Accurate Inverse-Compton Models Strongly Enhance Leptophilic Dark Matter Signals

The annihilation of TeV-scale leptophilic dark matter into electron-positron pairs (hereafter $e^+e^-$) will produce a sharp cutoff in the local cosmic-ray $e^+e^-$ spectrum at an energy matching the dark matter mass. At these high energies, $e^+e^-$ cool quickly due to synchrotron interactions with magnetic fields and inverse-Compton scattering with the interstellar radiation field. These energy losses are typically modelled as a continuous process. However, inverse-Compton scattering is a stochastic energy-loss process where interactions are rare but catastrophic. We show that when inverse-Compton scattering is modelled as a stochastic process, the expected $e^+e^-$ flux from dark matter annihilation is about a factor of $\sim$2 larger near the dark matter mass than in the continuous model. This greatly enhances the detectability of heavy dark matter.


I. INTRODUCTION
Dark matter particles that annihilate into electrons and positron pairs (hereafter, e + e − ) are expected to produce a sharp spectral cutoff in the local cosmic-ray e + e − spectrum at an energy that corresponds to the dark matter mass [1][2][3][4][5][6][7][8][9].While no such signal has been conclusively observed, the cosmic-ray e + e − fluxes have been measured to great precision at energies up to ∼1 TeV (e.g.AMS-02 [10,11], HESS [2,12]) and upcoming experiments are expected to reach energies of several tens to hundreds of TeV, e.g.Cherenkov Telescope Array (CTA) [13][14][15], AMS-100 [16] and HERD [17], expanding the parameter space of the search for dark matter signals to even higher energies with unprecedented precision.
Observations indicate that there are several important components of the local e + e − flux.At GeV energies, the dominant e + e − comes from the secondary interactions of other cosmic rays.At higher energies between ∼ 20 GeV to ∼ 400 GeV an unexpected excess was measured in the positron flux by PAMELA [18] and AMS-02 [19].While dark matter models of this excess have long-been explored [1][2][3][4][5][6], current studies indicate that this excess is best explained by nearby pulsars that produce e + e − pairs as they spin down (e.g.[20][21][22][23][24][25]).However, dark matter annihilation may produce a sub-dominant portion of the signal [7][8][9].
The propagation of the e + e − produced in the dark matter annihilation event must be properly modelled to predict the observed e + e − flux at Earth.During propagation, the e + e − lose energy through several processes.The most relevant are synchrotron interactions with Galactic magnetic fields and inverse-Compton scattering (ICS) interactions with interstellar radiation fields (ISRF).Typically, these energy losses have been calculated under the assumption that they are continuous over time.This approach is approximately correct for synchrotron losses.However, ICS interactions are rare and can remove a large fraction of the e + e − energy in just a single interaction.
In a previous paper [26], we have investigated the impact of correctly taking into account the stochastic effects of ICS en- The local e + e − flux expected from an annihilating dark matter particle with a mass of 100 TeV, for an energy loss model that approximates energy losses as either a continuous process (blue), or uses an exact stochastic formalism (orange).The e + e − flux sharpens at the energy near the dark matter mass by about a factor of 2.6 in the stochastic model compared to the continuous approximation.For reference, an astrophysical background is given by extrapolating HESS data [12] to higher energies.
ergy losses.Specifically, we have looked at the expected contribution from nearby pulsars to the positron flux, for which the continuous energy loss models predict sharp spectral features corresponding to the age of each pulsar.However, correctly treating the stochastic nature of ICS losses smooths out the local e + e − spectrum.This can be understood when looking at e + e − that are injected at the same initial energy and cool for the same amount of time: In the stochastic model, e + e − cool to a distribution of final energies, while in the continuous approximation the initial e + e − energy and pulsar age exactly determine the observed final energy.Importantly, the stochasticity of ICS does not only affect the spectrum of e + e − that originate from pulsars.It is relevant for any source that injects very-high-energy e + e − .One particularly interesting scenario involves TeV-scale leptophilic dark matter candidates, which has been extensively studied as potential solutions to the PAMELA positron excess [4], the DAMPE 1.4 TeV excess [27,28], as well as theoretically motivated models which have implications for, e.g., neutrino masses, the anomalous muon dipole moment, or wellmotivated dark sector extensions of the standard model [29][30][31][32][33].These scenarios also generically predict a sharp spectral feature in the observed e + e − flux at an energy that corresponds to the dark matter mass.This makes the local cosmicray e + e − fluxes a powerful tool to search for indirect dark matter signals.
In this paper, we show that the correct stochastic treatment of ICS strongly enhances the spectral peak observed in the local e + e − flux at an energy corresponding to the dark matter mass.This is due to the fact that in the stochastic case, e + e − remain at their injected energy for a long time before they undergo their first interaction, which then instantaneously removes a large fraction of their total energy.By contrast, the continuous case smears out these energy losses across all e + e − , smoothing out the spectrum near the dark matter mass.The amplitude of this effect sensitively depends on the magnetic field strength, which produces continuous synchrotron losses in each case.In Figure 1, we show this effect for a dark matter mass of 100 TeV and a magnetic field strength of 1 µG, finding an enhancement in the sharpness of the local e + e − spectrum by approximately a factor of 2.6.
This paper is structured as follows: In Section II, we give a detailed background about the energy loss processes and describe the continuous (II A) and stochastic (II B) energy loss models.In Section III, we present our results for different dark matter masses, final states and magnetic field strengths.We summarise and discuss our results in Section IV.

II. METHODOLOGY
In this section, we compute the relevant energy loss processes and describe our models for the calculation of the continuous ICS energy losses (Section II A) and the stochastic ICS energy losses (Section II B).
In both cases, synchrotron energy loss rate is given by: where u B is the energy density of the magnetic field, obtained from the magnetic field strength (in units of G) through u B = B 2 /(8π) × 6.24 × 10 11 eV/cm 3 .The average energy loss in a synchrotron interaction for an e + e − of energy E e is given by the critical energy which is ∼ 600 eV for a magnetic field strength of B = 1 µG and an electron energy of E e = 100 TeV.Even for strong magnetic fields of B = 3 µG and E e = 300 TeV, this is 16 keV, which is very small compared to the instrumental energy resolution.However, synchrotron interactions happen frequently and depending on the exact magnetic field strengths and ISRF components, synchrotron cooling typically exceeds ICS at energies exceeding ∼ 100 TeV, when ICS becomes highly suppressed by Klein-Nishina effects.
For ICS processes, on the other hand, a high-energy e + e − interacts with a photon from the interstellar radiation field, mostly from lower energy components such as the cosmic microwave background (CMB) or infrared (IR) radiation.The interaction cross section is given by [34][35][36]: where E γ is energy of the outgoing γ-ray photon, ν i is the initial energy of the photon, E e the initial energy of the e + e − , θ the scattering angle, r 0 the classical electron radius, z ≡ E γ /E e and b θ ≡ 2(1 − cos θ)ν i E e .This corresponds to a total energy loss rate for an e + e − that is given by [37] where σ T is the Thomson cross section, γ = E e /m e , n(ν) is the energy spectrum of the ISRF photons, and J (Γ) corresponds to the suppression of the Thomson cross-section due to Klein-Nishina effects and is given by: where Γ = 4νγ/m e and q = ν s /(Γ(γm − ν s )), where ν s is the energy of the scattered γ-ray photon.The function G(q, Γ) is given by: From these equations it can be seen that ICS processes are energy dependent, i.e., the interaction cross section decreases at high energies (Klein-Nishina suppression), which makes interactions especially rare, while at the same time a single interaction takes an increasingly large fraction of the e + e − energy.This means that ICS is a highly stochastic process, while synchrotron losses can be well-described as a continuous process.
Throughout this work, we use an interstellar radiation field similar to the model employed in the Galprop cosmicray propagation code [38], with four components: CMB, infrared (IR), optical/starlight and ultraviolet (UV) radiation.We assume the following energy densities and temperatures for each component: u UV = 0.1 eV/cm 3 , T UV = 20 000 K, u optical = 0.56 eV/cm 3 , T optical = 5 000 K, u IR = 0.41 eV/cm 3 , T IR = 20 K, u CMB = 0.26 eV/cm The energy loss factor as a function of electron energy for the specific ISRF components, as well as the three magnetic field strengths used throughout this work.The black line shows the total energy losses from all ISRF components combined (i.e., CMB, infrared, optical and ultraviolet).It can be seen that synchrotron losses start to dominate over ICS losses for energies above a few hundred TeV, depending on the magnetic field strength.
T CMB = 2.7 K. We note that for most of the dark masses considered here, only the CMB and IR energy densities have any effect on our results.Figure 2 shows the energy loss rates for the different ISRF components, as well as the energy losses for the three different magnetic field strengths (1 µG, 2 µG and 3 µG) that we consider throughout this paper.

A. Continuous ICS Energy Loss Model
In standard approaches, ICS energy losses are assumed to be continuous, taking an infinitesimal amount of energy from the e + e − in infinitesimal time steps according to the energy loss rates given in Equations 1 and 4 for synchrotron and ICS losses, respectively.
We model this by taking an e + e − with some initial energy and applying Equations 1 and 4 repeatedly for appropriately small times steps, until the desired cooling time has passed.

B. Stochastic ICS Energy Loss Model
In our stochastic model, we treat ICS interactions precisely as a probabilistic process, rather than approximating it as a continuous process.For this we create a Monte Carlo setup to determine if an e + e − undergoes an ICS interaction in a certain period of time, what the energy of the corresponding ISRF photon is, and how much energy is transferred in the interaction.
Specifically, we simulate the energy-loss evolution for each e + e − individually by applying the following steps, as also discussed in [26]: First, an e + e − is injected at some initial energy.Then we calculate a time step to be sufficiently small so that the energy loss due to synchrotron processes and the probability of having two ICS interactions is negligible within that time step.Based on the Klein-Nishina cross section (Equation 4), we use a Monte Carlo to determine if an interaction happens in the time step, and, if an interactions happens, the energy of the ISRF photon.Then, using another Monte Carlo, we determine the magnitude of the energy loss in that interaction.Finally, we subtract the energy losses from any ICS that happens as well as a continuous energy loss from synchrotron radiation (Equation 1) during that time step.We repeat this process until the e + e − have cooled for the desired amount of time.

C. Dark Matter Modeling
To obtain the flux produced by dark matter annihilation, we inject e + e − for uniformly distributed random cooling times, with a maximum value that exceeds the e + e − cooling times, in order to simulate continuous injection.
In this work, we consider two dark matter annihilation final states.In our main study, we assume that the dark matter particles annihilate directly into an e + e − pair.This means that all e + e − are injected at a single energy corresponding to the dark matter mass.
We further investigate a case where the dark matter particles annihilate into a µ + µ − pair, that subsequently produces e + e − .In this case, the e + e − are injected with a distribution of initial energies.To obtain this distribution, we use the injection spectra provided by DarkSUSY [39][40][41].Since Dark-SUSY only includes these spectra up to dark matter masses of a few TeV, we re-scale the injection spectra to match the heavier dark matter masses that are of interest here.This is possible because at these high energies, the muon mass (∼ 106 MeV) is negligible.

D. Electron and Positron Fluxes at Earth
After obtaining the e + e − fluxes from the simulations, we normalize them according to the dark matter annihilation rate.The rate of e + e − production from annihilating dark matter particles is given by where dn e /dt is the number density of e + e − per unit time, per unit volume, per unit energy, ρ 0 is the local dark matter energy density, m DM the mass of the dark matter particle, σv the thermally averaged dark matter annihilation cross section, and dN e /dE e is the energy spectrum of e + e − (number of e + e − per unit energy) from the dark matter annihilation.The factor 1/2 is necessary to not double-count dark matter annihilations.
The e + e − flux recorded at Earth follows then from  3. The e + e − flux expected from an annihilating dark matter particle with a mass of 300 TeV for a magnetic field strength of 1 µG.The result of the stochastic energy loss model is given in red, while the result of the continuous approximation model is given in blue.In the bottom, the enhancement of the sharp cutoff near the dark matter mass is given, showing that the feature is enhanced by about a factor of 2.3 in the stochastic model compared to the continuous treatment.The result of the stochastic energy loss model is given in red, while the result of the continuous approximation model is given in blue.In the bottom, the enhancement of the sharp cutoff near the dark matter mass is given, showing that the feature is enhanced by about a factor of 1.4 in the stochastic model compared to the continuous treatment.
E. Gamma-Ray Fluxes at Earth Furthermore, we keep track of the emitted γ-ray photons produced by e + e − in the energy loss interactions to investigate the impact of stochastic ICS of the γ-ray flux.
In the stochastic model, the γ-rays are readily obtained from the energy losses calculated in each interaction, since the ICS energy loss of a e + e − corresponds to the energy transferred to the photon.
On the other hand, the continuous model calculates the energy lost over a time step, which does not correspond to the energy transferred to a specific photon.Therefore, to compute the continuous γ-ray flux, we use our stochastic model, and at each time step, average over all possible energy losses.This gives the correct γ-ray spectrum and averages over the e + e − energy losses as in the continuous model.

F. Simulation Models
We compute the e + e − spectra for the following 5 dark matter masses: 10 TeV, 30 TeV, 50 TeV, 100 TeV and 300 TeV.Additionally, since the magnetic field strength is not well known, we consider 3 different magnetic field strengths 1 µG, 2 µG and 3 µG, which determine the impact of the synchrotron energy losses.In our standard scenario, we take m DM = 100 TeV and B = 1 µG.For each data set, we simulate sufficient e + e − to achieve statistically accuracy, which corresponds to about 300 000 particles per data set.
In the following, we present local e + e − fluxes normalised according to Equations 7 and 8.For illustrative purposes, we choose a dark matter annihilation cross section of σv = 10 −24 cm 3 /s, unless otherwise stated, that would produce a signal detectable at the expected CTA effective area and an expected energy resolution of ∼ 5% [15].However, we note that the flux can easily be re-scaled to a different annihilation cross section by applying Equation 7. Our results do not depend on the annihilation cross section, i.e., the relative flux between the stochastic and continuous model does not change.

A. Electron and Positron Spectra
Figure 1 shows a 100 TeV dark matter particle annihilating directly into an e + e − pair.The magnetic field strength is B = 1 µG.The upper panel shows the combined e + e − flux multiplied by E 2 as a function of e + e − energy, E, for the continuous approximation (blue) and the stochastic model (red).The lower panel shows the relative difference between the two models.At energies corresponding to the injection energy of the e + e − , the flux in the stochastic model exceeds the continuous approximation by a factor of 2.6.For reference, we show an astrophysical background extrapolated from HESS e + e − data [12] using a simple power law (of course, the true  5.The mean time it takes an e + e − to lose 10% (left panel) and 50% (right panel) of its initial energy for the different magnetic field strengths.Solid lines represent the continuous energy loss model and dashed lines the stochastic ICS model.The bottom panels show the ratio of the stochastic/continuous loss times.In the stochastic case, the low probability of ICS interactions, coupled with the significant energy loss per interaction, means that the mean energy-loss time is longer than in the continuous case.For larger total energy losses (e.g., 50%), this effect begins to fade and the mean energy loss times approach each other.astrophysical flux could be different).
Figures 3 and 4 show two more scenarios, representing an optimistic and a pessimistic case, respectively.In Figure 3, the dark matter mass is 300 TeV and the magnetic field strength 1 µG.The enhancement is 2.2, as stochastic ICS effects are strongest at higher energies, while the effects of synchrotron losses are minimized due to the small magnetic field strengths.Notably, in this case, the energy losses from individual ICS interactions become so large (as even ICS interactions with the CMB approach the Klein-Nishina limit), that we see a substantial dip in the electron spectrum at energies between ∼30-200 TeV due to electrons that lose nearly all of their energy in the first ICS interaction.
Conversely, in Figure 4, the dark matter mass is 10 TeV and the magnetic field strength 3 µG.Here, the enhancement is smaller, but still significant, reaching a peak of 1.4 near the dark matter mass.Figures for other dark matter masses and magnetic field strengths are shown in the Appendix.Across the various cases the enhancement of the local e + e − flux due to stochastic ICS at energies within 5% of the dark matter mass is about a factor of 2, and is higher for larger dark matter masses and weaker magnetic fields strengths.
Figure 5 shows the mean time it takes an e + e − to lose some fraction of its initial energy as a function of its initial energy (i.e., the dark matter mass).The left panel shows the mean energy loss time for a 10% energy loss, and the right panel for a 50% energy loss.It can be seen that the mean energy loss times are significantly longer (about a factor of 2 for a 10% energy loss) in the stochastic model compared to the continuous model, and this ratio is larger for larger injection energies.Additionally, the mean energy loss time in both the continuous and stochastic cases increases with decreasing magnetic field strength, as this reduces energy losses from synchrotron inter-actions.We note that, within the stochastic case, it is possible that the 10% energy loss occurs entirely due to synchrotron radiation before any ICS interactions have occurred.For the 50% energy loss case, the stochastic and continuous loss times are closer compared to the 10% energy loss case and only differ by about a factor of 3% at 100 TeV.This is due to the fact that the effect of stochasticity is smoothed out once multiple ICS interactions are likely.
Figure 6 shows the fractional energy lost in the first ICS interaction (i.e., the energy lost in the first ICS interaction divided by the initial energy) for the various initial e + e − energies and magnetic field strengths.This is an indication of the energy resolution required for telescopes to observe the enhanced feature.For smaller dark matter masses, the fractional energy lost becomes smaller, and a better energy resolution is required.For an e + e − with an initial energy of 0.1 TeV, the energy loss in the first ICS interaction is about 0.1 GeV, which would require an energy resolution of at least 0.1% to be experimentally observable.On the other hand, for an initial energy of 100 TeV, the energy loss is about 10 TeV, which corresponds to an energy resolution of 10%, which is feasible for upcoming experiments.
We note that Figures 5 and 6 justify our choice to ignore the effect of cosmic-ray diffusion on the local e + e − spectrum throughout this work.In particular, the time until an e + e − with an initial energy of 100 TeV loses 50% of its energy is t ∼ 70 kyr.The average spatial displacement at this energy is given by L = √ 6Dt, where D is the diffusion coefficient given by D = D 0 (E) δ .Using a typical normalization factor of D 0 = 2 × 10 28 cm 2 /s at 1 GeV, and a diffusion index δ = 0.4 [25,42], a 100-TeV e + e − displaces about a distance of ∼ 1700 pc.This is well below the Galactic halo size of ∼5 kpc, within which cosmic rays are contained, and  6.The average energy loss in the first ICS interaction after injection as a function of the initial e + e − energy, for the three different magnetic field strengths.To observe the enhancement of the dark matter spectral feature experimentally, the energy resolution should roughly be smaller than the energy loss in the first ICS interaction.e + e − energies, and the three different magnetic field models.The energy resolution is fixed to 5% -with a better energy resolution, enhancements at lower initial e + e − energies become stronger.also much smaller than the radius over which the dark matter density significantly varies [43].Even for a 100 GeV e + e − , where the time required to lose 50% of the initial energy is ∼ 3600 kyr, the average displacement becomes ∼ 3 kpc.Even if some e + e − escape the Galaxy, this happens at energies well below the injection energy, where our effect is observed.

Fractional Energy Lost in First ICS
In Figure 7 we summarize our results for the various dark matter masses and show the enhancement of the spectral cutoff in the stochastic model compared to the continuous model for the different magnetic field strengths.We choose an energy resolution of 5%, since this is a realistic value for upcoming experiments, but note that the enhancement for lower initial energies would be stronger for even better energy resolution (see Figure 6).

B. Local e + e − Spectrum from Dark Matter Annihilation to Muons
We also consider the case where dark matter particles annihilate into a µ + µ − pair that subsequently decays into e + e − .This smears out the initial e + e − injection spectrum from a delta function to a function that is approximately constant in dN dE .Figure 8 shows the e + e − spectrum for a 100 TeV dark matter particle and a magnetic field strength of 1 µG.Since the e + e − are now injected at a distribution of initial energies, rather than the same initial energy, the enhancement of the feature is smaller, about a factor of 1.2, in the stochastic model compared to the continuous model.
For the case of the dark matter annihilating into a τ + τ − pair (or a hadronic final state) that subsequently produces e + e − , the difference between the stochastic and continuous model would be even more reduced than in the µ + µ − case, since the e + e − injection spectrum would be even softer.

C. Gamma-Ray Spectra
When e + e − undergo an ICS interaction, the ISRF photon is converted to a high-energy γ-ray with an energy identical to the energy lost by the e + e − .Figure 9 shows the γ-ray flux expected from e + e − produced by a 100 TeV dark matter particle for the stochastic model (red) and the continuous model (blue) as a function of γ-ray energy.Similar to the e + e − spectra, the stochastic model shows an enhancement at the highest energies of almost a factor of 3.This is expected since in the stochastic model, a larger fraction of the e + e − energy can be lost in a single interaction compared to the continuous model, which means that a larger amount of energy is transferred to the photon when the e + e − has an energy near its initial value, resulting in higher energetic γ-rays.We note that while the relative enhancement of the γ-ray flux is similar to the enhancement in the local e + e − spectrum, the largest enhancement does not occur near the energetic peak of the γray emission, making this effect harder to observe practically with γ-ray telescopes.

IV. DISCUSSION AND CONCLUSIONS
In this paper, we have shown the importance of precisely treating energy losses from inverse-Compton interactions as a stochastic process instead of approximating them as a continuous process.Specifically, we have shown the impact of stochastic ICS losses on the expected signal from the annihilation of TeV-scale dark matter particles into e + e − pairs.When taking the stochasticity of ICS into account, the sharp cutoff at the dark matter mass is increased by about a factor of 2 compared to the continuous model.This effect is significant across TeV dark matter masses and different magnetic field strengths.This implies that the detectability of dark matter signals in local e + e − measurements is significantly greater than previously expected.This is important for current and near-future experiments, such as CTA, that are able to observe the local e + e − fluxes up to ∼100 TeV with energy resolutions of ∼10%.Our results are also relevant, but subdominant, at energies corresponding to the 1.4 TeV e + e − excess detected by DAMPE [27].We find that the amplitude of the e + e − feature would be enhanced by approximately 6%.This relatively small enhancement is due to the rather wide (10%) binning of the e + e − data by DAMPE.Several studies have indicated that a nearby dark matter clump would be capable of producing the DAMPE signal [9,[44][45][46], and our analysis can be considered to slightly increase the available parameter space of clumps that would be capable of producing the excess.
Our results are relevant for any dark matter models that directly annihilate at least partially into e + e − pairs.Enhancements from annihilations into µ + µ − exist but are less significant.Additionally, slightly more energetic γ-rays are expected in our stochastic models.However, this enhancement happens at γ-ray energies that are above the peak of the γ-ray emission, and may be difficult to detect.This result is particularly interesting in light of our recent work in Ref. [26], which found that the stochasticity of ICS smoothed out peaks in the the local e + e − spectrum from pulsars.This difference results from the fact that the e + e − injection from pulsars is highly peaked in time, but spectrally smooth, while the e + e − injection from dark matter is smooth in time, but spectrally peaked.Intriguingly, this breaks the degeneracy between the spectrally peaked features expected from dark matter annihilation and pulsar activity, re-affirming the status of a sharp feature in the local e + e − spectrum as unambiguous evidence for dark matter annihilation.
FIG. 1.The local e + e − flux expected from an annihilating dark matter particle with a mass of 100 TeV, for an energy loss model that approximates energy losses as either a continuous process (blue), or uses an exact stochastic formalism (orange).The e + e − flux sharpens at the energy near the dark matter mass by about a factor of 2.6 in the stochastic model compared to the continuous approximation.For reference, an astrophysical background is given by extrapolating HESS data[12] to higher energies.

FIG. 4 .
FIG.4.The e + e − flux expected from an annihilating dark matter particle with a mass of 10 TeV for a magnetic field strength of 3 µG.The result of the stochastic energy loss model is given in red, while the result of the continuous approximation model is given in blue.In the bottom, the enhancement of the sharp cutoff near the dark matter mass is given, showing that the feature is enhanced by about a factor of 1.4 in the stochastic model compared to the continuous treatment.
FIG.5.The mean time it takes an e + e − to lose 10% (left panel) and 50% (right panel) of its initial energy for the different magnetic field strengths.Solid lines represent the continuous energy loss model and dashed lines the stochastic ICS model.The bottom panels show the ratio of the stochastic/continuous loss times.In the stochastic case, the low probability of ICS interactions, coupled with the significant energy loss per interaction, means that the mean energy-loss time is longer than in the continuous case.For larger total energy losses (e.g., 50%), this effect begins to fade and the mean energy loss times approach each other.

FIG. 7 .
FIG.7.The enhancement of the dark matter cutoff in the stochastic loss model compared to the continuous model for the various initial e + e − energies, and the three different magnetic field models.The energy resolution is fixed to 5% -with a better energy resolution, enhancements at lower initial e + e − energies become stronger.

FIG. 8 .
FIG.8.The e + e − spectrum from a 100 TeV dark matter particle that annihilates into µ + µ − .For energies above ∼50 TeV, the stochastic model (orange) is enhanced by about a factor of 1.2 compared to the continuous approximation (blue).

FIG. 9 .
FIG.9.The γ-ray flux as a function of γ-ray energy from the annihilation of a 100 TeV dark matter particle into e + e − pairs that emit γ-rays when cooling, assuming a magnetic field of 2 µG.At the highest energies, the stochastic energy loss model (red) shows an enhancement of about a factor of 2 compared to the continuous energy loss model (blue).
FIG.10.The expected e + e − flux from the stochastic model (red) compared to the continuous approximation (blue), for a dark matter mass of 10 TeV (left panels) and 30 TeV (right panels) for the three different magnetic field strengths.
FIG. 11.The expected e + e − flux from the stochastic model (red) compared to the continuous approximation (blue), for a dark matter mass of 50 TeV (left panels) and 100 TeV (right panels) for the three different magnetic field strengths.
FIG.12.The expected e + e − flux from the stochastic model (red) compared to the continuous approximation (blue), for a dark matter mass of 300 TeV for the three different magnetic field strengths.