# Effects of heavy neutrinos on vacuum stability in two-Higgs-doublet model
with GUT scale supersymmetry

V Suryanarayana Mummidi<EMAIL_ADDRESS>Vishnu P. K
<EMAIL_ADDRESS>Ketan M. Patel<EMAIL_ADDRESS>Indian
Institute of Science Education and Research Mohali, Knowledge City, Sector 81,
S A S Nagar, Manauli 140306, India.

###### Abstract

We analyse the implications of right-handed neutrinos on the stability of the
electroweak vacuum in two-Higgs-doublet models with supersymmetry at high
scale. It is assumed that supersymmetry is broken at scale $M_{S}=2\times
10^{16}$ GeV and effective theory below $M_{S}$ is two-Higgs-doublet model of
type II with three generations of singlet neutrinos which induce small masses
for the standard model neutrinos through type I seesaw mechanism. We study the
high and low scale versions of seesaw mechanism. In both these cases, we show
that the presence of right-handed neutrinos significantly improves the
stability of electroweak vacuum if their Yukawa couplings with the SM leptons
are of ${\cal O}(1)$ or greater. However, this possibility is severely
constrained by the measured mass and couplings of Higgs and limits on the mass
of the charged Higgs from the flavour physics data. It is shown that the
stable or metastable electroweak vacuum and experimentally viable low energy
scalar spectrum require
$\tan\beta\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>2.5$
and the magnitude of neutrino Yukawa couplings smaller than ${\cal O}(1)$. The
results obtained in this case are qualitatively similar to those without
right-handed neutrinos.

##  I Introduction

An embedding of the Standard Model (SM) into supersymmetric grand unified
theories (GUTs) leads to an elegant and predictive framework which resolves
several technical problems obstructing the extendability of the SM gauge
theory to very short length scales. Supersymmetry (SUSY), close to TeV scale,
stabilizes the electroweak scale against the large radiative corrections that
arise due to higher fundamental scales present in the theory. TeV scale SUSY
provides precision gauge coupling unification at the scale $M_{\rm GUT}\approx
2\times 10^{16}$ GeV, although this can also be achieved if SUSY is broken at
very high scale but some of the super-partners remain as light as few hundred
GeV Giudice and Romanino (2004); Arkani-Hamed and Dimopoulos (2005). SUSY as a
local symmetry is an essential ingredient in the superstring theory which
provides a potential framework for unification of all the fundamental forces
Green  _et al._ (1988). However, it is generically expected that the SUSY
breaking scale in such a theory would be very close to the string scale. A
similar situation arises in supersymmetric GUTs constructed in five or six
dimensional spacetime in which the mechanisms used for breaking the unified
gauge symmetry often break also the supersymmetry Kitano and Li (2003);
Buchmuller  _et al._ (2015)111Note that in Kitano and Li (2003), breaking of
supersymmetry at high scale was prevented by introducing brane localized
$D$-terms. Supersymmetry is broken at the GUT scale in absence of such terms..
In these frameworks, the effective theory below the GUT scale is described by
non-supersymmetric SM which in some cases augmented by other possible light
states as remnants of an underlying ultraviolet complete theory.

The experimental data collected in the first and second runs of the Large
Hadron Collider (LHC) has shown no evidence for TeV scale supersymmetry Aaboud
_et al._ (2017); Sirunyan  _et al._ (2017). If this trend continues in the
future runs of LHC and other experiments then it would imply that
supersymmetry is not an underlying mechanism for stabilization of electroweak
scale. In this case, the assumption of existence of a weak scale supersymmetry
is no longer necessarily required. Following this, in this paper we assume
that SUSY exists in an underlying theory but it is broken at the scale much
above the electroweak scale. It is assumed that supersymmetry does not play
any role in stabilizing the electroweak scale and the electroweak scale
remains finely tuned or some new dynamics in the high energy theory take care
of the gauge hierarchy problem. Although an absence of SUSY at the low energy
seems to make it less interesting from the phenomenological point of view,
however its existence at high scale still leads to nontrivial consequences on
the low energy theory. For example, the scalar potential of an effective
theory below the SUSY breaking scale arises from the D-term potential of an
underlying ultraviolet supersymmetric theory. Therefore, the electroweak
symmetry breaking, stability of electroweak vacuum and mass spectrum of
scalars in the effective theory are constrained by the high scale SUSY. Such
consequences are already studied for an effective theory being only the SM
Giudice and Strumia (2012); Elias-Miro  _et al._ (2012); Draper  _et al._
(2014); Ellis and Wells (2017), SM with additional Higgs doublet Gorbahn  _et
al._ (2011); Lee and Wagner (2015), and SM with higgsinos and gauginos
Bagnaschi  _et al._ (2014).

It is observed that the SM alone as an effective theory cannot be matched to
its minimal supersymmetric version (MSSM) when SUSY breaking scale is higher
than $10^{11}$ GeV because of the vacuum stability constraints Giudice and
Strumia (2012); Elias-Miro  _et al._ (2012)222The quoted limit is obtained by
considering the mean value of the measured top quark mass. The limit is very
sensitive with respect to the choice made for the value of top quark mass
Draper  _et al._ (2014); Ellis and Wells (2017).. The SM with an additional
Higgs doublet close to the electroweak scale, known as two-Higgs-doublet model
(see Branco  _et al._ (2012) for a review), is another possible effective
theory with SUSY at high scale. The pair of scalar doublets in the two-Higgs-
doublet model (THDM) can be identified as the two Higgs doublets of the MSSM.
Such a matching between MSSM and THDM at high SUSY breaking scale has already
been considered in Gorbahn  _et al._ (2011); Lee and Wagner (2015); Bagnaschi
_et al._ (2016); Bhattacharyya  _et al._ (2017). It is found in Bagnaschi  _et
al._ (2016) that THDM can be consistently matched to MSSM with SUSY breaking
scale as high as the reduced Planck scale. This improvement over the SM is due
to the presence of an additional Higgs doublet which modifies the stability
conditions allowing more freedom in the effective potential. The stability or
metastability of the electroweak vacuum however puts stringent constraints on
the allowed values of $\tan\beta$ parameter which is the ratio of the vacuum
expectation value (VEV) of two Higgs doublets.

In this paper, we investigate the effects of the so-called right handed (RH)
neutrinos on the stability of scalar potential in THDM with supersymmetry
broken at GUT scale. The SM augmented with such singlet fermions provides
natural explanation for non-vanishing and tiny neutrino masses through type I
seesaw mechanism Minkowski (1977); Yanagida (1979); Gell-Mann  _et al._
(1979); Glashow (1980); Lazarides  _et al._ (1981); Schechter and Valle
(1980); Mohapatra and Senjanovic (1981). In the GUTs based on $SO(10)$ gauge
group the RH neutrinos reside, along with the SM fermions, in three copies of
16-dimensional irreducible representation of the gauge group. In these models,
if SUSY and gauge symmetry are both broken at the GUT scale leaving a pair of
MSSM Higgs doublets light then the effective theory below the GUT scale is
described by THDM with three generations of RH neutrinos. For example, this
possibility is naturally realized in the GUT models based on flux
compactification Buchmuller  _et al._ (2015); Buchmuller and Schweizer (2017);
Buchmuller and Patel (2018). Motivated by this, we assume that the MSSM is
broken at the scale $M_{S}=2\times 10^{16}$ GeV leaving THDM augmented with
three generations of RH neutrinos as an effective theory below $M_{S}$. The RH
neutrinos obtain their masses through lepton number violating interactions
which in turn induce tiny masses for the SM neutrinos through type I seesaw
mechanism. We consider two distinct possibilities in which the mass scale of
RH neutrinos is either close to $M_{S}$ or electroweak scale.

We find that RH neutrinos have considerable effects on the stability of
electroweak vacuum. In particular, if these neutrinos are strongly coupled
with the SM leptons then they lead to significant improvements in the
stability of the scalar potential in THDM. It is shown that the stable vacuum
can be achieved for almost any value of $\tan\beta$ if the magnitude of Yukawa
couplings of RH neutrinos are larger than that of the top quark Yukawa
coupling. However, the observed Higgs mass, measured couplings of Higgs with
the gauge bosons and limits on the charged Higgs mass from flavour physics
data severely constrain on this scenario.

The paper is organized as follows. We discuss the THDM framework with type I
seesaw mechanism in the next section. The procedure of renormalization group
(RG) evolution and matching at different scales have been discussed in section
III. Numerical analysis and their results are discussed in section IV. The
conclusion is presented in section V. Technical details related to
renormalization group equations, threshold corrections at the high scale,
extraction of the gauge and Yukawa couplings at top quark mass scale and
dependency of results on the choice of top quark mass are elaborated in the
Appendices.

##  II The Framework

The scalar potential of the most general THDM can be parametrized as

$\displaystyle V$ $\displaystyle=$ $\displaystyle
m_{1}^{2}\,H_{1}^{\dagger}H_{1}\,+\,m_{2}^{2}\,H_{2}^{\dagger}H_{2}\,-\,\left(m_{12}^{2}\,H_{1}^{\dagger}H_{2}\,+\,{\rm
H.c.}\right)$ (1) $\displaystyle+$
$\displaystyle\frac{\lambda_{1}}{2}\,(H_{1}^{\dagger}H_{1})^{2}\,+\,\frac{\lambda_{2}}{2}\,(H_{2}^{\dagger}H_{2})^{2}\,+\,\lambda_{3}\,(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})\,+\,\lambda_{4}\,(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})$
$\displaystyle+$
$\displaystyle\left(\frac{\lambda_{5}}{2}\,(H_{1}^{\dagger}H_{2})^{2}\,+\,\lambda_{6}\,(H_{1}^{\dagger}H_{1})(H_{1}^{\dagger}H_{2})\,+\,\lambda_{7}\,(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{2})\,+\,{\rm
H.c.}\right)\,,$

where $H_{1}$ and $H_{2}$ are two complex Higgs fields, each of them is a
doublet under $SU(2)_{L}$ and carries hypercharge $Y=1$. With an addition of
three copies of fermion singlet to the SM fermion spectrum, the most general
Yukawa Lagrangian of the model can be written as

$\displaystyle-{\cal L}_{Y}$ $\displaystyle=$
$\displaystyle\overline{Q}_{L}^{i}\left(Y_{d}^{ij}H_{1}+\tilde{Y}_{d}^{ij}H_{2}\right)d_{R}^{j}\,+\,\overline{Q}_{L}^{i}\left(\tilde{Y}_{u}^{ij}\tilde{H}_{1}+\,Y_{u}^{ij}\tilde{H}_{2}\right)u_{R}^{j}$
(2) $\displaystyle+$
$\displaystyle\,\overline{L}_{L}^{i}\left(Y_{e}^{ij}H_{1}+\tilde{Y}_{e}^{ij}H_{2}\right)e_{R}^{j}\,+\,\overline{L}_{L}^{i}\left(\tilde{Y}_{\nu}^{ij}\tilde{H}_{1}+\,Y_{\nu}^{ij}\tilde{H}_{2}\right)\nu_{R}^{j}\,+\,{\rm
H.c.}\,,$

where $i,j=1,2,3$ stand for three generations of fermions and
$\tilde{H}_{1,2}=i\sigma^{2}H^{*}_{1,2}$.

If THDM is assumed to be an effective theory, obtained from the MSSM after the
SUSY is broken at the scale $M_{S}$, then the scalar potential in Eq. (1) is
matched to the MSSM Higgs potential at $M_{S}$. The potential of the MSSM
Higgs doublets $H_{u}$ and $H_{d}$ (with hypercharge $Y=1$ and $Y=-1$,
respectively) contains the D-terms of superpotential and the soft
supersymmetry breaking terms. Setting $\tilde{H}_{1}=H_{d}$ and $H_{2}=H_{u}$,
tree level matching between the potentials leads to the following conditions
at $M_{S}$ Haber and Hempfling (1993); Lee and Wagner (2015):

$\lambda_{1}=\lambda_{2}=\frac{1}{4}\left(g_{2}^{2}+g_{Y}^{2}\right)\,,~{}~{}\lambda_{3}=\frac{1}{4}\left(g_{2}^{2}-g_{Y}^{2}\right)\,,~{}~{}\lambda_{4}=-\frac{1}{2}g_{2}^{2}\,,$
(3) $\lambda_{5}=\lambda_{6}=\lambda_{7}=0\,,$ (4)

where $g_{Y}=\sqrt{3/5}\,g_{1}$. Further, the terms involving
$H_{d}^{\dagger}$ and $H_{u}^{\dagger}$ in the Yukawa Lagrangian are absent at
the scale $M_{S}$. This implies

$\tilde{Y}_{d}^{ij}=\tilde{Y}_{u}^{ij}=\tilde{Y}_{e}^{ij}=\tilde{Y}_{\nu}^{ij}=0\,$
(5)

in Eq. (2) at the scale $M_{S}$. The conditions in Eqs. (4,5) imply that the
theory at $M_{S}$ is essentially type II THDM Branco  _et al._ (2012) with
additional boundary conditions on the scalar quartic couplings as given in Eq.
(3).

We assume that the supersymmetry breaking scale is very close to the GUT
scale. If the type I seesaw mechanism is considered as an underlying mechanism
to generate tiny masses for the SM neutrinos, it introduces new scales in the
theory, namely the mass thresholds of RH neutrinos. The Majorana masses for
the RH neutrinos can be written as

$-{\cal L}_{\nu_{R}}=\frac{1}{2}M_{R}^{ij}\,\nu_{R}^{Ti}{\cal
C}\nu_{R}^{j}\,+\,{\rm H.c.}\,.$ (6)

We denote the physical masses of RH neutrinos by $M_{R_{i}}$ with $i=1,2,3$
and adopt a convention in which $M_{R_{1}}\leq M_{R_{2}}\leq M_{R_{3}}$. It is
typically expected that $M_{R_{i}}$ lie in between the electroweak and GUT
scale. For $M_{R_{1}}\gg\langle H_{u}\rangle$, the effective light neutrino
mass matrix becomes

${\cal M}_{\nu}=-\langle
H_{u}\rangle^{2}\,Y_{\nu}\,M_{R}^{-1}\,Y_{\nu}^{T}\,.$ (7)

The $Y_{\nu}$ and $M_{R}$ cannot be fixed uniquely from the available
experimental information of neutrino masses and mixing parameters. This lack
of information is best parametrized by the Casas-Ibarra parametrization Casas
and Ibarra (2001) in which the Dirac neutrino Yukawa coupling matrix in the
diagonal basis of the charged lepton and RH neutrino mass matrices is
expressed as

$Y_{\nu}=\frac{1}{\langle H_{u}\rangle}\,U_{\rm PMNS}\,{\cal D}_{\nu}\,{\cal
R}\,{\cal D}_{R}\,.$ (8)

Here $U_{\rm PMNS}$ is the leptonic mixing matrix, ${\cal D}_{R}={\rm
Diag.}(\sqrt{M_{R_{1}}},\sqrt{M_{R_{2}}},\sqrt{M_{R_{3}}})$, ${\cal
D}_{\nu}={\rm
Diag.}(\sqrt{m_{\nu_{1}}},\sqrt{m_{\nu_{2}}},\sqrt{m_{\nu_{3}}})$ and
$m_{\nu_{i}}$ are the light neutrino masses. ${\cal R}$ is an unknown complex
orthogonal matrix which parametrize the freedom in choice of $Y_{\nu}$ allowed
by the seesaw formula, Eq. (7). We however do not consider this general case
but discuss two phenomenologically interesting limits as described in the
following.

###  II.1 High scale seesaw (HSS)

In $SO(10)$ based GUTs, all the quarks and leptons of a given generation are
embedded in a single ${\bf 16}$-dimensional irreducible spinorial
representation of the gauge group. In most of the situations, such a
unification leads to an approximate equality between $Y_{\nu}$ and $Y_{u}$.
For example, the renormalizable versions of supersymmetric $SO(10)$ models
with one or more ${\bf 10}$-plet Higgs in the Yukawa sector always imply
$Y_{\nu}=Y_{u}$ at the GUT scale Fritzsch and Minkowski (1975); Babu and
Mohapatra (1993). The exact equality between $Y_{u}$ and $Y_{\nu}$ at the GUT
scale is broken if the underlying model contains higher dimensional Higgs
representations, such as ${\bf\overline{126}}$ and/or ${\bf 120}$, or if the
corrections from higher order non-renormalizable operators are taken into
consideration. In many of these cases an approximate relation, $Y_{\nu}\approx
Y_{u}$, still holds (see Joshipura and Patel (2011) for example). There also
exists a possibility in which the hierarchy among the couplings in $Y_{\nu}$
is widely different from those in $Y_{u}$. This situation is known to arise
from the orbifolded $SO(10)$ GUTs in five or six spacetime dimensions Feruglio
_et al._ (2014, 2015); Buchmuller and Patel (2018). If fermions are kept in
the bulk and Higgs is localized on the brane, then the effective Yukawa
coupling matrix in four dimensional theory is given by, for example
$Y_{f}=F_{f_{L}}{\cal Y}_{f}F_{f_{R}}$ where ${\cal Y}_{f}$ is typically a
matrix with elements of order unity, and $F_{f_{L}}$, $F_{f_{R}}$ are diagonal
matrices with elements representing the values of profile factors at the given
four dimensional fixed point. The later decides the inter-generational mass
hierarchies in a given fermion sector. In general, if the GUT symmetry is
broken by orbifolding then $F_{Q_{L}}\neq F_{L_{L}}$ and $F_{u_{R}}\neq
F_{\nu_{R}}$. Therefore, the resulting $Y_{\nu}$ and $Y_{u}$ can have very
different hierarchical structure and/or relative strength of magnitude
Feruglio  _et al._ (2014, 2015).

To accommodate these possibilities, we generically parametrize $Y_{\nu}$ at
$M_{S}$ as:

$Y_{\nu}=\epsilon\,{\rm Diag.}\left(\xi^{N},\,\xi,\,1\right)\,Y_{u}\,,$ (9)

where $\epsilon$, $\xi$ and $N$ are real numbers which determine the relative
strength of Dirac Yukawa couplings of neutrinos with respect those of up type
quarks. Since the largest coupling in $Y_{u}$ is already of ${\cal O}(y_{t})$,
very large value of $\epsilon$ leads to non-perturbative $Y_{\nu}$. We
therefore consider $\epsilon\in[0.1-10]$, i.e. at most an order of magnitude
difference between $Y_{u}$ and $Y_{\nu}$. The parameters $\xi$ and $N$
determine the hierarchical structure of couplings in $Y_{\nu}$. For $\xi=1$,
one obtains the hierarchy in $Y_{\nu}$ same as that in $Y_{u}$. Different
hierarchical structure for $Y_{\nu}$ can be obtained using suitably chosen
values of $\xi$ and $N$. For the above values of $\epsilon$, Eq. (7) leads to
the masses of RH neutrinos in the range $10^{7}$ \- $10^{16}$ GeV. This case
is therefore named as the high scale seesaw (HSS) case. Since some of the
couplings in $Y_{\nu}$ are of ${\cal O}(y_{t})$ or large, one expects
considerable running effects from the RH neutrinos even though their masses
are close to $M_{S}$.

###  II.2 Low scale seesaw (LSS)

The running effects due to RH neutrinos are enhanced if the seesaw scale is
close to the electroweak scale and they are strongly coupled with the THDM. A
usual way to accommodate low seesaw scale is to consider $\epsilon\ll 1$ in
Eq. (9) which in turn decreases the masses of RH neutrinos by a factor of
$\epsilon^{2}$, as it can be seen from Eq. (7). Small $\epsilon$ however makes
RH neutrinos very weakly coupled with the THDM and their effects on the
running of couplings become negligible. There exists an alternate approach in
which the low seesaw scale can be realized with ${\cal O}(1)$ couplings in
$Y_{\nu}$ Kersten and Smirnov (2007); Lee  _et al._ (2013); Chattopadhyay and
Patel (2017). In this case, the smallness of the SM neutrino masses is
attributed to the flavour structure of $Y_{\nu}$ and $M_{R}$ instead of the
scale of RH neutrino masses or strength of couplings in $Y_{\nu}$. Due to the
matrix structure of the seesaw formula, it is possible to choose the form of
$Y_{\nu}$ and $M_{R}$ such that the Eq. (7) leads to vanishing ${\cal
M}_{\nu}$. In the diagonal basis of RH neutrinos, they can be written as

$Y_{\nu}=\left(\begin{array}[]{ccc}y_{1}&iy_{1}&0\\\ y_{2}&iy_{2}&0\\\
y_{3}&iy_{3}&0\end{array}\right)\,,~{}~{}~{}~{}M_{R}=\left(\begin{array}[]{ccc}M&0&0\\\
0&M&0\\\ 0&0&M_{3}\end{array}\right)\,.$ (10)

The above structures can be obtained from a global $U(1)$ symmetry Kersten and
Smirnov (2007) or from a class of discrete symmetries Chattopadhyay and Patel
(2017). One obtains ${\cal M}_{\nu}=0$ in this case irrespective of the values
of $y_{i}$, $M$ and $M_{3}$. It can be seen that the third RH neutrino does
not couple with the SM leptons. The first two generations of RH neutrinos are
strongly coupled if $y_{i}$s are chosen to be of order unity. Viable neutrino
masses can be generated by introducing perturbations to the above structure.
The strength of these perturbations are found to be very small Chattopadhyay
and Patel (2017) and therefore their contribution to RG effects are
negligible. The form of $Y_{\nu}$ and $M_{R}$ given in Eq. (10) therefore
provides a good description of low scale seesaw (LSS) with strongly coupled RH
neutrinos.

##  III RG evolution of the couplings and Constraints

The framework under consideration involves many hierarchically separated
scales. We perform renormalization group evolution of the couplings of
effective field theory between different scales and match their values at the
boundaries. The couplings are evolved using 2-loop RG equations and matching
at the thresholds are performed including 1-loop threshold corrections. The
2-loop RG equations are computed using a publicly available package SARAH
Staub (2014) and they are listed in Appendix A. In the following subsections,
we describe matching conditions, theoretical and phenomenological constraints
on the couplings at various scales.

###  III.1 Matching conditions at $M_{S}$

We assume that supersymmetry is broken at the scale $M_{S}=2\times 10^{16}$
GeV and the theory below $M_{S}$ is an effective THDM with or without RH
neutrinos at intermediate scales between $M_{S}$ and $M_{t}$. As it is
discussed earlier, a tree-level matching between MSSM and THDM leads to
relations given in Eqs. (3,4). The one-loop threshold corrections to these
matching conditions and to the Yukawa couplings, in the absence of RH
neutrinos, are given in Haber and Hempfling (1993); Lee and Wagner (2015).
These corrections depend on the sparticle spectrum at $M_{S}$ and also on the
values of trilinear couplings and $\mu$ parameter. For simplicity, we assume
that

$m_{\tilde{q}}=m_{\tilde{l}}=M_{1}=M_{2}=M_{3}=M_{S}\,,~{}A_{t}=A_{b}=A_{\tau}=0\,,~{}\mu\approx{\cal
O}(M_{S})\,,$ (11)

where $m_{\tilde{q}}$ and $m_{\tilde{l}}$ represent degenerate squark and
slepton masses respectively, $\mu$ is higgsino mass parameter, $M_{i}$ are
gaugino mass parameters and $A_{t,b,\tau}$ are the trilinear couplings of
squarks and sleptons with relevant MSSM Higgs fields. The above assumption is
realized in specific GUT based model Buchmuller  _et al._ (2015). With these
assumptions, the threshold corrections induced by squarks and sleptons are
suppressed by the degeneracy of their masses and also by vanishing trilinear
couplings. It can be seen from the expressions given in Lee and Wagner (2015),
the one-loop threshold corrections to Yukawa couplings vanish entirely for the
superpartner spectrum given in Eq. (11). The threshold corrections to the
quartic couplings, induced through one-loop box and triangle diagrams, depend
only on the Yukawa couplings and $\mu$ parameter in the limit of vanishing
trilinear couplings. We also estimate one-loop threshold corrections to
quartic couplings which arise from the Dirac Yukawa couplings of RH neutrinos
with the SM leptons. The expressions of threshold corrections, used in our
analysis, are listed in Appendix B. For the analysis presented in this paper,
we have chosen $\mu=0.1M_{S}$ for definiteness.

###  III.2 Constraints at intermediate scales

While evolving gauge, Yukawa and quartic couplings from $M_{S}$ to $M_{t}$, we
adopt the following procedure. If the mass scale $M_{R_{j}}$ of $j^{\rm th}$
RH neutrino $\nu_{R}^{j}$ appears below $M_{S}$, their running effects are
taken into account by appropriate RG evolution. It is expected that for
renormalization scale $Q<M_{R_{j}}$, the $\nu_{R}^{j}$ should be integrated
out from the spectrum and it should not contribute in the running of
couplings. We implement this decoupling of heavy neutrinos by switching off
the Yukawa couplings of $\nu_{R}^{j}$ with the SM leptons at the scale
$M_{R_{j}}$ and below. In other words, the values of elements of $j$th column
in the matrix $Y_{\nu}$ is put to zero after the running scale $Q$ crosses the
scale $M_{R_{j}}$. This procedure is carried out sequentially for all RH
neutrinos with masses between $M_{S}$ and $M_{t}$.

We also consider various theoretical constraints on the couplings which should
be satisfied at every scale. It is to be noted from Eq. (4) and RG equations
that the couplings $\lambda_{5,6,7}$ vanish at all the scales. This happens
because these couplings (as well as the Yukawa couplings in $\tilde{Y}_{f}$
for $f=u,d,e,\nu$) are protected by a softly broken $Z_{2}$ symmetry of an
effective THDM theory and therefore if they are zero at one scale then they
will not be generated by running333The threshold corrections at $M_{S}$ can
generate non-zero values for $\lambda_{5,6,7}$ and $\tilde{Y}_{f}$. However,
these threshold corrections are vanishing for the SUSY spectrum considered in
Eq. (11).. A stability of scalar potential in Eq. (1) would require the
following conditions to be satisfied by the remaining couplings Gunion and
Haber (2003)

$\displaystyle\lambda_{1}(Q)$ $\displaystyle>$ $\displaystyle 0\,,$
$\displaystyle\lambda_{2}(Q)$ $\displaystyle>$ $\displaystyle 0\,,$
$\displaystyle\lambda_{3}^{\prime}(Q)\equiv\lambda_{3}(Q)+\sqrt{\lambda_{1}(Q)\lambda_{2}(Q)}$
$\displaystyle>$ $\displaystyle 0\,,$
$\displaystyle\lambda_{4}^{\prime}(Q)\equiv\lambda_{4}(Q)+\lambda_{3}^{\prime}(Q)$
$\displaystyle>$ $\displaystyle 0\,,$ (12)

for $M_{t}\leq Q\leq M_{S}$. The above conditions are sufficient to provide
absolute stability for the electroweak vacuum. One may also consider a
phenomenologically allowed and a more conservative possibility in which the
electroweak vacuum is not completely stable but it is metastable with lifetime
greater than the age of universe $\sim 10^{10}$ years. This replaces the last
condition in Eq. (III.2) by a weaker condition Bagnaschi  _et al._ (2016)

$\lambda(Q)\equiv\frac{4\sqrt{\lambda_{1}(Q)\lambda_{2}(Q)}\,\lambda_{4}^{\prime}(Q)}{\lambda_{1}(Q)+\lambda_{2}(Q)+2\sqrt{\lambda_{1}(Q)\lambda_{2}(Q)}}-\lambda_{\rm
meta}\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>0\,,$
(13)

where

$\lambda_{\rm meta}=-\frac{2.82}{41.1+\log_{10}\left(\frac{Q}{\rm
GeV}\right)}\,.$ (14)

The derivation of the above condition involves probability of tunnelling into
the true vacuum which was estimated in case of single scalar field with
$\phi^{4}$ potential in Isidori  _et al._ (2001) including the quantum
effects. Following a similar approach, the metastability condition, Eq. (13),
was derived in Bagnaschi  _et al._ (2016) after mapping the THDM scalar
potential into single field potential using the first three conditions in Eq.
(III.2) with convinient choice of gauge and field basis. More details about
the derivation of Eq. (13) can be found in an Appendix of Bagnaschi  _et al._
(2016). For most of the cases studied here, it is found that the first three
conditions of Eq. (III.2) are always satisfied as a consequence of the
boundary values set by supersymmetry at $M_{S}$. Hence, the stability or
metastability of the electroweak vacuum is solely decided by values of
$\lambda_{4}^{\prime}$ and $\lambda$ at the intermediate scales.

###  III.3 Matching conditions and constraints at $M_{t}$

The RG equations determine the values of the couplings of effective THDM at
the scale $M_{t}$. At $M_{t}$, the gauge and Yukawa couplings are matched with
their experimentally measured values while the quartic couplings determine the
Higgs potential which is subject to the constraints imposed by consistent
electroweak symmetry breaking and measurement of Higgs properties.

The electroweak symmetry breaking is governed by the VEVs of two Higgs fields,

$\langle H_{i}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
v_{i}\end{array}\right)\,$ (15)

at the minimum of scalar potential. In our notation, $v_{1}=v_{d}$ and
$v_{2}=v_{u}$ and they define electroweak VEV $v$ and a parameter $\tan\beta$
as the following

$v\equiv\sqrt{v_{u}^{2}+v_{d}^{2}}\approx 246\,{\rm
GeV}\,,~{}~{}~{}\tan\beta\equiv\frac{v_{u}}{v_{d}}\,.$ (16)

The breaking of electroweak symmetry gives rise to five physical Higgs bosons
in the spectrum. These are two charged and CP-even ($H^{\pm}$), two neutral
and CP-even ($h$ and $H$), and a neutral and CP odd ($A$) scalars. At the
minimum of potential, the parameters $m_{1}^{2}$, $m_{2}^{2}$ and $m_{12}^{2}$
can be replaced by the following tree-level expressions Gunion and Haber
(2003):

$\displaystyle m_{12}^{2}$ $\displaystyle=$ $\displaystyle
M_{A}^{2}\,\sin\beta\cos\beta,$ $\displaystyle m_{1}^{2}$ $\displaystyle=$
$\displaystyle
M_{A}^{2}\sin^{2}\beta-\frac{1}{2}v^{2}\left(\lambda_{1}\cos^{2}\beta+(\lambda_{3}+\lambda_{4})\sin^{2}\beta\right)\,,$
$\displaystyle m_{2}^{2}$ $\displaystyle=$ $\displaystyle
M_{A}^{2}\cos^{2}\beta-\frac{1}{2}v^{2}\left(\lambda_{2}\sin^{2}\beta+(\lambda_{3}+\lambda_{4})\cos^{2}\beta\right)\,,$
(17)

where $M_{A}$ is the mass of pseudo-scalar Higgs in $\overline{\rm MS}$
renormalization scheme. With these replacements, the scalar potential given in
Eq. (1) is completely specified by $M_{A}$, $v$, $\tan\beta$ and the quartic
couplings.

The mass of charged Higgs is given by

$M_{H^{\pm}}^{2}=M_{A}^{2}-\frac{1}{2}\lambda_{4}\,v^{2}\,.$ (18)

The CP-even scalar states mix with each other and it is convenient to work in
so-called Higgs basis in which only one of the combinations of $H_{1}$ and
$H_{2}$, namely $h_{1}\equiv\cos\beta H_{1}+\sin\beta H_{2}$, acquires a non-
trivial VEV. The combination orthogonal to $h_{1}$ is identified as $h_{2}$
such that $\langle h_{2}\rangle=0$. In the basis $\\{h_{1},h_{2}\\}$, the
squared mass matrix of CP-even neutral scalars is given as Gunion and Haber
(2003)

${\cal M}^{2}=M_{A}^{2}\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)+v^{2}\left(\begin{array}[]{cc}g_{11}&g_{12}\\\
g_{12}&g_{22}\end{array}\right)\,,$ (19)

where

$\displaystyle g_{11}$ $\displaystyle=$
$\displaystyle\lambda_{1}\cos^{2}\beta+\lambda_{2}\sin^{2}\beta+2(\lambda_{3}+\lambda_{4})\sin^{2}\beta\cos^{2}\beta\,,$
$\displaystyle g_{12}$ $\displaystyle=$
$\displaystyle-\cos\beta\sin\beta\left(\lambda_{1}\cos^{2}\beta-\lambda_{2}\sin^{2}\beta-(\lambda_{3}+\lambda_{4})\cos
2\beta\right)\,,$ $\displaystyle g_{22}$ $\displaystyle=$
$\displaystyle(\lambda_{1}+\lambda_{2})\cos^{2}\beta\sin^{2}\beta-2(\lambda_{3}+\lambda_{4})\sin^{2}\beta\cos^{2}\beta\,.$
(20)

Performing another change of basis

$\left(\begin{array}[]{c}H\\\
h\end{array}\right)=U\left(\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\right)\,,~{}~{}U\equiv\left(\begin{array}[]{cc}\cos(\beta-\alpha)&-\sin(\beta-\alpha)\\\
\sin(\beta-\alpha)&\cos(\beta-\alpha)\end{array}\right)\,,$ (21)

such that

$U\,{\cal M}^{2}\,U^{\dagger}={\rm Diag.}(M_{H}^{2},m_{h}^{2})\,,$ (22)

where $m_{h}$ and $M_{H}$ are masses of physical CP-even neutral Higgs bosons.
These masses and the mixing angle $\beta-\alpha$ are computed from the above
diagonalization. We assume that $m_{h}\leq M_{H}$ and identify the lighter
state with the observed SM like Higgs. In order to make consistent matching
between theory and data, we convert the running mass $m_{h}$ evaluated at the
scale $M_{t}$ to the pole mass $M_{h}$ using the following formula

$M_{h}^{2}=m_{h}^{2}(M_{t})+\delta m_{h}^{2}(M_{t})\,,$ (23)

where $\delta m_{h}^{2}$ is the SM one-loop self-energy correction and its
expression in terms of $\overline{\rm MS}$ parameters is given in Draper  _et
al._ (2014); Lee and Wagner (2015). We do not include the contributions from
the other scalars in $\delta m_{h}^{2}$ and assume that they are sub-dominant
compared to the SM contributions. Numerically we find that the correction to
the Higgs mass induced by the second term in the above equation remains less
than $0.6$ GeV.

The spectrum of physical scalars and the angle $\beta-\alpha$ are subject to
several direct and indirect constraints. We consider the experimentally
measured value from Aad  _et al._ (2015) and allow a deviation of $\pm 3$ GeV
from the central value to account for theoretical uncertainty in estimating
the value of $M_{h}$. In THDM of type II, the charged Higgs with mass up to
580 GeV is disfavoured by $b\to s+\gamma$ measurements at $95\%$ confidence
level Misiak and Steinhauser (2017) for almost any value of $\tan\beta$.
Further, it can be seen from Eq. (21) that the couplings of $h$ with the weak
bosons is proportional to $\sin^{2}(\beta-\alpha)$. These couplings are
constrained by the signal strength of Higgs decaying into pair of vector
bosons. The results of a recent global fit of THDM parameters indicate that
the deviation from $\beta-\alpha=\pi/2$ cannot be larger than 0.055 in the
case of type II THDM Chowdhury and Eberhardt (2017). The above constraints on
the spectrum are summarized as

$\displaystyle M_{h}$ $\displaystyle=$ $\displaystyle(125\pm 3)\,{\rm GeV}\,$
$\displaystyle M_{H^{\pm}}$ $\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$
$\displaystyle 580\,{\rm GeV}\,$ $\displaystyle|\cos(\beta-\alpha)|$
$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ $\displaystyle 0.055\,.$ (24)

We find that the limit on $M_{H^{\pm}}$ also puts a lower bound on $M_{A}$,
from Eq. (18), and hence a lower bound on $M_{H}$ as well, since all the
quartic couplings are determined from the supersymmetry in this model. The
bounds on the masses of THDM scalars obtained in this way are more stringent
than the direct search bounds, see for example Chowdhury and Eberhardt (2017)
and references therein. We also investigate the effects of these scalars on
electroweak precision observables. For this, we estimate corrections to $W$
boson mass $M_{W}$ and effective weak mixing angle $\theta_{\rm eff}^{\rm
lept}$ in THDM and compare them with their measured values following the
procedure adopted in Broggio  _et al._ (2014). These constraints are found to
be always satisfied for the values of $M_{A}$ allowed by the constraints
listed in Eq. (III.3).

We use the experimental values of gauge couplings and fermion mass parameters
measured at different scales and evolve them to the scale $M_{t}$. This is
described in Appendix C. These values are listed in Table 1.

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---|---|---
$g_{1}$ | 0.46315 | $m_{u}$ | 1.21 MeV | $m_{d}$ | 2.58 MeV | $m_{e}$ | 0.499 MeV
$g_{2}$ | 0.65403 | $m_{c}$ | 0.61 GeV | $m_{s}$ | 52.74 MeV | $m_{\mu}$ | 0.104 GeV
$g_{3}$ | 1.1630 | $m_{t}$ | 163.74 GeV | $m_{b}$ | 2.72 GeV | $m_{\tau}$ | 1.759 GeV
Table 1: The values of the SM parameters, in ${\overline{\rm MS}}$ scheme, at
renormalization scale $M_{t}$ used in our analysis. More details are given in
Appendix C.

The Yukawa couplings are matched at $M_{t}$ using the following tree-level
relations in a convenient basis:

$\displaystyle Y_{u}(M_{t})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{v\sin\beta}\,{\rm
Diag.}\left(m_{u}(M_{t}),\,m_{c}(M_{t}),\,m_{t}(M_{t})\right)\,$
$\displaystyle Y_{d}(M_{t})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{v\cos\beta}\,V_{\rm CKM}\,{\rm
Diag.}\left(m_{d}(M_{t}),\,m_{s}(M_{t}),\,m_{b}(M_{t})\right)\,$
$\displaystyle Y_{e}(M_{t})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{v\cos\beta}\,{\rm
Diag.}\left(m_{e}(M_{t}),\,m_{\mu}(M_{t}),\,m_{\tau}(M_{t})\right)\,,$ (25)

where $V_{\rm CKM}$ is quark mixing matrix. For its elements, we use the
latest values from the PDG Patrignani  _et al._ (2016). In the standard
parametrization, this matrix is given in terms of three mixing angles and a
phase. We use their values: $\sin\theta_{12}^{q}=0.2251$,
$\sin\theta_{23}^{q}=0.041$, $\sin\theta_{13}^{q}=0.0036$ and $\delta_{\rm
CKM}=68.04^{\circ}$. In the next section, we discuss our procedure for solving
RG equations and present the obtained results in details.

##  IV Numerical results

We numerically solve the 2-loop RG equations for different cases and implement
matching conditions and constraints as discussed in the previous sections.
First, the gauge and Yukawa couplings are evolved from $M_{t}$ to $M_{S}$
using their values at $M_{t}$ and 1-loop RG equations. The running of these
couplings do not depend on quartic couplings at 1-loop. The values of quartic
couplings at $M_{S}$ are then obtained using conditions given in Eqs. (3, 4)
and one-loop threshold corrections listed in Appendix B. We then perform full
2-loop evolution of the gauge, Yukawa and quartic couplings form $M_{S}$ to
$M_{t}$ and compare the obtained values of gauge and Yukawa couplings with
their input values at $M_{t}$. The couplings are then evolved again from
$M_{t}$ to $M_{S}$ using 2-loop RG equations. This procedure is carried out
iteratively until the values of couplings converge to their input values at
$M_{t}$. During the running, we check that the stability or metastability
conditions, given in Eqs. (III.2, 13), are satisfied at all intermediate
scales. Once the convergence is obtained, we calculate the masses of physical
scalars using Eqs. (18,19) as function of input parameter $M_{A}$ and
$\tan\beta$.

Before we discuss the results of our numerical analysis, we outline our method
of obtaining the RH neutrino mass spectrum in the case of high scale seesaw.
The RH neutrino masses are evaluated using the boundary condition given in Eq.
(9) and seesaw formula Eq. (7) with the following replacement for ${\cal
M}_{\nu}$:

${\cal M}_{\nu}=U_{\rm PMNS}\,{\rm
Diag.}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})\,U_{\rm PMNS}^{T}\,,$ (26)

where $U_{\rm PMNS}$ is leptonic mixing matrix and it is parametrized by three
mixing angles and three CP phases in the standard parametrization Patrignani
_et al._ (2016). Since we consider a basis in which charged lepton Yukawa
matrix is diagonal444The charged lepton Yukawa matrix is taken diagonal at
$M_{t}$. Further, it remains almost diagonal at all the scales as the
magnitude of off-diagonal elements in $Y_{e}$ induced by RG evolution is very
small., the above form of ${\cal M}_{\nu}$ leads to realistic lepton mixing.
We assume normal ordering in the masses of neutrinos, and obtain the values of
$m_{\nu_{2}}$ and $m_{\nu_{3}}$ in terms of $m_{\nu_{1}}$ from the solar and
atmospheric squared mass differences, $\Delta m_{21}^{2}\equiv
m_{\nu_{2}}^{2}-m_{\nu_{1}}^{2}$ and $\Delta m_{31}^{2}\equiv
m_{\nu_{3}}^{2}-m_{\nu_{1}}^{2}$, respectively. The values used in our
analysis for leptonic mixing angles, CP phases and squared mass differences
are as follows:

$\sin^{2}\theta_{12}=0.307,\,\sin^{2}\theta_{23}=0.538,\,\sin^{2}\theta_{13}=0.02206,\,$
$\delta_{CP}=-\frac{\pi}{2},\,\alpha_{21}=\alpha_{31}=0,\,$ $\Delta
m_{21}^{2}=7.40\times 10^{-5}\,{\rm eV}^{2},\,\Delta m_{31}^{2}=2.494\times
10^{-3}\,{\rm eV}^{2}.$

The above values for mixing angles, $\delta_{\rm CP}$, $\Delta m_{21}^{2}$ and
$\Delta m_{31}^{2}$ are taken from the latest global fit of neutrino
oscillation data reported as NuFIT 3.2 (2018) Esteban  _et al._ (2017). We
also fix the lightest neutrino mass $m_{\nu_{1}}=0.001$ eV for definiteness.
Note that we use low energy values of neutrino masses and mixing parameters in
seesaw formula to determine the RH neutrino mass spectrum. It is assumed that
these parameters do not change significantly under RG evolution. Even if the
running effects are taken into consideration for these parameters, the change
in the masses of RH neutrinos obtained by Eq. (7) is small and therefore it
has negligible effects on the results of our analysis.

###  IV.1 Without seesaw

We first analyse a case without RH neutrinos. In this case, it is observed in
Bagnaschi  _et al._ (2016) that the quartic couplings $\lambda_{1}$,
$\lambda_{2}$ and $\lambda_{3}$ remain positive at all the scales because of
their boundary conditions at $M_{S}$ and hence the first three of the
conditions of Eq. (III.2) are always satisfied. $\lambda_{4}$ is negative at
$M_{S}$ and it remains negative at the intermediate scales which in turn
requires sufficiently large $\lambda_{1,2,3}$ in order to satisfy the last
condition of Eq. (III.2) or Eq. (13) for stability or metastability
respectively. For very small values of $\tan\beta$, it is observed that the
couplings $\lambda_{1}$ and $\lambda_{3}$ evolve slowly. The magnitude of
$\lambda_{2}$ however increases rapidly while running from $M_{S}$ to $M_{t}$
because of large and negative contribution to 1-loop beta function from the
top quark loop. In this case, it is large and positive $\lambda_{2}$ which
makes $\lambda^{\prime}_{4}(Q)>0$ and leads to a stable scalar potential. With
increasing $\tan\beta$ the top quark Yukawa coupling $y_{t}$ decreases which
in turn slows down the running of $\lambda_{2}$. In this case, $\lambda_{2}$
does not attain large enough value to make the potential stable. For very
large values of $\tan\beta$, the bottom quark Yukawa coupling $y_{b}$ becomes
as strong as $y_{t}$ and hence the running effect in $\lambda_{1}$ becomes as
strong as that in $\lambda_{2}$ and both are driven to positive and large
values at scales below $M_{S}$. Their combined contributions increase the
value of $\lambda^{\prime}_{4}(Q)$ which drive electroweak vacuum towards
metastability or stability. We obtain the values of quartic couplings at
$M_{t}$ and compute the scalar mass spectrum and impose the constraints given
in Eq. (III.3). The results are displayed in Fig. 1.

Figure 1: Left panel: Regions allowed in $M_{A}$-$M_{h}$ plane by absolute
stability (darker green) and metastability (green) of the scalar potential.
Right panel: Regions of absolute stability (unshaded), metastability (green)
and unstability (orange) of the scalar potential in $M_{A}$-$\tan\beta$ plane.
In the regions shaded by grey colour, the couplings become non-perturbative.
In both the panels, the region between two black lines correspond to the
values of $M_{h}$ in the range of 122-128 GeV while the dashed black line is
for $M_{h}=125$ GeV. The region on the left side of vertical dotted and dashed
red lines are disfavoured by the constraints, $M_{H^{\pm}}>580$ GeV and
$|\cos(\beta-\alpha)|<0.055$, respectively.

We find that the top Yukawa coupling becomes non-perturbative for
$\tan\beta\leq 1.36$ and hence the perturbative approach of RG evolution
breaks down. A stable scalar potential is achieved for $1.36<\tan\beta\leq
1.77$. As it is explained earlier, the large positive value of $\lambda_{2}$
makes $\lambda_{4}^{\prime}(Q)>0$ for all $Q$ between $M_{S}$ and $M_{t}$ in
this case. For $1.77<\tan\beta\leq 3.18$, the electroweak vacuum becomes
metastable while the region $3.18<\tan\beta\leq 38.5$ is disfavoured by an
unstable vacuum. The potential again becomes metastable for $\tan\beta>38.5$
because of large contribution of bottom quark in the running of $\lambda_{1}$.
It can be seen from Fig. 1 that the $b\to s+\gamma$ constraint on
$M_{H^{\pm}}$ implies
$M_{A}\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>580$
GeV. This together with constraint on Higgs mass restrict the values of
$\tan\beta$ in a very narrow range, i.e. $1.36<\tan\beta\leq 2.5$. We find
that our results are in qualitative agreement with the results obtained in
Bagnaschi  _et al._ (2016) but for $M_{S}=2\times 10^{17}$ GeV.

###  IV.2 Case: HSS

The effects of high scale seesaw mechanism on vacuum stability is studied by
incorporating RH neutrino thresholds in the RG evolution and by considering
the $SO(10)$ GUT inspired boundary conditions, Eq. (9), at the scale $M_{S}$.
The parametrization is chosen in such a way that $\xi$ and $N$ control inter-
generational hierarchy in $Y_{\nu}$. For example, the couplings in $Y_{\nu}$
are as hierarchical as those in $Y_{u}$ for $\xi\approx 1$. Further, from the
extrapolated values of couplings in $Y_{u}$ we find that all the three RH
neutrino species couple with the SM leptons with equal strength for $N\approx
2.07$ and $\xi\approx 340$. The parameter $\epsilon$ sets overall scale of
$Y_{\nu}$ and hence the scale of RH neutrinos. For example, the RH neutrino
masses obtained using Eqs. (7,9), as explained in the beginning of this
section, are displayed Fig. 2 for $\xi=340$ and $N=2.07$.

Figure 2: The mass spectrum of RH neutrinos with respect to the $\epsilon$
parameter, as obtained from the condition in Eq. (9) with $\xi=340$ and
$N=2.07$ and seesaw formula Eq. (7) and Eq. (26) for $\tan\beta=1.5$ (left
panel) and $\tan\beta=50$ (right panel). The horizontal line corresponds to
the value $M_{S}=2\times 10^{16}$ GeV.

It can be seen that for small values of $\tan\beta$, all the neutrinos
decouple from the effective theory for $\epsilon\geq 9$. For
$\epsilon\in[0.1,9]$, at least one neutrino remains in the spectrum below
$M_{S}$. We fix $N=2.07$ and evaluate the effect of neutrino(s) below $M_{S}$
on the vacuum stability in two different cases. First we fix $\epsilon=1$ and
vary $\xi$ in the range from $1$ to $400$ to investigate the effects of
different hierarchical structure of $Y_{\nu}$. In the second, we fix $\xi=340$
which corresponds to all the three neutrinos coupled with approximately equal
strength and vary $\epsilon$ in the range $0.1$ to $9$. The constraints on
$\tan\beta$ obtained by the consideration of vacuum stability for these cases
are displayed in Fig. 3.

Figure 3: The allowed values of $\tan\beta$ by an absolute stability
(unshaded) and metastability (green) of the scalar potential as function of
$\xi$ (in the left panel) and $\epsilon$ (in the right panel) parameters as
defined in Eq. (9) with $N=2.07$. The orange region corresponds to unstable
scalar potential while the grey region in the bottom is where the
perturbativity of the couplings is lost.

The presence of RH neutrinos below $M_{S}$ modifies the running of quartic
couplings in the following way. As it can be seen from the expressions of beta
functions given in Appendix A, RH neutrinos contribute to the running of
$\lambda_{2}$ with a term proportional to $-{\rm
Tr}(Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger})$. This contribution
increases the value of $\lambda_{2}$ from its already positive value at
$M_{S}$ while evolving from $M_{S}$ to $M_{R_{i}}$. As a result of this,
$\lambda_{2}$ takes relatively larger value at the scales between $M_{R_{i}}$
and $M_{t}$ in comparison to the case without RH neutrinos. This helps in
obtaining increased $\lambda^{\prime}_{4}(Q)$ and hence this effect can drive
the potential towards stability depending on the magnitude of couplings in
$Y_{\nu}$. As it can be seen in the right panel in Fig. 3, this happens only
when $\epsilon>1.5$ for which RH neutrinos are very strongly coupled with SM
leptons. One finds that all the values of $\tan\beta$ between 1.2 and 50 are
allowed by stability constraints for $\epsilon\in[1.5,8]$. For $\epsilon>8$
and large $\tan\beta$, the quartic coupling $\lambda_{2}$ suddenly becomes
negative resulting into an unstable region displayed in the upper right corner
of $\tan\beta$-$\epsilon$ plane in Fig. 3. At two-loop, the contribution from
RH neutrino Yukawa couplings in the running of $\lambda_{2}$ comes with
opposite sign in comparison to that of one-loop as it can be seen from the
expressions of beta functions given in Appendix A. For large $\epsilon$ and
just before the coupling $Y_{\nu}$ becomes non-perturbative, the two-loop beta
function dominates over the one-loop. This effect is further enhanced by large
$\tan\beta$ through large $Y_{d}$ and $Y_{e}$. This results into large
positive value of the total beta function which rapidly drives $\lambda_{2}$
from its initial positive value at $M_{S}$ to negative value at the scale
below $M_{S}$.

For $\epsilon\leq 1.5$, the heavy neutrinos do not significantly change the
stability constraints obtained in the case without RH neutrinos as it can be
seen from both the panels in Fig. 3. Although all the neutrinos are not
decoupled in this case, they do not have couplings strong enough to make any
significant change in the running of quartic couplings. Even for $\xi=340$,
$N=2.07$ which leads to all the three neutrinos with couplings of ${\cal
O}(y_{t})$, one obtains similar constraints on $\tan\beta$ as in the case
without RH neutrinos.

In order to check the viability of above results with Higgs mass and other low
energy constraints, we evaluate the scalar spectrum for three different
benchmark points. These are: (i) $\epsilon=\xi=1$, (ii) $\epsilon=1$,
$\xi=340$ and (iii) $\epsilon=4$, $\xi=340$. The choices (i) and (ii) are
motivated from a consideration that at least one or all the three neutrinos
have couplings as large as ${\cal O}(y_{t})$ while (iii) corresponds to very
strongly coupled neutrinos which lead to significant improvements in stability
constraints. The results are displayed in Fig. 4.

Figure 4: The details are same as given in the caption of Fig. 1. For all the
cases, $N=2.07$.

For $\epsilon=\xi=1$ (or equivalently $Y_{\nu}=Y_{u}$ at the GUT scale), the
results are almost identical to the ones obtained in the case without RH
neutrinos as it can be seen from Figs. 1 and 3. The results do not change
significantly also for $\epsilon=1$ and for any value of $\xi$ between 1 and
$340$. For these benchmark points, the stability or metastability of
electroweak vacuum and constraints on the masses of scalars imply
$1.2\leq\tan\beta\leq 2.2$ and $M_{A}>580$ GeV as viable ranges in which the
effective THDM theory can be extrapolated to $M_{S}$. $\epsilon>1.5$ leads to
stability of the scalar potential for almost all the values of $\tan\beta$,
however they face stringent constraints from $M_{h}$ and $M_{H^{\pm}}$ as it
can be seen from the bottom panel in Fig. 4. Strongly coupled neutrinos in
this case increase the magnitude of $\lambda_{2}$ at $M_{t}$ which results
into relatively higher Higgs mass for a given value of $M_{A}$. Hence to
obtain $M_{h}<128$ GeV, one needs lower $M_{A}$ which is already disfavoured
by the constraints on the charged Higgs mass. This disfavours the values of
$\epsilon>1$. The above results imply that an existence of strongly coupled
heavy neutrinos below $M_{S}$ improves the stability of THDM scalar potential,
however this scenario is very strongly constrained by the observed Higgs mass
and branching ratio of $b\to s+\gamma$.

###  IV.3 Case: LSS

We perform a similar analysis for a low scale seesaw scenario discussed in the
section II. For simplicity, we assume that $y_{1}=y_{2}=y_{3}\equiv y_{\nu}$
in Eq. (10). The third RH neutrino does not couple to THDM as it can be seen
from the flavour structure of $Y_{\nu}$ in Eq. (10). We take two sample values
for mass of the remaining degenerate heavy neutrinos, $M=10^{3}$ and $10^{9}$
GeV. The results are displayed in Fig. 5.

Figure 5: The allowed values of $\tan\beta$ by an absolute stability
(unshaded) and metastability (green) of the scalar potential as function of
$y_{1}=y_{2}=y_{3}\equiv y_{\nu}$ as defined (and for $M=10^{3}$ and $10^{9}$
GeV) in Eq. (10). The orange region corresponds to unstable scalar potential
while the grey region in the bottom is where the perturbativity of the
couplings is lost.

As it can be seen, strongly coupled neutrinos lead to significant changes in
the stability of the scalar potential. The quartic couplings receive
significant contributions from such neutrinos in this case which in turn helps
in making the scalar potential more stable. The stability constraints on
$y_{\nu}$ and $\tan\beta$ do not depend considerably on the mass scale of RH
neutrinos.

We analyse the low energy scalar spectrum for three benchmark values of
coupling: $y_{\nu}=0.1$, $0.3$ and $0.5$. The results are displayed in Fig. 6.

Figure 6: The details are same as given in the caption of Fig. 1.

For $y_{\nu}\leq 0.1$, the obtained constraints on $\tan\beta$ and $M_{A}$ are
very similar to those obtained in the case without RH neutrinos. In the regime
of strong $y_{\nu}$, these results change significantly. For $y_{\nu}=0.3$,
one always obtains $M_{h}>125$ GeV if constraint on the charged Higgs mass is
considered. Very strongly coupled neutrinos further push the value of $M_{h}$
on the higher side and such cases are disfavoured by limits on $M_{H^{\pm}}$
and $\beta-\alpha$.

In the analysis discussed above, we consider the top pole mas $M_{t}=173.5$
GeV. The change in top mass is known to have considerable effects on the RG
evolution. To evaluate such effects, we repeat the analysis carried out for
the case presented in Figs. 4 and 6 for $M_{t}=172.4$ GeV and $M_{t}=174.6$
GeV. These are values of top quark mass at $-1\sigma$ and $+1\sigma$ from the
mean value respectively. The results of such analysis are displayed in
Appendix D.

##  V Conclusion

We have investigated the viability of two-Higgs-doublet model, with type I
seesaw mechanism incorporated in it, as an effective theory below the GUT
scale in the presence of its minimal supersymmetric completion at the same
scale. Supersymmetry is assumed to be broken at the scale $M_{S}=2\times
10^{16}$ GeV leaving only the SM particles, three copies of RH neutrinos and
an additional Higgs doublet below $M_{S}$. It also leaves its imprints on the
effective scalar potential of THDM by relating quartic couplings with gauge
couplings at the SUSY breaking scale. Because of these constraints provided by
SUSY, the effective potential of THDM can be written in terms of only two free
parameters: the ratio of vacuum expectation value of two Higgs doublets
$\tan\beta$, and the mass of pseudo-scalar Higgs $M_{A}$. The same potential
governs the electroweak symmetry breaking and determines the masses of
physical scalars in the theory. The requirement that the electroweak vacuum is
stable or metastable (i.e. its lifetime is greater than the age of universe)
and constraints on the scalar spectrum from various direct and indirect
experimental searches restrict the allowed values of $\tan\beta$ and $M_{A}$.
In the absence of RH neutrinos, these constraints favour the values for
$\tan\beta$ in the range from $1.4$ to $2.5$ and
$M_{A}\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>580$
GeV.

Existence of RH neutrinos below the scale $M_{S}$ causes significant changes
in the above results if they are very strongly coupled with the SM leptons. We
have studied this possibility in the context of high and low scale type I
seesaw mechanism. In the case of high scale seesaw, we find that if one or
more neutrinos have Dirac type Yukawa couplings greater than $y_{t}$ at the
GUT scale, then they considerably modify the running of quartic couplings and
allow stable or metastable electroweak vacuum for almost all the values of
$\tan\beta$ for which the effective theory remains perturbative. The RH
neutrinos improve the stability of scalar potential in this case unlike in the
case of the SM without high scale SUSY in which the strongly coupled RH
neutrinos are known Elias-Miro  _et al._ (2012); Rodejohann and Zhang (2012);
Khan  _et al._ (2014); Chakrabortty  _et al._ (2013); Delle Rose  _et al._
(2015); Bambhaniya  _et al._ (2017); Coriano  _et al._ (2014, 2016) to
destabilize the scalar potential. The difference between the two cases arise
mainly because of the presence of SUSY at the high scale which ensures the
stability of scalar potential in ultraviolet completion. The difference is
also attributed to the presence of additional Higgs doublet in THDM which
modifies the stability conditions allowing more freedom in the effective
potential. We find that the observed Higgs mass and limit on the charged Higgs
mass from the flavour physics data put stringent constraints on $\tan\beta$
and $M_{A}$ when RH neutrinos are strongly coupled. Similar results are
obtained in case of low scale seesaw when neutrino Yukawa couplings are ${\cal
O}(1)$.

It is observed that if Yukawa couplings of RH neutrinos are of ${\cal
O}(y_{t})$ or smaller at the GUT scale then they do not have significant
impact on vacuum stability constraints in THDM. In the case of high scale
seesaw, particular cases have been explored in which it is assumed that the
neutrino Yukawa coupling matrix $Y_{\nu}$ is equal to up-type quark Yukawa
coupling matrix $Y_{u}$ or all the couplings in $Y_{\nu}$ are ${\cal
O}(y_{t})$ at the GUT scale. This class of boundary conditions are often
realized in $SO(10)$ based GUTs. It is found in these cases that RH neutrinos
have negligible effect on stability of the scalar potential and scalar
spectrum at low energy. A stable or metastable scalar potential consistent
with low energy constraints is obtained for $\tan\beta\in[1.2-2.5]$ in these
cases. These results are qualitatively similar to those obtained in case
without right-handed neutrinos in Bagnaschi  _et al._ (2016).

The results obtained in this paper are useful in constraining a class of
models in which supersymmetry is broken at the GUT scale and an effective
theory below this scale is THDM of type II with type I seesaw mechanism. It is
shown here that the stability or metastability of electroweak vacuum and a
consistent low energy scalar spectrum are achieved for only small $\tan\beta$
and for neutrino Yukawa couplings of ${\cal O}(y_{t})$ or smaller. In concrete
ultraviolet models, see for example the ones given in Buchmuller and Schweizer
(2017); Buchmuller and Patel (2018), the neutrino Yukawa couplings and/or
$\tan\beta$ are often determined from the enhanced symmetry structure of
underlying theory. Therefore, the results obtained in this paper can be used
to constrain such frameworks.

Our analysis provides a generic understanding of the effects of RH neutrinos.
The numerical results obtained in this paper are subject to change for
different choice of values of $M_{S}$ and/or GUT scale threshold corrections.
We have assumed a particular value of SUSY breaking scale, $M_{S}=2\times
10^{16}$ GeV, throughout our analysis. It is known that stability of vacuum
potential is considerably sensitive to the choice of such a scale Bagnaschi
_et al._ (2016). In particular, the constraints on the values of $\tan\beta$
imposed by unstability of potential become feeble (stronger) for relatively
smaller (larger) values of $M_{S}$. Another important issue is that the
precise unification of gauge couplings does not occur at the scale $M_{S}$ as
the effective theory below this scale is pure THDM with singlet neutrinos.
Exact gauge coupling unification may require either new fields at intermediate
scales (for example, a pair of TeV scale Higgsinos Bagnaschi  _et al._ (2016))
or sizeable threshold corrections at the GUT scale Ellis and Wells (2017). For
all these cases, a dedicated analysis would be required in order to derive
quantitative constraints on $\tan\beta$ and $M_{A}$. However, we anticipate
that the qualitative effects of RH neutrinos will remain the same. The
presence of one or more strongly coupled RH neutrinos below $M_{S}$ improves
the stability of vacuum in an underlying framework.

## Acknowledgements

This work was supported by Early Career Research Award (ECR/2017/000353) and
by a research grant under INSPIRE Faculty Award (DST/INSPIRE /04/2015/000508)
from the Department of Science and Technology, Government of India. KMP thanks
DESY Theory Group for the kind hospitality during the final stage of this
work.

##  Appendix A Renormalization group equations for THDM of type II with right
handed neutrinos

In this appendix, we provide 2-loop renormalization group equations for THDM
of type II. They are obtained using publicly available package SARAH Staub
(2014). Note that we use different convention for the quartic couplings
$\lambda_{1}$ and $\lambda_{2}$ in Eq. (1) in comparison to the one used in
SARAH. The RG equations listed below are therefore modified accordingly. The
same equations are also listed in Lee and Wagner (2015) but with considering
only the third generation of fermions.

The couplings evolve according to the following equation:

$\mu\,\frac{dC}{d\mu}=\frac{1}{16\,\pi^{2}}\,\beta_{C}^{(1)}+\Big{(}\frac{1}{16\,\pi^{2}}\Big{)}^{2}\,\beta_{C}^{(2)}\,,$
(27)

where $C$ represents gauge, Yukawa and quartic couplings and $\mu$ is the
renormalization scale. The one and two-loop beta functions for the different
couplings are as the following.

###  A.1 Gauge couplings

$\displaystyle\beta_{g_{1}}^{(1)}$ $\displaystyle=\frac{21}{5}\,g_{1}^{3}$
(28) $\displaystyle\beta_{g_{1}}^{(2)}$
$\displaystyle=\frac{1}{50}\,g_{1}^{3}\,\Big{(}180\,g_{2}^{2}+208\,g_{1}^{2}+440\,g_{3}^{2}-75\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{e}}\Big{)}-15\,\mbox{Tr}\Big{(}{Y_{\nu}^{\dagger}Y_{\nu}}\Big{)}$
$\displaystyle-25\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{d}}\Big{)}-85\,\mbox{Tr}\Big{(}{Y_{u}^{\dagger}Y_{u}}\Big{)}\Big{)}$
(29) $\displaystyle\beta_{g_{2}}^{(1)}$ $\displaystyle=-3\,g_{2}^{3}$ (30)
$\displaystyle\beta_{g_{2}}^{(2)}$
$\displaystyle=\frac{1}{10}\,g_{2}^{3}\,\Big{(}80\,g_{2}^{2}+120\,g_{3}^{2}+12\,g_{1}^{2}-5\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}-5\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-15\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}-15\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}\Big{)}$
(31) $\displaystyle\beta_{g_{3}}^{(1)}$ $\displaystyle=-7\,g_{3}^{3}$ (32)
$\displaystyle\beta_{g_{3}}^{(2)}$
$\displaystyle=-\frac{1}{10}\,g_{3}^{3}\,\Big{(}-11\,g_{1}^{2}+260\,g_{3}^{2}-45\,g_{2}^{2}+20\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+20\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}\Big{)}$
(33)

###  A.2 Yukawa couplings

$\displaystyle\beta_{Y_{u}}^{(1)}$
$\displaystyle=Y_{u}\,\Big{(}-8\,g_{3}^{2}-\frac{17}{20}\,g_{1}^{2}-\frac{9}{4}\,g_{2}^{2}+3\,\mbox{Tr}\Big{(}{Y_{u}^{\dagger}Y_{u}}\Big{)}+\mbox{Tr}\Big{(}{Y_{\nu}^{\dagger}Y_{\nu}}\Big{)}\Big{)}$
$\displaystyle+\frac{1}{2}\,\Big{(}3\,{Y_{u}Y_{u}^{\dagger}Y_{u}}+{Y_{d}Y_{d}^{\dagger}Y_{u}}\Big{)}$
(34) $\displaystyle\beta_{Y_{u}}^{(2)}$
$\displaystyle=Y_{u}\,\Big{(}\frac{1267}{600}\,g_{1}^{4}-\frac{9}{20}\,g_{1}^{2}g_{2}^{2}-\frac{21}{4}\,g_{2}^{4}+\frac{19}{15}\,g_{1}^{2}g_{3}^{2}+9\,g_{2}^{2}g_{3}^{2}-108\,g_{3}^{4}+\frac{3}{2}\,\lambda_{2}^{2}+\lambda_{3}^{2}$
$\displaystyle+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\frac{1}{8}\,\Big{(}160\,g_{3}^{2}+17\,g_{1}^{2}+45\,g_{2}^{2}\Big{)}\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+\frac{3}{8}\,\Big{(}5\,g_{2}^{2}+g_{1}^{2}\Big{)}\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-\frac{3}{4}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{27}{4}\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle+{Y_{d}Y_{d}^{\dagger}Y_{u}}\,\Big{(}-2\,\lambda_{3}+2\,\lambda_{4}+\frac{16}{3}\,g_{3}^{2}+\frac{33}{16}\,g_{2}^{2}-\frac{3}{4}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}-\frac{41}{240}\,g_{1}^{2}-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle+\frac{1}{80}\,{Y_{u}Y_{u}^{\dagger}Y_{u}}\Big{(}1280\,g_{3}^{2}-180\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+223\,g_{1}^{2}-540\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+675\,g_{2}^{2}-480\,\lambda_{2}\Big{)}$
$\displaystyle-\frac{1}{4}\,\Big{(}-6\,{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{u}}+{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{u}}+{Y_{u}Y_{u}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{u}}\Big{)}$
(35) $\displaystyle\beta_{Y_{d}}^{(1)}$
$\displaystyle=Y_{d}\,\big{(}-8\,g_{3}^{2}-\frac{1}{4}\,g_{1}^{2}-\frac{9}{4}\,g_{2}^{2}+3\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle+\frac{1}{2}\,\Big{(}3\,{Y_{d}Y_{d}^{\dagger}Y_{d}}+{Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}$
(36) $\displaystyle\beta_{Y_{d}}^{(2)}$
$\displaystyle=Y_{d}\,\Big{(}-\frac{113}{600}\,g_{1}^{4}-\frac{27}{20}\,g_{1}^{2}g_{2}^{2}-\frac{21}{4}\,g_{2}^{4}+\frac{31}{15}\,g_{1}^{2}g_{3}^{2}+9\,g_{2}^{2}g_{3}^{2}-108\,g_{3}^{4}+\frac{3}{2}\,\lambda_{1}^{2}+\lambda_{3}^{2}$
$\displaystyle+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\frac{5}{8}\,\Big{(}32\,g_{3}^{2}+9\,g_{2}^{2}+g_{1}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\frac{15}{8}\,\Big{(}g_{1}^{2}+g_{2}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle-\frac{27}{4}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}-\frac{3}{4}\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}\Big{)}$
$\displaystyle+\frac{1}{80}\,{Y_{d}Y_{d}^{\dagger}Y_{d}}\,\Big{(}1280\,g_{3}^{2}-180\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}+187\,g_{1}^{2}-540\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+675\,g_{2}^{2}-480\,\lambda_{1}\Big{)}$
$\displaystyle+\frac{1}{240}\,{Y_{u}Y_{u}^{\dagger}Y_{d}}\,\Big{(}1280\,g_{3}^{2}-180\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-480\,\lambda_{3}+480\,\lambda_{4}+495\,g_{2}^{2}-53\,g_{1}^{2}$
$\displaystyle-540\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}\Big{)}+\frac{1}{4}\,\Big{(}6\,{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{d}}-{Y_{d}Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}-{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}$
(37) $\displaystyle\beta_{Y_{\nu}}^{(1)}$
$\displaystyle=\frac{1}{2}\,\Big{(}3\,{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+{Y_{e}Y_{e}^{\dagger}Y_{\nu}}\Big{)}+Y_{\nu}\,\Big{(}3\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}-\frac{9}{20}\,\Big{(}5\,g_{2}^{2}+g_{1}^{2}\Big{)}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(38) $\displaystyle\beta_{Y_{\nu}}^{(2)}$
$\displaystyle=Y_{\nu}\,\Big{(}\frac{117}{200}\,g_{1}^{4}-\frac{27}{20}\,g_{1}^{2}g_{2}^{2}-\frac{21}{4}\,g_{2}^{4}+\frac{3}{2}\,\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\frac{1}{8}\,\Big{(}160\,g_{3}^{2}+17\,g_{1}^{2}$
$\displaystyle+45\,g_{2}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+\frac{3}{8}\,\Big{(}5\,g_{2}^{2}+g_{1}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}$
$\displaystyle-\frac{3}{4}\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}-\frac{27}{4}\,\mbox{Tr}\Big{(}{Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{u}}\Big{)}-\frac{9}{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle+\frac{1}{80}\,{Y_{\nu}Y_{e}^{\dagger}Y_{e}}\Big{(}-160\,\lambda_{3}+160\,\lambda_{4}+165\,g_{2}^{2}-180\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+21\,g_{1}^{2}$
$\displaystyle-60\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}\Big{)}+\frac{3}{80}\,{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}\Big{(}-180\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+225\,g_{2}^{2}-160\,\lambda_{2}$
$\displaystyle-60\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+93\,g_{1}^{2}\Big{)}-\frac{1}{4}\,\Big{(}-6\,{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}Y_{\nu}}+{Y_{\nu}Y_{\nu}^{\dagger}Y_{e}Y_{e}^{\dagger}Y_{\nu}}\Big{)}$
(39) $\displaystyle\beta_{Y_{e}}^{(1)}$
$\displaystyle=\frac{1}{4}\,\Big{(}6\,{Y_{e}Y_{e}^{\dagger}Y_{e}}+2\,{Y_{e}Y_{\nu}^{\dagger}Y_{\nu}}\Big{)}+Y_{e}\,\Big{(}3\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}-\frac{9}{4}\Big{(}g_{1}^{2}+g_{2}^{2}\Big{)}\Big{)}$
(40) $\displaystyle\beta_{Y_{e}}^{(2)}$
$\displaystyle=Y_{e}\,\Big{(}\frac{1449}{200}\,g_{1}^{4}+\frac{27}{20}\,g_{1}^{2}g_{2}^{2}-\frac{21}{4}\,g_{2}^{4}+\frac{3}{2}\,\lambda_{1}^{2}+\lambda_{3}^{2}+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}$
$\displaystyle+\frac{5}{8}\,\Big{(}32\,g_{3}^{2}+9\,g_{2}^{2}+g_{1}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\frac{15}{8}\,\Big{(}g_{1}^{2}+g_{2}^{2}\Big{)}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}-\frac{27}{4}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-\frac{9}{4}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}-\frac{3}{4}\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}\Big{)}$
$\displaystyle+\frac{3}{80}\,{Y_{e}Y_{e}^{\dagger}Y_{e}}\Big{(}129\,g_{1}^{2}-180\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+225\,g_{2}^{2}-160\,\lambda_{1}-60\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle+\frac{1}{80}\,{Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{(}-160\,\lambda_{3}+160\,\lambda_{4}+165\,g_{2}^{2}-180\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+33\,g_{1}^{2}$
$\displaystyle-60\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}+\frac{1}{4}\,\Big{(}6\,{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}Y_{e}}-{Y_{e}Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}-{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}$
(41)

###  A.3 Quartic couplings of scalars

$\displaystyle\beta_{\lambda_{1}}^{(1)}$
$\displaystyle=+\frac{27}{100}\,g_{1}^{4}+\frac{9}{10}\,g_{1}^{2}g_{2}^{2}+\frac{9}{4}\,g_{2}^{4}-\frac{9}{5}\,g_{1}^{2}\lambda_{1}-9\,g_{2}^{2}\lambda_{1}+12\,\lambda_{1}^{2}+4\,\lambda_{3}^{2}+4\,\lambda_{3}\lambda_{4}+2\,\lambda_{4}^{2}$
$\displaystyle+12\,\lambda_{1}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+4\,\lambda_{1}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}-12\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}-4\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}$
(42) $\displaystyle\beta_{\lambda_{1}}^{(2)}$
$\displaystyle=-\frac{3537}{1000}\,g_{1}^{6}-\frac{1719}{200}\,g_{1}^{4}g_{2}^{2}-\frac{303}{40}\,g_{1}^{2}g_{2}^{4}+\frac{291}{8}\,g_{2}^{6}+\frac{1953}{200}\,g_{1}^{4}\lambda_{1}+\frac{117}{20}\,g_{1}^{2}g_{2}^{2}\lambda_{1}-\frac{51}{8}\,g_{2}^{4}\lambda_{1}$
$\displaystyle+\frac{108}{10}\,g_{1}^{2}\lambda_{1}^{2}+\frac{108}{2},g_{2}^{2}\lambda_{1}^{2}-78\,\lambda_{1}^{3}+\frac{9}{5}\,g_{1}^{4}\lambda_{3}+15\,g_{2}^{4}\lambda_{3}+\frac{24}{5}\,g_{1}^{2}\lambda_{3}^{2}+24\,g_{2}^{2}\lambda_{3}^{2}-20\,\lambda_{1}\lambda_{3}^{2}$
$\displaystyle-16\,\lambda_{3}^{3}+\frac{9}{10}\,g_{1}^{4}\lambda_{4}+3\,g_{1}^{2}g_{2}^{2}\lambda_{4}+\frac{15}{2}\,g_{2}^{4}\lambda_{4}+\frac{24}{5}\,g_{1}^{2}\lambda_{3}\lambda_{4}+24\,g_{2}^{2}\lambda_{3}\lambda_{4}-20\,\lambda_{1}\lambda_{3}\lambda_{4}-24\,\lambda_{3}^{2}\lambda_{4}$
$\displaystyle+\frac{12}{5}\,g_{1}^{2}\lambda_{4}^{2}+6\,g_{2}^{2}\lambda_{4}^{2}-12\,\lambda_{1}\lambda_{4}^{2}-32\,\lambda_{3}\lambda_{4}^{2}-12\,\lambda_{4}^{3}$
$\displaystyle+\Big{(}80\,\lambda_{1}\,g_{3}^{2}+\frac{9}{2}\,\lambda_{1}^{2}+\frac{45}{2}\,g_{2}^{2}\lambda_{1}-\frac{9}{2}\,g_{2}^{4}+\frac{9}{10}\,g_{1}^{4}+\frac{1}{20}\,g_{1}^{2}\Big{(}50\,\lambda_{1}+108\,g_{2}^{2}\Big{)}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle-\frac{3}{20}\,\Big{(}30\,g_{1}^{4}-2\,g_{1}^{2}\Big{(}22\,g_{2}^{2}+25\,\lambda_{1}\Big{)}-20\,g_{2}^{2}\lambda_{1}+160\,\lambda_{1}^{2}+10\,g_{2}^{4}\Big{)}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}-6\,\Big{(}4\,\lambda_{3}^{2}$
$\displaystyle+4\,\lambda_{3}\lambda_{4}+2\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}-2\,\Big{(}4\,\lambda_{3}^{2}+4\,\lambda_{3}\lambda_{4}+2\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\Big{(}\frac{8}{5}\,g_{1}^{2}$
$\displaystyle-64\,g_{3}^{2}-3\,\lambda_{1}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}-9\,\lambda_{1}\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-\Big{(}\frac{25}{5}\,g_{1}^{2}+\lambda_{1}\Big{)}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle-3\,\lambda_{1}\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}+60\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}+12\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}$
$\displaystyle+20\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}+4\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{e}Y_{e}Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}}\Big{)}$
(43) $\displaystyle\beta_{\lambda_{2}}^{(1)}$
$\displaystyle=+\frac{27}{100}\,g_{1}^{4}+\frac{9}{10}\,g_{1}^{2}g_{2}^{2}+\frac{9}{4}\,g_{2}^{4}-\frac{9}{5}\,g_{1}^{2}\lambda_{2}-9\,g_{2}^{2}\lambda_{2}+12\,\lambda_{2}^{2}+4\,\lambda_{3}^{2}+4\,\lambda_{3}\lambda_{4}+2\,\lambda_{4}^{2}$
$\displaystyle+12\,\lambda_{2}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+4\,\lambda_{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-12\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-4\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
(44) $\displaystyle\beta_{\lambda_{2}}^{(2)}$
$\displaystyle=-\frac{3537}{1000}\,g_{1}^{6}-\frac{1719}{200}\,g_{1}^{4}g_{2}^{2}-\frac{303}{40}\,g_{1}^{2}g_{2}^{4}+\frac{291}{8}\,g_{2}^{6}+\frac{1953}{200}\,g_{1}^{4}\lambda_{2}+\frac{117}{20}\,g_{1}^{2}g_{2}^{2}\lambda_{2}-\frac{51}{8}\,g_{2}^{4}\lambda_{2}$
$\displaystyle+\frac{108}{10}g_{1}^{2}\lambda_{2}^{2}+54\,g_{2}^{2}\lambda_{2}^{2}-78\,\lambda_{2}^{3}+\frac{9}{5}\,g_{1}^{4}\lambda_{3}+15\,g_{2}^{4}\lambda_{3}+\frac{24}{5}\,g_{1}^{2}\lambda_{3}^{2}+24\,g_{2}^{2}\lambda_{3}^{2}-20\,\lambda_{2}\lambda_{3}^{2}$
$\displaystyle-16\,\lambda_{3}^{3}+\frac{9}{10}g_{1}^{4}\lambda_{4}+3\,g_{1}^{2}g_{2}^{2}\lambda_{4}+\frac{15}{2}\,g_{2}^{4}\lambda_{4}+\frac{24}{5}\,g_{1}^{2}\lambda_{3}\lambda_{4}+24\,g_{2}^{2}\lambda_{3}\lambda_{4}-20\,\lambda_{2}\lambda_{3}\lambda_{4}-24\,\lambda_{3}^{2}\lambda_{4}$
$\displaystyle+\frac{12}{5}\,g_{1}^{2}\lambda_{4}^{2}+6\,g_{2}^{2}\lambda_{4}^{2}-12\,\lambda_{2}\lambda_{4}^{2}-32\,\lambda_{3}\lambda_{4}^{2}-12\,\lambda_{4}^{3}$
$\displaystyle-6\,\Big{(}4\,\lambda_{3}^{2}+4\,\lambda_{3}\lambda_{4}+2\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}-2\,\Big{(}4\,\lambda_{3}^{2}+4\,\lambda_{3}\lambda_{4}+2\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}-\frac{171}{50}\,g_{1}^{4}+\frac{63}{5}\,g_{1}^{2}g_{2}^{2}-\frac{9}{2}\,g_{2}^{4}+\frac{17}{2}\,g_{1}^{2}\lambda_{2}+\frac{45}{2}\,g_{2}^{2}\lambda_{2}+80\,g_{3}^{2}\lambda_{2}-72\,\lambda_{2}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}-\frac{9}{50}\,g_{1}^{4}-\frac{3}{5}\,g_{1}^{2}g_{2}^{2}-\frac{3}{2}\,g_{2}^{4}+\frac{3}{2}\,g_{1}^{2}\lambda_{2}+\frac{15}{2}\,g_{2}^{2}\lambda_{2}-24\,\lambda_{2}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-9\,\lambda_{2}\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-3\,\lambda_{2}\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}-\frac{16}{5}\,g_{1}^{2}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}$
$\displaystyle-64\,g_{3}^{2}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-3\,\lambda_{2}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-\lambda_{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle+12\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{u}Y_{d}}\Big{)}+4\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}+60\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}$
$\displaystyle+20\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
(45) $\displaystyle\beta_{\lambda_{3}}^{(1)}$
$\displaystyle=+\frac{27}{100}\,g_{1}^{4}-\frac{9}{10}\,g_{1}^{2}g_{2}^{2}+\frac{9}{4}\,g_{2}^{4}-\frac{9}{5}\,g_{1}^{2}\lambda_{3}-9\,g_{2}^{2}\lambda_{3}+6\,\lambda_{1}\lambda_{3}+6\,\lambda_{2}\lambda_{3}+4\,\lambda_{3}^{2}+2\,\lambda_{1}\lambda_{4}$
$\displaystyle+2\,\lambda_{2}\lambda_{4}+2\,\lambda_{4}^{2}+6\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+2\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}+6\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}$
$\displaystyle+2\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-12\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-4\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}$
(46) $\displaystyle\beta_{\lambda_{3}}^{(2)}$
$\displaystyle=-\frac{3537}{1000}\,g_{1}^{6}+\frac{909}{200}\,g_{1}^{4}g_{2}^{2}+\frac{33}{40}\,g_{1}^{2}g_{2}^{4}+\frac{291}{8}\,g_{2}^{6}+\frac{27}{20}\,g_{1}^{4}\lambda_{1}-\frac{3}{2}\,g_{1}^{2}g_{2}^{2}\lambda_{1}+\frac{45}{4}\,g_{2}^{4}\lambda_{1}$
$\displaystyle+\frac{27}{20}\,g_{1}^{4}\lambda_{2}-\frac{3}{2}\,g_{1}^{2}g_{2}^{2}\lambda_{2}+\frac{45}{4}\,g_{2}^{4}\lambda_{2}+\frac{1773}{200}\,g_{1}^{4}\lambda_{3}+\frac{33}{20}\,g_{1}^{2}g_{2}^{2}\lambda_{3}-\frac{111}{8}\,g_{2}^{4}\lambda_{3}+\frac{36}{5}\,g_{1}^{2}\lambda_{1}\lambda_{3}$
$\displaystyle+36\,g_{2}^{2}\lambda_{1}\lambda_{3}-60\,\lambda_{1}^{2}\lambda_{3}+\frac{36}{5}\,g_{1}^{2}\lambda_{2}\lambda_{3}+36\,g_{2}^{2}\lambda_{2}\lambda_{3}-15\,\lambda_{2}^{2}\lambda_{3}+\frac{6}{5}\,g_{1}^{2}\lambda_{3}^{2}+6\,g_{2}^{2}\lambda_{3}^{2}-36\,\lambda_{1}\lambda_{3}^{2}$
$\displaystyle-36\,\lambda_{2}\lambda_{3}^{2}-12\,\lambda_{3}^{3}+\frac{9}{10}\,g_{1}^{4}\lambda_{4}-\frac{9}{5}\,g_{1}^{2}g_{2}^{2}\lambda_{4}+\frac{15}{2}\,g_{2}^{4}\lambda_{4}+\frac{12}{5}\,g_{1}^{2}\lambda_{1}\lambda_{4}+18\,g_{2}^{2}\lambda_{1}\lambda_{4}-4\,\lambda_{1}^{2}\lambda_{4}$
$\displaystyle+\frac{12}{5}\,g_{1}^{2}\lambda_{2}\lambda_{4}+18\,g_{2}^{2}\lambda_{2}\lambda_{4}-4\,\lambda_{2}^{2}\lambda_{4}-12\,g_{2}^{2}\lambda_{3}\lambda_{4}-16\,\lambda_{1}\lambda_{3}\lambda_{4}-16\,\lambda_{2}\lambda_{3}\lambda_{4}-4\,\lambda_{3}^{2}\lambda_{4}$
$\displaystyle-\frac{6}{5}\,g_{1}^{2}\lambda_{4}^{2}+6\,g_{2}^{2}\lambda_{4}^{2}-14\,\lambda_{1}\lambda_{4}^{2}-14\,\lambda_{2}\lambda_{4}^{2}-16\,\lambda_{3}\lambda_{4}^{2}-12\,\lambda_{4}^{3}$
$\displaystyle+\frac{1}{20}\,\Big{(}-120\,\Big{(}2\lambda_{3}^{2}+2\,\lambda_{1}\Big{(}3\,\lambda_{3}+\lambda_{4}\Big{)}+\lambda_{4}^{2}\Big{)}+225\,g_{2}^{2}\lambda_{3}$
$\displaystyle-45\,g_{2}^{4}+800\,g_{3}^{2}\lambda_{3}+9\,g_{1}^{4}+g_{1}^{2}\Big{(}25\lambda_{3}-54\,g_{2}^{2}\Big{)}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}-\frac{1}{20}\,\Big{(}15\,g_{2}^{4}+40\,\Big{(}2\lambda_{3}^{2}$
$\displaystyle+2\,\lambda_{1}\Big{(}3\,\lambda_{3}+\lambda_{4}\Big{)}+\lambda_{4}^{2}\Big{)}+45\,g_{1}^{4}-75\,g_{2}^{2}\lambda_{3}+g_{1}^{2}\Big{(}66\,g_{2}^{2}-75\,\lambda_{3}\Big{)}\Big{)}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}-\frac{171}{100}\,g_{1}^{4}-\frac{63}{10}\,g_{1}^{2}g_{2}^{2}-\frac{9}{4}\,g_{2}^{4}+\frac{17}{4}\,g_{1}^{2}\lambda_{3}+\frac{45}{4}\,g_{2}^{2}\lambda_{3}+40\,g_{3}^{2}\lambda_{3}-36\,\lambda_{2}\lambda_{3}-12\,\lambda_{3}^{2}$
$\displaystyle-12\,\lambda_{2}\lambda_{4}-6\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+\Big{(}-\frac{9}{100}\,g_{1}^{4}+\frac{3}{10}\,g_{1}^{2}g_{2}^{2}-\frac{3}{4}\,g_{2}^{4}+\frac{3}{4}\,g_{1}^{2}\lambda_{3}$
$\displaystyle+\frac{15}{4}\,g_{2}^{2}\lambda_{3}-12\,\lambda_{2}\lambda_{3}-4\,\lambda_{3}^{2}-4\,\lambda_{2}\lambda_{4}-2\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{27}{2}\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}-\frac{4}{5}\,g_{1}^{2}-64\,g_{3}^{2}+15\,\lambda_{3}\Big{)}\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-\frac{9}{2}\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}-\frac{12}{5}\,g_{1}^{2}+5\,\lambda_{3}\Big{)}\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}-\frac{27}{2}\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-\frac{9}{2}\,\lambda_{3}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle+12\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{d}}\Big{)}+24\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}+36\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}$
$\displaystyle+4\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}Y_{e}^{\dagger}Y_{e}}\Big{)}+8\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}+12\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}$
(47) $\displaystyle\beta_{\lambda_{4}}^{(1)}$
$\displaystyle=+\frac{9}{5}\,g_{1}^{2}g_{2}^{2}-\frac{9}{5}\,g_{1}^{2}\lambda_{4}-9\,g_{2}^{2}\lambda_{4}+2\,\lambda_{1}\lambda_{4}+2\,\lambda_{2}\lambda_{4}+8\,\lambda_{3}\lambda_{4}+4\,\lambda_{4}^{2}+6\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle+2\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}+6\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+2\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+12\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}$
$\displaystyle+4\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}$
(48) $\displaystyle\beta_{\lambda_{4}}^{(2)}$
$\displaystyle=-\frac{657}{50}\,g_{1}^{4}g_{2}^{2}-\frac{42}{5}\,g_{1}^{2}g_{2}^{4}+3\,g_{1}^{2}g_{2}^{2}\lambda_{1}+3\,g_{1}^{2}g_{2}^{2}\lambda_{2}+\frac{6}{5}\,g_{1}^{2}g_{2}^{2}\lambda_{3}+\frac{1413}{200}\,g_{1}^{4}\lambda_{4}+\frac{153}{20}\,g_{1}^{2}g_{2}^{2}\lambda_{4}$
$\displaystyle-\frac{231}{8}\,g_{2}^{4}\lambda_{4}+\frac{12}{5}\,g_{1}^{2}\lambda_{1}\lambda_{4}-7\,\lambda_{1}^{2}\lambda_{4}+\frac{12}{5}\,g_{1}^{2}\lambda_{2}\lambda_{4}-7\,\lambda_{2}^{2}\lambda_{4}+\frac{12}{5}\,g_{1}^{2}\lambda_{3}\lambda_{4}+36\,g_{2}^{2}\lambda_{3}\lambda_{4}$
$\displaystyle-40\,\lambda_{1}\lambda_{3}\lambda_{4}-40\,\lambda_{2}\lambda_{3}\lambda_{4}-28\,\lambda_{3}^{2}\lambda_{4}+\frac{24}{5}\,g_{1}^{2}\lambda_{4}^{2}+18\,g_{2}^{2}\lambda_{4}^{2}-20\,\lambda_{1}\lambda_{4}^{2}-20\,\lambda_{2}\lambda_{4}^{2}-28\,\lambda_{3}\lambda_{4}^{2}$
$\displaystyle+\frac{1}{20}\,\Big{(}225\,g_{2}^{2}\lambda_{4}-240\,\Big{(}\lambda_{1}\lambda_{4}+2\,\lambda_{3}\lambda_{4}+\lambda_{4}^{2}\Big{)}+800\,g_{3}^{2}\lambda_{4}+g_{1}^{2}\Big{(}108\,g_{2}^{2}+25\,\lambda_{4}\Big{)}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle+\frac{1}{20}\,\Big{(}3\,g_{1}^{2}\Big{(}25\,\lambda_{4}+44\,g_{2}^{2}\Big{)}+75\,g_{2}^{2}\lambda_{4}-80\,\Big{(}\lambda_{1}\lambda_{4}+2\,\lambda_{3}\lambda_{4}+\lambda_{4}^{2}\Big{)}\Big{)}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}\frac{63}{5}\,g_{1}^{2}g_{2}^{2}+\frac{17}{4}\,g_{1}^{2}\lambda_{4}+\frac{45}{4}\,g_{2}^{2}\lambda_{4}+40\,g_{3}^{2}\lambda_{4}-12\,\lambda_{2}\lambda_{4}-24\,\lambda_{3}\lambda_{4}-12\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}$
$\displaystyle+\Big{(}-\frac{3}{5}\,g_{1}^{2}g_{2}^{2}+\frac{3}{4}\,g_{1}^{2}\lambda_{4}+\frac{15}{4}\,g_{2}^{2}\lambda_{4}-8\,\lambda_{2}\lambda_{4}-8\,\lambda_{3}\lambda_{4}-4\,\lambda_{4}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-\frac{27}{2}\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}+\Big{(}\frac{4}{5}\,g_{1}^{2}+64\,g_{3}^{2}-24\,\lambda_{3}-33\,\lambda_{4}\Big{)}\mbox{Tr}\Big{(}{Y_{d}Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle-\frac{9}{2}\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}+\Big{(}\frac{12}{5}\,g_{1}^{2}-8\,\lambda_{3}-11\,\lambda_{4}\Big{)}\mbox{Tr}\Big{(}{Y_{e}Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle-\frac{27}{2}\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-\frac{9}{2}\,\lambda_{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-12\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}}\Big{)}$
$\displaystyle-12\,\mbox{Tr}\Big{(}{Y_{d}Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}-24\,\mbox{Tr}\Big{(}{Y_{d}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}}\Big{)}-4\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}}\Big{)}$
$\displaystyle-4\,\mbox{Tr}\Big{(}{Y_{e}Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}-8\,\mbox{Tr}\Big{(}{Y_{e}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}}\Big{)}$
(49)

##  Appendix B One loop threshold corrections at $M_{S}$

In this appendix, we list the one-loop threshold corrections to the scalar
quartic couplings considered in our analysis. These corrections arise from the
box and triangle diagrams in MSSM at the scale $M_{S}$ in $\overline{\rm MS}$
scheme Haber and Hempfling (1993); Lee and Wagner (2015). Note that the
following expressions are obtained for a particular choice of the SUSY
spectrum given in Eq. (11).

$\displaystyle\Delta\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{4}\,\left[-\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}-\frac{1}{6}\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\right]$
(50) $\displaystyle+$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\frac{g_{2}^{2}+g_{Y}^{2}}{4}\,\left[3\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\right]$
$\displaystyle\Delta\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{4}\,\left[-\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}-\frac{1}{6}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}\right]$
(51) $\displaystyle+$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\frac{g_{2}^{2}+g_{Y}^{2}}{4}\,\left[3\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}\right]$
$\displaystyle\Delta\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\left[\frac{3}{2}\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}+\frac{3}{2}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}+\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}\right]$
(52) $\displaystyle+$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\Big{(}1+\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\Big{)}\,\left[-\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}-\frac{1}{6}\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}\right]$
$\displaystyle+$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\frac{g_{2}^{2}-g_{Y}^{2}}{8}\,\left[3\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+3\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}\right]$
$\displaystyle\Delta\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\left[\frac{3}{2}\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}Y_{u}Y_{u}^{\dagger}}\Big{)}+\frac{3}{2}\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}Y_{d}Y_{d}^{\dagger}}\Big{)}+\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}Y_{e}Y_{e}^{\dagger}}\Big{)}\right]$
(53) $\displaystyle+$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\Big{(}1+\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\Big{)}\,\left[\frac{1}{2}\,\mbox{Tr}\Big{(}{Y_{d}^{\dagger}Y_{u}Y_{u}^{\dagger}Y_{d}}\Big{)}+\frac{1}{6}\,\mbox{Tr}\Big{(}{Y_{e}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{e}}\Big{)}\right]$
$\displaystyle-$
$\displaystyle\frac{1}{16\pi^{2}}\,\Big{(}\frac{\mu}{M_{S}}\Big{)}^{2}\,\frac{g_{2}^{2}}{4}\,\left[3\,\mbox{Tr}\Big{(}{Y_{u}Y_{u}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+3\,\mbox{Tr}\Big{(}{Y_{d}Y_{d}^{\dagger}}\Big{)}+\mbox{Tr}\Big{(}{Y_{e}Y_{e}^{\dagger}}\Big{)}\right]$

We assume $\mu=0.1M_{S}$ in evaluating the threshold corrections throughout
our analysis.

##  Appendix C Details of the gauge and Yukawa couplings at $M_{t}$

In this appendix, we briefly describe our procedure to evaluate the input
parameters at the scale $M_{t}$. We use experimental values of masses of the
quarks and leptons from PDG Patrignani  _et al._ (2016). The masses of light
quarks, namely $u$, $d$ and $s$, are given at 2 GeV while those of heavy
quarks $c$, $b$ and $t$ are determined at their respective pole masses. To
calculate the running quark masses at $M_{t}$ we consider 3-loop running of
gauge couplings and mass parameters in effective QCD theory and perform
appropriate matching at the intermediate thresholds with 2-loop threshold
corrections. We closely follow the notations and procedure given in Chetyrkin
_et al._ (2000); Xing  _et al._ (2008).

In a given effective QCD${}^{(n_{f})}$ theory, where $n_{f}$ is number of
quarks flavours lighter than the renormalization scale $\mu$, the running
strong coupling constant and quark masses can be calculated by solving the
following 3-loop RG equations.

$\displaystyle\mu^{2}\frac{d}{d\mu^{2}}\frac{\alpha_{s}^{(n_{f})}(\mu)}{\pi}$
$\displaystyle=$
$\displaystyle-\sum_{k=0}^{2}\beta_{k}^{(n_{f})}\,\left(\frac{\alpha_{s}^{(n_{f})}(\mu)}{\pi}\right)^{k+2}\,,$
(54) $\displaystyle\mu^{2}\frac{d}{d\mu^{2}}m_{q}^{(n_{f})}(\mu)$
$\displaystyle=$ $\displaystyle-
m_{q}^{(n_{f})}(\mu)\,\sum_{k=0}^{2}\gamma_{k}^{(n_{f})}\,\left(\frac{\alpha_{s}^{(n_{f})}(\mu)}{\pi}\right)^{k+1}\,.$
(55)

The explicit forms of coefficients $\beta_{i}$ and $\gamma_{i}$ are given in
Eq. (2) and Eq. (7) in Chetyrkin  _et al._ (2000). At the given mass threshold
the 2-loop corrected matching between the strong coupling constant and quark
masses is performed using

$\displaystyle\alpha_{s}(\mu)^{(n_{f}-1)}$ $\displaystyle=$
$\displaystyle\xi_{g}\,\alpha_{s}(\mu)^{(n_{f})}\,,$ $\displaystyle
m_{q}(\mu)^{(n_{f}-1)}$ $\displaystyle=$
$\displaystyle\xi_{m}\,m_{q}(\mu)^{(n_{f})}\,$ (56)

where $\xi_{g}$, $\xi_{m}$ are matching factors in $\overline{\rm MS}$ scheme
and we use their expressions as given in Eqs. (20,26) in Chetyrkin  _et al._
(2000).

We use 2-loop QED and 3-loop QCD corrections while evaluating the effect of
charged leptons and the strong coupling constant $\alpha_{s}$ on the running
of electromagnetic coupling constant $\alpha_{\rm em}$. The pole masses of
charged leptons obtained from PDG Patrignani  _et al._ (2016) are converted
into $\overline{\rm MS}$ running masses using the following relations

$\displaystyle\mu^{2}\frac{d\alpha_{em}(\mu)}{d\mu^{2}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{\rm
em}^{2}}{\pi}\left(\beta_{0}+\beta_{1}\,\frac{\alpha_{em}}{\pi}+\sum_{i=1,3}\,\rho_{i}\,\left(\frac{\alpha_{s}}{\pi}\right)^{i}\right)\,,$
$\displaystyle m_{l}(\mu)$ $\displaystyle=$ $\displaystyle
M_{l}\,\left(1-\frac{\alpha_{\rm
em}(\mu)}{\pi}\,\left(1+\frac{3}{2}\ln\left(\frac{\mu}{m_{l}(\mu)}\right)\right)\right)\,,$
(57)

where $l=e,\mu,\tau$ and $M_{l}$ is the pole mass of the corresponding charged
lepton. Detailed expressions for $\beta_{i}$ and $\rho_{i}$ can be found in
Eqs. (16,17) in Xing  _et al._ (2008).

The gauge couplings at $M_{t}$ are obtained using

$\displaystyle g_{1}(M_{t})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{5}{3}}\frac{\sqrt{4\pi\,\alpha_{\rm
em}(M_{t})}}{\cos\theta_{W}}$ (58) $\displaystyle g_{2}(M_{t})$
$\displaystyle=$ $\displaystyle\frac{\sqrt{4\pi\,\alpha_{\rm
em}(M_{t})}}{\sin\theta_{W}}$ (59) $\displaystyle g_{3}(M_{t})$
$\displaystyle=$ $\displaystyle\sqrt{4\pi\,\alpha_{s}(M_{t})}$ (60)

where $\theta_{W}$ is the weak mixing angle in $\overline{\rm MS}$ scheme. In
the above expression $\alpha_{\rm em}(M_{t})$ and $\alpha_{s}(M_{t})$ are
calculated using Eqs. (C) and (C) respectively. The values of the gauge
couplings and fermion masses obtained at $M_{t}$ in this way are listed in
Table 1 in section III.

##  Appendix D Effects of uncertainty in the measurement of $M_{t}$ on the
results

In the following we show the effects of uncertainty in the measurements of
$M_{t}$ on the results obtained by us in the case of high and low scale seesaw
displayed in Figs. 4 and 6 respectively. We carry out the similar analysis for
$M_{t}=172.4$ GeV and $M_{t}=174.6$ GeV which are values of top quark
threshold at $-1\sigma$ and $1\sigma$ respectively. The constraints on
$\tan\beta-M_{A}$ plane obtained for various cases and for the above values of
$M_{t}$ are displayed in Figs. 7 and 8.

Figure 7: The details are same as given in the caption of Fig. 1. For all the
cases, $N=2.07$.

Figure 8: The details are same as given in the caption of Fig. 1.

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