1 Introduction

IPMU14-0337
LCTS/2014-45

Non-abelian Dark Matter Solutions for Galactic

Gamma-ray Excess and Perseus 3.5 keV X-ray Line

[2.5cm] Kingman Cheung111cheung@phys.nthu.edu.tw, Wei-Chih Huang222wei-chih.huang@ucl.ac.uk, Yue-Lin Sming Tsai333yue-lin.tsai@ipmu.jp

[1cm]

Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan

Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Republic of Korea

Department of Physics and Astronomy, University College London, UK

Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

[1 cm] Abstract

[0.5cm]

We attempt to explain simultaneously the Galactic center gamma-ray excess and the 3.5 keV X-ray line from the Perseus cluster based on a class of non-abelian DM models, in which the dark matter and an excited state comprise a “dark” doublet. The non-abelian group kinetically mixes with the standard model gauge group via dimensions-5 operators. The dark matter particles annihilate into standard model fermions, followed by fragmentation and bremsstrahlung, and thus producing a continuous spectrum of gamma-rays. On the other hand, the dark matter particles can annihilate into a pair of excited states, each of which decays back into the dark matter particle and an X-ray photon, which has an energy equal to the mass difference between the dark matter and the excited state, which is set to be 3.5 keV. The large hierarchy between the required X-ray and -ray annihilation cross-sections can be achieved by a very small kinetic mixing between the SM and dark sector, which effectively suppresses the annihilation into the standard model fermions but not into the excited state.

## 1 Introduction

A gamma-ray excess around a few GeV near the Galactic center (GC) region, seen by the Fermi-LAT collaboration (see, for instance, the recent analysis by the collaboration [1]), has been widely discussed based on dark matter (DM) annihilations into standard model (SM) fermions  [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], which hadronize into neutral pions followed by , or electromagnetic bremsstrahlung. On the other hand, recent reports of the X-ray line [16, 17] from the XMM-Newton data have triggered many studies in the context of DM, for example, Refs. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. Roughly speaking, they can be classified into two categories: (i) DM undergoes upscattering into an excited stated followed by the decay back into DM and an X-ray photon; and (ii) decaying DM matter, such as a 7 keV sterile neutrino decaying into an active neutrino and the X-ray photon. The excited DM, however, has a advantage of explaining some null results on X-ray line searches due to a low local DM velocity as shown in Ref [66].

There is a very interesting connection between the -ray excess and X-ray line as follows. The GC -ray excess can be explained by annihilating DM with a mass from 10 to 60 GeV [2, 5, 6, 7, 8, 10, 11, 12], depending on the final state of the annihilation. On the other hand, due to the fact that the current DM velocity is around in the Perseus cluster, where the X-ray line is observed, the DM with a mass of 10 to 60 GeV coincidently has a kinetic energy of a few keV. It implies if there exists an excited state with a 3.5 keV mass splitting from the DM particle, then the DM particles can annihilate into the excited state, followed by the decay back into the DM particle with a photon accounting for the observed X-ray line.

In this work, we employ a class of non-abelian DM models proposed in Refs. [67, 18], where the DM particle and the excited state form an doublet with a 3.5 keV mass splitting. The kinetically mixes with the SM gauge group via dimension-5 operators, through which the SM particles can couple to the currents and the DM (and the excited state) couples to the SM currents. As mentioned above, the GC -ray excess comes from the DM annihilation into SM fermions accompanied by photon emission while the X-ray line is realized from the DM annihilation into the excited state followed by the subsequent decay. Besides, the annihilation into SM fermions, induced by the kinetic mixing, is suppressed compared to that into the excited state if the kinetic mixing is small. This suppression naturally explains the hierarchy between the required annihilation cross-sections for the -ray excess ( cmsec) and the X-ray line emission ( cmsec) as shown below. Note that similar ideas connecting the -ray and X-ray excess have been suggested in Refs. [39, 18] with intermediate states (instead of the SM fermion final state) while an effort connecting the keV line [68] and the GC -ray excess turns out to be negative [69].

This paper is organized as follows. In Sec. 2, we specify the model and divide into the Majorana and Dirac DM cases. In Sec. 3, we calculate the relevant cross-sections. In Sec. 4, we discuss the calculations of -ray and X-ray flux as well as the DM relic abundance. In Sec. 5, we present our numerical analysis with separation into the Majorana and Dirac cases. Finally, we conclude in Sec. 6.

## 2 Non-abelian Dark Matter Models

For the nonabelian DM model, we employ a “dark” gauge group with kinetic mixing with the SM gauge groups proposed in Refs. [67, 18]. We start with a doublet, which is comprised of the fields for the DM particle and an excited state. In the following we will discuss two cases: Majorana DM () with the Dirac excited state () and Dirac DM () with the Dirac excited state ().111In this work, we denote Majorana particles by and Dirac particles with for particles in the dark sector. As we shall see later, we have to make use of the resonance enhancement in order to achieve large annihilation cross-sections, especially for explaining the X-ray line. The resonance enhancement does not occur if both DM and the excited state are Majorana with nearly degenerate masses, as shown in Appendix A. On the other hand, the Dirac DM with the Majorana excited state will lead to a large -ray flux but a small X-ray one, in contradiction to the -ray and X-ray data Therefore, we will not discuss these two scenarios in this work. The Lagrangian of the model reads,

 L=LSM+LDM1+LDM2+Lmix, (2.1)

where is the SM Lagrangian. correspond to the DM sector, including the DM doublet and the dark gauge bosons,  (), and dark Higgs triplets/doublets, which are used to provide masses to s and s:

 LDM1=−14XμνaXaμν+(DXμΔ1)†(DXμΔ1)+(DXμΔ2)†(DXμΔ2), (2.2)

where is the covariant derivative of and are triplets, whose vacuum expectation values (VEVs) provide masses to dark gauge bosons. Note that one can play with the structure of to give different masses to . For example, with in the isospin basis (the first component has the highest isospin , the second with , and so on) are massive but remains massless.

### 2.1 Majorana DM

In the case of Majorana DM, the takes the form

 LDM2=iχ†DXμσμχ+i~χ†2∂μ¯σμ~χ2+(12λΔ(χ⋅Δ1⋅χ)+λh2(χ⋅hD)~χ2+h.c.), (2.3)

where the two-component Weyl spinor notation is employed. Here “” refers to the -invariant multiplication. is an doublet, consisting of two Weyl spinors, and : . In addition, is an scalar doublet. is a singlet under , which will be paired up with to form a Dirac fermion. The conversion between Dirac- and Weyl-spinors for , and is:

 ψ1 = (χ1χ†1), ψ2 = (χ2~χ†2). (2.4)

The corresponding -current in the Weyl and Dirac-spinor notation is given by

 L ⊃ gXX3μJμX=−gX2X3μχ†1¯σμχ1+gX2X3μχ†2¯σμχ2 (2.5) = −gX2X3μ¯ψ1γμ(−γ52)ψ1+gX2X3μ¯ψ2γμ(1−γ52)ψ2,

where the pre-factors come from the fact that has isospin .

In order to give a Majorana mass to , one can make use of the lowest isospin () component of , leaving VEVs of other components vanishing, i.e., in the isospin basis. The mass becomes . Similarly, with the lower isospin () of , the Dirac mass of and becomes , where is the VEV of the component of . Moreover, ’s masses, at phenomenological level, are considered independent since as mentioned above one can always use to give a mass to specific gauge boson(s).

The particle content in the dark sector and the relevant quantum numbers in this model are summarized in Table 1.

We would like to point out that the VEVs of and are used to give a mass to the particles of interest and induce the kinetic mixing between the SM and the dark sector. We simply assume that they are very heavy and play no roles in the context of GC gamma ray excess and the 3.5 keV X-ray line.

### 2.2 Dirac DM

In the case of Dirac DM, the takes the form

 LDM2=iχ†DXμσμχ+2∑i=1i~χ†i∂μ¯σμ~χi+(λh1(χ⋅hD1)~χ1+λh2(χ⋅hD2)~χ2+h.c.), (2.6)

where  () gives a Dirac mass to and  ( and ). We list the particle content and quantum numbers in Table 2. The conversion between Dirac- and Weyl-spinors for , and is:

 ψ1 = (χ1~χ†1), ψ2 = (χ2~χ†2), (2.7)

and the corresponding -current in the Weyl and Dirac-spinor notation is

 L ⊃ gXX3μJμX=−gX2X3μχ†1¯σμχ1+gX2X3μχ†2¯σμχ2 (2.8) = −gX2X3μ¯ψ1γμ(1−γ52)ψ1+gX2X3μ¯ψ2γμ(1−γ52)ψ2.

### 2.3 Kinetic Mixing

Finally, describes the mixing between the and SM gauge groups [67, 18] via dimension-5 (dim-5) operators:

 Lmix=2∑i=11ΛiΔaiXμνaYμν∼2∑i=1⟨Δai⟩ΛiXμνaYμν, (2.9)

where the corresponding mixes with the SM and once obtains a VEV. In this work, we choose to be

 Lmix=−sinχ2Xμν3Yμν−sinχ′2Xμν1Yμν, (2.10)

which implies and mixes with SM neutral gauge bosons at tree level. The reason why we include in the mixing is to enable the excited state to decay into the DM and a photon to explain the 3.5 keV X-ray line. Moreover, we assume for simplicity and neglect the effect of in diagonalizing the gauge boson mass matrix.222It is a legitimate assumption as long as the lifetime of the excited state is less than 1 sec, thus having no influence on Big-Bang nucleosynthesis. The relevant Lagrangian, with Lorentz indices suppressed, before and after diagonalizing the mass matrix of , and reads

 L⊃(AfZfX3f)⎛⎜⎝eJEMgJZgXJX⎞⎟⎠=(AmZmX3m)R⎛⎜⎝eJEMgJZgXJX⎞⎟⎠, (2.11)

where , and are , and gauge couplings, respectively. The subscript refers to the flavor states, denotes the mass and kinetic eigenstates, and s are currents.333To be more precise, , while for a fermion . is the left- (right-)handed projection operator. are defined in Eq. (2.5) and (2.8) for the Majorana and Dirac DM, respectively. is the rotation matrix connecting the flavor and mass basis of the gauge bosons [70]:

 R=⎛⎜⎝100−cosθwtanχsinζsinθwtanχsinζ+cosζsecχsinζ−cosθwtanχcosζsinθwtanχcosζ−sinζsecχcosζ⎞⎟⎠, (2.12)

where

 tan(2ζ) = 2δX(m2X3−m2Wsec2θw)(m2X3−m2Wsec2θw)2−δ2X, δX = −m2Wsinθwtanχcos2θw. (2.13)

It is clear that if . Note that the photon does not couple to at tree-level but the interaction will be induced at loop-level. From now on, we will suppress the subscript in the gauge bosons: , and refer to the mass and kinetic eigenstates, unless otherwise stated.

## 3 Relevant annihilation cross-sections

In this section, we calculate the DM annihilation cross-sections into SM fermions and the excited state . The first process will give rise to -rays via fragmentation of quarks and final state radiation from leptons, while the second one will yield X-rays when decays back into the DM and a photon via as shown in Fig. 1.

In this work, we focus on the regime, where , such that the GC gamma-ray excess and 3.5 keV X-ray line can be realized through DM annihilations into SM particles and excited , respectively. As we shall see below, we need a large resonance enhancement in the annihilation cross-section coming from the narrow width; therefore, to a very good approximation, we only include -exchange processes in the computation.

### 3.1 Majorana DM

For Majorana DM, we have the following relevant annihilation cross-sections:  (-wave) for -ray and the DM density,  (-wave),  (-wave) for the DM density. In order to account for the X-ray line, the mass splitting between and is set to be 3.5 keV, which in turn implies that the -wave is the dominant contribution to the DM abundance computation as opposed to the -ray excess and X-ray line, which arise from -wave processes due to axial-vector interactions of .

For annihilating into SM fermions of mass via , as shown in Fig. 2, the relevant interactions are 444Again, we use the Weyl spinor notation for Majorana .

 Lχ1χ1→¯ff⊃−12(gXsecχcosζ)X3μχ†1¯σμχ1+X3μ¯fγμ(gLPL+gRPR)f, (3.1)

where

 gL = −eQfcosθwtanχcosζ+(sinθwtanχcosζ−sinζ)gcosθw(I3−sin2θwQf), gR = −eQfcosθwtanχcosζ+(sinθwtanχcosζ−sinζ)gcosθw(−sin2θwQf),

in which is the fermion electric charge and is the isospin, associated with left-handed field.

The annihilation cross-section times the relative velocity is,

 (σv)χ1χ1→¯ff=∑f(gXsecχcosζ)2√s−4m2f48πm4X3s3/2((s−m2X3)2+Γ2X3m2X3)(λσ1+λσ2−λσ3), (3.3)

where

 s = 4m2χ11−v2/4, λσ1 = s2(m4X3(g2L+g2R)+6m2χ1m2f(gL−gR)2), λσ2 = 2m4X3m2χ1m2f(5g2L−18gLgR+5g2R), λσ3 = sm2X3(4m2χ1(m2X3(g2L+g2R)+3m2f(gL−gR)2)+m2X3m2f(g2L−6gLgR+g2R)).

Note that one has to sum over all different final states as denoted by . To simplify the expression, we employ the resonant limit of on the matrix element while keeping the kinetic part intact.555We apply this simplification to annihilation cross-sections below as well. The annihilation cross-section reads, up to the second order in ,

 (σv)χ1χ1→¯ff≃∑f(gXsecχcosζ)2√s−4m2f96π((s−m2X3)2+Γ2X3m2X3)mχ1(g1v2+g2v4), (3.4)

with

 g1 = (g2L+g2R)−m2f(g2L−6gLgR+g2R)4m2χ1, g2 = 18⎛⎝(g2L+g2R)+m2f(g2L−3gLgR+g2R)m2χ1⎞⎠.

Similarly, for , which is relevant for the relic abundance computation, we have, up to the first order in ,

 (σv)¯ψ2ψ2→¯ff≃∑f(gXsecχcosζ)2√s−4m2f64π((s−m2X3)2+Γ2X3m2X3)mψ2(h1+h2v2), (3.5)

with

 s = 4m2ψ21−v2/4, h1 = (g2L+g2R)−m2f(g2L−6gLgR+g2R)4m2ψ2, h2 =

In addition, the decay width is given by, including the channels into , and SM fermions,

 ΓX3=ΓX3→χ1χ1+ΓX3→¯ψ2ψ2+∑fΓX3→¯ff, (3.6)

where

 ΓX3→χ1χ1 = Θ(mX3−2mχ1)(gXsecχcosζ)2(m2X3−4m2χ1)3/296πm2X3, ΓX3→¯ψ2ψ2 = Θ(mX3−2mψ2)(gXsecχcosζ)2(m2X3−m2ψ2)(m2X3−4m2ψ2)1/296πm2X3, ΓX3→¯ff = Θ(mX3−2mf)√m2X3−4m2fm2X3(g2L+g2R)−m2f(g2L−6gLgR+g2R)24πm2X3.

On the other hand, with subsequent decay of  (or ) into and explaining the 3.5 keV X-ray line, as shown in Fig. 3, has the cross-section

 (σv)χ1χ1→¯ψ2ψ2=(gXsecχcosζ)4√s−4m2ψ2192πm4X3s3/2((s−m2X3)2+Γ2X3m2X3)(κσ1+κσ2), (3.8)

where

 κσ1 = 6s2m2χ1m2ψ2−12sm2X3m2χ1m2ψ2, κσ2 = m4X3(2m2χ1(5m2ψ2−2s)+s(s−m2ψ2)).

For , we have to a very good approximation, up to the second order in :

 (σv)χ1χ1→¯ψ2ψ2≃(gXsecχcosζ)4√s−4m2ψ21536π((s−m2X3)2+Γ2X3m2X3)mχ1(v2(3+v2)). (3.10)

### 3.2 Dirac DM

For Dirac DM, the distinctive feature compared to the Majorana case is that all relevant processes are -wave dominated due to the vector interactions of . The annihilation cross-section of is, up to the first order in ,

 (σv)¯ψ1ψ1→¯ff≃∑f(gXsecχcosζ)2√s−4m2f64π((s−m2X3)2+Γ2X3m2X3)mψ1(ω1+ω2v2), (3.11)

with

 s = 4m2ψ11−v2/4, ω1 = (g2L+g2R)−m2f(g2L−6gLgR+g2R)4m2ψ1, ω2 =

and for we have

 (σv)¯ψ1ψ1→¯ψ2ψ2≃(gXsecχcosζ)4√s−4m2ψ2393216π((s−m2X3)2+Γ2X3m2X3)mψ1(1152+336v2+83v4). (3.12)

Note that the partial decay width into becomes

 ΓX3→¯ψ1ψ1=Θ(mX3−2mψ1)(gXsecχcosζ)2(m2X3−m2ψ1)(m2X3−4m2ψ1)1/296πm2X3 (3.13)

Furthermore, the Dirac DM will have sizable DM-nucleon interactions in the context of direct detections. The effective DM-quark interaction reads,

 L⊃−(gXsecχcosζ)(gL+gR)8m4X3¯ψ1γμψ1¯fγμf, (3.14)

where and are defined in Eq. LABEL:eq:gLgR.

## 4 Observables

Based on the DM annihilation cross-sections into the excited state and SM fermions, we now describe how to compute the flux of -rays and -rays, and will comment on the DM relic density computation.

### 4.1 X-ray

Recently, a potential signal of a monochromatic photon line from the Perseus cluster at energy around has been identified from the XMM-Newton data [16, 17]. The flux of such a monochromatic photon line at the X-ray energy is measured to be ph cm [17]. Although the source of this X-ray line signal is still unclear, the DM annihilation (or decay) into photons is a well motivated possibility [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. Considering the Perseus Mass and the distance between the Perseus cluster and the solar system Mpc, the photon-line flux from DM annihilation can be written as

 Φγγcm−2s−1=2.08×10−3×[1GeVmDM]2×D⟨σv⟩γγ10−19cm3s−1, (4.1)

where is 1 for Majorana DM and 1/2 for Dirac DM. The monochromatic annihilation cross section is the relative velocity averaged with all the DM inside the Perseus cluster. Here, we adopt the relative velocity described by the Maxwell-Boltzmann distribution [71],

 f(vrel.)=v2rel.2√πv30exp[−v2rel.4v20], (4.2)

where we take the mean value of the velocity dispersion  [71]. One can see that a DM mass requires in order to explain the X-ray signal from the Perseus cluster.

It is worthy to mention that the information of Perseus mass, which is constrained by the velocity dispersion, can substantially reduce the uncertainties arising from halo inner slope. In Ref. [39], an overall uncertainty about a factor of 5 was obtained for the DM flux predicted in Eq. (4.1).

### 4.2 Gc γ-ray

A gamma-ray excess in the GC region, found in the Fermi-LAT data, has been widely studied in the context of DM annihilation [2, 3, 4, 5, 6, 7, 10, 11, 12]. Assuming spherical symmetry, the spatial distribution of such an excess can be explained by DM annihilation in the generalized Navarro-Frenk-White (gNFW, [72, 73]) profile,

 ρ(r)=ρs(r/rs)−γ(1+r/rs)3−γ. (4.3)

To explain the gamma-ray excess, the inner slope parameter requires  [11, 74]. In this work, we adopt this value together with the local density and .

The differential diffuse gamma-ray flux along a line-of-sight (l.o.s.) at an open angle relative to the direction of the GC is given by

 dNdE=⟨σv⟩γDπm2χdNγdE∫l.o.s.dsρ2(r(s,ψ)), (4.4)

where is 8 for Majorana DM but 16 for Dirac one. The is the velocity averaged annihilation cross section at the GC. However, the mean value of the velocity dispersion in Eq. (4.2) is at the GC region [71, 75].

The is the photon energy distribution per annihilation. All possible annihilation channels are included. The branching ratio of all the possible annihilation channels can be obtained by using Eq. (3.4) and (3.11). For each annihilation channel, the corresponding is taken from the numerical PPPC4 table [76].

One has to bear in mind that the background uncertainties for the GC gamma ray excess can significantly change the DM parameter space. Therefore, in order to include the background uncertainties, we use the central values and error bars in Fig. 17 from Ref. [77], where the systematic uncertainties coming from the Galactic diffuse emission have been properly included. Following Ref. [77], the inner Galactic central region described by the Galactic longitude and latitude is

 |ℓ|≤20∘and2∘≤|b|≤20∘. (4.5)

We conclude this section with Fig. 4 where the data on -ray spectrum is taken from Ref. [77] and the photon spectra are calculated using our best-fit points in both the Majorana (solid red line) and Dirac cases (dashed blue line), for which we include the -ray and X-ray data into fitting. One can see the GC -ray excess, a distinctive bump around a few GeV, can be well explained by DM annihilations into the SM fermions, which then fragment into photons. The continuous photon spectrum mainly comes from the decay of neutral pions, which are originated from the fragmentation of the quarks in the annihilation of the dark matter. In addition, the quarks can also fragment into charged pions, which subsequently decay into muons and eventually electrons. The dark matter can also directly annihilate into taus, muons, and electrons. The taus and muons will eventually decay into electrons. Although all these electrons undergo the inverse Compton scattering and bremsstrahlung, which can only give rise to photons at the lower photon energy, it does not effect the region of  [78]. As a result, we do not consider inverse Compton scattering and bremsstrahlung in this study.

### 4.3 DM relic abundance

The DM relic density can be obtained by solving the Boltzmann equation for the DM density evolution with the thermally-averaged annihilation cross-section into SM fermions. In this work, we assume that the thermal relic scenario such that the current relic density is determined by the DM annihilation and coannihilation of the excited state, and the number densities of these particles follow the Boltzmann distribution before freeze-out. Note that, in the context of the relic density calculation, one cannot simply assume , that is only valid in the -ray and -ray flux computation. Instead, one has to properly take into account the thermal average effect. Following Ref. [79], we compute the thermal relic density from the thermally averaged annihilation cross-section based on Eqs. (3.3), (3.5) and (3.11). However, the effective relativistic degrees of freedom are taken from the default numerical table of DarkSUSY  [80]. Also, we use the PLANCK result of  [81] together with the theoretical error to constrain the relic density.

A comment on the DM density computation is in order here. Due to a small mass splitting of 3.5 keV between the DM particle and excited state to account for the X-ray line, coannihilation processes involving the excited state have to be taken into account. As mentioned above, we focus on the scenario with the resonance enhancement via the exchange. As a result, the only relevant interactions are the DM annihilation and excited state annihilation into SM fermions. For the Majorana DM case, the dominant contribution to relic abundance comes from the excited state annihilation, , which is dominated by -wave due to the Dirac nature of , while is -wave suppressed because of being Majorana. Furthermore, the large resonance enhancement in the process required to explain the -ray line at current time () is no longer the case at the time of freeze-out, because during the freeze-out the relative velocity is much larger of order such that the annihilation deviates considerably away from the resonance region. Therefore, at freeze-out is much smaller than the current annihilation cross-section cmsec, which is the right size to accommodate the GC -ray excess.666One might think that -wave dominated interactions will have the same cross-section at freeze-out as the current one while -wave dominated ones have the larger cross-section at freeze-out due to the larger DM velocity () compared to the current velocity (  c). However, it is not always true, especially when the resonance enhancement takes place as we shall see later. Hence, alone cannot give rise to the correct relic density, which roughly requires an annihilation cross section of cmsec. For the Majorana DM case, this problem can be circumvented by the -wave process , which can give an annihilation cross section of order cmsec to explain the relic abundance.

In the Dirac DM case, however, both and are Dirac particles and all processes are -wave dominated. In the context of the DM density, annihilations of and those of contribute almost equally due to the nearly degenerate mass spectrum. However, the annihilation cross sections of both processes are much smaller than cmsec at freeze-out due to the deviation from the resonance region as explained above. Therefore, one has to involve an additional DM annihilation mechanism to reduce the relic abundance. A possible solution, for instance, is to embed into a larger multiplet such as such that coannihilations between and via is possible to bring down the relic density. As long as the mass difference between and is much larger than 3.5 keV, cannot be generated currently and thus the existence of is irrelevant to the X-ray line and -ray excess.

We summarize the discussion here with Table 3, where we show the cross-sections in orders of magnitude at the time of freeze-out and the current time. It is clear that only Majorana DM can accommodate the correct relic density due to the dominant contribution from the S-wave process .