Muonphilic Dark Matter explanation of gamma-ray galactic center excess: a comprehensive analysis

The Galactic center gamma-ray excess (GCE) is a long-standing unsolved problem. One of candidate solutions, the dark matter (DM) annihilation, has been recently tested with other astrophysical observations, such as AMS-02 electron-positron spectra, Fermi Dwarf spheroidal galaxies gamma-ray data, and so on. By assuming that the DM particles annihilate purely into a normal charged fermion pair, Di Mauro and Winkle (2021) claimed that only a muon-pair is compatible with the null detection of all the corresponding astrophysical measurements and can explain GCE simultaneously. On the other hand, a muonphilic DM model may also lead to a signal in the recent Fermilab muon $g-2$ measurement or be constrained by the latest PandaX-4T limit. In this work, we comprehensively study interactions between DM and muon, including various combinations of DM and mediator spins. In agreement with GCE (not only $2\mu$ but also $4\mu$ final states), we test these interactions against all the thermal DM constraints. Our results show that only the parameter space near the resonance region of mediator can explain GCE and relic density simultaneously, and larger parameter spaces are still allowed if other poorly-known systematic uncertainties are included. Regardless of the DM spin, only the interactions with the spin-0 mediator can explain the recent muon $g-2$ excess on top of GCE, relic density, and other DM and mediator constraints.


I. INTRODUCTION
Dark matter (DM) is a successful candidate to consistently explain many astrophysical and cosmological problems. Except for those known gravitational DM evidence, we are still seeking for any non-gravitational interaction between DM and the visible matter in order to pin-down the DM particle nature by means of collider experiments [1,2], DM direct detection (DD) [3,4] and indirect detection (ID) [5][6][7]. Among those non-gravitational detection, the Galactic center gamma-ray excess (GCE) reveals a possibility that the DM annihilation with the mass around 30 − 70 GeV in the Galactic center can well fit the shapes of the energy and spatial spectra [8][9][10][11][12]. However, the origin of this GCE has been a long-standing controversy. One possible astrophysical explanation is that some undetected millisecond pulsars are concentrated near the center of the Milky Way and in the bulge throughout the Galaxy [13][14][15][16][17]. Another explanation involves an outburst of leptons or hadrons accelerated by the supermassive black hole, known as the Sgr A * [18][19][20].
The systematic uncertainties of these GCE analyses are still unclear. It can be a challenge to discover or exclude the DM origin by only using GCE Fermi data. A strategy to test the DM origin is to cross-check against other astrophysical data, such as Fermi-LAT observations of dwarf spheroidal galaxies (dSphs; [21]) and AMS-02 cosmic-ray data [7]. Once all the above data do not support DM annihilation, we may abandon the DM explanation of GCE.
Motivated by such a consideration, a recent work [22] has performed a combined analysis by taking the γ-ray data of 48 dSphs and the latest AMS-02 positron and antiproton data into account. They focus on the DM annihilation to a pair of the Standard Model charged fermion f final state, namely DM + DM → f +f . When the final state particle mass is heavier than 80 GeV (for instance ZZ, W W , HH, and tt), the DM annihilation cannot generate a well-fit spectrum to GCE data. Although the pure hadronic or some mixture between leptonic and hadronic final states of DM annihilation can fit GCE data, their required vertical half-height of the diffusive zone for the AMS-02 antiproton data (L 2 kpc) are in tension with the fitted value of the radioactive cosmic ray and radio data (L = 4.1 +1.3 −0.8 kpc) [23] at a confidence level of ∼ 2 − 3σ. On the other hand, the final state e + e − of DM annihilation would be disfavoured by the AMS-02 e + data where the authors adopted semi-analytic propagation equation and the uncertainties from L and DM halo profiles were addressed. Finally, Ref. [22] claims that DM annihilation into the muon final state (called muonphilic DM hereafter) can reasonably explain GCE without violating other astrophysical constraints 1 . However, the astrophysical systematic uncertainties (both from propagation equation and unknown sources) involved in [22] could have been underestimated, see e.g. [10]. Thus, to further probe such a muonphilic DM model, implementing the explicit interaction terms confronting with other data are highly required.
We propose to verify muonphilic DM GCE signal by using the particle experimental constraints. There are at least four important motivations. First, the latest DM DD limit given by PandaX-4T [4] provides a severe constraint for DM and nucleon scattering. If DM would only couple to muon, it naturally generates a loop-suppressed DM-nucleon scattering cross section. Hence, it is not surprising that the muonphilic DM can explain the GCE and escape the constraint from the PandaX-4T detection. Second, the most recently reported excess of the muon g −2 measurement by the FermiLab E989 experiment is δa µ = (2.51±0.59)×10 −9 , which deviates from the standard model prediction at a confidence level of 4.2σ [24]. Although the sign of δa µ can be either positive or negative depending on the mediator (MED) nature, the combined result can restrict the parameter space of the muonphilic DM models.
Third, the relic density measurement with the thermal DM paradigm can further narrow down the parameter space. The interplay between the annihilation cross sections at the early and present time can be highly non-trivial. Conventionally, the annihilation cross section can be simply expanded by the power of relative velocity, namely σv a + bv 2 rel.
with dropping the higher order contribution. In the partial wave approach, one can define that the s-wave contribution is from a while the p-wave contribution is from bv 2 rel. . Thus, the relative velocities in the early universe for the relic density and the present universe for the GCE are very different. It is interesting to check whether the muonphilic DM explanation to the GCE is supported by the PLANCK relic density measurement. Finally, the muonphilic DM models with Z 2 -even mediators can easily escape the mono-photon and mono-jet constraints from LEP [25] and LHC [2,26] such that the electroweak scale DM is still allowed. Furthermore, the future muon colliders can be used to test these muonphilic DM models [27][28][29]. For the Z 2 -even MED, µ + µ − → µ + µ − process is powerful to directly search for the MED and the mono-γ process can be used to explore DM when a DM pair is produced form the on-shell MED. On the other hand, for the Z 2 -odd MED, the MED can 1 Note that the authors in [22] did not consider the mixture of charged fermion pairs and neutrino pairs in the final state. Since the final state neutrino does not affect the gamma-ray spectrum, this situation is somehow similar to the pure muon-pair case.
be searched for via its pair production process and the mono-γ process from t-channel DM pair production is again used to explore DM.
In this work, we comprehensively list all the possible renormalizable interactions by simply appending a DM and a MED to the standard model (SM). We restrict ourselves to only concern SM singlet DM and MED with the spins (s = 0, 1/2, 1). In total, we have 16 interaction types for Z 2 -even mediator while 7 interaction types for Z 2 -odd mediator. We will investigate all these 23 interaction types and eliminate some disfavoured ones by using a global analysis with the likelihoods from PLANCK relic density [30], Fermi GCE [31], PandaX-4T limits [32], the LEP limit [33], and δa µ [24].
The remainder of this paper is structured as follows. In Sec. II, we recap the explanation of the GCE by using the DM annihilation to 2µ scenario. Additionally, we include 4µ final state that can also mimic the signature of 2µ final state. In Sec. III, we summarize all the relevant experimental likelihoods used in our numerical work. After a comprehensive discussion of all the possible interaction types in Sec. IV, we can eliminate several disfavoured ones. In Sec. V and VI, we further evaluate the future detectability of the DD and muon g − 2 experiments, respectively. Finally, we summarize and conclude our results in Sec. VII. Some detailed formulas for calculations are included in three appendices.

II. THE MUONPHILIC DM EXPLANATION TO THE GCE
It is claimed in [22] that all hadronic and semi-hadronic annihilation channels can be excluded by the AMS-02 antiproton data, unless the height of the diffusion halo z h is smaller than 2 kpc which is however in tension with the radio data. Except for µ + µ − final state, these authors also found that DM annihilation to the leptonic channels can be ruled out by either the combined dSphs limits or AMS02 positron data. Therefore, it is concluded in [22] that the DM annihilation to a pair of muons with the mass around 60 GeV, decaying to electrons subsequently, can explain GCE via inverse Compton scattering (ICS) with starlight. The DM prompt γ emission, mainly from final state radiation, can also contribute to the gammaray fluxes at the higher energy range. The propagation of e ± , gamma-ray emission of ICS and prompt γ at the GC are summarized in Appendix A. The favoured annihilation cross sections (µ + µ − final state) and DM masses are [22] σv 2µ = 3.9 +0.5 −0.6 × 10 −26 cm 3 s −1 , and m D = 58 +11 −9 GeV.
where M is the mass of the mediator. The energy of electron and muon are E e and E µ , respectively. We can take the spectrum dNe [34,35] by using the where Θ is the Heaviside function. The maximum and minimum muon energy are Similarly, we can replace dNe (E µ , E γ ) to obtain the DM prompt gamma-ray contribution for 4µ case.  [37][38][39][40][41][42]. In addition, we take "non-cool-core" magnetic field model with the core radius of r c = 3 kpc, the central magnetic field of B 0 = 4.7 µG. For the DM density distribution, we choose Navarro-Frenk-White (NFW) profile [43,44] ρ(r) = ρ s r rs Milky Way galaxy model [47]. The average thermal electron number density is taken as n e ≈ 0.1 cm −3 [48]. We note that only b ICSL among the energy loss terms is sensitive to our conclusion.
In Fig. 1 with m D M/2, but the former is larger than the latter by a factor of 2. Thus, if requiring the same ICS gamma-ray fluxes to explain GCE, a twice higher annihilation cross section is needed, see the right panel of Fig. 1. Therefore, it will be difficult to explain the GCE and relic density measurement simultaneously in the sceanrio of DM + DM → MED + MED.

III. THE LIKELIHOODS
In this work, we mainly consider three important likelihoods. Although Fermi GCE and PLANCK relic density are based on the signal, the DM direct detection from PandaX-4T can set an upper limit on the interaction. In the below, we will present their χ 2 and the total χ 2 tot defined as the sum of individual χ 2 values of GCE, DM relic density, and DD cross section We hire emcee [49] based on Markov Chain Monte Carlo (MCMC) method to undertake the task of sampling the parameter space with the likelihood ∝ exp(−χ 2 tot /2). We use Feynrules [50] to implement the models, and then import them to MicroMEGAS [51] for DM relic density calculation. The number of samples for each model in 2σ and 3σ ranges are about 3 × 10 5 and 4.5 × 10 5 respectively.
• Fermi GCE: We accommodate the GCE reduced χ 2 as where dN dE i , dN 0 dE i and σ i are predicted gamma-ray spectra, GCE spectra extracted from Fermi-LAT data after background modeling and their errors [12]. Here, we simply ignore the orange error bars and systematic uncertainties (gray and orange bands) in Fig. 1 as well as the correlation between energy bins. Therefore, the total number of data bins used for our analysis (blue error bars in Fig. 1) are 19. The predicted gamma-ray spectrum is where the annihilation fraction BR 2µ describes the portion of the 2µ annihilation final state.
In Ref. [22], they obtained a minimum reduced χ 2 of 5.47 for the 2µ final state. This quoted value is not located at around one because some uncertainties such as the model uncertainties of the Galactic gas and the interstellar radiation field are not taken into account 3 . However, it is unable to reach a consensus about the precise uncertainties, e.g., the gray and orange bands in Fig. 1. When we ignore these poorlyknown systematic uncertainties, a minimum reduced χ 2 red of 5.15 (4.34) for the 2µ (4µ) final state are obtained. However, the fact that the reduced χ 2 is not more or less equal to 1 implies that some systematic uncertainties may be overlooked. Therefore, we adopt two statistical approaches to demonstrate our results in two colors in our plots. In the first approach (green layer in all the scatter plots), according to Ref. [52] (see Sec. 5.2 there), if the minimum reduced chi-square χ 2 red is larger but not much greater than 1, then we can enlarge our errorbars with square-root of the minimal reduced χ 2 red , namely where σ i,GCE is used in our study but σ i is the same as defined in Eq. (8) of Ref. [12].
Apparently, the new overall GCE reduced chi-squares are smaller and the new minimum value of them is exactly equal to 1. Thus, our analysis to examine the allowed parameter space is more conservative than the one directly computed by Eq. (8). In the second approach (gray layer in all the figures), we will also perform a comparison with the allowed spectra in agreement with the gray region as shown in Fig. 1. For the most conservative way, the second approach allows us to project the systematic uncertainties from propagation and source models to our particle model parameter space.
• PLANCK relic density: The DM PLANCK relic density χ 2 is described as a Gaussian distribution where µ t is predicted from the theoretical value, µ 0 is an experimental central value, and theoretical uncertainty σ theo = τ µ t . We use PLANCK 2018 data [30] to constrain our predicted relic density Ωh 2 . Their reported central value with statistical error is On the other hand, we may also need to address the uncertainties from the Boltzmann equation solver and the entropy table in the early universe. Hence, we conservatively introduce τ = 10% based on our prediction.
• PandaX-4T σ SI χp : The estimation of χ 2 for the DM-nucleus spin-independent (SI) direct detection cross section χ 2 DD is  In this section, we summarize all representative interaction types for two DM particles annihilating into a pair of muons and a pair of MEDs at tree level. By taking a Z 2 symmetry to prevent DM decay, we introduce Z 2 -even mediators for s-channel while Z 2 -odd mediators for t-channel annihilation. As presented in Table I, both DM and MED can be scalar (spin-0), fermionic (spin-1 2 ), or vector (spin-1). In this work, we will discuss self-conjugate and not self-conjugate DM fields, i.e., (i) real and complex scalar DM, (ii) Majorana and Dirac DM, and (iii) real and complex vector DM. For the sake of simplicity, we only concern a self-conjugate field for Z 2 -even mediator in this study. However, a complex field is required for Z 2 -odd mediator.

A. Z 2 -even mediator
In Table II, all representative interaction types between Z 2 -even mediator, DM, and µ are listed. Here, we use the notation as defined in Table I to present the spin nature of DM and mediator. The column σv 2µ indicates the velocity dependence of the cross section for DM annihilating to µ + µ − final state at the present time. However, σv 4µ is for the process DM + DM → MED + MED and then each mediator decays to a pair of muons successively.
The last column of Table II shows the equation number of the DM-nuclei elastic scattering cross section whose formula is given in Appendix B. We will discuss the DM direct detection in Sec. V. The sign "-" in the last column means that the cross section is negligible.
Note that some cross sections contain both s-and p-wave contributions but their a/b ratio is non-trivial. Therefore, we divide them into several cases and discuss them below.
First, for 2µ final state, we can simplify the analytical expressions of σv [53] near resonance where C 0 (in GeV −2 ) and C 1,2 are positive coefficients. The resonance parameter R is defined is to be kinematically allowed and R ≤ 2 is for a physical mass M .
If 0 < R ≤ 2, we can see σv is with the largest value at v = 0. When R approaches zero from above and the coefficient of v 2 is dominant over the first term, the total σv with a relativistic speed is smaller than s-wave sole component as the blue line in the left panel of Fig. 2. This can explain DM relic density and GCE simultaneously. When 0 R, the second term is v suppressed and the cross section is s-wave that is not able to fit both DM relic density and GCE data. On the other hand, the condition R < 0 implies that annihilating DM needs some kinetic energies to hit the resonance. Therefore, we can see the resonance with a small velocity and negative R for L 6 and L 11 as shown in Fig. 2. In these negative R regions, one can find a solution for a correct relic density and GCE by tweaking the decay width. For 2µ final state, we summarize the correlation between R and a/b as follows.
• Case (i): The sign of R and a/b is always opposite. When we enhance the value of |R|, the absolute ratio |a/b| is also increased. The value of |a| and |b| can be comparable Z 2 even mediator types Lagrangian σv 2µ σv 4µ DD a + bv 2 a + bv 2 χ and φ The columns σv 2µ and σv 4µ show the cross sections of DM annihilation to 2µ and 4µ final states.
We define As mentioned previously, these interactions can not explain GCE and DM relic density at the same time.
In the left panel of Fig. 2, we choose four benchmark interaction types (L 1 , L 3 , L 6 , and  Table II. early universe and present Milky Way. In the right panel of Fig. 2, we qualitatively show three different mediator decay widths: 10 −5 GeV (solid line), 10 −3 GeV (dashed line), and 10 −2 GeV (dotted line). We find that the peak is so sensitive to the decay width. By adjusting the resonance width and height of DM annihilation to 2µ, it is possible to make the annihilation at the present time higher than the early time.
In  Table II. Therefore, to explain both signals (GCE and relic density), the very fine-tuning parameter space, namely resonance region, is needed, regardless of s, p or s + p wave, unless the GCE systematic uncertainties have been taken into account.   The charged mediator such as slepton suffers from the stringent lower mass limit 103.5 GeV from LEP [33]. When the charged mediator mass is heavier than LEP limit, it implies that the larger coupling g D is needed to fit the DM relic density and GCE cross section. Consequently, such a considerable coupling may violate the PandaX 4T and XENON1T limit. For the Z 2 -odd mediator scenario, we perform the parameter scans in the following range 20 GeV < m D < 200 GeV, m D < M < 1000 GeV, 10 −6 < g D < 2.  Fig. 6. First, we consider the Z 2 -even mediator case and define the general lepton current as lΓ l l. Following Ref. [54], the one loop contributions are nonzero only for vector and tensor lepton currents, namely Γ l = γ µ , σ µν . Therefore, only L 5,7,11,15 can generate one loop contributions to the DM-nuclei elastic scattering [55][56][57][58] as shown in the left panel of Fig. 6.
For the scalar lepton current, Γ l = 1, the one loop contribution vanishes since a scalar current cannot couple to a vector current. The DM-quark interaction can only be induced at two loop level for L 1,3,9,13 [57,58], as depicted in the middle panel of Fig. 6.
For pseudo-scalar and axial-vector lepton currents Γ l = γ 5 , γ µ γ 5 , the diagrams vanish to all loop orders. The interaction with γ 5 gives either zero or a fully anti-symmetric tensor αβµν . Since there are only three independent momenta in the 2 → 2 scattering process, two indices can be contracted with the same momentum and a zero amplitude square is obtained. Therefore, we mark "-" in the last column of Table. II for L 2,4,6,8,10,12,14,16 . 90% upper limit [32], the blue dashed lines are expected PandaX-4T 90% upper limit [61], and the black dotted lines are neutrino floor [62]. The color coding is same as Fig. 3.
For the case of Z 2 -odd mediator, the DM-nuclei scattering cross sections are suppressed for the self-conjugate DM, namely real scalar, Majorana fermion, and real vector. As given in Ref. [54], the self-conjugate DM couples to a single photon in t-channel simplified models only through the anapole moment. This leads to that DM-quark scattering amplitude is suppressed in the non-relativistic limit as for L 19,20,21,23 . On the other hand, if the muonphilic DM are complex scalar, Dirac fermion and complex vector, the one loop induced DM-quark interactions cannot be ignored [54,59,60], and the Feynman diagram shown in the right panel of Fig. 6.
In Fig. 7, we plot 2σ distributions on the (m D , σ SI χp ) plane for five models L 7,9,11,13,15 where L 9 and L 13 are fairly similar to each other. The red thick lines are the 90% upper limits of present PandaX-4T while its future sensitivity [61] is presented by blue thick dashed lines. The muon anomalous magnetic moment a µ has been recently announced by E989 at Fermilab. By combining the new data with the previous measurement from Brookhaven National Lab (BNL) [63], they found a deviation δa µ = (2.51 ± 0.59) × 10 −9 with 4.2σ significance [24] from the value of the SM prediction.
Since the mediator couples to muon lepton, we intuitively check whether the mediator in the loop can contribute to the experimentally measured muon g − 2 excess. In the Appendix C, we provide four analytical formulas for scalar, pseudo-scalar, vector, and axialvector mediators. The contributions from pseudo-scalar and axial-vector mediator are negative at one loop level. For vector mediator, δa µ is too small to reach 2σ region. Thus, only the contributions form scalar mediators are shown in Fig. 8. When muon-muon-mediator coupling g f is around the order of ∼ 0.1, muon g − 2 excess can be explained. Therefore, as long as the E989 result can be confirmed in the near future, only L 3 (fermionic DM), L 9 (scalar DM) and L 13 (vector DM) are allowed to explain the correct DM relic density, GCE and muon g − 2 excess simultaneously.

VII. CONCLUSIONS
The GCE is a well-known anomaly and the DM annihilation explanation is one of the popular solutions. If this DM annihilation explanation is correct, the same DM mass and cross section shall be found in other astrophysical observations such as Fermi dSphs γ data and AMS-02 e + andp data. Especially, Ref. [22] has suggested that χχ → µµ annihilation can explain all the astrophysical observations consistently. Motivated by such a claim, we perform a comprehensive analysis for the muonphilic DM. We attempt to probe the muonphilic DM from the particle physics point of view. We first build 16 interaction types for Z 2 -even mediator (s-channel and contact interaction annihilation) while 7 interaction types for Z 2 -odd mediator (t-channel annihilation). We hire the measurement from Fermi GCE, PLANCK relic density, PandaX-4T σ SI χp , and LEP new charged particle search in order to pin down the favoured interaction types and their parameter space. After we find the allowed interaction types and parameter space, we discuss their predictions of muon g − 2 by comparing with the recently reported excess from the FermiLab E989 experiment.
In this work, we demonstrate our results by using two types of GCE systematic uncertainties. The first type ("normal approach") is to simply enlarge the error bars given in [22] with the square-root of the minimum reduced chi-square. This trick enables us to find out the potentially overlooked systematic uncertainties. The normal approach is presented by green color. However, we also presented the results with the second type GCE systematic uncertainties as a more conservative error band (gray band) shown in Fig. 1. The results using the second type uncertainties are called "conservative approach" and presented by gray color.
We summarize our findings for muonphilic DM models with Z 2 -even mediators. First, we find that all of 16 interaction types with Z 2 -even mediators, apart from four-point interactions, can provide correct DM relic density, an explanation to GCE but a tiny σ SI χp to DM DD. For normal approach, only the narrow phase spaces of resonances are remained to accommodate both GCE and DM relic density. Because the favoured σv for GCE is higher than the one from the relic density, both the velocity-independent cross section (s-wave) and the v 2 -dependent cross section (p-wave) cannot be the solutions. Additionally, the annihilation to 4µ final state requires a cross section σv 4µ ∼ 8 × 10 −26 cm 3 s −1 to explain GCE.
Such a cross section is higher than the one required by the correct DM relic density. Therefore, if considering normal approach, we are unable to find any solution for M < m D where DM annihilate to a pair of mediators. On the other hand, for M > m D , we still find some solutions with two mechanisms in a subtle way. The first mechanism is resonance based on the condition that the mediator is lighter than twice DM mass. By tuning the decay width, one can simply tweak the resonance position to correctly obtain the early and present DM cross sections. The second mechanism happens only for the condition 2m D > M . Under this condition, one can find some cancellation between s-wave and p-wave so that σv at the early universe time can be suppressed. Clearly, if the annihilation cross section only contains the p-wave contribution, the allowed parameter space only owes to the first resonance mechanism. However, for conservative approach, the constraints are less tight. A sizable amount of non-resonant samples appears around m D ∼ 60 GeV in interactions L 3,4,8,9,10,13,14 and the DM masses can be as higher as 200 GeV.
As for the muonphilic DM models with Z 2 -odd mediators, because a Z 2 -odd mediator must carry an electric charge in these models, the condition m D < M is held to maintain the electrically neutral universe. To explain the DM annihilation cross sections required by the GCE and PLANCK relic density measurement, it results a lighter mediator and a larger coupling. However, the PandaX-4T σ SI χp upper limit already rules out most of the interaction types due to their larger couplings required to explain GCE. Only the interaction L 21 and L 23 are survival from the PandaX-4T constraints but the required mediator masses of L 21 are still lower than the LEP new charge particle mass limit. Therefore, all of interaction types with Z 2 -odd mediators are excluded based on normal approach. For conservative approach, L 23 is still survived.
Although the muonphlic DM can only scatter with proton via loop contributions, the current PandaX-4T σ SI χp upper limit is still sensitive to them. If considering those dimensional couplings M Dφ , even the two loop contribution can be largely probed before reaching the neutrino floor. Besides L 7,9,11,13,15 , the rest interaction types are still hidden below the neutrino floor. Similarly, if muon g − 2 result from E989 can be confirmed, only the scalar mediator is allowed and the possible interaction types are L 3 (fermionic DM), L 9 (scalar DM) and L 13 (vector DM). Among these three models, only L 3 cannot be tested by future DD experiments.
Finally, we would like to comment the GeV anti-proton excess in the AMS-02 data and searches of the allowed muonphilic DM models at the future muon collider. In this work we solely follow the argument of Ref. [22] on the absence of the anti-proton excess. The antiproton created by muon DM annihilation final state can be neglected. If there was a GeV anti-proton excess as argued in [64,65], the GCE as well as the g − 2 anomaly would also shed valuable light on the possible DM origin [66]. On the other hand, the proposed 3 TeV muon collider [67] is an ideal machine to test the MED and DM for the survival parameter space in this study. The Z 2 -even (Z 2 -odd) MED can be searched for via µ + µ − → µ + µ − (MED pair production) process. The mono-γ process is used to explore DM for both Z 2even and Z 2 -odd MED models. We will return to a detailed study of the muon collider in a future work.
where ∂ne ∂E is the equilibrium electron density in the interval d 3 rdE. The diffusion coefficient and the energy loss rate are defined as D(E, r) and b(E, r), respectively. The electron source term, Q(E, r), can be expressed as Here σv is DM averaged annihilation cross section, ρ(r) is DM density profile, m D is DM mass, and dNe dEe is e ± injection spectrum per DM annihilation. If DM is not self-conjugated, σv will be replaced by σv /2.
The local emissivity of γ rays is defined as and the ICS power P ICS (E, E γ ) is given by where is the energy of target starlight, n( ) is photon number density. E and E γ are the energy of electrons/positions and upscattered photons, respectively. The ICS cross section σ(E γ , , E) can be written as which is the so-called Klein-Nishina formula. Here σ T ∼ 0.665 barn is the Thomson cross section, and G(q, λ) is given by [68] G(q, λ) = 2q ln q + (1 + 2q)(1 − q) + (2q) 2 (1 − q) 2(1 + Γq) , where Finally, the approximation of integrated flux density for ICS at energy E γ for a small region with much greater distance than size is in which D A is the angular diameter distance.
In this work, we adopted simplified power-law diffusion coefficient  Here we show all non-zero DM-nucleus SI cross sections for the simplified muonphilic DM models in Table II and III [54][55][56][57][58] : where α, m N , Z, A and m µ are fine structure constant, target nucleus's mass, target nucleus's charge, target nucleus's mass number and muon's mass respectively. The velocity of DM near the earth is v ∼ 10 −3 , reduced mass of DM-nucleus system is µ N = m N m D m N +m D , and the cut-off scale assumed to be Λ = M √ g D g f . The coefficient σ 0 = α 2 Z 2 µ 2 N π 3 A 2 M 4 .