The Poker Face of the Majoron Dark Matter Model: LUX to keV Line

The Poker Face of the Majoron Dark Matter Model: LUX to keV Line

Farinaldo S. Queiroz fdasilva@ucsc.edu    Kuver Sinha kusinha@syr.edu Department of Physics and Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA
Department of Physics, Syracuse University, Syracuse, NY 13244, USA
Abstract

We study the viability of pseudo Nambu-Goldstone bosons (Majorons) arising in see-saw models as dark matter candidates. Interestingly the stability of the Majoron as dark matter is related to the scale that sets the see-saw and leptogenesis mechanisms, while its annihilation and scattering cross section off nuclei can be set through the Higgs portal. For Majorons, we compute observables such as the abundance, scattering cross section, Higgs invisible decay width, and emission lines and compare with current data in order to outline the excluded versus still viable parameter space regions. We conclude that the simplest Majoron dark matter models coupling through the Higgs portal, except at the Higgs resonance, are excluded by current direct detection data for Majorons lighter than  GeV and future runnings are expected to rule out decisively the 1GeV-1TeV window. Lastly, we point out that light keV-scale Majorons whose relic density is set by thermal freeze-in from sterile neutrinos can account for the keV line observed by XMM-Newton observatory in the spectrum of 73 galaxy clusters, within a see-saw model with a triplet Higgs.

pacs:
95.35.+d, 14.60.Pq, 98.80.Cq, 12.60.Fr

I Introduction

The identity of dark matter constitutes one of the most exciting puzzles in current science. Interestingly, dark matter is often connected to other paradigms of fundamental physics, with WIMPs (Weakly Interacting Massive Particles) in supersymmetric theories being the most studied ones. The dark sector can also be connected to other important phenomena such as the generation of the neutrino masses, leptogenesis, and baryogenesis Tsuyuki:2014aia (); Kashiwase:2013uy (); seesaw (). The Majoron dark matter model is an example which occurs in see-saw models of neutrino mass generation.

In see-saw models, the lepton number might be explicitly broken by the Majorana masses of right-handed neutrinos. If the lepton number is instead broken spontaneously by the vev of a complex scalar field (a singlet “Higgs”), one has a new pseudo-scalar gauge singlet Nambu-Goldstone boson (the Majoron). In such models, the Majoron is a natural decaying dark matter candidate ernestma (); CMBmajoron (); Gelmini:1984pe (); Berezinsky:1993fm (). The Majoron lifetime is determined by its decay into Standard Model (SM) neutrinos which is suppressed by the scale of lepton number violation. For sufficiently high lepton violation scale the Majoron is cosmologically stable. This scale also sets the heavy right handed neutrinos masses in the see-saw type I setup. Therefore, the viability of the Majoron dark matter is connected to the see-saw mechanism responsible for generating the SM neutrinos masses and the scale of leptogenesis which occurs through the decays of the heavy right-handed neutrinos Covi:2009xn ().

As for the mass of the Majoron, it can arise from explicit soft terms, or from quantum gravitational effects that explicitly break lepton number. In the former case, the mass can be hundreds of GeV. From an effective field theory point of view, nothing prevents a coupling of the Majoron to the Higgs scalar potential at tree level. Majoron models of this category are thus a particular UV realization of the effective Higgs-portal scalar models studied by Ref.Burgess:2000yq (). In the case where the mass is due to quantum gravitational effects, on the other hand, the Majoron is expected to be very light. A particularly well-motivated scenario is a keV Majoron, which can satisfy the thermal relic density.

We study dark matter observables in both cases. In the former, we compute observables such as the abundance, scattering cross section, Higgs invisible decay width, and emission lines and compare with current data in order to outline the excluded versus still viable parameter space regions in the mass range. We find that LUX bounds on the dark matter scattering cross section, along with relic density requirements and Higgs invisible decay width limit, effectively rule out thermal dark matter below GeV in the this scenario (except near the Higgs resonance when the Majoron mass is GeV). Furthermore, future direct detection running coming from XENON1T and LUX are forecasted to rule out the entire GeV-TeV mass range. A way out would be the inclusion of new particles with masses close to the Majoron in order to exploit co-annihilation channels. Additionally, a non-thermal history for the Majoron, where the initial number density of Majorons is fixed without further annihilation, is possible Allahverdi:2010rh ().

In the case of a Majoron, we point out that it may be possible to accommodate the recently observed keV line from the XMM-Newton observatory KeVline (), for appropriate choice of parameters in a see-saw model with a triplet (as well as the singlet) Higgs scalar. The branching to photons arises at loop level from the projection of the Majoron along the doublet (Standard Model) Higgs. The relic density can be set by thermal freeze-in from the right-handed neutrinos. We provide order-of-magnitude estimates as a proof of concept that the observed keV line can be obtained in this class of models.

The plan of the paper is as follows. In Section II, we describe the Majoron dark matter model in the context of type I see-saw. In Section III, we discuss the dark matter observables of the model. In Section IV, we discuss the case of a light Majoron and the recently observed keV line. We end with our Conclusions.

Ii Model

We will study a model in which the leptogenesis conditions, the see-saw mechanism and existence of a dark matter candidate are connected. The model is comprised of a neutral singlet scalar and three right-handed neutrinos. Therefore the Lagrangian of the model might be written as,

(1)

where and are the SM standard model lepton and Higgs doublets respectively, whereas are the right-handed neutrinos and is the singlet scalar. According to Eq.1 we notice that the right-handed neutrinos carry a lepton number of 1, as well as the SM particles, whereas the neutral scalar carries a lepton number of . The general scalar potential of the fields is found to be,

(2)

where is the term that softly breaks the global lepton number given by,

(3)

In the scenario where , the spontaneous symmetry breaking of the global lepton number, caused by the vev of the real component of the neutral scalar , will induce a massless Majoron in the theory. Conversely when is as defined above, the Majoron will acquire mass proportional to as we will see further. Additional soft terms such as the presence of Majorana mass terms for the right-handed neutrino may be evoked in more general settings. In this work we will assume the existence of a discrete symmetry with the fields transforming as: and , where f stands for all fermions including the right-handed neutrinos with the remaining fields transforming trivially under . Notice that the Lagrangians aforementioned are invariant under this symmetry. Throughout our study we will keep and to guarantee the potential bounded from below. Since is a complex scalar, we can write it as,

(4)

Here u refers to the scale at which the lepton number is violated. In this work we assume u to lie at the GUT scale for leptogenesis and dark matter purposes. Anyway, after the spontaneous symmetry breaking mechanism, we find ernestma ()

(5)

where we have used the standard definition for the Higgs doublet with and identified the following relations,

(6)
(7)
(8)

From Eq.5 we conclude the Majoron (J) is decoupled from the other scalars of the model. Additionally, due to the high hierarchy between assumed GUT scale vev u the electroweak vev (v), the mass matrix in Eq.5 yields ernestma (),

(9)

ii.1 See-saw Type I

See-saw type I also known as canonical see-saw refers to the mechanism where the standard model neutrinos acquire masses through the insertion of the heavy right-handed neutrinos as in our model. After the spontaneous symmetry breaking Eq.(1) turns into,

(10)

which can be rewritten as

(11)

and yields,

(12)

where according to Eq.(1) we find,

(13)

For the standard model neutrino masses can lie at the eV range naturally small with being at the weak scale. The leptogenesis mechanism is not the focus in this work, however it is important to point out that the presence of heavy right-handed neutrinos coupled to the left-handed SM neutrinos through the interactions given in the first term of Eq.10 offers an important connection to leptogenesis because the decay of the right-handed neutrinos and their annihilations into SM particles can generate a lepton asymmetry in non-CP conserving setups Gu:2009hn (). For  GeV and  GeV, the desired CP asymmetry is induced as long as the annihilations of the right-handed neutrinos into the massless Majorons go out of equilibrium before the sphaleron process is over Gu:2009hn ().

Now we will turn our focus on the dark matter observables.

Iii Abundance, Direct and Indirect Detection and Higgs Bounds

Viable dark matter candidates are either truly stable or cosmologically stable in the sense that their lifetime is much bigger than the age of the universe. In our model the Majoron belongs to the latter case as we will explain further. After the spontaneous symmetry breaking Eq.(1) turns into,

(14)

where J is the pseudo-Majoron and the dark matter candidate of the model. We can clearly see that in the mass regime of interest that is , the Majoron will decay into SM particles via virtual right-handed neutrinos which are expected to be super heavy due to the large value of the vev u. The decay width is dominated by the two-body decay into SM neutrinos according to,

(15)

which gives

(16)

Hence for the mass range of interest a singlet Majoron is cosmologically stable and in principle a viable dark matter candidate. In our model the Majoron is thermally produced via the Higgs portal according to the interactions,

(17)

Therefore the parameter play a crucial role in this model because it connects the dark and visible sector of the model. Moreover, this coupling between pseudo-Majoron to the SM Higgs sets the relic abundance as well as the scattering cross section. At the end of the day we are left with two free parameters only which are and the Majoron mass. Note for a sizeable , the present symmetry breaking pattern, i.e. , requires a large cancellation between and so that can be of the order of ernestma ().

First we would like to derive Higgs bounds on the Majoron model. For a Majoron lighter than half of the Higgs mass, the h is kinematically available and therefore it alters the measured invisible width of Higgs. The current limit on the branching ration into invisible particles is around Belanger:2013xza (); Ellis:2013lra (). A projected bound of at  TeV LHC after has been claimed Peskin:2012we (). We use the latter one as reference with no impact on our conclusions. In this Majoron model the Higgs branching ratio into Majoron is found to be,

(18)

where  MeV for  GeV and,

(19)

With this in mind we have derived the bound shown in Fig.1 (pink shaded region). There we show that has to be smaller than . It is important to point out that there is a lower bound of for the Majorons to get thermalized down to the electroweak epoch Burgess:2000yq ().

Regarding the dark matter observables we have computed the relic abundance and the scattering cross section of the Majoron as a function of the two relevant parameters and the Majoron mass. The relic density of singlet Majoron is driven by the s-channel annihilation into SM particles and sub-dominantly determined by the annihilation into hh, through the quartic scalar interaction in Eq.17 and by the t-channel Majoron exchange. In Fig.1 we exhibit the under-abundant (light gray) and the over-abundant (dark gray) parameter space, as well as region determined by the green points which yields the right abundance. 111There is a small resonance in the Top quark mass. This is due to our numerical calculation using micrOMEGAS micromegas (). Micromegas computes tree-level calculations into two-body final states only, except when the gauge bosons are in the final state. Therefore micrOMEGAS does not provide a very accurate approximation close to the threshold for producing gauge boson pairs as they miss the 3- and 4-body final states from virtual decays as well as QCD corrections for quarks in the final state. However, the inclusion of these corrections induce only mild differences in the abundance for the model of interest. The reader can easily see that our results agree well with the results of Ref.Cline:2013gha (); Cline:2012hg () where such processes have been accounted for. Therefore, our conclusions are unaffected by the insertion of these corrections. See Refs. for similar studies of the Higgs portal hportalmodels ().

In the left panel of Fig.1 we have zoomed in the resonance region in order to clearly show that the Higgs invisible decay width bound requires assuming a . If we had assumed a 10% limit instead the would have found . Additionally, the green fine line between the light and dark gray regions reproduces the right abundance according to PLANCK data PLACKcoll ().

In the right panel of Fig.1 we display in blue the region ruled out by the LUX bound based on the 2013 data LUXbound (). One can clearly conclude that a thermally produced singlet Majoron is ruled out by direct detection data for  GeV, unless we are sitting at the Higgs resonance. Moreover, future direct detection running coming from XENON1T and LUX are forecasted to exclude the entire GeV-TeV mass range. In order to circumvent this result one might need to evoke the presence of new particles in order to use them as potential co-annihilation channels and consequently suppress the abundance. In other words, pushing down the parameter space which yields the right abundance. Conversely, one could invoke a non-thermal Majoron where the number density is set by a decaying modulus without further annihilation.

Figure 1: The viable excluded parameter space of the Majoron dark matter model. The light gray scatter points on the top yield , whereas the dark gray ones towards the bottom produces . The fine green line between the two gray regions sets . The pink shaded region is ruled out by the Higgs invisible decay width taking the face value . The blue region is excluded by LUX 2013 data. The parameter is the coupling which connects the dark and visible sector through the Higgs portal. In the left panel we zoomed in the Higgs resonance region and the invisible decay bound, so one can clearly see the latter bounds requires . In the right panel we included the LUX constraint in blue. Apart from the resonance region we conclude that the simplest thermal Majoron model studied in the previous section is excluded by direct detection data for  GeV. Moreover, future XENON1T and LUX running are expected to exclude the entire 1GeV-TeV mass range, except the Higgs resonance.

In addition to the abundance and direct detection bounds, it is important to also consider indirect detection bounds in this model. Late decay of Majorons to neutrinos would produce too much power at large scales, through the late integrated Sachs-Wolfe effect, thus spoiling the CMB anisotropy spectrum. WMAP third year data can be used to constrain CMBmajoron (),

(20)

From Eq.16 one may notice for the mass range of interest ( TeV) this limit is easily obeyed as long as a huge lepton number violation scale is assumed. Furthermore, bounds coming from measurements of the atmospheric neutrinos background flux might be important as well Covi:2009xn (); Esmaili:2012us (), but for Majorons lighter than  TeV the LUX bounds are the most relevant because of the the cutoff in the energy flux. See Fig.2 of Covi:2009xn (). Because in this section we are studying the simplest see-saw Majoron model, no radiative decay is present and therefore no bounds from x- or -ray apply. However, in the next section we briefly discuss the case of light Majoron dark matter in more general seesaw models, which can possibly explain the recent keV line observed by KeVline () and show the most current constraints on x- and -rays lines in Fig.2.

Iv Light Majorons and the keV Line

In this section, we discuss light Majorons with mass keV as dark matter candidates. We will assume that the coupling to the Higgs portal is small enough to evade invisible decay width and other bounds described in the previous section.

The relic density of light Majoron dark matter at the present time is determined by its number density relative to photons at the time of decoupling, and should also account for the finite decay width in Eq.20. In general one might write the relic density as

(21)

for the case of thermal freeze-out at the time of decoupling (and ). Clearly, a Majoron of mass keV satisfies the relic density constraint via thermal freeze-out. In general, however, higher masses are possible - for example, thermal freeze-in of the Majoron through the sterile neutrino portal can accommodate masses between 1 keV and 3 MeV Frigerio:2011in (). Various non-thermal mechanisms of Majoron production have also been considered in the literature Rothstein:1992rh (), which can give a heavier Majoron in the keV range to satisfy the relic density.

In the simplest Majoron model described in the previous sections, there is no coupling to photons. However, a coupling like is natural at loop level in see-saw models with the presence of scalar that are triplets under Berezinsky:1993fm ().

Following Berezinsky:1993fm (), one extends the Lagrangian in Eq. 1 by also adding the term , where has hypercharge 1 and lepton number and is the relevant Yukawa coupling. In a calculation similar to that in the previous sections, one finds that the Majoron now has a non-zero component along the Higgs doublet, and it is given by

(22)

where and are the singlet, doublet, and triplet Higgs vevs respectively, arranged to satisfy the relation . The Majoron is thus still mostly along the singlet direction, although the doublet projection will be vital for us.

The decay rate to neutrinos (which controls the lifetime and cosmological stability of the Majoron) is controlled mainly by the profile of the Majoron along the singlet Higgs direction in accordance with Eq. 15. This can be seen by writing out the coupling of the Majoron to the mass eigenstate neutrinos explicitly to leading order, most conveniently by utilizing the symmetry properties of the Higgs potential

(23)

The profile along the triplet direction may also lead to interesting physics, such as the possible decay of neutrinos, as elucidated in Berezinsky:1993fm (). Models with the triplet Higgs , independently of the Majoron, have been widely studied in the context of Higgs physics, LHC searches, and Left-Right models triplethiggs ().

The leading decay width of the Majoron is to neutrinos. But now there is also a coupling of the Majoron to charged fermions coming from the Higgs projection, whose strength goes like

(24)

where and are the mass and weak isospin of the charged fermions. This further induces at loop level a coupling to photons, with a decay width given by

(25)

where . Note that and are the electric charge and color factor of the SM fermions that couple to the Majoron CMBmajoron ().

In the analysis of KeVline (), the requirements to explain the keV line observed by XMM-Newton observatory in the spectrum of 73 galaxy clusters are

(26)

The required decay width can be obtained for GeV, with TeV The triplet vev in this range is compatible with electroweak precision constraints Erler:1999ub (). We note that this value of also satisfies the constraint on the leading order decay to neutrinos, Eq. 20. Moreover, this result is also consistent x-rays bounds coming from a variety of sources. In Fig.2 we show the exclusion regions as well as the result predicted in the Majoron model with a Higgs triplet discussed above.

Since we will be interested in keV, the relic density constraint requires a mechanism that is different from thermal freezeout. We mention a few more points about satisfying the relic density in the scenario above. As stated before, thermal freeze-in is an option to satisfy the relic density with a keV Majoron. In this scenario, the sterile neutrinos annihilate into the Majoron with a rate that is too low for the Majoron to thermalize. The under-abundant Majoron density reaches a plateau when the temperature mass of the sterile neutrinos. The process thus depends on the annihilation cross section of the sterile neutrinos into the Majoron, which depends on the mass and the Yukawa coupling of the Majoron with the sterile neutrinos, as in Eq. 1. In fact, for the Majoron mass keV needed to satisfy the x-ray signal, we require for thermal freeze-in to give the correct relic density Frigerio:2011in ().

In summary, a Majoron produced by thermal freeze-in or non-thermally might be a potential explanation to this 3.5 keV line recently observed while still being consistent with other searches. Alternative solutions to this keV line have been put forth KeVmodels ().

Figure 2: We show the predicted line emission in the Majoron model with Higgs triplets. Besides we exhibit the line emission constraints on the decay rate into two monoenergetic photons from left to right: CHANDRA LETG NGC3127 (orange) CMBmajoron (), Milky Way (blue)M31bound (), HEAO observations of the diffuse x-ray background (green) HEAO (), INTEGRAL diffuse background INTEGRAL2 (), COMPTEL and EGRET Telescopes (pink) COMPTELEGRET (), M31 (dark gray) BULLETCluster (), XMM observations of the Milky Way (red) XMMbound (), Fermi-LAT search (yellow) fermi (),INTEGRAL SPI line search in the Milky Way halo (light gray) INTEGRAL ().

V Conclusions

Majorons are Nambu-Goldstone bosons arising from the spontaneous breaking of lepton number symmetry by a complex scalar. The Majoron mass is model-dependent, although it is expected to be small, due to explicit lepton symmetry breaking by quantum gravity effects. On the other hand, explicit soft terms can be introduced to make the Majoron mass large.

These scalar dark matter models have several interesting features:

they are examples of decaying dark matter, with the decay being mainly to neutrinos, suppressed by the scale of lepton symmetry breaking.

in the heavy (GeV - TeV) Majoron case, this model is a UV realization of the effective Higgs-portal scalar dark matter framework. With this in mind we have computed the Higgs invisible decay width into Majorons, abundance, direct and indirect detection observables and compared them with the most current data available. The Majoron has a somewhat large parameter space that can reproduce the right abundance as shown in Fig.1. In case the Majoron is sitting at the Higgs resonance the model is consistent with all current bounds. Otherwise the recent constraint on the scattering cross section reported by LUX in 2013 decisively rules out the thermal Majoron window for  GeV in the simplest Majoron model which has a singlet scalar and heavy-right handed neutrinos. Additionally, future direct detection running coming from XENON1T and LUX are expected to rule out the entire GeV-TeV mass range. Therefore either non-thermal production mechanisms or the inclusion of new particle to exploit co-annihilation channels are required to circumvent this conclusion. Another alternative would be the inclusion of new particles to play to role of the mediator in new annihilations channels.

We have also discussed indirect detections bounds. In particular the late decay of Majorons may distort the CMB power spectrum, therefore a bound of is required. This enforces the Majoron mass to not be much greater than the TeV scale for a large scale where the lepton number is softly broken. If one pushes down the latter scale the upper bound on Majoron is rapidly strengthened according to Eq.16.

lastly, we discussed a Majoron model which has a Higgs triplet in its spectrum in light of the recent 3.5 keV line and concluded that the required signal could be obtained for a lepton number violation scale TeV. We noted that the thermal freeze-out scenario does not address such a line because the mass of the Majoron cannot be larger than  keV. However, if one uses thermal freeze-in through the sterile neutrino portal to set the Majoron relic density (alternatively, a non-thermal mechanism may also work), then it is possible to have the Majoron as the possible candidate to this  keV signal while being consistent with other x-ray searches according to Fig.2. We provided a proof of concept that the observed keV line can be obtained in this class of models, for plausible choices of the model parameters. The relation between light decaying dark matter, line emission bounds, and thermal freeze-in certainly warrants further study farinaldokuver ().

Acknowledgement

FQ is partly supported by US Department of Energy Award SC0010107 and the Brazilian National Counsel for Technological and Scientific Development (CNPq). KS is supported by NASA Astrophysics Theory Grant NNH12ZDA001N. The authors would like to thank Will Shepherd and Patrick Draper for useful discussions.

References

  • (1) R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. c44, 912 (1980);
  • (2) S. Kashiwase and D. Suematsu, Eur. Phys. J. C 73, 2484 (2013) [arXiv:1301.2087 [hep-ph]].
  • (3) T. Tsuyuki, arXiv:1403.5053 [hep-ph].
  • (4) P. -H. Gu, E. Ma and U. Sarkar, Phys. Lett. B 690, 145 (2010) [arXiv:1004.1919 [hep-ph]].
  • (5) F. Bazzocchi, M. Lattanzi, S. Riemer-Sorensen and J. W. F. Valle, JCAP 0808, 013 (2008) [arXiv:0805.2372 [astro-ph]]; M. Lattanzi, S. Riemer-Sorensen, M. Tortola and J. W. F. Valle, Phys. Rev. D 88, 063528 (2013) [arXiv:1303.4685 [astro-ph.HE]].
  • (6) G. Gelmini, D. N. Schramm and J. W. F. Valle, Phys. Lett. B 146, 311 (1984).
  • (7) V. Berezinsky and J. W. F. Valle, Phys. Lett. B 318, 360 (1993) [hep-ph/9309214].
  • (8) L. Covi, M. Grefe, A. Ibarra and D. Tran, JCAP 1004, 017 (2010) [arXiv:0912.3521 [hep-ph]].
  • (9) C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B 619, 709 (2001) [hep-ph/0011335].
  • (10) R. Allahverdi, B. Dutta and K. Sinha, Phys. Rev. D 83, 083502 (2011) [arXiv:1011.1286 [hep-ph]]; R. Allahverdi, B. Dutta and K. Sinha, Phys. Rev. D 82, 035004 (2010) [arXiv:1005.2804 [hep-ph]].
  • (11) E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein, et al. (2014), arxiv:1402.2301; A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, and J. Franse (2014), arxiv:1402.4119.
  • (12) P. -H. Gu and U. Sarkar, Eur. Phys. J. C 71, 1560 (2011) [arXiv:0909.5468 [hep-ph]].
  • (13) G. Belanger, B. Dumont, U. Ellwanger, J. F. Gunion and S. Kraml, Phys. Rev. D 88, 075008 (2013)
  • (14) J. Ellis and T. You, JHEP 1306, 103 (2013) [arXiv:1303.3879 [hep-ph]].
  • (15) M. E. Peskin, arXiv:1207.2516 [hep-ph].
  • (16) J. M. Cline, K. Kainulainen, P. Scott and C. Weniger, Phys. Rev. D 88, 055025 (2013) [arXiv:1306.4710 [hep-ph]].
  • (17) J. M. Cline and K. Kainulainen, JCAP 1301, 012 (2013) [arXiv:1210.4196 [hep-ph]].
  • (18) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 185, 960 (2014) [arXiv:1305.0237 [hep-ph]]; G. Belanger, F. Boudjema, P. Brun, A. Pukhov, S. Rosier-Lees, P. Salati and A. Semenov, Comput. Phys. Commun. 182, 842 (2011) [arXiv:1004.1092 [hep-ph]].
  • (19) E. C. F. S. Fortes and M. D. Tonasse, arXiv:1312.2837 [hep-ph]; J. K. Mizukoshi, C. A. de S.Pires, F. S. Queiroz and P. S. Rodrigues da Silva, Phys. Rev. D 83, 065024 (2011) [arXiv:1010.4097 [hep-ph]]; P. V. Dong, T. P. Nguyen and D. V. Soa, Phys. Rev. D 88, 095014 (2013) [arXiv:1308.4097 [hep-ph]]; J. D. Ruiz-Alvarez, C. A. de S.Pires, F. S. Queiroz, D. Restrepo and P. S. Rodrigues da Silva, Phys. Rev. D 86, 075011 (2012) [arXiv:1206.5779 [hep-ph]]; K. P. Modak, D. Majumdar and S. Rakshit, arXiv:1312.7488 [hep-ph]; D. Cogollo, A. X. Gonzalez-Morales, F. S. Queiroz and P. R. Teles, arXiv:1402.3271 [hep-ph]; M. S. Boucenna and S. Profumo, Phys. Rev. D 84, 055011 (2011) [arXiv:1106.3368 [hep-ph]]; A. Djouadi, A. Falkowski, Y. Mambrini and J. Quevillon, Eur. Phys. J. C 73, 2455 (2013) [arXiv:1205.3169 [hep-ph]]; A. Djouadi, O. Lebedev, Y. Mambrini and J. Quevillon, Phys. Lett. B 709, 65 (2012) [arXiv:1112.3299 [hep-ph]]; Y. Mambrini, J. Phys. Conf. Ser. 375, 012045 (2012) [arXiv:1112.0011 [hep-ph]]. J. C. Montero and B. L. Sanchez-Vega, Phys. Rev. D 84, 055019 (2011) [arXiv:1102.5374 [hep-ph]]; A. G. Dias and V. Pleitez, Phys. Rev. D 73, 017701 (2006) [hep-ph/0511104]; A. G. Dias, C. A. de S.Pires and P. S. Rodrigues da Silva, Phys. Lett. B 628, 85 (2005) [hep-ph/0508186].
  • (20) P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].
  • (21) D. S. Akerib et al. [LUX Collaboration], arXiv:1310.8214 [astro-ph.CO].
  • (22) A. Esmaili, A. Ibarra and O. L. G. Peres, JCAP 1211, 034 (2012) [arXiv:1205.5281 [hep-ph]].
  • (23) J. F. Gunion, R. Vega and J. Wudka, Phys. Rev. D 42, 1673 (1990); N. G. Deshpande, J. F. Gunion, B. Kayser and F. I. Olness, Phys. Rev. D 44, 837 (1991); C. Englert, E. Re and M. Spannowsky, Phys. Rev. D 88, 035024 (2013) [arXiv:1306.6228 [hep-ph]]; K. Agashe, A. Azatov, A. Katz and D. Kim, Phys. Rev. D 84, 115024 (2011) [arXiv:1109.2842 [hep-ph]]; H. E. Logan and M. -A. Roy, Phys. Rev. D 82, 115011 (2010) [arXiv:1008.4869 [hep-ph]];
  • (24) C. R. Watson, J. F. Beacom, H. Yuksel and T. P. Walker, Phys. Rev. D 74, 033009 (2006) [arXiv:astro-ph/0605424]; A. Boyarsky, D. Iakubovskyi, O. Ruchayskiy and V. Savchenko, arXiv:0709.2301 [astro-ph].
  • (25) A. Boyarsky, A. Neronov, O. Ruchayskiy and M. Shaposhnikov, Mon. Not. Roy. Astron. Soc. 370, 213 (2006) [arXiv:astro-ph/0512509]; A. Boyarsky, A. Neronov, O. Ruchayskiy, M. Shaposhnikov and I. Tkachev, Phys. Rev. Lett. 97, 261302 (2006) [arXiv:astro-ph/0603660].
  • (26) A. Boyarsky, D. Malyshev, A. Neronov, and O. Ruchayskiy, Mon. Not. R. Astron. Soc. 387, 1345 (2008).
  • (27) H. Yuksel and M. D. Kistler, Phys. Rev. D 78, 023502 (2008).
  • (28) A. Boyarsky, O. Ruchayskiy and M. Markevitch, Astrophys. J. 673, 752 (2008) [arXiv:astro-ph/0611168]
  • (29) A. Boyarsky, J. Nevalainen and O. Ruchayskiy, Astron. Astrophys. 471, 51 (2007) [arXiv:astro-ph/0610961].
  • (30) M. Ackermann et al., Phys. Rev. D 86, 022002 (2012).
  • (31) A. Boyarsky, D. Malyshev, A. Neronov and O. Ruchayskiy, arXiv:0710.4922 [astro-ph]; H. Yuksel, J. F. Beacom and C. R. Watson, arXiv:0706.4084 [astro-ph].
  • (32) J. Erler and P. Langacker, Phys. Lett. B 456, 68 (1999) [hep-ph/9903476].
  • (33) M. Frigerio, T. Hambye and E. Masso, Phys. Rev. X 1, 021026 (2011) [arXiv:1107.4564 [hep-ph]].
  • (34) I. Z. Rothstein, K. S. Babu and D. Seckel, Nucl. Phys. B 403, 725 (1993) [hep-ph/9301213].
  • (35) N. -E. Bomark and L. Roszkowski, arXiv:1403.6503 [hep-ph]; C. Kolda and J. Unwin, arXiv:1403.5580 [hep-ph]; M. Cicoli, J. P. Conlon, M. C. D. Marsh and M. Rummel, arXiv:1403.2370 [hep-ph]; S. Baek and H. Okada, arXiv:1403.1710 [hep-ph]; K. Nakayama, F. Takahashi and T. T. Yanagida, arXiv:1403.1733 [hep-ph]; R. Krall, M. Reece and T. Roxlo, arXiv:1403.1240 [hep-ph]; H. M. Lee, S. C. Park and W. -I. Park, arXiv:1403.0865 [astro-ph.CO]; T. Higaki, K. S. Jeong and F. Takahashi, arXiv:1402.6965 [hep-ph]; D. P. Finkbeiner and N. Weiner, arXiv:1402.6671 [hep-ph]; M. Frandsen, F. Sannino, I. M. Shoemaker and O. Svendsen, arXiv:1403.1570 [hep-ph]; S. P. Liew, arXiv:1403.6621 [hep-ph]; R. Allahverdi, B. Dutta and Y. Gao, arXiv:1403.5717 [hep-ph]. P. Ko, Z. kang, T. Li and Y. Liu, arXiv:1403.7742 [hep-ph]. K. Kong, J. -C. Park and S. C. Park, arXiv:1403.1536 [hep-ph]; A. G. Dias, A. C. B. Machado, C. C. Nishi, A. Ringwald and P. Vaudrevange, arXiv:1403.5760 [hep-ph].
  • (36) Farinaldo Queiroz and Kuver Sinha, work in progress.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
234118
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description