Exploring the hyperchargeless Higgs triplet model up to the Planck scale

The European Physical Journal C, Apr 2018

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 \(\mathbb {Z}_2\)-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.

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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. 92. S.C. Park, S. Yamaguchi, JCAP 0808, 009 (2008). arXiv:0801.1722 [hep-ph] 93. R.N. Lerner, J. McDonald, Phys. Rev. D 80, 123507 (2009). arXiv:0909.0520 [hep-ph] 94. O. Lebedev, H.M. Lee, Eur. Phys. J. C 71, 1821 (2011). arXiv:1105.2284 [hep-ph] 95. G.K. Chakravarty, S. Mohanty, Phys. Lett. B 746, 242 (2015). arXiv:1405.1321 [hep-ph] 96. G.K. Chakravarty, G. Gupta, G. Lambiase, S. Mohanty, Phys. Lett. B 760, 263 (2016). arXiv:1604.02556 [hep-ph] 97. G.K. Chakravarty, U.K. Dey, G. Lambiase, S. Mohanty, Phys. Lett. B 763, 501 (2016). arXiv:1607.06904 [hep-ph] 98. J. Ellis, arXiv:1702.05436 [hep-ph] 99. F. Kahlhoefer, J. McDonald, JCAP 1511(11), 015 (2015). arXiv:1507.03600 [astro-ph.CO] 100. P.A.R. Ade et al., Planck Collaboration. Astron. Astrophys. 594, A13 (2016). arXiv:1502.01589 [astro-ph.CO] 101. P.A.R. Ade et al., Planck collaboration. Astron. Astrophys. 594, A20 (2016). arXiv:1502.02114 [astro-ph.CO] 102. F. Staub, Comput. Phys. Commun. 185, 1773 (2014). arXiv:1309.7223 [hep-ph] 103. F. Staub, arXiv:1503.04200 [hep-ph] 104. K. Kannike, Eur. Phys. J. C 72, 2093 (2012). arXiv:1205.3781 [hep-ph] 1. G. Aad et al., ATLAS Collaboration. Phys. Lett. B 716 , 1 ( 2012 ). arXiv: 1207 .7214 [hep-ex] 2. S. Chatrchyan et al., CMS Collaboration. Phys. Lett. B 716 , 30 ( 2012 ). arXiv: 1207 .7235 [hep-ex] 3. P.P. Giardino , K. Kannike , I. Masina, M. Raidal , A. Strumia , JHEP 1405 , 046 ( 2014 ). arXiv: 1303 .3570 [hep-ph] 4. G. Aad et al., ATLAS and CMS Collaborations. JHEP 1608 , 045 ( 2016 ). arXiv: 1606 .02266 [hep-ex] 5. D. Buttazzo , G. Degrassi, P.P. Giardino , G.F. Giudice , F. Sala , A. Salvio , A. Strumia , JHEP 1312 , 089 ( 2013 ). arXiv: 1307 . 3536 6. G. Degrassi , S. Di Vita , J. Elias-Miro , J.R. Espinosa , G.F. Giudice , G. Isidori , A. Strumia , JHEP 1208 , 098 ( 2012 ). arXiv: 1205 .6497 [hep-ph] 7. I. Masina , Phys. Rev. D 87 , 053001 ( 2013 ). arXiv: 1209 .0393 [hepph] 8. J. Elias-Miro , J.R. Espinosa , G.F. Giudice , G. Isidori , A. Riotto , A. Strumia , Phys. Lett. B 709 , 222 ( 2012 ). arXiv: 1112 .3022 [hepph] 9. V. Branchina , E. Messina, Phys. Rev. Lett . 111 , 241801 ( 2013 ). arXiv: 1307 .5193 [hep-ph] 10. V. Branchina , E. Messina , A. Platania , JHEP 1409 , 182 ( 2014 ). arXiv: 1407 .4112 [hep-ph] 11. V. Branchina , E. Messina, M. Sher , Phys. Rev. D 91 , 013003 ( 2015 ). arXiv: 1408 .5302 [hep-ph] 12. V. Branchina , E. Messina , D. Zappala , EPL 116 ( 2 ), 21001 ( 2016 ). arXiv: 1601 .06963 [hep-ph] 13. V. Branchina , E. Messina, EPL 117 ( 6 ), 61002 ( 2017 ). arXiv: 1507 .08812 [hep-ph] 14. E. Bentivegna , V. Branchina , F. Contino , D. Zappal , JHEP 1712 , 100 ( 2017 ). arXiv: 1708 .01138 [hep-ph] 15. N. Khan , S. Rakshit, Phys. Rev. D 90 , 113008 ( 2014 ). arXiv: 1407 .6015 [hep-ph] 16. N. Khan , S. Rakshit, Phys. Rev. D 92 , 055006 ( 2015 ). arXiv: 1503 .03085 [hep-ph] 17. A. Datta , N. Ganguly , N. Khan and S. Rakshit , arXiv: 1610 .00648 [hep-ph] 18. L. Basso , O. Fischer , J.J. van Der Bij , Phys. Lett. B 730 , 326 ( 2014 ). arXiv: 1309 .6086 [hep-ph] 19. O. Fischer , arXiv: 1607 .00282 [hep-ph] 20. T. Blank , W. Hollik, Nucl. Phys. B 514 , 113 ( 1998 ). arXiv:hep-ph/9703392 21. J.R. Forshaw , A. Sabio Vera , B.E. White , JHEP 0306 , 059 ( 2003 ). arXiv:hep-ph/0302256 22. M.C. Chen , S. Dawson , C. B . Jackson , Phys. Rev. D 78 , 093001 ( 2008 ). arXiv: 0809 .4185 [hep-ph] 23. M.C. Chen , S. Dawson , T. Krupovnickas, Phys. Rev. D 74 , 035001 ( 2006 ). arXiv:hep-ph/0604102 24. M.C. Chen , S. Dawson , T. Krupovnickas, Int. J. Mod. Phys. A 21 , 4045 ( 2006 ). arXiv:hep-ph/0504286 25. P.H. Chankowski , S. Pokorski , J. Wagner , Eur. Phys. J. C 50 , 919 ( 2007 ). arXiv:hep-ph/0605302 26. J.R. Forshaw , D.A. Ross , B.E. White , JHEP 0110 , 007 ( 2001 ). arXiv:hep-ph/0107232 27. Z.U. Khandker , D. Li , W. Skiba , Phys. Rev. D 86 , 015006 ( 2012 ). arXiv: 1201 .4383 [hep-ph] 28. P. Fileviez Perez , H.H. Patel , M.J. Ramsey-Musolf , K. Wang , Phys. Rev. D 79 , 055024 ( 2009 ). arXiv: 0811 .3957 [hep-ph] 29. L. Wang , X.F. Han , JHEP 1403 , 010 ( 2014 ). arXiv: 1303 .4490 [hep-ph] 30. N. Khan , B. Mukhopadhyaya , S. Rakshit , A. Shaw , arXiv: 1608 .05673 [hep-ph] 31. T. Araki , C.Q. Geng , K.I. Nagao , Phys. Rev. D 83 , 075014 ( 2011 ). arXiv: 1102 .4906 [hep-ph] 32. S. Y. Ayazi , S. M. Firouzabadi , arXiv: 1501 .06176 [hep-ph] 33. S.Y. Ayazi , S.M. Firouzabadi , JCAP 1411 , 005 ( 2014 ). arXiv: 1408 .0654 [hep-ph] 34. F.X. Josse-Michaux , E. Molinaro, Phys. Rev. D 87 , 036007 ( 2013 ). arXiv: 1210 .7202 [hep-ph] 35. W. B. Lu , P. H. Gu , JCAP 1605 ( 05 ), 040 ( 2016 ). arXiv: 1603 .05074 [hep-ph] 36. O. Fischer , J.J. van der Bij , Mod. Phys. Lett. A 26 , 2039 ( 2011 ) 37. O. Fischer , J.J. van der Bij , JCAP 1401 , 032 ( 2014 ). arXiv: 1311 .1077 [hep-ph] 38. K.A. Olive et al., Particle Data Group Collaboration. Chin. Phys. C 38 , 090001 ( 2014 ) 39. B.W. Lee , C. Quigg , H.B. Thacker , Phys. Rev. Lett . 38 , 883 ( 1977 ) 40. B.W. Lee , C. Quigg , H.B. Thacker , Phys. Rev. D 16 , 1519 ( 1977 ) 41. Y.P. Yao , C.P. Yuan , Phys. Rev. D 38 , 2237 ( 1988 ) 42. H.G.J. Veltman , Phys. Rev. D 41 , 2294 ( 1990 ) 43. H.J. He et al., Phys. Rev. Lett . 69 , 2619 ( 1992 ) 44. S. Kanemura et al., Phys. Lett. B 313 , 155 - 160 ( 1993 ) 45. A. Arhrib, arXiv:hep-ph/0012353 46. M.E. Peskin , T. Takeuchi, Phys. Rev. D 46 , 381 ( 1992 ) 47. M. Baak et al., Gfitter Group Collaboration. Eur. Phys. J. C 74 , 3046 ( 2014 ). arXiv: 1407 .3792 [hep-ph] 48. G. Belanger, B. Dumont , U. Ellwanger , J.F. Gunion , S. Kraml, Phys. Rev. D 88 , 075008 ( 2013 ) 49. A. Djouadi, Phys. Rept . 459 , 1 ( 2008 ). arXiv:hep-ph/0503173 50. G. Aad et al., ATLAS Collaboration. Phys. Rev. D 90 , 112015 ( 2014 ). arXiv: 1408 .7084 [hep-ex] 51. V. Khachatryan et al., CMS Collaboration. Eur. Phys. J. C 74 , 3076 ( 2014 ). arXiv: 1407 .0558 [hep-ex] 52. K.G. Chetyrkin , M.F. Zoller , JHEP 1206 , 033 ( 2012 ). arXiv: 1205 .2892 [hep-ph] 53. M. F. Zoller , arXiv: 1209 .5609 [hep-ph] 54. K.G. Chetyrkin , M.F. Zoller , JHEP 1304 , 091 ( 2013 ). arXiv:1303.2890 [hep-ph]. [Erratum-ibid. 1309 , 155 ( 2013 )] 55. M. Zoller, PoS EPS-HEP2013 , 322 ( 2014 ) arXiv: 1311 .5085 [hepph] 56. M. Cirelli , N. Fornengo , A. Strumia , Nucl. Phys. B 753 , 178 ( 2006 ). arXiv:hep-ph/0512090 57. M. Cirelli , A. Strumia , New J. Phys. 11 , 105005 ( 2009 ). arXiv: 0903 .3381 [hep-ph] 58. G. Belanger, B. Dumont , U. Ellwanger , J.F. Gunion , S. Kraml, Phys. Rev. D 88 , 075008 ( 2013 ). arXiv: 1306 .2941 [hep-ph] 59. K. Griest , D. Seckel , Phys. Rev. D 43 , 3191 ( 1991 ) 60. A. Alloul , N.D. Christensen , C. Degrande , C. Duhr , B. Fuks , Comput. Phys. Commun . 185 , 2250 ( 2014 ). arXiv:1310 . 1921 [hep-ph] 61. G. Belanger , F. Boudjema , P. Brun , A. Pukhov , S. Rosier-Lees , P. Salati , A. Semenov , Comput. Phys. Commun . 182 , 842 ( 2011 ). arXiv: 1004 .1092 [hep-ph] 62. G. Belanger , F. Boudjema , A. Pukhov , A. Semenov , Comput. Phys. Commun . 185 , 960 ( 2014 ). arXiv: 1305 .0237 [hep-ph] 63. P.A.R. Ade et al. [Planck Collaboration]. arXiv:1303 . 5076 [astroph .CO] 64. E. Aprile et al., XENON100 Collaboration. Phys. Rev. Lett . 107 , 131302 ( 2011 ). arXiv:1104.2549 [astro-ph.CO] 65. E. Aprile et al., XENON100 Collaboration. Phys. Rev. Lett . 109 , 181301 ( 2012 ). arXiv:1207.5988 [astro-ph.CO] 66. D.S. Akerib et al., LUX Collaboration. Phys. Rev. Lett . 112 , 091303 ( 2014 ). arXiv:1310.8214 [astro-ph.CO] 67. D.S. Akerib et al., Results from a search for dark matter in the complete LUX exposure . Phys. Rev. Lett . 118 ( 2 ), 021303 ( 2017 ) 68. E. Aprile et al., XENON Collaboration. Phys. Rev. Lett . 119 , 181301 ( 2017 ). arXiv:1705.06655 [astro-ph.CO] 69. J.A. Casas , J.R. Espinosa , M. Quiros , Phys. Lett. B 342 , 171 ( 1995 ). arXiv:hep-ph/9409458 70. G. Altarelli, G. Isidori, Phys. Lett. B 337 , 141 ( 1994 ) 71. J.A. Casas , J.R. Espinosa , M. Quiros , A. Riotto , Nucl. Phys. B 436 , 3 ( 1995 ). arXiv:hep-ph/9407389 [Erratum-ibid. B 439 , 466 ( 1995 )] 72. J.A. Casas , J.R. Espinosa , M. Quiros , Phys. Lett. B 382 , 374 ( 1996 ). arXiv:hep-ph/9603227 73. M. Quiros , arXiv:hep-ph/9703412 74. C. Ford , I. Jack , D.R.T. Jones , Nucl. Phys. B 387 , 373 ( 1992 ). arXiv:hep-ph/0111190 [Erratum-ibid. B 504 , 551 ( 1997 )] 75. S.P. Martin , Phys. Rev. D 65 , 116003 ( 2002 ). arXiv:hep-ph/0111209 76. A. Sirlin , R. Zucchini , Nucl. Phys. B 266 , 389 ( 1986 ) 77. F. Bezrukov , M.Y. Kalmykov , B.A. Kniehl , M. Shaposhnikov , JHEP 1210 , 140 ( 2012 ). arXiv: 1205 .2893 [hep-ph] 78. G. Isidori, G. Ridolfi , A. Strumia , Nucl. Phys. B 609 , 387 ( 2001 ). arXiv:hep-ph/0104016 79. S. R. Coleman , Phys. Rev. D 15 , 2929 ( 1977 ) [Erratum-ibid . D 16 , 1248 ( 1977 )] 80. S.R. Coleman , F. De Luccia, Phys. Rev. D 21 , 3305 ( 1980 ) 81. G. Isidori , V.S. Rychkov , A. Strumia , N. Tetradis , Phys. Rev. D 77 , 025034 ( 2008 ). arXiv: 0712 .0242 [hep-ph] 82. L. Delle Rose , C. Marzo , A. Urbano , JHEP 1605 , 050 ( 2016 ). arXiv: 1507 .06912 [hep-ph] 83. J.R. Espinosa , M. Quiros , Phys. Lett. B 353 , 257 ( 1995 ). arXiv:hep-ph/9504241 84. F.L. Bezrukov , M. Shaposhnikov , Phys. Lett. B 659 , 703 ( 2008 ). arXiv: 0710 .3755 [hep-th] 85. F. Bezrukov , D. Gorbunov , M. Shaposhnikov , JCAP 0906 , 029 ( 2009 ). arXiv: 0812 .3622 [hep-ph] 86. F.L. Bezrukov , A. Magnin , M. Shaposhnikov , Phys. Lett. B 675 , 88 ( 2009 ). arXiv: 0812 .4950 [hep-ph] 87. F. Bezrukov , M. Shaposhnikov , JHEP 0907 , 089 ( 2009 ). arXiv: 0904 .1537 [hep-ph] 88. A.O. Barvinsky , A.Y. Kamenshchik , A.A. Starobinsky , JCAP 0811 , 021 ( 2008 ). arXiv: 0809 .2104 [hep-ph] 89. A.O. Barvinsky , A.Y. Kamenshchik , C. Kiefer , A.A. Starobinsky , C. Steinwachs , JCAP 0912 , 003 ( 2009 ). arXiv: 0904 .1698 [hepph] 90. A. De Simone , M.P. Hertzberg , F. Wilczek , Phys. Lett. B 678 , 1 ( 2009 ). arXiv: 0812 .4946 [hep-ph] 91. J. Garcia-Bellido , D.G. Figueroa , J. Rubio , Phys. Rev. D 79 , 063531 ( 2009 ). arXiv: 0812 .4624 [hep-ph]


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Najimuddin Khan. Exploring the hyperchargeless Higgs triplet model up to the Planck scale, The European Physical Journal C, 2018, 341, DOI: 10.1140/epjc/s10052-018-5766-4