Radiatively scotogenic type-II seesaw and a relevant phenomenological analysis

When a small vacuum expectation value of Higgs triplet ($v_\Delta$) in the type-II seesaw model is required to explain neutrino oscillation data, a fine-tuning issue occurs on the mass-dimension lepton-number-violation (LNV) scalar coupling. Using the scotogenic approach, we investigate how a small LNV term is arisen through a radiative correction when an $Z_2$-odd vector-like lepton ($X$) and an $Z_2$-odd right-handed Majorana lepton ($N$) are introduced to the type-II seesaw model. Due to the dark matter (DM) direct detection constraints, the available DM candidate is the right-handed Majorana particle, whose mass depends on and is close to the $m_X$ parameter. Combing the constraints from the DM measurements, the $h\to \gamma\gamma$ decay, and the oblique $T$-parameter, it is found that the preferred range of $v_\Delta$ is approximately in the region of $10^{-5}-10^{-4}$ GeV; the mass difference between the doubly and the singly charged Higgs is less than 50 GeV, and the influence on the $h\to Z\gamma$ is not significant. Using the constrained parameters, we analyze the decays of each Higgs triplet scalar in detail, including the possible three-body decays when the kinematic condition is allowed. It is found that with the exception of doubly charged Higgs, scalar mixing effects play an important role in the Higgs triplet two-body decays when the scalar masses are near-degenerate. In the non-degenerate mass region, the branching ratios of the Higgs triplet decays are dominated by the three-body decays.


I. INTRODUCTION
An extension of the standard model (SM) is necessary due to the observed massive neutrinos. If the origin of neutrino masses arises from a similar Brout-Englert-Higgs mechanism in the SM [1][2][3], where the W ± and Z gauge bosons, the quarks, and the charged leptons obtain their masses through a Higgs doublet (H), it is natural to introduce a Higgs triplet (∆) to the SM as a neutrino mass source. Hereafter, we call the Higgs triplet model the type-II seesaw model [4][5][6][7][8]. Since only the left-handed leptons couple to the Higgs triplet, neutrinos are the Majorana particles.
In addition to the Yukawa couplings, the neutrino masses are associated with the vacuum expectation value (VEV) of the Higgs triplet. In the minimal type-II seesaw model, it is known that the ∆ VEV indeed is dictated by the lepton-number softly breaking term µ ∆ H T iτ 2 ∆ † H, which appears in the scalar potential. Thus, a fine-tuning issue on µ ∆ is caused when the condition of µ D O(m W ) is required to explain the neutrino mass [9][10][11].
From the astrophysical observation, dark matter (DM) is introduced to explain more than 80% of non-baryonic matter. If DM is a kind of weakly interacting massive particle (WIMP), a radiatively scotogenic mechanism for generating the neutrino masses can be applied [12,13], where the particles in the dark sector are the mediators in the loop Feynman diagrams. Various applications of scotogentic models can be found in .
In order to naturally obtain a small µ D parameter in the type-II seesaw model, in this study, we consider that µ ∆ H T iτ 2 ∆ † H is suppressed at the tree level due to the leptonnumber symmetry; then, the necessary µ ∆ term is radiatively induced through the scotogenic mechanism [39][40][41]. Since the minimal type-II seesaw model does not include any particles that belong to the invisible side, we inevitably have to add new dark representations to the type-II seesaw model. Because the Higgs triplet cannot couple to singlet fermions, the minimum representation that directly couples to the Higgs triplet is the SU (2) L doublet fermion (X). Due to H and X being the SU (2) L doublets, in order to form a gauge invariant interaction, we can add one more singlet fermion (N ) into the model such that the H, X, and N coupling can generate the µ ∆ term through the one-loop level.
If the new representation set is assumed to be a minimal choice, due to the gauge anomaly free condition, the new doublet fermion can be a vector-like lepton doublet, and the singlet fermion can be a right-handed Majorana lepton without carrying any SM gauge quantum numbers. In addition, to have a stable DM candidate, we impose a Z 2 -symmetry to the vector-like lepton doublet and right-handed singlet; that is, X and N belong to the dark representations. Thus, the loop-induced µ ∆ term indeed arises from the lepton-number soft breaking effects in the invisible sector.
The main characteristics in the simple extension of the type-II seesaw model can be sum- In addition, we analyze the constraints from the Higgs diphoton decay and the oblique T parameter [42]; as a result, |m H ±± −m H ± | 50 GeV is allowed and the new physics influence on the h → Zγ decay is not significant.
In addition to the DM candidate and the origin of the neutrino masses, similar to the conventional type-II seesaw model, it is of interest to explore and probe the new scalars of the Higgs triplet at the LHC, especially the search for H ±± . With an integrated luminosity of 12.9 fb −1 at √ s = 13 TeV, CMS reports that the bounds on m H ±± through the ± ± ( = e, µ), ± τ ± , and τ ± τ ± channels are between 800 and 820 GeV, between 643 and 714

II. THE MODEL
In addition to the SM particles, we add one Higgs triplet ∆, one vector-like lepton doublet X R,L , and one SU (2) singlet heavy neutrino into the SM, where their representations in SU (2) L × U (1) Y are given in Table I. In order to avoid the Dirac neutrino mass term, we require that X and N are Z 2 -odd states and that the others are Z 2 -even; therefore, the lightest neutral particles of X and N could be the DM candidate. In addition, in order to dynamically generate the finite dimension-3 lepton-number violating term in the scalar potential, we assign that X L(R) , N and ∆ carry the lepton numbers as 0(1), 0 and 2, respectively, where the lepton number symmetry is softly broken by the X Dirac mass term.
The detailed charge assignments of the introduced particles are shown in Table I.
Based on the chosen representations and charge assignments, the gauge invariant Yukawa couplings can be written as: where the flavor indices are suppressed; C = iγ 2 γ 0 is charge conjugation matrix; H is the SM Higgs doublet,H = iτ 2 H * , τ 2 is the Pauli matrix, and L T = (ν, ) is the SM lepton doublet. It can be seen that the lepton number symmetry is explicitly broken by the m X dimension-3 terms. The Higgs doublet, vector-like lepton doublet, and Higgs triplet are respectively expressed as:

in which
v h and v ∆ are the VEVs of the Φ 0 and ∆ 0 fields, respectively. The VEVs and scalar masses are determined by the scalar potential.

A. Heavy Majorana masses
Because of the X L HN and X R ∆X R couplings, it is found that the Dirac-type X 0 not only mixes with Majorana particle N but also has a Majorana mass, which is related to v ∆ X T R CX R when ∆ 0 obtains a VEV. Thus, using the basis of (X R , X C L , N ), the Majoranatype heavy fermion mass matrix is written as: Since v ∆ is induced from one-loop in this study, it is expected that m 0 m N,X . It is found that the M M eigenvalues can be approximately expressed as follows: For m N > m X , where we use N i as the Majorana particle eigenstates, and e N,X and e δ are obtained as: For m N < m X , they are: where the corresponding e N,X and e δ are given as: Based on the obtained eigenvalues, the 3 × 3 orthogonal matrix elements (O ij ), which transform the (X R , X C L , N ) state to the (N 1 , N 2 , N 3 ) state, can be formulated as: where N 2 i = k O 2 ik are the normalization factors.

B. Gauge couplings of Z 2 -odd particles
If we define the Majorana states χ i as χ i = N i + N C i = χ C i , which satisfy P R χ i = N i and P L χ i = N C i , the charged current interactions of the heavy fermions can be expressed as: where the mixing matrix elements O ij for the neutral Z 2 -odd particles are included. The neutral current interactions of the Z-gauge boson and the photon with the Z 2 -odd particles can be obtained as: where c W = cos θ W and c 2W = cos 2θ W with Weinberg angle θ W ; X − includes X − R and X − L , Q X = −1 is the X − electric charge, and c Z ij show the FCNC effects and are defined as: From Eq. (10), it can be seen that the Z-boson coupling to the Z 2 -odd particle is through axial-vector currents; therefore, it will lead to the SD DM-nucleon elastic scattering.
When N 1 (χ 1 ) is the DM candidate, in order to satisfy the DM direct detection constraints, we must require c Z 11 to be small enough. From Eq. (8), if we drop the m 0 and y X v h / √ 2 effects, it can be seen that c Z 11 = 0. However, the case leads to m N 1 = m N 2 and c Z 12 = 1, where the DM-nucleon scattering occurs through χ 1R χ 2R Z coupling (or X 0 X 0 Z coupling). Hence, in addition to the c Z 11 magnitude, we have to take proper m 0 and y X v h / √ 2 in such a way that the mass splitting between N 1 and N 2(3) is large enough, so that the DM scattering off the nucleon through N 1 N 2,3 Z coupling can be kinematically suppressed. If we take m N > m X , the mass splitting between N 1 and N 2 can be found to be ∆m 12 = e X + e δ ≈ e N , and the c Z 11 coefficient can be expressed as: If N 3 (χ 3 ) is the DM candidate, because c Z 33 is small, we will show that the SD DM-nucleon scattering cross section is under the current PICO-60 [70] and Xenon1T [71] upper limits.

III. SCALAR POTENTIAL AND YUKAWA SECTOR
According to the convention in [68,79], we write the gauge invariant scalar potential as: where we take µ 2 , λ > 0 for the purpose of spontaneously breaking the electroweak gauge symmetry. It can be seen that due to the lepton-number conservation, the dimension-3 H T iτ 2 ∆ † H term is suppressed at the tree level. Without this term, the Higgs triplet cannot obtain a VEV and the SM neutrinos are still massless. In order to generate the finite dimension-3 term, we require that the right-handed Z 2 -odd lepton doublet only couples to the Higgs triplet by assigning proper lepton numbers to X R and X L , which are shown in Table. ?? Thus, the finite H T iτ 2 ∆ † H term can be dynamically generated through a fermion loop, in which the m X lepton number violating effect is involved. The associated Feynman diagram is shown in Fig. 1, where the cross symbols denote the mass insertions of the N and X leptons. Thus, the resulting dimension-3 term can be expressed as: where the µ ∆ coefficient is obtained as: For clarity, we show the contours of µ ∆ as a function of y X and y R in Fig. 2(a), where m X = 80 GeV and m N = 400 GeV are used. Clearly, we can easily obtain µ ∆ < 10 −2 GeV without extremely fine-tuning the y R and y X parameters. For comparison, we make a contour plot with m X = 800 GeV and m N = 700 GeV in Fig. 2(b). We will show that the former and latter plots correspond to the cases for which χ 1 and χ 3 are the DM candidates, respectively.
Combining Eqs. (13) and (14), the minimum of the scalar potential can be obtained through ∂V /∂v h = 0 and ∂V /∂v ∆ = 0, and the minimum conditions can be written as: Because we focus on the case of µ ∆ < 10 −2 GeV, i.e., v ∆ 1 GeV, when we neglect the small µ ∆ v ∆ and v 2 ∆ effects, the VEVs of Φ 0 and ∆ 0 can be respectively obtained as v h ≈ 4µ 2 /λ and To obtain v ∆ > 0, we require µ ∆ > 0, which is equivalent to y R > 0. Because of v ∆ 1 GeV, the influence on the electroweak ρ-parameter can be neglected. We note that in addition to µ ∆ and M ∆ , v ∆ also depends on the λ 1,4 parameters. We will discuss the correlation between v ∆ and λ 1,4 when the constraints on the λ 1,4 parameters are studied.
The vacuum stability of scalar potential has been studied in the literature [77][78][79]. Following the results in [79], the conditions for the scalar potential bounded from below in our notations can be written as: and For the sake of satisfying perturbativity, we take λ, |λ i | ≤ 4π before we find the stricter constraints.

A. Scalar mass spectra and scalar couplings
In addition to the SM-like Higgs boson, the type-II seesaw model has two doubly and two singly charged Higgs, and one CP-even and one CP-odd scalar. The scalar mass spectra and the scalar-scalar couplings can be obtained from the scalar potential. Since the doubly charged Higgs does not mix with the other scalars, its mass can be easily obtained as: where the minimal conditions in Eq. (16) have been applied in the second line. The masssquare matrices for (G − , ∆ − ), (G 0 , Im∆ 0 ), and (ReΦ 0 , Re∆ 0 ) can be respectively derived as: It can be easily verified that the determinants of the mass-square matrices in Eqs. (21) and (22) vanish; that is, there exists a massless boson, which corresponds to the Goldstone boson, in each matrix. The detailed eigenvalues of the mass-square matrices and the associated mixing angles are shown in Appendix A.
Because the off-diagonal elements in Eq. (23) are much smaller than v 2 h µ ∆ /( √ 2v ∆ ), the mixing effect between ReΦ 0 and Re∆ 0 can be approximately neglected if we only concentrate on the scalar spectrum. Thus, from the mass-square matrices, the mass squares for the physical bosons, such as the charged scalar H ± , the CP-odd pseudoscalar A 0 , and the two CP-even H 0 and h, can be written as: and m 2 h ≈ λv 2 h /2, respectively, where h is the SM-like Higgs boson. If we ignore the small v ∆ and µ ∆ effects, it can be found that: where the mass splittings in the Higgs triplet components can be constrained by the electroweak oblique parameters [42]. From Eq. (25), we have the mass ordering m H 0 (A 0 ) > m H ± > m H ±± when λ 4 > 0; however, the order is reversed when λ 4 < 0.
In order to study the Higgs precision measurement constraint, we write the Higgs trilinear couplings to the triplet scalars as: The Higgs triplet couplings to the gauge bosons can be obtained from the kinetic terms, written as: where the covariant derivative of the Higgs triplet is given as: The detailed trilinear couplings to gauge bosons can be found in Appendix B.

B. Yukawa couplings and neutrino masses
Using the heavy Majorana flavor mixing matrix in Eq. (8), the scalar Yukawa couplings to the heavy Z 2 -odd fermions can be straightforwardly obtained as: In addition to the SM lepton coupling to the Higgs doublet, the SM left-handed leptons also couple to the Higgs triplet. When we derive the lepton couplings to the Higgs triplet in physical states, we have to simultaneously consider the y and y ∆ terms in Eq. (1). In terms of the components of the Higgs doublet and triplet, the relevant Yukawa couplings of Z 2 -even leptons are written as: where we have neglected the small µ ∆ and v ∆ effects. To diagonalize the charged lepton and Majorana neutrino mass matrices, we introduce the unitary matrices for which the transformations are defined as: Eq. (30) with respect to the lepton physical states can be written as: where the diagonal mass matrices are given as: In order to explain the neutrino data, it is necessary to have v ∆ h ∼ 10 −2 eV. It will be shown that the partial decay widths of the Higgs triplet scalars decaying to leptons are sensitive to v ∆ , which is dictated by the parameters, such as M ∆ , λ 1 , λ 4 , and µ ∆ .

IV. THE CONSTRAINTS
In this section, we discuss the constraints, such as the neutrino mass data, the observed DM relic density, the DM direct detections, the T-parameter, and the Higgs to diphoton precision measurement. It will be found that the χ 1 -DM candidate will be excluded by the upper limits of the DM-nucleon scattering cross sections. Since the cross section upper limit of the SD DM-neutron scattering in Xenon1T [71] is smaller than that of the SD DMproton scattering in PICO-60 [70], we take the Xenon1T data as the upper limit of the SD DM-nucleon scattering cross section and use it to bound the parameters.
A. Constraint from the neutrino data From Eq. (32), the matrix elements of h can be written as: where the sum in k for all active light neutrinos is indicated. It can be seen that the h ij magnitudes strongly depend on the v ∆ value. Using the PMNS matrix parametrized as [80]: where s ij ≡ sin θ ij , c ij ≡ cos θ ij ; δ is the Dirac CP violating phase, and α 21,31 are Majorana CP violating phases, and the experimental data through the neutrino oscillation measurements can be given as [80]: where ∆m 2 ij ≡ m 2 i − m 2 j , and ∆m 2 32 > 0 and ∆m 2 32 < 0 denote the normal ordering (NO) and inverted ordering (IO), respectively. The uncertain sign in m 2 32 originates from the undetermined neutrino mass ordering. Since the neutrino oscillation experiments cannot detect the Majorana CP phases, for simplicity, we take α 31,32 = 0 in the following numerical estimates.
According to the recent results obtained by a global fit analysis, the central values of θ ij , δ, and ∆m 2 ij are given as [81]: where m 1(3) = 0 for NO (IO) is taken. Using these results, the corresponding h ij Yukawa matrix element values are shown in Table II, where the values are in units of 10 −3 eV/(2v ∆ ).
When v ∆ is fixed, the h ij values then can be determined. With v ∆ ∼ 10 −4 GeV, it can be seen that the h ij magnitudes can be in the range of ∼ (0.1, 1) × 10 −7 . Due to the small Yukawa couplings, it can be expected that the lepton-flavor violating effects will be small. In this model, the DM candidate could be an χ 1 or χ 3 Majorana fermion. Regardless of which one is the DM candidate, it is necessary to examine that whether the involved couplings can produce the current correct DM relic abundance (Ω DM h 2 ), which is observed as in [87]: Since the DM relic density is inversely proportional to the product of the DM annihilation cross section and its velocity, i.e. < σv >, in addition to the thermal effects in the early time of the universe, we have to consider the DM annihilation and co-annihilation to the SM particles in the final states. In order to deal with the thermal effects and to calculate the Z 2 -odd particle annihilation processes, we employ micrOmegas [88] with a choice of a unitary gauge. For clarity, we separately discuss the situations of χ 1 -and χ 3 -DM in the following analysis. Although DM couples to the Higgs triplet, since we take the associated y R parameter to be O(10 −2 ), the effects indeed are small. Thus, we neglect the Higgs triplet contributions to the DM relic density.
When the DM candidate is the χ 1 Majorana particle, because its origin is the SU (2) lepton doublet, and it has a large coupling to the SM gauge bosons, we require that the DM mass satisfies m χ 1 > 45 GeV due to the invisible Z decay constraint. To avoid obtaining too large of a DM annihilation rate, the massive gauge boson pair production should be suppressed; that is, χ 1 cannot be too heavy. In order to understand the correlation between Ω DM h 2 and the m N,X and y X parameters, the scanned parameter regions are chosen as: where we require that the resulting Ω DM satisfies 0.09 < Ω DM h 2 < 0.15. We note that, in order to get more sampling points for illustration, the region of Ω DM h 2 is taken slightly wider than the observed Ω DM h 2 . We show the allowed parameter space as a function of m N and m X and as a function of y X and m X in Fig. 3  In addition to the DM relic density, we have to examine whether the same parameter space, which can fit Ω obs DM h 2 , is excluded by the DM direct detection experiments. In the model, it is found that the SI DM scattering off a nucleon is dictated by the Higgs mediation, whereas the SD scattering is through the Z-mediated effects. According to the interactions in Eq. (10) and Eq. (29), the relevant four-Fermi effective interactions for χ 1 and the SM particles can be expressed as: Accordingly, the h-mediated SI DM-nucleon scattering cross section can be written as [73]: where f N ≈ 0.3, and µ χ 1 n = m χ 1 m n /(m χ 1 + m n ) is the DM-nucleon reduced mass. The Z-mediated DM-nucleon scattering cross-section can be expressed as [74] σ SD Z ≈ 3µ 2 where the quark spin fractions of the nucleon are taken as ∆ n u = 0.84, ∆ n d = −0.43, and ∆ n s = −0.08 [88]. Using Eq. (40) and Eq. (41), we show σ SI h and σ SD Z as a function of m χ 1 in Fig. 5(a) and (b), respectively. A comparison with the results in Fig. 4 clearly shows that the allowed parameter regions, which can fit the observed Ω DM h 2 , are excluded by the current Xenon1T SI and SD measurements [69,71]. Thus, it can be concluded that χ 1 cannot be the DM candidate due to the strict constraints from the direct detection experiments.
Next, we discuss χ 3 as the DM candidate. Since χ 3 originates from an SU (2) singlet righthanded lepton, without the y X coupling, it can a heavy Z 2 -odd sterile neutrino and doesn't couple to the SM particles. Therefore, the χ 3 effects are all related to the y X parameter and the main interactions are through the Higgs couplings, i.e. the χ i χ 3 h couplings shown in Eq. (29). Similar to the χ 1 case, to understand the correlation between Ω DM h 2 and the m N,X and y X parameters, we choose the scanned parameter regions to be: and the resulting Ω DM h 2 is required to be in the region of 0.09 < Ω DM h 2 < 0.15. As a result, the correlations between m N and m X and between m N and y X are shown in Fig. 6(a) and (b), respectively. From the plots, it can be seen that when χ 3 is the DM candidate, the DM mass prefers to be heavy, and y X is of the order of 0.1. In addition, according to the result shown in Fig. 6(a), it can be seen that the allowed maximum m N follows an approximate relation with m X as m X − m N ∼ 100 GeV. Based on the results, we show Ω DM h 2 as a function of m χ 3 in Fig. 7, where m X = 800 GeV is fixed, and the solid, dashed, and dotted lines denote the results of y X = 0.06, 0.08, and 0.10, respectively. It can be seen that m χ 3 ∼ (680, 670, 650) GeV with y X ∼ (0.06, 0.08, 0.1) can fit the observed Ω DM h 2 . As mentioned earlier, the maximum of m N is close to 700 GeV when m X = 800 GeV is taken; therefore, the three lines end at m χ 3 ≈ 700 GeV. Due to m χ 3 > m Z,h , we can evade the constraints from the invisible Z and h decays.
Similar to the χ 1 case, χ 3 can contribute to the SI and SD DM-nucleon scatterings through the h and Z mediation, respectively. To estimate the elastic scattering cross sections, we can use the formulas in Eqs. (40) and (41) by replacing c h,Z 11 and µ χ 1 n with c h,Z 33 and µ χ 3 n = m χ 3 m n /(m χ 3 + m n ). Accordingly, we show the SI and SD χ 3 -nucleon scattering cross sections as a function of m χ 3 in Fig. 8(a) and (b), where m X = 800 GeV is used, and the solid, dashed, and dotted lines denote the results of y X = 0.06, 0.08, and 0.1, respectively. A comparison with the results shown in Fig. 7 reveals clearly that σ SI h and σ SD Z at the m χ 3 value, which is determined by Ω obs DM h 2 , are all under the Xenon1T upper limits [69,71]. That  [76], the T -parameter, which arises from the Higgs triplet, can be formulated as [76]: Basically, the mass splitting in the vector-like lepton doublet can also contribute to the T-parameter, where the mass difference is dictated by e N . Using y X = 0.1, m X = 800 GeV, and m χ 3 = 700 GeV, we obtain e N ≈ 3.2 GeV, where the resulting T can be estimated to be T ≈ 0.8 × 10 −3 [38]. Since the influence on T -parameter is not significant, we drop the vector-like lepton doublet contribution in this study.
where Γ SM γγ ≈ 6.50 − i0.02; Q H ±± = 2 and Q H ± = 1; A 0 (τ ) = τ (1 − τ f (τ )), and the loop function is defined as: Thus, we can write the signal strength for pp → h → γγ as: For numerical estimates, we take the Higgs width in the SM as Γ SM ≈ 4.07 MeV [83]. From Eqs. (15) and (17), it is known that in addition to the m X,N and y X,R parameters, v ∆ also depends on the λ 1,4 constraints. Since the DM candidate in this model is χ 3 , and its mass is determined to be m χ 3 ∼ 680 GeV when m X ∼ 800 GeV is used, in order to simplify the study on the λ 1,4 constraints, we fix m N (X) = 700 (800)

V. PHENOMENOLOGICAL ANALYSIS
After analyzing the potential constraints, in this section, we study the relevant phenomenology in detail, such as the h → Zγ and H ±± , H ± , and H 0 (A 0 ) decays. From the earlier analysis, since m X is taken to be 800 GeV, the processes, in which the Higgs triplet decays to the vector-like leptons, are kinematically suppressed when we focus on the study with m ∆ < 1 TeV; therefore, we only consider the SM particles in the final states, where the three-body decays are also included when the kinematic condition is allowed. When the final states are all leptons, for simplicity, we sum up all possible lepton flavors. In addition, since the neutrino constraints from the NO and IO are similar in most lepton Yukawa couplings, hereafter, we only use the NO constraint as the inputs.

A. Signal strength for h → Zγ
We have shown that the Higgs to diphoton measurement can bound the Higgs couplings to H ±± and H ± , which is dominated by the λ 1 parameter. Since the same couplings can also contribute to the loop-induced h → Zγ, with the constrained parameters, we can predict the h → Zγ in the model. Thus, similar to the case in h → γγ, the signal strength of h → Zγ can be expressed as: where the h production cross section is dominated by the SM effects in the model, and the current upper limit is µ Zγ < 6.6 [80].
Based on the results in [57,82,[89][90][91][92], we write the partial decay rate for h → Zγ as: where the SM and Higgs triplet contributions can be expressed as [82,92]: Here, N C = 3 is the color number; τ i h(Z) = 4m 2 i /m 2 h(Z) , Q f is the electric charge of f fermion; I f 3 is the third component of weak isospin of f fermion, and the charged Higgs couplings to h and Z bosons are given as: The detailed loop functions A h 0,1/2,1 can be found in Appendix C. Accordingly, we show the µ Zγ contours as a function of λ 1 and λ 4 in Fig. 10(a) and (b) for M ∆ = 400 GeV and M ∆ = 800 GeV, respectively, where the T -parameter and µ γγ constraints shown in Fig. 9 are included. From the plots, it can be seen that the influence from the Higgs-triplet charged particles is ∆µ Zγ = |µ SM Zγ − µ Zγ | 4% and is not significant. According to the introduced gauge and Yukawa couplings, the two-body H ±± partial decay rates can be expressed as: where y W = m 2 W /m 2 H ±± , S ii = 1/2, and S ij = 1 for i = j. For λ 4 < 0, m H ±± is the heaviest Higgs triplet; then, the three-body partial decay rate for H ++ → H + W + * can be expressed as: with y H ± = m 2 H ± /m 2 H ±± , s min = 0, and s max = (1 − √ y H ± ) 2 . The phase space integral can be simplified as: with λ(a, b) = 1 + a 2 + b 2 − 2a − 2b − 2ab. If we assume that the main H ++ decay modes are W + W + , + i + j , and H + W + * , the relative BRs as a function of λ 4 can be shown in Fig. 11

C. Singly charged Higgs decays
In addition to the H + direct couplings to the SM particles, the singly charged Higgs can also decay through mixing with the SM charged-Goldstone boson (G + ), where the relation between the mixing angle φ + and the v ∆ parameter is shown in Appendix A. Thus, if the direct H + couplings to the SM particles are proportional to v ∆ , the mixing effects with G + become important. We find that with the exception of + ν mode, the decay channels, such as tb, hW + , ZW + , and γW + , are all related to the mixing angle φ + . Hence, the partial decay rates for the fermionic H + decays can be expressed as: with s φ + (c φ + ) = sin φ + (cos φ + ) and y t = m 2 t /m 2 H ± . Since the G + coupling to a quark is proportional to the quark mass [82], we only consider the tb mode and the m b effect is It is found that in addition to the G + hW − coupling, H + can decay to the hW + final state through the mixing between ReΦ and Re∆, where the mixing effect is dictated by the mixing angle α shown in Eq. (66). Using the gauge couplings in Eq. (67) and the φ + and α mixing effects, the partial decay rates for the H + diboson decays can then be formulated as: It is known that the λ 4 parameter determines the order of the Higgs triplet masses. Therefore, it is expected that H + can decay to H ++ and H 0 (A 0 ) through the three-body decay when λ 4 > 0 and λ 4 < 0, respectively. Similar to the H ++ → H + W + * decay, we write the partial decay rates for H + → (H ++ W − * , H 0 (A 0 )W + * ) as: with S = H 0 (A 0 ).
Based on the partial decay rate formulations, we show the BR for each decay mode as a function of λ 4 in Fig. 12 Table III. In addition, in order to understand the scalar mixing influence on the BRs, we show the BRs with φ + = α = 0 in Fig. 13, where M ∆ = 400 GeV and λ 1 = 2.5 are used. It can be seen that without the φ + and α mixing effects, the contributions to the tb and hW + modes vanish, and the BR order follows BR(H + → + ν) > BR(H + → ZW + ) > BR(H + → γW + ).

FIG. 12:
This legend is the same as that shown in Fig. 11 with the exception of the H + decays.  Higgs, it can be found that H 0 can further decay to tt and that A 0 can decay to tt and hZ.
Therefore, according to the introduced Yukawa and gauge couplings, the partial decay rates of the fermionic H 0 /A 0 decays can be expressed as: whereas the H 0 /A 0 diboson decays are given as: with z i = m 2 i /m 2 S . When H 0 (A 0 ) is the heaviest scalar, i.e. λ 4 > 0, similar to the cases in the H + and H ++ decays, the three-body decays H 0 (A 0 ) → H + W − * , H − W + * are open and the partial decay rates are written as: Using the obtained partial decay rates, we show the BR for each decay channel as a function of λ 4 in Fig. 14 which is close to BR(H 0 → ZZ). Accordingly, we see that the BR of H 0 → W + W − obtains a destructive contribution from the α mixing effect. When φ 0 = α = 0, A 0 only can decay to νν in the region of λ 4 < 0; therefore, we do not explicitly show the situation for the A 0 decay.  GeV, and λ 1 = 2.5 are used.

VI. CONCLUSION
Using the scotogenic approach, we studied the radiatively induced lepton-number violation dimension-3 term µ ∆ H T iτ 2 ∆ † H in the base of the type-II seesaw model, where the introduced dark vector-like doublet lepton X and dark right-handed singlet Majorana lepton N are the mediators in the loop. It was found that the dynamically induced Higgs triplet VEV is limited in the region of 10 −5 − 10 −4 GeV when the relevant parameters satisfy the constraints from the DM measurements. Due to the DM direct detection constraints, only the singlet Majorana lepton can be the DM candidate in the model, and the DM mass depends on and is close to the m X parameter.
In the model, the Higgs triplet VEV, v ∆ , depends not only on the µ ∆ and M ∆ parameters, but also on the λ 1,4 parameters in the scalar potential, which dictate the SM Higgs couplings to the doubly and singly charged Higgses. Moreover, the mass ordering of the Higgs triplet scalars is dictated by the λ 4 sign. We showed that the Higgs diphoton decay and the oblique T -parameter can further bound the λ 1,4 parameters. As a result, we obtain |m H ±± −m H ± | 50 GeV.
We did not explicitly study the collider signatures in this work. Rather, we analyzed the decay channels of each Higgs triplet scalar and estimated the associated branching ratios in detail. We found that the scalar mixing effects have an important influence on the partial decay rates of the singly charged-Higgs, CP-even scalar, and CP-odd pseudoscalar in the near degenerate masses (i.e. λ 4 1). In the non-degenerate mass region, the branching ratios of the Higgs triplet scalar decays are dominated by the three-body decays when they are kinematically allowed.

Appendix A SCALAR MASS SQUARES AND MIXING ANGLES
The symmetric mass-square matrices in Eqs. (21), (22), and (23) where the 2×2 symmetric matrix can be diagonalized using an orthogonal matrix U through A dia = U AU T with the parametrization: It can be found that the two eigenvalues A L and A H and the mixing angle φ can be expressed as: A L(H) = a 11 + a 22 2 ∓ 1 2 (a 11 − a 22 ) 2 + 4a 2 12 , Since the (G + , ∆ + ) and (G 0 , Im∆ 0 ) states have massless Goldstone bosons, their physical mass squares can be straightforwardly obtained by taking traces of the mass-square matrices, i.e. m 2 H + = TrA G + ∆ + and m 2 A 0 = TrA G 0 Im∆ 0 . From Eq. (63), the corresponding mixing angles for diagonalizing A G + ∆ + and A G 0 Im∆ 0 shown in Eqs. (21) and (22) are given as: Clearly, if v ∆ v h , the mixing angles are small. In the case of the (ReΦ 0 , Re∆ 0 ) states, we do not have a simple way to obtain their eigenvalues. If we use h and H 0 to denote the light and heavy scalars, their eigenvalues m h(H 0 ) and mixing angles should follow Eq. (63), where the associated matrix elements are: