Non-abelian Dark Matter Solutions for Galactic Gamma-ray Excess and Perseus 3.5 keV X-ray Line

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 $SU(2)$ DM models, in which the dark matter and an excited state comprise a"dark"$SU(2)$ 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 $\gamma$-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.


Introduction 2 Non-abelian Dark Matter Models
For the nonabelian DM model, we employ a "dark" SU (2) X gauge group with kinetic mixing with the SM gauge groups proposed in Refs. [13,60]. We start with a SU (2) X doublet, which is comprised of the fields for the DM particle and an excited state. In the following we will discuss two cases: (i) Majorana DM (χ 1 ) with the Dirac excited state (ψ 2 ) and (ii) Dirac DM (ψ 1 ) with the Dirac excited state (ψ 2 ). 1 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 γ-ray flux but a small X-ray one, in contradiction to the γ-ray and X-ray data Therefore, we will not discuss these two scenarios in this work. The Lagrangian of the model reads, where L SM is the SM Lagrangian. L DM 1,2 correspond to the DM sector, including the DM doublet and the dark SU (2) gauge bosons, X a (a = (1, 2, 3)), and dark Higgs triplets/doublets, which are used to provide masses to χs and Xs: where D X µ is the covariant derivative of SU (2) X and ∆ 1,2 are SU (2) triplets, whose vacuum expectation values (VEVs) provide masses to dark gauge bosons. Note that one can play with the structure of ∆ i to give different masses to X a . For example, with ∆ 2 = (0, v, 0) T in the isospin basis (the first component has the highest isospin I X 3 = 1, the second with I X 3 = 0, and so on) X 1,2 are massive but X 3 remains massless.

Majorana DM
In the case of Majorana DM, the L DM 2 takes the form where the two-component Weyl spinor notation is employed. Here "·" refers to the SU (2)invariant multiplication. χ is an SU (2) X doublet, consisting of two Weyl spinors, χ 1 and χ 2 : χ = (χ 2 χ 1 ) T . In addition, h D is an SU (2) X scalar doublet.χ 2 is a singlet under SU (2) X , which will be paired up with χ 2 to form a Dirac fermion. The conversion between Dirac-and Weyl-spinors for χ 1 , χ 2 andχ 2 is: (2.4) The corresponding X 3 -current in the Weyl and Dirac-spinor notation is given by where the pre-factors ±1/2 come from the fact that χ 2(1) has SU (2) X isospin 1/2 (−1/2). In order to give a Majorana mass to χ 1 , one can make use of the lowest isospin (I X 3 = −1) component of ∆ 1 , leaving VEVs of other components vanishing, i.e., ∆ 1 = (0, 0, v −1 ) T in the isospin basis. The χ 1 mass becomes λ ∆ v −1 . Similarly, with the lower isospin (I 3 = −1/2) of h D , the Dirac mass of χ 2 andχ 2 becomes λ h v −1/2 , where v −1/2 is the VEV of the component of I X 3 = −1/2. Moreover, X a 's masses, at phenomenological level, are considered independent since as mentioned above one can always use ∆ 2 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 ∆ 1,2 and h D 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.

Dirac DM
In the case of Dirac DM, the L DM 2 takes the form gives a Dirac mass to χ 1 andχ 1 (χ 2 andχ 2 ). We list the particle content and quantum numbers in Table 2. The conversion between Diracand Weyl-spinors for χ 1 , χ 2 andχ 2 is: and the corresponding X 3 -current in the Weyl and Dirac-spinor notation is

Kinetic Mixing
Finally, L mix describes the mixing between the SU (2) X and SM gauge groups [13,60] via dimension-5 (dim-5) operators: where the corresponding X a mixes with the SM γ and Z once ∆ a i obtains a VEV. In this work, we choose L mix to be which implies X 1 and X 3 mixes with SM neutral gauge bosons at tree level. The reason why we include X 1 in the mixing is to enable the excited state ψ 2 to decay into the DM and a photon to explain the 3.5 keV X-ray line. Moreover, we assume sin χ sin χ for simplicity and neglect the effect of sin χ in diagonalizing the gauge boson mass matrix. 2 The relevant Lagrangian, with Lorentz indices suppressed, before and after diagonalizing the mass matrix of γ, Z and X 3 reads where e, g and g X are U (1) EM , SU (2) L and SU (2) X gauge couplings, respectively. The subscript f refers to the flavor states, m denotes the mass and kinetic eigenstates, and Js are currents. 3 R is the rotation matrix connecting the flavor and mass basis of the gauge bosons [63]: It is clear that R = 1 3×3 if sin χ = 0. Note that the photon does not couple to J X at treelevel but the interaction will be induced at loop-level. From now on, we will suppress the subscript m in the gauge bosons: A, Z and X 3 refer to the mass and kinetic eigenstates, unless otherwise stated.

Relevant annihilation cross-sections
In this section, we calculate the DM annihilation cross-sections into SM fermions and the excited state ψ 2 . The first process will give rise to γ-rays via fragmentation of quarks and final state radiation from leptons, while the second one will yield X-rays when ψ 2 decays back into the DM and a photon via sin χ X µν 1 Y µν as shown in Fig. 1. In this work, we focus on the regime, where m X a > m DM , 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 χ 2 , respectively. As we shall see below, we need a large resonance enhancement in the annihilation cross-section coming from the X 3 narrow width; therefore, to a very good approximation, we only include X 3 -exchange processes in the computation. Figure 1: χ 2 decays into χ 1 and a photon through a dim-5 operator, X µν 1 Y µν . Figure 2: χ 1 χ 1 annihilate into SM particles that fragment into photons, which are responsible for GC gamma rays.

Majorana DM
For Majorana DM, we have the following relevant annihilation cross-sections: χ 1 χ 1 →f f (Pwave) for γ-ray and the DM density, χ 1 χ 1 →ψ 2 ψ 2 (P -wave),ψ 2 ψ 2 →f f (S-wave) for the DM density. In order to account for the X-ray line, the mass splitting between m χ 1 and m ψ 2 is set to be 3.5 keV, which in turn implies that the S-waveψ 2 ψ 2 →f f is the dominant contribution to the DM abundance computation as opposed to the γ-ray excess and X-ray line, which arise from P -wave processes due to axial-vector interactions of χ 1 . For χ 1 annihilating into SM fermions f of mass m f via X 3 , as shown in Fig. 2, the relevant interactions are 4 where g L = −eQ f cos θ w tan χ cos ζ + (sin θ w tan χ cos ζ − sin ζ) g cos θ w in which Q f is the fermion electric charge and I 3 is the isospin, associated with left-handed field. The annihilation cross-section times the relative velocity v is, where s = 4m 2 Note that one has to sum over all different final states as denoted by f . In the limit of the resonance region m χ 1 = 1 2 m X 3 and v 1, the annihilation cross-section becomes Similarly, forψ 2 ψ 2 →f f , which is relevant for the relic abundance computation, we have In addition, the X 3 decay width Γ X 3 is given by, including the channels into χ 1 , ψ 2 and SM fermions, Figure 3: χ 1 χ 1 annihilation intoψ 2 ψ 2 responsible for the X-ray line. where On the other hand, χ 1 χ 1 →ψ 2 ψ 2 with subsequent decay of ψ 2 (orψ 2 ) into χ 1 and γ explaining the 3.5 keV X-ray line, as shown in Fig. 3, has the cross-section (3.9) In the limit of the resonance region, m χ 1 = 1 2 m X 3 m ψ 2 and v 1, we have to a very good approximation: (3.10)

Dirac DM
For Dirac DM, the distinctive feature compared to the Majorana case is that all relevant processes are S-wave dominated due to the vector interactions of ψ 1 . The annihilation crosssection ofψ 1 ψ 1 →f f in the limit of the resonance region m χ 1 = 1 2 m X 3 and v 1 is Note that the X 3 partial decay width intoψ 1 ψ 1 becomes Furthermore, the Dirac DM ψ 1 will have sizable DM-nucleon interactions in the context of direct detections. The effective DM-quark interaction reads, where g L and g R are defined in Eq. 3.2.

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

X-ray
Recently, a potential signal of a monochromatic photon line from the Perseus cluster at energy around 3.56 keV has been identified from the XMM-Newton data [11,12]. The flux of such a monochromatic photon line at the X-ray energy E γ = 3.56 keV is measured to be Φ γγ = 5.2 +3.70 −2.13 × 10 −5 ph cm −2 s −1 [12]. Although the source of this X-ray line signal is still unclear, the DM annihilation (or decay) into photons is a well motivated possibility . Considering the Perseus Mass 1.49 × 10 14 M and the distance between the Perseus cluster and the solar system 78 Mpc, the photon-line flux from DM annihilation can be written as where D is 1 for Majorana DM and 1/2 for Dirac DM. The monochromatic annihilation cross section σv γγ is the relative velocity averaged with all the DM inside the Perseus cluster. Here, we adopt the relative velocity v rel. described by the Maxwell-Boltzmann distribution [64], where we take velocity dispersion v 0 ∼ 10 −3 c. One can see that a DM mass m DM ∼ 10 GeV requires σv γγ ∼ 2.5 × 10 −19 cm 3 s −1 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. [33], an overall uncertainty about a factor of 5 was obtained for the DM flux predicted in Eq. (4.1).

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][8][9][10]. Assuming spherical symmetry, the spatial distribution of such an excess can be explained by DM annihilation in the generalized Navarro-Frenk-White (gNFW, [65,66]) profile, To explain the gamma-ray excess, the inner slope γ parameter requires γ = 1.2 [9,67]. In this work, we adopt this value together with the local density ρ(8.5 kpc) = 0.4 GeV/cm 3 and r s = 20 kpc. 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 where D is 8 for Majorana DM but 16 for Dirac one. The σv γ is the velocity averaged annihilation cross section at the GC. However, the velocity dispersion v 0 in Eq. (4.2) is ∼ 10 −4 c at the GC region. The dNγ dE is the photon energy distribution per annihilation, obtained by using PPPC4 numerical table [68].
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. [69], where the systematic uncertainties coming from the Galactic diffuse emission have been properly included. Following Ref. [69], the inner Galactic central region described by the Galactic longitude l and latitude b is (4.5) Figure 4: GC γ-ray excess spectrum taken from Ref. [69]. We also show the corresponding photon spectra obtained for the Majorana (solid red line) and Dirac (dashed blue line) case.
We conclude this section with Fig. 4 where the data on γ-ray spectrum is taken from Ref. [69] 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 γ-ray and X-ray data into fitting. One can see the GC γ-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. 5

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. We compute the thermal relic density based on the formulas presented in Ref. [70]. However, the effective relativistic degrees of freedom are taken from the default numerical table of DarkSUSY [71]. Also, we use the PLANCK result of Ωh 2 = 0.120 [72] together with the 10% 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 X 3 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, ψ 2ψ2 →f f , which is dominated by S-wave due to the Dirac nature of ψ 2 , while χ 1 χ 1 →f f is P -wave suppressed because of χ 1 being Majorana. Furthermore, the large resonance enhancement in the process χ 1 χ 1 →f f required to explain the X-ray line at current time (v ∼ 10 −3 c) is no longer the case at the time of freeze-out, because during the freeze-out the relative velocity is much larger of order ∼ 1 3 c such that the annihilation deviates considerably away from the resonance region. Therefore, σv (χ 1 χ 1 →f f ) at freeze-out is much smaller than the current annihilation crosssection 10 −26 cm 3 sec −1 , which is the right size to accommodate the GC γ-ray excess. 6 Hence, χ 1 χ 1 →f f alone cannot give rise to the correct relic density, which roughly requires an annihilation cross section of 3 × 10 −26 cm 3 sec −1 . For the Majorana DM case, this problem can be circumvented by the S-wave process ψ 2ψ2 →f f , which can give an annihilation cross section of order 10 −26 cm 3 sec −1 to explain the relic abundance.
In the Dirac DM case, however, both ψ 1 and ψ 2 are Dirac particles and all processes are S-wave dominated. In the context of the DM density, annihilations of ψ 1 and those of ψ 2 contribute almost equally due to the nearly degenerate mass spectrum. However, the annihilation cross sections of both processes are much smaller than 10 −26 cm 3 sec −1 at freezeout 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 (χ 2 , χ 1 ) T into a larger multiplet such as (χ 4 , χ 2 , χ 1 , χ 3 ) T such that coannihilations between χ 1 and χ 3 via X 1,2 is possible to bring down the relic density. As long as the mass difference between m χ 3 and m χ 1 is much larger than 3.5 keV, χ 3 cannot be generated currently and thus the existence of χ 3 is irrelevant to the X-ray line and γ-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 processψ 2 ψ 2 →f f .

Results
In order to employ the resonance enhancement, we rewrite m X 3 = (2 − δ)m DM with m DM = m χ 1 (m ψ 1 ) in the Majorana (Dirac) DM case. Therefore, at phenomenological level we choose m χ 1 , g X , δ and sin χ as 4 independent input parameters to investigate if the proposed nonabelian DM models can simultaneously account for the GC γ-ray excess and the 3.5 keV X-ray line, and thermally reproduce the correct relic abundance. Before moving into the numerical analysis, we would like to comment on the region of interest for sin χ. For illustration, we choose the Majorana DM case but the Dirac DM case exhibits the same feature. As mentioned above, we aim for σv χ 1 χ 1 →ψ 2 ψ 2 ∼ 10 −19 cm 3 sec −1 to explain the 3.5 keV X-ray line and σv χ 1 χ 1 →f f ∼ 10 −26 cm 3 sec −1 to realize the GC γ-ray excess. On the other hand, we have from Eq. (3.4) and (3.10) in the limit of m X 3 2m χ 1 , v 1 and m f 0, where we have suppressed the kinematics factors and the coupling constant g X , which do not affect the argument. It is clear that in order to achieve σv X-ray / σv γ ∼ 10 7 , one must have sin 2 χ smaller than 10 −7 . It implies that in the denominator of the cross-section, In other words, we saturate the resonance enhancement since the DM velocity becomes dominant in the denominator and any further decrease in sin χ will not affect the cross-section. In Fig. 5, we can clearly see that for both the Majorana and Dirac case, sin χ is located in the saturated area, i.e., sin χ v. Furthermore, the reasons why the required mixing is so small, sin 2 χ 10 −7 , are because first, σv χ 1 χ 1 →ψ 2 ψ 2 has a large kinematical suppression factor, s − 4m 2 ψ 2 , compared to σv χ 1 χ 1 →f f and second, the v for X-rays in the Perseus cluster is larger than the v for γ-rays in the GC. Subsequently, σv χ 1 χ 1 →ψ 2 ψ 2 is much smaller than σv χ 1 χ 1 →f f with sin χ ∼ 1. It indicates that one actually needs sin 2 χ 10 −7 in order to fulfill σv X-ray / σv γ ∼ 10 7 .
For the fitting procedure, we make use of the minimum chi-squared method. Since the likelihoods for the relic density, X-ray line, and GC γ-rays data are well Gaussian-distributed, and the 2σ and 3σ confidence limits in two-dimensional contour plots correspond to δχ 2 = 5.99 and δχ 2 = 11.83, respectively.

Majorana case
In this section, we present the results in the Majorana DM case with the Majorana DM χ 1 and the Dirac excited state ψ 2 . Throughout this (and also next) section, the way we present the results is to project confidence regions into planes of parameters or observables. 7 In the figures, inside the legend: "GC+Perseus" means that the confidence regions are obtained from the fit with only the GC γ-ray and X-ray data, while "GC+Perseus+Ωh 2 " indicates that the DM relic density is also included in the fitting in addition to the γ-and X-ray data. In Fig. 6, we show the confidence regions in terms of the DM annihilation cross-section for the X-ray and γ-ray versus the DM mass. In the upper panels, we include the GC γ-ray excess and X-ray line only while the DM density is also included in the lower panels. The green (blue) area corresponds to the 2σ (3σ) confidence region while the red star represents  We show on the left panels: σv X−ray versus m χ 1 for X-rays, and on the right panels: σv γ versus m χ 1 for γ-rays.
the best-fit point. Furthermore, the unitarity bound [73] denoted by the red line comes from We would like to make the following comments.
• The γ-ray spectrum coming from χ 1 χ 1 →f f certainly depends on the final states. In this model, the final states include both quarks and leptons, and the final state composition is fixed according to Eq. (3.1) and (3.2). In general, a quark final state demands a higher DM mass due to soft photon spectra compared to a leptonic one. Therefore, m χ 1 will lie between that of the purely b-quark case and that of purely τ case.
• For X-ray plots (left panels), there exists a sharp cut-off close to the best-fit point on the top of the confidence regions. It is due to the perturbative limit: g X ≤ 4π as we shall see later the best-fit point has g X quite close to 4π. In contrast, the best-fit points for γ-ray plots (right panels) are located near the central area of the confidence region, which comes from the fact the σv γ can be enlarged by increasing the mixing sin χ between the SM and dark sector without varying g X as shown in Fig. 5 while σv X-ray is insensitive to sin χ in the saturated area.
• All plots exhibit a sharp cut-off on 2σ regions especially on the left-hand side. It is due to the fact that GC γ-ray bump shown in Fig. 4 has a sharp drop around 0.5 GeV compared to the milder change on the right hand side around 20 GeV. Consequently, the bump of the predicted photon spectrum will not coincide with that of the GC excess, leading to a surge in chi-square, once m χ 1 becomes much smaller than the best-fit value.
• As explained in Section 4.3, the Majorana DM case can accommodate the correct DM density with the S-wave process ψ 2 ψ 2 →f f being the main contribution. Including the DM relic constraint reduces the confidence region significantly; only in the region of 25 m χ 1 40 GeV can the model yield the correct DM density.
In Fig. 7, we show g X versus m χ 1 in the upper panels and sin χ versus m χ 1 in the lower panels. For the left panels, only the γ-ray and X-ray data are included in the fits while the DM density is also included in the right panels. Note that both χ 1 χ 1 →f f responsible for the GC γ-ray excess and χ 1 χ 1 →ψ 2 ψ 2 for the X-ray line are P-wave suppressed by the small DM velocity in the current Universe. To compensate for the velocity suppression, one needs large g X in addition to the resonance enhancement to realize the very large σv , which is proportional to g 4 X , for the X-ray line. It turns out that g X is close to the perturbativity limit 4π for the best-fit point. In contrast, as we shall see later, the Dirac case features S-wave dominated cross-sections, i.e., without the velocity suppression, where g X can be much smaller (∼ 1). The mixing between the SM and the dark sector is roughly of order 10 −7 but with a dip for m χ 1 = 1 2 m Z . It comes from large ζ defined in Eq.  Figure 7: The 2-and 3-σ confidence-level regions in the planes of (m χ 1 , g X ) (upper panels) and (m χ 1 , sin χ) (lower panels) obtained in the fits with (i) GC gamma-ray and Perseus X-ray data (left panels) and (ii) also the DM relic density (right panels) for the Majorana DM case.
2m χ 1 ) m Z , leading to large g L,R (∼ sin ζ) defined in Eq. (3.2) and large σv γ . On the other hand, σv X-ray (∼ cos ζ) does not change dramatically for m χ 1 = 1 2 m Z . So as to maintain σv X-ray / σv γ ∼ 10 7 , smaller sin χ is needed to suppress the γ-ray flux with respect to the X-ray one. Note that when m X 3 ∼ m Z , the electroweak precision data put a stringent bound on the SM-dark sector mixing, sin χ 5 × 10 −3 [74], which is however too weak to constrain any relevant parameter space of the model under consideration. We conclude this section with Fig. 8, in which we project the confidence regions, including the X-ray and γ-ray data only, into the DM relic density and m χ 1 plane. It is clear that only for 25 ≤ m χ 1 ≤ 40 GeV, the correct DM density can be reproduced. Furthermore, the bestfit point corresponds to the slightly lower relic density, which results in a minor shift in the best-fit point when the DM relic density is included into the fit, as can be seen from Fig. 6 and 7.

Dirac case
Here we show the results of the Dirac DM case. As argued in Sec. 4.3, all relevant processes responsible for the X-ray line, GC γ-ray excess and DM relic density are S-wave dominated. Moreover, the large cross-section needed to account for the X-ray line requires the large resonance enhancement with the help of the very small X 3 decay width. The large DM velocity (∼ 1 3 c) at freeze-out implies a considerable deviation from the resonance when DM decouples from the thermal universe. In order to achieve σv γ ∼ 10 −26 cm 3 sec −1 at current time, the cross-section ofψ 1 ψ 1 →f f at freeze-out will be much smaller than 3 × 10 −26 cm 3 sec −1 , the size required to reproduce the DM density. Therefore, we do not include the DM relic density constraint into the fits here. Notice that we have to take into account the stringent bounds on the spin-independent DM-nucleon cross-section [75] due to vector-current interactions, but it hardly has any impact on the analysis since sin χ of interest is extremely small, leading to a large suppression on the DM-nucleon cross-section. In Fig. 9, we show σv versus m ψ 1 for X-rays (left panel) and γ-rays (right panel). Unlike the Majorana case in Fig. 6, the best-fit point in σv X-ray is near the central area of the confidence region since g X is much smaller than that of the Majorana case. The steep shrink on the 2σ confidence region, which is observed at the Majorana case as well, around m ψ 1 ≈ 29 GeV is again due to the sharp change on the GC γ-ray spectrum around 0.5 GeV shown in Fig. 4.
In Fig. 10, we show g X , sin χ and δ versus m ψ 1 , respectively. All processes of interest are S-wave dominated without the DM velocity suppression. It implies that the resonance enhancement from the narrow X 3 decay width alone is sufficient to achieve the large σv X-ray without resorting to large g X . Therefore, g X is of O(0.6) in this case, compared to g X ∼ 10 for Majorana χ 1 . Similar to the Majorana DM case explained above, around m ψ 1 ( m X 3 ) ≈ 1 2 m Z , ζ defined in Eq. (2.13) becomes large, resulting in large g L,R (∼ sin ζ) and σv γ . It is then offset by the decrease in sin χ as shown in the left panel of Fig. 10. At the same time, from Eq. (3.11), large ζ implies that σv X-ray (∼ (g X cos ζ) 4 ) becomes smaller. So g X has to increase to achieve σv X-ray ∼ 10 −19 cm 3 sec −1 for the X-ray line, as seen from the middle panel. In addition, larger g X implies larger σv γ , which can be reduced by larger δ (and a larger deviation away from the resonance region) as in the right-panel. 8 Note that for the Majorana case, we do not spot this behavior for g X and δ since g X is constrained to be less than 4π.   To summarize, we present in Table 4 the chi-squares (χ 2 ) and p-values for the best-fit points, and also the 3σ confidence regions for both the Majorana and Dirac DM cases. We would like to emphasize again that first, both cases can explain the γ-ray and X-ray data but only the Majorana DM can reproduce the correct DM density. Second, the best-fit point shifts toward the lower m χ 1 region when including the DM relic density into the fits. Finally, g X is close to 4π in the Majorana case because of P -wave velocity suppression as opposed to the Dirac case where g X is of O(0.6).

Conclusions
In this work, we have attempted to explain simultaneously the GC γ-ray excess and the 3.5 keV X-ray line, as well as fulfilling the DM relic abundance in the context of non-abelian DM models, and we have success in the case of Majorana DM with a Dirac excited state. We employed a "dark" SU (2) X gauge group with a SU (2) X doublet consisting of the DM particle and the excited state with a mass-splitting of 3.5 keV. The SU (2) X sector talks to the SM gauge groups via kinetic mixing, characterized by sin χ. We have studied two cases: Majorana DM (χ 1 ) and Dirac DM (ψ 1 ), and both with the Dirac excited state ψ 2 . The X-ray line results from χ 1 χ 1 (ψ 1 ψ 1 ) →ψ 2 ψ 2 , followed by the decay of ψ 2 back to the DM and a photon. On the other hand, DM annihilations into SM fermions, which then emit photons, can explain the GC γ-ray excess but this process is suppressed by the aforementioned SU (2) X -SM kinetic mixing. In order to account for the γ-and X-ray data, one would need σv X-ray ∼ 10 −19 and σv γ ∼ 10 −26 cm 3 sec −1 . We employ the resonance enhancement to fulfill the large σv X-ray . Additionally, the large hierarchy between two cross-sections σv X-ray / σv γ (∼ 10 7 ) can be realized if the kinetic mixing is very small (sin χ 10 −7 ).
For the Majorana DM case, both the γ-ray and X-ray excess can be accommodated really well. However, all χ 1 -involved processes are P -wave suppressed due to the Majorana nature. As a result, the SU (2) X gauge coupling g X is driven close to the perturbativity limit 4π to counterbalance the velocity suppression. Regarding the DM relic density, the relatively large DM velocity (∼ 1 3 c) at freeze-out compared to the current one (∼ 10 −3 c) implies a large deviation from the resonance region. Therefore, the cross-section of χ 1 χ 1 →f f at freeze-out is much smaller than the current value 10 −26 cm 3 sec −1 demanded to explain the GC γ-ray excess, as well as much smaller than 3×10 −26 cm 3 sec −1 , the size of the cross section to achieve the correct relic density. The solution comes from the S-wave dominated processψ 2 ψ 2 →f f , which should be included as an coannihilation process at the freeze-out in light of the tiny mass splitting between χ 1 and ψ 2 . The coannihilation can reach a level of 3 × 10 −26 cm 3 sec −1 for certain m ψ 2 to reproduce the correct relic density. The allowed m χ 1 (also m ψ 2 ) ranges from 25 to 40 GeV. This coannihilation cross section is much larger than the cross section of the P -wave process χ 1 χ 1 →f f .
In the Dirac DM case, the model can explain both γ-and X-ray data for 16 m ψ 1 56 GeV, with much smaller g X (∼ 0.6) compared to the Majorana DM case, since all processes involved are S-wave dominated without the velocity suppression. Nevertheless, it cannot yield the proper DM relic density because both ψ 1 and ψ 2 annihilations into SM fermions are of the same order at freeze-out and both annihilations are away from the resonance region due to the large DM velocity. As a consequence, they are much smaller than 10 −26 cm 3 sec −1 , the current value forψ 1 ψ 1 →f f , associated with the GC γ-ray. An additional mechanism has to be introduced to increase the annihilation cross-section and lower the relic density. In addition, the direct search bounds hardly constrain our model since the SU (2) X -SM mixing sin χ is very small such that the DM-nucleon cross-section is negligible.
Finally, we would like to point out that we have taken a phenomenological approach, without justifying the smallness of the mass splitting between the DM and the excited state, as well as and the tiny kinetic mixing. Both are basically determined based on the γ-and X-ray data. The concrete model building is beyond the scope of this work.  Figure 11: The cross-section for the purely Majorana case (blue) and Dirac case (red). We assume a 3.5 keV mass splitting between DM and the excited state, Γ X = m X 10 = 10 GeV and v ∼ 3 × 10 −3 c.
resonance. In contrast, in the Dirac case the numerator is unsuppressed, which is characteristic of S-wave. In Fig. 11, we show the comparison between the two cases, in which we assume a 3.5 keV mass splitting between the DM and the excited state, Γ X = m X 10 = 10 GeV, and v ∼ 3×10 −3 c. It is clear that the pure Majorana case has the resonance "cancellation" instead of enhancement while the S-wave Dirac case features the resonance behavior as expected.