Explaining the AMS positron excess via Right-handed Neutrinos

We have witnessed in the past decade the observation of a puzzling cosmic-ray excess at energies larger than $10$ GeV. The AMS-02 data published this year has new ingredients such as the bump around $300$ GeV followed by a drop at $800$ GeV, as well as smaller error bars. Adopting the background used by the AMS-02 collaboration in their analysis, one can conclude that previous explanations to the new AMS-02 such as one component annihilating and decaying dark matter as well as pulsars seem to fail at reproducing the data. Here, we show that in the right-handed neutrino portal might reside the answer. We discuss a decaying two-component dark matter scenario where the two-body decay products are right-handed neutrinos that have their decay pattern governed by the type I seesaw mechanism. This setup provides a very good fit to data, for example, for a conservative approach including just statistical uncertainties leads to $\chi^2/d.o.f \sim 2.3$ for $m_{DM_1}=2150$ GeV with $\tau_{1}=3.78 \times 10^{26}$ s and $m_{DM_2}=300$ with $\tau_{2}=5.0 \times 10^{27}$ s for $M_N=10$ GeV, and, in an optimistic case, including systematic uncertainties, we find $\chi^2/d.o.f \sim 1.12$, for $M_N = 10$ GeV, with $m_{DM_1}=2200$ GeV with $\tau_{1}=3.8 \times 10^{26}$ s and $m_{DM_2}=323$ GeV with $\tau_{2}=1.68 \times 10^{27}$ s.


I. INTRODUCTION
The observation of cosmic-rays have boosted our understanding of astrophysical phenomena that undergo diffusion and energy loss processes in the intergalactic medium. Historically, in 2008 the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) surprisingly announced the first evidence of a rise in the cosmic-ray positron fraction at GeV energies with high statistics [1]. Fermi-LAT confirmed this cosmic-ray anomaly much later in 2011. Taking advantage of the absent onboard magnet, they could distinguish electrons from positrons by exploiting the Earth's shadow, which is offset in opposite directions for opposite charges due to the Earth's magnetic field. With this technique they were able to indeed observe a positron fraction rise for energies between 20 and 200 GeV [2]. With much better statistics, the AMS mission measured the positron fraction up to 350 GeV [3], and reported a flat positron fraction for energies above 150 GeV.
That has triggered a number of works which were able to explain the AMS excess of events. Some attempts focused on annihilating dark matter [4,5], but the annihilation cross section needed to fit the excess was too large to be in agreement with gamma-ray observations in the direction of the galactic center and dwarf spheroidal galaxies [6,7], and cosmic microwave background data [8,9]. Interpretations in terms of decaying dark matter were also put forth, where a lifetime of the order of 10 27 s for µμ final states could provide a reasonable fit to data [10][11][12][13][14]. Alternatively, nearby astrophysical objects presented themselves as good candidates [15][16][17]. That was the whole story until the new AMS data and HAWC observations came into light.
The new AMS data has new ingredients [18]: (i) features much smaller error bars at low energies and a rise at ∼ 10 GeV; (ii) the previously observed flat spectrum for en- * Electronic address: farinaldo.queiroz@iip.ufrn.br † Electronic address: csiqueira@iip.ufrn.br ergies larger than 150 GeV now exhibits a bump-like feature with a peak around 300 GeV; (ii) a sharp drop for energies above 400 GeV is visible. These new ingredients significantly harden the shape of the spectrum making a dark matter interpretation difficult, especially adopting the single component scenario. Moreover, the High-Altitude Water Cherenkov Observatory (HAWC) observed the presence of energetic electrons and positrons from nearby pulsars and from that the diffusion parameters were inferred. The diffusion parameters derived are inconsistent with the one observed by AMS-02 though, thus ruling out such pulsars as the origin of the AMS excess [19]. In conclusion, the new AMS data begs for a new interpretation [20].
In this work, we attempt to explain the positron excess in terms of two-component dark matter comprised of two scalars. Such scalars decay into two right-handed neutrinos that decay into Standard Model particles according to the type-I seesaw mechanism [21,22]. This scenario appears in Majoron-inspired models, for instance [23][24][25][26][27][28][29][30][31][32][33]. We emphasize that in the canonical Majoron model, the decay into righthanded neutrinos is not dominant. Decays into left-handed neutrinos are instead more relevant, and they lead to an interesting phenomenology explored elsewhere [32]. In this work, we are investigating the possibility of fitting the AMS-02 data with a two-component dark matter setup where each component decays into two right-handed neutrinos. We are not interested in a explicit theoretical realization of this scenario but we do emphasize that having a two-component decay dark matter model requires going beyond the vanilla Majoron models and other type I seesaw model incarnations. Our idea is simply to assess whether one could get a reasonable fit to the AMS-02 data if such decays are dominant, without having an specific model at hand.
That said, we perform a chi-squared analysis choosing different masses for the right-handed neutrino (10 GeV, 50 GeV and 80 GeV) and leaving the DM mass and the decay rate as free parameters to get the best fit to the data. In addition, we choose two different set of propagation parameters which are known as medium (MED) and maximum (MAX) diffusion models, using the Navarro-Frenk-White (NFW) profile.
Moreover, we carry out all this procedure including only statistical errors, and statistical plus systematic errors to really assess the impact of the systematic effects on our conclusions. Including only the statistical uncertainties we find the best-fit of χ 2 /d.o.f ∼ 2.3 for m DM1 = 300 with τ 1 = 1.67 × 10 27 s and m DM2 = 2000 GeV with τ DM2 = 4 × 10 26 s for M N = 10 GeV, and, for the optimistic case, including systematic uncertainties, we get τ 1 = 1.68 × 10 27 s and τ DM2 = 3.8 × 10 26 s, for m DM1 = 323 GeV, and m DM2 = 2200 GeV respectively with M N = 10 GeV, yielding Lastly, we put our results into perspective with gamma-rays observations [34,35]. We start our reasoning discussing below how we obtain the positron flux.

II. POSITRON FLUX
The positron flux reported by AMS seems to be compatible with a background, which is given by a diffuse flux at low energies and a new source at high energies. So, the collaboration interpreted the whole signal as a background plus a new source term as follows, In this work, we choose decaying dark matter particles to be responsible for this new source flux, Φ source , described above. For this purpose, it is necessary to compute the decaying DM positron flux, which is given by, where E is the positron energy after propagation and E s is the positron energy at production, ρ = 0.4 GeV/cm 3 is the DM density in the location of the Sun, m DM is the DM mass, Γ is the decay rate of DM particle, BR f is the branching ratio for a given final state f and dN e + f dE (E s ) is the number of positrons per energy produced after decay before the propagation. The parameter b(E) is the called energy loss function, which takes into account the possible energy losses via synchrotron radiation and inverse Compton scattering.
For the purpose of being conservative, we choose the same diffuse flux as reported by the collaboration which includes contributions from the interaction between galactic cosmic rays with the intergalactic medium, where the values for the parameters reported by the collaboration were: E 1 = 7 GeV,Ê(E) = E + ϕ e + , with ϕ e + = 1.10±0.03 GeV, c d = (6.51±0.14)×10 −2 (m 2 sr s GeV) −1 , γ d = −4.07 ± 0.06, where we use the central values for the parameters E 1 and ϕ e + , while the values for c d and γ d were chosen within 3σ contour in order to provide the best-fit to the data. Furthermore, the halo function I(E, E s ), computed using the numerical package PPPC4DMID, appears as a solution to the diffusion equation, and it is dependent on the loss energy function (b(E)), on the DM profile (here we choose the NFW), on the diffusion parameters K 0 = 0.0112 kpc 2 / Myr and δ = 0.70 for medium (MED) and K 0 = 0.0765 kpc 2 / Myr and δ = 0.46 for the maximum (MAX) propagation models.
In the next, we will compute the fluxes for the model considered here.

III. RESULTS
The scenario involves two DM particles decaying into two right handed neutrinos (RHN) pairs. We assume each DM candidate composing 50% of the DM abundance of the Universe, and of course these values can be easily changed by rescaling the decay rate accordingly. These RHN couples to standard model particles via Higgs and gauge bosons, leading to the following RHN decay pattern N R → W +/− + l −/+ , N R → Z + ν l , and N R → H + ν l , in principle, l can be the three leptonic flavors, but in our case, for simplicity, we choose the l = e. In addition, we impose three different values for the RHN mass, M N = 10 GeV, M N = 50 GeV, and M N = 80 GeV.
Fixing the RHN masses, we compute the positron flux in Eq. (2) using the PPPC4DMID code which computes the halo function I(E, E s ), and the Pythia 8 package to obtain the positron spectrum for each right-handed neutrino mass. Then we left as free parameters the DM masses M DM1 and M DM2 and the decay rates in order to fit the data reported by the AMS collaboration, namely, for each RHN mass we found a combination of DM mass versus decay rate which provides the Total Flux (MN = 10 GeV) best-fit for the data.
To be conservative, in these first analyses, we compute the goodness of the fit, χ 2 /d.o.f , using only the statistical uncertainties provided by the collaboration. For each scenario, we chose the best values within 3σ error for the parameters c d and γ d to get the best values for the fit, according to the Table I. In Figs. 1, 2 and 3, we present the computed fluxes including that predicted by each decaying DM component (continuous and dashed lines), the background contribution (gray lines) and the sum over all components (black lines). The AMS data [18] is also shown for comparison.
In Fig. 1, we show our results for M N = 10 GeV for two different propagation models, MED and MAX (top and bottom, respectively). We found the best fit value equal to χ 2 /d.o.f. = 3.57 for the MED propagation model and 3 for the MAX propagation. In Table I, we include the background parameters c d and γ d adopted for each scenario.
In Fig. 2, we show our results taking M N = 50 GeV Total Flux (MN = 50 GeV)   . 3).   Summarizing, in Table II we show the best-fit values found for the parameters DM mass and lifetime for each DM candidate in order to get the best fit to the data. In Figs. 4 (MED propagation) and 5 (MAX propagation), we present the 1σ, 2σ and 3σ contours for both DM1 (top) and DM2 (bottom) candidates following the same color pattern for the RHN masses described above.
As we can see, the larger the right handed neutrino mass the worst is the fit to data. This is due to the change in the shape of the spectrum. Although their shapes seems to be quite similar, minimum modifications in the tale (lower energies) provide a significant impact on the χ 2 /d.o.f. as a result of the smallness of the error bars at lower energies.
In the same way, the MED propagation model yields smaller fluxes than MAX propagation one. Hence, we can play with the decay rate (or lifetime) in order to obtain similar fits for both propagation models. For example, taking M N = 10 GeV the best fit is found for τ 2 = 1.67×10 27 s and M DM2 = 300 GeV for MED propagation while for MAX we need 5.0 × 10 27 s (see Table II). As the MAX models gives rise to the steeper energy spectrum, we need to increase the lifetime to find a similar fit. The combination of two different candidates can provide an excellent agreement with AMS excess, including one of them with mass around hundreds of GeV and another with mass of a few TeV. It is worth emphasizing that the choice 50%-50% for each DM candidate is arbitrary, in a way that modification of this percentage results simply in a re-scaling of the lifetime.

A. Including Systematic Uncertainties
The previous analysis included just statistical uncertainties which can be considered conservative as the interpretation of the AMS data is dominated by systematics. Here, we include systematic uncertainties in order to verify its impact in the limits. We concluded that the main impact occurs at lower energies which features rather small error bars. Therefore, the impact in the χ 2 /d.o.f. can be large but usually decreases by a factor of a few. In our study, we choose the MED propagation, with systematic uncertainties provided by the collaboration [18]. One can easily realize that the choice for MED or MAX propagation model does not result in significant changes to our conclusions and for this reason we picked the focused on the MED model in this particular analysis. We emphasize that our conclusions would still apply for the MAX propagation model. We repeat the procedure above and assume that each dark matter particle contributes to 50% of the dark matter density. In the Fig. 6, we present the fluxes that yield the best-fit for M N = 10 GeV. As shown in the Fig. 6  We have explicitly shown that our benchmark scenarios provide a good fit to data and now display the best-fit contours (1σ (continuous lines), 2σ (dashed lines) and 3σ (dotted lines) contours) in terms of the lifetime and dark matter mass in the Fig. 9 for each setup discussed where both statistical and systematic errors are included.
One could find that the best-fit is found for M N = 50 GeV, however, since the AMS data is driven by systematic errors is reasonable to conclude that all of them provide an equally  Thus, we did find a good fit to the data. However, we would like to stress that the statistical method used is not of utmost importance because the AMS-02 data is driven by systematic.

IV. DISCUSSION
The two component dark matter scenario where a scalar (or vector) decays into right-handed neutrino pair was motivated by scalar models which embed the type I seesaw mechanism. In the type I seesaw mechanism the right-handed neutrinos are typically very heavy, however we found that for masses heavier than ∼ 100 GeV the fit to the AMS data becomes quite poor. This can be understood via the energy spectrum. When right-handed neutrinos are heavier than 100 GeV, the decay channels into Z and W bosons are open leading to significant changes in the energy spectrum, and as we checked, it provides a poor fit to data. That said, even in the type I seesaw mechanism we can easily assume right-handed neutrino masses between 10-80 GeV by tuning the Yukawa couplings, bringing no changes to the branching ratio pattern, which justifies our analysis.
Another aspect of our study is the compatibility with limits stemming from gamma-ray data, because our decay channels also produce gamma-rays. Our setup involved dark matter decaying into right-handed neutrino pairs where each righthanded neutrino might decay into leptons and quarks via offshell W, Higgs and Z bosons. Thus, as we have not fixed a final decay channel it is not so simple to compare out with other existing limits in the literature. Sifting the energy spectra produced by DM decay into SM particles, we realized that the gamma-ray spectrum produced by a direct DM decay into W W and W though different, yield the closest shape to the energy spectra produced by our setup. Thus we can compare the energy spectra and notice by how much different they are, and then re-scale our energy spectra by a given amount to match the energy spectra of the W W and W channels. In this way, we may roughly estimate whether our benchmark points are in agreement with existing gamma-ray limits [36]. We concluded that taking into account the facts that we have a two component dark matter setup and the uncertainties involved in the gamma-ray limits our benchmark points are consistent with the existing gamma-ray bounds. Although, we highlight that there are no existing gamma-ray limits directly applicable to our model, and that required an extra effort from our side to somehow compare our results with gamma-ray probes that feature a similar energy spectrum. We will prolong this discussion in the Appendix.
In summary, we have shown that such two component dark matter via the right-handed neutrino portal offer a good fit to data for right-handed neutrino masses between 10-80 GeV with the inclusion or not of systematic errors in the analysis. Within this mass range, the precise mass of the right-handed neutrino does not change much the lifetime and dark matter mass that best fit the data, but do change the χ 2 /d.o.f by a factor of two. In addition to that, the change from MED to MAX propagation model does not bring significant changes to our study, despite the MAX propagation being recently favored by recent observations of the Boron-to-Carbon ratio [37]. In our study we concluded that masses around 300 GeV and 2 TeV with lifetime of 4 × 10 26 s and 2 × 10 27 s respectively, are favored and marginally consistent with current bounds rising from gamma-ray observations.

V. CONCLUSIONS
The positron excess provided by the AMS collaboration [18] remains an open question. In this work we assessed a scenario where two decaying dark matter candidates may constitute an answer to the observed excess via the right-handed neutrino portal.
We have shown that DM particles decaying into right handed neutrino pairs which couples to SM particles through Z, W and Higgs bosons, inspired by the type I seesaw mechanism provide a very good fit to data. For example, for a conservative approach including just statistical uncertainties we got Such benchmark points are consistent with existing gamma-ray bounds for lighter DM but in tension for heavier DM, however, as described in the appendix, due to the large uncertainties in the gamma-ray limits, we may argue that our benchmarks are in agreement with gamma-ray data. It is important to emphasize that this is an estimate, and a careful analysis is needed. In addition, our benchmarks are significantly modified by changing the propagation model from MED to MAX. Knowing that the AMS results are dominated by systematics our best-fit points might alter for different assumption for the background. In our work we adopted the background recommended by the AMS collaboration.
In summary, we presented a plausible explanation to the puzzling AMS data via the right-handed neutrino portal. 2016/01343-7. CS thanks UFRN and MEC for the financial support. We thank the High Performance Computing Center (NPAD) at UFRN for providing computational resources.

VI. APPENDIX
The decay into right-handed neutrinos also produces gamma-rays, thus we need to check if our scenario is in agreement with existing gamma-ray observations. Although, there is no gamma-ray limit in the literature for dark matter decaying into right-handed neutrinos. Thus in order to estimate if our best-fit points are then consistent gamma-ray bounds we looked after the popular decay channels to check which ones produce similar gamma-ray spectra. They are all different, but the ones that resemble most our case are the decay into WW and W . To explicitly show our procedure we chose a benchmark scenario with M N = 10, 50, 80 GeV, where the best fit to the positron data is given by M DM1 2000 GeV and M DM2 = 300 GeV. In order to get a comparable limit, we need to re-scale the spectrum according to the Fig. 10 below. We highlight that the W W and W spectra were re-scale by different a constant factor to approximate their spectra to ours. For example, for M DM1 2000 GeV and M N = 10 GeV we had to multiply the our spectrum by six. Therefore, the limits provided by [36] need to be suppressed also by a factor of six times to be applicable to our setup. Moreover, an additional factor 1/2 should be included due to the DM density since our case we have DM components. That said, the gamma-ray limit from [36] at face value reads 3.6 × 10 28 s, but it should be read as 3 × 10 27 s. While for M DM2 = 300 GeV, at face value the limit reads 4.8 × 10 27 s [36], but taking into account the factors provides 1.2 × 10 27 s.
Using these estimates, we conclude that the lighter DM candidate is in agreement with the limits while the heavier not by a factor of a few. Having in mind that the limits in [36] are optimistic due to the profile and target selected (inner galaxy) these gamma-ray bounds are subject to large uncertainties, we may argue that our best fit points are marginally in agreement with the gamma-ray bounds. A similar reasoning could be applied to different benchmark points.
In addition, we emphasize that any assessment of the bestfit points rely on the background model assumed for the positron secondary production resulted from the collision of primary cosmic rays with the interstellar medium. In our work, we adopt the background model used by AMS-02 collaboration in their data release [18], thus our conclusions are based on that. There are other competitive gamma-ray bounds in the literature [38] which can be also relaxed in a similar way.