{\pm}X\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} m_{Q}&=$800\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\\\ M_{Z^{\prime}}&=$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} \sqrt{s}&=$14\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\\\ \mathcal{L}&=$110\text{\,}\mathrm{f}\mathrm{b}^{-1}$\end{aligned}$ | $M_{U}<$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$
[137] |  $\\!\\!\\!\\!\\!\\!\begin{aligned} pp&\to U^{\mp\mp}U^{\pm\pm}\\\ &\to e^{\mp}e^{\mp}\mu^{\pm}\mu^{\pm}\end{aligned}$ | $\\!\\!\\!\\!\\!\\!M_{Z^{\prime}}=$2.6\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} \sqrt{s}&=$14\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\\\ \mathcal{L}&=$1\text{\,}\mathrm{f}\mathrm{b}^{-1}$\end{aligned}$ | $M_{U}<$700\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\hskip 2.27621pt*$
[138] |  $\\!\\!\\!\\!\\!\\!\begin{aligned} pp&\to U^{++}U^{--}X\\\ &\to\mu^{+}\mu^{+}\mu^{-}\mu^{-}X\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} m_{Q}&=$800\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\\\ M_{Z^{\prime}}&=$3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} \sqrt{s}&=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\\\ \mathcal{L}&=$100\text{\,}\mathrm{f}\mathrm{b}^{-1}$\end{aligned}$ | $M_{U}<$1.1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$
[139, 140] |  $\\!\\!\\!\\!\\!\\!\begin{aligned} pp&\to U^{++}U^{--}jj\\\ &\to(4\ell)jj\end{aligned}$ |  $\\!\\!\\!\\!\\!\\!\begin{aligned} m_{Q}&=$1700\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\\\ M_{Z^{\prime}}&=$3.229\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$\end{aligned}$ | $\sqrt{s}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ | $M_{U}\approx$874\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\hskip 2.27621pt*$

References [139, 140] qualitatively check the reach of the LHC at
$\sqrt{s}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ to observe scalar and
vector bileptons for a fixed single benchmark choice. In the first work, the
authors consider double bilepton production together with two jets, $pp\to
U^{++}U^{--}jj\to\ell^{+}\ell^{+}\ell^{-}\ell^{-}jj$, and choose a single
benchmark point for the complete m331 Model, including every exotic heavy
quark and scalar parameter, which is coherent with every known quantity and
the current phenomenological standing. This benchmark tests the doubly-charged
vector bilepton mass of
$M_{U}=$873.3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. They carry a profound
analysis of the possible discriminant observables to enhance signal strength,
and predict that it is able to overcome the $ZZjj$ SM background and that a
$873.3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ $U^{\pm\pm}$, in the single
point chosen in parameter space, could not only be excluded, but discovered.
The second work extends this investigation, considering also the contributions
of the doubly-charged scalars of the model and vetoing final state jets.

We compile the strongest bounds of each reviewed paper in Table 5.1 for easy
consult. One characteristic that is shared by all these works is that they
ignore the mixing among leptons induced by the $V_{U}$ matrix in their
interaction. Even if the processes that are described involve diagonal
interactions, $U$ could still couple to different flavours with different
strength.

### 5.2 Process and background

In this work, the mass of every possibly contributing exotic particle other
than the $U$ will be deemed heavy. In the m331, for instance, this first
simplification amounts to the reasonable assumption $m_{J},m_{j_{i}}\gg
M_{U}$. In a process with two initial state particles, with the heavy quarks
forbidden to participate, it is clear that the most economical way to probe
the $U$ come from signal final states with four charged leptons. The
literature we have just reviewed considers lepton flavour conserving reactions
or lepton flavour violating ones but with only diagonal $U$ interactions. The
former are summarized, at parton level, by the simplest

$q\bar{q}\to\ell^{+}\ell^{+}\ell^{-}\ell^{-}.$ (5.1)

This group has background, and SM contributions for this type of process
appear on the diagrams of Figure 5.1.

Figure 5.1: Diagrams contributing to the production of a (lepton flavour
conserving) final state of 4 charged leptons in the SM.

To test the importance of this irreducible background, we evaluate the cross
section at $\sqrt{s}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, with the
usual kinematic cuts, from the individual contributions, for the specialized
$4e$ final state $pp\to e^{+}e^{-}e^{+}e^{-}$. The results are

$\sigma_{t:q}=$5.3\text{\,}\mathrm{f}\mathrm{b}$,$ (5.2)

for the cross section containing $t$-channel exchange of a quark and

$\sigma_{s:\ell}=$0.83\text{\,}\mathrm{f}\mathrm{b}$,$ (5.3)

for the cross section that features one $s$-channel exchange of a lepton. The
total SM prediction for the rate of this process, featuring interference
effects between the two types of contributions above, is given by

$\sigma_{\mathrm{total}}=$6.2\text{\,}\mathrm{f}\mathrm{b}$.$ (5.4)

With the luminosity usually associated with the LHC on phenomenological
analysis in the present, $140\text{\,}\mathrm{f}\mathrm{b}^{-1}$, this
corresponds to about 900 events. Now, it has been shown that the distributions
of observables such as invariant mass of pairs of leptons, transverse momentum
and others hint that such SM contributions may be clearly distinguished from
exotic physics. Nonetheless, a background production rate of this magnitude is
an enourmous hindrance on the discovery potential of any weak scale physics.

Now, in general, the interaction of the $U^{\pm\pm}$ with leptons allows for a
CKM-like matrix that mix different flavours. In particular, in the m331, such
a matrix is almost unavoidable as a consequence of the fact that the lepton
masses come from two distinct sources: $G^{\eta}$ and $G^{s}$. This prompts us
to investigate CLFV processes, and if they could provide a possible smoking
gun. Specifically, since these particles are more easily observable, our
signal is comprised mostly by muons in the form of the trimuon process

$pp\to\mu^{\pm}\mu^{\pm}\mu^{\mp}e^{\mp}.$ (5.5)

As an aside, in the late seventies some events resulting from neutrino and
anti-neutrino exposure ending with two or three muons have been found [141,
142, 143, 144] and the latter were called trimuon as well. No further events
of this type appear to have been detected.

The reason this process, in principle possible through $U^{\pm\pm}$ mediation,
represents a strong road to information gain is because such reactions, as any
with charged lepton flavour violation, is forbidden to all orders in
perturbation theory in the SM (more on this in the next section). The $U$
contributions to this exotic process that interest us are given by the
diagrams in Figure 5.2. This signal is free of irreducible backgroud. By
completeness, Figure 5.3 shows the contributions that would be given by the
charge $-4/3$ and $5/3$ exotic quarks of the m331. These are assumed
negligible by our condition on the masses of any additional exotic particles.

Figure 5.2: m331 contributions to the trimuon process. Allowed permutations in
the outgoing legs must also be considered.

Figure 5.3: $t$-channel exchange of exotic quarks that contribute to any four
lepton final state at the LHC in the m331.

### 5.3 Parametrizing and numerically defining the necessary interactions

We shall start with perfectly general interactions and then justify every
numerical choice. Although our results are not fully model independent, they
should cover an interesting sector of theory space. The most general form for
the $U\ell\ell$ interaction that may be inferred from Lorentz and
electromagnetic invariance is

$\mathcal{L}_{U\ell\ell}=\sum_{a}g_{U}\overline{\ell_{a\phantom{c}L}^{\prime
c}}\gamma^{\mu}\ell^{\prime}_{aL}\thinspace
U_{\mu}^{++}+g_{U}\overline{\ell^{\prime}_{aL}}\gamma^{\mu}\ell^{\prime
c}_{a\phantom{c}L}\thinspace U_{\mu}^{--}.$ (5.6)

The Lagrangian above is diagonal on symmetry eigenstates, the primed fields,
because it supposedly comes from a minimally coupled kinetic term. In fact, if
the underlying, ultraviolet complete model is not lepton universal, then there
could be relative factors between flavours, but in this work we shall ignore
this possibility. Note that a hand mirrored term
$g^{\prime}_{U}\overline{\ell_{a\phantom{c}R}^{\prime
c}}\gamma^{\mu}\ell^{\prime}_{aR}U_{\mu}^{++}$ encompass all the same degrees
of freedom. In fact, $g^{\prime}_{U}\overline{\ell_{a\phantom{c}R}^{\prime
c}}\gamma^{\mu}\ell^{\prime}_{aR}U_{\mu}^{++}=-g^{\prime}_{U}\overline{\ell_{a\phantom{c}L}^{\prime
c}}\gamma^{\mu}\ell^{\prime}_{aL}U_{\mu}^{++}$, so that adding the right-
handed term would amount to a mere redefinition of the coupling (we shall come
back to this and the following facts in the coming chapters).

The biunitary transformation whith rotates symmetry to mass eigenstates is
defined by

$\begin{split}\ell^{\prime}_{L}&\equiv V^{\ell}_{L}\ell_{L}\\\
\ell^{\prime}_{R}&\equiv V^{\ell}_{R}\ell_{R},\end{split}$ (5.7)

from which one obtains

$\begin{split}\overline{\ell^{\prime}_{L}}&=\overline{\ell_{L}}V_{L}^{\ell\dagger}\\\
(\ell^{\prime c})_{L}&=V_{R}^{*}(\ell^{c})_{L}\\\ \overline{(\ell^{\prime
c})_{L}}&=\overline{(\ell^{c})_{L}}V_{R}^{T},\end{split}$ (5.8)

and similarly for right-handed fields. With this, we write the $U\ell\ell$
interaction for mass eigenstates as

$\mathcal{L}_{U\ell\ell}=\sum_{a,b}g_{U}\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}(V_{U})_{ab}\ell_{b}\thinspace
U_{\mu}^{++}+g_{U}\bar{\ell_{a}}\gamma^{\mu}P_{L}(V_{U}^{\dagger})_{ab}\ell_{b}^{c}\thinspace
U_{\mu}^{--},$ (5.9)

where we have defined the general mixing matrix

$V_{U}\equiv V_{R}^{T}V_{L}.$ (5.10)

.

In Eq. (5.9), $g_{U}$ is an absolutely arbitrary coupling, except for possible
unitarity bounds. Our first benchmark corresponds to using the m331 value for
this parameter. This choice is natural, as this model is the only motivated
TeV scale theory to feature such particle. Consider, in this theory, the
algebra valued electroweak gauge field with the non-diagonal, charged sector
in the physical basis

$\begin{split}-i&g_{3L}\lambda^{i}W_{\mu}^{i}-ig_{X}XB_{\mu}\equiv-
ig_{3L}\mathcal{M}\\\
&=-i\frac{g_{3L}}{2}\begin{pmatrix}W_{\mu}^{3}+\frac{1}{\sqrt{3}}W_{\mu}^{8}+2t_{X}XB_{\mu}&\sqrt{2}W_{\mu}^{+}&\sqrt{2}V_{\mu}^{-}\\\
\sqrt{2}W_{\mu}^{-}&-W_{\mu}^{3}+\frac{1}{\sqrt{3}}W_{\mu}^{8}+2t_{X}XB_{\mu}&\sqrt{2}U_{\mu}^{--}\\\
\sqrt{2}V_{\mu}^{+}&\sqrt{2}U_{\mu}^{++}&-\frac{2}{\sqrt{3}}W_{\mu}^{8}+2t_{X}XB_{\mu}\end{pmatrix}.\end{split}$
(5.11)

This matrix appears in the kinetic term of the left-handed leptons
$-ig_{3L}\bar{L^{\prime}}\mathcal{M}L^{\prime}$, from which the form of the
$U\ell\ell$ interaction may be extracted to be

$\mathcal{L}_{U\ell\ell}=-i\frac{g_{3L}}{\sqrt{2}}\overline{\ell_{L}^{c}}\gamma^{\mu}V_{U}\ell_{L}U^{++}_{\mu}+\mathrm{H.C.},$
(5.12)

in the mass basis. From this, we set

$g_{U}=\frac{g_{3L}}{\sqrt{2}}.$ (5.13)

Another required interaction is the one which connects a pair of $U$ to the
known $Z$ boson, which should generically be of the form

$\begin{split}\mathcal{L}_{UUZ}&=f(g,v)\left\\{U^{++}_{\mu}\left[U^{--}_{\alpha}(\partial_{\mu}Z_{\alpha})-Z_{\alpha}(\partial_{\mu}U^{--}_{\alpha})\right]\right.\\\
&\hskip
51.21495pt{}+U^{--}_{\nu}\left[Z_{\alpha}(\partial_{\nu}U^{++}_{\alpha})-U^{++}_{\alpha}(\partial_{\nu}Z_{\alpha})\right]\\\
&\left.\hskip
55.19832pt{}+Z_{\alpha}\left[U^{++}_{\mu}(\partial_{\alpha}U^{--}_{\mu})-U^{--}_{\mu}(\partial_{\alpha}U^{++}_{\mu})\right]\right\\}.\end{split}$
(5.14)

Above, $f(g,v)$ is a dimensionless effective coupling which, when the
interaction above descends from an ultraviolet complete model, is a function
of the various VEVs and gauge couplings of the theory. There are two natural
options for the numerical definition of this quantity: (i) Setting
$f(g,v)=g_{2L}c_{W}$ reproduces the $WWZ$ vertex of the SM. This represents an
interesting choice as it would correspond to a preferred practice in the
parametrizations of LHC searches; (ii) Another possibility is to make use of
the m331 prediction once more. In the m331 constrained by the Solution to the
closure of Eq. (4.51), the $f$ factor is given by
$f(g,v)=(4s_{W}^{2}-1)/c_{W}$. Notice, in particular, that in this
configuration the $U$ is $Z$-phobic since $s_{W}^{2}\approx 0.23$. Only this
latter alternative will be treated.

It must be noted that to achieve the selected expression as the $UUZ$ vertex
strength in the m331 the following implicit simplification must be made

$t_{X}^{2}=\frac{s_{W}^{2}}{1-4s_{W}^{2}}.$ (5.15)

This relation is ubiquitously utilized in m331 phenomenological studies, not
always sensibly. This matter will be discussed in the last part of this
thesis.

A second triple gauge vertex that we must fixate corresponds to the $UU\gamma$
interaction. This time we are able to give it in a truly model independent way
as

$\begin{split}\mathcal{L}_{UU\gamma}&=2Q_{e}(g)\left\\{U^{++}_{\mu}\left[U^{--}_{\alpha}(\partial_{\mu}A_{\alpha})-A_{\alpha}(\partial_{\mu}U^{--}_{\alpha})\right]\right.\\\
&\hskip
52.63759pt{}+U^{--}_{\nu}\left[A_{\alpha}(\partial_{\nu}U^{++}_{\alpha})-U^{++}_{\alpha}(\partial_{\nu}A_{\alpha})\right]\\\
&\left.\hskip
56.62096pt{}+A_{\alpha}\left[U^{++}_{\mu}(\partial_{\alpha}U^{--}_{\mu})-U^{--}_{\mu}(\partial_{\alpha}U^{++}_{\mu})\right]\right\\}.\end{split}$
(5.16)

It is easy to be convinced that the corresponding interaction must be of this
form in any given model. This is because, from the point of view of the photon
(which carries the force corresponding to a conserved symmetry), the $U$ is
identical to the $W$, but with double the charge. As consequence, the
Lagrangian must be analogous but with $Q_{e}\to 2Q_{e}$. The $Q_{e}(g)$ factor
represents the fundamental electric charge of the theory as given in terms of
the gauge couplings – in the SM, $e=g_{2L}g_{Y}/\sqrt{g_{Y}^{2}+g_{2L}^{2}}$.

Now, the $U$ could also couple to the SM Higgs, which would give an extra
contribution to the process at hand. It is expected, however, that the vector
mediated contributions dominate over the scalar ones. Usual arguments that
lead to this conclusion are based upon the fact that there are less
possibilities in spin space for the scalar channel, and upon the Goldstone
Equivalence Theorem [145]. Nonetheless, since there is no reason to assume
that the $UUH$ generic coupling strength is suppressed, let us take a closer
look at how these contributions could be taken into account and at their
importance within the m331.

Even within this model, a strong benchmark is necessary, which amounts to the
fixing of the unitary rotation $O$ between symmetry ($x_{i}^{0}$) and mass
($h_{i}^{0}$ – unless otherwise stated, we denote the SM Higgs by $h^{0}_{0}$)
eigenstates of $\mathit{CP}$-even neutral scalars,
$\operatorname{Re}x_{i}^{0}=O_{ij}h_{j}^{0}$. As the simplest construction to
be used as avatar, consider the Higgs to correspond exactly to the
$\mathit{CP}$-even part of one of the neutral symmetry eigenstates of the
theory that originate from the triplets: $\chi^{0}$, $\eta^{0}$ or $\rho^{0}$.
In order to find the best choice we mind each in turn, considering first the
$\chi^{0}$. It cannot be an important $h^{0}$ component because it is
absolutely neutral, sterile, from the SM perspective, and its only role is to
break the 3-3-1 symmetry at a high scale. As for the $\eta^{0}$, consider the
portion of the Lagrangian as parametrized in the ultraviolet completion that
could originate the $UU\eta^{0}$ interaction. It comes from the scalar kinetic
terms and reads

$(-ig\mathcal{M}^{\mu}\eta)^{\dagger}(-ig\mathcal{M}_{\mu}\eta)\propto\begin{pmatrix}\hphantom{\sqrt{2}U_{\mu}^{--}}&\hphantom{\sqrt{2}U_{\mu}^{--}}&\hphantom{\sqrt{2}U_{\mu}^{--}}\\\
\hphantom{\sqrt{2}U_{\mu}^{--}}&\hphantom{\sqrt{2}U_{\mu}^{--}}&\sqrt{2}U_{\mu}^{--}\\\
\hphantom{\sqrt{2}U_{\mu}^{--}}&\sqrt{2}U_{\mu}^{++}&\hphantom{\sqrt{2}U_{\mu}^{--}}\end{pmatrix}\begin{pmatrix}\frac{v_{\eta}+\text{Re}\,\eta^{0}+\text{Im}\,\eta^{0}}{\sqrt{2}}\\\
\hphantom{\frac{v_{\eta}+\text{Re}\eta^{0}+\text{Im}\eta^{0}}{\sqrt{2}}}\\\
\hphantom{\frac{v_{\eta}+\text{Re}\eta^{0}+\text{Im}\eta^{0}}{\sqrt{2}}}\end{pmatrix}=0.$
(5.17)

In brief, a particle which is mostly $\operatorname{Re}\eta^{0}$ does not
interact with the $U$. The remaining alternative must then be the chosen one,
and we identify $h_{0}^{0}=\rho^{0}$. The interaction may be read from
$g^{2}(\mathcal{M}^{\mu}\eta)^{\dagger}(\mathcal{M}_{\mu}\eta)$ to be

$\mathcal{L}_{UUH}=\frac{g^{2}}{2}v_{\rho}HU_{\mu}^{++}U^{--\mu}.$ (5.18)

Now, as a (very) crude estimate of the relation of importance of this and the
other contributions, we evaluate the widths of the channels $H\to
U^{++}U^{--}$ and $\gamma\to U^{++}U^{--}$. Directly to the point, the $H$
involving process is more than 8 orders of magnitude smaller. With such a
disparity, we may safely discard any Higgs involving contribution to such
bilepton processes, at least to the current precision.

Finally, the effective model appropriate to investigate the prospect of
bilepton discovery at the LHC, through the trimuon processes, is given by

$\mathcal{L}_{e3\mu}\equiv\mathcal{L}_{\mathrm{SM}}+\mathcal{L}_{U^{\pm\pm}}+\mathcal{L}_{U\ell\ell}+\mathcal{L}_{UUA}+\mathcal{L}_{UUZ}+\mathcal{L}_{UUH}.$
(5.19)

where

$\mathcal{L}_{U\ell\ell}=-\frac{1}{2}(\partial_{\mu}U^{++}_{\nu}-\partial_{\nu}U^{++}_{\mu})(\partial^{\mu}U^{--\nu}-\partial^{\nu}U^{--\mu})+M_{U}^{2}U^{++}U^{--},$
(5.20)

and the other pieces may be verified in Eqs. (5.12) through (5.18).

As a last, possibly obvious, remark, we stress that the Lagrangian above is
not unitary or gauge invariant. In particular, we do not include the
$Z^{\prime}$ in the analysis, usually added to the effective model to preserve
unitarity in some bilepton processes. Although this is good practice, we make
the choice to focus on the sole $U$ contributions. The dangerous high energy
behaviour should not cause any problems at the fixed partonic energies, and
considering the $U$ alone simply turns our predictions into conservative
bounds.

### 5.4 Parameter space and simulation

There are essentially ten free parameters that must be dealt with in
$\mathcal{L}_{\mathrm{e3\mu}}$: $M_{U}$ plus the nine degrees of freedom of
$V_{U}$. It would be tempting to assume that this number is, in fact, much
smaller because only the elements
$(V_{U})_{\mu\mu},(V_{U})_{e\mu},(V_{U})_{\mu e}$ enter the hard process. All
the other ones, however, enter the calculation of the total width
$\Gamma_{U}$, which does influence the results. In any case, the number must
be reduced, and we start by considering $V_{U}$ real, ignoring the five phases
of the entire unitary $V_{U}$. This turns $V_{U}$ into an orthogonal matrix
and we are left with exactly four real parameters. Now an ansatz is needed to
further cut this number down.

By its apparent reasonableness, we chose that of considering a symmetric
$V_{U}$. To understand the interplay between this and the unitarity
requirement over the degrees of freedom of a general $3\times 3$ matrix, let
us put forward a simple argument. First notice that orthogonality guarantees
that there are four possible eigenvalue signatures, the ‘sum’ of which
generates our entire space. Let us analyse each signature in turn: (i)
$\\{1,1,1\\}$: in this case there are no degrees of freedom as there is a
single corresponding matrix – the identity. (ii) $\\{-1,-1,-1\\}$: Analogous
to (i) but with $V_{U}=-\mathbb{1}$. (iii) $\\{1,1,-1\\}$ or $\\{-1,-1,1\\}$:
Here there exists a two- and a one-dimensional eigenspaces. Now, the symmetry
condition assures orthogonal diagonalization, i.e., that the two eigenspaces
are orthogonal. Hence, defining the one-dimensional eigenspace also fixes the
two-dimensional one. This shows that a symmetric, orthogonal $3\times 3$
matrix features two degrees of freedom: the coordinates of a point in the unit
sphere.

Our target parameter space may now be fully defined. $M_{U}$ is the most
important quantity to be scanned over, and $V_{U}$ is reducible to two free
elements, which are arbitrarily chosen to be $(V_{U})_{11},(V_{U})_{12}$,
i.e., $(V_{U})_{ee}$ and $(V_{U})_{e\mu}$. All the other matrix elements
become numerically defined by a choice of these two through the orthogonality
conditions (constrained by the symmetry requirement) given by

$\begin{split}&(V_{U})_{ee}^{2}+(V_{U})_{e\mu}^{2}+(V_{U})_{e\tau}^{2}=1\\\
&(V_{U})_{ee}(V_{U})_{e\mu}+(V_{U})_{e\mu}(V_{U})_{\mu\mu}+(V_{U})_{e\tau}(V_{U})_{\mu\tau}=0\\\
&(V_{U})_{ee}(V_{U})_{e\tau}+(V_{U})_{e\mu}(V_{U})_{\mu\tau}+(V_{U})_{e\tau}(V_{U})_{\tau\tau}=0\\\
&(V_{U})_{e\mu}^{2}+(V_{U})_{\mu\mu}^{2}+(V_{U})_{\mu\tau}^{2}=1\\\
&(V_{U})_{e\mu}(V_{U})_{e\tau}+(V_{U})_{\mu\mu}(V_{U})_{\mu\tau}+(V_{U})_{\mu\tau}(V_{U})_{\tau\tau}=0\\\
&(V_{U})_{e\tau}^{2}+(V_{U})_{\mu\tau}^{2}+(V_{U})_{\tau\tau}^{2}=1.\end{split}$
(5.21)

We explore this parameter space focusing on the bi-dimensional subspace
composed by $M_{U}\times(V_{U})_{e\mu}$. Although the existing literature
already constrains masses of up to $1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$,
this work is innovative in that it takes into account the usually ignored
mixing. Because of this, the scan in $M_{U}$ is effected starting from low
masses. Masses are generated in the range
$M_{U}\in\left($100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,$1200\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\right)$,
in steps of $50\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, and $(V_{U})_{e\mu}$
is varied from 0.001 to 0.9 – with the exception of when $(V_{U})_{ee}=0.9$,
in which case it stops at $0.4$ – through 12 strategically chosen points..
This 2D region is scanned four times, one for each $(V_{U})_{ee}$ among
$(V_{U})_{ee}=0.001,0.01,0.1,0.9$, which is the last free parameter to be
dealt with.

To avoid the inconvenience and computational strain of manually doing the
convolution of the hard partonic cross sections against the proton Parton
Distribution Functions (PDF), we employ a Monte-Carlo generator. The
$\mathcal{L}_{e3\mu}$ model is implemented through FeynRules [146, 147], which
generates a Universal FeynRules Output that is then interfaced with the Monte-
Carlo package MadGraph [148]. We generate $10^{4}$ events in each of the 1035
points in parameter space, subjected to the kinematic cuts

$$1500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$>p_{T_{\ell}}>$30\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,\;\;\;|\eta_{\ell}|<2.5,\;\;\;\Delta
R_{\ell\ell}>0.4.$ (5.22)

$p_{T_{\ell}}$ is the transverse momentum (relative to the beam axis) of any
charged lepton, and is limited from below to guarantee that the particles
reach the detector and from above mainly to avoid (possibly uncertain)
outliers. $\eta_{\ell}$ is the pseudorapidity and, in particle physics is
defined as

$\eta\equiv-\ln\tan\left(\frac{\theta}{2}\right)=\frac{1}{2}\ln\frac{|\mathbf{p}|+p_{L}}{|\mathbf{p}|-p_{L}},$
(5.23)

where $\theta$ is the polar angle with respect to the beam axis and
$|\mathbf{p}|$ and $p_{L}$ are the magnitude of the momentum of the given
particle and its longitudinal component, respectively. In other words,
pseudorapidity is a pure measure of the polar angle and is limited from above
by the familiar reason of avoiding that the species is lost along the track of
the beam. Finally, $\Delta R_{\ell\ell}$ is the angular distance between any
pair of leptons, defined as

$\Delta R\equiv\sqrt{\Delta\eta^{2}+\Delta\phi^{2}},$ (5.24)

where $\Delta$ indicates the difference in the corresponding property between
the two particles and $\phi$ is the azimutal angle. This definition uses
$\Delta\eta$ over $\Delta\theta$ because it is invariant under longitudinal
boosts. The same is clearly true for $\Delta\phi$ too, causing $\Delta R$ to
be invariant under longitudinal Lorentz transformations as well. Finally, the
reason for its lower limit is simply that exaggeratedly collinear particles
are hard to distinguish.

### 5.5 Statistics

Now, our process has no background. This frees us from having to find a
discriminant observable; arguing over resolution; and doing much formal
statistics. The works we have reviewed in the beginning of this Chapter are
separed by as much as 26 years, and also span a large range of choices for
discovery criteria. There is one study which defines 25 events as necessary to
announce a discovery, and another which seeks a single signal event as
sufficient for an exclusion. Although we escaped from much of the statistics
requirements, in order to at least motivate an exclusion criteria, let us
perform a short, simplistic analysis (for amazing introductions to statistics
in particle physics, see [149, 150]).

In the border of the exclusion contours of a process without background, the
number of events is, by construction, low. This forces us to employ a Poisson
ansatz instead of a Gaussian one, which could work well for a large amount of
events. The probability of an independent counting experiment (a single bin,
if we were effecting a more complex analysis over a distribution in our case),
with average count number $\mu$, to see $n$ events is given by

$P(n;\mu)=\frac{e^{-\mu}\mu^{n}}{n!}.$ (5.25)

The most simple hypothesis testing method – and recall that such methods are
somewhat arbitrary, i.e., although their motivations are easy to understand,
they are not ‘proven’ and their efficiency is not a mathematical fact – would
be to exclude models that fit the observation only outside their $95\%$
compatibility region. Specifically, assuming the separate expectations of
signal ($\mu_{s}$) and background ($\mu_{b}$), we could then find what is the
number of events $N_{c}$ corresponding to the threshold below which only $5\%$
($\beta=0.05$) of measurements would fall. In a binned analysis, this comes
from

$\beta=\sum\limits_{N=0}^{N_{c}}P(N;\mu_{B}+\mu_{S})=\sum\limits_{N=0}^{N_{c}}\frac{e^{-(\mu_{B}+\mu_{S})}(\mu_{B}+\mu_{S})^{N}}{N!}.$
(5.26)

This so-called $\mathit{CL}$ method, however, is oversensitive to downward
flutuations of the background or, more simply, tends to exclude alternate
hypothesis too strictly in cases with, for instance, small background. To
remedy this obstacle, the $\mathit{CL}_{s}$ method is popularly used. It
modifies the $\mathit{CL}$ by dividing it by the null hypothesis confidence
level in the same distribution point, thus lowering the value of the threshold
$N_{c}$. The formula becomes

$\beta=\frac{\sum\limits_{N=0}^{N_{c}}P(N;\mu_{B}+\mu_{S})}{\sum\limits_{N=0}^{N_{c}}P(N;\mu_{B})}=\frac{\sum\limits_{N=0}^{N_{c}}\frac{e^{-(\mu_{B}+\mu_{S})}(\mu_{B}+\mu_{S})^{N}}{N!}}{\sum\limits_{N=0}^{N_{c}}\frac{e^{-\mu_{B}}\mu_{B}^{N}}{N!}}=\frac{\sum\limits_{N=0}^{N_{c}}\frac{e^{-\mu_{S}}(\mu_{B}+\mu_{S})^{N}}{N!}}{\sum\limits_{N=0}^{N_{c}}\frac{\mu_{B}^{N}}{N!}}.$
(5.27)

To make it clear, $\beta$ is a predefined measure of certainty or, more
specifically, relates to the confidence level as $\beta=1-\mathit{CL}$.

We are interested in deriving exclusion, rather than discovery, contours on
the $M_{U}\times(V_{U})_{e\mu}$, and the recipe for that in particle physics
suggests a 95% confidence level (or $2\sigma$ in a gaussian analysis) band.
Instead of inputting an expected $\mu_{s}$ and finding the threshold $N_{c}$,
we invert the logic, to obtain what would be the bounds on $\mu_{s}$ if
nothing unusual was observed, i.e., with $N_{c}=\mu_{b}$. Finally, it is
enough to numerically solve

$0.05=\frac{\sum\limits_{N=0}^{\epsilon}\frac{e^{-s}s^{N}}{N!}}{\sum\limits_{N=0}^{\epsilon}\frac{\epsilon^{N}}{N!}},$
(5.28)

for $s$ to obtain an estimate of the excluded sector on parameter space.
Above, we have estimated $\mu_{s}$ by the result of the Monte-Carlo
pseudoexperiments $s$, and an arbitrarily small number $\epsilon$ is replaced
for $\mu_{b}$. In the end, we have

$s_{c}=-\ln 0.05\approx 3,$ (5.29)

so that the points that are excluded by negative results at the LHC are those
that predict more than three events. Technically, we have basically defined a
single event as necessary for exclusion in a no-background situation,
analogously to what is done in [136], with the difference that they perform a
bayesian analysis while our techniques are frequentist.

### 5.6 Results

The results are presented in Figure 5.4. The number of events in each point is
calculated as

$N=\sigma_{\text{T}}\times\mathcal{L}_{\mathrm{int}},$ (5.30)

and the points to the left of the curves are excluded at 95% confidence level
through $\mathcal{L}_{\mathrm{int}}=$140\text{\,}\mathrm{f}\mathrm{b}^{-1}$$
of integrated luminosity (a more detailed distribution of the number of events
appears in Figure 5.5).

Figure 5.4: 95% exclusion contour on the $M_{U}\times(V_{U})_{e\mu}$ plane
that would be implied by the non-observation of trimuon events in an LHC run
of $\mathcal{L}_{\mathrm{int}}=$140\text{\,}\mathrm{f}\mathrm{b}^{-1}$$. Four
benchmark $(V_{U})_{ee}$ are chosen, which arguably span the entire natural
range for this parameter. In every $(V_{U})_{e\mu}$ point, every other $V_{U}$
element becomes fixed by the orthogonality and ordinary matrix symmetry
constraints.

The first claim to be inferred from the plot is that the countour, in this
specific arrangement, is close to independent from the choice of
$(V_{U})_{ee}$. To understand this, we note that, near the resonance, the
amplitudes giving rise to the signal are proportional to

$\mathcal{M}\left(q\bar{q}\to
U^{++}U^{--}\to\mu^{\pm}\mu^{\pm}\mu^{\mp}e^{\mp}\right)\propto\frac{(V_{U})_{\mu\mu}(V_{U})_{e\mu}}{\Gamma_{U^{\pm\pm}}}.$
(5.31)

The leptonic part of the $U$ decay width, at least above the GeV scale, is
proportional to

$\Gamma_{U^{\pm\pm}}\propto(V_{U})_{ee}^{2}+(V_{U})_{\mu\mu}^{2}+(V_{U})_{\tau\tau}^{2}+(V_{U})_{e\mu}^{2}+(V_{U})_{e\tau}^{2}+(V_{U})_{\mu\tau}^{2}.$
(5.32)

Together with the constraints of Equation (5.21), this causes Eq. (5.31) to be
resistant against a change in $(V_{U})_{ee}$.

Finally, our results indicate that the $(V_{U})_{e\mu}$ which delivers optimal
resolution is given by $(V_{U})_{e\mu}\sim 0.52$, and that the largest mass
that can be excluded is around
$\sim$1100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

For hierarchic matrices which are diagonal in the $\mu$ sector (or, possibly,
favour the $\mu-\tau$ mixture to some extent), this process is essentially
useless in constraining the $U^{\pm\pm}$ parameters, as can be seen in the
lower end of the contours. This means that one should consider flavour
diagonal $U$ processes simultaneously in order to be able to ‘corner’ the
excluded region into a high mass range.

Figure 5.5: Event thresholds in the exclusion panel, for a more detailed
analysis.

Our process is free from irreducible background, but we have ignored the
unavoidable source of noise constituted by reducible one. The most obvious
example would be the failure to identify the missing energy in
$pp\to\mu^{+}\mu^{+}\mu^{-}e^{-}+\nu_{\mu}\nu_{\mu}\bar{\nu}_{\mu}\bar{\nu}_{e}$
(or the charge mirrored reaction). A more thorough analysis of the LHC physics
involved in actually detecting our signal is thus the first way this study
could be improved. Another style of analysis would be the hard adoption of a
model, allowing us to fully employ its predictions, such as effective
couplings and additional exotic particles. In the m331, this means the three
heavy quarks $j_{i},J$ and the $Z^{\prime}$. A few benchmarks could be chosen
for the parameters of these particles and a more reliable – although, then,
completely model dependent – result for the bounds could be achieved. Finally,
although the $V_{U}=V_{U}^{T}$ condition seems a reasonable simplification and
a great place to start, other ansatz for the $V_{U}$ matrix could be
considered as well. Or, in the same ansatz, our result showed how the results
are nearly independent of the choice of $(V_{U})_{ee}$, which gives room to
fix this parameter to an arbitrary value and deepen the analysis in some other
aspect.

##  6 Model independent constraints on exotic particles from flavour violating
lepton decays: Preparations

Last section we began our investigations of the $U$ phenomenology by focusing
on the regularly neglected $V_{U}$. In this section, we continue with the same
goal, but with a much larger scope and greater complexity. The source of the
constraints we seek to derive, now, are the purely leptonic decays
$\tau^{+}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k}$, where $\ell_{i,j,k}=e,\mu$,
and $\mu^{+}\to e^{+}e^{-}e^{+}$. Without ceasing to pay attention to the
mixing matrix, we now consider, in a model independent way, two exotic
particles at a time, taken among the three species that can, at tree level,
contribute to our channels. Our priority remains the $U$, and our primary
results comprise the most conservative model independent lower bounds over its
mass, given by the CLFV decays, in models where it is accompanied by an exotic
neutral or doubly-charged scalar. In this chapter, the calculations and
operational method are detailed, whereas the results are presented in the next
one.

### 6.1 Phenomenological status of the 3-body lepton decays

The LHC is, without contestation, the experimental particle apparatus with the
farthest energy reach. Moreover, the bulk of data that it is capable to
produce within each subsequent run increases at a fast pace. This sets the
expectation of it eventually becoming a precision machine of the low TeV
scale. Nonetheless, this chapter will test an alternative source of data to
check if it can complement (or even overcome the usefulness of) the
phenomenology arising from current LHC processes for the doubly-charged vector
bilepton.

To initiate the qualitative discussion it should be noted that, in comparison
with simple leptonic decays, the LHC introduces several complications:
hadronic physics, which might relevantly change by the existence of new quarks
or other alternative concepts; Detector properties and a large set of
reducible background which causes the need for a profound analysis of the
observables and definition of triggers and cuts; And the overall process
‘noise’ that an LHC run inevitably produces.

In the minimal SM, the processes of the form
$\ell_{0}^{+}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k}$, with $\ell_{0}$ heavy
enough, are forbidden to all orders of perturbation theory. They are possible,
however, in the SM with neutrino masses through PMNS mixing. The diagrams
appear in Figure 6.1. Although there have been claims that the branching
ratios (BR) associated with these processes could be calculated to give
exceedingly high values of $10^{-14}\sim 10^{-16}$ [151, 152], the classical
(and robust) result is that these BR stand well below $10^{-50}$ [153, 154,
155, 156]. One could thus say that the products of lepton decays are ‘clean’
both in the experimental and theoretical sense, i.e., there is still no
relevant irreducible background and the reducible one is exaggeratedly smaller
than that present in the LHC (although there is still beam induced background
as we shall mention briefly).

Figure 6.1: Lowest order contributions to the 3-body lepton decays in the SM
with neutrino masses. In this theory, this process is made possible thanks to
the $\ell\bar{\nu}_{\ell^{\prime}}W$ vertex, which relies on one factor of
$(V_{\mathrm{PMNS}})_{\ell\ell^{\prime}}$.

Besides highly efficient, processes with charged lepton flavour violation
(CLFV) represent a great prospect for the inspection of BSM physics because it
is, in one form or another, predicted by most kinds of exotic models, such as
Supersymmetry [157, 158, 159], two-Higgs-doublet (2HDM) [160, 161, 162] and
3-3-1 models [163, 164, 165, 166].

Table 6.1: Current experimental limits on every 3-body lepton decay channel. Process | BR
---|---
$\mu^{+}\to e^{+}e^{-}e^{+}$ | $<1.0\times 10^{-12}$
$\tau^{+}\to e^{+}e^{-}e^{+}$ | $<2.7\times 10^{-8}$
$\tau^{+}\to e^{+}\mu^{-}\mu^{+}$ | $<2.7\times 10^{-8}$
$\tau^{+}\to\mu^{+}e^{-}e^{+}$ | $<1.8\times 10^{-8}$
$\tau^{+}\to\mu^{+}\mu^{-}\mu^{+}$ | $<2.1\times 10^{-8}$
$\tau^{+}\to\mu^{+}e^{-}\mu^{+}$ | $<1.7\times 10^{-8}$
$\tau^{+}\to e^{+}\mu^{-}e^{+}$ | $<1.5\times 10^{-8}$

The most sensitive experiments – which are equipped to generate the most
stringent bounds – are the ones relative to the simplest decay: $\mu^{+}\to
e^{+}e^{+}e^{-}$. Antimuons are used to avoid negative muon capture by nuclei,
and because of the low, MeV scale energies involved, the detector must have
excellent trackers to record the trajectories. The first source of reducible
background comes from the unmeasured missing energy in $\mu^{+}\to
e^{+}e^{-}e^{+}\nu_{e}\bar{\nu}_{\mu}$. The second results from the
coincidence of two or three muon decays, whose occurrence can be reduced by
the use of a continuous beam. The standing bound is already more than 35 years
old, and was given by the SINDRUM collaboration to be $\mathrm{BR}(\mu^{+}\to
e^{+}e^{-}e^{+})<10^{-12}$ [167]. The Mu3e experiment at the Paul Scherrer
Institute is the centerpiece of the global efforts to improve this limit, and
intends to strengthen it by four orders of magnitude to $10^{-16}$ [168].

All 3-body purely leptonic decays of the $\tau$ are much less sensible.
Notwithstanding, the $\tau$ is unstable and has a rich hadronic decay
spectrum, which complicates its experimental dealing. The current bound on
every one of the six possible channels was obtained by [169] and appear in
Table 6.1. Again, these limits are naturally much less sensitive and provide a
much smaller prospect for the testing of exotic hypotheses than the $\mu\to
3e$ one. For additional discussions on the state of the art and improving
sensitivity of the CLFV decay experiments see [170, 171, 172, 173, 174, 175].

### 6.2 Exotic contributions and model independent interactions

Our main goal is to draw relevant exclusion contours on the masses of three
species of exotic particles, constraining theory space by taking advantage of
the simultaneous data of 3-body lepton decays. Despite the large disparity
between the bound on the muon decay and the other channels, considering all
the bounds together will prove useful. The examined particles are the ones
which can contribute to the relevant processes: Doubly-charged vector
bileptons $U^{\pm\pm}$; Doubly-charged scalars $Y^{\pm\pm}$; And flavour
violation mediating neutral scalars $s$. The diagrams that these species
generically induce to a given channel are displayed in Figure 6.2.

Figure 6.2: Explicit diagrams contributing to the 3-body lepton decays through
exchange of an exotic particle. The red arrow defines the direction of the
Arbitrary Fermion Flow, and its function will become apparent soon.

Since we claim to, now, take every contributing species into account, it is an
appropriate moment to recall an important remark made last chapter and expand
on it. Besides the aforementioned particles, a neutral vector boson
$Z^{\prime}$ of the type contained in the m331 could also contribute to our
processes. This, however, can only happen in non-democratic underlying models,
where distinct lepton families constitute different representations of the
gauge group, otherwise the a priori diagonal kinetic terms result in a mixing
matrix of the form $\mathcal{O}_{Z^{\prime}}=V_{L}^{\dagger}V_{L}=\mathbb{1}$.
Because of this, since we avoid focusing on specific models and, furthermore,
non-democratic leptonic sectors being rare, we overlook the possible role of
an exotic neutral vector boson.

Besides simply checking the constraints that the 3-body lepton decays may
conspire to impose, a crucial aspect that we wish to investigate is the
dynamics of interference. In particular, we shall undertake the question of to
what extent destructive interference between different species may weaken the
bounds implied by the data over their exotic parameters. It is for this reason
that we strive to take two species into account at once.

Again, one of the most substantial challenges of a study that seeks to impose
limits, as model independent as possible, on the new particle parameters is an
appropriate parametrization of every relevant interaction. As in the last
study, we shall tackle each Lagrangian in turn, starting with the $U\ell\ell$
interaction. Commencing again from the intuitive form of the most general
Lorentz and electromagnetic invariant Lagrangian that descends from an
unknown, lepton universal, gauge invariant kinetic term:

$\mathcal{L}_{U\ell\ell}=\sum_{a}g_{U}\overline{\ell_{a\phantom{c}L}^{\prime
c}}\gamma^{\mu}\ell^{\prime}_{aL}\thinspace
U_{\mu}^{++}+g_{U}\overline{\ell^{\prime}_{aL}}\gamma^{\mu}\ell^{\prime
c}_{a\phantom{c}L}\thinspace U_{\mu}^{--},$ (6.1)

or, in the mass basis,

$\mathcal{L}_{U\ell\ell}=\sum_{a,b}g_{U}\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}(V_{U})_{ab}\ell_{b}\thinspace
U_{\mu}^{++}+g_{U}\bar{\ell_{a}}\gamma^{\mu}P_{L}(V_{U}^{\dagger})_{ab}\ell_{b}^{c}\thinspace
U_{\mu}^{--}.$ (6.2)

Now, as an example, inspect the $U^{++}$ part of the interaction above.
Additionally, isolate the terms which pertain to the $Ue\mu$ interaction,
which read111For now we will be specially clear regarding the order in which
operations should be performed, e.g., in $\overline{e^{c}}$ one first takes
the charge conjugate and then the Lorentz conjugate.

$\mathcal{L}_{Ue\mu}=g_{U}\left[\overline{e^{c}}\gamma^{\mu}P_{L}(V_{U})_{e\mu}\mu+\overline{\mu^{c}}\gamma^{\mu}P_{L}(V_{U})_{\mu
e}e\right]U_{\mu}^{++}.$ (6.3)

Unlike in any interaction which conserves lepton number222Depending on the
scalar potential of the theory, it is still possible to assign a lepton number
to the entire representation content of the theory in such a manner to keep
the theory invariant under this generalized $L$., both terms in Eq. (6.3)
feature the same degrees of freedom, i.e., contain operators which annihilate
(and create) the same states. This means that both contribute to the vertex,
and in order for the rules to be actually derived, it is convenient to put the
two spinor chains into the same form. To perform these transformations, let us
start decomposing Lagrangian (6.2) into diagonal and non-diagonal parts

$\mathcal{L}_{U\ell\ell}=\sum_{a\neq
b}g_{U}\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}(V_{U})_{ab}\ell_{b}\thinspace
U_{\mu}^{++}+\sum_{a}g_{U}\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}(V_{U})_{aa}\ell_{a}\thinspace
U_{\mu}^{++}+\mathrm{H.C.},$ (6.4)

and proceed to treat the non-diagonal portion. For each $(a,b)$ pair, an
ansatz is needed regarding the order in which the particles appear, and we
chose to set the heaviest particle in the end – that is, right – of the chain.
For this, it is enough to transpose each term of the pairs analogous to Eq.
(6.3) which does not obey this requirement. The transposition may be performed
without consequence as these terms are Lorentz scalars, and follows

$\begin{split}\bar{\ell_{b}^{c}}\gamma_{\mu}P_{L}(V_{U})_{ba}\ell_{a}&=\left[\bar{\ell_{b}^{c}}\gamma_{\mu}P_{L}(V_{U})_{ba}\ell_{a}\right]^{T}\\\
&=-(V_{U})_{ba}\,\ell_{a}^{T}P_{L}^{T}\gamma_{\mu}^{T}\left(\ell_{b}^{T}C\right)^{T}\\\
&=-(V_{U})_{ba}\,\ell_{a}^{T}P_{L}^{T}\gamma_{\mu}^{T}C^{T}\ell_{b}\\\
&=-(V_{U})_{ba}\,\ell_{a}^{T}C^{-1}CP_{L}^{T}C^{-1}C\gamma_{\mu}^{T}C^{-1}\ell_{b}\\\
&=(V_{U})_{ba}\,\ell_{a}^{T}C^{-1}P_{L}\gamma_{\mu}\ell_{b}\\\
&=-(V_{U})_{ba}\bar{\ell_{a}^{c}}P_{L}\gamma_{\mu}\ell_{b}\\\
&=-(V_{U})_{ba}\bar{\ell_{a}^{c}}\gamma_{\mu}P_{R}\ell_{b},\end{split}$ (6.5)

where we have used $\overline{\ell_{b}^{c}}=\ell_{b}^{T}C$,
$C^{-1}\gamma_{\mu}C=-\gamma_{\mu}^{T}$, $C^{-1}\gamma_{5}C=\gamma_{5}^{T}$
and $C=C^{*}=-C^{T}=-C^{\dagger}=-C^{-1}$ (most of these facts are basis
independent, but some of the last equalities are specific to the Dirac
representation).Using this we may rewrite the most general $U\ell\ell$
interaction as

$\begin{split}\mathcal{L}_{U\ell\ell}&=\sum_{b>a}g_{U}\left\\{\bar{\ell_{a}^{c}}\gamma^{\mu}[P_{L}(V_{U})_{ab}-P_{R}(V_{U})_{ba}]\ell_{b}\thinspace
U_{\mu}^{++}+\bar{\ell_{a}}\gamma^{\mu}[P_{L}(V_{U}^{\dagger})_{ab}-P_{R}(V_{U}^{\dagger})_{ba}]\ell_{b}^{c}\thinspace
U_{\mu}^{--}\right\\}\\\
&+\sum_{a}g_{U}\left\\{\bar{\ell_{a}^{c}}\gamma^{\mu}[P_{L}(V_{U})_{aa}]\ell_{a}\thinspace
U_{\mu}^{++}+\bar{\ell_{a}}\gamma^{\mu}[P_{L}(V_{U}^{\dagger})_{aa}]\ell_{a}^{c}\thinspace
U_{\mu}^{--}\right\\},\end{split}$ (6.6)

where $a,b$ are generation labels. Not rarely, these manipulations cause some
confusion and the Lagrangian above is rarely achieved. Notice that a property
of the vector bilepton interaction is that the diagonal vertices are purely
axial:
$\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}\ell_{a}=-\bar{\ell_{a}^{c}}\gamma^{\mu}\frac{\gamma^{5}}{2}\ell_{a}$.

Now we turn to the doubly-charged scalar. While the previous interaction was
originated from the covariant derivative of an unknown higher symmetry, this
one is assumed to stem from Yukawa Lagrangians. We have as the most general
effective interaction

$\mathcal{L}_{Y\ell\ell}=-\sum_{a,b}g_{Y}\left\\{\bar{\ell^{c}_{a}}(\mathcal{O}_{Y})_{ab}P_{L}\ell_{b}\thinspace
Y^{++}+\bar{\ell_{a}}(\mathcal{O}_{Y}^{\dagger})_{ab}P_{R}\ell_{b}^{c}\thinspace
Y^{--}\right\\}.$ (6.7)

In the Lagrangian above, the fermions are already mass eigenstates and the
interaction mixing matrix is arbitrary: it is related to one a priori
(arbitrary) Yukawa matrix $G_{Y}$ as $\mathcal{O}_{Y}=V_{R}^{\ell
T}G_{Y}V^{\ell}_{L}$. Again, it is not necessary to add a second handedness
term. We perform the same transformations as in the $U\ell\ell$ Lagrangian to
arrange any term involving a like-sign pair $\ell_{a}-\ell_{b}$ of leptons
into a reference spinor chain. The Lagrangian in its most useful form is then
given by

$\begin{split}\mathcal{L}_{Y\ell\ell}=&-\sum_{b>a}g_{Y}\left\\{\bar{\ell^{c}_{a}}\left[(\mathcal{O}_{Y})_{ab}+(\mathcal{O}_{Y})_{ba}\right]P_{L}\ell_{b}\thinspace
Y^{++}+\bar{\ell_{a}}\left[(\mathcal{O}_{Y}^{\dagger})_{ab}+(\mathcal{O}_{Y}^{\dagger})_{ba}\right]P_{R}\ell_{b}^{c}\thinspace
Y^{--}\right\\}\\\
&-\sum_{a=b}g_{Y}\left\\{\bar{\ell^{c}_{a}}(\mathcal{O}_{Y})_{aa}P_{L}\ell_{a}\thinspace
Y^{++}+\bar{\ell_{a}}(\mathcal{O}_{Y}^{\dagger})_{aa}P_{R}\ell_{a}^{c}\thinspace
Y^{--}\right\\}.\end{split}$ (6.8)

The interaction that is missing is that of the neutral scalar with leptons.
Lorentz and electric charge invariance dictates it must be simply

$\begin{split}\mathcal{L}_{s\ell\ell}&=-g_{s}\bar{\ell}\mathcal{O}_{s}P_{L}\ell\thinspace
s-g_{sL}\bar{\ell}\mathcal{O}_{s}^{\dagger}P_{R}\ell\thinspace s\\\
&=-\sum_{a,b}g_{s}\bar{\ell_{a}}\left[(\mathcal{O}_{s})_{ab}P_{L}+(\mathcal{O}^{\dagger}_{s})_{ab}P_{R}\right]\ell_{b}\thinspace
s,\end{split}$ (6.9)

where $\mathcal{O}_{s}$ is arbitrary and related to a Yukawa matrix as
$\mathcal{O}_{s}=V_{R}^{\ell\dagger}G_{s}V^{\ell}_{L}$.

### 6.3 On the free parameters

In order for the free particles to be defined, our effective model must also
feature the kinetic terms

$\begin{split}\mathcal{L}_{\text{kin}}=&{}-\frac{1}{2}U_{\mu\nu}^{\dagger}U^{\mu\nu}+M_{U}^{2}U^{++}U^{--}\\\
&{}+\partial_{\mu}Y^{++}\partial^{\mu}Y^{--}-M_{Y}^{2}Y^{++}Y^{--}\\\
&{}+\frac{1}{2}\partial_{\mu}s\thinspace\partial^{\mu}s-\frac{1}{2}M_{s}^{2}\thinspace
s^{2},\end{split}$ (6.10)

where $U_{\mu\nu}=\partial_{\mu}U^{++}_{\nu}-\partial_{\nu}U^{++}_{\mu}$.
Three 2-particle Scenarios will be considered, each with a pair of exotic
species that interfere. The corresponding Lagrangians are

$\begin{split}\mathcal{L}_{U-s}&=\mathcal{L}_{\text{kin}}+\mathcal{L}_{U\ell\ell}+\mathcal{L}_{s\ell\ell}\\\
\mathcal{L}_{U-Y}&=\mathcal{L}_{\text{kin}}+\mathcal{L}_{U\ell\ell}+\mathcal{L}_{Y\ell\ell}\\\
\mathcal{L}_{Y-s}&=\mathcal{L}_{\text{kin}}+\mathcal{L}_{Y\ell\ell}+\mathcal{L}_{s\ell\ell}.\end{split}$
(6.11)

Now, notice that $g_{s}$ and $g_{Y}$ could be absorbed into their
corresponding mixing matrices, and although we write them explicitly on
analytical expressions (mostly for book keeping purposes), they will be
effectively set to $1$ in all numerical evaluations. Notice also, checking Eq.
(6.8), that any element of $\mathcal{O}_{Y}$ only appears together with its
symmetric partner, so that this effective mixing matrix may be taken
symmetric.

Since to effect numerical optimization with the number of free parameters that
exists when considering the general case is impractical, we considerably
reduce this number by restricting the analysis to real matrices. $V_{U}$ then
becomes an orthogonal matrix, whose determinant may be chosen to be $1$
without loss of generality, and which we parametrize with Euler Angles:

$V_{U}=\begin{pmatrix}\cos\psi\cos\phi-\cos\theta\sin\phi\sin\psi&\cos\psi\sin\phi+\cos\theta\cos\phi\sin\psi&\sin\theta\sin\psi\\\
-\sin\psi\cos\phi-\cos\theta\sin\phi\cos\psi&-\sin\psi\sin\phi+\cos\theta\cos\phi\cos\psi&\sin\theta\cos\psi\\\
\sin\theta\sin\phi&-\sin\theta\cos\phi&\cos\theta\end{pmatrix}.$ (6.12)

The number of free parameters in each Scenario, which includes masses and
degrees of freedom of the applicable matrices, may be checked to be as appears
in Table 6.2.

Table 6.2: Number of parametric degrees of freedom contained within each Scenario if the mixing matrices are regarded as complex and real. Scenario | Complex | Real
---|---|---
$U-Y$ | 23 | 11
$U-s$ | 29 | 14
$s-Y$ | 32 | 17

A few comments are now in order regarding the parametric structure of these
interactions and how it relates to our objectives. It is true that if the
elements of the 3 mixing matrices parametrizing the interaction Lagrangians
could be arbitrarily small, any experimental constraint could be easily met;
however, if these particles do exist (i) elements too small are not desirable
because of matters such as naturalness and; (ii) more importantly,
orthogonality of the $V_{U}$ matrix is powerful in inducing exclusion
contours.

Lastly, we should clarify the role of this matrix. Considering now the
complete unitary case and referring to its definition $V_{U}\equiv
V_{R}^{T}V_{L}$, there are two situations in which this mixing can be ignored
in a natural way: (i) If $V_{R}$ and $V_{L}$ could be set to $\mathbb{1}$.
This can occur whenever the lepton mass matrix and every leptonic interaction
can be simultaneously diagonalized, which is not general and relates to a
small part of theory space. In fact, it is not the case of the m331, where the
mass matrix receives a couple of different contributions. (ii) A more general
possibility is $V_{R}=V_{L}^{*}$. This implies that the mass matrix is
diagonalized by an orthogonal transformation instead of by a biunitary one. In
our special case of $V_{U}$ orthogonal, the condition becomes $V_{R}=V_{L}$. A
squared mass matrix diagonalizable by a transformation of this type is
symmetric and, therefore, not general. We consider a non-diagonal orthogonal
$V_{U}$, which, apart from the missing phases, should be consistent with the
general case.

### 6.4 Feynman rules and diagrams from Lagrangians with explicit charge
conjugation

To correctly derive amplitudes from Lagrangians with explicit charge
conjugation can be troublesome. The issue arises because when these fields
make up the interaction there is generally more than a way to contract a
spinor chain with initial and final states. In this case, simply writing the
vertices with an explicit charge conjugation matrix is not by itself a well
defined and unambiguous process. Therefore, we briefly discuss how to arrive
at the amplitudes corresponding to the doubly-charged vector and scalar boson
mediation, which suffer from this complication. We follow the algorithm and
refer to the description of Refs. [176, 177], but focus on the matter of
dealing with Lagrangians with explicit charge conjugation, in the form as
would naturally emerge from a renormalizable fundamental gauge theory, and
strive to be didactic regarding the procedure instead of focusing on why this
algorithm works.

We begin defining how to write down the spinor structure. Each spinor line in
a diagram will come together with an Arbitrary Fermion Flow Arrow (AFFA) –
recall that the true fermion flow is not continuous in this type of graph.
With reference to this arbitrarily drawn line and the true fermion flow arrow,
the rules for external fermion lines are

Now to read the vertices off of the Lagrangians (6.6) and (6.8). Considering
always incoming bosons, we have a first set of vertices, corresponding to the
case in which the AFFA ends on the heaviest fermion, for each charge of the
boson:

$\includegraphics[height=65.44142pt,valign={c}]{06-diagrama.pdf}\equiv\Gamma^{U^{++}}_{ba}=g_{U}\gamma^{\mu}[P_{L}(V_{U})_{ab}-P_{R}(V_{U})_{ba}]$
(6.13)

$\includegraphics[height=65.44142pt,valign={c}]{08-diagrama.pdf}\equiv\Gamma^{U^{--}}_{ba}=g_{U}\gamma^{\mu}[P_{L}(V_{U}^{\dagger})_{ab}-P_{R}(V_{U}^{\dagger})_{ba}],$
(6.14)

The formulas above are valid even when $a=b$, which can be seen symmetrizing
the diagonal part of Lagrangian (6.6) as
$\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}\ell_{a}=\frac{1}{2}\left[\bar{\ell_{a}^{c}}\gamma^{\mu}P_{L}\ell_{a}-\bar{\ell_{a}^{c}}\gamma^{\mu}P_{R}\ell_{a}\right]$.
333Notice again that the vector part of this interaction dies out.The
remaining relative factor of $1/2$ is compensated in the rule by a factor of
$2$ due to the identical particles. These vertices are called regular.

The seemingly innocuous choice of leaving the heaviest fermion on the right in
the Lagrangians made in Section 6.2 is what leads to the definition of the
vertices above as the regular ones.

There is a second set of vertices for the $U^{\pm\pm}$, corresponding to
graphs with the AFFA ending on the lightest lepton. The rule is obtained
conjugating the original vertex by the charge conjugation matrix like
$\Gamma^{\prime}=C\Gamma C^{-1}$ – this recipe comes directly by transposition
and manipulation of the reference spinor chain. In our case, this calculation
gives

$C\gamma^{\mu}[P_{L}(V_{U})_{ab}-P_{R}(V_{U})_{ba}]C^{-1}=\gamma^{\mu}[P_{L}(V_{U})_{ba}-P_{R}(V_{U})_{ab}],$
(6.15)

so that the new vertex rule is $\Gamma^{\prime}_{ab}=\Gamma_{ba}$. For
completeness, we write the reversed vertices below (recall that we chose
$\ell_{b}$ to symbolize the heaviest of the 2 leptons)

$\includegraphics[height=65.44142pt,valign={c}]{07-diagrama.pdf}=\Gamma^{\prime
U^{++}}_{ab}=ig_{U}\gamma^{\mu}[P_{L}(V_{U})_{ba}-P_{R}(V_{U})_{ab}]$ (6.16)

$\includegraphics[height=65.44142pt,valign={c}]{09-diagrama.pdf}=\Gamma^{\prime
U^{++}}_{ab}=ig_{U}\gamma^{\mu}[P_{L}(V_{U}^{\dagger})_{ba}-P_{R}(V_{U}^{\dagger})_{ab}].$
(6.17)

As an example up to this point, we write the rule corresponding to the two
different choices of AFFA for the subdiagrams (and not vertex representations)
below

$\includegraphics[height=65.44142pt,valign={c}]{14-diagrama.pdf}=\bar{v}_{\mu}\Gamma^{U^{--}}_{\mu
e}u_{e}$ (6.18)

$\includegraphics[height=65.44142pt,valign={c}]{15-diagrama.pdf}=\bar{v}_{e}\Gamma^{\prime
U^{--}}_{e\mu}u_{\mu}.$ (6.19)

The complete set of vertices of the doubly-charged scalar read

$\includegraphics[height=65.44142pt,valign={c}]{10-diagrama.pdf}=\Gamma^{Y^{++}}_{ba}=-ig_{Y}\left[(\mathcal{O}_{Y})_{ab}+(\mathcal{O}_{Y})_{ba}\right]P_{L}$
(6.20)

$\includegraphics[height=65.44142pt,valign={c}]{12-diagrama.pdf}=\Gamma^{Y^{--}}_{ba}=-ig_{Y}\left[(\mathcal{O}_{Y}^{\dagger})_{ab}+(\mathcal{O}_{Y}^{\dagger})_{ba}\right]P_{R}$
(6.21)

$\includegraphics[height=65.44142pt,valign={c}]{11-diagrama.pdf}=\Gamma^{\prime
Y^{++}}_{ab}=-ig_{Y}\left[(\mathcal{O}_{Y})_{ba}+(\mathcal{O}_{Y})_{ab}\right]P_{L}$
(6.22)

$\includegraphics[height=65.44142pt,valign={c}]{13-diagrama.pdf}=\Gamma^{\prime
Y^{--}}_{ab}=-ig_{Y}\left[(\mathcal{O}_{Y}^{\dagger})_{ba}+(\mathcal{O}_{Y}^{\dagger})_{ab}\right]P_{R}.$
(6.23)

We emphasize one last time that what defines if a vertex is regular or
reversed is the direction of the AFFA with respect to fermion generation –
which, in turn, is a consequence of the conventional form of the Lagrangian.

With the vertices and how to write the exotic spinor chains now understood,
the missing ingredient is the ability to find the relative sign between
diagrams. This is the greatest reason for the necessity of an algorithm that
substitutes the mere explicit use of the charge conjugation matrix. Within the
algorithm, to find the relative signs amounts to simply comparing particle
“order” – more precisely, the order in which spinors appear in the chain –
with respect to the AFFA and identifying the order of the relating
permutation. For a direct example, refer to our real diagrams of Figure 6.2.
The particle orders are (we label different particles by the momenta)

$\begin{split}R(\mathcal{M}_{U})&=(p,k_{3},k_{1},k_{2})\\\
R(\mathcal{M}_{Y})&=(p,k_{3},k_{1},k_{2})\\\
R(\mathcal{M}_{s1})&=(p,k_{1},k_{3},k_{2})\\\
R(\mathcal{M}_{s2})&=(p,k_{2},k_{3},k_{1}).\end{split}$ (6.24)

One may identify that the only ordered set related to the referential
$R(\mathcal{M}_{U})$ by an odd permutation is $R(\mathcal{M}_{s1})$, so that
this amplitude comes attached to an extra minus sign.

This concludes a sufficient description of how our amplitudes can be obtained
from the given Lagrangians without having to appeal to an explicit analysis of
the possible Wick contractions involved in the correlator.

### 6.5 Amplitudes, phase space and method of evaluation

Refer to Figure 6.2 for the diagrams that contribute to
$\ell^{+}_{l}(p)\to\ell^{+}_{i}(k_{1})\ell^{+}_{j}(k_{2})\ell^{-}_{k}(k_{3})$
at tree level. Following the rules and algorithm explained above, it is easy
to write down the amplitudes:

$\begin{split}i\mathcal{M}_{U}=\left(ig_{U}\right)^{2}&\bar{v}_{\ell_{l}}(p)\gamma^{\mu}(V_{Ukl}P_{L}-V_{Ulk}P_{R})v_{\ell_{k}}(k_{3})\frac{-ig_{\mu\nu}}{(k_{1}+k_{2})^{2}-M_{U}^{2}}\\\
\times&\bar{u}_{\ell_{i}}(k_{1})\gamma^{\nu}(V_{Uij}P_{L}-V_{Uji}P_{R})v_{\ell_{j}}(k_{2})\end{split}$
(6.25)

$\begin{split}i\mathcal{M}_{Y}=\left(-ig_{Y}\right)^{2}&\bar{v}_{\ell_{l}}(p)(\mathcal{O}_{Ylk}+\mathcal{O}_{Ykl})P_{R}v_{\ell_{k}}(k_{3})\frac{i}{(k_{1}+k_{2})^{2}-M_{Y}^{2}}\\\
\times&\bar{u}_{\ell_{i}}(k_{1})(\mathcal{O}_{Yij}+\mathcal{O}_{Yji})P_{L}v_{\ell_{j}}(k_{2})\\\
\end{split}$ (6.26)

$\begin{split}i\mathcal{M}_{s1}=(-1)\left(-ig_{s}\right)^{2}&\bar{v}_{\ell_{l}}(p)(\mathcal{O}_{sli}P_{L}+\mathcal{O}_{sil}P_{R})v_{\ell_{i}}(k_{1})\frac{i}{(k_{2}+k_{3})^{2}-M_{s}^{2}}\\\
\times&\bar{u}_{\ell_{k}}(k_{3})(\mathcal{O}_{skj}P_{L}+\mathcal{O}_{sjk}P_{R})v_{\ell_{j}}(k_{2})\\\
\end{split}$ (6.27)

$\begin{split}i\mathcal{M}_{s2}=\left(-ig_{s}\right)^{2}&\bar{v}_{\ell_{l}}(p)(\mathcal{O}_{slj}P_{L}+\mathcal{O}_{sjl}P_{R})v_{\ell_{j}}(k_{2})\frac{i}{(k_{1}+k_{3})^{2}-M_{s}^{2}}\\\
\times&\bar{u}_{\ell_{k}}(k_{3})(\mathcal{O}_{ski}P_{L}+\mathcal{O}_{sik}P_{R})v_{\ell_{i}}(k_{1}).\\\
\end{split}$ (6.28)

Nevertheless, we check the amplitudes above by generating them through
FeynRules [178, 179] in association with the FeynArts [180] and FeynCalc
package [181, 182].

Figure 6.3: Simple representation of the 3-body decay kinematics in the rest
frame of the parent system. The labels represent the 3-momentum and mass of
the particle, respectively.

The kinematics of the final state with three particles is immensely more
complicated than the more usual $2\to 2$ phase space, and we make an aside to
briefly discuss it. For a standard assessment of the usual parametrization,
corresponding to the theory of Dalitz plots, see, for instance, the PDG [72] –
but let us try to find a more intuitive route to describe these quantities.
Our goal is to understand the kinematics in order to write the invariant phase
space element of the final state in a $1\to 3$ process (refer to Figure 6.3).
The first step is to favour one of the three end state particles and find the
absolute allowed range for its energy. Focusing on particle 1, the minimum
allowed energy is clearly obtained when this (massive) body is at rest, giving

$E_{1}^{\mathrm{min}}=m_{1}.$ (6.29)

Now, energy conservation enforces $M=E_{1}+E_{2}+E_{3}$, and momentum
conservation implies $\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}}=0$. The
analysis is facilitated by replacing particles 2 and 3 by a single system $Q$
with mass $m_{Q}$ and momentum $\mathbf{k_{Q}}=-\mathbf{k_{1}}$. The maximum
$\mathbf{k_{1}}$ (which corresponds to maximum $E_{1}$) is attained at minimum
$m_{Q}$. Since this mass is invariant, we may inspect its inequivalent
configurations (which corresponds to inequivalent $k_{2},k_{3}$
configurations) in the $Q$ rest frame. In this system, it becomes clear that
the smallest energy (corresponding to smallest $m_{Q}$) is $m_{Q}=m_{2}+m_{3}$
and occurs for particles 2 and 3 at rest. In turn, in the parent rest frame
this corresponds to both particles moving at the same velocity and opposite
direction to particle 1. With this information, energy conservation gives

$\sqrt{m_{1}^{2}+\left\lvert\mathbf{k_{1}}^{\mathrm{max}}\right\rvert^{2}}+\sqrt{(m_{2}+m_{3})^{2}+\left\lvert\mathbf{k_{1}}^{\mathrm{max}}\right\rvert^{2}}=M,$
(6.30)

which, after some algebra, delivers

$E_{1}^{\mathrm{max}}=\frac{M^{2}+m_{1}^{2}-(m_{2}+m_{3})^{2}}{2M}.$ (6.31)

There is still one energy degree of freedom to be integrated over. This time
we only need to find the limits on $E_{2}$ for a fixed $E_{1}$. Obviously
enough, the minimum (maximum) $E_{2}$ corresponds to $k_{3}$ parallel (anti-
parallel) to $k_{2}$. The conservation laws then give

$\sqrt{E_{2}^{*2}-m_{2}^{2}}=\sqrt{E_{1}^{2}-m_{1}^{2}}\pm\sqrt{(M-E_{1}-E_{2}^{*})^{2}-m_{3}^{2}},$
(6.32)

where $E_{2}^{*}=E_{2}^{\mathrm{max(min)}}$ for the plus (minus) sign. Now, by
conformity, define the usual energy fractions $x\equiv\frac{2E_{1}}{M}$,
$y\equiv\frac{2E_{2}}{M}$ and $z\equiv\frac{2E_{3}}{M}$, and the squared mass
ratios $\xi_{i}=m_{i}^{2}/M^{2}$. It is easy to show that the positive
solution of each form of Eq. (6.32) is also a solution to the quadratic
equation

$(1-x+\xi_{1})y^{2}+[x(2+\lambda-x)-2\lambda]y+x(x+\xi_{2}x-2\lambda)+\lambda^{2}-4\xi_{1}\xi_{2}=0.$
(6.33)

This can now be solved directly and we find

$\begin{split}y^{\mathrm{min}}(x)&=\frac{(2-x)(r_{1}+\xi_{2}-\xi_{3})-\sqrt{x^{2}-4\xi_{1}}\sqrt{\lambda(r_{1},\xi_{2},\xi_{3})}}{2r_{1}}\\\
y^{\mathrm{max}}(x)&=\frac{(2-x)(r_{1}+\xi_{2}-\xi_{3})+\sqrt{x^{2}-4\xi_{1}}\sqrt{\lambda(r_{1},\xi_{2},\xi_{3})}}{2r_{1}},\end{split}$
(6.34)

where we have defined $r_{1}\equiv 1+\xi_{1}-x$ and $\lambda$ is the Källén
function, $\lambda(x,y,z)=(x-y-z)^{2}-4yz$.

Coming back now to the configuration
$\ell^{+}_{l}(p)\to\ell^{+}_{i}(k_{1})\ell^{+}_{j}(k_{2})\ell^{-}_{k}(k_{3})$,
the angular inclusive differential partial width may be written in terms of
the invariant phase space as

$\begin{split}d\Gamma(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})&=\frac{1}{64\pi^{3}m_{l}}|\overline{\mathcal{M}}(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})|^{2}dE_{1}\,dE_{1}\\\
&=\frac{m_{l}}{256\pi^{3}}|\overline{\mathcal{M}}(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})|^{2}dxdy.\end{split}$
(6.35)

The final inclusive width is obtained as

$\Gamma(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})=\frac{m_{l}}{256\pi^{3}}\int_{x^{\mathrm{min}}}^{x^{\mathrm{max}}}dx\int_{y^{\mathrm{min}}}^{y^{\mathrm{max}}}|\overline{\mathcal{M}}(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})|^{2}\,dy.$
(6.36)

For completeness, we show the expressions for all the invariants in terms of
our parametrization444We associate the labels as $i\leftrightarrow 1$,
$j\leftrightarrow 2$ and $k\leftrightarrow 3$.

$\begin{split}k_{1}\cdot
k_{2}&=\frac{m_{l}^{2}}{2}(-1+x+y-\xi_{1}-\xi_{2}+\xi_{3})\\\ k_{1}\cdot
k_{3}&=\frac{m_{l}^{2}}{2}(1-y-\xi_{1}+\xi_{2}+\xi_{3})\\\ k_{2}\cdot
k_{3}&=\frac{m_{l}^{2}}{2}(1-x+\xi_{1}-\xi_{2}-\xi_{3}).\end{split}$ (6.37)

Now, we are going to perform constrained optimization in order to find the
best points in parameter space, which correspond to those which allow the
smallest $M_{U}$, $M_{s}$ and $M_{Y}$ – i.e., those which do not exclude light
particles. This procedure is quite demanding computationally, and to render it
feasible we require an analytic expression for each BR. However, because of
the complicated $1\to 3$ phase space,
$|\overline{\mathcal{M}}(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})|^{2}$
is an enormous rational expression in terms of $x,y$. The conclusion is that
some approximations are required, and we describe the operational method
including them below.

The hierarchy between mass scales in the processes we consider is

$M_{U},M_{s},M_{Y}\gg m_{\tau}\gg m_{\mu}\gg m_{e}.$ (6.38)

The denominators of the squared matrix elements are of the form
$M_{1}^{2}-m_{\ell}^{2}f(x,y)$, where $M_{1}$ is any of the heavy boson masses
and $m_{\ell}$ any of the lepton ones, and $f(x,y)$ is a small polynomial on
the energy fractions. This hints that we could start ignoring the lepton
masses on natural denominators of
$|\overline{\mathcal{M}}(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})|^{2}$.
The integration in $y$ is performed with this simplified form of
$\mathcal{M}$:

$|\overline{\mathcal{M}}|_{y}^{2}\equiv\int_{y^{\mathrm{min}}}^{y^{\mathrm{max}}}|\overline{\mathcal{M}}(\ell^{+}_{l}\to\ell^{+}_{i}\ell^{+}_{j}\ell^{-}_{k})|^{2}\,dy.$
(6.39)

Because of the complicated dependence of $y^{\mathrm{min}(\mathrm{max})}$ on
$x$, this is a rational expression whose subsequent $x$ integration cannot be
effected with ease. To remedy this situation we expand
$|\overline{\mathcal{M}}|_{y}^{2}$ around $x=0.5$ up to third order

$\begin{split}|\overline{\mathcal{M}}|_{y}^{2}(x)&=A_{0}(M_{1},m_{\ell})+A_{1}(M_{1},m_{\ell})(x-0.5)+A_{2}(M_{1},m_{\ell})(x-0.5)^{2}\\\
&\quad+A_{3}(M_{1},m_{\ell})(x-0.5)^{3}+O\left((x-0.5)^{4}\right).\end{split}$
(6.40)

With this, the $x$ integration may be analytically performed. To validate the
approximation above, we compare its results with those obtained by numerically
performing the integration

$\int_{x^{\mathrm{min}}}^{x^{\mathrm{max}}}|\overline{\mathcal{M}}|_{y}^{2}(x)\,dx,$
(6.41)

with the exact $|\overline{\mathcal{M}}|_{y}^{2}(x)$. This comparison is made
for 100 points strategically spread across parameters space, and lead us to
conclude that the approximation may be safely maintained without any
repercussions.

##  7 Model independent constraints on exotic particles from flavour violating
lepton decays: Results

Finally, in this chapter we present the results for each 2-particle model,
dubbed Scenario. The ranges for the non-massive parameters that are allowed in
principle are

$\begin{split}0&\leq\phi,\,\psi<2\pi\\\ 0&\leq\theta<\pi\\\
-1&<\mathcal{O}_{Yij},\,\mathcal{O}_{sij}<1.\end{split}$ (7.1)

These are the limits that exhaust the $V_{U}$ space and keep the scalar
interactions perturbative. As for the masses, we investigate

$M_{U},M_{Y},M_{s}>500.$ (7.2)

This is enough because smaller masses are extremely unlikely (to say the
least) and their analysis would not change the qualitative aspects of the
study.

Our objective is to find a solution, i.e., a value for each of the free
parameters (except masses) that allow the smallest possible masses to be
phenomenologically possible. In the Scenarios where the $U$ is present, we
search for solutions that prioritize its mass – i.e., we seek sets of numbers
which minimize $M_{U}$, with every other parameter, including $M_{s},M_{Y}$,
free. We consider additional benchmark conditions to fix the lower bound on
the modulus of matrix elements and/or impositions on the diagonal couplings of
the mixing matrices. Very restrictive, these constraints are designed to check
what are the lowest possible masses depending on of the level of naturalness
of the model that correctly describes nature.

(a) $|V_{Uij}|>10^{-3}$

(b) $|V_{Uij}|>10^{-4}$

(c) $|V_{Uij}|>10^{-5}$

(d) $|V_{Uij}|>10^{-5}$ and $V_{11}=0.85$

Figure 7.1: Exclusion ranges for $M_{U}$ generated by the experimental bounds
on the various branching rations for leptonic decays, for each of our 4
benchmark cases. The allowed region is painted green.

The optimal points in the multi-dimensional parameters spaces are found
through a numerical constrained global optimization routine, and the stability
of the results are checked through its repetition with $10^{2}$ random seeds.

### 7.1 Pure $U$ Scenario

In order to set the stage for the 2-particle analysis, we begin examining the
constraints resulting from a doubly-charged vector bilepton alone. The results
are presented in Figure 7.1 and the solutions in Table 7.1. We show the
contours in the $M_{U}\times M_{s}$ plane (even though there is no $M_{s}$
dependence) to facilitate comparison with the Scenario below.

As mentioned, we seek for solutions which minimize the allowed $M_{U}$ with
additional requirements on the absolute values of every matrix element. These
are meant to limit hierarchies and force interesting solutions to be kept over
strongly diagonal ones. In the most liberal case, we allow for matrix elements
as small as $10^{-5}$, corresponding to a hierarchy already as large as that
of the SM quark sector. Observing the results, we recognize that to allow for
masses of the order of $1100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ (see Fig.
7.1(b)) we need a hierarchy111Note that, in an orthogonal matrix, small
elements imply the need for large ones. within $V_{U}$ of four orders of
magnitude, such that we approach a non-natural parameter regime. But the
greatest feature to observe is that with small general hierarchy (Fig. 7.1(a))
or with large but not maximal diagonal coupling (Fig. 7.1(d)) the constraints
are strong, demanding $M_{U}>$3200\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
and $M_{U}>$6900\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, respectively. It
must be recognized that, in each instance, the contours result from a
specific, not always evident, interplay between one or various BR bounds and
the unitarity conditions of $V_{U}$.

Table 7.1: Solutions of the pure $U$ Scenario, corresponding to the plots of
Fig. 7.1. Pure $U$: $\;|V_{Uij}|>10^{-3}$
---
$M_{U}$ |  $3380$ |  $\psi$ |  $1.48108$
$\phi$ |  $3.03983$ |  $\theta$ |  $2.66989$
Pure $U$: $\;|V_{Uij}|>10^{-4}$
$M_{U}$ |  $1100$ |  $\psi$ |  $0.78535$
$\phi$ |  $5.49774$ |  $\theta$ |  $0.00014$
Pure $U$: $\;|V_{Uij}|>10^{-5}$
$M_{U}$ |  $500$ |  $\psi$ |  $0.72066$
$\phi$ |  $0.72067$ |  $\theta$ |  $3.13801$
Pure $U$: $\;|V_{Uij}|>10^{-5}$, $V_{U11}=0.85$
$M_{U}$ |  $6830$ |  $\psi$ |  $4.74012$
$\phi$ |  $4.68301$ |  $\theta$ |  $0.55504$

An important remark is that the $V_{U}=\mathbb{1}$ choice made in the
phenomenological literature reviewed before may correspond, here, to the case
of high hierarchy $|V_{Uij}|>10^{-5}$. This claim is not necessarily true, and
to confirm that the solution to this benchmark imposition can be compared to
the results of those studies, it must be checked that it is of the form
$V_{U}\sim\mathbb{1}$ – which is indeed the case. Now, the strongest bounds
from the diagonal LHC literature are capable of excluding bilepton masses
$M_{U}\lesssim$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. Our conclusion is
that for the sector of theory space with a $V_{U}$ hierarchy of $10^{4}$ or
lower, the CLFV lepton decays bounds should be considered, while for the
flavour conserving sector, numerically and casually equivalent to tolerant
hierarchies of $10^{5}$ or higher, the more energetic LHC phenomenology should
be more appropriate.

Notice that the argument above is general: for not exceedingly low, improbable
masses, the case of no mixing, $V_{U}=\mathbb{1}$ (which can be contained in a
natural way within a theory as discussed in Sec. 6.3) in which our processes
do not occur at all, is represented by the most permissive case of high
hierarchy $|V_{Uij}|>10^{-5}$.

Figure 7.2: Mnemonic device which quickly shows that this Scenario features
interference. Time flows from left to right and from top to bottom. The first
step is choosing a handedness arrangement for the initial state, and we look
at $\mu^{-}_{L}\to e^{-}_{L}e^{-}_{R}e^{+}_{L}$ (notice that, without any
harm, we look at a negative decaying muon now). In the calculation of the
squared matrix element, one of the amplitudes must be conjugated – here, we
choose the $s$ contribution. We then take advantage of $\mathit{CPT}$
invariance to transform the graphs in order to assure that one of them has
only outgoing and the other only incoming lines. Finally, we attempt to join
corresponding lines. This is possible for the $U$ and $s$ contributions, hence
interference is present (the point in which lines are joined is highlighted by
a red blob – the lines do not touch at other crossings).

### 7.2 $U-s$ Scenario

The Scenario which postulates a model that contains a vector bilepton
$U^{\pm\pm}$ and a neutral scalar $s$ is the most interesting one because it
features interference and the influence of the unitary $V_{U}$. As a shortcut
to realize that interference is present, we may employ the mnemonic device of
checking if two diagrams can be ‘glued’ together – see Figure 7.2.

(a) $|O_{sij},V_{Uij}|>10^{-3}$

(b) $|O_{sij},V_{Uij}|>10^{-4}$

(c) $|O_{sij},V_{Uij}|>10^{-5}$

(d) $|O_{sij},V_{Uij}|>10^{-5}$ and $O_{s11}=O_{s22}=1$

Figure 7.3: Exclusion contours on the $M_{U}\times M_{s}$ plane generated by
the bounds on 3-body Lepton Flavour Violating decays. The allowed region is
painted green. Table 7.2: Solutions of the $U-s$ Scenario, respective of the
plots shown in Fig. 7.3 $U-s$: $\;|V_{Uij},\mathcal{O}_{sij}|>10^{-3}$
---
$M_{U}$ |  $2650$ |  $\mathcal{O}_{s11}$ |  $3.11564\times 10^{-3}$ |  $\mathcal{O}_{s22}$ | $1.43763\times 10^{-3}$
$M_{s}$ |  $500$ |  $\mathcal{O}_{s12}$ |  $3.01898\times 10^{-3}$ |  $\mathcal{O}_{s23}$ | $4.06257\times 10^{-2}$
$\phi$ |  $6.26303$ |  $\mathcal{O}_{s13}$ |  $4.20596\times 10^{-2}$ |  $\mathcal{O}_{s31}$ | $-1.32129\times 10^{-1}$
$\psi$ |  $1.55218$ |  $\mathcal{O}_{s21}$ |  $3.19852\times 10^{-3}$ |  $\mathcal{O}_{s32}$ | $1.51094\times 10^{-1}$
$\theta$ |  $2.90919$ |  |  |  |
$U-s$: $\;|V_{Uij},\mathcal{O}_{sij}|>10^{-4}$
$M_{U}$ |  $840$ |  $\mathcal{O}_{s11}$ |  $\hphantom{-}2.57667\times 10^{-3}$ |  $\mathcal{O}_{s22}$ | $-2.14209\times 10^{-3}$
$M_{s}$ |  $500$ |  $\mathcal{O}_{s12}$ |  $-3.29400\times 10^{-3}$ |  $\mathcal{O}_{s23}$ | $-6.05650\times 10^{-2}$
$\phi$ |  $1.45901$ |  $\mathcal{O}_{s13}$ |  $\hphantom{-}4.05147\times 10^{-1}$ |  $\mathcal{O}_{s31}$ | $-4.81308\times 10^{-2}$
$\psi$ |  $1.45911$ |  $\mathcal{O}_{s21}$ |  $-3.34363\times 10^{-3}$ |  $\mathcal{O}_{s32}$ | $-1.44602\times 10^{-1}$
$\theta$ |  $3.13998$ |  |  |  |
$U-s$: $\;|V_{Uij},\mathcal{O}_{sij}|>10^{-5}$
$M_{U}$ |  $<500$ |  $\mathcal{O}_{s11}$ |  $1.00000\times 10^{-5}$ |  $\mathcal{O}_{s22}$ | $1.00000\times 10^{-5}$
$M_{s}$ |  $<500$ |  $\mathcal{O}_{s12}$ |  $1.00000\times 10^{-5}$ |  $\mathcal{O}_{s23}$ | $1.00000\times 10^{-5}$
$\phi$ |  $0.72067$ |  $\mathcal{O}_{s13}$ |  $1.00000\times 10^{-5}$ |  $\mathcal{O}_{s31}$ | $1.00000\times 10^{-5}$
$\psi$ |  $0.72066$ |  $\mathcal{O}_{s21}$ |  $1.00000\times 10^{-5}$ |  $\mathcal{O}_{s32}$ | $1.00000\times 10^{-5}$
$\theta$ |  $3.13801$ |  |  |  |
$U-s$: $\;|V_{Uij},\mathcal{O}_{sij}|>10^{-5}$,
$\mathcal{O}_{s11}=\mathcal{O}_{s22}=1$
$M_{U}$ |  $1800$ |  $\mathcal{O}_{s11}$ |  $1.00000$ |  $\mathcal{O}_{s22}$ | $1.00000$
$M_{s}$ |  $580$ |  $\mathcal{O}_{s12}$ |  $-1.12685\times 10^{-5}$ |  $\mathcal{O}_{s23}$ | $-8.60549\times 10^{-4}$
$\phi$ |  $0.00020$ |  $\mathcal{O}_{s13}$ |  $-1.19611\times 10^{-3}$ |  $\mathcal{O}_{s31}$ | $1.83141\times 10^{-4}$
$\psi$ |  $0.00024$ |  $\mathcal{O}_{s21}$ |  $-1.13022\times 10^{-5}$ |  $\mathcal{O}_{s32}$ | $2.55491\times 10^{-3}$
$\theta$ |  $3.09077$ |  |  |  |

The exclusion contours are shown in Figure 7.3 and the solutions appear on
Table 7.2. From the contours, we learn that to allow for bilepton masses of
the order of $M_{U}<$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, at least some
effective couplings $g_{\text{eff}}\sim g_{U}V_{Uij}$ must be set as low as
$<10^{-4}$, meanwhile the entire parameter space is possible if the matrix
elements are allowed to become as small as $\sim 10^{-5}$, showing, again, the
complementarity between the phenomenology of CLFV decays and LHC processes,
which cover the non-diagonal and diagonal $V_{U}$ models, respectively.
Additionally, Figure 7.3(d) shows that if
$\mathcal{O}_{s11}=\mathcal{O}_{s22}=1$, is enforced the bound on $M_{U}$ is
strengthened from $M_{U}>$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ to
$M_{U}>$1500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ while virtually
unchanging the bound on $M_{s}$. This just reasserts that the vector
contribution is indeed dominant.

We would also like to answer the question of weather destructive interference
can save interesting sectors of theory space (with light exotic vector
particles) from exclusion. In practice, this question is translated into the
need to investigate how the presence of a second particle may relieve naive
constraints derived from single exotic particle Lagrangians. For this we show,
in Figure 7.4, a density plot of the ratio

$\frac{\text{BR}_{U-s}+\text{BR}_{s}}{\text{BR}_{U}},$ (7.3)

where $\text{BR}_{X-Y}$ is the contribution of the interference between $X$
and $Y$ and $\text{BR}_{X}$ is the pure $X$ contribution to the BR. We notice
that, even if the scalar contribution is significantly smaller, it allows the
solution to enhance destructive interference, which causes the distortion on
the inferior left corner of the contour and contributes to, in the
$|O_{sij},V_{Uij}|>10^{-3}$ case, rendering constraints softer by 19% on
$M_{U}$.

Figure 7.4: Density plot of the extra contributions to $\mu^{+}\to
e^{+}e^{-}e^{+}$ caused by the addition of the neutral scalar $s$ to a model
with the doubly-charged vector bilepton $U^{\pm\pm}$.

### 7.3 $U-Y$ Scenario

The Scenario with a doubly-charged vector and scalar bileptons is much simpler
as a consequence of the absence of relevant interference. The contributions of
interference between the $U^{\pm\pm}$ and the $Y^{\pm\pm}$ to this averaged
fermionic process is proportional to the lepton masses (see Fig. 7.5). In
particular, this means that interference effects are absolutely negligible in
the formation of the most powerful bound from $\mu\to 3e$. This indicates that
possible solutions for this Scenario involve vector bilepton parameters
identical to those of the pure $U^{\pm\pm}$ Scenario with $Y^{\pm\pm}$-related
parameters very small in modulus – the least allowed by the benchmark
conditions. This guarantees that the $Y^{\pm\pm}$ contribution is rendered
insignificant and does not affect the exclusion contour, turned similar to
those of Figure 7.1.

Figure 7.5: Verification of the interference structure between the $U$ and $Y$
contributions. Now, the diagrams can only be joined together through $e$ mass
terms. Figure 7.6: Exclusion contour on the $M_{U}\times M_{Y}$ plane, showing
that the solution subjected to the $|O_{Yij},V_{Uij}|>10^{-3}$ condition,
prioritizing $M_{U}$, is analogous to the one of the pure $U^{\pm\pm}$ case,
with negligible scalar contributions (except for the distortion at exceedingly
low $M_{s}$).

To illustrate the point above, we show, in Figure 7.6 the plot corresponding
to the solution of Fig. 7.1(a) together with $O_{Yij}=10^{-3}$. The results
confirm the simple numerical thesis that, prioritizing the $U$ and forcing the
scalar contributions to be as small as possible, the optimal bound is
analogous to that of the pure $U$ Scenario.

### 7.4 $Y-s$ Scenario

The double scalar Scenario is even less involved. The interference is, again,
proportional to $m_{e}$ for the strongest source of bounds and, in this case,
there is no unitary mixing. Consequently, the structure of the solution is
such that lower masses become allowed with diminishing couplings. We enforce
that, in the optimal point, scalar masses are nearly degenerate. Then we see,
from Figure 7.7, that couplings of the order $g_{\text{eff}}\sim O_{sij}\sim
10^{-2}$ allow for scalar masses of the order of
$2.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$, while couplings as small as
$10^{-3}$ are permissive of low masses. It is easy to notice that, in this
case, since there is neither conditions tying different matrix elements
together nor interference, the strongest bound, i.e., that of $\mu\to 3e$, is
the only one that matters.

(a) $|O_{Yij},O_{sij}|>10^{-2}$

(b) $|O_{Yij},O_{sij}|>10^{-3}$

Figure 7.7: Exclusion contours on the $M_{Y}\times M_{s}$ plane for bosons
constrained to have similar masses on the optimal point.

### 7.5 Analysis and conclusions

A crucial aspect in the search for new physics is knowing where to look, which
turns the constraining of theory space through its exotic parameters a central
objective. The crown of this line of work is usually the obtaining of lower
bounds on masses of hypothetical particles. The high explorable energies and
the excited stage of high amount of data collection reached by the LHC make it
one ideal tool in this quest. As we have exhaustively discussed, it has been
used to derive constraints on $M_{U}$, the mass of a doubly-charged vector
bilepton, rare feature of BSM models,whose collective result is approximately
well described by the bound
$M_{U}\gtrsim$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$.

Now, in any general phenomenological research, if purely leptonic processes
(such as the 3-body decays) are able to provide for useful bounds, their
operational advantages are manifest. In the specific case of the CLFV
$U^{\pm\pm}$ physics, although the 3-body phase space is more computationally
demanding than the LHC counterpart, the hadron physics and background analysis
needed in LHC phenomenology are an immensely heavier complication.

We have showed that the simple bounds on the branching ratios of the 3-body
lepton decays produce strong constraints on the bilepton mass and allow to
explore general mixing matrices, regularly neglected. A fine representative of
this is the pure $U^{\pm\pm}$ Scenario, in which we predict
$M_{U}>$3200\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ if the hierarchy within
$V_{U}$ is of the order or lesser than $10^{3}$. We conclude that the this
data complement the LHC flavour-diagonal phenomenology, and is, furthermore,
considerably more effective in the case of finite mixing if compared with our
results of Chapter 5, which considers a CLFV LHC process treating $V_{U}$
through a simplified construction.

We do not primarily intend to achieve new specific mass bounds for the
scalars: the interactions these particles participate in are not governed by
unitary mixing, and there are concrete (model-dependent) experimental bounds
over the doubly-charged scalars [183, 184], and the neutral scalar is a well
known and common particle, analogous to the Higgs boson, so that its
phenomenology is well understood in most models where it is present [185, 186,
2, 187]. Nevertheless, we considered a pure scalar $Y-s$ Scenario and what we
find is that for low masses to be possible after enforcement of the CLFV
bounds, the effective coupling must be of order of $10^{-3}$. For comparison,
the corresponding SM $He\bar{e}$ and $H\tau\bar{\tau}$ couplings have strength
$g_{He\bar{e}}\sim 2.07\times 10^{-6}$ and $g_{H\tau\bar{\tau}}\sim 7.24\times
10^{-3}$, indicating that it is certainly reasonable for an exotic flavour
violating $C\\!P$-even neutral scalar, generally associated with higher
characteristic mass scales, to possess interactions parametrized by effective
couplings of the order necessary for its mass to be low.

The addition of the scalar bosons to our analysis is mainly intended to aid us
understand the part that secondary, non-dominant, particles can play altering
the naive (single particle) exclusion contours of dominant degrees of freedom,
which, in the present context, occurs when it is considered together with the
$U^{\pm\pm}$. We observe that the balance between the $U^{\pm\pm}$ and $s$
contributions follows a trend in which, in the optimal interference region,
the lower bound on the mass of the $U^{\pm\pm}$ is relieved by 20%. Although
it could be argued that such phenomenon can only happen in small, fine-tuned
regions of parameter space, this behavior can happen in a general multi-
particle scenario, specially in ones where a subset of parameters is
constrained by exterior phenomenological or theoretical input, like the
fitting of well measured distinct masses or mixing parameters.

##  8 The 331 perturbative regime

We have now deepened our understanding of the BSM particle $U^{\pm\pm}$
through two distinct phenomenological inquiries: first an analysis of its
contribution to the LHC trimuon process [166], focusing on the role of the
unitary mixing. In turn, this motivated a study on how it is affected by the
limits on the 3-body CLFV lepton branching ratios [188], and whether this
older data could supersede the utility of the LHC bounds under some
conditions. Although all we have done is model independent to a high degree
(specially the study over the 3-body decays), our investigations are highly
relevant to the m331 and vice versa. This last part of the thesis takes a
parallel path, explicitly focusing on this model.

The m331 is understood to present arbitrary growth in the evolution of its
$U(1)_{X}$ coupling (sometimes called a Landau pole) at characteristic
energies of around $4\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ [189, 190, 191,
192]. If unchanged by additional theoretical mechanisms such as emerging
states in the theory, this fact severely limits the usefulness of the model,
as it implies that perturbativity is lost at nearly experimentable low TeV
scales. This fact is usually derived through an approximation which considers
the SM symmetry at intermediate scales. We escape this simplified method and
explicitly perform the 1-loop renormalization of the heavy contributions to
the running of constants. The results present a much safer model regarding its
perturbativity regime.

### 8.1 Symmetry Matching Conditions and Symmetric RGE

Consider the breaking pattern $G_{1}\times G_{2}\times\cdots\times G_{n}\to
G$, where all factors are simple, with respective coupling constants
$g_{1},g_{2},\cdots g_{n}$ and $g$. In this case, at the breaking scale, the
matching conditions for $g$ usually fall into one of the following two
categories [193]:

  1. 1.

$G\subset G_{1}$

If the embedding is such that the lower group is entirely contained within a
simple higher group factor, the condition becomes simply

$g=g_{1}.$ (8.1)

  2. 2.

$G\subset G_{1}\times G_{2}\times\cdots G_{n}$

Suppose, more generally, that $G$ is contained within the non-simple product
and specialize to the case $G=U(1)$. The embedding may be parametrized
as111Clearly, the product of simple groups in which $G$ is embedded could be a
proper subset of the complete symmetry above the threshold, in which case
$p_{i}=0$ for some $i$.

$Z=\sum_{i=1}^{n}p_{i}T_{i},$ (8.2)

where $Z$ is the $U(1)$ generator and $T_{i}$ collectively denote every
generator of $G_{1}\times G_{2}\times\cdots G_{n}$. In this case, the matching
condition reads222Notice that, in the sum above, every distinct $i$-term which
corresponds to generators of a same group exhibits the same coupling factor,
i.e., $g_{i}=g_{j}$ if $T_{i},T_{j}$ belong to the algebra of the same simple
group.

$\frac{1}{g^{2}}=\sum_{i=1}^{n}\frac{p_{i}^{2}}{g_{i}^{2}}.$ (8.3)

These conditions are not analogous to the finding of Wilson coefficients in
the matching of effective theories at mass thresholds. They are, rather,
simply a consequence of requiring that the theory can be described by
Lagrangians with the expected symmetry and corresponding couplings at each
energy range, and that at the breaking scale the theories coincide, as
required by continuity. A simple example is the SM relation

$\frac{1}{e^{2}}=\frac{1}{g_{2L}^{2}}+\frac{1}{g_{Y}^{2}},$ (8.4)

which is a consequence of the definition of electric charge and that may, at
the level of the lagrangian, be read from

$Q=T_{3}+Y.$ (8.5)

Once the different couplings, defined to exist at distinct energy ranges, are
correctly matched at the symmetry transition scales, they may be evolved
through the RGE between thresholds or towards infinity according to

$\beta(g)=\mu\frac{dg}{d\mu}.$ (8.6)

The $\beta$-function in a fully gauge-symmetric theory may be written as

$\beta(g)=-\frac{g^{3}}{(4\pi)^{2}}b_{1},$ (8.7)

where the $\beta$-function coefficient at one loop may be readily found from
its related group theoretical quantities [194]

$b_{1}=\frac{11}{3}C_{2}(\text{Gauge})-\frac{4}{3}\kappa
S_{2}(\text{Fermion})-\frac{1}{6}\eta S_{2}(\text{Scalar}),$ (8.8)

where $C_{2}(R)$ and $S_{2}(R)$ are, respectively, the Casimir and Dynkin
Index invariants of the representation $R$, and $\kappa=1/2(1)$ for
Weyl(Dirac) components and $\eta=1(2)$ for real(complex) scalars.

### 8.2 Exotic mass scales and rotations in the m331

In order to more clearly define our framework, let us recall a few facts of
the m331 model already laid down in Chapter 4. To start, we have found that,
after diagonalization, the electrically charged part of the spin-1 sector of
the model is comprised by new singly and doubly charged bosons, written in
terms of the symmetry eigenstates, along with the $W$-boson, simply as

$\begin{split}W_{\mu}^{\pm}&=(W_{\mu}^{1}\mp iW_{\mu}^{2})/\sqrt{2}\\\
V_{\mu}^{\pm}&=(W_{\mu}^{4}\pm iW_{\mu}^{5})/\sqrt{2}\\\
U_{\mu}^{\pm\pm}&=(W_{\mu}^{6}\pm iW_{\mu}^{7})/\sqrt{2},\\\ \end{split}$
(8.9)

with masses

$\begin{split}M_{W}^{2}&=\frac{1}{4}g_{3L}^{2}(v_{\eta}^{2}+v_{\rho}^{2}+2v_{s}^{2})\equiv\frac{1}{4}g_{3L}^{2}v_{W}^{2}\\\
M_{V}^{2}&=\frac{1}{4}g_{3L}^{2}(v_{\eta}^{2}+2v_{s}^{2}+v_{\chi}^{2})\\\
M_{U}^{2}&=\frac{1}{4}g_{3L}^{2}(v_{\rho}^{2}+2v_{s}^{2}+v_{\chi}^{2}).\end{split}$
(8.10)

Note that, to numerically fit the $W$ mass,
$v_{\eta}^{2}+v_{\rho}^{2}+2v_{s}^{2}$ must sum to the Higgs VEV $v_{W}^{2}$.

The last addition with respect to the SM is an extra neutral vector boson
$Z^{\prime}$. The diagonalization of the neutral sector is immensely more
complicated and given by

$(W_{3\mu},W_{8\mu},B_{\mu})^{T}=\mathcal{O}\cdot(Z_{\mu},Z^{\prime}_{\mu},A_{\mu})^{T},$
(8.11)

where the $\mathcal{O}$ matrix (with basis ordered as above) is given exactly
and in general by

$\mathcal{O}=\begin{pmatrix}-N_{1}a_{1}&-N_{2}a_{2}&\frac{t_{X}}{\sqrt{4t_{X}^{2}+1}}\\\
-\sqrt{3}N_{1}b_{1}&-\sqrt{3}N_{2}b_{2}&-\frac{\sqrt{3}t_{X}}{\sqrt{4t_{X}^{2}+1}}\\\
2t_{X}\left(1-\bar{v}_{\rho}^{2}\right)N_{1}&2t_{X}(1-\bar{v}_{\rho}^{2})N_{2}&\frac{1}{\sqrt{4t_{X}^{2}+1}}\end{pmatrix},$
(8.12)

with

$\begin{split}a_{1(2)}&=3m_{2(1)}^{2}+\bar{v}_{\rho}^{2}-2\bar{v}_{W}^{2}\\\
b_{1(2)}&=m_{2(1)}^{2}+\bar{v}_{\rho}^{2}-\frac{2}{3}\bar{v}_{W}^{2}-\frac{2}{3},\end{split}$
(8.13)

where the overbar indicates the ratio by $v_{\chi}$ as
$\bar{v}_{\alpha}\equiv\frac{v_{\alpha}}{v_{\chi}}$, and
$t\equiv\tan\theta_{X}\equiv\frac{g_{X}}{g_{3L}}$. The normalization factors
are given by

$\begin{split}N_{1}^{-2}&=3\left(2m_{2}^{2}+\bar{v}_{\rho}^{2}-\frac{4}{3}\bar{v}_{W}^{2}-\frac{1}{3}\right)+(\bar{v}_{\rho}^{2}-1)^{2}(4t_{X}^{2}+1)\\\
N_{2}^{-2}&=3\left(2m_{1}^{2}+\bar{v}_{\rho}^{2}-\frac{4}{3}\bar{v}_{W}^{2}-\frac{1}{3}\right)+(\bar{v}_{\rho}^{2}-1)^{2}(4t_{X}^{2}+1).\end{split}$
(8.14)

We have defined the factors

$\begin{split}A&=\frac{1}{3}\left[3t_{X}^{2}\left(\bar{v}_{\rho}^{2}+1\right)+\bar{v}_{W}^{2}+1\right]\\\
R&=\left\\{1-\frac{1}{3A^{2}}\left(4t_{X}^{2}+1\right)\left[\bar{v}_{W}^{2}\left(\bar{v}_{\rho}^{2}+1\right)-\bar{v}_{\rho}^{4}\right]\right\\}^{1/2},\end{split}$
(8.15)

and the dimensionless masses

$\begin{split}m_{1}^{2}&=\frac{2M^{2}_{Z_{1}}}{g_{3L}^{2}v_{\chi}^{2}}=A(1-R)\\\
m_{2}^{2}&=\frac{2M^{2}_{Z_{2}}}{g_{3L}^{2}v_{\chi}^{2}}=A(1+R).\end{split}$
(8.16)

The mass matrices of the exotic quarks are of the form

$M^{J}_{ab}=\frac{v_{\chi}}{\sqrt{2}}y^{J},\;\;\;\;M^{j_{i}}_{ab}=\frac{v_{\chi}}{\sqrt{2}}K^{j}_{ab},$
(8.17)

which are proportional to the large $v_{\chi}$ and can be made arbitrarily
massive by the free (besides possible phenomenological and unitarity
constraints) Yukawa couplings $y^{J},K^{j}_{ab}$. As for the scalars, the
model contains, in total, four singly-charged, three doubly-charged, five
$\mathit{CP}$-even neutral and three $\mathit{CP}$-odd physical scalars. The
sheer amount of exotic parameters in the scalar sector together with the
absence of TeV scale phenomenological input allow scalar masses to be large as
well. Henceforth, the words exotic and heavy will be used interchangeably to
qualify all the particles in the set

$\left\\{Z^{\prime},\;V^{\pm},\;U^{\pm\pm},\;j_{1}^{-4/3},\;j_{2}^{-4/3},\;J^{5/3},\;\text{15
exotic scalars}\right\\}.$ (8.18)

### 8.3 Approximating the m331 by the SM symmetry

####  8.3.1 Matching and strategy

The usual manner to investigate the running structure of the gauge couplings
takes advantage of the fact that, below the scale of importance of the exotic
particles (all heavier), the m331 must be approximated by the SM. This is a
consequence of the natural decoupling of the heavier particles and of the
apparent absence of new physics up to the TeV scale. Furthermore, the SSB of
the model automatically splits into two processes, a fact which can be used to
describe the approximation mentioned above at a given energy range. From this
perspective, the breaking pattern may be written as

$\begin{split}SU(&3)_{c}\times SU(3)_{L}\times U(1)_{X}\\\ &\hskip
22.76219pt\xrightarrow[E_{\text{high}}]{\langle\chi\rangle}SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}\\\ &\hskip 31.29802pt\hskip
28.45274pt\xrightarrow[E_{\text{low}}]{\langle\eta\rangle,\langle\rho\rangle}SU(3)_{c}\times
U(1)_{\text{EM}}.\end{split}$ (8.19)

The schematics above represents that, at the unknown $E_{\text{high}}$ scale,
the 3-3-1 symmetry is broken down to the SM one, after which, at
$E_{\text{low}}$ (to be identified with the electroweak scale), the breakdown
to the conserved sector occurs.

Referring to this, the full process induced by the RGE transformations is
understood as follows: (i) Nature is assumed to be well described by the
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ symmetry immediately above an
$E_{\text{low}}$ scale, below which the theory is supposed to be broken to the
conserved $SU(3)_{c}\times U(1)_{\text{EM}}$; At this threshold, the couplings
are fixed to their numerically known values; (ii) $g_{2L},g_{Y}$ are evolved,
according to the appropriate particle content (see below and in the next
section), with increasing energy, up to $E_{\text{high}}$; (iii) The second
matching is performed to replace $g_{2L},g_{Y}$ by the emergent
$g_{3L},g_{X}$; (iv) The couplings are evolved again, now up to the pole,
according to a second structure corresponding to the higher symmetry and
larger set of particles suitable to the new energy range. Notice that
$E_{\text{high}}$ is the scale in which the effects of the heavy particles
become important, which causes the apparent symmetry to change.

At $E_{\text{low}}$, generically taken to be at the $Z$-pole from now on,
$E_{\text{low}}=M_{Z}=$91.1876\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, the
known SM couplings are simply fixed to their well measured numerical values
[72]:

$g_{2L}(E_{\text{low}})=0.62977,\;\;\;\;\;g_{Y}(E_{\text{low}})=0.34537.$
(8.20)

Table 8.1: $\beta$-function coefficients of the m331 couplings in the two energy ranges. Two configurations for the representation content are emphasized: ‘Full’ refers to the entire m331 content, whereas only the SM degrees of freedom contribute to ‘SM’. |  $g_{Y(X)}$: Full |  $g_{2L(3L)}$: Full |  $g_{Y}$: SM |  $g_{2L}$: SM
---|---|---|---|---
$E_{\text{low}}<\mu<E_{\text{high}}$ | $38$ | $-2$ | 41/6 | -19/6
$\mu>E_{\text{high}}$ | $22$ | $-17/3$ |  |

At $E_{\text{high}}$, the two exotic couplings may be matched to the SM ones
according to the formalism reviewed in Sec. 8.1. Regarding $g_{3L}$, it is
enough to realize that the relevant embedding structure obeys
$SU(2)_{L}\subset SU(3)_{L}$, causing the necessary condition to be simply

$g_{3L}(E_{\text{high}})=g_{2L}(E_{\text{high}}).$ (8.21)

For $g_{X}$, we recall

$Y=-\sqrt{3}T_{8}+X,$ (8.22)

which is a consequence of Eq. (4.13). Of inserting this information into
formula (8.3), results

$\frac{1}{g_{Y}(E_{\text{high}})^{2}}=\frac{3}{g_{3L}(E_{\text{high}})^{2}}+\frac{1}{g_{X}(E_{\text{high}})^{2}},$
(8.23)

or, plugging Eq. (8.21),

$g_{X}(E_{\text{high}})=\frac{g_{2L}(E_{\text{high}})\,g_{Y}(E_{\text{high}})}{\sqrt{g_{2L}(E_{\text{high}})^{2}-3g_{Y}(E_{\text{high}})^{2}}}.$
(8.24)

Once more, it must be emphasized that these boundary conditions on the
breaking scales are merely consistency requirements to guarantee the supposed
symmetry structure and continuity. In fact, they obviously could be found –
and usually are – by explicit brute force comparison of the two lagrangians
with correct symmetry properties around the threshold.

The $\beta$-function coefficients are straightforward to calculate through Eq.
(8.8) and are shown in Table 8.1. The values corresponding to the m331 with
the exotic degrees of freedom removed are also presented. Note that removing
the entire exotic sector within the 3-2-1 (SM symmetry) energy range amounts
to eliminating closed multiplets, keeping the theory symmetric. This assures
that the exact RGE may be solved directly, with the $\beta$-function
coefficients given by the classic result of Eq. (8.8).

####  8.3.2 Results

If the entire particle content is naively kept from very low energies, the
explicit $g_{X}$ dependence on energy is given by

$\begin{split}g_{X}^{\text{full}}(\mu)=2\pi(3&0.1-11.0\log{E_{\text{high}}}+22.0\log{E_{\text{low}}}-11.0\log{\mu})^{-1/2},\;\;\;\;\mu>E_{\mathrm{high}},\end{split}$
(8.25)

with the immediate result that for
$E_{\text{high}}\gtrsim$370\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
$g_{X}^{\text{full}}$ becomes imaginary in the entire higher range. This
signifies that, for these choices of $E_{\mathrm{high}}$, the mandatory
matching conditions are impossible to fulfil, rendering the theory senseless
in the higher symmetry regime. The upper bound on $E_{\text{high}}$ then reads

$E_{\text{high}}\lesssim$370\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.$ (8.26)

However, such small values for the breaking scale are clearly not
phenomenologically viable.

This result does not come as a surprise, since in $\overline{\text{MS}}$ the
decoupling of heavy particles is not automatic as happens in physical
renormalization schemes, and must be introduced by hand [195]. If enforced,
and only the known particles are kept at energies $\mu<E_{\text{high}}$, with
the heavy degrees of freedom integrated out and only included above this
threshold, the coupling depends on energy as

$\begin{split}g_{X}^{\text{SM}}(\mu)=2\pi(3&0.1+2.83\log{E_{\text{high}}}+8.17\log{E_{\text{low}}}-11.0\log{\mu})^{-1/2},\;\;\;\;\mu>E_{\mathrm{high}}.\end{split}$
(8.27)

The corresponding behaviour for several benchmark $E_{\text{high}}$ is shown
in Figure 8.1. It may be observed that, for breaking scales obeying
$E_{\text{high}}\gtrsim$3770\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, the
$g_{X}$ coupling is highly vertical, originating larger than $4\pi$ and
rapidly diverging, and there is no effectively perturbative window. The upper
bound with the heavy particles decoupled becomes

$E_{\text{high}}\lesssim$3770\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.$ (8.28)

Unfortunately, even for the optimal
$E_{\text{high}}=$3770\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ perturbativity
is quickly harmed at around
$\sim$4600\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. This result is usually
quoted without the corresponding choice of $E_{\text{high}}$, which, in turn,
is usually thought to define the $v_{\chi}$ parameter. In summary, assuming
the position for the pole in $g_{X}$ automatically induces phenomenological
consequences.

Figure 8.1: Running of $g_{Y}$ (solid) and $g_{X}$ (dashed) in the SM
approximation for five $E_{\text{high}}$ benchmarks.

The conclusions of this section could be qualitatively understood by noting
that, in the matching condition,

$\frac{1}{g_{X}^{2}}=\frac{1}{g_{Y}^{2}}-\frac{3}{g_{2L}^{2}},$ (8.29)

the RHS is smaller than one ($\sim 0.84$) already at the electroweak scale,
and its reciprocal very sensitive to small changes in $g_{Y}$ and $g_{2L}$,
which are increasing and decreasing, respectively.

### 8.4 Effective approach

####  8.4.1 Matching and strategy

The conclusion from the last section is that, to match an exact 3-2-1 – the SM
as embedded in the m331 – to an exact 3-3-1 theory is only possible in a very
low regime. Such attempts are justified by the reasoning that, if the exotic
particles are ignored, interactions should be approximately parametrized as
within the SM. This, in turn, should be possible within a model like the m331
because they are constructed with an SM embedding in mind, which guarantees
that its good predictions are reproduced. To use the SM as a low energy
description of an extension, however, is only an approximation (to be made
around a given characteristic energy scale), and does not translate well to an
RGE analysis. At zero temperature all that matters are the form and content of
the non decoupled interactions, and the definition of an active intermediate
symmetry, as in Eq. (8.19), is only a convenient manner to describe them in a
theory with the embedded SM and every exotic effect decoupled. This, however,
is not necessarily possible at an arbitrary mass regime. Note, in this sense,
that in a general model it is not strictly necessary for the SSB pattern to
decompose as in the m331: the VEV acquisition for distinct scalars (or
subgroups of them) could each trigger the direct breaking to the
electromagnetism or to arbitrary intermediate groups. Nonetheless, in
principle, such a theory could still produce all the electroweak predictions
of the SM.

We shall now take on a more precise approach, in which the 3-3-1
parametrization is assumed from the start. What can be gained from this is the
circumventing of the symmetry matching process (which is artificial and an
approximation), and the downside is the need for a brute force, explicit
approach, as Eq. (8.8) is no longer valid when the heavy degrees of freedom
are integrated out.

To describe the coupling evolution process, notice that $g_{2L},g_{Y}$ are now
undefined, and $g_{3L},g_{X}$ hold unrelated new values, to be numerically
matched at $E_{\text{low}}$ with experimentally obtained quantities. For that,
match the strength of the $\gamma e\bar{e}$ and $We\bar{\nu}$ interactions to
their experimental counterpart. The requirements read

$\begin{split}\Gamma_{\mu}^{\gamma
e\bar{e}}&=i\gamma_{\mu}\left(-\frac{g_{3L}}{2}\mathcal{O}_{13}P_{L}+\frac{g_{3L}}{2\sqrt{3}}\mathcal{O}_{23}P_{L}+\frac{g_{3L}}{\sqrt{3}}\mathcal{O}_{23}P_{R}\right)\overset{!}{=}-i|e|\gamma_{\mu}\\\
\Gamma_{\mu}^{We\bar{\nu}}&=i\frac{g_{3L}}{\sqrt{2}}\gamma_{\mu}\overset{!}{=}i\frac{g_{2L}}{\sqrt{2}}\gamma_{\mu},\end{split}$
(8.30)

where $\mathcal{O}$ rotates mass to symmetry eigenstates like
$(W_{3\mu},W_{8\mu},B_{\mu})^{T}=\mathcal{O}\cdot(Z_{\mu},Z^{\prime}_{\mu},A_{\mu})^{T}$,
and is given explicitly in Eq. (8.12). In the end, the matching amounts to
setting

$g_{3L}(M_{Z})=0.63,\;\;\;\;g_{X}(M_{Z})=1.1.$ (8.31)

Let us now discuss how heavy particle decoupling may be implemented. In the
last section, the theory that remained after elimination of the heavy
particles continued symmetric in the SM approximation, and it was enough to
modify the $\beta$-function coefficients to encompass the light multiplets
alone. This was made possible because the SM projection of the m331 contains
every exotic degree of freedom as a singlet, which may be removed without
spoiling the symmetry. An analogous situation occurs if one inspects the RGE
of the SM without the third generation of quarks. In the other hand, if the
gauge symmetry is broken by disregarding a few particles, some brute force
method becomes necessary. This is exemplified by the exclusion of the top
quark from the SM, and corresponds to our problem at hand since the
$SU(3)_{L}$ triplets and octet are broken once the heavy particles are
removed. The theory is no longer symmetric below $E_{\mathrm{high}}$, and an
explicit, specific calculation cannot be escaped.

The procedure amounts to subtracting from the complete, symmetric
$\beta$-functions the part resulting from the 1-loop contributions of the
exotic particles (indicated by an $\mathscr{H}$ superscript), $\delta
Z^{\mathscr{H}}_{g_{3L}},\delta Z^{\mathscr{H}}_{g_{X}}$, only activated above
the $E_{\mathrm{high}}$ threshold. For that end, we construct a completely
determined system of equations from which the counterterms of the couplings
may be obtained. In order to build such a system, the counterterms for at
least two vertex functions are required, and are conveniently chosen to be the
$Wu\bar{d}$ and $\gamma u\bar{u}$ ones.

To summarize, the evolution process within the current construction follows
as: (i) At $E_{\text{low}}$, we match $g_{3L},g_{X}$ to fit $Z$-pole
interactions as in Eq. (8.30); (ii) We initially evolve the couplings with
increasing energy according to an effective representation content where every
exotic particle has been integrated out through some procedure; (iii) At a
parametrically free scale $E_{\text{high}}$, referring now to the one in which
their effects become important, we include the heavy particles back into the
theory, match to the model below, and evolve the couplings up to the pole.
Notice that only $SU(3)_{L}\times U(1)_{X}$ quantities are ever mentioned.

####  8.4.2 Renormalization framework

We renormalize the various quantities by making333Note the distinction between
our convention and the common one which renormalizes the gauge vertex as
$\Gamma_{0}^{Vf_{1}\bar{f_{2}}}=Z_{g}\Gamma^{Vf_{1}\bar{f_{2}}}$. In that
alternative definition, one has
$g_{0}=\frac{Z_{g}}{Z_{V}^{1/2}Z^{1/2}_{f_{1}}Z^{1/2}_{f_{2}}}$ [196].

$f_{0}=Z_{f}^{1/2}f,\;\;\;\;V_{0}=Z_{V}^{1/2}V,\;\;\;\;g_{0}=Z_{g}g,$ (8.32)

where $f$ is any fermion, $V$ any vector boson field and $g$ any gauge
coupling. The vertex functions to be renormalized read

$\begin{split}\Gamma_{m_{1}m_{2}\mu}^{Wu\bar{d}}&=i\frac{g_{3L}}{\sqrt{2}}\delta_{m_{1}m_{2}}\gamma_{\mu}P_{L}\\\
\Gamma_{m_{1}m_{2}\mu}^{\gamma
u\bar{u}}&=i\frac{g_{3L}}{2}\delta_{m_{1}m_{2}}\gamma_{\mu}\left[\left(\mathcal{O}_{13}-\frac{\mathcal{O}_{23}}{\sqrt{3}}-\frac{2\mathcal{O}_{33}}{\sqrt{3}}\right)P_{L}+\frac{4t_{X}\mathcal{O}_{33}}{3}P_{R}\right],\end{split}$
(8.33)

where $m_{1},m_{2}$ are color indices.

At this point, in order to solve for $\delta Z^{\mathscr{H}}_{g_{3L}},\delta
Z^{\mathscr{H}}_{g_{X}}$, the set of functions

$\left\\{\delta\Gamma^{\mathscr{H}}_{Wu\overline{d}},\delta\Gamma^{\mathscr{H}}_{\gamma
u\bar{u}},\delta Z^{\mathscr{H}}_{u},\delta Z^{\mathscr{H}}_{d},\delta
Z^{\mathscr{H}}_{W},\delta Z^{\mathscr{H}}_{A}\right\\},$ (8.34)

must be evaluated. Since the dependence of all these quantities on their
individual 1-loop contributions is additive, they may be calculated referring
only to the exotic diagrams, shown in Fig. 8.4, and there are no crossed
effects between those and the contributions of the pure SM. From Eq. (8.33)
the relations among the vertex function counterterms
$\delta\Gamma^{\mathscr{H}}$ and the other ones may be found and read
(omitting the Lorentz and color indices in the LHS for clarity)

$\begin{split}\delta\Gamma^{\mathscr{H}}_{Wu\bar{d}}&=i\frac{g_{3L}}{2\sqrt{2}}\delta_{m_{1}m_{2}}\gamma_{\mu}P_{L}\left(2\delta
Z^{\mathscr{H}}_{g_{3L}}+\delta Z^{\mathscr{H}}_{d}+\delta
Z^{\mathscr{H}}_{u}+\delta Z^{\mathscr{H}}_{W}\right)\\\
\delta\Gamma^{\mathscr{H}}_{\gamma
u\bar{u}}&=i\frac{\gamma_{\mu}}{2}\delta_{m_{1}m_{2}}\left[g_{3L}P_{L}\left(\mathcal{O}_{13}-\frac{\mathcal{O}_{23}}{\sqrt{3}}\right)\delta_{\mathrm{E}}Z^{\mathscr{H}}_{g_{3L}}\right.\\\
&\left.{}\hskip
62.59605pt+g_{X}\left(\frac{4\mathcal{O}_{33}}{3}P_{R}-\frac{2\mathcal{O}_{33}}{\sqrt{3}}P_{L}\right)\delta_{\mathrm{E}}Z^{\mathscr{H}}_{g_{3L}}\right],\end{split}$
(8.35)

where we have defined the ‘effective’ coupling counterterms

$\begin{split}\delta_{\text{E}}Z^{\mathscr{H}}_{g_{3L}}&\equiv\delta
Z^{\mathscr{H}}_{g_{3L}}+\delta Z^{\mathscr{H}}_{u}+\frac{\delta
Z^{\mathscr{H}}_{A}}{2}\\\
\delta_{\text{E}}Z^{\mathscr{H}}_{g_{X}}&\equiv\delta
Z^{\mathscr{H}}_{g_{X}}+\delta Z^{\mathscr{H}}_{u}+\frac{\delta
Z^{\mathscr{H}}_{A}}{2},\end{split}$ (8.36)

where $Z_{x}\equiv 1+\delta Z_{x}$.

The calculated wave function counterterms read

$\begin{split}\delta
Z^{\mathscr{H}}_{u}&=-\frac{g_{3L}^{2}}{64\pi^{2}\epsilon}\\\ \delta
Z^{\mathscr{H}}_{d}&=-\frac{g_{3L}^{2}}{64\pi^{2}\epsilon}\\\ \delta
Z^{\mathscr{H}}_{W}&=\frac{5g_{3L}^{2}(2\mathcal{O}_{12}^{2}+1)}{48\pi^{2}\epsilon}\\\
\delta
Z^{\mathscr{H}}_{A}&=\frac{3g_{3L}^{2}(5\mathcal{O}_{13}^{2}+9\mathcal{O}_{23}^{2})+16\sqrt{3}g_{3L}g_{X}\mathcal{O}_{23}\mathcal{O}_{33}-126g_{X}^{2}\mathcal{O}_{33}^{2}}{144\pi^{2}\epsilon}.\end{split}$
(8.37)

With this, together with the vertex function counterterms, the system in Eq.
(8.35) may be solved for $\delta Z^{\mathscr{H}}_{g_{X}},\delta
Z^{\mathscr{H}}_{g_{3L}}$. Instead of giving
$\delta\Gamma^{\mathscr{H}}_{Wu\bar{d}},\delta\Gamma^{\mathscr{H}}_{\gamma
u\bar{u}}$, we write down the solutions for the couplings directly:

$\begin{split}\delta
Z^{\mathscr{H}}_{g_{3L}}&=-\frac{3g_{3L}^{2}(53\mathcal{O}_{12}^{2}+\mathcal{O}_{22}^{2}+25)+4\sqrt{3}g_{3L}g_{X}\mathcal{O}_{22}\mathcal{O}_{32}+4g_{X}^{2}\mathcal{O}_{32}^{2}}{576\pi^{2}\epsilon}\\\
\delta
Z^{\mathscr{H}}_{g_{X}}&=\frac{\frac{28}{3}g_{X}^{3}F_{1}+\frac{2}{3}g_{3L}g_{X}^{2}F_{2}-g_{3L}^{2}g_{X}F_{3}-g_{3L}^{3}F_{4}}{192\pi^{2}g_{X}\mathcal{O}_{33}\epsilon},\end{split}$
(8.38)

where,

$\begin{split}F_{1}&=\mathcal{O}_{33}(\mathcal{O}_{32}^{2}+9\mathcal{O}_{33}^{2})\\\
F_{2}&=2\mathcal{O}_{32}\mathcal{O}_{33}(3\mathcal{O}_{12}-\sqrt{3}\mathcal{O}_{22})+3\mathcal{O}_{13}(2\mathcal{O}_{32}^{2}+63\mathcal{O}_{33}^{2})-\sqrt{3}\mathcal{O}_{23}(2\mathcal{O}_{32}^{2}+79\mathcal{O}_{33}^{2})\\\
F_{3}&=\mathcal{O}_{12}\left[6\mathcal{O}_{13}\mathcal{O}_{32}-2\sqrt{3}(\mathcal{O}_{22}\mathcal{O}_{33}+\mathcal{O}_{23}\mathcal{O}_{32})\right]-4\sqrt{3}\mathcal{O}_{13}(\mathcal{O}_{22}\mathcal{O}_{32}-4\mathcal{O}_{23}\mathcal{O}_{33})\\\
&\quad+\mathcal{O}_{33}\left[\mathcal{O}_{22}^{2}+2\mathcal{O}_{23}^{2}+10\mathcal{O}_{13}^{2}+3(\mathcal{O}_{12}^{2}+1)\right]+4\mathcal{O}_{22}\mathcal{O}_{23}\mathcal{O}_{32}\\\
F_{4}&=-3\mathcal{O}_{13}(-5\mathcal{O}_{13}+28\mathcal{O}_{12}^{2}-\sqrt{3}\mathcal{O}_{12}\mathcal{O}_{22}+\mathcal{O}_{22}^{2}-9\mathcal{O}_{23}^{2}+23)+{}\\\
&\quad+\mathcal{O}_{23}\left[-3\mathcal{O}_{12}\mathcal{O}_{22}+\sqrt{3}(28\mathcal{O}_{12}^{2}+\mathcal{O}_{22}^{2}-9\mathcal{O}_{23}^{2}-5\mathcal{O}_{13}^{2}+35)\right].\end{split}$
(8.39)

To understand how these formulae can be used to find $\beta^{\mathscr{H}}$,
start from the dimensionally regularized Eq. (8.32), or (with $g_{X}$ as
representative)

$\ln g_{X_{0}}=\ln Z_{g_{X}}g_{X}\tilde{\mu}^{\epsilon}.$ (8.40)

Where, above, $Z_{g_{X}}=Z_{g_{X}}(g_{X},g_{3L})$. Exchanging the
renormalization factor by the counterterm correction, taking the
$\ln\tilde{\mu}$ derivative and multiplying by $g_{X}$, one obtains

$0=\left(1+g_{X}\delta Z_{g_{X}}^{(g_{X})}\right)\frac{\partial
g_{X}}{\partial\ln\tilde{\mu}}+g_{X}\delta Z_{g_{X}}^{(g_{3L})}\frac{\partial
g_{3L}}{\partial\ln\tilde{\mu}}+g_{X}\epsilon,$ (8.41)

where, here, a $(y)$ superscript indicates $\partial/\partial y$. Assuming the
counterterms to be perturbatively defined, the equation above gives, after
linearization

$\frac{\partial
g_{X}}{\partial\ln\tilde{\mu}}=-g_{X}\epsilon\left(1-g_{X}\delta
Z_{g_{X}}^{(g_{X})}-g_{3L}\delta Z_{g_{X}}^{(g_{3L})}\right).$ (8.42)

Denoting the Laurent expansion of the counterterms as

$\delta Z_{g_{X}}=\sum_{n=1}^{\infty}\frac{G_{X,n}}{\epsilon^{n}},$ (8.43)

the $\beta$-function is the part of the RHS of Eq. (8.42) finite at
$\epsilon\to 0$444The $\beta$-function results from requiring perturbative
consistency at order $\epsilon^{0}$. Demanding the same consistency at every
remaining (negative) power of the expansion give an infinite number of
recursive relations.. Explicitly,

$\beta_{g_{X}}=g_{X}^{2}G_{X,1}^{(g_{X})}+g_{X}g_{3L}G_{X,1}^{(g_{3L})}.$
(8.44)

Specializing to the $\beta$-function contribution of the heavy particles and
adding the formula for $g_{3L}$, finally

$\begin{split}\beta^{\mathscr{H}}_{g_{X}}=g_{X}^{2}G_{X,1}^{{\mathscr{H}}(g_{X})}+g_{X}g_{3L}G_{X,1}^{{\mathscr{H}}(g_{3L})}\\\
\beta^{\mathscr{H}}_{g_{3L}}=g_{3L}^{2}G_{{3L},1}^{{\mathscr{H}}(g_{3L})}+g_{X}g_{3L}G_{{3L},1}^{{\mathscr{H}}(g_{X})}.\end{split}$
(8.45)

With these formulae, the running is obtained, in the non-symmetric regime
below the heavy particles threshold $E_{\mathrm{high}}$, by numerically
solving the coupled system of differential equations

$\displaystyle\frac{\partial
g_{3L}(\mu)}{\partial\ln\mu}-\frac{g_{3L}(\mu)^{3}}{(4\pi)^{2}}b_{g_{3L}}$
$\displaystyle-\left[-\beta^{\mathscr{H}}_{g_{3L}}\left(g_{3L}(\mu),g_{X}(\mu)\right)\right]=0$
(8.46) $\displaystyle\frac{\partial
g_{X}(\mu)}{\partial\ln\mu}-\frac{g_{X}(\mu)^{3}}{(4\pi)^{2}}b_{g_{X}}$
$\displaystyle-\left[-\beta^{\mathscr{H}}_{g_{X}}\left(g_{3L}(\mu),g_{X}(\mu)\right)\right]=0.$
(8.47)

An important remark is that the exotic scalars are not included in the
calculation of the counterterms (see Fig. 8.4). This is because they come in
fully exotic triplets, and can be correctly eliminated from the theory through
the subtraction of their contributions to $b_{g_{X}},b_{g_{3L}}$. The
exception is the SM scalar doublet, which must be kept. We consider it to be
the one projected by the $\rho$-triplet, $\rho^{(2)}$ from Table 4.2, and
conserve its contribution to the coefficients. The $b_{g_{X}},b_{g_{3L}}$ to
be plugged into the symmetric term of the equations above, already accounting
for the removal of the scalars other than $\rho^{(2)}$, are

$b_{g_{X}}=\frac{62}{3},\;\;\;\;\;b_{g_{3L}}=-\frac{41}{6}.$ (8.48)

####  8.4.3 Parametric structure

In order for Eqs. (8.46,8.47) to be solved for $g_{X}$ below
$E_{\mathrm{high}}$, the quantities $v_{\rho},v_{\eta},v_{\chi}$ must be
fixed. We use the solution to the closure mentioned near Eq. (4.51) to set
$v_{\rho}=$54\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$v_{\eta}=$240\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. Additionally, to fix
the values of the exotic quarks masses, we set their three Yukawa eigenvalues
to $0.5$, i.e.,

$y^{J}=0.5,\;\;\;\;K^{j}=0.5\,\mathbb{1}.$ (8.49)

With this, we examine four benchmarks for $v_{\chi}$, including the resulting
masses for the exotic fermionic and vector boson particles

$\begin{split}&\mathrm{B1}:\quad
v_{\chi}=$3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$,\,M_{U}=$945\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,\,M_{V}=$948\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,\,M_{Z^{\prime}}=$3476\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,\\\
&\hphantom{\mathrm{B1}:\quad
v_{\chi}=$3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$,}\hskip
5.69054ptM_{j}=$1061\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,\,M_{J}=$1061\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$;\\\\[2.20001pt]
&\mathrm{B2}:\quad