# Type-III Seesaw: Phenomenological Implications of the Information Lost in
Decoupling from High-Energy to Low-Energy

Saiyad Ashanujjaman<EMAIL_ADDRESS>Kirtiman Ghosh<EMAIL_ADDRESS>Institute of Physics, Bhubaneswar, Sachivalaya Marg, Sainik School,
Bhubaneswar 751005, India  Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400085, India

###### Abstract

The type-III seesaw seems to explain the very minuteness of neutrino masses
readily and naturally. The high-energy see-saw theories usually involve larger
number of effective parametres than the physical and measurable parametres
appearing in the low-energy neutrino phenomenology. Casas-Ibarra
parametrisation facilitates to encode the information lost in integrating the
heavy fermions out in an arbitrary complex orthogonal matrix. The CMS
collaboration has already searched for triplet fermions in the type-III seesaw
model with only one generation of triplet fermion flavour democratically
decaying into SM leptons. We reinterpret this CMS search in the context of a
realistic type-III seesaw model with two or three generations of triplet
fermions, and endeavour to comprehend the implications of the foregoing matrix
on the $95\%$ CL lower limit on the mass of the triplet fermions. We also
discuss the phenomenological implication of the aforesaid matrix in view of
charged lepton flavour violating observables and displaced decays of the
triplet fermions at colliders.

Type-III Seesaw, Multilepton Final States, Lepton Flavour Violation, Displaced
Decays.

††preprint: IP/BBSR/2021-0X

##  I Introduction

Notwithstanding Standard Model(SM)’s mightily success, it falls short in
addressing the issue of non-zero tiny neutrino masses and mixing. Although
plausible, it seems flimsy that the tiny neutrino masses are generated by dint
of the usual Brout, Englert and Higgs mechanism as it entails minute Yukawa
couplings which render philosophical and aesthetic displeasure. Diversely, the
three UV completions corresponding to the so-called Weinberg operator
seesaw_gut_2  at tree level seesaw_gut_5  — type-I seesaw1_1 ; seesaw1_2 ;
seesaw1_3 ; seesaw1_4 ; seesaw1_5 , type-II seesaw2_1 ; seesaw2_2 ; seesaw2_3
; seesaw2_4 ; seesaw2_5 ; seesaw2_6  and type-III seesaw3_1  seesaw seem to
accommodate the observed sub-eV neutrino masses more readily and naturally.
Though the fields which mediate neutrino masses in classical seesaw are
naturally motivated to have very high scale masses, nothing preclude them to
have masses at collider testable TeV scale if balanced with a small Yukawa
coupling. In this work, we focus on the type-III seesaw model typeIII_1 ;
typeIII_2 ; typeIII_3 ; Aguila ; Aguila2 ; Hambye ; TaoHan  which employs TeV
scale $SU(2)_{L}$ triplet chiral fermions with zero hypercharge.

Given that the lightest neutrino mass is still undetermined, the low-energy
effective theory involves 9(7) physical and measurable parametres for non-zero
(zero) lightest neutrino mass, whereas the high-energy seesaw theory involves
15(9) effective parametres in 3RHN(2RHN) case111We refer the scenario when the
SM is extended by three(two) generations of right-handed triplet fermions as
3RHN(2RHN) case.(see Section II). A number of parametres get lost in
integrating the heavy fields out. Casas-Ibarra parametrisation Casas_Ibarra ;
Ibarra_Ross  facilitates to encode these otherwise lost information in an
arbitrary complex orthogonal matrix. To paraphrase it, the gap between the
low- and high-energy is bridged via an arbitrary complex orthogonal matrix.
Mostly, in the literature Aguila ; Aguila2 ; Hambye ; Deepa ; Bpriyo , this
matrix has been assumed imprudently to be identity ignoring the possibility of
its non-triviality and hence its phenomenological implications. Also, most of
the studies, especially the experimental searches ex0 ; ex1 ; ex2 ; ex3 ; ex4
; ex5 ; ex6 ; cms_137 , considered a simplified type-III scenario with only
one generation of triplet fermions flavour democratically decaying into
leptons, albeit a realistic type-III scenario requires two or more generations
of triplet fermions. In this work, we inspect the phenomenological
implications of the said matrix in view of lepton flavour violation, displaced
decays and a recent CMS search cms_137  for type-III seesaw triplets at the
Large Hadron Collider (LHC) with $\sqrt{s}=13$ TeV and 137.1 fb-1 integrated
luminosity.

##  II  The Type-III Seesaw Model

The field content of the SM is extended by at least two generations of right-
handed $SU(2)_{L}$ triplet fermions ($\Sigma_{R}$) with zero hypercharge

$\Sigma_{R}=\left(\begin{array}[]{cc}\Sigma^{0}_{R}/\sqrt{2}&\Sigma^{+}_{R}\\\
\Sigma^{-}_{R}&-\Sigma^{0}_{R}/\sqrt{2}\end{array}\right)~{}.$

The relevant triplet Yukawa and mass terms which generates Weinberg operator
at tree level are given by

$-{\cal L}\supset M_{\Sigma}^{ij}{\rm
Tr}\left(~{}\overline{\tilde{\Sigma}^{i}_{R}}\Sigma_{R}^{j}\right)+\sqrt{2}~{}Y_{\Sigma}^{i\alpha}~{}\tilde{H}^{\dagger}\overline{\Sigma_{R}^{i}}L^{\alpha}~{}+\rm{h.c.}~{},$

where $i,j$ and $\alpha$ are the generation indices – $i$ and $j$ run over 1
and 2(1,2 and 3) for 2RHN(3RHN) case, and $\alpha$ runs over $e,~{}\mu$ and
$\tau$; $M_{\Sigma}$ is the Majorana mass matrix for $\Sigma_{R}$,
$\tilde{\Sigma}_{R}$ denotes the charge-conjugation of $\Sigma_{R}$, i.e.
$\tilde{\Sigma}_{R}=C(\overline{\Sigma_{R}})^{T}$ with $C$ being the charge-
conjugation matrix, $Y_{\Sigma}$ is the Yukawa matrix (in general, complex)
with $Y_{\Sigma}^{1\alpha},Y_{\Sigma}^{2\alpha}$ and $Y_{\Sigma}^{3\alpha}$
being affiliated to first, second and third generation of triplet fermions,
$L=\left(\nu_{L},\ell_{L}\right)^{T}$ is the left-handed SM lepton doublet,
and $H$ is the SM Higgs doublet. After the electroweak symmetry breaking
(EWSB), the above Lagrangian leads to mixing between the SM leptons and the
heavy triplet leptons which, in turn, leads to the following seesaw formula
for light neutrinos:

$m_{\nu}\approx-\frac{v^{2}}{2}Y_{\Sigma}^{T}M_{\Sigma}^{-1}Y_{\Sigma}~{},$
(1)

where $v$ is the vacuum expectation value for the SM Higgs. Further, $m_{\nu}$
can be diagonalised using the so-called Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
matrix, $U$, as $U^{T}m_{\nu}U=D_{m_{\nu}}={\rm diag}(m_{1},m_{2},m_{3})$,
where $U$ is usually parametrised by three (real) angles
($\alpha_{12},\alpha_{23}$ and $\alpha_{13}$), one Dirac phase ($\delta$) and
two Majorana phases ($\phi$ and $\phi^{\prime}$) with
$U\times\mathrm{diag}(e^{i\phi/2},e^{i\phi^{\prime}/2},1)~{}=$

$\left[\begin{array}[]{ccc}1&0&0\\\ 0&c^{23}&s^{23}\\\
0&-s^{23}&c^{23}\end{array}\right]\left[\begin{array}[]{ccc}c^{13}&0&s^{13}e^{-i\delta}\\\
0&1&0\\\
-s^{13}e^{i\delta}&0&c^{13}\end{array}\right]\left[\begin{array}[]{ccc}c^{12}&s^{12}&0\\\
-s^{12}&c^{12}&0\\\ 0&0&1\end{array}\right],$

where $c^{ij}\left(s^{ij}\right)=\cos\alpha_{ij}\left(\sin\alpha_{ij}\right)$.
We take $\delta,\phi$ and $\phi^{\prime}$ to be zero for simplicity throughout
this work222We take these phases to be zero simply because either they are
poorly measured or not measured at all. However, a non-zero choice for the
phases may significantly alter the phenomenology.. For normal hierarchy (NH),
$m_{2}=\sqrt{\Delta m_{21}^{2}+m_{1}^{2}}$, $m_{3}=\sqrt{\Delta
m_{31}^{2}+m_{1}^{2}}$, and for inverted hierarchy (IH), $m_{1}=\sqrt{|\Delta
m_{32}^{2}+\Delta m_{21}^{2}-m_{3}^{2}|}$, $m_{2}=\sqrt{|\Delta
m_{32}^{2}-m_{3}^{2}|}$; where $\Delta m_{ij}^{2}$’s are the experimentally
measured mass-squared differences, and $m_{1}(m_{3})$ is the obscured lightest
neutrino mass for NH(IH) subject to the the current bound from cosmology
$\sum_{i}m_{i}<0.12$ eV planck . The best fit values for the neutrino
oscillation parametres used in this work are taken from Ref. nu_osci_param .

Note that the low-energy neutrino oscillation data is completely described in
terms of 3(2) neutrino masses, 3 mixing angles and 3(2) phases, i.e. 9(7)
physical and measurable parametres, whereas the high-energy seesaw theory
involves 15(9) effective parametres in 3RHN(2RHN) case333In the basis where
$M_{\Sigma}$ is diagonal, $M_{\Sigma}$ is determined by 3(2) real parametres
and $Y_{\Sigma}$, being a $3\times 3(2\times 3)$ complex matrix, contains
18(12) real parametres in 3RHN(2RHN) case. However, 3 redundant phases in
$Y_{\Sigma}$ can be eliminated by a redefinition of $L$, and 3(2) more
parametres can be reduced in view of scaling symmetry of Eq. (1), thus, we end
up with only 15(9) effective parametres in 3RHN(2RHN) case.. In order to
ensure the consistency between the high-energy seesaw theory and the low-
energy neutrino phenomenology, proper parametrisation of the Yukawa matrix
($Y_{\Sigma}$) is required. In the basis where $M_{\Sigma}$ is diagonal, the
most general texture of $Y_{\Sigma}$ resulting into measured values of the
neutrino oscillation parametres is given by Casas_Ibarra ; Ibarra_Ross ,

$Y_{\Sigma}=i\frac{\sqrt{2}}{v}\sqrt{M_{\Sigma}}R\sqrt{D_{m_{\nu}}}U^{\dagger}~{},\hskip
28.45274pt(i=\sqrt{-1})$ (2)

where $M_{\Sigma}={\rm diag}(M_{1},M_{2},M_{3})[(M_{1},M_{2})]$ in 3RHN[2 RHN]
case, and $R$ is an arbitrary complex orthogonal matrix444Strictly speaking,
in 2RHN case, $R$ is not an orthogonal matrix, withal it satisfies
$R^{T}R=\mathbb{1}$. which encodes the residual freedom in $Y_{\Sigma}$ once
the other parametres are are determined from the low-energy neutrino
oscillation data. To comprehend the phenomenological consequences of $R$, we
parametrise the same by three(one) complex angles in 3RHN(2RHN) case. For 2RHN
case Ibarra_Ross ,

$R=\begin{cases}\left(\begin{array}[]{ccc}0&\cos\theta&\zeta\sin\theta\\\
0&-\sin\theta&\zeta\cos\theta\end{array}\right)\mathrm{for~{}NH}~{},\\\ \\\
\left(\begin{array}[]{ccc}\cos\theta&\zeta\sin\theta&0\\\
-\sin\theta&\zeta\cos\theta&0\end{array}\right)\mathrm{for~{}IH}~{}.\end{cases}$

where $\zeta=\pm$1555Throughout this analysis, we take $\zeta=+1$ as
$\zeta=-1$ will not yield any physically different texture of $Y_{\Sigma}$.
and the complex angle $\theta=x+iy$ with $x,y\in\mathbb{R}$. Note that
$|(Y_{\Sigma})_{ij}|$’s are invariant separately under ${\it(i)}x\to x+\pi$
and ${\it(ii)}y\to-y$666To be precise, $|(Y_{\Sigma})_{ij}|$’s are invariant
under $y\to-y$ and $\delta\to-\delta$, but, as mentioned earlier, we take
$\delta$ to be zero for simplicity.. Ergo, it suffices to restrict to
$x\in[0,\pi]~{}\mathrm{and}~{}y\geq 0$. For 3RHN case Casas_Ibarra ,

$R=\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}\\\
0&1&0\\\
-s_{13}&0&c_{13}\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right)~{},$

where
$c_{ij}\left(s_{ij}\right)=\cos\theta_{ij}\left(\sin\theta_{ij}\right)$777Note
that this parametrisation of $R$ is not the most general one as it does not
include reflections; however, the uncounted forms of $R$ do not produce any
physically different textures of $Y_{\Sigma}$. and three complex angles
$\theta_{23}=x_{1}+iy_{1}$, $\theta_{13}=x_{2}+iy_{2}$ and
$\theta_{12}=x_{3}+iy_{3}$ with $x_{1,2,3},y_{1,2,3}\in\mathbb{R}$. Because of
large number of parametres in this case, the effect of $R$ on the Yukawa
matrix $Y_{\Sigma}$ is quite intricate unlike 2RHN case, however, it can be
reckoned that $x$’s lead to oscillatory nature of $Y_{\Sigma}$ and $y$’s
enhance $Y_{\Sigma}$ exponentially. Note that $|(Y_{\Sigma})_{ij}|$’s are
invariant under $x_{1}\to x_{1}+\pi$. Therefore, it suffices to restrict to
$x_{1}\in[0,\pi]$ and $x_{2,3}\in[0,2\pi]$ while keeping $y_{1,2,3}$
unbounded888Ref. Arindam_Sanjoy  take $y_{1,2,3}\in[-\pi,\pi]$ in their
analysis. We accentuate that, in general, $y_{1,2,3}$ are unbounded; there is
no reason for $y$’s to be bounded in the range $[-\pi,\pi]$..

Note that $R_{1j}\to R_{2j}$ ($j=1,2,3$) as $\theta\to\theta+\pi/2$ for 2RHN
case, whereas $R_{2j}\to R_{3j}$ ($j=1,2,3$) as
$\theta_{23}\to\theta_{23}+\pi/2$ for 3RHN case. This is a consequence of the
orthogonality of the $R$ matrix. In fact, this in turn has a phenomenological
consequence as we will see later.

##  III Lepton flavour violation

The texture of the triplet Yukawa couplings (see Eq. (2)) necessary to
generate flavour mixing in the neutrino sector, also gives rise to SM charged
lepton flavour violating processes (CLFVPs) like $\mu\to e\gamma$ and
$\tau\to\ell\gamma$ ($\ell=e,\mu$) at one-loop level, $\mu\to 3e$ and $\tau\to
3\ell$ at tree level, etc. The constraints on the matrix $R$ resulting from
the experimental limits on the CLFVPs are discussed in the following.

For convenience, it is customary to define
$|\epsilon_{\alpha\beta}|=\frac{v^{2}}{2}|Y_{\Sigma}^{\dagger}M_{\Sigma}^{-2}Y_{\Sigma}|_{\beta\alpha}$.
Among all the CLFVPs, the charge-changing $\mu^{-}\to e^{+}$ conversion rate
to total nucleon muon capture rate ratio on ${}^{48}_{22}\rm Ti$ for the
ground state transition from SINDRUM-II collaboration at PSI mueti  puts the
most stringent bound999Note that this limit is better than what is quoted in
Ref. Biggio_et_al . Somehow they missed the most improved limit from SINDRUM-
II collaboration mueti . Also, note the relation between the notations used in
Ref. Biggio_et_al  and the present work: $|\epsilon_{e\mu}|=2|\eta_{\mu e}|$.
on $|\epsilon_{e\mu}|<3.7\times 10^{-7}$ at $2\sigma$. Diversely, $\tau$
decays into three leptons — $e\mu\mu$ and $\mu ee$ belle  set the most
constraining bounds on $|\epsilon_{e\tau}|<6.0\times 10^{-4}$ and
$|\epsilon_{\mu\tau}|<5.0\times 10^{-4}$ Biggio_et_al , respectively. Because
of many free parametres in this scenario, the expressions for
$\epsilon_{\alpha\beta}$ become perplexing; hence, we avert to write them down
explicitly. Instead, we perform a scanning over the parametres of $R$, namely
the $x$’s and $y$’s, and put constraints on them using the aforecited upper
bounds on $|\epsilon_{\alpha\beta}|$. We find that, in 2RHN case, for
$M_{\Sigma}\sim$ TeV, $y\lesssim\mathcal{O}(9)$ are allowed for all values of
$x$101010This bound is consistent with Ref. srubabati_2019 .. Whereas in 3RHN
case, for $M_{\Sigma}\sim$ TeV, we find the following constraints:

  1. 1.

$y_{1,2,3}\lesssim\mathcal{O}(11)$ and
$y_{1}+y_{2}+y_{3}\lesssim\mathcal{O}(14)$ for all values of $x_{1,2,3}$
except for $x_{2}\sim 3\pi/2$.

  2. 2.

$y_{1,3}\lesssim\mathcal{O}(45)$, $y_{2}\lesssim\mathcal{O}(10)$ and
$y_{1}+y_{2}+y_{3}\lesssim\mathcal{O}(90)$ for $x_{2}\sim 3\pi/2$.

These bounds get stronger with decreasing $M_{\Sigma}$ as well as increasing
the lightest neutrino mass as
$|\epsilon_{\alpha\beta}|\propto\mathcal{O}(m_{\nu}/M_{\Sigma})$. Also, the
bounds on $y$’s are bit more constraining in IH case than in NH one. Note that
the bounds from perturbativity as well as electroweak precision data are much
liberal than those obtained from CLFVPs.

##  IV Production and Decay Modes of Fermion Triplets

The TeV scale triplet fermions are copiously pair produced typeIII_1 ;
typeIII_2 ; typeIII_3 ; Aguila ; Aguila2 ; Hambye ; TaoHan  at the LHC
typeIII_1 ; typeIII_2 ; typeIII_3 ; Aguila ; Aguila2 ; Hambye ; TaoHan  by
quark-antiquark annihilation via $s$-channel $\gamma/Z$ and $W^{\textpm}$
exchanges, namely the Drell-Yan processes111111Also, the charged triplet
fermions can be produced via $t/u$-channel photon-photon fusion process.
However, their production through such process is sub-dominant for the mass
range of our interest..After being produced, the triplets undergo two types of
decays — the heavy state transitions due to the radiative mass-
splitting121212$\Delta M=M_{\Sigma^{+}}-M_{\Sigma^{0}}\sim 166~{}\rm MeV$
cirelli  for $M_{\Sigma}\gg M_{Z,W}$. between $\Sigma^{\pm}$ and $\Sigma^{0}$
and the SM two-body final state decays induced by their mixing with the SM
leptons. The corresponding partial decay widths are given by

$\displaystyle\Gamma(\Sigma^{\pm}_{i}\to\Sigma^{0}_{i}\pi^{\pm})=\frac{2G_{\rm
F}^{2}V_{ud}^{2}f_{\pi}^{2}(\Delta
M)^{3}}{\pi}\sqrt{1-\frac{m_{\pi}^{2}}{(\Delta M)^{2}}}~{},$
$\displaystyle\Gamma(\Sigma^{\pm}_{i}\to\Sigma^{0}_{i}\ell^{\pm}\raisebox{4.30554pt}{\hbox{\tiny(}}\overline{\nu}\raisebox{4.30554pt}{\hbox{\tiny)}}_{\ell})=\frac{2G_{F}^{2}(\Delta
M)^{5}}{15\pi^{3}}\sqrt{1-\frac{m_{\ell}^{2}}{(\Delta
M)^{2}}}~{}P\left(\frac{m_{\ell}}{\Delta M}\right)~{},$
$\displaystyle\Gamma(\Sigma^{0(\pm)}_{i}\to
XY^{(\prime)})=\kappa\frac{M_{i}}{64\pi}|(Y_{\Sigma})_{i\alpha}|^{2}f\left(\frac{M_{X}}{M_{i}}\right)~{},$
(3)

where $G_{F}$ is the Fermi coupling constant, $|V_{ud}|\approx 0.974$ Vud ,
$f_{\pi}\simeq 130$ MeV fpi  is the pion decay constant, $m_{\pi}$ and
$m_{\ell}$ are, respectively, the charged pion and lepton ($\ell=e,\mu$) mass,
$P(r)=1-\frac{9}{2}r^{2}-4r^{4}+\frac{15r^{4}}{2\sqrt{1-r^{2}}}{\rm
tanh}^{-1}\sqrt{1-r^{2}}$ , $XY\ni
h\raisebox{4.30554pt}{\hbox{\tiny(}}\overline{\nu}\raisebox{4.30554pt}{\hbox{\tiny)}}_{\alpha},Z\raisebox{4.30554pt}{\hbox{\tiny(}}\overline{\nu}\raisebox{4.30554pt}{\hbox{\tiny)}}_{\alpha},W^{\pm}\ell^{\mp}_{\alpha}$,
$XY^{\prime}\ni
h\ell^{\pm}_{\alpha},Z\ell^{\pm}_{\alpha},W^{\pm}\raisebox{4.30554pt}{\hbox{\tiny(}}\overline{\nu}\raisebox{4.30554pt}{\hbox{\tiny)}}_{\alpha}$,
$f(r)=(1-r^{2})^{2}(1+\kappa^{\prime}r^{2})$, and
$\kappa(\kappa^{\prime})=1,1,2(0,2,2)$ for $X=h,Z,W^{\pm}$, respectively. The
total decay width of $\Sigma^{0,\pm}_{i}$ are given by131313The contribution
of the heavy state transitions to the total decay width of $\Sigma^{\pm}$
turns out to be $\sim 3.5\times 10^{-15}$ GeV.:

$\displaystyle\Gamma_{i,\rm
T}^{0}\approx\frac{M_{i}}{16\pi}\sum_{\alpha}|(Y_{\Sigma})_{i\alpha}|^{2}=\frac{M_{i}^{2}}{8\pi
v^{2}}\sum_{j}m_{j}\left|R_{ij}\right|^{2}{\rm~{}and}$

$\Gamma_{i,\rm T}^{\pm}\approx\Gamma_{i,\rm T}^{0}+3.5\times
10^{-15}\mathrm{~{}GeV}$. The relevant expressions for $\Gamma_{i,\rm
T}^{0}$’s in terms of the parametres in the arbitrary orthogonal complex
matrix $R$, namely the $x$’s and $y$’s, are presented in the following. For
2RHN case, the total decay widths for the neutral component of the first and
second generation triplets are given by,

$\displaystyle\Gamma_{1,\rm T}^{0}$
$\displaystyle\approx\frac{M_{1}^{2}}{16\pi
v^{2}}\times\begin{cases}\left[m_{2}f^{+}+m_{3}f^{-}\right]~{}{\rm~{}for~{}NH}~{},\\\
\left[m_{1}f^{+}+m_{2}f^{-}\right]~{}{\rm~{}for~{}IH}~{},\end{cases}$ (4)
$\displaystyle\Gamma_{2,\rm T}^{0}$
$\displaystyle\approx\frac{M_{2}^{2}}{16\pi
v^{2}}\times\begin{cases}\left[m_{2}f^{-}+m_{3}f^{+}\right]~{}{\rm~{}for~{}NH}~{},\\\
\left[m_{1}f^{-}+m_{2}f^{+}\right]~{}{\rm~{}for~{}IH}~{};\end{cases}$ (5)

For 3RHN case,

$\displaystyle\Gamma_{1,\rm T}^{0}\approx\frac{M_{1}^{2}}{32\pi
v^{2}}\left[m_{1}f^{+}_{2}f^{+}_{3}+m_{2}f^{+}_{2}f^{-}_{3}+2m_{3}f^{-}_{2}\right],$
(6)

where $f^{\pm}_{(k)}=\cosh 2y_{(k)}\pm\cos 2x_{(k)}$ with $k=1,2,3$. One can
obtain similar expressions for the total decay widths of the neutral component
of the second and third generation triplets for 3RHN case.

We follow from the above expressions that $\Gamma_{1,\rm
T}^{0}/M_{1}^{2}\to\Gamma_{2,\rm T}^{0}/M_{2}^{2}$ as $f^{\pm}\to f^{\mp}$,
i.e. $\theta\to\theta+\pi/2$ for 2RHN case. A similar relation, i.e.
$\Gamma_{2,\rm T}^{0}/M_{2}^{2}\to\Gamma_{3,\rm T}^{0}/M_{3}^{2}$ as
$\theta_{23}\to\theta_{23}+\pi/2$, holds for 3RHN case. This is an upshot of
the fact that, as mentioned at the closing of Section II, $R_{1j}\to
R_{2j}$($R_{2j}\to R_{3j}$) with $j=1,2,3$ as
$\theta\to\theta+\pi/2$($\theta_{23}\to\theta_{23}+\pi/2$) for 2RHN(3RHN)
case. Thereupon, it suffices to anatomise the decays of only the first(first
two) generation(s) of triplets for 2RHN(3RHN) case.

##  V Displaced decays

Dependence of the decay widths on the free parametres of the model, namely
$x^{\prime}s$, $y$’s and the lightest neutrino mass, suggests that it is quite
plausible that the heavy fermions of this model are long-lived for a certain
region in the parametre space, and if so, disappearing track signature or
other displaced vertex signals141414The disappearing track signature arsing
from $\Sigma^{\pm}$ decaying to $\Sigma^{0}$ and $\pi^{\pm}$ can be searched
at the LHC if $\Sigma^{0}$ is long-lived enough to pass through the detector,
i.e. $c\tau^{0}\gtrsim\mathcal{O}(10)$ m, whereas such a displaced vertex
signature can be observed at $ep$-colliders like FCC-he and LHeC if
$c\tau^{\pm}\in[10^{-5},10^{-1}]$ m. Meanwhile, $\Sigma^{0}$ can be observed
by MATHUSLA if $c\tau^{0}>\mathcal{O}(100)$ m. arising from them can be probed
at collider. Ref. sudip  and Ref. Arindam_Sanjoy  discussed this possibility
in detail. Here, we briefly mention our findings in this regard.

For 2RHN case, the decay lengths151515For $M_{i}\gg M_{W,Z,h}$, the total
decay lengths of $\Sigma^{0,\pm}_{i}$ are given by
$c\tau^{0,\pm}_{i}\approx\times 1.975\times 10^{-16}\times\left(\Gamma_{i,\rm
T}^{0,\pm}\right)^{-1}~{}{\rm GeV~{}m}$. can not be arbitrarily large, withal
the decay lengths can be as large as $\mathcal{O}(10^{-4})$ m when $y\sim 0$,
see Eq. (4) and Eq. (5). While one doesn’t expect any disappearing track
signature arising from $\Sigma^{\pm}_{i}$ at the LHC, the same can be observed
at FCC-he and LHeC.

For 3RHN case, the decay length for the neutral heavy fermions can be
arbitrarily large as can be reckoned from Eq. (6), whereas that of the charged
ones could be as large as $\sim 5.6$ cm, and this has important implications
of probing this scenario in various colliders. We briefly summarise the
results for the first generation triplet fermion for NH in the following.

  1. 1.

For $m_{1}\lesssim\mathcal{O}(10^{-8})$ eV, $x_{2,3}\sim 0,\pi$ and
$y_{2,3}\sim 0$, one gets $c\tau^{0}\gtrsim\mathcal{O}(100)$ m and
$c\tau^{+}\sim 5.6$ cm. This is ideal for the MATHUSLA detector.

  2. 2.

For $m_{1}\lesssim\mathcal{O}(10^{-7})$ eV, $x_{2,3}\sim 0,\pi$ and
$y_{2,3}\sim 0$, $c\tau^{0}\gtrsim\mathcal{O}(10)$ m and $c\tau^{+}\sim 5.6$
cm. This is ideal for the disappearing track signature searches at the LHC.

  3. 3.

For $m_{1}\lesssim\mathcal{O}(10^{-1})$ eV, $y_{2}\lesssim\mathcal{O}(1)$ and
$y_{3}\lesssim\mathcal{O}(2)$, $c\tau^{+}$ lies in the range
$[10^{-5},10^{-1}]$ m for all values of $x_{2}$ except
$x_{2}\sim\pi/2,3\pi/2$. For $x_{2}\sim\pi/2,3\pi/2$, one would expect
$c\tau^{+}$ to lie in the same range for $y_{2}\sim 0$ and
$y_{3}\lesssim\mathcal{O}(40)$. This is ideal for the displaced vertex
signature searches at FCC-he and LHeC.

We see that for a certain region in the parametre space of
$x_{1,2,3},y_{1,2,3}$ and $m_{1}$, the first generation of triplet fermion
could be long-lived and, possibly, can be observed by the detectors. Likewise,
the second (and hence third in view of the symmetry stated at the end of
Section II) generation of triplet fermion could also be long-lived for a
certain region in the parametre space of $x_{1,2,3},y_{1,2,3}$ and
$m_{1}$161616Note that for the most trivial choice $R=\mathbb{1}$ (i.e. all
$x,y$’s $\sim 0$), only $\Sigma^{0}_{1}(\Sigma^{0}_{3})$ has arbitrarily large
decay length as $m_{1}(m_{3})\to 0$ for NH(IH); whereas for a non-trivial
choice of $R$, $\Sigma^{0}_{2,3}(\Sigma^{0}_{1,2})$ could also have
arbitrarily large decay length as $m_{1}(m_{3})\to 0$ for NH(IH). For example,
a choice like $y_{1,3}\sim 0$, $x_{1}\sim 0[\pi/2]$ and $x_{3}\to\pi/2,3\pi/2$
leads to arbitrarily large decay length for $\Sigma^{0}_{2}[\Sigma^{0}_{3}]$
as $m_{1}\to 0$ for NH; whereas a choice like $x_{2}\sim\pi/2,3\pi/2$ and
$y_{2}\sim 0$ leads to arbitrarily large decay length for $\Sigma^{0}_{1}$ as
$m_{3}\to 0$ for IH.. A similar discussion follows for IH case also.

Though the quantity $\tilde{m}_{i}=\sum_{j}m_{j}\left|R_{ij}\right|^{2}$ can
be inferred, as argued in Ref. Hambye , by measuring the displaced vertices if
the triplet fermions are long-lived, the lightest neutrino mass can not be
estimated unless the complex orthogonal matrix, $R$, is known beforehand.
Since we do not know any viable way to measure the complex angles in $R$ yet,
we can’t infer the lightest neutrino mass even if we could measure the decay
lengths. In fact, to do so, we have to know the $R$ matrix very precisely,
specifically the values of the $y$’s. This is because the decay lengths have
exponential dependence on the $y^{\prime}s$. Thus, even for hardly different
values of $y$, the same $c\tau$ value may corresponds to completely different
values of the lightest neutrino mass. However, one can extract some
information regarding the complex angles in $R$ by measuring the decay lengths
at the MATHUSLA or the LHC. For example, a positive search result at the
MATHUSLA detector or the LHC would infer that $x_{1}\sim 0,\pi/2$, $x_{2}\sim
0,\pi$, $x_{3}\sim 0,\pi/2,\pi,3\pi/2$ and $y_{1,2,3}\sim 0$.

##  VI Multilepton final states search

Production of the triplet fermions in pair and their subsequent decays into SM
two body final states lead to a variety of final state signatures including
the multilepton final states signatures at the LHC. The CMS collaboration has
recently published a multilepton final states search cms_137  with an
integrated luminosity of $137.1$ fb-1 of pp collisions at $\sqrt{s}=13$ TeV.
Taking into account all possible multilepton final states in the type-III
seesaw model with one generation of triplet fermions, the CMS analysis
excluded them for masses below 880 GeV at $95\%$ confidence level (CL) in the
flavour democratic scenario. Note that a realistic type-III seesaw scenario
requires at least two generations of triplets; also, their decays are not
necessarily flavour democratic. This is why the CMS observed 95$\%$ CL upper
limits cms_137  on the total pair production cross-sections of triplet
fermions are not straightforwardly applicable for such a realistic type-III
seesaw model. To this end, we extend this search in the context of a realistic
type-III seesaw model with two or three generations of triplet fermions, and
set forth to explore the collider implications of the arbitrary complex
orthogonal matrix, $R$. Thereupon, we restrict ourselves in the case where the
heavy fermions produced at the LHC decay promptly inside the detector, and to
ensure that we assume $m_{1}(m_{3})>10^{-3}$ eV for NH(IH) in 3RHN case.

We have implemented the model in SARAH sarah ; sarah2  to generate UFO
modules. The signal events are simulated using MadGraph mg5  with the
NNPDF23LO nnpdf ; nnpdf2  parton distribution function, and showered with
PYTHIA pythia , and then DELPHES delphes  is used for detector simulation.
Since the relevant SM backgrounds in our analysis are exactly same as that of
Ref. cms_137 , we do not simulate the backgrounds, instead we use
distributions of expected SM backgrounds and observed events given in figure 3
and 4 of Ref. cms_137  and also in cms137hepdata . We closely follow the
multilepton search strategies of Ref. cms_137  for object reconstruction and
selection, defining signal regions and event selection. This search considers
only the final states with three or more leptons ($e,\mu$). Based on the
number of leptons, number of opposite-sign same-flavour (OSSF) lepton pairs
and the invariant mass of OSSF pair, $\rm{M_{OSSF}}$, the events are
categorised into seven signal regions (SRs), namely 4LOSSF0, 4LOSSF1, 4LOSSF2,
3LOSSF0, 3L below-Z, 3L on-Z and 3L above-Z171717The names of the 4 (4L) or
more lepton signal regions are self explanatory. The events with exactly three
leptons containing an OSSF lepton pair are categorised based on
$\rm{M_{OSSF}}$, and they are labelled by 3L below-Z, 3L on-Z and 3L above-Z
when $\rm{M_{OSSF}}$ is below, within and above the Z-boson mass window
($\rm{M_{Z}\pm 15}$), respectively. The detailed description of object
reconstructions, event selections and signal region definitions can be found
in Ref. cms_137 . Whereas, our implementation and validation of the CMS
multilepton search strategies in Ref. cms_137  were presented in Ref. project1
.. The signal regions are further classified into 40 statistically independent
signal bins using $\rm{L_{T}+p_{T}^{miss}}$ or the transverse mass181818The
transverse mass is defined as $M_{T}=\sqrt{2p_{T}^{\rm
miss}p_{T}^{\ell}[1-\cos(\Delta\phi_{\vec{p}_{T}^{\rm
miss},\vec{p}_{T}^{\ell}})]}~{},$ where $\vec{p}_{T}^{\ell}$ is the transverse
momentum vector of the lepton which is not a part of the on-Z pair, and
$\Delta\phi_{\vec{p}_{T}^{\rm miss},\vec{p}_{T}^{\ell}}$ is the azimuthal
separation between $\vec{p}_{T}^{\rm miss}$ and $\vec{p}_{T}^{\ell}$. as the
primary kinematic discriminant where $\rm{L_{T}}$ is the scalar sum of
transverse momentum of all charged leptons, $\rm{p_{T}^{miss}}$ is the missing
transverse momentum. We use a hypothesis tester named ‘Profile Likelihood
Number Counting Combination’ bayes  to estimate CL. See project1  for details.

For convenience, we define total branching ratios of neutral and charged
component of the $i^{\rm th}$ generation triplet fermion ($\Sigma^{0}_{i}$ and
$\Sigma^{\pm}_{i}$) to the lepton flavour $\ell_{\alpha}(=e,\mu,\tau)$ as:
$\mathrm{BR}^{0}_{i,\alpha}=\mathrm{BR}\left(\Sigma^{0}_{i}\to\ell_{\alpha}^{\pm}W^{\mp}\right)$
and
$\mathrm{BR}^{\pm}_{i,\alpha}=\mathrm{BR}\left(\Sigma^{\pm}_{i}\to\ell_{\alpha}^{\pm}Z\right)+\mathrm{BR}\left(\Sigma^{\pm}_{i}\to\ell_{\alpha}^{\pm}h\right)$,
subject to $\sum_{\alpha}{\rm BR}^{0,\pm}_{i,\alpha}=1$. For $M_{i}\gg
M_{W,Z,h}$, $\mathrm{BR}^{0}_{i,\alpha}\approx\mathrm{BR}^{\pm}_{i,\alpha}$
(see Eq. (3)); henceforward, we will not distinguish between
$\mathrm{BR}^{0}_{i,\alpha}\mathrm{and~{}BR}^{\pm}_{i,\alpha}$. Moreover, it
is not possible to distinguish among the degenerate (in mass) copies of
triplets at the LHC unless a big conspiracy makes these copies very different
from one another. Therefore, in case of degenerate copies, instead of
branching ratios for individual copies, we use the average branching ratio
defined as:

$\mathrm{BR}_{\alpha}^{\rm
avg}=\frac{1}{N}\sum_{i=1}^{N}\mathrm{BR}_{i,\alpha}=\frac{1}{N}\sum_{i=1}^{N}\frac{|(Y_{\Sigma})_{i\alpha}|^{2}}{\sum_{\alpha}|(Y_{\Sigma})_{i\alpha}|^{2}}~{},$
(7)

where $N=2(3)$ for 2RHN(3RHN) case. Noting that CMS has high reconstruction
and identification efficiency for both electrons and muons, for the sake of
simplicity, we assume equal reconstruction and identification efficiency for
both electrons and muons. This assumption does not affect the estimated limits
significantly191919For example, for ${\rm BR}_{e}={\rm BR}_{\mu}=50\%$, the
$95\%$ CL lower limit on the triplet mass is 1000 GeV (see Figure 1 and Figure
2); if one considers ${\rm BR}_{e}=100\%$ (${\rm BR}_{\mu}=100\%$), the limit
would be relaxed(improved) by not more than 20(10) GeV which is well within a
few percent level.. Therefore, enduring the margin of a few percent, the
$95\%$ CL limits doesn’t depend on ${\rm BR}_{e}$ and ${\rm BR}_{\mu}$
exclusively but on ${\rm BR}_{e}+{\rm BR}_{\mu}=1-{\rm BR}_{\tau}$.

Bearing the aforesaid discussions in mind, we estimate $95\%$ CL upper limit
on the total production cross section of triplet fermionic pairs for different
branching ratio to the $\tau$-lepton final state (see Figure 1) following the
approach prescribed in Ref. project1 . The theoretical predictions for the
total triplet production cross section at LO+NLO202020The cross sections at
LO+NLO are taken from Ref. cms_137 . are also presented in Figure 1.

Figure 1: $95\%$ CL upper limit on total production cross section of heavy
fermions for different values of ${\rm BR}_{\tau}$. Solid red line shows
theoretical prediction for the total cross section at LO+NLO for one
generation of triplet fermions.

Figure 2 shows $95\%$ CL lower limit on the (degenerate) mass of the triplet
fermions, $\rm{M_{excl}}$, as a function of (average) branching ratio to the
$\tau$-lepton final state for one, two and three degenerate copies of triplet
fermions. Since the CMS analysis cms_137  is dedicated to search for multi-
electron/muon final states, the most restrictive bound on $\rm{M_{excl}}$ is
found for $\tau$-phobic scenario, i.e. $\rm{BR_{\tau}^{\rm(avg)}}=0$, whereas
the least restrictive bound corresponds to $\tau$-philic scenario, i.e.
$\rm{BR_{\tau}^{\rm(avg)}}=1$.

Figure 2: $95\%$ CL lower limit on the degenerate mass of the triplet fermions
as a function of $\rm{BR_{\tau}^{\rm avg}}$.

The remainder of this letter is devoted to comprehend the consequence of the
free parametres in $R$ on the $95\%$ CL lower limit on the triplet masses in a
realistic type-III seesaw with two or three generations of triplet fermions.
As stated at the closing of Section II and Section IV, because of the symmetry
$R_{1j}\to R_{2j}$ ($R_{2j}\to R_{3j}$) with $j=1,2,3$ as
$\theta\to\theta+\pi/2$ ($\theta_{23}\to\theta_{23}+\pi/2$) for 2RHN (3RHN)
case, it suffices to anatomise the decays of only the first (first two)
generation(s) of triplets for 2RHN (3RHN) case. Noting this, we explicitly
study only the first (first two) generation(s) of triplets for 2RHN (3RHN)
case.

###  VI.1 2RHN case

For 2RHN case, there are two real free parametres in $R$ —
$x(\in[0,\pi])~{}\mathrm{and}~{}y(\geq 0)$. Left(middle) panel in Figure 3
shows $95\%$ CL lower limit on the mass ($\rm{M_{excl}}$) of the first
generation of triplet fermions in the $x$-$y$212121For large enough $y$
($y>\mathcal{O}(1)$), $|\cos\theta|\approx|\sin\theta|\approx e^{y}/2$ and
thus, the normalised branching ratios and hence, the resulting collider bound,
$M_{\rm excl}$, become independent of both $x$ and $y$ — $\rm BR_{e}\approx
6\%(49\%)$, $\rm BR_{\mu}\approx 50\%(22\%)$ and $\rm BR_{\tau}\approx
43\%(29\%)$ for NH(IH). plane for NH(IH) assuming that the second generation
of the same are too heavy to be produced so that effectively only the first
generation fermions are produced at the LHC. The bound on $\rm{M_{excl}}$
ranges over 565–1000 (775–1000) GeV for NH(IH). For NH, the most(least)
restricted bound is found for $x\sim 23^{\circ}$ and $y\sim 0$ ($x\sim
158^{\circ}$ and $y\sim 0$) which corresponds to ${\rm BR}_{\tau}\sim
89\%(0\%)$. Whereas, for IH, the most(least) restricted bound is found for
$x\sim 26^{\circ}$ and $y\sim 0$ ($x\sim 116^{\circ}$ and $y\sim 0$) which
corresponds to ${\rm BR}_{\tau}\sim 58\%(0\%)$. For the second generation of
triplet fermions, the bounds on $\rm{M_{excl}}$ are exactly same as that of
the first generation ones in Figure 3 with the $x$-coordinates replaced by
$x+\pi/2$222222This is a consequence of the symmetry $R_{1j}\to R_{2j}$
($j=1,2,3$) as $\theta\to\theta+\pi/2$..

Figure 3: To the left(middle): $95\%$ CL lower limit on the mass, shown by the
colour gradient, of the first generation of triplet fermions in GeV in the
$x$-$y$ plane for NH(IH). To the right: $95\%$ CL lower limit on the
degenerate mass of the triplet fermions in GeV in the $x$-$y$ plane for NH.

Right panel in Figure 3 shows $95\%$ CL lower limit on the degenerate mass of
triplet fermions for NH. The bound on $\rm{M_{excl}}$ ranges over 990–1130
GeV. The most(least) restricted bound is found for $x\sim 18^{\circ}$ and
$y\sim 0$ ($x\sim 72^{\circ}$ and $y\sim 0$) which corresponds to ${\rm
BR}_{\tau}\sim 60\%(28\%)$. For IH, the corresponding lower limit is $\sim
1050$ GeV for all values of $x$ and $y$232323For IH, the first and second
generation of light neutrinos becomes almost degenerate, i.e. $m_{1}\approx
m_{2}$, and Eq. (7) reduces to $\mathrm{BR}_{\alpha}^{\rm
avg}\approx\frac{1}{2}(1-|U_{\alpha 3}|^{2})$. Thus, the branching ratios
become independent of both $x$ and $y$ and so is the $95\%$ CL limit..

###  VI.2 3RHN case

For 3RHN case, there are six real free parametres in $R$ —
$x_{1}(\in[0,\pi]),x_{2,3}(\in[0,2\pi])~{}\mathrm{and}~{}y_{1,2,3}$. For the
sake of simplicity, we assume that $R$ is real, i.e. $y_{1,2,3}\sim
0$242424Implications of large $y$’s are discussed at the end of this section..
The normalised branching ratios for the first generation of triplet fermions
are independent of $x_{1}$ (see Eq. (2) and (3)), and so are the $95\%$ CL
lower limits on the mass. Left panel in Figure 4 shows $95\%$ CL lower limit
on the mass of the first generation of triplet fermions in GeV in the
$x_{2}$-$x_{3}$ plane for $m_{1}(m_{3})=0.1$ eV for both NH and IH252525Note
that for the lightest neutrino mass of $\mathcal{O}(0.1)$ eV, the light
neutrinos become almost degenerate in mass making the hierarchy irrelevant.
This has been reflected in Figure 4(left panel) and also in Figure 5(left
panel). assuming that effectively only the first generation of triplet
fermions are produced at the LHC. Likewise, middle(right) panel in Figure 4
shows the same for $m_{1}(m_{3})=10^{-3}$ eV for NH(IH). In all these cases,
depending on the values of $x_{2}$ and $x_{3}$, the limits range over 435–1000
GeV.

Figure 4: $95\%$ CL lower limit on the mass of the first generation of triplet
fermions in GeV in the $x_{2}$-$x_{3}$ plane. The left panel corresponds to
$m_{1}(m_{3})=0.1$ eV for both NH and IH. The middle(right) panel correspond
to $m_{1}(m_{3})=10^{-3}$ eV for NH(IH).

Unlike the first generation, the normalised branching ratios for the second
generation of triplet fermions, and hence the $95\%$ CL lower limit on the
mass of the same depend on all the three parametres — $x_{1,2,3}$. So, in the
second generation case, we estimate least restrictive $95\%$ CL lower limit on
the mass in the $x_{2}$-$x_{3}$ plane. To rephrase it, for a given set of
$x_{2}$ and $x_{3}$, we vary $x_{1}$ to find the most conservative limit. The
right panel in Figure 5 shows least restrictive $95\%$ CL lower limit on the
mass of the second generation of triplet fermions in the $x_{2}$-$x_{3}$ plane
for $m_{1}(m_{3})=0.1$ eV for NH(IH). Likewise, the middle(right) panel in
Figure 5 shows the same for $m_{1}(m_{3})=10^{-3}$ eV for NH(IH). In Figure 5,
the contours, in the $x_{2}$-$x_{3}$ plane, show the values of $x_{1}$ for
which the corresponding limit on the mass is least restrictive. Figures 5 and
4 clearly show that the type-III seesaw triplets with mass as low as 435 GeV
is still allowed in certain parts of $x_{1}$–$x_{2}$–$x_{3}$ parameter space.
Note that this limit is substantially smaller than the limit derived in Ref.
cms_137  in the context of simplified type-III seesaw.

Figure 5: Least restrictive $95\%$ CL lower limit on the mass of the second
generation of triplet fermions in GeV along with the contours for $x_{1}$ in
the $x_{2}$-$x_{3}$ plane. The left panel corresponds to $m_{1}(m_{3})=0.1$ eV
for both NH and IH. The middle(right) panel correspond to
$m_{1}(m_{3})=10^{-3}$ eV for NH(IH).

Now, we consider the scenario when three degenerate copies of triplet fermions
are produced at the LHC. For $m_{1}(m_{3})\sim 0.1$ eV for NH(IH), i.e. for
degenerate mass spectrum for the light neutrinos, the average branching ratio
to different leptonic final states become flavour universal: Eq. 7 simplifies
to ${\rm BR}_{\alpha}^{\rm avg}\approx 1/3$ with $\alpha=e,\mu,\tau$. In such
so-called flavour democratic scenario, the $95\%$ CL lower limit on the mass
of the triplet fermions is found to be $\sim$ 1110 GeV. At this point, we
accentuate that the flavour democratic scenario does not necessarily be an ad-
hoc assumption, rather it could be a consequence of degenerate mass spectrum
for both the light and heavy neutrinos 262626We briefly mention about two
other special cases which also give rise to flavour democratic scenario: (i)
For the most trivial choice of $R$, i.e. $R=\mathbb{1}$, the branching ratio
of $\Sigma^{0,\pm}_{i}$ to $\ell_{\alpha}$ becomes $|U_{\alpha i}|^{2}$. If
one assumes the triplet fermions to be degenerate, then this gives rise to
flavour democratic scenario. (ii) For degenerate mass spectrum of the light
neutrinos, if one identifies the complex angles in $R$ as the mixing angles in
the PMNS matrix (i.e. $R=U$), the Yuakawa matrix ($Y_{\Sigma}$) becomes
flavour diagonal. If one further assumes that the triplet fermions are
degenerate, then this reduces to flavour democratic scenario..

For smaller values of the lightest neutrino mass, flavour universality of the
the average branching ratio of the triplets to different leptonic final states
breaks down. For brevity, we do not show any plots for this case. However, we
note that for $m_{1}(m_{3})=10^{-3}$ eV in NH(IH), the limit on the degenerate
triplet mass ranges over 935-1150 (1045-1150) GeV.

Lastly, we briefly mention the implications of large $y$’s. For large enough
$y$’s, $|\cos\theta$’s$|\approx|\sin\theta$’s$|\approx\frac{1}{2}e^{y}$’s.
Thus, the normalised branching ratios become independent of all $x$’s and
$y$’s, and so are the limits. For NH(IH) with $m_{1}(m_{3})=0.1$ eV, the limit
is 890 GeV corresponding to $\rm BR_{\tau}\sim 29\%$ for both the first and
second generation of triplets. For NH(IH) with $m_{1}(m_{3})=10^{-3}$ eV, the
limit is 845(890) GeV corresponding to $\rm BR_{\tau}\sim 42\%(29\%)$ for both
the first and second generation of triplets. Likewise, in the case of three
degenerate copies of triplet fermions, the limit is 1125 GeV corresponding to
$\rm BR_{\tau}\sim 28\%$ for NH(IH) with $m_{1}(m_{3})=0.1$ eV, and for NH(IH)
with $m_{1}(m_{3})=10^{-3}$ eV, the limit is 1080(1125) GeV corresponding to
$\rm BR_{\tau}\sim 42\%(28\%)$.

##  VII Summary and Outlook

The high-energy see-saw theory involves 15(9) effective parametres, whereas
the low-energy neutrino phenomenology involves 9(7) physical and measurable
parametres in 3RHN(2RHN) case. A number of parametres get lost in integrating
the heavy fields out. The well-known Casas-Ibarra parametrisation facilitates
to encode the information lost in the decoupling of the heavy fermions in an
arbitrary complex orthogonal matrix ($R$) with 6(2) real parametres in
3RHN(2RHN) case. We have explored the phenomenological implications of the
said matrix in view of lepton flavour violation, displaced decays and a recent
multilepton final states search by the CMS collaboration. We have seen that
for a certain region in the parametre space of $x_{1,2,3},y_{1,2,3}$ , all
generations of triplet fermions could be long-lived; and possibly, a limited
region of that parametre space can be probed by the detectors like MATHUSLA,
LHC or FCC-he and LHeC. This is in contrast to the most trivial choice of $R$
($=\mathbb{1}$) in which only the first(third) generation of triplet fermions
could be long-lived. We accentuate that the lightest neutrino mass can not be
inferred by measuring the displaced vertices unless the orthognal matrix is
known beforehand. However, one can extract some information regarding the
complex angles in R by measuring the decay lengths at the MATHUSLA or the LHC.
For example, a positive search result at the MATHUSLA detector or the LHC
would infer that $x_{1}\sim 0,\pi/2$, $x_{2}\sim 0,\pi$, $x_{3}\sim
0,\pi/2,\pi,3\pi/2$ and $y_{1,2,3}\sim 0$. We have explored collider
implications of the arbitrary complex orthogonal matrix using a recent
multilepton final states search by the CMS collaboration. Specifically, we
have explored the consequence of the free parametres in $R$ on the $95\%$ CL
lower limit on the mass of the triplet fermions in view of the multilepton
final states search for both 2RHN and 3RHN case. We note that degenerate mass
spectrum for both the light and heavy neutrinos lead to a flavour democratic
scenario willy-nilly. In such-scenario, the $95\%$ CL lower limit on the
degenerate mass of triplet fermions is found to be $\sim$1110 GeV. Depending
on the choice for the said matrix, we find that the bounds could be notably
contrasting than that obtained by the CMS search.

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