Confronting cosmic ray electron and positron excesses with hybrid triplet Higgs portal dark matter

The existence of dark matter (DM) is well-established by cosmological and astrophysical evidence [1]. However, the nature of DM remains mysterious. Cosmic ray electrons and positrons (CRE) originating from adjacent galactic sources are smoking gun probes to the signature of DM annihilation or decay. Different terrestrial and satellite based experiments, such as HESS [2,3], Fermi-LAT [4,5], VERITAS [6,7], and AMS-02 [8–10], measure cosmic ray electron and positron flux up to TeV scale energy. The AMS-02 experiment shows distinctive features of positron flux data with a low energy excess around $ E\sim 25 $ GeV and a peak excess at approximately 300 GeV compared to the power-law background, which can be described well by the background with an extra positron source [11]. The DAMPE satellite borne experiment [12] also reported measurements of the cosmic ray electron/position spectrum ranging from 25 GeV to 4.6 TeV. DAMPE CRE data [13] report a spectral break around 0.9 TeV, which cannot be fitted properly by the single power-law background, and an excess peak around 1.5 TeV, which is expected to have a possible origin from DM annihilation and/or decay. The most convincing explanation for this 1.5 TeV excess is based on the assumption that the excess peak is due to DM annihilation (and/or decays) in nearby sub-halos enriched with DM. Various studies have been extensively performed to explain the aforementioned electron/position excesses and the possible flavor composition of DM annihilation final states [14–24].

In the present work, we consider a scalar DM accompanied by a type-II seesaw motivated triplet Higgs extension of the Standard Model (SM), in order to explain the AMS-02 and DAMPE cosmic ray electron and positron spectra. Triplet scalars dominantly decay into leptons with their vacuum expectation values (VEVs) less than $ 10^{-5} $ GeV, therefore they could play a perfect role as a leptonic portal [25–27] and provide a viable neutrino mass origin mechanism simultaneously. The scalar DM candidate is protected by unbroken $ Z_2 $ symmetry and couples to the Higgs triplet via quartic interaction, which provides the necessary leptonic annihilations. Earlier attempts to explain CRE obtained from AMS-02 and DAMPE data are carried out with single triplet Higgs extension of the SM with scalar DM [27–30]. In the present work, we consider a hybrid triplet Higgs extension of the SM. We show that such hybrid triplet Higgs portal DM models can provide a better fit to recent AMS-02 positron excess compared with the single triplet Higgs extension of the SM. We consider the DAMPE data with a single power-law background and that there exists a significant excess cosmic ray electron/positron spectrum ranging from 20 GeV to 1.5 TeV along with the excess peak-like feature. We show that in the hybrid triplet Higgs portal model, the reported excesses in AMS-02/DAMPE data are well fitted when electron/positron excess from the Milky Way (MW) and nearby DM sub-halo is taken into account. We find that there is a common parameter space to fit both AMS-02 and DAMPE data. We perform our analysis considering all possible neutrino mass hierarchies (normal, inverted, and quasi-degenerate), and the corresponding flavor structure is explored in detail.

The paper is organized as follows. We first briefly discuss both single and hybrid triplet Higgs portal models and neutrino Yukawa couplings derived for the different mass ordering of neutrinos (normal, inverted, and quasi-degenerate) by neutrino oscillation experimental data. In the next section, we describe the production and propagation of cosmic ray electron and positron flux within the galaxy. We then perform detailed analysis of AMS-02 positron excess data with single and hybrid triplet Higgs models separately for different neutrino mass hierarchies and acquire the parameter space for fitting the AMS-02 data. We show that hybrid triplet Higgs portal model can provide better explanation for the AMS-02 results. In the next section, we provide the fit results of DAMPE excess arising from the MW and DM sub-halo nearby and a common parameter space to fit both AMS-02 and DAMPE data. Finally, we summarize the work with concluding remarks.

In addition to the SM fields, an extra ${ {SU(2)_L}}$ triplet Higgs field is introduced, providing a solution to the neutrino mass origin, which is the so called type-II seesaw mechanism. We denote the triplet complex scalar fields Δ with hypercharge $ Y = 2 $ and SM doublet Higgs Φ as

where $\Delta^0 = (\delta + {\rm i} \eta)/\sqrt{2}$ and $ \phi^0 = (\phi_R + i \phi_I)/\sqrt{2} $. The most general scalar potential involving Δ and Φ can be written as

where $\tilde{\Phi} = {\rm i}\tau_2 \Phi^*$. The VEVs of the Higgs fields are given by

Without loss of generality, we assumed all parameters and VEVs are real. The triplet Higgs VEV is naturally chosen as $ v_{\Delta} \sim {\rm{keV}} $, well below the constraints from the electroweak precision test [31]. The conditions for minimization of potential are

After the Higgs fields Δ and Φ acquire VEVs due to the spontaneous symmetry breaking, the mass squared matrices of CP-even Higgs fields $ (\phi_R,\delta) $, CP-odd Higgs fields $ (\phi_I,\eta) $, singly-charged Higgs fields $ (\phi^+,\Delta^+) $, and double-charged Higgs fields $ \Delta^{\pm\pm} $ are given respectively by

By diagonalizing the above mass squared matrices, we obtain seven physical states: CP-even Higgs bosons h and H, CP-odd Higgs boson A, singly-charged Higgs boson $ H^{\pm} $, and double-charged Higgs boson $ H^{\pm\pm} $. The coupling parameters in the scalar potential are expressed in terms of the masses of physical particles and mixing angle α between two CP-even Higgses,

where we denote $ c_\alpha\equiv\cos \alpha $ and $ s_\alpha\equiv\sin \alpha $. The triplet field Δ can couple with a pair of lepton doublets $ L_L^i = (\nu_L^i,\ell_L^i)^T $ through the Yukawa terms, which are written as

where C denotes the charge conjugation. When the triplet Higgs field Δ acquires VEV, the lepton number is broken, and the neutrino mass matrix is then generated from Eq. (16)

The Yukawa coupling matrix $ Y_{\Delta} $ is fixed by the three neutrino masses and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix $ U_{{\rm{PMNS}}} $

where $ M_{\nu}^{{\rm{diag}}} $ denotes the diagonal neutrino mass matrix. The PMNS matrix can be parameterized as [1]

where $ c_{ij}\equiv \cos\theta_{ij} $, $ s_{ij}\equiv\sin\theta_{ij} $, and δ is the the Dirac CP phase. For simplicity, we have set the Majorana phases in the PMNS matrix to be zero. The values of neutrino oscillation data we used are listed in Table 1. The cosmological upper limit on the sum of three light neutrino masses is given by the Planck data at $ 95{\text{%}} $ confidence level, $ \sum m_{i} \leq 0.12\ {\rm{eV}} $ [33]. The exact values of each neutrino mass and even the mass ordering still remain unknown. In this work, we have set the lightest neutrino mass to be zero for both normal hierarchy (NH) and inverted hierarchy (IH), and then, the other two neutrino masses can be fixed by the oscillation data. For the quasi-degenerate hierarchy (DH) case, we assume that the sum of the three neutrinos masses is 0.12 eV. After confirming the neutrino masses, the Yukawa coupling matrices in Eq. (18) for three different hierarchy cases are given respectively by

Quantity	Normal ordering	Inverted ordering
$\theta_{12} /(^{\circ})$	33.44	33.45
$\theta_{23} /(^{\circ})$	49.0	49.3
$\theta_{13} /(^{\circ})$	8.57	8.61
$\delta_{CP} /(^{\circ})$	195	286
$\Delta m_{21}^2 /{\rm{eV} }^2$	$ 7.42 \times 10^{-5} $	$ 7.42 \times 10^{-5} $
$\Delta m_{31}^2 /{\rm{eV} }^2$	$ 2.514 \times 10^{-3} $	$ -2.497 \times 10^{-3} $

Table 1.  Global fit of neutrino oscillation parameters, taken from [32].

The kinetic terms in the Lagrangian for the SM doublet Φ and triplet Δ are given by

where $D_{\mu}\Phi = \partial_{\mu}\Phi-\dfrac{{\rm i}}{2}gW_{\mu}^a\tau^a \Phi - \dfrac{{\rm i}}{2}g^{\prime}B_{\mu}\Phi$ and $D_{\mu}\Delta = $$ \partial_{\mu}\Delta-\dfrac{{\rm i}}{2}g[W_{\mu}^a\tau^a, \Delta] - g^{\prime}B_{\mu}\Delta$. The doubly charged scalar $ (H^{\pm\pm}) $ can decay into a singly charged scalar and a W boson $ (H^{\pm}W^{\pm}) $, a pair of leptons $ (\ell^{\pm}\ell^{\pm}) $, or a pair of W bosons $ (W^{\pm}W^{\pm}) $. The channel $ H^{\pm\pm} \to H^{\pm} W^{\pm} $ can be kinematically forbidden by taking $ m_{H^{\pm\pm}} \simeq m_{H^{\pm}} $. The branching ratio of channel $ H^{\pm\pm} \to \ell^{\pm}\ell^{\pm} $ and channel $ H^{\pm\pm} \to W^{\pm}W^{\pm} $ depends on the value of the triplet Higgs VEV, and the $ H^{\pm\pm} $ dominantly decays into leptons when the triplet Higgs VEV $ v_{\Delta} < 10^{-5} $ GeV [34]. With the assumption of $ m_{H^{\pm}} \simeq m_{H} \simeq m_{A} $ and $ v_{\Delta} < 10^{-5} $ GeV, $ H^{\pm} $, H and A dominantly decay into $ \ell^{\pm}\nu $, $ \nu\nu $ and $ \nu\nu $, respectively.

In this work, we consider a hybrid triplet portal model with two Higgs triplets to explain cosmic ray electron and positron excesses and compare this scenario with the single triplet Higgs model. Models with two triplet Higgs fields have been proposed to address the neutrino mass matrix pattern and leptogenesis [35–37] and were explored for the context of collider experiments [36,38]. We denote two complex triplet Higgs fields $ \Delta_1 $ and $ \Delta_2 $ with hypercharge $ Y = 2 $ as

where $\Delta_{1,2}^0 = (\delta_{1,2}^0 + {\rm i} \eta_{1,2})/\sqrt{2}$. The general scalar potential of these two triplet Higgs fields $ \Delta_1 $ and $ \Delta_2 $ and the SM doublet Higgs fields Φ can be written as

where $ k,l = 1,2 $. The VEVs of the Higgs fields are given by

The orders of magnitude for the parameters in the potential are assumed:

For convenience, we define some matrices as

With the approximation of $ v_1,v_2 \ll v $, the conditions for minimization of potential are

The mass squared matrices of CP-even Higgs $ (\phi_R,\delta_1, \delta_2) $, CP-odd Higgs $ (\phi_I,\eta_1, \eta_2) $, singly-charged Higgs $ (\phi^+,\Delta_1^+, \Delta_2^+) $, and double-charged Higgs $ (\Delta_1^{++}, \Delta_2^{++}) $ are given respectively by

After diagonalization of these matrices, we obtain three Goldstone bosons and thirteen physical states. The transformations are parametrized by

The Yukawa terms for triplet Higgses $ \Delta_1, \Delta_2 $ coupled to leptons are given by

When the triplet Higgs fields acquire VEVs, the neutrino mass matrix is generated,

Compared with the single triplet Higgs portal case, the coupling matrices $ Y_{\Delta_1} $ and $ Y_{\Delta_2} $ are more flexible after fixing the VEVs of triplet Higgses. Gauge interactions of the SM Higgs and triplet Higgses are governed by

The charged Higgs $ H_{1,2}^{\pm} $ and $ H_{1,2}^{\pm\pm} $ dominantly decay into leptons in the scenario with the parameters set as follows: small VEVs of triplet fields $ v_{1,2} < 10^{-5} $ GeV, small mass splitting between two triplet Higgses $ m_{H_{i}^{\pm\pm}} \simeq m_{H_{i}^{\pm}} \simeq $$ m_{H_{i}} \simeq m_{A_{i}}\; (i = 1,2) $, and weak mixing between $ \Delta_1 $ and $ \Delta_2 $.

In addition to the triplet Higgs fields, we introduce a singlet real scalar field χ as a DM candidate. To ensure the candidate stability, the scalar field χ is assumed to be odd under $ Z_2 $ symmetry, while all SM fields and the triplet Higgs fields are assumed to be even. The scalar potential relevant to DM χ is given by

where $ i, j = 1, 2 $ for the hybrid triplet Higgs portal model, and $ \lambda_{{ij}} = \lambda $ for the single triplet Higgs portal case. The coupling $ \lambda_{\Phi} $ is suppressed by DM direct detection experimental results. With the assumption of $ \lambda_{\Phi} \ll \lambda_{ij} $, a pair of dark matter scalar fields χ dominantly annihilate into $ H_i^{\pm\pm}H_j^{\pm\pm} $, $ H_i^{\pm}H_j^{\pm} $, $ H_iH_j $, and $ A_iA_j $.

It has been generally postulated that the Milky Way galaxy is located in a larger spherical DM halo. For the DM spatial distribution in the galaxy, we considered a generalized NFW profile [39,40] to describe,

where $ \rho_s $ and $ r_s $ are typical density and typical radius respectively, whose values can fixed by the DM density at the location of the Sun (local density) and the total DM mass of the Milky Way, and γ is an undetermined parameter which mainly affects the DM density around the center of the Milky Way. According to the astrophysical observations to the rotation curve of the Milky Way, the local density of DM is roughly in the range of $ 0.2-0.6\ {\rm{GeV/cm^3}} $ [41–45], and the total DM mass contained in 60 kpc is estimated to be $ M_{60} = 4.7 \times 10^{11} M_{\odot} $ [46]. With the original NFW profile $ \gamma = 1 $, we can fix the typical density and typical radius to be $ \rho_s = 0.184\, {\rm{GeV/cm^3}} $ and $ r_s = 24.42 $ kpc for local DM density $ \rho_{\odot} = 0.3\; {\rm{GeV/cm^3}} $ at $ r_{\odot} = 8.5 $ kpc. In our model, the DM particles can self-annihilate into charged leptonic final states and generate electron/positron flux. The propagation of the $ e^{\pm} $ number density energy spectrum $f\equiv {\rm d}N/{\rm d}E$ in the Milky Way can be described by the diffusion-loss equation,

where $ {\cal{K}}(E, \vec{x}) $ is the diffusion coefficient function, which describes the propagation through the turbulent magnetic fields. Assuming the turbulent magnetic fields to be homogeneous and isotropic, the diffusion coefficient function can be parameterized as $ {\cal{K}}(E, \vec{x}) = {\cal{K}}_0 (E/ {\rm{GeV}})^{\delta} $. $ b(E, \vec{x}) $ is the energy loss coefficient function, which describes energy loss due to inverse-Compton scattering of cosmic microwave background and starlight photons as well as synchrotron radiation. It can be parameterized as $ b(E, \vec{x}) = b_0 (E/{\rm{GeV}})^2 $ [47]. The electron/positron source term $ Q(E, \vec{x}) $ is given as

where $ \rho(\vec{x}) $ is the DM density distribution, $ M_{{\rm{DM}}} $ is the mass of DM, and $ \langle \sigma v \rangle $ is the thermal averaged cross section. Considering a steady-case, $ \partial f/ \partial t = 0 $, and introducing Green's function formalism, the number density energy spectrum of a given point in the Galaxy can be written as

where Green’s function $ G(E, \vec{x}; E_s, \vec{x}_s) $ stands for the probability of injected electron or positron at $ x_s $ with the energy $ E_s $ to reach a location x with the degraded energy E. Green's function satisfies the following equation,

The equation has a general solution without boundary conditions [48],

where the diffusion length λ is given by

We solve Eq. (43) numerically in the shape of a flat cylinder with height $ 2L $ in the z direction and radius $ R = 20\,{\rm{kpc}} $ in the r direction, while considering the number density energy spectrum f vanished on the surface of the cylinder. The comparison of solving with and without boundary conditions is described in Ref. [49]. With the number density energy spectrum $ f(E,\vec{x}) $, the electron and positron flux is given as

where $ v(E) $ is the velocity of electron and positron with energy E and is set as c because of $ E \gg m_e $.

According to the latest analysis about cosmic ray positrons collected by the Alpha Magnetic Spectrometer (AMS-02) [11] on the International Space Station, a significant excess starting from $ 25\, {\rm{GeV}} $ to $ 1\,{\rm{TeV}} $ is reported, which possibly originates from DM particles annihilation. We choose the same background formula as the latest report by the AMS Collaboration [11],

where $ \hat{E} = E+\varphi_{e^+} $, with the effective solar potential $ \varphi_{e^+} $ accounting for the solar effects. We implement the model in FeynRules [50] to generate the model files and put it in micrOMEGAs 5.0.8 [51] to generate the positron flux from DM annihilation. For the propagation of flux, we set the diffusion coefficient $ {\cal{K}}_0 = 0.0112\, {\rm{kpc}}^2/{\rm{Myr}} $, $ \delta = 0.7 $, and the energy loss coefficient $ b_0 = 10^{-16}\, {\rm{s}}^{-1} $.

With the DM mass ($ m_{{\rm{DM}}} $) and triplet Higgs mass ($ m_{\Delta} $) fixed, the DM thermal relic density only depends on the coupling $ \lambda_{ij} $ in Eq. (38). In other words, the couplings $ \lambda_{ij} $ can be fixed by the observed DM thermal relic of $ \Omega h^2 = 0.12 $ [33], and then, the thermal averaged cross sections in Eq. (41) are also fixed. However, the thermal averaged cross sections obtained from observed DM thermal relic $ \langle\sigma v \rangle_0 $ are usually much smaller than the required value $ \langle\sigma v \rangle $ to account for the excess. In order to fit the AMS-02 data, we need enhancement to the DM annihilation cross sections. The enhancement is usually parameterized by a so-called boost factor (BF) $\kappa_{{\rm{BF}}} = $$ \langle\sigma v\rangle / \langle\sigma v \rangle_0$. The boost factor may originate from small-scale inhomogeneities of the DM density distribution [27,52] or Sommerfeld/Breit-Wigner enhancement of DM annihilation [53–59].

To describe the goodness of the fit, we introduce the chi-square function $ \chi^2 $, which defined as

where $ f^{{\rm{th}}}_i $ are the theoretical predictions, $ f^{{\rm{exp}}}_i $ are the central values of the experimental data, and $ \delta f_i $ are combined errors of theoretical and experimental data. For the fit to AMS-02 data, we have only considered the experimental errors and used all $ E > 1 {\rm{GeV}} $ AMS-02 data (71 data points). In our model, a pair of DM can annihilate to a pair of singly charged triplet Higgs $ H^{\pm} $ or doubly charged triplet Higgs $ H^{\pm\pm} $, and these particles will further decay into two or four leptonic final states. The Yukawa coupling matrices are fixed by the neutrino mass and oscillation data, and the matrix elements are given by Eqs. (20)–(22) for different neutrino mass hierarchies. As shown in Table 2, the doubly charged Higgs $ H^{\pm\pm} $ dominantly decays into μ and τ leptons for the normal neutrino mass hierarchy, while for the inverted hierarchy, it dominantly decays to electrons, and for the quasi-degenerate case, the three flavors are almost democratic. When the difference value of DM mass and triplets mass is small ($ m_{{\rm{DM}}} \simeq m_{\Delta} $), the leptons generated from triplet Higgs decays have a roughly monochromatic energy spectrum. When $ m_{{\rm{DM}}} \gg m_{\Delta} $, since the triplet Higgses are highly boosted, the leptons will have a continuous energy spectrum. The generated muons and taus will further decay into positrons and contribute to the cosmic ray positron flux. The energy spectrum of the positron directly produced by the triplet Higgs is sharp and drop around energy $ E = m_{{\rm{DM}}}/2 $, while for the positrons generated from the muon- and tau-cascade decays, the energy spectrum will be softer, and the peak is lower.

Hierarchy	$H^{\pm\pm}$ decays							$H^{\pm}$ decays
	$ee$	$e\mu$	$e\tau$	$\mu \mu$	$\mu \tau$	$\tau \tau$		$e\nu$	$\mu \nu$	$\tau \nu$
NH	0.49%	0.67%	4.36%	37.37%	34.90%	22.21%		3.00%	55.14%	41.86%
IH	47.53%	1.32%	0.84%	9.84%	23.74%	16.73%		49.27%	50.72%	0%
DH	25.26%	0.32%	0.28%	37.83%	1.88%	34.43%		24.79%	25.51%	49.70%

Table 2.  Branching ratios of charged Higgs leptonic decays for different neutrino mass hierarchies.

For the single triplet Higgs portal model, we scan over the mass range from 500 GeV to 2 TeV for both DM and triplet Higgs and assume the triplet Higgs mass splitting is negligible ($ m_{\Delta} \simeq m_{H} \simeq m_{A}\simeq m_{H^{\pm}} \simeq m_{H^{\pm\pm}} $). As shown in Fig. 1(a), we present the favored region for $ \chi^2/{\rm{d.o.f.}} $ in the plane of the DM mass $ m_{{\rm{DM}}} $ versus the triplet Higgs mass $ m_{\Delta} $. In the right panel of Fig. 1(b), we plot the mass difference $ (m_{{\rm{DM}}}-m_{\Delta}) $ against the triplet Higgs mass for the NH scenario for different $ \chi^2/{\rm{d.o.f.}} $ values. The black data points in both Fig. 1(a) and (b) are corresponding to the best fit data points obtained from $ \chi^2/{\rm{d.o.f.}} $ analysis for the NH case. Similar plots are generated for the DH case and depicted in Fig. 1(c-d) along with the best fit data points. A comparison between the NH and DH scenarios reveals that the NH case has a larger range of DM mass of approximately (1–1.8) TeV; for the chosen range of $ \chi^2/{\rm{d.o.f.}}\in (1.0-2.5) $, the corresponding range is restricted into $ (1-1.4) $ TeV for the DH case. Figs. 1(b) and 1(d) show that the mass splitting between DM mass and the triplet Higgs in the case of NH is larger than that in the DH case. Various parameters for the best fit to AMS-02 data with single triplet Higgs portal DM model are listed in Table 3 along with the contribution ratios from different DM annihilation channels. It is to be noted for the IH case that $ \chi^2/{\rm{d.o.f.}} $ is always found to be larger than 5 for the chosen range of DM and triplet Higgs mass. Therefore, we do not present the results for the IH scenario.

Figure 1.  (color online) Favored region of $ \chi^2/{\rm{d.o.f.}} $ for the single triplet portal case. The upper and lower panel are dedicated to the normal and degenerate neutrino mass hierarchies, respectively.

Best fit	$m_{\text{DM}}$/GeV	$m_{\Delta}$/GeV	$\rm{Br}(H_1 H_1/A_1 A_1)$	$\rm{Br}(H^+_1 H^-_1/H^{++}_1 H^{--}_1)$	$\lambda_{\Delta}$	BF
NH	1340	660	$16\,$%	$34\, $%	0.18	1758
DH	980	600	$17\,$%	$33\, $%	0.12	608

Table 3.  Best fit data points to AMS-02 positron excess in the single triplet DM model.

In Fig. 2, we plot positron flux spectrum using the best fit data points obtained for the NH and DH cases (Table 3). One can notice that the best fit data do not provide a good fit to the positron flux tail at the high energy end. The reason is that, in the single triplet Higgs portal DM model, the flavor structure of the leptonic final states generated by the DM annihilation is fixed by the neutrino oscillation data. The positron flux spectrum range from 10 GeV to 300 GeV indicates that the positrons are dominantly generated by the muon- and/or tau-cascade decays but not for the case when the energy is beyond 300 GeV. This is also the reason that there are more favored ranges for the NH case compared with the IH case. In the IH case, the charged triplet Higgses dominantly decay to electrons (as demonstrated in Table 2). In the following, we investigate the hybrid triplet Higgs portal DM, and we find that the fit gets much better, especially for the IH case.

Figure 2.  (color online) The best-fit of the positron flux observed by the AMS-02 in the single triplet portal DM model.

We now fit the AMS-02 data by the hybrid triplet portal DM, with a light triplet Higgs and a heavy triplet Higgs final states as the DM main annihilation channels. We find that the positron flux can be best fit when the light triplet Higgs mainly decay to muons and taus, whereas the heavy one mainly decay to electrons. In order to generate the high energy region positrons by the heavier triplet Higgs decays, we open the two annihilation channels $ \chi \chi \to \Delta_1 \Delta_1 $ and $ \chi \chi \to \Delta_1 \Delta_2 $, while the channel $ \chi \chi \to \Delta_2 \Delta_2 $ is forbidden by setting $ m_{\Delta_1}+m_{\Delta_2} < 2m_{{\rm{DM}}} < 2m_{\Delta_2} $. Without loss of generality, as a benchmark, we set $2m_{{\rm{DM}}}- $$ (m_{\Delta_1}+m_{\Delta_2}) = 10$ GeV. We scan over the light triplet Higgs mass $ m_{\Delta_1} $ ranging from 500 GeV to 1.5 TeV while the heavier triplet Higgs mass is varied within the range from 1.5 TeV to 3.5 TeV. In Fig. 3 (a-c), we plot the allowed range of $ m_{\Delta_1} $ versus $ m_{\Delta_2} $ parameter space, which fits the AMS-02 data with $ 0.8\leq \chi^2/{\rm{d.o.f.}} \leq 1 $ for the NH, IH, and DH scenarios, respectively.

Figure 3.  (color online) Favored region with $ \chi^2/{\rm{d.o.f.}} $ for the hybrid triplet portal DM, for the case of the mass difference taken as $ 2m_{{\rm{DM}}}-(m_{\Delta_1}+m_{\Delta_2}) = 10\ {\rm{GeV}} $.

The favored regions for the masses of DM and the triplet Higgses are shown in Fig. 3, and the best fit data sets are given in Table 4, with the corresponding positron flux results shown in Fig. 4. One intriguing feature of the hybrid triplet portal DM is that it can provide a pretty good fit to the AMS-02 data for the IH case, unlike the single triplet portal DM model discussed earlier. Apart from the above, we also find that the hybrid triplet portal DM can provide better fit for the excess at high energy region for the energy above 300 GeV. This unique feature of the hybrid triplet model is due to that the heavy triplet Higgses produce hard positrons, while the light triplet Higgses generate soft positrons. Finally, we make a direct comparison between the single triplet Higgs portal DM model and the hybrid triplet Higgs portal DM model. The $ \chi^{2}/{\rm{d.o.f.}} $ of the best fit points in varied scenarios are listed in Table 5, which clearly indicates that the hybrid triplet Higgs portal model provides a much better fit to the AMS-02 data. Especially, the fit is significantly improved for the neutrino mass IH scenario.

Best fit	$m_{\Delta1}$/GeV	$m_{\Delta2}$/GeV	$\rm{Br}(H_1 H_1/A_1 A_1)$	$\rm{Br}(H^+_1 H^-_1/H^{++}_1 H^{--}_1)$	$\rm{Br}(H_1 H_2/A_1 A_2)$	$\rm{Br}(H^+_1 H^-_2/H^{++}_1 H^{--}_2)$	$\lambda_{11}$	$\lambda_{12}$	BF
NH	780	2720	$12\,$%	$25\, $%	$4\, $%	$9\, $%	0.10	0.10	2100
IH	650	2370	$15\,$%	$30\, $%	$2\, $%	$3\, $%	0.14	0.09	1126
DH	590	2210	$13\,$%	$29\, $%	$3\, $%	$5\, $%	0.12	0.10	1267

Table 4.  Best fit data points to AMS-02 positron excess for hybrid triplet Higgs portal DM model. The "Branch ratios" indicate the branch ratios of the DM annihilation channels. "BF" denotes the needed enhancement factors.

Figure 4.  (color online) The best fit to the positron flux observed by the AMS-02 in the hybrid triplet portal DM model.

Note that the DM annihilation also produces gamma-rays. We compared the gamma-rays produced by the DM annihilation with the isotropic diffuse gamma-ray background (IGRB) observed by the Fermi-LAT experiment [60], and we find that the predicted gamma-rays are slightly constrained by the Fermi-LAT observation in the energy region beyond hundreds of GeV. An earlier study about the single triplet Higgs portal model reached a similar conclusion [28]. A feasible way to release the inconsistency between the DM origin of cosmic positron excesses and the IGRB constraints is to introduce a dark matter disk instead of a simple dark matter halo as the dark matter density distribution [61]. More studies on the compatibility between the DM origin of cosmic positron excesses and gamma-ray limits are reported in Refs. [62–66].

In this section, we attempt to explain the cosmic ray electron and positron excesses observed by the DAMPE satellite. Different from AMS-02, DAMPE does not distinguish the charge of electron events and collect the sum of electron and positron events. The DAMPE data have been considered to be the electron and positron flux associated with a smoothly broken power law spectrum [12]. The signals have also been considered as an excess in the range (0.6–1.5) TeV from DM sources with single broken power-law background and double broken power-law background [21, 23, 67] or excess in the range of (0.1–1.5) TeV with single power law background [20]. Here, we assume that there exists a smooth excess in the energy range (0.1–1.5) TeV along with an excess peak at 1.5 TeV. We consider that the continuous excess of electron and positron flux is attributed to the DM annihilation within the Milky Way (MW), and the sharp peak at 1.5 TeV originates mainly from a nearby DM sub-halo. Meanwhile, we take a simple single power law background to fit the background for $ E<100 $ GeV and $ E>1.5 $ TeV, and the background formula of the electron and positron flux is expressed as

where we set the free parameter: $C = 450\ {\rm (GeV\cdot m^2\cdot s\cdot $$ \rm sr)^{-1} }$ and $ \gamma = 3.26 $ by fitting the first two points and the four points between 1.5 TeV and 2.5 TeV of the DAMPE data.

To explain the peak around 1.5 TeV in the DAMPE data, the heavy triplet Higgs mass $ m_{\Delta_2} $ is required to be set around 3 TeV, and the heavy charged triplet Higgs mainly decays to the electron flavor. From the results of fitting AMS-02 data, we learn that the heavy triplet Higgs can be chosen as high as 3 TeV only in the normal neutrino mass hierarchy scenario; therefore, we choose the NH case to fit the DAMPE electron and position excess in the hybrid triplet Higgs portal model. We scan over the light triplet Higgs mass $ m_{\Delta_1} $ ranging from 500 GeV to 1.5 TeV and the heavy triplet Higgs mass $ m_{\Delta_2} $ ranging from 1.5 TeV to 4 TeV. We obtain the allowed range of $ m_{\Delta_1} $ versus $ m_{\Delta_2} $ that fits both the AMS-02 data with $ 0.8 \leq \chi^2/ $$ {\rm{d.o.f.}} \leq 1 $ and the DAMPE data with $ 0.2 \leq \chi^2/{\rm{d.o.f.}} \leq 0.3 $ for the NH case, as shown in Fig. 5. For the dark matter sub-halo profile, we adopt the NFW density profile with $ \rho_s = 100\ {\rm{GeV/cm^3}} $, $ r_s = 0.1 $ kpc, and $ \gamma = 0.5 $, and the distance between the Sun and the center of the sub-halo is set as $ d_s = 0.2 $ kpc.

Figure 5.  (color online) (a): The allowed parameter space of $ m_{\Delta_1} $ versus $ m_{\Delta_2} $ that fits both the AMS-02 data with $ 0.8 \leq \chi^2/{\rm{d.o.f.}} \leq 1 $ and the DAMPE data with $ 0.2 \leq \chi^2/{\rm{d.o.f.}} \leq 0.3 $ for the NH case. (b): The best fit to the DAMPE CRE spectrum with $ m_{\Delta_1} = 810\ {\rm{GeV}} $, $ m_{\Delta_2} = 3000\ {\rm{GeV}} $, and $ \chi^2/{\rm{d.o.f.}} = 0.2 $. The blue line indicates the contribution from DM annihilation in the Milky Way DM halo, and the green line includes the contribution of the nearby sub-halo.

In this study, we investigate a hybrid triplet Higgs portal DM model by extending the SM with two triplet scalar fields and a singlet scalar DM, which can generate neutrino mass by a hybrid Type-II seesaw mechanism and provide a DM candidate. In this framework, the DM particles can self-annihilate into triplet Higgses, which further cascade decay into leptons. We calculate the energy spectrum of the electron and positron flux from the DM annihilation in the Galactic halo to explain the flavor structure of both recent AMS-02 positron data and DAMPE electron/positron data. We revisit the single triplet Higgs portal DM model to address the AMS-02 positron excess and obtain the favored parameters region for different neutrino mass hierarchies. We then use the hybrid triplet Higgs portal DM to fit the AMS-02 data. We find that the hybrid triplet Higgs portal model can provide a much better fit, especially for the inverted hierarchy neutrino mass scenario. We also perform a fit to the DAMPE electron/positron data using the hybrid triplet Higgs portal model and find that the NH scenario has a favored region for fitting both the AMS-02 and the DAMPE signals.

Here, we present the model parameter values corresponding to the best fit points (BFPs) in Table 3 and Table 4.

BFP 1: Input parameters for the NH case in Table 3:

BFP 2: Input parameters for the DH case in Table 3:

BFP 3: Input parameters for the NH case in Table 4:

With these parameter values, the particle masses are determined as

BFP 4: Input parameters for the IH case in Table 4:

With these parameter values, the particle masses are determined as

BFP 5: Input parameters for the DH case in Table 4:

With these parameter values, the particle masses are determined as