New Interpretation of the Recent Result of AMS-02 and Multi-component Decaying Dark Matters with non-Abelian Discrete Flavor Symmetry

Recently the AMS-02 experiment has released the data of positron fraction with much small statistical error. Because of the small error, it is no longer easy to fit the data with a single dark matter for a fixed diffusion model and dark matter profile. In this paper, we propose a new interpretation of the data that it originates from decay of two dark matter. This interpretation gives a rough threshold of the lighter DM component. When DM decays into leptons, the positron fraction in the cosmic ray depends on the flavor of the final states, and this is fixed by imposing non-Abelian discrete symmetry in our model. By assuming two gauge-singlet fermionic decaying DM particles, we show that a model with non-Abelian discrete flavor symmetry, e.g. $T_{13}$, can give a much better fitting to the AMS-02 data compared with single dark matter scenario. Few dimension six operators of universal leptonic decay of DM particles are allowed in our model since its decay operators are constrained by the $T_{13}$ symmetry. We also show that the lepton masses and mixings are consistent with current experimental data, due to the flavor symmetry.


Introduction
The latest experiment of Planck [1] tells us that about 26.8 % of energy density of the universe consists of Dark Matter (DM). Many experiments are being performed to search DM signatures. The recent result of the indirect detection experiment of AMS-02 [2] is in favor of the previous experiments such as PAMELA [3,4] and Fermi-LAT [5], which had reported the excess of positron fraction in the cosmic ray. Moreover, it smoothly extends the anomaly line of positron fraction with energy up to about 350 GeV with small statistics error compared with the previous experiments. These observations can be, in general, explained by scattering and/or decay of the GeV/TeV-scale DM particles. In addition leptophilic DM is preferable since PAMELA observed no anti-proton excess [6]. Along this line of thought, several papers have been released [7][8][9][10][11][12][13][14][15]. Due to the smallness of statistics error of AMS-02, it became difficult to fit to the data as same as the previous experiments like PAMELA [14].
In this paper, we show that we can obtain a better fitting to the data with two component decaying DM. We introduce two kinds of fermionic DM particles with mass of O(100) GeV and O (1) TeV into the framework of T 13 flavor symmetric model [16]. In our model, the flavor symmetry T 13 works at least in two ways: (i) It constrains interactions between DM and the Standard Model (SM) particles. DM particles which are gauge singlet fermion X and X ′ couple with leptons by dimension six operatorsLELX ( ′ ) /Λ 2 due to the T 13 symmetry, thus these are leptophilic. DM particles decay into leptons via these operators with the suppression factor Λ ∼ 10 16 GeV, giving desired lifetime of DM particles, Γ −1 ∼ ((TeV) 5 /Λ 4 ) −1 ∼ 10 26 s [17,18]. (ii) Flavor of the final states of DM decay is determined by the T 13 symmetry. We give a concrete example of the universal final states X/X ′ → ν e e + e − /ν µ µ + µ − /ν τ τ + τ − . Due to a specific selection rule by the flavor symmetry mentioned above, we show that two-component DM model is preferable for explanation of the precise AMS-02 result. In addition to that, we find a set of parameters that is consistent with the observed lepton masses and their mixings especially somewhat large angle of θ 13 recently reported by several experiments [19][20][21][22][23][24][25].
This paper is organized as follows. In Section 2, we briefly mention the T 13 symmetric model and construct mass matrices of the lepton sector in definite choice of T 13 assignment of the fields. We show that there exists a consistent set of parameters. In section 3, we show that desirable dimension six DM decay operators are allowed by T 13 symmetry and that leptonic decay of the two DM particles by those operators shows good agreement with the cosmic-ray anomaly experiments. Section 4 is devoted to the conclusions.

Lepton masses and mixings with T 13 flavor symmetry
First of all, we briefly review our model based on the non-Abelian discrete group T 13 , which is isomorphic to Z 13 ⋊ Z 3 [16,[26][27][28]. The T 13 group is a subgroup of SU(3), and known as the minimal non-Abelian discrete group having two complex triplets as the irreducible representations, see ref. [16] for details.
Lepton masses and mixings are derived from the setup shown in Table 1 Here notice that X and X ′ have dimension six operatorsLELX,LELX ′ , and mass terms m X XX, m X ′X ′ X ′ . For the matter content and the T 13 assignment given in Table 1, the charged-lepton and neutrino masses are generated from the T 13 invariant operators where H c = ǫH * , and LH(3 2 )LH(3 1 ) is T 13 invariant in two different products, corresponding to b ν and c ν . The fundamental scale Λ = 10 11 GeV is needed for the certain neutrino mass scale 16 GeV is required to obtain the desired lifetime of DM, where λ is coupling constant of DM decay operators as we will discuss later). After the electroweak symmetry breaking, the 1 All the assignment and particle contents are the same as our previous work [16] except the DM sector.
where the vacuum expectation values (VEVs) of the Higgs bosons are defined as H( and the mixing matrices are given by which are all consistent with the present experimental data [29,30]. In particular in the case of  of anti-proton and secondary positron by scattering with nucleon and interstellar medium. With the notation L i = (ν i , ℓ i ) = (U eL ) iα (ν α , ℓ α ) and E i = (U eR ) iβ E β (i = 1, 2, 3, α, β = e, µ, τ ), the four-Fermi decay interaction is explicitly written as where the factor ω 2(i−1) is only for the case of X ′ decay because of the multiplication rule of the T 13 flavor symmetry. As seen from Eq. (3.1), decay mode of the DM particles X and X ′ depends on the mixing matrices U eL and U eR , which are given in Eq. (2.6).
Next, we consider the decay width of the decaying DM through the T 13 invariant interaction Eq. (3.1). Due to the particular generation structure, the DM particles X and X ′ decay into several tri-leptons final state with the mixing-dependent rate. The decay width of DM X per each flavor is , and the decay width Γ αβγ is calculated as

2)
to be enough tiny. The most stringent constraint process is X ′ → X, h that comes from H † HXX ′ , where h is the standard model Higgs whose mass is 126 GeV [36]. We find that its coupling should be less than O(10 −18 ) in order to conservatively satisfy the no excess constraint of the antiproton with the lifetime of DM to be longer than O(10 28 ) s. where The decay width of X ′ named Γ ′ αβγ is obtained by replacing X → X ′ . The differential decay width is written as This is required to calculate the energy distribution function of injected e ± from DM decay, dN e ± /dE. Here we have neglected the masses of charged leptons in the final states. In both X and X ′ DM cases, the flavor dependent factor U αβγ gives a factor three when one takes the sum of flavor indices α, β and γ. That is not by a particular choice of parameters Eq. (2.4), but by the T 13 symmetry. Therefore, the branching fraction of each decay mode is given The DM mass m X and the total decay width Γ X = α,β,γ Γ αβγ are chosen to be free parameters in the following analysis since it can be always tuned with the coupling λ X and the cut-off scale Λ.
Given the differential decay width and the branching ratios, the primary source term of the positron and electron coming from DM decay at the position r of the halo associated with our galaxy is expressed as where (dN e ± /dE) f is the energy distribution of e ± coming from the DM decay with the final state f , and E is the energy of injected e ± . We use the PYTHIA 8 [37] to evaluate the energy distribution function. Although it is often assumed that the relic density of the DM is thermally determined, non-thermal production of the DM dark matter is also possible [38]. We thus do not specify the origin of the relic DM in the following analysis, and assume that the number densities of X and X ′ are the same for simplest case. The non-relativistic DM number density n X (r) is rewritten by n X (r) = ρ X (r)/m X with the DM profile ρ(r). In this work, we adopt the Navarro-Frank-White (NFW) profile [39], where ρ ⊙ ≃ 0.40 GeV/cm 3 is the local DM density at the solar system, r is the distance from the galactic center whose special values r ⊙ ≃ 8.5 kpc and r c ≃ 20 kpc are the distance to the solar system and the core radius of the profile, respectively. The diffusion equation must be solved to evaluate the e ± flux observed at the Earth, and it depends on diffusion model. The observable e ± flux at the solar system dΦ e ± /dE which is produced by DM decay is given by where b(E) is a space-independent energy loss coefficient written as b(E) = E 2 /(τ ⊙ · 1GeV) with τ ⊙ = 5.7 × 10 15 s, and I ⊙ (E, E ′ ) is the reduced halo function at the solar system which is expressed by Fourier-Bessel expansion [40]. A fitting function for the reduced halo function I(λ D ) is given in ref. [40] as a function of a single parameter λ D which is called diffusion length. The diffusion length λ D is given by where we use the following diffusion parameters: δ = 0.70, K 0 = 0.0112 kpc 2 /Myr which is called MED. In addition, the diffusion zone is considered as a cylinder that sandwiches the galactic plane with height of 2L and radius R where L = 4 kpc and R = 20 kpc.
As seen from Eq. (2.6) and (3.2), the DM decays into e ± as well as µ ± and τ ± in the equal rate.
As a result, pure leptonic decays give dominant contributions, and it is consistent with no anti-proton excess of the PAMELA results [6]. We may take into account the gamma-ray constraint since a lot of gamma-ray is produced by the hadronization of τ ± . As we see below, the obtained lifetimes of DM particles X and X ′ are roughly τ X , τ X ′ 5 × 10 26 s. Thus we do not need to consider the gamma-ray constraint seriously as long as comparing with ref. [41].

Result for AMS-02
We use 31 data points of AMS-02 which are higher than 20 GeV for chi-square analysis. The only statistics error is taken into account as the experimental errors here [2]. The positron fraction for the scenario of the leptonically decaying DM with T 13 symmetry is depicted in Figure 1 with

Conclusions
We revisited a decaying DM model with a non-Abelian discrete symmetry T 13 , and extended it to the two-component DM scenario by adding an extra DM X ′ . We have shown that our model is consistent with all the observed masses and mixings in the lepton sector. Due to also the specific selection rule of T 13 , we have found that both of DM particles have the universal decay coming from dimension six operators that gives a promising model in current indirect detection searches of DM.
Fitting to the positron fraction with a single DM under the assumption of MED diffusion model and NFW DM profile can no longer give a good interpretation of the positron excess by DM decay because of the precise measurement of AMS-02. However taking into account two-component DM as our model gives much better fitting to AMS-02 observation. The obtained parameters are m X =208 GeV with Γ −1 X = 1.8 × 10 27 s and m X ′ =1112 GeV with Γ −1 X ′ = 4.7 × 10 26 s, assuming that X and X ′ have equal number density.