Muon $g-2$ in a two-Higgs-doublet model with a type-II seesaw mechanism

We study the two-Higgs-doublet model with type-II seesaw mechanism. In view of constraints from the Higgs data, we consider the aligned two-Higgs-doublet scheme and its effects on muon anomalous magnetic dipole moment, $a_{\mu}$, including both one-loop and two-loop Barr-Zee type diagrams. Thanks to a sizable trilinear scalar coupling, the Barr-Zee type diagrams mediated by the Higgs triplet fields have a dominant effect on $a_{\mu}$. In particular, unlike the usual two-Higgs-doublet models that require exotic Higgs bosons light in mass, the masses of the corresponding particles in the model are of ${\cal O}(100)$~GeV. The doubly-charged Higgs boson presents a different decay pattern from the usual Higgs triplet model and thus calls for a new collider search strategy, such as multi-$\tau$ searches at the LHC.


I. INTRODUCTION
A long-standing anomaly in particle physics is the muon anomalous magnetic dipole moment (dubbed the muon g − 2 anomaly) denoted by a µ ≡ (g − 2) µ /2, where the data and Standard Model (SM) show an over 3σ disagreement. The E821 experiment at Brookhaven National Lab (BNL) has presented a precision measurement of: with an uncertainty of 0.54 ppm [1]. The current theoretical estimate of a µ within the SM has also reached a comparable precision of 0.369 ppm, and is shown to be [2]: The deviation between the experiment and the SM is ∆a µ = a exp µ − a SM µ = 279(76) × 10 −11 with an achievement of 3.7σ. The new muon g − 2 measurement performed in the E989 Run 1 experiment at Fermilab, designed to have a precision of 0.14 ppm, reports its first measurement as [3]: Combining all available measurements on the quantity, we now have a 4.2σ deviation between experiment and SM expectation 1 , accentuating the muon g − 2 anomaly.
On the other hand, since the discovery of Higgs boson at the LHC in 2012 summer, measurements of the Higgs signal strengths, commonly used as a measure of deviations from the SM, have been improving over the years. They are found to be quite consistent with the SM expectations and, hence, models with extensions in the scalar sector are severely constrained. One possibility for a new physics (NP) model to achieve such a good agreement with the SM in the Higgs couplings while having exotic Higgs bosons of mass at O(100) GeV scale is when the model shows the so-called alignment limit [59][60][61].
In this work, we study the contributions of a model with an extended scalar sector to the muon g − 2 when the relevant theoretical and experimental constraints are taken into account. One purpose is to revisit the two-Higgs-doublet models (2HDMs), where the earlier studies can be found in Refs. . It is known that to explain the muon g − 2 in this 1 The latest lattice QCD calculation for the leading hadronic vacuum polarization from the BMW collaboration is obtained as a LO−HVP µ = 707.5(5.5) × 10 −10 , which leads to a larger a µ , can be found in [4].
framework, the new scalar or pseudoscalar boson are required to be as light as O(10) GeV.
Although such a parameter space is still allowed by the current data, it is of interest to probe the scenarios where the new scalar masses can be more relaxed and of ∼ O(100) GeV by further extending the scalar sector. More importantly, such a new extension should also address some other unsolved issues, such as the origin of neutrino mass, that the simple 2HDMs cannot accommodate.
To achieve the above-mentioned goals, we consider the 2HDM with type-II seesaw mechanism [26,27]. In addition to the SM Higgs doublet, the scalar sector contains another complex doublet and a complex triplet. Moreover, we will consider the so-called aligned two-Higgs-doublet scheme (A2HDS), where the Yukawa couplings of the two Higgs doublets to the SM fermions are proportional to each other and one of the neutral physical Higgs boson is SM-like. The A2HDS has the interesting feature that it reduces to various 2HDM types by taking proper limits on the alignment parameters. With a small vacuum expectation value (VEV) induced by electroweak symmetry breaking from the two Higgs doublets, the Higgs triplet in the model provides Majorana mass to neutrinos through the so-called type-II seesaw mechanism [33][34][35][36][37][38].
It is found that rather than a simple combination of the 2HDM and the type-II seesaw model (also called the Higgs triplet model or HTM), the model presents several interesting features: 1. The coupling between the heavier neutral Higgs boson in the 2HDM and the doublycharged Higgs boson in the HTM can significantly enhance the muon g − 2 through two-loop Barr-Zee type diagrams [39,40], even when the heavier neutral Higgs mass is ∼ O(100) GeV.
2. The Higgs triplet VEV is now determined by three lepton number-violating parameters instead of just one in the simple HTM. As a result of the extra freedom, these parameters are not necessarily of the same order as the Higgs triplet VEV [26].
3. With a sizable Higgs triplet VEV, the doubly-charged Higgs boson shows a richer decay pattern. As a result, the doubly-charged Higgs boson can evade the recent ATLAS lower bound of 350 GeV in pair production [41]. In addition to the like-sign diboson channel, the doubly-charged Higgs boson can also be probed via channels involving the light charged Higgs boson.
The paper is organized as follows. In Sec. II, we derive the Yukawa couplings in the A2HDS and show the relations between the scheme and the various types of 2HDMs with Z 2 symmetry. The mass-square relations of the triplet Higgs bosons are discussed, and the CPeven neutral Higgs couplings with the charged Higgses are given. In Sec. III, we discuss the results of one-loop and the dominant two-loop Barr-Zee type diagrams. Using the bounded parameters, we present the detailed numerical analysis and discussion in Sec. IV. Sec. V summarizes our findings in this work. The full scalar mass matrices and their approximations in the limit of neglecting v ∆ are given in appendix A.

II. MODEL AND INTERACTIONS
We consider a model where the scalar sector is extended with a doublet with Y = 1/2 and a complex triplet with Y = 1. In the following, we discuss the general Yukawa interactions and scalar potential in this model.

A. Scalar potential and the trilinear scalar couplings
Since the scalar sector is an extension of 2HDM or of type-II seesaw, in the following, we briefly discuss the essential parts for our analysis. First, as we will assume negligibly small mixing between the doublet fields and the triplet field, it is useful to go to the Higgs basis in the usual 2HDM, defined by: where 246 GeV. Written in terms of field components, the Higgs doublets H 1,2 and triplet ∆ are: will respectively mix. The only nonmixing state is the doubly-charged Higgs, where from Eq. (9), its mass can be expressed as: It can be seen that the new doublet-triplet couplings terms shift the δ ±± mass.
The detailed discussions for the scalar, pseudoscalar, and charged scalar mass matrix are given in appendix A. We summarized the characteristics as follows: from Eqs. (A2) and (A7), it can be seen that m 2 G 0 and m 2 GeV, their values can be dropped. If µ 3 = tan 2β(µ 1 − µ 2 ) is required, we can find that the In comparison with the mass-square elements of other massive particles, their mixing effects are small. Although the small mixing effects can have important influence on some processes, e.g. δ ±± → W ± H ± can be induced, their influence on the muon g − 2 can be indeed neglected. Hence, when we numerically estimate the muon g − 2, we take h(H), H ± (δ ± ) and A 0 as the physical states; however, for other processes, one can take the mixing effects into account if necessary.
The new doublet-triplet couplings can cause the triplet scalar mass splittings, and the mass differences can be found as: It can be seen that the mass split can be or be less than O(100) GeV .
The trilinear interactions among a neutral Higgs boson and two charged Higgs bosons are given by: where the couplings are written as: In the alignment limit of c β−α = 0, the h and H trilinear terms can be easily obtained by the replacement of H 0 1 → h and H 0 2 → −H. Therefore, the corresponding trilinear couplings have the relations: We note that the pseudoscalar A 0 does not couple to the charged scalars in the CP-conserving case.

B. Yukawa interactions
The most general Yukawa couplings in the model are given by: where the flavor indices are suppressed, y ν is a symmetric matrix, Q L (L) denotes the quark (lepton) doublets, q R ( R ) denotes the quark (lepton) singlets, Y f 1,2 with f = u, d, are respectively the Yukawa matrices for the up-type quarks, down-type quarks, and charged leptons, C is the charge conjugation operator, andΦ i ≡ iτ 2 Φ * i with τ 2 being the Pauli matrix. Since Φ 1 and Φ 2 simultaneously couple to each type of fermions, flavor-changing neutral currents (FCNCs) naturally arise at tree level. The FCNC effects are usually suppressed by introducing, for example, a Z 2 discrete symmetry [43]. In this case, the 2HDM can be categorized into Type-I [6,44], Type-II [44,45], Type-X, and Type-Y [46][47][48][49]. See Ref. [50] for a detailed review. In addition to the above-mentioned schemes in 2HDM, the tree-level FCNCs can also be avoided by imposing a certain relation between Y f 1 and Y f 2 , where f = u, d, and . The A2HDS assumes the relation Y f 2 = ξ f Y f 1 , where ξ f is a proportionality constant [28]. Alternatively, one may also impose the condition [51][52][53], where the possible N I matrices can be found in Ref. [51]. In this work, we are considering the A2HDS.
With the assumed VEVs of Φ i , Y f 1 and Y f 2 in Eq. (18) can be linearly combined to form two matrices: so that X f (Z f ) is associated with the doublet H 1 (2) , and the fermion mass matrix can be Moreover, M f can be diagonalized by the unitary matrices 1 and Y f 2 are two linearly independent matrices and cannot be diagonalized simultaneously, then tree-level FCNCs can arise due to the Z f couplings because its off-diagonal elements cannot be removed when X f is diagonalized.
When the A2HDS relation Y f 2 = ξ f Y f 1 is taken, the Yukawa matrix can be related to the mass matrix as: As a result, both Y f 1 and Y f 2 now can be diagonalized simultaneously, and the H 0 1,2 FCNCs are suppressed at the tree level.
For simplicity, we only concentrate on the CP-conserving case and assume ξ f to be real, though they can generally be complex. Using Eqs. (4) and (6) and the notations used in [28], the mass terms and Yukawa interactions with h, H, A 0 , and H ± are found to be: where s d, = +1, s u = −1; P R(L) = (1 ± γ 5 )/2 are the chirality projection operators, [56,57], and with t β = tan β. In general, ζ f can be complex numbers as ξ f [28], and their magnitudes can be large without requiring a large tan β. In this study, we only focus on the CP-conserving case. The A 0 and H ± Yukawa couplings do not depend on c β−α (s β−α ). For comparison, we show in Table I the vanishing and non-vanishing Y f 1,2 for various 2HDM types and the associated ζ f . In particular, ζ f in Type-I, -II, -X, and -Y can be obtained from the A2HDS by taking an appropriate limit of ξ f and thus ζ f : Since the SM-like Higgs couplings generally depend on c β−α (s β−α ) and ζ f , the current Higgs production and decay measurements put stringent constraints on the value of c β−α .
Here we simply take the alignment limit with As a result, the H and A 0 couplings to the SM fermions have the same magnitude and are dictated by ζ f . In this work, we demonstrate how a large ζ can affect the muon g − 2 when m H > m h and the Hδ ++ δ −− and Hδ + δ − couplings are present.
Using the component fields of the Higgs triplet shown in Eq. (5), the neutrino mass and lepton Yukawa interactions with the triplet fields are given by: where f C = Cγ 0 f * and M ν = y ν v ∆ / √ 2 is the neutrino mass matrix. In order to fit the neutrino data, the values of (M ν ) ij has to be of O(10 −3 − 10 −2 ) eV [26,62,63]. In the type-II seesaw model with the assumed triplet VEV v ∆ ∼ O(10 −3 − 10 −4 ) GeV, the Yukawa couplings y ν ij are very small, O(10 −7 ). Therefore, δ ± and δ ±± of O(10 2 ) GeV mass have negligible effects on most lepton processes.
As we will numerically show below, the Yukawa couplings as well as the trilinear scalar couplings Hδ −−(−) δ ++(+) and HH − H + , arising from the scalar potential, play important roles in producing a sizable correction to the muon g − 2.
III. ONE-AND TWO-LOOP MUON g − 2 The electromagnetic interaction of a lepton can be written as: The lepton anomalous magnetic dipole moment is then defined by Since the magnetic moment is associated with dipole operator, the lepton g − 2 originates from radiative quantum corrections. In the model, the one-loop corrections from new physics are induced by the mediation of H, A 0 , and H ± , where the associated Feynman diagrams are shown in Fig. 1(a) and (b). Moreover, it is known that the two-loop Barr-Zee type diagrams can have important contributions to the magnetic dipole moment due to a large coupling enhancement [39,40]. The potentially large two-loop diagrams mediated by heavy fermions, including top, bottom, and τ , are shown in Fig. 1(c). The essential mechanism contributing to the muon g − 2 in the model is the two-loop with Barr-Zee type diagram mediated by the charged scalars, including δ ++ , δ + , and H + , as shown in Fig. 1(d). In addition to the lepton Yukawa coupling, such diagrams further enjoy the enhancement of the electric charges associated with the charged scalars.
The one-loop corrections to the anomalous magnetic dipole moment in the 2HDM have been studied long time ago [5][6][7][8]. Using the Yukawa couplings shown in Eq. (21), the muon g − 2 from Fig. 1(a) and (b) can be expressed as: where From the expressions, it can be seen that the induced muon g − 2 is proportional to m 4 µ . One of the four factors of m µ comes from the definition in Eq. (25), another comes from the mass insertion for chirality flip, and the rest two enter through the two Yukawa interaction vertices, each of which is proportional to the muon mass. Thus, if the intermediate scalar mass is of O(100) GeV, the resulting muon g −2 is far below 10 −9 . To get ∆a µ up to ∼ 10 −9 , the mediating particle has to be as light as tens of GeV. This was the observation previously found for the 2HDM in the literature.
Following the results shown in Ref. [16], the two-loop Barr-Zee type diagrams with fermion and charged scalars can be written as: where N f C is the number of color for the fermion f , Q P (P = f, S) is the electric charge of the particle, and the loop functions are given by: , The two-loop results are proportional to m 2 µ because there is only one muon Yukawa coupling involved. It can be seen that when y H/A u,d are strictly bounded by the experimental data, their contributions become subleading, and ∆a 2,S µ is the dominant effect.

IV. NUMERICAL ANALYSIS
In this section, we present how we choose the parameters in our model, how they affect ∆a µ at one-loop and partial two-loop levels, and how the doubly-charged Higgs phenomenology at the LHC is modified.

A. Parameter choice
Among the parameters in the Yukawa and scalar sectors, most relevant ones for the muon g − 2 are combinations of the Yukawa matrix elements, the quartic scalar couplings, and t β that appear in various couplings. More explicitly, the relevant parameters are: c β−α (s β−α ), , and λ HH − H + . We will show how they contribute the muon g − 2.
Before a numerical analysis, we first need to find the allowed parameter space for the model. All potential constraints from experimental measurements, including various flavor physics processes, Higgs data, and electroweak precision observables, and theoretical bounds, such as perturbative unitarity and positivity of the scalar potential, have to be taken into account. Recently, such a global fit, considering the theoretical constraints has been done in the A2HDS [64]. In this work, we will follow the global fit results in Ref. [64] when the parameter values are taken for the numerical estimations.
Two scenarios, the light scenario and the heavy scenario, are discussed by in Ref. [64], where the former refers to the case with m H > m h and the latter has m H < m h . Since we are interested in the heavy scalar boson contribution to ∆a µ , we will concentrate on the light scenario.
The values of parameters used in our numerical analysis are described below. Using the experimental data at 2σ errors, the global fit gives |c β−α | < 0.04. Thus, we will take the alignment limit of c β−α = 0. Under this limit, the HW − W + and HZZ couplings vanish identically. It is found that ζ u,d have to be of the same sign and their values are restricted to small-value regions when |ζ | approaches the boundary of maximum, i.e., |ζ | = 100.
Moreover, the sign of ζ cannot be determined by the global fit, and it always appears in the product along with other parameters in ∆a µ , e.g., ζ λ Hδ −− δ ++ . We can thus fix the value of ζ and let the associated parameter vary. In numerical calculations, we take: Since the maximally allowed value of |ζ | in the negative region is larger than that in positive region, we assume ζ to be negative. The signs of ζ u,d are taken to fit the positive ∆a µ .
From Eq. (16), it is seen that the trilinear couplings have involved relations with the parameters in the scalar potential. If we assume that the 2HDM with a Z 2 symmetry contributes little to the HH − H + coupling, the λ 6,7 parameters in A2HDS become the dominant source, i.e., Since the constrained λ 6,7 values allow λ 6 −3.5 and λ 7 −2.5, we will take λ HH − H + ≈ 1.5 to estimate the muon g − 2.
Global fits cannot determine the masses of the involved new scalar particles. Nevertheless, their mass differences are strongly correlated and constrained. In our numerical analysis, we take m δ ± = m δ ±± + 100 GeV according to Eq. (14). When discussing the CP-even or CP-odd scalar effects, we take m H(A 0 ) as a free parameter. When combining the effects of H and A 0 together, we take m H = m H ± and m A 0 = m H ± 50 GeV.
The general transition matrix element for τ → ν ν τ can be written as [71]: where κ = S, V, T denotes the type of interaction, (λ) = R, L is the lepton chirality, and the chirality of m(n) can be determined when κ and (λ) are fixed. In the SM, due to the V − A interaction, we only have g V LL = −1. Since the involved couplings in the A2HDM are scalar and vector types and the H ± -Yukawa coupling is proportional to the lepton mass, we only need to consider the muon mode and the effective couplings g S RR and g V LL . Thus, the Michel parameters of η µ and ξ µ are expressed as: where g S RR and g V LL with one-loop corrections [67] in the model are given by: with the loop function F defined by To illustrate the constraints from the measured Michel parameters, we show the contours of

B.
Muon g − 2 In the following, we divide our discussion of muon g − 2 into three contributing parts.  In the A 0 -mediated part given in Fig. 4(b), similar to the case mediated by H, the top-quark and bottom-quark effects are small. However, the bottom-quark and τ -lepton contributions interchange sign, and the latter becomes the dominant effect. The sign difference arises from the loop functions J H τ (z) and J A 0 τ (z) shown in Eq. (30). For the region of m A 0 > m h , it can be seen from Fig. 4(b) that its two-loop effect on ∆a µ is smaller than 10 −9 .
Nevertheless, when A 0 is lighter than the SM-like Higgs, its contribution to ∆a µ increases significantly. When the negative one-loop contribution is included, one observes that the A 0 contribution to ∆a µ can reach 2 × 10 −9 when m A 0 ≈ 55 GeV. Following the global analysis in the A2HDS [64], when m H > m h , such a light pseudoscalar boson is not excluded by the current experimental data. Hence, the conclusion is consistent with that obtained in the 2HDM type-X [15,17].

Barr-Zee contribution from the triplet extension
It has been shown that when the CP-even and CP-odd scalar masses are heavier than the observed Higgs mass, the H-mediated and A 0 -mediated effects in the 2HDM become ineffective to accommodate the measured ∆a µ . In the following analysis, we discuss the new contributions from the doubly-and singly-charged Higgs bosons derived from the Higgs triplet extension. In the analysis, we focus on the m H > m h scenario, following the parameter constraints given in Ref. [64].
These results demonstrate that the measured muon g − 2 can be readily achieved even when the exotic Higgs bosons in our model have mass of a few ×100 GeV. Besides the enhancement from the two units of electric charge, it is interesting to note the other enhancement factor associated with the doubly-charged Higgs boson by comparing the result with that induced from the τ -loop. Because the H and A 0 couplings to muon are the same in the alignment limit, if we further set m H = m A 0 = m X the only different factors come from the couplings, A 0 τ τ and HH − H + , and from the loop integrals, J A 0 f (z) and J S (z) defined in Eq. (30). For simplicity, we use z χ J χ to represent the effect for the τ -loop and the δ ++ -loop, where z χ and J χ are the associated coupling factor and integral function, respectively. We show z χ J χ as a function m X in Fig. 6, where ζ = −100, m δ ++ = 350 GeV, and λ Hδ −− δ ++ = 3 are applied. It can be seen that once m X > 160 GeV, the δ ++ -loop contribution is larger than the τ -loop.  2. When the H-and A 0 -mediated two-loop diagrams are combined, A 0 with a mass of 70 GeV can lead to ∆a µ ∼ 2 × 10 −9 , similar to the situation of 2HDM type-X [15]. However, when m A 0 > m h , the 2HDM contribution is below 10 −9 .
3. The H-mediated δ ++(+) -loop contribution is independent of m A 0 , and it can play an important role on ∆a µ in a wide range of m H and m δ ±± , particularly when the exotic Higgs masses are a few hundred GeV.
Before closing this subsection, we make a brief remark about the implication of the model on the anomalous magnetic dipole moment of the electron (electron g − 2). Applying the accurate measurements of the fine structure constant from 133 Cs and 87 Rb to the theoretical calculations [72,73], the differences in the electron g − 2 between the experiments and the SM expectation are found to be: ∆a e ( 133 Cs) = −(8.8 ± 3.6) × 10 −13 [74] , ∆a e ( 87 Rb) = (4.8 ± 3.0) × 10 −13 [75] , i.e., having −2.4σ and 1.6σ deviations, respectively. Their weighted average is ∆a e = where g is the SU (2) L gauge coupling. It can be seen that the δ ++ H − H − and δ ++ W − W − couplings are dictated by the factor of v ∆ . That is, for the doubly-charged-Higgs decays, the mixing of O(v ∆ ) among G ± , H ± , and δ ± has to be taken into account though its effects are small in the muon g −2. If we take µ 3 = t 2β (µ 1 −µ 2 ) and λ 8 c 2 β +λ 9 s 2 β +λ 12 s 2β ∼ −2(2m ∆ /v) 2 in Eq. (A7), it is found that m 2 With such parameters and dropping O(v 2 ∆ ) terms, the relevant charged scalar mass-square matrix can be simplified as a 2 × 2 matrix: where the matrix can be diagonalized by an SO(2) transformation, similar to the expression given in Eq. (A8) but using the θ ± mixing angle instead of α. Due to the fact that v ∆ v, we can ignore the influence of m 2 H − δ + on m 2 H ± and m 2 δ ± , and the physical states and the small mixing angle can be simply expressed as: Taking λ 8 − λ 9 = 1, λ 12 = 1, m δ ± = 350 GeV, m H ± = 180 GeV, v ∆ = 5 × 10 −4 and c β = s β = 1/ √ 2, the mixing angle value is estimated to be θ ± 4.8 × 10 −7 . Since H ±

1(2)
only carries a tiny component of δ ± (H ± ), in the following analysis we keep using H ± (δ ± ) instead of H ± 1 (H ± 2 ). Because of the introduction of θ ± , in addition to the W ± W ± and H ± H ± modes, the doubly-charged Higgs can also decay into H ± W ± . In the case an on-shell decay to H ± H ± is impossible, we consider the decay channel H ± H ± * , i.e., one of H ± is off-shell, with the assumption that the H ± → τ ± ν is dominant. The partial decay widths of these dominant modes are explicitly expressed as: where For the purpose of comparison, we include the δ ++ → + i + j channels. According to the δ ++ Yukawa couplings to − i − j shown in Eq. (24), the decay rate is given by: where δ ij is the Kronecker delta. We use the (M ν ) ij values determined by a global fit to the neutrino data and given in [76].
For illustration purposes, we show in Fig. 9 their branching ratios as functions of µ 1 with the parameter choice of t β = 1, |ζ | = 100, ∆ = v ∆ = 5 × 10 −4 GeV, m δ ±± = 300 (400) GeV The mixing angle θ ± could induce a sizable effect if λ 8 − λ 9 = 2 is used. Note also that λ 12 does not contribute to the coupling λ δ ++H − H − when t β = 1 is taken. In Fig. 10, we also show the branching ratios of δ ++ as functions of m δ ++ with µ 1 = 0.01 GeV, where the other parameter values are the same as those shown in Fig. 9. We find that the H + H +( * ) mode becomes dominant when m δ ±± > 2m H + and that the branching ratios of W + W + and H + W + slightly increase with  Finally we discuss signals from doubly-charged Higgs boson production at the LHC. The δ ++ δ −− pair can be produced via electroweak interactions in proton-proton collision process.
In our scenario, the produced δ ±± dominantly decay into W ± W ± and/or H ± H ±( * ) mode, depending on the value of µ 1 as discussed above. Here we focus on the H ± H ±( * ) mode as it is a special channel in our model. The scenario when the W ± W ± mode is more dominant is equivalent to the Higgs triplet model with v ∆ > O(10 −4 ) GeV. The singly-charged Higgs H ± dominantly decays into τ ν mode since the Yukawa interaction between H ± and leptons is enhanced by the large ζ factor that is required for a sizable muon g − 2 contribution. Therefore, the signature from δ ++ δ −− production is 4τ + / E T in our scenario. Note that we can relax the bound on m δ ±± of ∼ 350 GeV [41] because the analysis assumes that the W ± W ± mode is dominant and considers only the muons/electrons in the final state.
It would be difficult to reconstruct the doubly-charged Higgs mass because of the missing transverse energy carried away by neutrinos from H ± decays. Our signature could be tested in multi-tau searches in future LHC experiments.

V. SUMMARY
In this work, we have studied an extension of the Standard Model only in the scalar sector, with the addition of one Higgs doublet and one complex Higgs triplet, rending a two-Higgs-doublet model (2HDM) with the type-II seesaw mechanism. For the 2HDM part, we consider the aligned two-Higgs-doublet scheme (A2HDS) to avoid undesired flavor-changing neutral currents induced by the two Higgs doublet fields and to satisfy the current Higgs data constraints. The Higgs triplet field obtains a small vacuum expectation value (VEV) induced by the electroweak symmetry breaking and gives Majorana mass to neutrinos through Yukawa couplings.
We have examined how the model can accommodate the measured muon g − 2 deviation.
Simple 2HDMs usually require CP-even and -odd Higgs bosons (H and A 0 ) to be sufficiently light (about a few ×O(10) GeV) and rely on the contributions of Barr-Zee type diagrams to account for the muon g − 2 anomaly, ∆a µ . In our model, the Barr-Zee type diagrams get additional contributions from a large Hδ ++(+) δ −−(−) coupling, an enhanced coupling between charged leptons and the charged Higgs boson (H ± ) in 2HDM, and the electric charges of the charged Higgs bosons (δ ± and δ ±± ) from the Higgs triplet field, independent of the mass of CP-odd Higgs boson. In fact, the mass of the exotic Higgs bosons is allowed to have a wider range, up to a few hundred GeV.
Owing to the new interactions with the other charged Higgs and W bosons, the doublycharged Higgs boson presents a different decay pattern than the usual Higgs triplet model.
With the assumed Higgs triplet VEV, v ∆ ∼ 5 × 10 −4 GeV, the doubly-charged Higgs boson may dominantly decay into like-sign charged Higgs bosons in the 2HDM rather than likesign W bosons, when the magnitudes of the trilinear couplings µ 1,2,3 are greater than 10 −3 (10 −2 ) GeV for both (one of the) charged Higgs bosons being on-shell. Therefore, pair productions of the doubly-charged Higgs bosons will lead to the signature of 4 τ -leptons and missing energy at the LHC.
From Eqs. (B2)-(B4), the doubly-charged Higgs mass is obtained as: The mass splittings in the Higgs triplet are: The trilinear couplings of the neutral scalars to the charged Higgs scalars with c β−α = 0 are given by: where the couplings: