We see that the contribution coming from renormalisable part ($Z$-mediated
case) dominates over the dimension six contribution when mixing angle is
$\tilde{\theta}=10^{-3}$ and because of this black dotted and thick blue line
coincides in each panel.

Figure 19: Partial decay width corresponds to the decay mode
$\Gamma\left(N_{i}\to\nu_{j}\,d_{\alpha}\,\bar{d}_{\alpha}\right)$ of RHN mass
ranging from 10 GeV to 80 GeV. The color code is same as in Fig. 18 except the
orange dot-dashed line now stand for four-Fermi operators.

  * •

$\bf{\Gamma\left(N_{i}\to\nu_{j}\,\nu\,\bar{\nu}\right)}$

The RHNs can also decay to pure active neutrino states. The decay can be
mediated either via off-shell $Z$ decay or the Fermi operators
$\mathcal{O}_{NN}$ and $\mathcal{O}_{NNNN}$ that arise in the dimension six
set up. However the relevant terms coming from these dimension six operators
are proportional to cubic power of the mixing angle along with the inverse of
the quadratic $\Lambda$ suppression. Hence, once can ignore those terms for
practical purposes. The partial decay is then can be written as

$\displaystyle\Gamma\left(N_{i}\to\nu_{j}\nu\bar{\nu}\right)$
$\displaystyle=\frac{G^{2}_{F}M^{5}_{N}}{96\pi^{3}}|\tilde{\theta}|^{2}.$ (35)

In Fig. 20, we present the corresponding decay width for different choices of
mixing angles. As the EFT operators does not participate in the process the
change in the cut-off scale is irrelevant in this calculation.

Figure 20: Partial decay width corresponds to the decay mode
$\Gamma\left(N_{i}\to\nu_{j}\,\nu\,\bar{\nu}\right)$ of RHN mass ranging from
10 GeV to 80 GeV. The black dotted and red dashed line stands for mixing angle
$\tilde{\theta}=10^{-3}$ and $\tilde{\theta}=10^{-6}$, respectively.

In Fig. 21 we illustrate the the total decay width of the RHN fields while
taking into account all possible decay modes in the mass range
$10~{}\text{GeV}<M_{N}<80~{}\text{GeV}$. The red dot dashed line and the blue
dashed line correspond to mixing angle $\tilde{\theta}=10^{-6}$ and
$\tilde{\theta}=10^{-3}$ respectively. On the other hand, the black thick line
and the brown dotted line signifies the total decay width corresponds to full
$d=6$ $N_{R}$-EFT, where the cut-off scale $\Lambda$ = 4 TeV and 500 GeV
respectively. In the EFT calculation we set the active-sterile mixing angle to
be $\tilde{\theta}=10^{-3}$. The value of the total decay width suggests that
the RHN field would behave as a prompt particle in both the EFT benchmark
points as well as $\tilde{\theta}=10^{-3}$ case for entire range of 10 GeV
$<M_{N}<$ 80 GeV.

Figure 21: The total decay width correspond to RHN field where the
$\tilde{M}_{N}$ is ranging from 10 GeV to 80 GeV. The blue dashed and red dot-
dashed line stands for the mixing angle $\tilde{\theta}=10^{-3}$ and
$\tilde{\theta}=10^{-6}$, respectively. The black solid line represents the
total decay width of the $N$ field under full $d=6$ $N_{R}-$EFT where the cut
off scale $\Lambda$ is set to be 4 TeV. Similarly the brown dotted line stands
for the total decay width of the $N$ field with cut off scale $\Lambda$ = 500
GeV.

We now present the branching ratio for the benchmark points BP1 ($\Lambda=500$
GeV) and BP2 ($\Lambda=4$ TeV) while setting active-sterile mixing angle to be
$\tilde{\theta}=10^{-3}$. In upper panel of Fig. 22, we present the
corresponding plots for these scenarios. For both these cases the BR value for
the $\ell_{j}ud$ channel will be maximum for the entire mass range as shown by
the black dotted curve. Similarly, the BR value for the $\nu_{j}\nu\bar{\nu}$
mode is minimum. But the relative difference between the other possible
channels vary depending on the underlying benchmark choice. For comparison
purpose we also present the branching ratio in the lower panel of Fig. 22
while considering only the renormalisable part of the EFT Lagrangian. The
important change in dimension six $N_{R}$-EFT is the enhancement of the BR
value of pure leptonic three body modes. In renormalisable level the BR for
$N_{i}\to\nu_{j}q_{\alpha}\overline{q}_{\alpha}$ channels dominate over entire
range of 10 GeV$<M_{N}<M_{W}$. However, the interplay between the operators
$\mathcal{O}_{LNLe}$ and $\mathcal{O}_{HNe}$ alter these result and BR of
$N_{i}\to\ell_{k}\ell_{k}\nu_{k}/\ell_{j}\ell_{k}\nu_{k}$ (Grey solid and
brown dotted) dominates over $N_{j}\to\nu_{j}q_{\alpha}q_{\alpha}$ mode for
both the benchmark scenarios.

Figure 22: Upper Panel: The branching ratios of RHN mass ranging from from 10
GeV to 80 GeV in various three body modes for the benchmark points BP1
($\Lambda$ = 500 GeV) and BP2 ($\Lambda$ =4 TeV). In both cases, we have
considered the active sterile mixing angle $\tilde{\theta}=10^{-3}$. Lower
Panel: We calculate the branching ratio of the RHN mass ranging from 10 GeV to
80 GeV in different three body decay modes while considering renormalisable
part of the Lagrangian.

##  6 Prospective Multi-Lepton Final States

We focus on producing multi-lepton final states that arise from single/pair
production of RHN fields and their subsequent decays. We have calculated
various three-body decay modes of the RHNs. In Fig. 22, we show that the $N$
decaying to purely leptonic final states have branching ratios of 20$\%$ to
25$\%$, depending on the cut-off scale $\Lambda=500$ GeV and 4 TeV. With these
results, one can estimate the expected number of events in the heavy neutrino
production processes at $pp$, $ep$ and $e^{+}e^{-}$ colliders, which we show
in Fig. 23. We assume in all these cases that the RHN decays purely
leptonically.

Fig. 23 shows the number of events expected at the HL-LHC with c.m. energy
$\sqrt{s}=14$ TeV and $\mathcal{L}=3000\,\rm{fb}^{-1}$. The solid black line
represents the number of events for the $4\ell+MET$ final state, which results
from the pair-production of $N$’s and their subsequent decay to leptonic final
states. $N$-pair production is either mediated via Higgs decay, where the
Higgs boson is produced through gluon fusion, or via four-fermion operators.
The $Z$ boson, produced via the Drell-Yan process, can give rise to the same
final state. However, this process is suppressed by the mixing angle
$\tilde{\theta}$ and fails to provide any appreciable contribution. As seen
from the figure, the total number of events for this channel is as large as
$\mathcal{O}(10^{6})$.

Instead, the blue dashed line shows the $2\ell+MET$ final state, resulting
from single $N$ production and its subsequent leptonic decay. Similar to the
previous channel, the $N$ in this case can be produced either via Higgs decay
or via four-fermion operators. Additionally, there is a Drell-Yan contribution
via $Z$-mediated contributions. These production modes, however, depend on the
active-sterile mixing angle. Hence, the total number of events for this
process is small compared to the $4\ell+MET$ final state, where the four-
fermion $\mathcal{O}^{5}_{2}$ operators can give mixing angle independent,
unsuppressed contributions.

The $4\ell+2j+MET$ and $2\ell+2j+MET$ final states are represented by the
orange dot-dashed line and the grey dotted line, respectively. These final
states are produced via Higgs and $Z$ boson-mediated VBF production modes. The
above two final states can appear depending on the rates for the decay
$h/Z\to\nu N/NN$ and subsequent leptonic decay of $N$. Amongst these two
processes, the former is dominant, and many signal events can be expected
$\sim\mathcal{O}(10^{5})$.

The $3\ell+MET$ final state, shown as red dashed line in Fig. 23, which arises
from the $qq^{{}^{\prime}}\to W\to\ell N$ production mode can provide signal
events ranging between $\mathcal{O}(10^{4})$ to $\mathcal{O}(10^{3})$.

Figure 23: The left panel denotes the number of events that can be obtained at
HL-LHC with c.m.energy 14 TeV and $\mathcal{L}=3000\,\rm{fb}^{-1}$. The right
panel shows achievable number events that can be obtained at $e^{+}e^{-}$ and
$e^{-}p$ colliders for different c.m.energies.

In 23 we present the number of signal events one can obtain at $e^{+}e^{-}$
and $e^{-}p$ colliders. For the $e^{+}e^{-}$ collider, we consider a c.o.m
energy of 91 GeV and 3 TeV, respectively. On the other hand, for $e^{-}p$
colliders, the c.o.m energy has been chosen to be 3.46 TeV. In all these
cases, we have considered the luminosity $\mathcal{L}=1\,\rm{ab}^{-1}$. The
grey dashed line denotes the $4\ell+MET$ final state for $\sqrt{s}=91$ GeV.
There can be different processes contributing to the final state, among which
the $Z$ mediated diagram generating two $N$s followed by $N\to\ell\ell\nu$
decay give rise to a significant number of events $N\sim\mathcal{O}(10^{5})$.
One can obtain $2\ell+MET$ final states via $Z$ boson decay or four-fermi
operators. Although, the relevant production vertex is suppressed by the
mixing angle $\tilde{\theta}$, the presence of $Z-$pole leads to substantial
number of signal events.

The number of signal events associated with $4\ell+MET$ final state is shown
for $e^{+}e^{-}$ collider with c.o.m. energy of 3 TeV. This process can arise
via VBF production or four-fermion contact operators, which give rise to a
sizeable number of events $N\sim\mathcal{O}(10^{4})$.

For $e^{-}p$ colliders, the only process which can generate a significant
number of signal events is the $e^{-}p\to jN\to 2l+j+MET$ channel. The number
of signal events corresponding to this channel is represented by the red
dotted curve and is $N\sim\mathcal{O}(10^{4})$.

##  7 Summary and Conclusion

The $N_{R}$-EFT framework can provide a suitable phenomenological description
of the BSM physics. In this work we have systematically established this EFT
up to dimension six level and shown the detailed construction of all possible
operators that one can construct at $d=5$ and $d=6$ respectively. At each
order these higher dimensional operators either modify different $d=4$
couplings involving RHN field or introduce new set of interactions relevant to
neutrino phenomenology.

The interaction of the RHN with SM fields leads to new decay modes and hence
modifies the decay width of $h/W/Z$. However, the existing experimental
observations restrict such modifications and impose meaningful bounds on
mixing angle $\tilde{\theta}$, Wilson coefficients and the associated cut off
scale $\Lambda$. One can further tighten these constraints while incorporating
different direct search limits and determine allowed range of EFT parameters.
Our analysis deduce in case of cut off $\Lambda$ = 4 TeV one is allowed to fix
the mixing angle $\tilde{\theta}=10^{-3}$, the Wilson coefficients
$\alpha^{(5)}_{2}\leq 0.1$ $\alpha_{HN},\alpha_{HNe}\leq 1$,
$\alpha_{LNW/B}\leq 0.1$ and coefficients correspond to four-Fermi operators
to be $\alpha_{\text{four Fermi}}\leq 0.5$. Similarly, for cut off scale
$\Lambda$ = 500 GeV one needs to rescale these _Wilson_ coefficients
accordingly.

We have presented the viable production mode of the RHN fields at 14 TeV $pp$,
as well as _future_ $e^{+}e^{-}$ and $e^{-}p$ colliders. In case of $pp$
collider pair production of $N$ field via $gg\to h\to NN$ mode remains
dominant production mode for the mass range $M_{N}<\frac{m_{h}}{2}$ with the
cross section roughly around $\mathcal{O}(10)$ pb. Above this mass range RHN
pair production through four-Fermi contact interaction serves as a dominant
production mode with the cross section ranging between $\mathcal{O}(10^{-1})$
pb to $\mathcal{O}(10^{-3})$ pb for the mass range 100 GeV$<M_{N}<$ 1 TeV. All
the operators which can generate $NN$ pair can also invoke single $N$
production. However the cross section correspond to single RHN field
production is comparatively low _w.r.t_ RHN pair production as this is active-
sterile mixing suppressed. For completeness we also present RHN production
through VBF as well as Drell-Yan mechanism. Apart from RHN production via $W$
boson decay (where $W$ is produced via Drell-Yan process), for all other cases
the cross section is significantly low. The cross section for RHN production
via Drell-Yan $W$ mode is large (roughly around $\mathcal{O}(10^{-1})$ pb to
$\mathcal{O}(10^{-3})$ pb) due to the significant contribution coming from
$d=6$ operator $\mathcal{O}_{HNe}$. In case of $ep$ collider $ep\to jN$ mode
stands as a viable production channel for the RHN field. This process can
occur either via t-channel $W$-mediated process or via contact interaction
through different four-Fermi operators and corresponding cross section lies
within the range $\mathcal{O}(10^{-1})$ pb to $\mathcal{O}(10^{-3})$ pb. In
case of $e^{+}e^{-}$ collider we have studied the feasible production modes
for $c.m.$ energies $\sqrt{s}=$ 91 GeV and 3 TeV respectively. For $\sqrt{s}=$
91 GeV, RHN pair production via s-channel $Z$ boson is the dominant production
mode. The presence of $Z$ pole in the underlying process significantly
increases the production cross section to 10 pb. In contrast to that for
$\sqrt{s}$ = 3 TeV, this process is primarily governed by different four-Fermi
operators ($\mathcal{O}_{eN}$, $\mathcal{O}_{LN}$ _etc._) and the
corresponding cross section lies roughly around $\mathcal{O}(0.5)$ pb for
$M_{N}$ ranging between 100 GeV to 1 TeV.

Apart from the production mechanism, we evaluate the decay width as well as
branching ratio of RHN fields while considering both renormalisable part of
effective Lagrangian as well as relevant higher dimension operators. Depending
upon RHN masses it can either decay via two body or three body decay modes
where we assume decay among RHN fields are forbidden due to mass degeneracy.
In case of two body decay calculation, the $N$ fields can decay into $\nu h$,
$\nu Z$ and $\ell W$ modes if we only consider $d=4$ Lagrangian. The presence
of EFT operators $\mathcal{O}^{(5)}_{3}$, $\mathcal{O}_{LNB}$ and
$\mathcal{O}_{LNW}$ invoke an additional decay mode $\nu\gamma$ for the RHN
field. The branching ratio correspond to $N\to\ell W$ mode remains dominant
for entire $M_{N}$ mass range in both $d=4$ as well as $d=6$ set up and the
contributions from $\mathcal{O}_{HNe}$ significantly enhance BR$(N\to\ell W)$
_w.r.t_ the $d=4$ result. The BR correspond to $N\to\nu h$ mode remains
minimum for entire RHN mass range in both these set up.

In case of $M_{N}<M_{W}$, the RHN fields can decay via different leptonic as
well as semileptonic three body modes in addition to the two body $\nu\gamma$
mode. Upto dimension five these three body decay modes can only occur due to
the off shell decay of $W$ and $Z$ bosons. In dimension six the presence of
different four-Fermi operators significantly alters this situation. These
four-Fermi operators can contribute to the three body modes through contact
interaction and enhance the corresponding partial width. For example, operator
$\mathcal{O}_{LNLe}$ would participate in different leptonic channels
$N_{i}\to\ell_{j}\ell_{k}\nu_{k}/\ell_{k}\ell_{k}\nu_{k}/\nu_{j}\ell_{k}\ell_{k}$
and increases partial width. The $d=6$ operator $\mathcal{O}_{HNe}$ noticeably
increases off shell $W$ mediated contribution but the off shell $Z$ boson
decay does not receive any significant enhancement from the EFT operators as
the relevant couplings are either mixing angle $\tilde{\theta}$ suppressed or
loop suppressed. We also present comparative analysis between the branching
ratio for two EFT benchmark scenarios ($\Lambda=$ 500 GeV, 4 TeV) as well as
renormalisable level Lagrangian. Our analysis dictates the BR value for
$\ell_{j}u_{\alpha}d_{\beta}$ mode is maximum and BR value for
$\nu_{j}\nu_{\alpha}\nu_{\alpha}$ remains minimum in all these scenarios. The
important aspect of dimension six $N_{R}$-EFT is the enhancement of the BR
value of pure leptonic three body modes. In renormalisable level the BR for
$N_{i}\to\nu_{j}q_{\alpha}\overline{q}_{\alpha}$ channels dominate over entire
range of 10 GeV$<M_{N}<M_{W}$. However, the interplay between the operators
$\mathcal{O}_{LNLe}$ and $\mathcal{O}_{HNe}$ alter these result and BR of
$N_{i}\to\ell_{j}\ell_{k}\nu_{k}/\ell_{k}\ell_{k}\nu_{k}$ dominates over
$N_{j}\to\nu_{j}q_{\alpha}q_{\alpha}$ (red dashed line) mode for both the
benchmark scenarios.

The major advantage of the $N_{R}$-EFT set up is the enhancement in the BR
value for different three body pure leptonic decay modes. This motivates us to
propose few _golden_ channel signatures for the RHN field in multi-leptonic
final states. Our study suggest $4\ell+MET$ channel can serve as an optimal
discovery mode for the RHN field at both 14 TeV HL-LHC and future $e^{+}e^{-}$
(with c.m $\sqrt{s}=$ 91 GeV and 3 TeV) machine. In case of $ep$ collider the
viable discovery mode would be $2\ell+j+MET$ final state. We will investigate
these channels for a detailed cut-based analysis in our future studies.

###### Acknowledgements.

The work of S.M. is supported by KIAS Individual Grants (PG086001) at Korea
Institute for Advanced Study. M.M and A.S acknowledge the support from the
Indo-French Centre for the Promotion of Advanced Research (Grant no: 6304-2).
M.M acknowledges the support from IPPP DIVA Programme. RP acknowledges the
support of Fulbright-Nehru Doctoral Research Fellowship, 2022-2023.

##  Appendix A : Useful Transformation Rules between Flavour and Mass Basis

In Eq. 8 and Eq. 13 we established the relation between the gauge basis
$\\{\nu_{L},N^{c}_{R}\\}$ and the mass basis $\\{\nu_{L,m},N^{c}_{R,m}\\}$ in
dimension five and dimension six framework respectively. Furthermore, for the
explicit coupling extraction one need other possible gauge basis definitions
like $\nu^{c}_{L}$, $N_{R}$ _etc._ In Eq. 36, and Eq. 37 we illustrate
different relationship between the gauge basis and the mass basis states of
both the active and sterile neutrinos.

$\begin{split}\nu_{L}&\simeq U_{\text{PMNS}}\nu_{L,m}+\theta
N^{c}_{R,m},~{}~{}~{}~{}~{}\overline{\nu_{L}}\simeq\overline{\nu}_{L,m}U^{\dagger}_{\text{PMNS}}+\overline{N}^{c}_{R,m}\theta^{\dagger}\\\
\nu_{L}^{c}&\simeq
U^{*}_{\text{PMNS}}\nu^{c}_{L,m}+\theta^{*}N_{R,m},~{}~{}~{}~{}~{}~{}~{}\overline{\nu}^{c}_{L}\simeq\overline{\nu}^{c}_{L,m}U^{\dagger}_{\text{PMNS}}+\overline{N}_{R,m}\theta^{T}.\end{split}$
(36)

$\begin{split}N^{c}_{R}&\simeq-\theta^{T}\nu_{L,m}+\kappa
N^{c}_{R,m},~{}~{}~{}~{}~{}\overline{N_{R}}^{c}\simeq-\theta^{*}\overline{\nu}_{L,m}+\kappa^{\dagger}\overline{N}^{c}_{R,m}\\\
N_{R}&\simeq-\theta^{\dagger}\nu^{c}_{L,m}+\kappa^{*}N_{R,m},~{}~{}~{}~{}~{}\overline{N}_{R}\simeq-\theta\overline{\nu}^{c}_{L,m}+\kappa^{T}\overline{N}_{R,m}.\end{split}$
(37)

In the above we present all these relations for dimension five set up. At
dimension six, one should replace the mixing angle $\theta$ with
$\tilde{\theta}$.

##  Appendix B : Evaluating Couplings Upto Dimension five set up

The RHN fields couple to different SM fields even in the minimal seesaw
Lagrangian. At renormalisable level these couplings are governed by the mixing
angle between the active and sterile neutrinos. In $d=5$ $N_{R}$-EFT one can
write three more operators $\mathcal{O}^{(5)}_{1}$, $\mathcal{O}^{(5)}_{2}$
and $\mathcal{O}^{(5)}_{3}$ which can modify the existing couplings as well as
introduce new couplings involving RHN fields. Here we present the explicit
expansion correspond to the relevant part of the renormalisable Lagrangian as
well as dimension five operators.

$\begin{split}\bar{L}\gamma_{\mu}D_{\mu}L+h.c.&\subset\frac{g}{\sqrt{2}}\bar{\nu}_{L}\gamma^{\mu}W^{+}_{\mu}\ell_{L}+\bar{\nu}_{L}\gamma^{\mu}\left(\frac{g}{2}W^{3}_{\mu}-\frac{g^{{}^{\prime}}}{2}B_{\mu}\right)\nu_{L}+h.c.\\\
&\subset\frac{g}{\sqrt{2}}\bar{\nu}_{L}\gamma^{\mu}W^{+}_{\mu}\ell_{L}+\frac{g}{2c_{w}}\bar{\nu}_{L}\gamma^{\mu}Z_{\mu}\nu_{L}+h.c.\\\
&\subset\frac{gU}{\sqrt{2}}[\overline{\nu}_{m}\gamma_{\mu}P_{L}+\overline{N}_{m}\gamma_{\mu}P_{L}]\ell_{m}+\frac{g}{2c_{w}}Z_{\mu}[U^{\dagger}U\overline{\nu}_{m}\gamma^{\mu}P_{L}\nu_{m}+U^{\dagger}\theta\overline{\nu}_{m}\gamma^{\mu}P_{L}N_{m}\\\
&+U\theta^{\dagger}\overline{N}_{m}\gamma^{\mu}P_{L}\nu_{m}+\theta^{\dagger}\theta\overline{N}_{m}\gamma^{\mu}P_{L}N_{m}]+h.c.\end{split}$
(38)

The lepton doublets are charged under the SM $SU(2)_{L}\times U(1)_{Y}$ gauge
group. As a result the left-handed $\nu_{L}$ field couples to electroweak
gauge bosons through the covariant derivative term as illustrated in Eq. 38.
Using the transformation rule as discussed in Appendix. A one can rewrite the
$\nu_{L}$ fields in terms of mass basis $\\{\nu_{m},N_{m}\\}$. As a result one
obtain different three point couplings between the RHN fields and SM
electroweak gauge bosons. One can notice, the coupling involving RHN field is
mixing angle suppressed as it does not directly couple to $W/Z$ boson in the
renormalisable Lagrangian.

$\displaystyle Y_{\nu}\bar{L}\tilde{H}N_{R}$
$\displaystyle=\frac{Y_{\nu}v}{\sqrt{2}}\bar{\nu}_{\ell}\left(1+\frac{h}{v}\right)N_{R}\subset\frac{Y_{\nu}}{\sqrt{2}}h\bar{\nu}_{\ell}N_{R}$
$\displaystyle=\frac{Y_{\nu}}{\sqrt{2}}h\left[U^{\dagger}\theta^{\dagger}\overline{\nu}_{m}P_{R}\nu_{m}-U^{\dagger}\kappa^{*}\overline{\nu}_{m}P_{R}N_{m}+\theta^{\dagger}\theta^{\dagger}\overline{N}_{m}P_{R}\nu_{m}-\theta\kappa^{*}\overline{N}_{m}P_{R}N_{m}\right]$
(39)

In renormalisable part of the Lagrangian the neutrino fields can achieve mass
through usual Yukawa term. This will invoke different couplings between both
active, sterile neutrinos with Higgs boson. In Eq. 39 we present the explicit
evaluation of these couplings.

$\begin{split}\mathcal{O}^{\left(5\right)}_{1}&=\left(\frac{\alpha^{(5)}_{1}}{\Lambda}\right)\left(\overline{L}^{c}\tilde{H}^{\dagger}\tilde{H}L\right)=\left(\frac{\alpha^{(5)}_{1}}{\Lambda}\right)\overline{\nu_{\ell}}^{c}\frac{\left(v+h\right)^{2}}{2}\nu_{\ell}\\\
&\subset\left(\frac{\alpha^{(5)}_{1}v}{\Lambda}\right)h\left[U^{T}U\overline{\nu}_{m}P_{L}\nu_{m}+\theta^{T}U\overline{N}_{m}P_{L}\nu_{m}+U^{T}\theta\overline{\nu}_{m}P_{L}N_{m}+\theta^{T}\theta\overline{N}_{m}P_{L}N_{m}\right]\end{split}$
(40)

The Higgs-neutrino couplings receive further modification through the
dimension five operators $\mathcal{O}^{(5)}_{1}$ and $\mathcal{O}^{(5)}_{2}$.
In Eq. 40 and Eq. 41 we present these corrections in detailed manner.

$\begin{split}\mathcal{O}^{\left(5\right)}_{2}&=\left(\frac{\alpha^{(5)}_{2}}{\Lambda}\right)\left(\overline{N_{R}}^{c}N_{R}\right)\left(H^{\dagger}H\right)=\left(\frac{\alpha^{(5)}_{2}}{\Lambda}\right)\left(\overline{N_{R}}^{c}N_{R}\right)\frac{\left(v+h\right)^{2}}{2}\\\
&\subset\left(\frac{\alpha^{(5)}_{2}v}{\Lambda}\right)h\left[\theta^{*}\theta^{\dagger}\overline{\nu}_{m}P_{R}\nu_{m}-\kappa^{\dagger}\theta^{\dagger}\overline{N}_{m}P_{R}\nu_{m}-\theta^{*}\kappa^{*}\overline{\nu}_{m}P_{R}N_{m}+\kappa^{\dagger}\kappa^{*}\overline{N}_{m}P_{R}N_{m}\right]\end{split}$
(41)

In the above, we only have presented the three point vertices involving RHN
fields and SM Higgs field. Apart from the operators involving Higgs field, one
can write another operator $\mathcal{O}^{(5)}_{3}$ involving $U(1)_{Y}$ field-
strength tensor in $d=5$ $N_{R}$-EFT.

$\begin{split}\mathcal{O}^{\left(5\right)}_{3}&=\left(\frac{\alpha^{(5)}_{3}}{\Lambda}\right)\left(\overline{N_{R}}^{c}\sigma_{\mu\nu}N_{R}\right)B_{\mu\nu}=\left(\frac{\alpha^{(5)}_{3}}{\Lambda}\right)\left(\overline{N_{R}}^{c}\sigma_{\mu\nu}N_{R}\right)\left(\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}\right)\\\
&=\left(\frac{\alpha^{(5)}_{3}}{\Lambda}\right)\left(\overline{N_{R}}^{c}\sigma_{\mu\nu}N_{R}\right)\left(-\sin\theta_{w}\\{\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}\\}+\cos\theta_{w}\\{\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\\}\right)\\\
&=c_{w}A_{\mu\nu}\left(\frac{\alpha^{(5)}_{3}}{\Lambda}\right)\left[\theta^{*}\theta^{\dagger}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}-\kappa^{\dagger}\theta^{\dagger}\overline{N}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}-\theta^{*}\kappa^{*}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}N_{m}+\kappa^{\dagger}\kappa^{*}\overline{N}_{m}\sigma_{\mu\nu}P_{R}N_{m}\right]\\\
&-s_{w}Z_{\mu\nu}\left(\frac{\alpha^{(5)}_{3}}{\Lambda}\right)\left[\theta^{*}\theta^{\dagger}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}-\kappa^{\dagger}\theta^{\dagger}\overline{N}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}-\theta^{*}\kappa^{*}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}N_{m}+\kappa^{\dagger}\kappa^{*}\overline{N}_{m}\sigma_{\mu\nu}P_{R}N_{m}\right]\end{split}$
(42)

Here we define $A_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$Z_{\mu\nu}=\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}$. In above equation,
we present the explicit expansion of this operator. The $B_{\mu}$ field
correspond to $U(1)_{Y}$ gauge group couples to RHN field through this
operator. One can re-define this $B_{\mu}$ field as linear sum of neutral
gauge boson $Z_{\mu}$ and the photon field $A_{\mu}$. As a consequence the RHN
fields would couple to both the $Z$ and $\gamma$. The coupling between the $N$
fields and the photon is a direct consequence of the underlying $N_{R}$-EFT.

##  Appendix C : Evaluating Coupling Using Dimension Six Operators

In this section we present the explicit expansion of various operators which
leads to different couplings that involves heavy $N_{R}$ fields. The dimension
six set up allows one to construct different class of operators which are
$\psi^{2}H^{3}$, $\psi^{2}H^{2}D$, $\psi^{2}H^{2}X$ and four fermi
respectively. The Higgs to neutrino couplings get substantial modification due
to the operator $\mathcal{O}_{LNH}$ which comes under $\psi^{2}H^{3}$ class.
The $\tilde{H}\left(H^{\dagger}H\right)$ part of the operator invokes
sufficient change in the neutrino mass matrix once the Higgs field acquires
_vev_ and break the electroweak symmetry. As an outcome of the EWSB, this
operator also introduce an Yukawa interaction between Higgs and neutral
leptons. Transforming the gauge basis to mass basis using the prescriptions
which is mentioned in Appendix. A, one can deduce the precise form of
interaction terms. The parameter $\tilde{\theta}$ is the redefined active
sterile mixing. In our presentation of different interaction terms involving
Higgs field, we stick to three point vertices.

$\begin{split}\mathcal{O}_{LNH}&=\frac{\alpha_{LNH}}{\Lambda^{2}}\left(\overline{L}N_{R}\right)\tilde{H}\left(H^{\dagger}H\right)+h.c.\\\
&\subset\frac{3v^{2}\alpha_{LNH}}{2\sqrt{2}\Lambda^{2}}h\left[-U^{\dagger}\tilde{\theta}^{\dagger}\overline{\nu}_{m}P_{R}\nu_{m}-\tilde{\theta}^{\dagger}\tilde{\theta}^{\dagger}\overline{N}_{m}P_{R}\nu_{m}+U^{\dagger}\kappa^{*}\overline{\nu}_{m}P_{R}N_{m}+\tilde{\theta}^{\dagger}\kappa^{*}\overline{N}_{m}P_{R}N_{m}\right]+h.c.\end{split}$
(43)

In Eq[43], we present the precise form of
$\mathcal{C}^{(6)}_{h\overline{\nu}\nu}$, $\mathcal{C}^{(6)}_{h\overline{N}N}$
and $\mathcal{C}^{(6)}_{h\left(\overline{\nu}N+\overline{N}\nu\right)}$
couplings respectively. Under the class of $\psi^{2}H^{2}D$ there are two
operators $\mathcal{O}_{HNe}$ and $\mathcal{O}_{HN}$. Here also we focus only
on the three point vertices involving gauge boson fields. The operator
$\mathcal{O}_{HNe}$ invokes coupling between the $W$ boson and the heavy
neutrinos.

$\displaystyle\mathcal{O}_{HNe}$
$\displaystyle=\frac{\alpha_{HNe}}{\Lambda^{2}}\left(\overline{N}_{R}\gamma^{\mu}e_{R}\right)\left(\tilde{H}^{\dagger}iD_{\mu}H\right)+h.c.\subset\frac{\sqrt{2}\alpha_{HNe}M^{2}_{W}}{g\Lambda^{2}}W^{+}_{\mu}\left[\overline{N}_{R}\gamma^{\mu}P_{R}e_{R}\right]+h.c.$
(44)
$\displaystyle=\frac{\sqrt{2}\alpha_{HNe}M^{2}_{W}}{g\Lambda^{2}}W^{+}_{\mu}\left[-\tilde{\theta}\overline{\nu}_{m}\gamma^{\mu}P_{R}e_{m}+\kappa^{T}\overline{N}_{m}\gamma^{\mu}P_{R}e_{m}\right]+h.c.$

In addition to that, this operator also generate additional term to the
charged gauge boson and SM leptons. The presence of right handed chiral matrix
suggest non standard interaction which indicates a distinct deviation from the
SM $SU(2)_{L}$ gauge properties. Using explicit definition of the
$H^{\dagger}\overleftrightarrow{D}H$ _i.e._
$H^{\dagger}D_{\mu}H-\left(D_{\mu}H\right)^{\dagger}H$ one can see the $N_{R}$
fields only couple the the neutral gauge bosons via three point interactions.

$\begin{split}\mathcal{O}_{HN}&=\frac{\alpha_{HN}}{\Lambda^{2}}\left(\overline{N}_{R}\gamma^{\mu}N_{R}\right)\left(iH^{\dagger}\overleftrightarrow{D}_{\mu}H\right)=-\left(\frac{M_{Z}v\alpha_{HN}}{\Lambda^{2}}\right)\left(\overline{N}_{R}\gamma^{\mu}N_{R}\right)Z_{\mu}\left[v+h\right]^{2}\\\
&\subset-\left(\frac{M_{Z}\alpha_{HN}v}{\Lambda^{2}}\right)Z_{\mu}\left[\tilde{\theta}\tilde{\theta}^{\dagger}\overline{\nu}_{m}\gamma^{\mu}P_{R}\nu_{m}-\kappa^{T}\tilde{\theta}^{\dagger}\overline{N}_{m}\gamma^{\mu}P_{R}\nu_{m}-\tilde{\theta}\kappa^{*}\overline{\nu}_{m}\gamma^{\mu}P_{R}N_{m}+\kappa^{T}\kappa^{*}\overline{N}_{m}\gamma^{\mu}P_{R}N_{m}\right]\end{split}$
(45)

In Eq[45], we present the explicit form of these couplings. One can notice
that, this operator modifies the coupling between the $Z$ boson and active
neutrino pair. This coupling substantially change the total decay width of the
$Z$ boson which is precisely measured. However, apart from the dimension six
operator co-efficient this coupling is also dependent on the square of the
mixing angle. Hence the bounds coming from the EWPO can be evaded by suitably
fixing the value of $\frac{\alpha_{HN}}{\Lambda^{2}}$ and $\tilde{\theta}$.
Similar to dimension five one can write down two more operators that include
stress-energy tensors. We begin with the operator which consider the
$B_{\mu\nu}$ tensor associated with the abelian $U(1)_{Y}$ group. In Eq[46],
we present the explicit form of this operator.

$\begin{split}\mathcal{O}_{NB}&=\frac{\alpha_{LNB}}{\Lambda^{2}}\overline{L}\sigma_{\mu\nu}N_{R}\tilde{H}B_{\mu\nu}+h.c.\subset\frac{\alpha_{LNB}v}{\sqrt{2}\Lambda^{2}}\left[c_{w}A_{\mu\nu}-s_{w}Z_{\mu\nu}\right]\left(\overline{\nu}_{L}\sigma_{\mu\nu}N_{R}\right)+h.c.\\\
&\subset\frac{\alpha_{LNB}v}{\sqrt{2}\Lambda^{2}}\left[c_{w}A_{\mu\nu}-s_{w}Z_{\mu\nu}\right][-U^{\dagger}\tilde{\theta}^{\dagger}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}-\tilde{\theta}^{\dagger}\tilde{\theta}^{\dagger}\overline{N}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}\\\
&+U^{\dagger}\kappa^{*}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}N_{m}+\tilde{\theta}^{\dagger}\kappa^{*}\overline{N}_{m}\sigma_{\mu\nu}P_{R}N_{m}]\end{split}$
(46)

From SM, one can rewrite the gauge field as the linear sum of photon and $Z$
boson fields. Using that one can express $B_{\mu\nu}$ as
$\left[c_{w}A_{\mu\nu}-s_{w}Z_{\mu\nu}\right]$. Hence this operator generates
different three point vertices between the leptons and the neutral gauge
bosons both massless and massive. The dimension six also allows us to build
operator that involve non-Abelian stress energy tensor. Depending on the
generators the operator $\mathcal{O}_{NW}$ can be re-expressed as the sum of
$\mathcal{O}^{\pm}_{NW}$ and $\mathcal{O}^{0}_{NW}$.

$\begin{split}\mathcal{O}_{NW}&=\mathcal{O}^{\pm}_{NW}+\mathcal{O}^{0}_{NW}=\left(\frac{\alpha_{LNW}}{\Lambda^{2}}\right)\left(\overline{L}\sigma_{\mu\nu}N_{R}\right)\tau^{I}\tilde{H}W^{I}_{\mu\nu}+h.c.\end{split}$
(47) $\displaystyle\mathcal{O}^{\pm}_{NW}$
$\displaystyle=\frac{v\alpha_{LNW}}{\sqrt{2}\Lambda^{2}}W^{\pm}_{\mu\nu}\left[-\tilde{\theta}^{\dagger}\overline{e}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}+\kappa^{*}\overline{e}_{m}\sigma_{\mu\nu}P_{R}N_{m}\right]+h.c.$
(48)
$\begin{split}\mathcal{O}^{0}_{NW}&=\frac{v\alpha_{LNW}}{\sqrt{2}\Lambda^{2}}\left[s_{w}A_{\mu\nu}+c_{w}Z_{\mu\nu}\right][-U^{\dagger}\tilde{\theta}^{\dagger}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}-\tilde{\theta}^{\dagger}\tilde{\theta}^{\dagger}\overline{N}_{m}\sigma_{\mu\nu}P_{R}\nu_{m}\\\
&+U^{\dagger}\kappa^{*}\overline{\nu}_{m}\sigma_{\mu\nu}P_{R}N_{m}+\tilde{\theta}\kappa^{*}\overline{N}_{m}\sigma_{\mu\nu}P_{R}N_{m}]\end{split}$
(49)

In Eq[47] and Eq[49] we present the analytic form of these operators in the
mass basis.

Now we turn our discussion to different four fermi operators. The operator
$\mathcal{O}_{QuNL}$ comes under the class of
$\left(\overline{L}R\right)\left(\overline{R}L\right)$.

$\displaystyle\mathcal{O}_{QuNL}$
$\displaystyle=\frac{\alpha_{LNQ}}{\Lambda^{2}}\left(\overline{Q}u_{R}\right)\left(\overline{N}_{R}L\right)+h.c.=\frac{\alpha_{LNQ}}{\Lambda^{2}}\begin{bmatrix}\overline{u}_{L}u_{R}&\overline{d}_{L}u_{R}\end{bmatrix}\begin{bmatrix}\overline{N}_{R}\nu_{\ell}\\\
\overline{N}_{R}e_{L}\end{bmatrix}+h.c.$ (50)
$\displaystyle=\frac{\alpha_{LNQ}}{\Lambda^{2}}\left(\overline{u}_{m}P_{R}u_{m}\right)\left(-\tilde{\theta}U\overline{\nu}_{m}P_{L}\nu_{m}+\kappa^{T}U\overline{N}_{m}P_{L}\nu_{m}-\tilde{\theta}\tilde{\theta}\overline{\nu}_{m}P_{L}N_{m}+\kappa^{T}\tilde{\theta}\overline{N}_{m}P_{L}N_{m}\right)$
$\displaystyle+\frac{\alpha_{LNQ}}{\Lambda^{2}}\left(\overline{d}_{m}P_{R}u_{m}\right)\left(-\tilde{\theta}\overline{\nu}_{m}P_{L}e_{m}+\kappa^{T}\overline{N}_{m}P_{L}e_{m}\right)+h.c.$

The operator $\mathcal{O}_{fN}$ that comes under
$\left(\overline{R}R\right)\left(\overline{R}R\right)$ class signifies
different scenario for different $f$. The label $f$ stands for various right
handed gauge singlet SM quarks or charged leptons.

$\displaystyle\left(\overline{R}R\right)\left(\overline{R}R\right)$
$\displaystyle\subset\mathcal{O}_{fN}=\frac{\alpha_{fN}}{\Lambda^{2}}\left(\overline{f}_{R}\gamma^{\mu}f_{R}\right)\left(\overline{N}_{R}\gamma_{\mu}N_{R}\right)$
(51)
$\displaystyle=\frac{\alpha_{fN}}{\Lambda^{2}}\left(\overline{f}_{m}\gamma^{\mu}P_{R}f_{m}\right)\left\\{\tilde{\theta}\tilde{\theta}^{\dagger}\overline{\nu}_{m}\gamma^{\mu}P_{R}\nu_{m}-\kappa^{T}\tilde{\theta}\overline{N}_{m}\gamma^{\mu}P_{R}\nu_{m}-\tilde{\theta}\kappa^{*}\overline{\nu}_{m}\gamma^{\mu}P_{R}N_{m}+\kappa^{T}\kappa^{*}\overline{N}_{m}\gamma^{\mu}P_{R}N_{m}\right\\}$

In addition to that one can construct another operator $\mathcal{O}_{duNe}$
which comes under $\left(\overline{R}R\right)\left(\overline{R}R\right)$ which
takes the following explicit form.

$\displaystyle\mathcal{O}_{duNe}$
$\displaystyle=\frac{\alpha_{duNe}}{\Lambda^{2}}\left(\overline{d}_{R}\gamma^{\mu}u_{R}\right)\left(\overline{N}_{R}\gamma_{\mu}e_{R}\right)+h.c.$
$\displaystyle=\frac{\alpha_{duNe}}{\Lambda^{2}}\left(\overline{d}_{m}\gamma^{\mu}P_{R}u_{m}\right)\left[-\tilde{\theta}\overline{\nu}_{m}\gamma_{\mu}P_{R}e_{m}+\kappa^{T}\overline{N}_{m}\gamma^{\mu}P_{R}e_{m}\right]+h.c.$
(52)

The $\mathcal{O}_{FN}$ stands for two different operators which takes the
generic form $\left(\overline{L}L\right)\left(\overline{R}R\right)$. The $F$
represents various SM-like quarks and lepton fields which transforms as a
doublet under $SU(2)_{L}$ gauge group.

$\begin{split}\left(\overline{L}L\right)\left(\overline{R}R\right)&\subset\mathcal{O}_{FN}=\frac{\alpha_{FN}}{\Lambda^{2}}\left(\overline{f}^{(1)}_{L}\gamma^{\mu}f^{(1)}_{L}+\overline{f}^{(2)}_{L}\gamma^{\mu}f^{(2)}_{L}\right)\left(\overline{N}_{R}\gamma_{\mu}N_{R}\right)\\\
&=\frac{\alpha_{FN}}{\Lambda^{2}}\left(\overline{f}^{(1)}_{m}\gamma^{\mu}P_{L}f^{(1)}_{m}+\overline{f}^{(2)}_{m}\gamma^{\mu}P_{L}f^{(2)}_{m}\right)\\{\tilde{\theta}\tilde{\theta}^{\dagger}\overline{\nu}_{m}\gamma^{\mu}P_{R}\nu_{m}-\kappa^{T}\tilde{\theta}\overline{N}_{m}\gamma^{\mu}P_{R}\nu_{m}\\\
&-\tilde{\theta}\kappa^{*}\overline{\nu}_{m}\gamma^{\mu}P_{R}N_{m}+\kappa^{T}\kappa^{*}\overline{N}_{m}\gamma^{\mu}P_{R}N_{m}\\}\end{split}$
(53)

The $f^{(1)}_{L}$ and $f^{(2)}_{L}$ denotes the up and down components of the
left-handed doublet fermions. We conclude this section with the remaining
three operators that comes under
$\left(\overline{L}R\right)\left(\overline{L}R\right)$. The $\epsilon_{ij}$ in
Eq[54] to Eq[56] is the $2\times 2$ matrix which is equal to $i\sigma_{2}$,
where $\sigma_{2}$ is the second Pauli matrix. The operator
$\mathcal{O}_{LNLe}$ is constructed only the lepton fields whereas other two
take both quarks and leptons into account.

$\begin{split}\mathcal{O}_{LNLe}&=\left(\frac{\alpha_{LNLe}}{\Lambda^{2}}\right)\left[\overline{L}_{i}N_{R}\right]\epsilon_{ij}\left[\overline{L}_{j}e_{R}\right]+h.c.\\\
&=\left(\frac{\alpha_{LNLe}}{\Lambda^{2}}\right)\left[\overline{\nu}_{L}N_{R}\right]\left[\overline{e}_{L}e_{R}\right]-\left(\frac{\alpha_{LNLe}}{\Lambda^{2}}\right)\left[\overline{e}_{L}N_{R}\right]\left[\overline{\nu}_{L}e_{R}\right]+h.c.\\\
&=\left(\frac{\alpha_{LNLe}}{\Lambda^{2}}\right)\left[-U^{\dagger}\tilde{\theta}^{\dagger}\overline{\nu}_{m}P_{R}\nu_{m}-\tilde{\theta}\tilde{\theta}\overline{N}_{m}P_{R}\nu_{m}+U^{\dagger}\kappa^{*}\overline{\nu}_{m}P_{R}N_{m}+\tilde{\theta}\kappa^{*}\overline{N}_{m}P_{R}N_{m}\right]\left(\overline{e}_{m}P_{R}e_{m}\right)\\\
&-\left(\frac{\alpha_{LNLe}}{\Lambda^{2}}\right)\left[-\tilde{\theta}^{\dagger}\overline{e}_{m}P_{R}\nu_{m}+\kappa^{*}\overline{e}_{m}P_{R}N_{m}\right]\left[U^{\dagger}\overline{\nu}_{m}P_{R}e_{m}+\tilde{\theta}\overline{N}_{m}P_{R}e_{m}\right]+h.c.\end{split}$
(54)
$\begin{split}\mathcal{O}_{LNQd}&=\left(\frac{\alpha_{LNQd}}{\Lambda^{2}}\right)\left[\overline{L}_{i}N_{R}\right]\epsilon_{ij}\left[\overline{Q}_{j}d_{R}\right]+h.c.\\\
&=\left(\frac{\alpha_{LNQd}}{\Lambda^{2}}\right)\left[\overline{\nu}_{L}N_{R}\right]\left[\overline{d}_{L}d_{R}\right]-\left(\frac{\alpha_{LNQd}}{\Lambda^{2}}\right)\left[\overline{e}_{L}N_{R}\right]\left[\overline{u}_{L}d_{R}\right]+h.c.\\\
&=\left(\frac{\alpha_{LNQd}}{\Lambda^{2}}\right)\left[-U^{\dagger}\tilde{\theta}\overline{\nu}_{m}P_{R}\nu_{m}-\tilde{\theta}^{\dagger}\tilde{\theta}^{\dagger}\overline{N}_{m}P_{R}\nu_{m}+U^{\dagger}\kappa^{*}\overline{\nu}_{m}P_{R}N_{m}+\tilde{\theta}\kappa^{*}\overline{N}_{m}P_{R}N_{m}\right]\left(\overline{d}_{m}P_{R}d_{m}\right)\\\
&-\left(\frac{\alpha_{LNQd}}{\Lambda^{2}}\right)\left[\kappa^{*}\overline{e}_{m}P_{R}e_{m}-\tilde{\theta}\overline{e}_{m}P_{R}e_{m}\right]\left(\overline{u}_{m}P_{R}d_{m}\right)\end{split}$
(55)
$\begin{split}\mathcal{O}_{LdQN}&=\left(\frac{\alpha_{LdQN}}{\Lambda^{2}}\right)\left[\overline{L}_{i}d_{R}\right]\epsilon_{ij}\left[\overline{Q}_{j}N_{R}\right]+h.c.\\\
&=\left(\frac{\alpha_{LdQN}}{\Lambda^{2}}\right)\left[\overline{\nu}_{L}d_{R}\right]\left[\overline{d}_{L}N_{R}\right]-\left(\frac{\alpha_{LdQN}}{\Lambda^{2}}\right)\left[\overline{e}_{L}d_{R}\right]\left[\overline{u}_{L}N_{R}\right]+h.c.\\\
&=\left(\frac{\alpha_{LdQN}}{\Lambda^{2}}\right)\left[U^{\dagger}\overline{\nu}_{m}P_{R}d_{m}+\tilde{\theta}^{\dagger}\overline{N}_{m}P_{R}d_{m}\right]\left[-\tilde{\theta}\overline{d}_{m}P_{R}\nu_{m}+\kappa^{*}\overline{d}_{m}P_{R}N_{m}\right]\
\\\
&-\left(\frac{\alpha_{LdQN}}{\Lambda^{2}}\right)\left[\overline{e}_{m}P_{R}d_{m}\right]\left[-\tilde{\theta}^{\dagger}\overline{u}_{m}P_{R}\nu_{m}+\kappa^{*}\overline{u}_{m}P_{R}N_{m}\right]\end{split}$
(56)

##  Appendix D Integration Formula

The phase space integrals which are required to evaluate the three body decay
modes of RHN field. The $\lambda$ here stands for the usual Kellen function of
the form $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$.

$\displaystyle\mathcal{I}_{1}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z}\left(z-x^{2}_{a}-x^{2}_{b}\right)\left(1+x^{2}_{c}-z\right)\lambda^{\frac{1}{2}}\left(1,z,x^{2}_{c}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{I}_{2}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=-\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z}x_{c}\left(z-x^{2}_{a}-x^{2}_{b}\right)\lambda^{\frac{1}{2}}\left(1,z,x^{2}_{c}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{I}_{3}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z^{2}}\\{x_{b}x_{c}\left(z+x^{2}_{a}-x^{2}_{b}\right)\left(1-x^{2}_{c}+z\right)-\frac{3}{2}x_{a}x_{a}\left(z-x^{2}_{a}+x^{2}_{b}\right)\left(1-x^{2}_{c}-z\right)\\}$
$\displaystyle~{}~{}~{}~{}\times\lambda^{\frac{1}{2}}\left(1,z,x^{2}_{c}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{I}_{4}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z^{2}}\\{\frac{3}{2}x_{c}x_{a}\left(z+x^{2}_{b}-x^{2}_{a}\right)\left(1-x^{2}_{c}+z\right)-2x_{b}\left(z+x^{2}_{a}-x^{2}_{b}\right)\left(1-x^{2}_{c}-z\right)\\}$
$\displaystyle~{}~{}~{}~{}\times\lambda^{\frac{1}{2}}\left(1,z,x^{2}_{c}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{I}_{5}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=-\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z}x_{c}x_{a}x_{b}\lambda^{\frac{1}{2}}\left(1,z,x^{2}_{c}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{H}_{1}\left(x_{a},x_{b}\right)$
$\displaystyle=\int^{4x^{2}_{b}}_{\left(1-x_{a}\right)^{2}}\frac{dz}{z}x^{2}_{b}\left(1+x^{2}_{a}-z\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},z\right)\lambda^{\frac{1}{2}}\left(z,x^{2}_{b},x^{2}_{b}\right)$
$\displaystyle\mathcal{H}_{2}\left(x_{a},x_{b}\right)$
$\displaystyle=\int^{4x^{2}_{b}}_{\left(1-x_{a}\right)^{2}}\frac{dz}{z}x_{b}\left(1-x^{2}_{a}-z\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},z\right)\lambda^{\frac{1}{2}}\left(z,x^{2}_{b},x^{2}_{b}\right)$
$\displaystyle\mathcal{H}_{3}\left(x_{a},x_{b}\right)$
$\displaystyle=-\int^{4x^{2}_{b}}_{\left(1-x_{a}\right)^{2}}\frac{dz}{z}x_{a}x_{b}\left(1-x^{2}_{a}+z\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{a},z\right)\lambda^{\frac{1}{2}}\left(z,x^{2}_{b},x^{2}_{b}\right)$
$\displaystyle\mathcal{G}_{1}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z^{2}}x_{a}\left(1-x^{2}_{c}-z\right)\left(z-x^{2}_{a}+x^{2}_{b}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{c},z\right)\lambda^{\frac{1}{2}}\left(z,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{G}_{2}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=-\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z^{2}}x_{c}x_{a}\left(1-x^{2}_{c}+z\right)\left(z-x^{2}_{a}+x^{2}_{b}\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{c},z\right)\lambda^{\frac{1}{2}}\left(z,x^{2}_{a},x^{2}_{b}\right)$
$\displaystyle\mathcal{G}_{3}\left(x_{a},x_{b},x_{c}\right)$
$\displaystyle=-\int^{\left(1-x_{c}\right)^{2}}_{\left(x_{a}+x_{b}\right)^{2}}\frac{dz}{z}x_{a}x_{b}\left(1+x^{2}_{c}-z\right)\lambda^{\frac{1}{2}}\left(1,x^{2}_{c},z\right)\lambda^{\frac{1}{2}}\left(z,x^{2}_{a},x^{2}_{b}\right)$

##  Appendix E Collider Signatures For RHN fields

In this section we will present different possible signatures involving RHN
field for $pp$, $e^{+}e^{-}$ and $e^{-}p$ colliders. In Section. 6, we
discussed few of the multi-lepton final states where the RHN fields produce
either via single or via pair production mechanism and then subsequently decay
to three body pure leptonic channels. In addition to those channels one can
also look for the RHN fields in other final states. In Table. 10, we present
different production mode for the RHN fields for $pp$ colliders. The first row
and first column of the Table. 10 illustrate the flavour label of the final
state leptons and quarks.

Production Decay | $N_{i}\to\ell_{j}\ell_{k}\nu_{k}$ | $N_{i}\to\ell_{k}\ell_{k}\nu_{k}$ | $N_{i}\to\nu_{j}\ell_{k}\ell_{k}$ | $N_{i}\to\ell_{j}u_{\alpha}d_{\beta}$ | $N_{i}\to\nu_{j}q_{\alpha}q_{\alpha}$ | $N_{i}\to\nu_{j}\nu_{k}\nu_{k}$
---|---|---|---|---|---|---
$gg\to h\to\nu_{\delta}N_{i}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $\ell+2q+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$
$qq^{\prime}\to W\to\ell_{\delta}N_{i}$ | $3\ell+\not{E}_{T}$ | $3\ell+\not{E}_{T}$ | $3\ell+\not{E}_{T}$ | $2\ell+2q$ | $\ell+2q+\not{E}_{T}$ | $\ell+\not{E}_{T}$
$q\bar{q}\to Z\to\nu_{\delta}N_{i}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $\ell+2q+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$
$pp\to Zh$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $3\ell+2q$ | $2\ell+2q$ | $2\ell+\not{E}_{T}$
$\to\ell_{\rho}\ell_{\rho}\nu_{\delta}N_{i}$ |  |  |  | $\not{E}_{T}$ | $\not{E}_{T}$ |
$pp\to Zh$ | $2\ell+2b$ | $2\ell+2b$ | $2\ell+2b$ | $\ell+2b$ | $2b+2q$ | $2b+\not{E}_{T}$
$\to\nu_{\delta}N_{i}2b$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$ |
$pp\to\nu_{\delta}N_{i}$ | $2\ell$ | $2\ell$ | $2\ell$ | $\ell+2q$ | $2q$ | $\ell$
via Four-Fermi | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$
VBF $\to Wqq^{\prime}\to\ell_{\delta}N_{i}qq^{\prime}$ | $3\ell+2q+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $2\ell+4q$ | $\ell+4q+\not{E}_{T}$ | $\ell+2q+\not{E}_{T}$
VBF $\to Zqq^{\prime}\to\nu_{\delta}N_{i}qq^{\prime}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $\ell+4q+\not{E}_{T}$ | $4q+\not{E}_{T}$ | $2q+\not{E}_{T}$
VBF $\to hqq^{\prime}\to\nu_{\delta}N_{i}qq^{\prime}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $\ell+4q+\not{E}_{T}$ | $4q+\not{E}_{T}$ | $2q+\not{E}_{T}$
Table 10: Different collider signatures for the single RHN production at $pp$
collider. Apart from the $\text{VBF}\to
Wqq^{\prime}\to\ell_{\delta}N_{i}qq^{\prime}$ and Drell-Yan $qq^{\prime}\to
W\to\ell_{\delta}N_{i}$ process all the other channels can be used for the
pair production of the RHN fields. For details regarding the final states
correspond to $N$-pair production see the text.

Similarly in Table. 11 we tabulate different single production of $N$ fields
at $e^{+}e^{-}$ colliders. The commonalities between Table. 10 and Table. 11
is that we only highlight the single $N$ field production. Majority of the
channels which are presented in these two Tables can serve as the pair
production mode for the RHN fields. For example in case of $pp$ collider apart
from $\text{VBF}\to Wqq^{\prime}\to\ell_{\delta}N_{i}qq^{\prime}$ and Drell-
Yan $qq^{\prime}\to W\to\ell_{\delta}N_{i}$ process all the other modes can
potentially produce pair of RHN fields. On other hand in case of $e^{+}e^{-}$
collider apart from the t-channel $W$ boson mediated process all the other
process can generate pair of $N$ fields. Using Table. 12 one can evaluate the
final state signature for these modes. In Table. 12 we only illustrate the
final states that can arise due to the subsequent decay of both $N_{i}$ and
$N_{j}$. Moreover, at the production level other SM particle can arise in
association with the RHN fields. Hence one need to put suitable particles in
place of $X$ as mentioned in the first row of Table. 12.

Production Decay | $N_{i}\to\ell_{j}\ell_{k}\nu_{k}$ | $N_{i}\to\ell_{k}\ell_{k}\nu_{k}$ | $N_{i}\to\nu_{j}\ell_{k}\ell_{k}$ | $N_{i}\to\ell_{j}u_{\alpha}d_{\beta}$ | $N_{i}\to\nu_{j}q_{\alpha}q_{\alpha}$ | $N_{i}\to\nu_{j}\nu_{k}\nu_{k}$
---|---|---|---|---|---|---
$e^{+}e^{-}\to\nu_{\delta}N_{i}$ | $2\ell$ | $2\ell$ | $2\ell$ | $\ell+2q$ | $2q$ | $\ell$
via Four-Fermi | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$
$e^{+}e^{-}\to Zh$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $3\ell+2q$ | $2\ell+2q$ | $2\ell+\not{E}_{T}$
$\to\ell_{\rho}\ell_{\rho}\nu_{\delta}N_{i}$ |  |  |  | $\not{E}_{T}$ | $\not{E}_{T}$ |
$e^{+}e^{-}\to Zh$ | $2\ell+2b$ | $2\ell+2b$ | $2\ell+2b$ | $\ell+2b$ | $2b+2q$ | $2b+\not{E}_{T}$
$\to\nu_{\delta}N_{i}2b$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$ |
$e^{+}e^{-}\to Z\to\nu_{\delta}N_{i}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $\ell+2q+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$
VBF $\to Z\nu_{\rho}\nu_{\delta}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $\ell+2q$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$
$\to N_{i}\nu_{\sigma}\nu_{\rho}\nu_{\delta}$ |  |  |  | $+\not{E}_{T}$ |  |
VBF $\to W\ell_{\delta}\nu_{\rho}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $3\ell+2q$ | $2\ell+2q$ | $2\ell+\not{E}_{T}$
$\to\ell_{\delta}\nu_{\rho}N_{i}\ell_{\sigma}$ |  |  |  | $\not{E}_{T}$ | $\not{E}_{T}$ |
$e^{+}e^{-}\to W\to\ell_{\delta}N_{i}$ | $3\ell+\not{E}_{T}$ | $3\ell+\not{E}_{T}$ | $3\ell+\not{E}_{T}$ | $2\ell+2q$ | $\ell+2q$ | $\ell+\not{E}_{T}$
t channel |  |  |  | \+ $\not{E}_{T}$ | \+ $\not{E}_{T}$ |
Table 11: Different collider signatures for the single RHN production at
$e^{+}e^{-}$ collider. Apart from the t-channel $W$ boson mediated process all
the other modes can serve as the pair production mode for RHN fields. For
details regarding the final states correspond to $N$-pair production see the
text.

We like to elaborate this point with two suitable examples. For the process
$gg\to h\to N_{i}N_{j}$ the possible final states are the states that are
mentioned in this Table and one does not need to put any SM fields in place of
$X$. On the other hand, in case of $pp\to
Zh\to\ell_{\rho}\ell_{\rho}N_{i}N_{j}$, the $X$ would be replaced with
$\ell_{\rho}\ell_{\rho}$ and to obtain the final states correspond to this
process one needs to add $2\ell$ in each entires of Table.12.

Possible Production Mode |  $pp\to N_{i}N_{j}X$ and $e^{+}e^{-}\to N_{i}N_{j}X$
---|---
$N_{j}$ Decay $N_{i}$ Decay | $N_{i}\to\ell_{j}\ell_{k}\nu_{k}$ | $N_{i}\to\ell_{k}\ell_{k}\nu_{k}$ | $N_{i}\to\nu_{j}\ell_{k}\ell_{k}$ | $N_{i}\to\ell_{j}u_{\alpha}d_{\beta}$ | $N_{i}\to\nu_{j}q_{\alpha}q_{\alpha}$ | $N_{i}\to\nu_{j}\nu_{k}\nu_{k}$
$N_{j}\to\ell_{a}\ell_{b}\nu_{b}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+\not{E}_{T}$
$N_{j}\to\ell_{b}\ell_{b}\nu_{b}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+\not{E}_{T}$
$N_{j}\to\ell_{a}\ell_{a}\nu_{b}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $4\ell+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+\not{E}_{T}$
$N_{j}\to\ell_{a}u_{\rho}d_{\delta}$ | $3\ell+2q+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $3\ell+2q+\not{E}_{T}$ | $2\ell+4q+\not{E}_{T}$ | $\ell+4q+\not{E}_{T}$ | $\ell+2q+\not{E}_{T}$
$N_{j}\to\nu_{a}q_{\rho}q_{\rho}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $2\ell+2q+\not{E}_{T}$ | $\ell+4q+\not{E}_{T}$ | $4q+\not{E}_{T}$ | $2q+\not{E}_{T}$
$N_{j}\to\nu_{a}\nu_{b}\nu_{b}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $2\ell+\not{E}_{T}$ | $\ell+2q+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $\not{E}_{T}$
Table 12: Different possible final states that can arise due to the subsequent
three body decays of RHN pair. The $X$ in the first row signifies the SM
fields that can generate due the underlying production mechanism in
association with RHN pair. For explicit evaluation one should replace $X$ with
appropriated field content.

In Table. 13, we present different RHN production modes relevant for $e^{-}p$
collider. Here we will restrict ourselves to single production of the $N$
fields. In principle one can generate more than one RHN fields in this
collider. However, the processes related to these would involve more than one
EFT vertices. Hence the cross section would receive higher order cut off scale
suppression (see Fig. 8 for details).

Production Decay | $N_{i}\to\ell_{j}\ell_{k}\nu_{k}$ | $N_{i}\to\ell_{k}\ell_{k}\nu_{k}$ | $N_{i}\to\nu_{j}\ell_{k}\ell_{k}$ | $N_{i}\to\ell_{j}u_{\alpha}d_{\beta}$ | $N_{i}\to\nu_{j}q_{\alpha}q_{\alpha}$ | $N_{i}\to\nu_{j}\nu_{k}\nu_{k}$
---|---|---|---|---|---|---
$ep\to W\to q_{\delta}N_{i}$ | $2\ell+q+\not{E}_{T}$ | $2\ell+q+\not{E}_{T}$ | $2\ell+q+\not{E}_{T}$ | $\ell+3q$ | $3q+\not{E}_{T}$ | $q+\not{E}_{T}$
via t- channel |  |  |  |  |  |
$ep\to q_{\delta}N_{i}$ | $2\ell+q+\not{E}_{T}$ | $2\ell+q+\not{E}_{T}$ | $2\ell+q+\not{E}_{T}$ | $\ell+3q$ | $3q+\not{E}_{T}$ | $q+\not{E}_{T}$
via Four-Fermi |  |  |  |  |  |
VBF $\to h\nu_{\rho}q_{\delta}$ | $2\ell+q+\not{E}_{T}$ | $2\ell+q+\not{E}_{T}$ | $2\ell+q+\not{E}_{T}$ | $\ell+3q+\not{E}_{T}$ | $3q+\not{E}_{T}$ | $q+\not{E}_{T}$
$\to\nu_{\sigma}N_{i}\nu_{\rho}q_{\delta}$ |  |  |  |  |  |
$ep\to tbN_{i}\to N_{i}+2b$ | $2\ell+2b$ | $2\ell+2b$ | $2\ell+2b$ | $2\ell+2b$ | $\ell+2b$ | $\ell+2b$
$+q_{\delta}+q_{\sigma}$ | $+2q+\not{E}_{T}$ | $+2q+\not{E}_{T}$ | $+2q+\not{E}_{T}$ | $+4q+\not{E}_{T}$ | $2q+\not{E}_{T}$ | $+\not{E}_{T}$
$ep\to tbN_{i}\to N_{i}+2b$ | $3\ell+2b$ | $3\ell+2b$ | $3\ell+2b$ | $2\ell+2q$ | $\ell+2q$ | $\ell+2b$
$+\ell_{\delta}\nu_{\rho}$ | +$\not{E}_{T}$ | +$\not{E}_{T}$ | \+ $\not{E}_{T}$ | $2b+\not{E}_{T}$ | $2b+\not{E}_{T}$ | $\not{E}_{T}$
Table 13: Different collider signatures for the single RHN production at
$e^{-}p$ collider.

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