#### Exploring the hyperchargeless Higgs triplet model up to the Planck scale

Eur. Phys. J. C
Exploring the hyperchargeless Higgs triplet model up to the Planck scale
Najimuddin Khan 0
0 Discipline of Physics, Indian Institute of Technology Indore , Khandwa Road, Simrol, Indore 453 552 , India
We examine an extension of the SM Higgs sector by a Higgs triplet taking into consideration the discovery of a Higgs-like particle at the LHC with mass around 125 GeV. We evaluate the bounds on the scalar potential through the unitarity of the scattering matrix. Considering the cases with and without Z2-symmetry of the extra triplet, we derive constraints on the parameter space. We identify the region of the parameter space that corresponds to the stability and metastability of the electroweak vacuum. We also show that at large field values the scalar potential of this model is suitable to explain inflation.
1 Introduction
The revelation of the Higgs boson [
1–3
] in 2012 at the Large
Hadron Collider (LHC) confirmed the existence of all the
Standard Model (SM) particles and showed the Higgs
mechanism to be responsible for electroweak symmetry breaking
(EWSB). So far, the LHC, operated with pp collision energy
at √s ∼ 8 and 13 TeV, has not found any signature of new
physics beyond the standard model (BSM). However,
various theoretical issues, such as the hierarchy problem related
to the mass of the Higgs, mass hierarchy and mixing patterns
in the leptonic and quark sectors, suggest the need for new
physics beyond the SM. Different experimental observations,
such as the non-zero neutrino mass, the baryon–antibaryon
asymmetry in the Universe, the mysterious nature of dark
matter (DM) and dark energy, and inflation in the early
Universe indicate the existence of new physics. Moreover, the
measured properties of the Higgs boson with mass ∼125
GeV are consistent with those of the scalar doublet as
predicted by the SM. However, the experimental data [
4
] still
comfortably allow for an extended scalar sector, which may
also be responsible for the EWSB.
The present experimental values of the SM parameter
of the Lagrangian indicate that if the validity of the SM is
extended up to the Planck mass (MPl = 1.2 × 1019 GeV), a
second, deeper minimum is located near the Planck mass such
that the EW vacuum is metastable. The transition lifetime of
the EW vacuum to the deeper minimum is finite τEW ∼ 10300
years [
5–16
]. The EW vacuum remains metastable even after
adding extra scalar particles to the SM, which has been
discussed in Refs. [
15–19
].
In this work, we add a real hypercharge Y = 0 scalar
triplet to the SM. In the literature, this model is termed the
hyperchargeless Higgs triplet model, HTM (Y = 0) [
20
].
We consider both the neutral C P-even component of the SM
doublet and the extra scalar triplet take part in the EWSB.
Including radiative corrections, we check the validity of the
parameters of the model up to the Planck mass MPl. We
review various theoretical and experimental bounds of this
model. In this work, we especially discuss the unitary bounds
of the quartic couplings of the scalar potential. To the best
of our knowledge, the unitary bounds of this model were not
discussed in the literature. Next, we impose a Z2-symmetry
such that an odd number of scalar particles of the triplet do
not couple with the SM particles. The lightest neutral scalar
particle does not decay and becomes stable. This scalar field
can be taken as a viable DM candidate which may fulfill
the relic abundance of the Universe. In this context, it is
instructive to explore whether these extra scalars can also
prolong the lifetime of the Universe. In this model, we find
new regions in the parameter space of this model in which
the EW vacuum remains metastable. We also consider that
the extra neutral scalar field (also compatible as a viable dark
matter candidate) can act as an inflaton. We show that this
scalar field is able to explain the inflationary observables.
A detailed study of the HTM (Y = 0) parameter space,
which is valid up to 1 TeV, has been performed in Refs.
[
21
]. Two different renormalization schemes, electroweak
precision, and decoupling of Higgs triplet scenario have been
discussed in Ref. [
22
]. Using the electroweak precision test
(EWPT) data and a one-loop correction to the ρ parameter,
the Higgs mass range has been predicted in Refs. [
23–27
].
The detailed structure of the vacuum of the scalar potential
at tree level has been studied in Ref. [
28
]. The constraints
on the parameter spaces from the recent LHC μγ γ and μZγ
data have been discussed in Ref. [
29
]. The LHC and future
collider experiments with high luminosity can be used as an
useful tool to detect these extra scalar particles through vector
bosons scatterings [
30
]. More recently, the inert scalar triplet
has been investigated in the context of dark matter direct
and indirect detection [
31–33
]. The heavier inert fields can
decay through one loop via extra Majorana fermions [
34,35
].
This model has the required ingredients to realize a
successful leptogenesis which can explain the matter asymmetry in
the Universe [
34,35
]. Multi-component dark matter has been
investigated [
36,37
] in HTM with extra scalar multiplets of
the SU (2) representation.
The paper is organized as follows. Section 2 starts with
a detailed description of the HTM (Y = 0) model. We
discuss detailed constraints in Sect. 3. Considering the
lightest Z2-odd neutral particle as a viable DM, we analyze the
scalar potential up to the Planck mass and identify regions of
parameter space corresponding to the stable and metastable
EW vacuum in Sect. 4. We explain inflation as well in Sect. 5.
Finally we conclude in Sect. 6.
2 Model
We consider a model with a real Higgs doublet, , and a
real, isospin I = 1, hypercharge Y = 0 triplet T . The extra
scalar triplet consists of a pair of singly charged fields and a
C P-even neutral scalar field. The doublet and triplet scalar
are conventionally written as [
22
]
=
G+
√12 (v1 + h10 + i G0)
,
⎛ η+ ⎞
T = ⎝ v2 + η0 ⎠ .
−η−
The kinetic part of the Lagrangian is given by
Lk = | Dμ
|2 + 21 | DμT |2 ,
where the covariant derivatives are defined by
Dμ
where Wμa (a = 1, 2, 3) are the SU (2)L gauge bosons,
corresponding to three generators of SU (2)L group and Bμ is
the U (1)Y gauge boson. The σ a (a = 1, 2, 3) are the Pauli
matrices, and ta can be written as follows:
1 ⎛ 0 1 0 ⎞ 1 ⎛ 0 −i
t1 = √2 ⎝ 1 0 1 ⎠ , t2 = √2 ⎝ i 0
0 1 0 0 i
0 ⎞
i
− ⎠ ,
0
The scalar potential is such that both the neutral C P-even
component of the SM doublet and the extra scalar triplet
receive vacuum expectation values (VEVs) and thus take
part in the EWSB. After EWSB, one of the linear
combinations of charged scalar fields of the scalar doublet and the
triplet is eaten by the W boson, which becomes massive,
and other orthogonal combinations of these fields become
massive charged scalar fields. Similarly, a pseudoscalar of
the scalar doublet becomes the longitudinal part of the
massive Z gauge boson. This scalar may give rise to a signature
through the scattering of vector bosons [
30
] in collider
experiments. The spontaneous EWSB generates masses for the W
and Z bosons, thus:
M W2 = g422 v12 + 4v22 , and
g2
MZ2 = 4c2θ2 v12 ,
where cW ≡ cos θW = g2/ g12 + g22 and sW ≡ sin θW . The
scalar doublet VEV v1 and the triplet VEV v2 are related to
the SM VEV by vSM(≡ 246.221 GeV) = v12 + 4v22.
One can see that this model violates custodial symmetry
at tree level,
ρ =
M W2 v2
MZ2 cW2 = 1 + 4 v122 .
The experimental value of ρ is 1.0004 ± 0.00024 [
38
] at 1σ .
Hence, δρ ≈ 0.0004 ± 0.00024 and we will adopt the bound
δρ ≤ 0.001. This puts stringent constraints on v2 and we find
that v2 should be less than 4 GeV.
The tree-level scalar potential with the Higgs doublet and
the real scalar triplet is invariant under SU (2)L × U (1)Y
transformation. This is given by
2
V ( , T ) = μ1 |
λ3
+ 2 |
|2 + μ222 | T |2 +λ1 |
|4 + λ42 | T |4
|2| T |2 +λ4
†σ a
Ta .
We have the following minimization conditions of the
treelevel scalar potential:
μ12 = 21 {2λ4v2 − (2λ1v12 + λ3v22)},
2 1 2 2
μ2 = 2v2 {λ4v1 − λ3v1 v2 − 2λ2v23}.
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
h
H
G±
H ±
where
sγ =
sβ =
=
=
cγ sγ
−sγ cγ
cβ sβ
−sβ cβ
.
In the large μ22 and small v2 limits, one can express sin γ
and sin β as
After electroweak symmetry breaking, the squared mass
matrix can be expressed as 6 × 6 for the scalar fields
(G1±, η±, η0 and h0). This matrix is composed of three 2 × 2
submatrices with bases, (G1+, η+), (G1−, η−) and (h0, η0).
After rotating these fields into the mass basis, we get four
physical mass eigenstates (H ±, h, H ). The remaining two
states (G±) and G0 become massless Goldstone bosons.
The physical masses of the particles are given by
1
Mh2 = 2
1
M H2 = 2
M H2± = λ4
where
(B + A) +
(v12 + 4v22) ,
2v2
(B + A) −
(B − A)2 + 4C 2 ,
(B − A)2 + 4C 2 ,
2
A = 2λ1v1 , B =
C = −λ4v1 + λ3v1v2.
The mixing between the doublet and triplet in the charged
and C P-even scalar sectors are, respectively, given by
(2.9)
(2.10)
(2.11)
(2.12)
In these limits, the quartic λ1,2,3 and λ4 can be written as
λ1( ) ≥ 0, λ2( ) ≥ 0, λ3( ) ≥ −2
M 2
λ1 = 2vh12 , λ2 =
2(M H2 − M H2± ) ,
2 2
v1 sβ
λ3 = 2(M H2± − v(s12γ /sβ )M H2 ) , λ4 = sβ Mv1H2± . (2.13)
In the same limits, if MH± and MH are very heavy
compared with Mh , then MH± and MH become degenerate (see
Eqs. (2.9) and (2.10)). If the mass difference between MH±
and MH is large, then the quartic couplings λ2,3 will violate
the perturbativity and unitarity bounds (see Sects. 3.2 and
3.3).
The SM gauge symmetry, SU (2)L , prohibits direct
coupling of the SM fermions with the scalar fields of the triplet.
The couplings of the new scalar fields (H, H ±) with SM
fermions are generated after the EWSB. The strengths of
H f¯ f (the f are the up, down quarks and charged leptons)
are proportional to sin γ . The couplings H +ν¯ll− and H +u¯d
are proportional to sin β.
3 Constraints on the hyperchargeless Higgs triplet
model
The parameter space of this model is constrained by
theoretical considerations like the absolute vacuum stability,
perturbativity, and unitarity of the scattering matrix. In the
following, we will discuss these theoretical bounds and the
constraints of the Higgs to diphoton signal strength from the
LHC and the electroweak precision measurements.
3.1 Vacuum stability bounds
A necessary condition for the stability of the vacuum comes
from requiring that the scalar potential is bounded from
below, i.e., it should not approach negative infinity along
any direction of the field space for large field values. For
h0, η0,± v1,2, the quadratic terms μ12| |2, μ222 |T |2 and
λ4 †σ a Ta of the scalar potential in Eq. (2.6) are
negligibly small compared with the other quartic terms, so the scalar
potential is given by
1
V (h0, η0, η±) = 4
λ1(h0)4 + λ2(η2 + 2η+η−)2
+λ3(h0)2(η2 + 2η+η−) .
(3.1)
The potential can be written in a symmetric matrix with basis
{(h0)2, (η0)2, η−η+}. Using the copositivity criteria [104],
one can calculate the required conditions for the absolute
stability/bounded from below of the scalar potential. The
treelevel scalar potential V ( , T ) ≡ V (h0, η0, η±) is absolutely
stable if
λ1( )λ2( ).
(3.2)
The coupling constants are evaluated at a scale using
RGEs. In this study, we use the SM RGEs up to three loops
which have been given in Refs. [
52–55
]. The triplet
contributions are taken up to two loops which are presented in
Appendix A. If the quantum corrections are included to the
scalar potential, then there is a possibility to form a minimum
along the Higgs field direction near the Planck mass MPl. For
negative λ1( ) the minimum at the energy scale becomes
deeper than the EW minimum and vice versa. In these
situations, the above conditions in Eq. (3.2) become more
complicated. These modifications will be shown in Sect. 4.2. As λ3
gives a positive contribution to the running of λ2, λ2 remains
positive up to the Planck mass MPl. Hence, it is clear that no
extra minimum will develop along the new scalar field
directions. The sign and the value of λ3 can change the Higgs
diphoton signal strength and the stability of the EW
vacuum. The importance of the sign of λ3 will be discussed in
Sects. 3.5 and 4.3.
3.2 Perturbativity bounds
To ensure that the radiatively improved scalar potential
V ( , T ) remains perturbative at any given energy scale ( ),
one must impose the following conditions:
λ4
4π.
| λ1,2,3 |
4π and
3.3 Unitarity bounds
The unitarity bound on the extended scalar sectors can be
calculated from the scattering matrix (S-matrix) of different
processes. The technique was developed in Refs. [
39,40
] for
the SM and it can also be applied to the HTM (Y = 0). The
S-matrix for the HTM (Y = 0) consists of different scalar–
scalar, gauge boson–gauge boson, gauge boson–scalar
scattering amplitudes. Using the Born approximation, the
scattering cross-section for any process can be written as
σ =
16π ∞
s
l=1
(2l + 1)|al (s)|2,
where s = 4E C2 M is the Mandelstam variable, and EC M
is the center of mass energy of the incoming particles. The
al are the partial wave coefficients corresponding to specific
angular momenta l. This leads to the following unitarity
constraint: Re(al ) < 21 . At high energy the dominant
contribution to the amplitude al of the two-body scattering processes
a, b → c, d comes from the diagram involving the quartic
couplings. Far away from the resonance, the other
contributions to the amplitude from the scalar mediated s-, t -, and
u-channel processes are negligibly small. Also, in the high
energy limit, the amplitude of scattering processes
involving longitudinal gauge bosons can be approximated by the
scalar amplitude in which gauge bosons are replaced by their
corresponding Goldstone bosons. For example, the
amplitude of the WL+WL− → WL+WL− scattering is equivalent to
G+G− → G+G−. This is known as the equivalence
theorem [
40–43
]. So to test the unitarity of HTM (Y = 0),
we construct the S-matrix which consists of only the scalar
quartic couplings.
(3.3)
(3.4)
The scalar quartic couplings in the physical bases G±, G0 ,
H ±, h and H are complicated functions of λ’s, γ , β. The
hhhh vertex is 6(λ1 cos4 γ + λ3 cos2 γ sin2 γ + λ2 sin4 γ ).
It is difficult to calculate the unitary bounds in the physical
bases. One can consider the non-physical scalar fields bases,
i.e., G1±, η±, G0, h0 and η0 before the EWSB. Here the
crucial point is that the S-matrix, which is expressed in terms
of the physical fields, can be transformed into a S-matrix
for the non-physical fields by making a unitary
transformation [
44,45
].
Different quartic couplings in non-physical bases are
obtained by expanding the scalar potential of Eq. (2.6) which
are given by
{G0 G0 G0 G0} = 6λ1,
G1+ G1+ G1− G−
1
= 4λ1,
G1+ G1− h0 h0
= 2λ1, {G0 G0 η0 η0} = λ3,
{h0 h0 η0 η0} = λ3,
G0 G0 η+ η−
= λ3,
h0 h0 η+ η−
= λ3,
G0 G0 G1+ G−
1
= 2λ1,
{G0 G0 h0 h0} = 2λ1, {h0 h0 h0 h0} = 6λ1,
G1+ G1− η0 η0
= λ3, {η0 η0 η0 η0} = 6λ2,
G1+ G1− η+ η−
= λ3,
η0 η0 η+ η−
= 2λ2,
(3.5)
η+ η+ η− η−
The full set of these non-physical scalar scattering processes
can be expressed as a 16 × 16 S-matrix. This matrix is
composed of three submatrices of dimensions 6 × 6, 5 × 5, and
5 × 5 which have different initial and final states.
The first 6 × 6 sub-matrix, M1, corresponds to
scattering processes whose initial and final states are one of
h0 G1+, G0 G1+, η0 G1+, h0 G1+, G0 η+, and η0 η+.
Using the Feynman rules in Eq. (3.5), one can obtain M1
= diag( 2λ1, 2λ1, 2λ1, λ3, λ3, λ3).
The sub-matrix M2 corresponds to scattering processes
with one of the following initial and final states: h0 G0,
G1+ η−, η+ G1−, η0 G0, and h0 η0. Similarly, one can
calculate M2 = diag( 2λ1, λ3, λ3, λ3, λ3).
The third sub-matrix, M3, corresponds to scattering fields
(G1+ G1−, η+ η−, G√02G0 , h√0h20 , and η√0η20 ). The factor √12
has appeared due to the statistics of identical particles. M3
is given by
Eigenvalues of M3 are 2λ1, 2λ1, 2λ2, and 21 (6λ1 + 5λ2±
Unitary constraints of the scattering processes demand
that the eigenvalues of the S-matrix should be less than 8π .
3.4 Bounds from electroweak precision experiments
Electroweak precision data has imposed severe bounds on
new physics models via the Peskin–Takeuchi parameters,
S, T , U [
46
]. The additional contributions from this model
are given by [
21,26
]
S
T
M H4 log
1
U = − 3π
M H2 + M H2±
MZ2
M H2
M H2±
log
( M )2
MZ2
,
M H2
M H2±
(3M H2± − M H2 )
(M H2 − M H2± )3
+
5(M H4 + M H4± ) − 22M H2± M H2
6(M H2 − M H2± )2
M
3π MH±
,
(3.7)
(3.8)
(3.9)
where M = MH± − MH . S is proportional to sin β. The
experimental value of the parameter ρ demands the triplet
VEV v2 to be less than 4 GeV [
38
]. Hence, the contributions
to the S parameter from the triplet scalar fields are negligible.
MH± and MH are almost degenerate for MH±,H Mh . The
contributions to the parameters T and U from this model are
also negligibly small [
47
].
3.5 Bounds from LHC diphoton signal strength
As the dominant production cross-section of h at LHC is
coming through gluon fusion, the Higgs to diphoton signal
strength μγ γ can be written as
μγ γ =
=
σ (gg → h → γ γ )H T M
σ (gg → h → γ γ )SM
σ (gg → h)H T M Br (h → γ γ )HTM
σ (gg → h)SM Br (h → γ γ )SM
.
We use the narrow width approximation as htotal/Mh → 0.
The Higgs h to f f¯ and V V (V stands for vector bosons)
couplings are proportional to cos γ , so they μγ γ can be
simplified as
(3.10)
μγ γ = cos2 γ
total
h,SM
total
h,HTM
(h → γ γ )HTM
(h → γ γ )SM
.
The charged Higgs H ± will alter the decay width of h → γ γ ,
Z γ through one loop, which implies (h → γ γ , Z γ )
total. Also, if the mass of the extra scalar particles (H T =
h
H, H ±) happen to be lighter than Mh /2, then they might
contribute to the invisible decay of the Higgs boson. Using the
global fit analysis [
48
] we see that such an invisible
branching ratio is less than ∼ 20%. In Eq. (3.11), the first ratio
provides a suppression of ∼0.8–1. For MH,H± > Mh /2, the
total
h,HTM ≈ cos12γ . Hence, the Higgs to diphoton
ratio becomes toht,aSlM
signal strength can be written as
(3.11)
(3.12)
μγ γ ≈
(h → γ γ )HTM
(h → γ γ )SM
.
In HTM, the additional contributions to (h → γ γ ) at one
loop due to the H ± is given by [
49
]
α2 M 3
h
(h → γ γ )HTM = 256π 3v2
f
N cf Q2f y f F1/2(τ f )
2
(3.13)
+yW F1(τW ) + Q2H± vμ2hMHH2+±H− F0(τH± )
where τi = Mh2/4M 2. Q f , Q H± denote the electric charges
i
of the corresponding particles. N cf is the color factor. y f
and yW denote the Higgs couplings to f f¯ and W +W −.
μh H+ H− = {2λ4sin βcos βcos γ + cos β2(λ3v1cos γ +
4λ2v2sin γ ) + sin β2(λ4sin γ + λ1v1cos γ + λ3v2sin γ )} ≈
λ3vSM stands for the coupling constant of the h H + H −
vertex. The loop functions F(0, 1/2, 1) can be found in Ref [
49
].
Recently, the ATLAS [
50
] and CMS [
51
] collaborations
have measured the ratio of the prediction of the diphoton
rate μγ γ of the observed Higgs to the SM prediction. The
present combined value of μγ γ is 1.14−+00..1189 from these
experiments [
4
].
In (h → γ γ )HTM (see Eq. (3.13)), a positive λ3 leads
to a destructive interference between H T and SM
contributions and vice versa. One can see from Eq. (3.13) that the
contribution to the Higgs diphoton channel is proportional to
MλH23± . If the charged scalar mass is greater than 300 GeV, then
the contribution of H ± to the diphoton signal is negligibly
small.
Now we present our results for the central values of the
SM parameters such as the Higgs mass Mh = 125.7 GeV, the
top mass Mt = 173.1 GeV, the Z boson mass MZ = 91.1876
GeV, and the strong coupling constant αs = 0.1184. We take
the triplet vev v2, λ4 and the other quartic couplings λ1,2,3
as input parameters. Hence, depending on these parameters
the mixing angle γ can vary in between 0 and π/2. The
triplet scalar masses also become arbitrarily heavy. Here,
we assume that no new physics shows up below the Planck
mass MPl. We examine the renormalization group (RG) flow
of all couplings and establish bounds on the heavy scalar
masses under the assumption that the parameters are valid
up to the Planck mass MPl. In this calculation, we use the
SM RGEs up to three loops [
52–55
] and the triplet
contributions up to two loops. We first calculate all couplings at
Mt . To find their values at Mt , one needs to take into account
different threshold corrections up to Mt [
5,6,15,16,76,77
].
Using the RGEs, we evolve all the coupling constants from
Mt to the Planck mass MPl. By this procedure we obtain
new parameter regions which are valid up to the Planck mass
MPl.
We show the allowed region (green) in the MH± –MH
plane for this model in Fig. 1. We demand that the EW
vacuum of the scalar potential remain absolutely stable and do
not violate the perturbative unitarity up to the Planck mass
MPl. One can also obtain the parameter spaces,
corresponding to the metastable EW vacuum, which are seen to be small
in this plane. Furthermore, we impose the EWPT constraints
on the parameters so that the region between the black-dashed
lines survives.
In Fig. 1, we show the allowed region for fixed central
values of all the SM parameters. In the left panel, we present
the plot for the choice of the quartic couplings λ2,3 = 0.1
and triplet VEV v2 = 3 GeV. In the right panel, we use the
value of the triplet VEV v2 = 1 GeV. We vary the quartic
coupling λ1 and dimensionful mass parameter λ4 to calculate
the neutral C P-even Higgs mass MH , the charged Higgs
mass MH± and the mixing angle γ . These scalar masses
increase, whereas the mixing angle decreases with λ4. We
find that the EW vacuum becomes unbounded from below
for λ1 0.128. The theory also violates unitarity bounds for
λ1 0.238 before the Planck mass MPl. One can see from
Fig. 1a, the allowed region becomes smaller for the larger
values of heavy scalar masses. In most of the parameter space
the running couplings either violate unitary or perturbativity
bounds before the Planck mass MPl.
As λ2,3 stabilize the scalar potential, we will get a wider
green region for smaller scalar masses, but this will violate
the unitarity bound in the higher mass region. We find that the
EW vacuum becomes unbounded from below for the values
of the quartic couplings λ1 0.027 and λ2,3 = 0.285. We
also check that the choice of the quartic couplings λ1
0.05 and λ2,3 = 0.285 will violate unitary and perturbativity
bounds before the Planck mass MPl. One can also understand
from the expressions of Eq. (2.13) that if we decrease the
value of v2, the area of the allowed region from the stability,
unitary and perturbativity bounds will increase. We show the
plot in Fig. 1b for the choice of v2 = 1 GeV.
If the vacuum expectation value of the scalar triplet
becomes zero, then the minimization condition of the scalar
potential given in Eq. (2.8) is no longer valid. The mass
parameter μ2 becomes free and the parameter λ4 does not
play any role in the stability analysis. In the next section, we
will show the detailed stability analysis in the presence of
extra Z2-symmetry in this model.
4 Dark matter in HTM (Y = 0)
We impose a Z2 symmetry on this model such that the scalar
triplet is odd under this transformation, i.e., T → −T ,
whereas the SM fields are even under this transformation. In
the literature, the HTM including the Z2-symmetry is known
as the inert triplet model (ITM) [
31
]. In this model, the term
λ4 H †σ a Ta is absent in the scalar potential in Eq. (2.6),
which implies λ4 = 0. The Z2-symmetry prevents the triplet
scalar to acquire a VEV, i.e., v2 = 0. The potential can have
a minimum along the Higgs field direction only. The EWSB
is driven by the SM Higgs doublet. The scalar fields of the
triplet do not mix with the scalar fields of SM doublet. After
the EWSB, the scalar potential in Eq. (2.6) is then given by
V (h, H, H ±)
1
= 4
2μ21(h + v)2 + λ1(h + v)4 + 2μ22(H 2 + 2 H + H −)
+ λ2(H 2 + 2 H + H −)2 + λ3(h + v)2(H 2 + 2 H + H −) .
Here, v ≡ vSM and the masses (see Eq. (2.9)) of the scalar
fields1 h, H and H ± are given by
Mh2 = 2λ1v2,
M H2 = μ22 + λ23 v2,
M H2 ± = μ22 + λ23 v2. (4.2)
At tree level the mass of the neutral scalar H and the charged
particles H ± are degenerate. If we include a one-loop
1 For v2 = 0, the notation in Eq. (2.1) H ≡ η0 and H ± ≡ η± are the
physical scalar fields.
(4.1)
mass for the portal coupling λ3(MZ ) = 0.10. b The relic density h2 as
a function of the DM mass MDM(≡ MH ) (red line) for λ3(MZ ) = 0.10
radiative correction, the charged particles become slightly
heavier [
56, 57
] than the neutral ones. The mass difference
between them is given by
M = (MH ± − MH )1-loop
= α4MπH f MMWH
− cW2 f
MZ
MH
,
(4.3)
with f (x ) = − x4 2x 3 log(x )+(x 2−4) 23 log x2−2−x2√x2−4 .
It has been shown in Refs. [
56, 57
] that the mass
splitting between charged and neutral scalars remains ∼ 150
MeV for MH = 0.1–5 TeV. In Fig. 2a, we show the
variation M (green line) with the MH (≡ MDM) mass. As
the Z2-symmetry also prohibits the couplings of an odd
number of scalar fields of the triplet with the SM particles,
H can serve as a viable DM candidate which may
saturate the measured DM relic density of the Universe. In this
work, we use the software package FeynRules [60] along
with micrOMEGAs [
61, 62
] to calculate the relic density of
the DM. As M is very small, the effective annihilation
cross-section is dominated by the co-annihilation channels
H H ± → SM particles [59]. Although it is dominated by
the co-annihilation channel, we need a very small Higgs
portal coupling λ3 to obtain the correct relic density. The
effective annihilation cross-section (see the black line in Fig. 2a)
decreases rapidly with M for the DM mass below 500 GeV
and becomes ∼ 10−26 cm3s−1 around MDM = 2000 GeV.
We obtain the relic density in the right ballpark.
V1SM+IT(h) = V1SM(h) + V1IT(h),
where [
69–73
]
V1SM(h) =
5
i=1
ni Mi2(h)
64π 2 Mi4(h) ln μ2(t ) − ci .
ni is the number of degrees of freedom and Mi2(h) =
κi (t ) h2(t ) − κi (t ). ni , ci , κi and κi can be found in Eq. (4) in
Ref. [
69
]. t is a dimensionless parameter which is expressed
in terms of the running parameter μ(t ) = MZ ex p(t ).
The contributions to the effective Higgs potential from
the new scalars (H, H ±) of the inert scalar triplet are given
by [
21
]
V1IT(h) =
j=H,H+,H−
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
In Fig. 2b, we present the plot for the relic density as a
function of the DM mass for the fixed Higgs portal coupling
λ3(MZ ) = 0.10. The light-red band is excluded from the
Higgs invisible decay width [
58
]. There are two deep regions
in the relic density band (red line). The first one is situated
near the DM mass MDM ≈ 45 GeV. It is due the resonance
of the s-channel H H ± → SM fermions processes, mediated
by the vector bosons W ±. The second one is situated near
the DM mass MDM ≈ Mh /2 for the Higgs-mediated H H →
SM fermions processes. There is another, shallower region
located around the DM mass MDM = 100 GeV, which is due
to the dominant contributions coming from H H ±, H H →
gauge bosons channels.
For 500 GeV, we find that the total cross-section σ v ∼
10−25 cm3s−1, so the relic density becomes ∼ 0.01. In this
region, the dominant channels are H, H ± → Z W ±, γ W ±
(∼ 35, ∼ 10%) and H ±, H ± → Z W ± (∼ 25%). We
also check that the smaller dark matter mass along with the
Higgs portal coupling λ3 (within the perturbative limit) does
alter the relic density only in the t hir d decimal place. If
we increase the DM masses, then the effective annihilation
cross-section decreases. This is mainly due to the mass
suppression. We get a DM relic density in the right ballpark for
DM masses greater than 1.8 TeV.
One can see that the mass splitting M attains saturation
for MDM > 700 GeV. Hence, the relic density is mainly
regulated by the Higgs-mediated s-channel processes, although
the contributions are small. We check that the Higgs portal
coupling λ3 can be varied in between 0 to 1 for the DM mass
1850–2200 GeV to get the right relic density. For example,
we obtain the relic density h2 = 0.1198 for λ3 = 0.001
and MDM = 1894.5 GeV. We get the same relic density for
λ3 = 0.8 and MDM = 2040 GeV. However, the running
couplings will violate the unitary and perturbativity bounds for
λ3 0.6.
The non-observation of DM signals in direct detection
experiments at XENON 100 [
64,65
], LUX [66] and
LUX2016 [
67
] put severe restrictions [
33
] on the Higgs portal
coupling λ3 for a given DM mass. In this model, we check the
parameter regions which are satisfying the relic density and
are allowed by the recent LUX-2016 [
67
] and
XENON1T2017 [
68
] data.
4.1 Metastability in ITM (Y = 0)
As in the SM the EW vacuum is metastable, it is important to
explore if ITM has any solution in its reserve. As the scalar
WIMP H protected by Z2-symmetry can serve as a viable
DM candidate, it is interesting to explore if they help prolong
the lifetime of the Universe. The effective Higgs potential
gets modified in the presence of these new extra scalars.
The one-loop effective Higgs potential in ms scheme and
the Landau gauge is given by
where M 2j(h) = 21 λ j (t ) h2(t ) + μ22(t ), with λH,H± (t ) =
λ3(t ). In the present work, in the Higgs effective potential,
SM contributions are taken up to two-loop level [
5,6,74,75
]
and the IT scalar contributions are considered up to one loop
only [21].
For h v, the quantum corrections to the Higgs potential
are reabsorbed in the effective running coupling λ1,eff , so that
the effective potential becomes
VeSffM+IT(h)
with
λ1,eff (h)
,
λ1,eff (h) = λ1S,Meff (h) + λI1T,eff (h) ,
where the expression of λ1S,Meff (h) up to two-loop
quantum corrections can be found in Ref. [
5
] and λI1T,eff (h) =
e4 (h) 2536λπ23 2 ln λ23 − 23 , with (h) = Mh t γ (μ) d
ln μ. The wave function renormalization of the Higgs field is
taken into account by the anomalous dimension γ (μ). Here,
all running coupling constants are evaluated at μ = h,
ensuring that the potential remains within the perturbative domain.
We first calculate all couplings with the threshold
corrections [
5,6,15,16,76,77
] at Mt . Then we evolve all the
couplings up to the Planck mass MPl using our own computer
codes incorporating the RG equations. Here, the SM effects
in the RGEs are taken up to three loops [
52–55
] and IT
contributions are considered up to two loops (see Appendix A).
We choose a specific benchmark point MDM(≡ MH ) =
1897 GeV, Mh = 125.7 GeV and αs (MZ ) = 0.1184, so
that it can give the right DM density of the Universe. The
corresponding values of all quartic couplings λ1,2,3 at Mt =
173.1 GeV and MPl = 1.2 × 1019 GeV are presented in
Table 1. For this benchmark point, we show the evolution of
the running of the quartic couplings (λ1,2,3) in Fig. 3. We
find that this specific choice of benchmark point with the
top mass2 Mt = 173.1 GeV and the central values of other
SM parameters leads to a metastable EW vacuum. It implies
that the βfunction of the Higgs quartic coupling λ1 becomes
zero at very high energy scale and remains positive up to the
Planck mass MPl. We find that a deeper minimum is situated
at that high energy scale before the Planck mass MPl. We also
check that the EW vacuum remains metastable (one-sided)
for the quartic coupling λ2 ≤ 0.1, the Higgs portal coupling
λ3 ≤ 0.15 and the DM mass MDM ≥ 1900 GeV. We obtain
the stable EW vacuum (> 99.99% confidence level,
onesided) for the choice of the parameters λ2 = 0.1, λ3 = 0.3
and MDM = 1915 GeV. The running couplings will violate
the unitary and perturbativity bounds for λ3 0.6. In the
following subsections, we will discuss the metastability of
the EW vacuum of the scalar potential.
4.2 Tunneling probability
Using the experimentally measured values of the SM
parameters at the EW scale, when analyzing the SM scalar potential
at higher energy scales, one encounters the so-called
metasta6yt4
2 As the βfunction of the Higgs quartic coupling, λ1 contains − 16π2
(see Eq. (A1)), the values of the Higgs quartic couplings λ1 at very high
energies are extremely sensitive to Mt .
bility of the EW vacuum [
5–7,15,16
]. Since a second (true)
minimum, deeper than the EW minimum, is situated near the
Planck mass, there exists a non-zero probability that the EW
minimum will tunnel into the second minimum. The
tunneling probability of the EW vacuum to the true vacuum at the
present epoch can be expressed as [
5,78,79
]
where S( B ) is the minimum action of the Higgs potential
of a bounce of size R = −B1 and is given by
(4.9)
(4.10)
It becomes minimum when λ1( B ) is minimum, i.e.,
βλ1 ( B ) = 0. In this work, we neglect loop [
78
] and
gravitational corrections [
80,81
] to the action as in Refs. [
15,16
].
A finite temperature also affects to EW vacuum
stability [
78,82,83
]. In this work, we consider field theory in the
zero-temperature limit.
In the ITM, the additional scalar fields give a positive
contribution to βλ1 (see Eqs. (A1) and (A2). Due to the
presence of these extra scalars, a metastable EW vacuum
goes towards the stability, i.e., the tunneling probability P0
becomes smaller. We first calculate the minimum value of
λ1,eff of Eq. (4.8). Putting this minimum value in Eq. (4.10),
we compute the tunneling probability P0. As the stability
of the EW vacuum is very sensitive to the top mass Mt , we
show the variation of the tunneling probability P0 as a
function of Mt in Fig. 4a. The right band in Fig. 4a corresponds
to the tunneling probability for our benchmark point. We
present P0 for the SM as the left band to see the effect of
the additional IT scalar. We also display 1σ error bands in
αs (light-gray) and Mh (light-red). One can see from this
figure that the effect of αs on the tunneling probability is larger
than the effect of Mh . To see the effect of the ITM parameter
spaces, we plot P0 as a function of the Higgs portal coupling
λ3(MZ ) in Fig. 4b for different choices of λ2(MZ ). We keep
the fixed central values of all SM parameters. Here, the DM
mass MDM is also varied with λ3 to get the DM relic density
h2 = 0.1198.
The additional IT scalar fields in the IT model improve
the stability of the EW vacuum as follows:
• If 0 > λ1( B ) > λ1,min( B ), then the vacuum is
metastable.
• If λ1( B ) < λ1,min( B ), then the vacuum is unstable.
• If λ2 < 0, the potential is unbounded from below along
the H and H ±-direction.
• If λ3( I) < 0, the potential is unbounded from below
along a direction in between H and h and also H ± and h.
matter constraints are respected for these specific choice of parameters.
The light-green band stands for Mt at ±1σ . b P0 is plotted against the
Higgs DM coupling λ3(MZ ) for different values of λ2(MZ )
In the above λ1,min( B ) = 1−0.0−0908.066l4n8(8v/ B ) and I
represents any energy scale for which λ1 is negative [
15,16
].
4.3 Phase diagrams
In order to show the explicit dependence of the electroweak
stability for different parameters of the ITM, we present
various kinds of phase diagrams.
In Fig. 5a, we calculate the confidence level for our bench
mark points MDM = 1897 GeV, λ2(MZ ) = 0.10 and
λ3(MZ ) = 0.10 by drawing an ellipse passing through the
stability line λ = βλ = 0 in the Mt –Mh plane. If the area of
the ellipse is χ times the area of the ellipse, it represents the
1σ error in the same plane. This factor χ is the confidence
level of the stability of EW vacuum. We develop a proper
method to calculate this factor and the tangency point for the
stability line. In this case, the confidence level of
metastability is decreased (one-sided) with αs(MZ ), i.e., the EW
vacuum moves towards the stability region. We obtain the
similar factor in the αs(MZ )–Mt plane. In this case, the
confidence level decreases with Mh . One can see from the phase
diagrams in Fig. 5 that the stable EW vacuum is excluded at
1.2 σ (one-sided).
If the ITM is valid up to the Planck mass, which also
saturates the DM abundance of the Universe, then the
confidence level vs. λ3(MZ ) phase diagram becomes important to
realize where the present EW vacuum is residing. In Fig. 6,
we vary the DM mass with λ3(MZ ) to keep the relic
density at h2 = 0.1198. One can see that the EW vacuum
approaches the stability with larger values of λ2,3(MZ ). The
EW vacuum becomes absolutely stable for λ3(MZ ) ≥ 0.154
and λ2(MZ ) ≈ 0.10 (see the blue line in Fig. 6). We show
this phase diagram for central values of the SM parameters.
Moreover, if we increase the top mass and/or decrease the
Higgs mass along with αs(MZ ), then the size of the region
corresponding to the metastable EW vacuum will increase.
We see that the conditions of a DM mass MDM ≥ 1912 GeV,
λ3(MZ ) ≥ 0.31 and λ2(MZ ) ≥ 0.1 are required to stabilize
the EW vacuum for Mt = 174.9 GeV, Mh = 124.8 GeV and
αs(MZ ) = 0.1163.
In Fig. 7, we show the allowed parameter spaces in the
λ3(MZ )–MH± plane for central values of the SM
parameters and λ2(MZ ) = 0.1. The lower (red) region is excluded
since the scalar potential becomes unbounded from below
along the direction in between H ± and h. In this region,
the effective Higgs quartic coupling is negative and at the
same time λ3 remains negative up to the Planck mass MPl.
We obtain the parameter space with negative λ3(MZ ), which
is also allowed from metastability. In this case, λ3 becomes
positive at the scale B and remains positive up to the Planck
mass MPl. The EW vacuum is absolutely stable in the green
region. The upper red region violates unitary bounds. The
right side of the black dotted line is allowed from μγ γ at 1σ .
5 Inflation in HTM (Y = 0)
Observations of super-horizon ansiotropies in the CMB
data, measured by various experiments such as WMAP and
Planck, have established that the early Universe underwent
resent error ellipses at 1, 2 and 3σ . The three boundary lines (dotted,
solid and dotted red) correspond to αs(MZ ) = 0.1184 ± 0.0007
Fig. 6 Dependence of confidence level at which the EW vacuum
stability is excluded (one-sided) or allowed on λ3(MZ ) and λ2(MZ ) in
ITM. Regions of absolute stability (green) and metastability (yellow)
of EW vacuum are shown for λ2(MZ ) = 0.1
a period of rapid expansion. This is known as inflation.
This can solve a number of cosmological problems, such
as the horizon problem, the flatness problem and the
magnetic monopole problem of the present Universe. If the
electroweak vacuum is metastable, then the Higgs is unlikely to
play the role of an inflaton [
84–92
] in the SM. Therefore,
Fig. 7 Phase diagram in λ3(MZ )–MH± plane in ITM. The right side
of the black dotted line is allowed from the signal strength ratio of
μγ γ within 68% confidence level and the left side is excluded at 1σ . In
the metastable region, the Higgs portal coupling λ3(MZ ) is negative;
however, beyond the scale B it is greater than zero
extra new degrees of freedom are needed in addition to the
SM ones to explain inflation in the early Universe [93–98].
Here, we study an extension of the Higgs sector with a real
triplet scalar T in the presence of large couplings ζh,H to the
Ricci scalar curvature R. This theory can explain inflation
in the early Universe at the large field values in the scale
invariance Einstein frame.
In this model, the action of the fields in a Jordan frame is
given by
S j =
! √
1 1
−gd4x LSM + 2 (∂μ )†(∂μ ) + 2 (∂μT )†(∂μT )
−ζh R| |2 − ζH R|T |2 − V ( , T ) .
(5.1)
In the present work, we consider H as an inflaton. The Higgs
h can also act as an inflaton for the stable EW vacuum. In
order to calculate the inflationary observables such as the
tensor-to-scalar ratio r , the spectral index ns and the running
of the spectral index nrs, we perform a conformal
transformation from Jordan frame to Einstein frame, so that the
nonminimal coupling ζH of the scalar field to the Ricci scalar
disappears.
The transformation is given by [99]
g˜μν =
2gμν ,
=
"
1 + ζH MP2l .
H 2
The action of Eq. (5.1) in the Einstein frame can be written
! √−gd4x
.
.
We plot this potential in Fig. 8 for the choice of the bench
mark point ζH = 1 and λ2 = 10−9. One can also get the same
plot for the parameters ζH = 104 and λ2 = 0.1. However,
this choice of the parameters violates the unitary bound. One
can see that the potential has the ability to explain slow-roll
inflation.
One can define the slow-roll parameters , η and ζ in terms
of the potential by
1
= 2
1 dV
V dχ
2
1 d2V 1 dV d3V
, η = V dχ 2 , and ζ = V 2 dχ dχ 3 .
The inflationary observable quantities such as the
tensor-toscalar ratio r , the spectral index ns and the running of the
spectral index nrs are defined as
r = 16 , ns = 1 − 6 + 2η, and
nrs = −2ζ − 24 2 + 16η
(5.3)
(5.4)
(5.5)
(5.6)
Fig. 8 Inflation potential in the Planck unit for ζH = 1 and λ2 = 10−9
and the number of e-folds is given by
N =
! χend
V
χstart dV /dχ
dχ
where χstart (χend) is the initial (final) value when inflation
starts (ends). At χstart, is 1. We calculate the χend from
Eq. (5.7) for N = 60.
At the end of inflation, we get
r = 0.0037, ns = 0.9644, and nrs = −6.24 × 10−4,
which is allowed by the present experimental data at 1σ [100,
101]. Hence, the neutral component of the triplet scalar can
simultaneously serve as an inflaton and dark matter particle
as well.
(5.7)
(5.8)
6 Discussion and conclusions
The measurements of the properties of the Higgs-like scalar
boson detected at the Large Hadron Collider on 4th July 2012
are consistent with the minimal choice of the scalar sector.
But the experimental data of the Higgs signal strengths and
the uncertainties in the measurement of other standard model
parameters still allow for an extended scalar sector. We have
taken an extra hyperchargeless scalar triplet as new physics.
First, we have assumed that the extra neutral C P-even
component of the scalar triplet has also participated in the EWSB.
We have shown the detailed structure of the tree-level scalar
potential and mixing of the scalar fields. We have also
discussed the bounds on the VEV (v2) of the neutral C P-even
component of the scalar triplet from the ρ-parameter. To the
best of our knowledge the full expressions of the unitary
bounds on the quartic couplings of the scalar potential in
this model have not yet been presented in the literature. We
have shown these unitary bounds in this model. As the SM
gauge symmetry SU (2)L prohibits the coupling of SM
neutrinos with the neutral C P-even component (η0) of the scalar
triplet, the model does not lead to neutrino masses. But the
model is still interesting, as it can play a role in improving the
stability of the Higgs potential. We have taken into account
various threshold corrections to calculate all the couplings
at Mt . Then using three-loop SM RGEs and two-loop triplet
RGEs, we have evolved all the couplings up to the Planck
mass MPl. We have shown the allowed region in the MH± –
MH plane. We have demanded that the EW vacuum of the
scalar potential remain absolutely stable and do not violate
the perturbative unitarity up to the Planck mass MPl. We have
discussed the constraints on the parameter spaces from the
recent LHC μγ γ and μZγ data. Furthermore, only a very
small region of the parameter space is shown to survive on
imposing the EWPT constraints.
Astrophysical observations of various kinds, such as
anomalies in the galactic rotation curves and gravitational
lensing effects in bullet clusters, have indicated the existence
of DM in the Universe. In the ITM, the extra scalar fields
are protected by a discrete Z2-symmetry which ensures the
stability of the lightest neutral particle. We have verified that
the mass of the neutral scalar particle (H ) is slightly lighter
than the mass of the charged particle (H ±) so that the
contributions coming from co-annihilation between H and H ±
play a significant role in the relic density calculation. In the
low mass region, the co-annihilation rates are quite high so
that the dark matter density is found to be much smaller than
the right relic density h2 = 0.1198 ± 0.0026 of the
Universe. We have obtained the relic density in the right ballpark
for a DM mass greater than 1.8 TeV. In this context, we have
shown how the presence of an additional hyperchargeless
scalar triplet improves the stability of the Higgs potential. In
this study, we have used state of the art next-to-next-to
leading order (NNLO) for the SM calculations. We have used
the SM Higgs scalar potential up to two-loop quantum
corrections which is improved by three-loop renormalization
groups of the SM couplings. We have taken into account
the contributions to the effective Higgs potential of the new
scalars at one loop only. These contributions are improved
by two-loop renormalization groups of the new parameters.
In this paper, we have explored the stability of the EW
minimum of the new effective Higgs potential up to the Planck
mass MPl. We have presented new modified stability
conditions for the metastable EW vacuum. We have also shown
various phase diagrams in various parameter spaces to show
the explicit dependence of the EW (meta)stability on
various parameters. For the first time, we have identified new
regions of parameter space that correspond to the stable and
metastable EW vacuum, which also provides the relic
density of the DM in the Universe as measured by the WMAP
and Planck experiments. In the present paper, we have also
shown that the extra neutral scalar field H can play a role in
inflation and can serve as a dark matter candidate. The scalar
potential can explain inflation for large scalar field values.
We have obtained the inflationary observables as observed
by the experiments.
Acknowledgements This work is supported by a fellowship from the
University Grants Commission. This work is partially supported by
a Grant from the Department of Science and Technology, India, via
Grant no. EMR/2014/001177. I would like to thank Subhendu Rakshit,
Amitava Raychaudhuri, Amitava Datta, Subhendra Mohanty and Girish
K. Chakravarty for useful discussions.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Two-loop beta functions for IT model
In this study, we use the SM RGEs up to three loops which
have been given in Refs. [
52–55
]. The triplet contributions
(λ2,3) are taken up to two loops which have been generated
using SARAH [102,103].
In the HTM (Y = 0), the RGEs of the couplings (χi =
g1,2,3, λ1,2,3 and Yl,u,d ) and dimensionful mass parameters
(μ1,2 and λ4) are defined as
∂χi 1 1
βχi = ∂ ln μ = 16π 2 βχ(1i) + (16π 2)2 βχ(2i) .
For μ > MH , the RGEs of the scalar quartic couplings
λ1,2,3 and the mass parameter λ4 are given by
β(1) 27 4 9 2 2 9 4
λ1 = + 200 g1 + 20 g1 g2 + 8 g2 − 59 g12λ1 − 9g22λ1 + 24λ21
+ 23 λ32 + 12λ1Tr Yd Yd† + 4λ1Tr Yl Yl†
+ 12λ1Tr Yu Yu† − 6Tr Yd Yd†Yd Yd†
− 2Tr Yl Yl†Yl Yl†
− 6Tr Yu Yu†Yu Yu† ,
(A1)
βλ(21) = − 23040101 g16 − 1460707 g14g22 − 38107 g12g24 + 21767 g26 + 1280807 g14λ1
+ 12107 g12g22λ1 − 289 g24λ1 + 1058 g12λ21
+ 108g22λ12 − 312λ31 + 5g24λ3 + 12g22λ32 − 15λ1λ32 − 2λ33
1
+ 20 − 5 64λ1 − 5g32 + 9λ1 − 90g22λ1 + 9g24
+ 9g14 + g12 50λ1 + 54g22
Tr Yd Yd†
+ 1603 g12g22Tr YuYu† − 49 g24Tr YuYu†
+ 127 g12λ1Tr YuYu† + 425 g22λ1Tr YuYu†
+ 80g32λ1Tr YuYu† − 144λ21Tr YuYu†
(A5)
Tr Yl Yl†
9 27
− 2 λ3Tr Yl Yl†Yl Yl† − 2 λ3Tr YuYu†YuYu† ,
β(1) † †
λ4 = 2λ4Tr Yl Yl + 4λ1λ4 + 4λ3λ4 + 6λ4Tr Yd Yd
For μ < MH , βλ1 = βλ1 (λ2,3 = 0) and βλ2,3,4 = 0, where
Yu = yu, yc, yt are the Yukawa couplings of up, charm and
top quark, Yd = yd , ys, yb for down, strange and bottom
quark. Yl represents the Yukawa couplings for the charged
leptons. In our work, we have included the contribution only
from the top quark. Since the other Yukawa couplings are
very small, they do not alter our result. We have also taken
into account the contributions to the beta functions of the
gauge couplings g1,2,3 of the new physics. The importance
of the mass parameters μ1,2 and λ4 are found to be negligible
in the stability analysis.
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