Exotic Sterile Neutrinos and Pseudo-Goldstone Phenomenology
We study the phenomenology of a light (GeV scale) sterile neutrino sector and the pseudo-Goldstone boson (not the majoron) associated with a global symmetry in this sector that is broken at a high scale. Such scenarios can be motivated from considerations of singlet fermions from a hidden sector coupling to active neutrinos via heavy right-handed seesaw neutrinos, effectively giving rise to a secondary, low-energy seesaw framework. Such scenarios allow for rich phenomenology with observable implications for cosmology, dark matter, and direct searches, involving novel sterile neutrino dark matter production mechanisms from the pseudo-Goldstone-mediated scattering or decay, modifications of BBN bounds on sterile neutrinos, suppression of canonical sterile neutrino decay channels at direct search experiments, late injection of an additional population of neutrinos in the Universe after neutrino decoupling, and measurable dark radiation at BBN or CMB decoupling.
The most straightforward explanation of tiny neutrino masses is the seesaw mechanism, involving Standard Model (SM) singlet (sterile) right-handed neutrinos at a heavier scale. GUT (grand unified theory) scale seesaw models Minkowski (1977); Mohapatra and Senjanovic (1981); Yanagida (1980); Gell-Mann et al. (1979); Schechter and Valle (1980) accomplish this with couplings with heavy sterile neutrinos at GeV. However, the seesaw mechanism is also consistent with light sterile neutrinos below the electroweak scale, which finds additional motivation from dark matter (DM) and leptogenesis considerations as in the Neutrino Minimal Standard Model (MSM) Asaka et al. (2005); Asaka and Shaposhnikov (2005); Asaka et al. (2007), and is also attractive due to rich phenomenology, offering signals in cosmology, indirect detection, as well as direct searches Abazajian et al. (2012); Drewes et al. (2017).
Drastic departures from this phenomenology, which is dictated by the mixing between the active and sterile neutrino sectors as necessitated by the seesaw mechanism, is possible if additional structure (such as additional symmetries or particles) exists in the sterile neutrino sector beyond the basic elements of the seesaw framework (see e.g. discussions in Ma (2017); Alanne et al. (2017); Sayre et al. (2005)). Charges under an additional symmetry are plausibly necessary for sterile neutrinos to be light, since the Majorana mass of a pure singlet fermion is expected to lie at the ultraviolet (UV) cutoff scale of the theory (such as the GUT or Planck scale). A well-motivated choice is to draw a connection to lepton number, tying the sterile neutrino masses with a low scale of lepton number breaking Chikashige et al. (1981); Gelmini and Roncadelli (1981); Georgi et al. (1981); Schechter and Valle (1982); Gelmini et al. (1982); Lindner et al. (2011); Escudero et al. (2017); rich phenomenology ensues from the existence of additional scalars Maiezza et al. (2015); NemevÅ¡ek et al. (2017); Dev et al. (2017) and massive gauge bosons Mohapatra and Senjanovic (1980, 1981); Heeck and Teresi (2016) or a (pseudo-) Goldstone boson, the majoron Chikashige et al. (1981); Gelmini and Roncadelli (1981); Georgi et al. (1981); Schechter and Valle (1982); Gelmini et al. (1982); Lindner et al. (2011); Escudero et al. (2017).
It is, however, also possible that the protecting symmetry is confined entirely to the sterile neutrino sector. This can occur, for instance, if the sterile neutrinos originate from a separate hidden sector. In the next section, we will see that even with a GUT-scale realization of the seesaw mechanism, exotic fermions from hidden sectors that couple to the GUT scale right-handed neutrinos develop couplings to the SM neutrinos, mimicking a low energy seesaw setup, thus effectively acting as light sterile neutrinos akin to those studied in, e.g. the MSM.
In this letter, we study the phenomenology associated with a pseudo-Goldstone boson of a global symmetry confined to, and spontaneously broken in, a light (GeV scale) exotic sterile neutrino sector. Since GeV scale seesaw sterile neutrinos play an important role in the early Universe — they equilibrate with the thermal bath and dominate the energy density before big bang nucleosynthesis (BBN) Asaka et al. (2006) — their interplay with can lead to novel cosmological scenarios. The phenomenology can be very different from the more familiar majoron phenomenology, as the scale of symmetry breaking, lepton number breaking, and sterile neutrino masses are all different. This could enable the and the sterile neutrinos to have similar masses. This freedom of scale separation and coincidence of masses gives rise to many rich possibilities for cosmology, dark matter, and direct searches that are not possible in the majoron framework.
Ii Charged-Singlet Seesaws
The canonical seesaw mechanism involves three SM-singlet, right-handed neutrinos with the following Dirac and Majorana masses:
and are the SM lepton doublet and Higgs fields, and are dimensionless Yukawa couplings. The hierarchy (where is the Higgs vacuum expectation value (vev)) leads to the familiar seesaw mechanism, resulting in active and sterile neutrino masses , with an active-sterile mixing angle sin . GeV produces the desired neutrino masses for , whereas GeV requires .
A global or gauged or symmetry for Chikashige et al. (1981); Gelmini and Roncadelli (1981); Georgi et al. (1981); Schechter and Valle (1982); Gelmini et al. (1982); Lindner et al. (2011); Escudero et al. (2017) allows for the Dirac mass term but precludes the Majorana mass term until the symmetry is broken. In this case, the lagrangian is instead
with the singlet Higgs field appropriately charged under the lepton or symmetry. A vev breaks the symmetry and gives rise to sterile neutrino masses . If the symmetry is global, a physical light degree of freedom, the Goldstone boson, the majoron, emerges Chikashige et al. (1981); Gelmini and Roncadelli (1981).
In this paper, we consider instead a global symmetry, for instance a , that is confined to the sterile neutrinos and does not extend to any of the SM fields. Such a symmetry forbids both the Dirac and Majorana mass terms in Eq. 1. Nevertheless, a scalar field carrying the opposite charge to enables the higher dimensional operator , where is a UV-cutoff scale.
ii.1 “Sterile neutrinos” from a hidden sector with a heavy right-handed neutrino portal
We start with the original seesaw motivation of pure singlet, heavy (scale , possibly close to the GUT scale) right-handed neutrinos that couple through Yukawa terms . If the also act as portals to a hidden sector
If the hidden sector scalar acquires a vev , this can be rewritten as
where we have defined , and . Here, the first term accounts for the active neutrino masses from the primary seesaw involving integrating out the pure singlet neutrinos . The latter two terms give a similar contribution to the active neutrino masses from the secondary seesaw resulting from integrating out the fermions (note the analogy between Eq. 4 and Eq. 1).
The mixing angle between the active neutrinos and these hidden sector singlets is approximately
which is precisely the relation expected from a seesaw framework. Therefore, light sterile neutrinos that appear to satisfy the seesaw relation could have exotic origins in a hidden sector connected via a high scale neutrino portal, be charged under symmetries unrelated to the SM, and themselves obtain light masses via the seesaw mechanism, which is active at a much higher scale .
ii.2 Pseudo-Goldstone Boson
The spontaneous breaking of the global by gives rise to a massless Goldstone boson, which we will call the -boson. It is conjectured that non-perturbative gravitational effects explicitly break global symmetries, leading to a pseudo-Goldstone boson mass of order via an operator of the form Rothstein et al. (1993); Akhmedov et al. (1993).
Next, we draw the distinction between the -boson and the more familiar majoron Chikashige et al. (1981); Gelmini and Roncadelli (1981); Georgi et al. (1981); Schechter and Valle (1982); Gelmini et al. (1982); Lindner et al. (2011); Escudero et al. (2017). For both the majoron and , couplings to (both active and sterile) neutrinos are proportional to the neutrino mass suppressed by the scale of symmetry breaking, as expected for Goldstone bosons, hence several phenomenological bounds on the majoron symmetry breaking scale Gelmini et al. (1983, 1982); Jungman and Luty (1991); Pilaftsis (1994); Garcia-Cely and Heeck (2017); Drewes et al. (2017) are also applicable to . However, the majoron is associated with the breaking of lepton number — a symmetry shared by the SM leptons as well as the sterile neutrinos — and the sterile neutrino mass scale approximately coincides with the scale of lepton number breaking. This results in the majoron being much lighter that the sterile neutrinos. In addition, this scaling results in specific relations between majoron couplings and sterile neutrino masses, which drives many of the constraints on majoron parameter space Gelmini et al. (1983, 1982); Jungman and Luty (1991); Pilaftsis (1994); Garcia-Cely and Heeck (2017); Drewes et al. (2017).
In contrast, these energy scales are distinct in the framework: the symmetry breaking scale (i.e., the scale of breaking) is independent of the breaking of lepton number, which occurs at a much higher scale , and is also distinct from the sterile neutrino mass scale (), which, as discussed above, can itself be suppressed by a seesaw mechanism. The ability to vary them independently opens up phenomenologically interesting regions of parameter space. Furthermore, the sterile neutrino masses can be comparable to the -boson mass (if ); this coincidence of mass scales can have important implications for cosmology and DM, as we will see later.
Iii Framework and Phenomenology
We focus on a low-energy effective theory containing three sterile neutrinos (which we have reset to the label rather than ), and the pseudo-Goldstone boson .
We treat , and as independent parameters. We assume GeV scale, and are correspondingly small in a natural way that matches the measured and mixings among the light active neutrinos. We will consider the possibility that the lightest sterile neutrino is DM, since this is an interesting and widely studied case, and especially appealing given recent claims of a 3.5 keV X-ray line Bulbul et al. (2014); Boyarsky et al. (2014) compatible with decays of a 7 keV sterile neutrino DM particle. We also assume that so that the breaking singlet scalar is decoupled and irrelevant for phenomenology.
Lifetime: The lifetime is controlled by decay rates into (both active and sterile) neutrinos. For instance,
where eV is the active neutrino mass scale. For decay channels and involving the sterile neutrinos, the in the above formula is replaced by and respectively.
Fig. 1 shows the lifetime as a function of , with GeV and keV, for two different values of . Depending on the scale and the available decay channels, a range of interesting lifetimes are possible: can decay before or after BBN (and before/after Cosmic Microwave Background (CMB) decoupling), or live longer than the age of the Universe, providing a potential DM candidate (for studies of majoron DM, see Rothstein et al. (1993); Berezinsky and Valle (1993); Gu et al. (2010); Frigerio et al. (2011); Queiroz and Sinha (2014); Lattanzi et al. (2014); Boucenna et al. (2014); Boulebnane et al. (2017); Heeck and Teresi (2017)).
A pseudo-Goldstone coupling to neutrinos faces several constraints Choi and Santamaria (1990); Pastor et al. (1999); Kachelriess et al. (2000); Hirsch et al. (2009); Garcia i Tormo et al. (2011). However, many of these constraints weaken/become inapplicable if the pseudo-Goldstone is heavy or can decay into sterile neutrinos. We remark that these constraints are generally not very stringent in the parameter space of interest in our framework.
Cosmology: In the early Universe, GeV scale sterile neutrinos (but not the DM candidate , which has suppressed couplings to neutrinos) are in equilibrium with the thermal bath due to their mixing with active neutrinos, decouple while still relativistic at GeV Asaka et al. (2006), can grow to dominate the energy density of the Universe, and decay before BBN Scherrer and Turner (1985); Bezrukov et al. (2010); Asaka et al. (2006).
On the other hand, couples appreciably only to the sterile neutrinos, and is produced via sterile neutrino annihilation processes (see Fig.2 (a)) or decay (if kinematically open). The annihilation process, although -wave suppressed, is nevertheless efficient at high temperatures . One can estimate the magnitude of for such annihilations to be rapid compared to Hubble expansion by comparing the annihilation cross section Garcia-Cely et al. (2014); Gu and Sarkar (2011) with the Hubble rate at
For GeV, this process is efficient for GeV, and produces an abundance comparable to the abundance. For larger values of , the annihilation process is feeble, and a small abundance will accumulate via the freeze-in process instead Chung et al. (1999); Hall et al. (2010).
Dark Matter Production: can also mediate interactions between the sterile neutrinos (Fig.2 (b)), which enables a novel DM production mechanism . One can analogously estimate the scale below which this production cross section Chacko et al. (2004) is efficient: This would generate an abundance comparable to relativistic freezeout, which generally overcloses the Universe, hence this scenario is best avoided. Likewise, decays can also produce DM if . By comparing rates, we find that production from such decays dominates over the annihilation process provided , which generally holds over most of our parameter space. Additional DM production processes, such as annihilation and decays via an off-shell , are always subdominant and therefore neglected. The novel production processes discussed here do not rely on mixing with active neutrinos, which is particularly appealing since this canonical (Dodelson-Widrow) production mechanism Dodelson and Widrow (1994) is now ruled out by various constraints Boyarsky et al. (2006a); Boyarsky et al. (2007); Boyarsky et al. (2006b, 2008a, 2008b); Seljak et al. (2006); Asaka et al. (2007); Boyarsky et al. (2009); Horiuchi et al. (2014).
Next, we discuss various interesting cosmological histories that are possible with this framework. Our purpose is not to provide a comprehensive survey of all possibilities, which is beyond the scope of this note, but simply to point out some interesting features. Since available decay channels and lifetimes are crucial to the subsequent evolution of these particles, we find it useful to organize our discussion into three different regimes.
When , all decay channels to sterile neutrinos are open, and decays rapidly, long before BBN. If is rapid, maintains an equilibrium distribution at , and the decay generates a freeze-in abundance of , which can be estimated to be Hall et al. (2010); Shakya (2016); Roland and Shakya (2016); Shakya and Wells (2017); Merle et al. (2014); Adulpravitchai and Schmidt (2015); Kang (2015); Roland et al. (2015a); Merle and Totzauer (2015)
The observed DM abundance is produced, for instance, with GeV, GeV, and keV.
If the annihilation process is feeble, a freeze-in abundance of is generated instead, and its decays produce a small abundance of . The yield is suppressed by the branching fraction BR(. The resulting abundance is much smaller than from Eq. 9 and cannot account for all of DM unless .
In addition to annihilation processes, can now also be produced directly from heavy sterile neutrino decay when . Ignoring phase space suppression, the decay rate is
If this width is sufficiently large, this exotic decay channel can compete with the traditional sterile neutrino decay channels induced by active-sterile mixing Gorbunov and Shaposhnikov (2007). In Fig. 3, we plot (blue curve) the scale below which this decay channel dominates (assuming the standard seesaw relations). This region carries important implications for collider and direct searches for sterile neutrinos (such as at DUNE Adams et al. (2013) and SHiP Jacobsson (2016)), as the new dominant channel suppresses the traditionally searched-for decay modes, rendering the sterile neutrinos invisible at such detectors (unless also decays in the detector, as can occur if it is not DM and participates in the seesaw instead).
It is well known that are constrained by several recombination era observables Kusenko (2009); Hernandez et al. (2014); Vincent et al. (2015) and generally are required to decay before BBN, necessitating and consequently MeV in the standard seesaw formalism. The above decay channel , if sufficiently large, can reduce the sterile neutrino lifetime, thus allowing lighter masses to be compatible with BBN. In Fig. 3, the red dashed line shows the scale below which the sterile neutrino decays before BBN. For GeV, even lighter (MeV scale) sterile neutrinos are compatible with the seesaw as well as BBN constraints, in stark contrast to the standard seesaw implications.
Depending on parameters, can decay before or after BBN (see Fig. 1), but its dominant decay channel is , which can be DM. If decay dominantly into , or if thermalizes with , the abundance is comparable to that from relativistic freezeout, which would likely overclose the Universe. Viable regions of parameter space instead involve a small fraction of decaying into , which subsequently decays to . In this case, the heavy sterile neutrinos undergo relativistic freezeout, and their branching ratio into gets converted completely to ; one can thus derive the following approximate relation for to account for the observed DM abundance for GeV:
For instance, keV requires GeV.
Here, DM () is produced from late decays of heavier particles ( and ) and can be warm. Such late production of warm DM can carry interesting cosmological signatures and structure formation implications, but a detailed study lies beyond the scope of this paper.
The most interesting aspect of the hierarchy is that all sterile neutrinos (including the DM candidate ) can decay into . In particular, a new very long-lived DM decay channel exists. Since subsequently decays into two neutrinos, this can provide distinct signatures at neutrino detectors such as IceCube, Borexino, KamLAND, and Super-Kamiokande. Note that, unlike the standard decay channel, this has no gamma ray counterpart.
Another intriguing possibility with this hierarchy is that is extremely long-lived, and if sufficiently light, can contribute measurably to dark radiation at BBN or CMB Chikashige et al. (1980); Garcia-Cely et al. (2013, 2014). Ref. Weinberg (2013) pointed out that a Goldstone that freezes out above 100 MeV contributes to at CMB; this can be matched in the current setup if the sterile neutrino annihilation to is efficient or if sterile neutrinos decay dominantly to . A particularly interesting variation occurs if decays after neutrino decoupling, resulting in a late injection of energetic neutrinos, particularly around the eV era, imparting an additional radiation energy density in the CMB Chacko et al. (2004).
Finally, if is sufficiently long-lived and heavy, it can also account for part or all of DM. The phenomenology in this case is similar to that of the majoron Rothstein et al. (1993); Berezinsky and Valle (1993); Gu et al. (2010); Queiroz and Sinha (2014); Frigerio et al. (2011); Lattanzi et al. (2014); Boucenna et al. (2014); Boulebnane et al. (2017); Heeck and Teresi (2017), with neutrino lines as an interesting signal Garcia-Cely and Heeck (2017).
We studied phenomenological implications of a pseudo-Goldstone boson associated with a spontaneously broken global symmetry in a light (GeV scale) sterile neutrino sector. The presence of sterile neutrinos and at similar mass scales gives rise to many rich possibilities for cosmology, DM, and direct searches. Primary among these are novel sterile neutrino DM production mechanisms from -mediated scattering or decay. can also provide new decay channels for heavy sterile neutrinos, which can alleviate BBN bounds and suppress traditional search channels at direct search experiments, or for DM, which can provide distinct signals at neutrino detectors. Likewise, can contribute measurably to dark radiation at BBN or CMB, inject a late population of SM neutrinos from its late decays, or account for DM.
We have only touched upon various interesting phenomenological possibilities in this framework, and several directions could be worthy of further detailed study. A late decay of into energetic neutrinos could lead to interesting CMB observables. It would also be interesting to study how the presence of affects a proper realization of leptogenesis Asaka et al. (2005); Asaka and Shaposhnikov (2005); Asaka et al. (2007); Josse-Michaux and Molinaro (2011); Gu and Sarkar (2011). Likewise, a more careful study of the flavor structure and mixing angles from the hidden sector interpretation could reveal interesting differences from the canonical seesaw mechanism. We leave these and other intriguing aspects for future work.
Acknowledgements: The authors are supported in part by the DoE under grants DE-SC0007859 and DE-SC0011719. BS acknowledges support from the University of Cincinnati and thanks the CERN and DESY theory groups, where part of this work was conducted, for hospitality. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293.
- Such operators have been studied in the context of supersymmetry Cleaver et al. (1998); Langacker (1998); Arkani-Hamed et al. (2001, 2000); Wells (2005), including the freeze-in production of sterile neutrino DM Roland et al. (2015a, b); Roland and Shakya (2016).
- For recent studies of right-handed neutrinos acting as portals to a hidden/dark sector, see Falkowski et al. (2009, 2011); Pospelov (2011); Pospelov and Pradler (2012); Cherry et al. (2014); Berryman et al. (2017); Batell et al. (2017); Schmaltz and Weiner (2017).
- This setup holds similarities with extended seesaw models Chun et al. (1995); Ma and Roy (1995); Zhang (2012); Barry et al. (2011); Boulebnane et al. (2017), which also employ a seesaw suppression for sterile neutrino masses to naturally accommodate an eV scale sterile neutrino.
- An explicit breaking Goldstone mass term is also possible. A small mass is also generated from the Yukawa coupling Frigerio et al. (2011), but is negligible for the parameters we are interested in.
- P. Minkowski, Phys. Lett. 67B, 421 (1977).
- R. N. Mohapatra and G. Senjanovic, Phys. Rev. D23, 165 (1981).
- T. Yanagida, Prog. Theor. Phys. 64, 1103 (1980).
- M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. C790927, 315 (1979), eprint 1306.4669.
- J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227 (1980).
- T. Asaka, S. Blanchet, and M. Shaposhnikov, Phys. Lett. B631, 151 (2005), eprint hep-ph/0503065.
- T. Asaka and M. Shaposhnikov, Phys. Lett. B620, 17 (2005), eprint hep-ph/0505013.
- T. Asaka, M. Laine, and M. Shaposhnikov, JHEP 01, 091 (2007), [Erratum: JHEP02,028(2015)], eprint hep-ph/0612182.
- K. N. Abazajian et al. (2012), eprint 1204.5379.
- M. Drewes et al., JCAP 1701, 025 (2017), eprint 1602.04816.
- E. Ma, Mod. Phys. Lett. A32, 1730007 (2017), eprint 1702.03281.
- T. Alanne, A. Meroni, and K. Tuominen, Phys. Rev. D96, 095015 (2017), eprint 1706.10128.
- J. Sayre, S. Wiesenfeldt, and S. Willenbrock, Phys. Rev. D72, 015001 (2005), eprint hep-ph/0504198.
- Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, Phys. Lett. 98B, 265 (1981).
- G. B. Gelmini and M. Roncadelli, Phys. Lett. 99B, 411 (1981).
- H. M. Georgi, S. L. Glashow, and S. Nussinov, Nucl. Phys. B193, 297 (1981).
- J. Schechter and J. W. F. Valle, Phys. Rev. D25, 774 (1982).
- G. B. Gelmini, S. Nussinov, and M. Roncadelli, Nucl. Phys. B209, 157 (1982).
- M. Lindner, D. Schmidt, and T. Schwetz, Phys. Lett. B705, 324 (2011), eprint 1105.4626.
- M. Escudero, N. Rius, and V. Sanz, JHEP 02, 045 (2017), eprint 1606.01258.
- A. Maiezza, M. NemevÅ¡ek, and F. Nesti, Phys. Rev. Lett. 115, 081802 (2015), eprint 1503.06834.
- M. NemevÅ¡ek, F. Nesti, and J. C. Vasquez, JHEP 04, 114 (2017), eprint 1612.06840.
- P. S. B. Dev, R. N. Mohapatra, and Y. Zhang, Nucl. Phys. B923, 179 (2017), eprint 1703.02471.
- R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
- J. Heeck and D. Teresi, Phys. Rev. D94, 095024 (2016), eprint 1609.03594.
- T. Asaka, M. Shaposhnikov, and A. Kusenko, Phys. Lett. B638, 401 (2006), eprint hep-ph/0602150.
- G. Cleaver, M. Cvetic, J. R. Espinosa, L. L. Everett, and P. Langacker, Phys. Rev. D57, 2701 (1998), eprint hep-ph/9705391.
- P. Langacker, Phys. Rev. D58, 093017 (1998), eprint hep-ph/9805281.
- N. Arkani-Hamed, L. J. Hall, H. Murayama, D. Tucker-Smith, and N. Weiner, Phys. Rev. D64, 115011 (2001), eprint hep-ph/0006312.
- N. Arkani-Hamed, L. J. Hall, H. Murayama, D. Tucker-Smith, and N. Weiner (2000), eprint hep-ph/0007001.
- J. D. Wells, Phys. Rev. D71, 015013 (2005), eprint hep-ph/0411041.
- S. B. Roland, B. Shakya, and J. D. Wells, Phys. Rev. D92, 113009 (2015a), eprint 1412.4791.
- S. B. Roland, B. Shakya, and J. D. Wells, Phys. Rev. D92, 095018 (2015b), eprint 1506.08195.
- S. B. Roland and B. Shakya (2016), eprint 1609.06739.
- A. Falkowski, J. Juknevich, and J. Shelton (2009), eprint 0908.1790.
- A. Falkowski, J. T. Ruderman, and T. Volansky, JHEP 05, 106 (2011), eprint 1101.4936.
- M. Pospelov, Phys. Rev. D84, 085008 (2011), eprint 1103.3261.
- M. Pospelov and J. Pradler, Phys. Rev. D85, 113016 (2012), [Erratum: Phys. Rev.D88,no.3,039904(2013)], eprint 1203.0545.
- J. F. Cherry, A. Friedland, and I. M. Shoemaker (2014), eprint 1411.1071.
- J. M. Berryman, A. de GouvÃªa, K. J. Kelly, and Y. Zhang, Phys. Rev. D96, 075010 (2017), eprint 1706.02722.
- B. Batell, T. Han, D. McKeen, and B. Shams Es Haghi (2017), eprint 1709.07001.
- M. Schmaltz and N. Weiner (2017), eprint 1709.09164.
- E. J. Chun, A. S. Joshipura, and A. Yu. Smirnov, Phys. Lett. B357, 608 (1995), eprint hep-ph/9505275.
- E. Ma and P. Roy, Phys. Rev. D52, R4780 (1995), eprint hep-ph/9504342.
- H. Zhang, Phys. Lett. B714, 262 (2012), eprint 1110.6838.
- J. Barry, W. Rodejohann, and H. Zhang, JHEP 07, 091 (2011), eprint 1105.3911.
- S. Boulebnane, J. Heeck, A. Nguyen, and D. Teresi (2017), eprint 1709.07283.
- I. Z. Rothstein, K. S. Babu, and D. Seckel, Nucl. Phys. B403, 725 (1993), eprint hep-ph/9301213.
- E. K. Akhmedov, Z. G. Berezhiani, R. N. Mohapatra, and G. Senjanovic, Phys. Lett. B299, 90 (1993), eprint hep-ph/9209285.
- M. Frigerio, T. Hambye, and E. Masso, Phys. Rev. X1, 021026 (2011), eprint 1107.4564.
- G. B. Gelmini, S. Nussinov, and T. Yanagida, Nucl. Phys. B219, 31 (1983).
- G. Jungman and M. A. Luty, Nucl. Phys. B361, 24 (1991).
- A. Pilaftsis, Phys. Rev. D49, 2398 (1994), eprint hep-ph/9308258.
- C. Garcia-Cely and J. Heeck, JHEP 05, 102 (2017), eprint 1701.07209.
- E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein, et al., Astrophys.J. 789, 13 (2014), eprint 1402.2301.
- A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, and J. Franse, Phys.Rev.Lett. 113, 251301 (2014), eprint 1402.4119.
- V. Berezinsky and J. W. F. Valle, Phys. Lett. B318, 360 (1993), eprint hep-ph/9309214.
- P.-H. Gu, E. Ma, and U. Sarkar, Phys. Lett. B690, 145 (2010), eprint 1004.1919.
- F. S. Queiroz and K. Sinha, Phys. Lett. B735, 69 (2014), eprint 1404.1400.
- M. Lattanzi, R. A. Lineros, and M. Taoso, New J. Phys. 16, 125012 (2014), eprint 1406.0004.
- S. M. Boucenna, S. Morisi, Q. Shafi, and J. W. F. Valle, Phys. Rev. D90, 055023 (2014), eprint 1404.3198.
- J. Heeck and D. Teresi, Phys. Rev. D96, 035018 (2017), eprint 1706.09909.
- K. Choi and A. Santamaria, Phys. Rev. D42, 293 (1990).
- S. Pastor, S. D. Rindani, and J. W. F. Valle, JHEP 05, 012 (1999), eprint hep-ph/9705394.
- M. Kachelriess, R. Tomas, and J. W. F. Valle, Phys. Rev. D62, 023004 (2000), eprint hep-ph/0001039.
- M. Hirsch, A. Vicente, J. Meyer, and W. Porod, Phys. Rev. D79, 055023 (2009), [Erratum: Phys. Rev.D79,079901(2009)], eprint 0902.0525.
- X. Garcia i Tormo, D. Bryman, A. Czarnecki, and M. Dowling, Phys. Rev. D84, 113010 (2011), eprint 1110.2874.
- R. J. Scherrer and M. S. Turner, Phys. Rev. D31, 681 (1985).
- F. Bezrukov, H. Hettmansperger, and M. Lindner, Phys. Rev. D81, 085032 (2010), eprint 0912.4415.
- C. Garcia-Cely, A. Ibarra, and E. Molinaro, JCAP 1402, 032 (2014), eprint 1312.3578.
- P.-H. Gu and U. Sarkar, Eur. Phys. J. C71, 1560 (2011), eprint 0909.5468.
- D. J. H. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D60, 063504 (1999), eprint hep-ph/9809453.
- L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, JHEP 03, 080 (2010), eprint 0911.1120.
- Z. Chacko, L. J. Hall, T. Okui, and S. J. Oliver, Phys. Rev. D70, 085008 (2004), eprint hep-ph/0312267.
- S. Dodelson and L. M. Widrow, Phys. Rev. Lett. 72, 17 (1994), eprint hep-ph/9303287.
- A. Boyarsky, A. Neronov, O. Ruchayskiy, M. Shaposhnikov, and I. Tkachev, Phys. Rev. Lett. 97, 261302 (2006a), eprint astro-ph/0603660.
- A. Boyarsky, J. Nevalainen, and O. Ruchayskiy, Astron. Astrophys. 471, 51 (2007), eprint astro-ph/0610961.
- A. Boyarsky, A. Neronov, O. Ruchayskiy, and M. Shaposhnikov, Mon. Not. Roy. Astron. Soc. 370, 213 (2006b), eprint astro-ph/0512509.
- A. Boyarsky, D. Iakubovskyi, O. Ruchayskiy, and V. Savchenko, Mon. Not. Roy. Astron. Soc. 387, 1361 (2008a), eprint 0709.2301.
- A. Boyarsky, D. Malyshev, A. Neronov, and O. Ruchayskiy, Mon. Not. Roy. Astron. Soc. 387, 1345 (2008b), eprint 0710.4922.
- U. Seljak, A. Makarov, P. McDonald, and H. Trac, Phys. Rev. Lett. 97, 191303 (2006), eprint astro-ph/0602430.
- A. Boyarsky, J. Lesgourgues, O. Ruchayskiy, and M. Viel, JCAP 0905, 012 (2009), eprint 0812.0010.
- S. Horiuchi, P. J. Humphrey, J. Onorbe, K. N. Abazajian, M. Kaplinghat, and S. Garrison-Kimmel, Phys. Rev. D89, 025017 (2014), eprint 1311.0282.
- B. Shakya, Mod. Phys. Lett. A31, 1630005 (2016), eprint 1512.02751.
- B. Shakya and J. D. Wells, Phys. Rev. D96, 031702 (2017), eprint 1611.01517.
- A. Merle, V. Niro, and D. Schmidt, JCAP 1403, 028 (2014), eprint 1306.3996.
- A. Adulpravitchai and M. A. Schmidt, JHEP 01, 006 (2015), eprint 1409.4330.
- Z. Kang, Eur. Phys. J. C75, 471 (2015), eprint 1411.2773.
- A. Merle and M. Totzauer, JCAP 1506, 011 (2015), eprint 1502.01011.
- D. Gorbunov and M. Shaposhnikov, JHEP 10, 015 (2007), [Erratum: JHEP11,101(2013)], eprint 0705.1729.
- C. Adams et al. (LBNE) (2013), eprint 1307.7335.
- R. Jacobsson (SHiP Collaboration), Tech. Rep. CERN-SHiP-PROC-2016-007, CERN, Geneva (2016), URL https://cds.cern.ch/record/2130433.
- A. Kusenko, Phys. Rept. 481, 1 (2009), eprint 0906.2968.
- P. Hernandez, M. Kekic, and J. Lopez-Pavon, Phys. Rev. D90, 065033 (2014), eprint 1406.2961.
- A. C. Vincent, E. F. Martinez, P. HernÃ¡ndez, M. Lattanzi, and O. Mena, JCAP 1504, 006 (2015), eprint 1408.1956.
- Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, Phys. Rev. Lett. 45, 1926 (1980).
- C. Garcia-Cely, A. Ibarra, and E. Molinaro, JCAP 1311, 061 (2013), eprint 1310.6256.
- S. Weinberg, Phys. Rev. Lett. 110, 241301 (2013), eprint 1305.1971.
- F.-X. Josse-Michaux and E. Molinaro, Phys. Rev. D84, 125021 (2011), eprint 1108.0482.