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. This framework involves 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.


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of the pseudo-Goldstone boson η of this broken symmetry. GeV scale sterile neutrinos can equilibrate with the thermal bath and dominate the energy density of the Universe before big bang nucleosynthesis (BBN) [27] -their interplay with η can therefore give rise to novel cosmological scenarios. The η 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, which can enable several new possibilities for cosmology, dark matter, and direct searches that are not possible in the majoron framework.

Charged-singlet seesaws
The canonical seesaw mechanism involves three SM-singlet, right-handed neutrinos N i , with: A global or gauged U(1) lepton or U(1) B−L symmetry for N i [15][16][17][18][19][20][21] precludes the Majorana mass term; the lagrangian is instead A vev for the exotic Higgs field φ, appropriately charged under the lepton or B − L symmetry, breaks the symmetry and produces sterile neutrino masses M i ∼ x φ . If the symmetry is global, a physical light degree of freedom, the Goldstone boson, known as the majoron, emerges [15,16].
In this paper, we consider instead a global symmetry, for instance a U(1) , that is confined to the sterile neutrinos and does not extend to any SM field. Such a symmetry forbids both terms in eq. (2.1). However, a scalar field φ carrying the opposite U(1) charge to N i enables the higher dimensional operator 1 Λ LhN φ, where Λ is a UV-cutoff scale. 1 A φ vev breaks the U(1) and produces the Yukawa interaction term from eq. (2.1) with the effective Yukawa coupling y ∼ λ 1 φ /Λ; thus such an operator also provides a natural explanation for the tiny Yukawas in terms of the hierarchy between the two scales φ and Λ. Next, we discuss a UV completion of this setup in terms of singlet fermions from a hidden sector that couple to heavy right-handed seesaw neutrinos.
2.1 "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 M , possibly close to the GUT scale) right-handed neutrinos that couple to SM neutrinos through Yukawa

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terms y ij L i hN j . If the N j also act as portals to a hidden sector, 2 this invokes the generic prospect of an analogous Yukawa term y ij L i h N j , where L i h is a singlet combination of hidden sector fields analogous to L i h. Integrating out the N i produces the following dimension-5 operators connecting the visible and hidden sectors: 3 In the above we have ignored flavor structure and dropped indices for simplicity, assuming all y ij (y ij ) are roughly the same, so that the above terms should only be taken as approximate. If the hidden sector scalar acquires a vev v , the above can be rewritten as where we have defined Λ −1 eff ≡ y 2 /M , y eff ≡ yy v /M , and M eff ≡ y 2 v 2 /M . Here, the first term accounts for the active neutrino masses y 2 v 2 /M from the primary seesaw involving integrating out the pure singlet neutrinos N i . The latter two terms give a similar contribution to the active neutrino masses from the secondary seesaw resulting from integrating out the L i fermions (note the analogy between eq. (2.4) and eq. (2.1)).
The mixing angle between the active neutrinos and these hidden sector singlets L is approximately which is 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, with symmetries unrelated to the SM, and themselves obtain light masses via the seesaw mechanism. 4 We will henceforth ignore the integrated out "true" right-handed seesaw neutrinos and work with the effective field theory (EFT) in eq. (2.4), switching the notation N i to refer to these light sterile states L , whose phenomenology we will pursue in this paper.

Pseudo-Goldstone boson
The spontaneous breaking of the global U(1) by φ ≡ f gives rise to a massless Goldstone boson, which we will call the η-boson. It is conjectured that non-perturbative gravitational effects explicitly break global symmetries, leading to a pseudo-Goldstone boson mass of order m 2 η ∼ f 3 /M P l via an operator of the form φ 5 M P l [49,50]. 5 For generality, we treat m η as a free parameter, but this approximate mass scale should be kept in mind.

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Next, we draw the distinction between the η-boson and the more familiar majoron [15][16][17][18][19][20][21]. For both, 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 [11,19,[52][53][54][55] are also applicable to η. However, the majoron is associated with the breaking of lepton number -a symmetry shared by the SM leptons as well as the sterile neutrinosand 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. Furthermore, this scaling leads to specific relations between majoron couplings and sterile neutrino masses, which drives many of the constraints on majorons [11,19,[52][53][54][55].
In contrast, these energy scales are distinct in the η framework: the symmetry breaking scale f (i.e., the scale of U(1) breaking) is independent of the breaking of lepton number (at the much higher real seesaw scale M ) and is also distinct from the sterile neutrino mass scale (M eff ∼ f 2 /M ), which, as discussed above, is suppressed by a seesaw mechanism. The ability to vary them independently opens up phenomenologically interesting regions of parameter space. Furthermore, the sterile neutrino masses ; this coincidence of mass scales can carry important implications for cosmology and DM, as we will see later.

Framework and phenomenology
We focus on the low-energy effective theory containing three sterile neutrinos (which we have reset to the label N i rather than L ), and the pseudo-Goldstone boson η. We treat m N i , f , and m η as independent parameters. We assume m N i ∼ GeV scale, and y ij are correspondingly small in a natural way that matches the measured ∆m 2 ν and mixings among the light active neutrinos. We will consider the interesting and widely studied possibility that the lightest sterile neutrino N 1 is DM, which is especially appealing given recent claims of a 3.5 keV X-ray line [56,57] compatible with decays of a 7 keV sterile neutrino DM particle. We also assume f v; the U(1) breaking singlet scalar is then decoupled and irrelevant for phenomenology.
Lifetime. The η lifetime is controlled by decay rates into (both active and sterile) neutrinos. For instance, where m ν ∼ 0.1 eV is the active neutrino mass scale. For the decay channels η → N i ν and η → N i N i involving the sterile neutrinos, m ν is replaced by √ m N i m ν and m N i respectively. A pseudo-Goldstone coupling to neutrinos faces several constraints [64][65][66][67][68]. 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 N 2,3 (but not the DM candidate N 1 , which has suppressed couplings to neutrinos) are in equilibrium with the thermal bath due to their mixing with active neutrinos, decouple while relativistic at T ∼ 20 GeV [27], can grow to dominate the energy density of the Universe, and decay before BBN [27,69,70].
η couples appreciably only to the sterile neutrinos, and is produced via sterile neutrino annihilation N i N i → ηη (see figure 2 (a)) or decay (if kinematically open). The annihilation process, despite p-wave suppression, is efficient at high temperatures T m N 2,3 . The magnitude of f for such annihilations to be rapid can be estimated by comparing the annihilation cross section [71,72] with the Hubble rate at T ∼ m N 2,3 For m N 2,3 ∼ GeV, this process is efficient for f 10 5 GeV, and produces an η abundance comparable to the N 2,3 abundance. For f > 10 5 GeV, the annihilation process is feeble, and a small η abundance will accumulate via the freeze-in process instead [73,74].

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Dark Matter production. η can also mediate N i N i → N j N j interactions between the sterile neutrinos ( figure 2 (b)), which enables a novel DM production mechanism N i N i → N 1 N 1 . One can analogously estimate the scale f below which this process [75] is efficient: f ∼ √ m N 1 (M P l m N 2,3 ) 1/4 . This would generate an N 1 abundance comparable to relativistic freezeout, which generally overcloses the Universe, hence this scenario is best avoided. Likewise, η decays can also produce DM if m η > 2m N 1 . By comparing rates, we find that production from such decays dominates over the annihilation process provided m η > m 3 N 2,3 /f 2 , which generally holds over most of our parameter space. Additional DM production processes, such as η annihilation and N 2,3 decays via an off-shell η, are always subdominant and therefore neglected. The novel production processes discussed here do not rely on N 1 mixing with active neutrinos, which is particularly appealing since this canonical (Dodelson-Widrow) production mechanism [76] is now ruled out by various constraints [9,[77][78][79][80][81][82][83][84].
Next, we discuss various cosmological histories that are possible within this framework. Our purpose is not to provide a comprehensive survey of all possibilities, but simply to highlight some novel and interesting features that can be realized. Since available decay channels and lifetimes are crucial to the cosmological history, we find it useful to organize our discussion into the following three different regimes.
Heavy regime: m η > m N i . All η decay channels to sterile neutrinos are open, and η decays rapidly, long before BBN. If N i N i → ηη is rapid, η maintains an equilibrium distribution at T m η , and the decay η → N 1 N 1 generates a freeze-in abundance of N 1 , estimated to be [33,35,74,[85][86][87][88][89][90] The observed DM abundance is produced, for instance, with f ∼ 10 5 GeV, m η ∼ 10 GeV, and m N 1 ∼ 10 keV. If the N i N i → ηη annihilation process is feeble, a freeze-in abundance of η is generated instead, and its decays produce a small abundance of N 1 . The N 1 yield is suppressed by . The resulting abundance is much smaller than Y eq from eq. (3.3) and cannot account for all of DM unless m N 1 ∼ m N 2 ,N 3 .
Intermediate regime: m N 2,3 > m η > m N 1 . In addition to annihilation processes, η can now also be produced directly from heavy sterile neutrino decay. Ignoring phase space suppression, the decay rate is If sufficiently large, this exotic decay channel can compete with the standard sterile neutrino decay channels induced by active-sterile mixing [91]. In figure 3, we plot (blue curve) the scale f below which this channel dominates (assuming standard seesaw relations). In this region, the traditionally searched-for decay modes are suppressed, rendering the sterile neutrinos invisible at detectors such as at DUNE [92] and SHiP [93] (unless N 1 also decays in the detector, as can occur if it is not DM). N 2,3 are generally required to decay before BBN due to constraints from several recombination era observables [94][95][96], necessitating τ N 2,N 3 1 s and consequently m N 2,N 3 O(100) MeV in the standard seesaw formalism. The new decay channel N i → ην, if dominant, can reduce the sterile neutrino lifetime, allowing lighter masses to be compatible with BBN. In figure 3, the red dashed line shows the scale f below which the sterile neutrino decays before BBN. For f 10 6 GeV, even lighter (MeV scale) sterile neutrinos are compatible with the seesaw as well as BBN constraints, in stark contrast to the standard seesaw requirements.
Depending on parameters, η can decay before or after BBN (figure 1), but its dominant decay channel is to the DM candidate η → N 1 N 1 . If N 2,3 decay dominantly into η, or if N 1 thermalizes with N 2,3 , the N 1 relic density is overabundant for DM. Viable regions of parameter space instead involve a small fraction of N 2,3 decaying into η, which subsequently decays to N 1 . In this case, N 1 accounts for the observed DM abundance (for m N 2,3 = 1 GeV) for f ≈ 10 9 GeV m N 1 GeV . For instance, m N 1 = 7 keV requires f ∼ 10 6 GeV. Here, DM (N 1 ) is produced from late decays of heavier particles (η and N 2,3 ) and can be warm. Such late production of warm DM can carry interesting cosmological signatures and structure formation implications, which lie beyond the scope of this paper.
Light regime: m N i > m η > m ν . All sterile neutrinos can now decay into η. In particular, a new, very long-lived DM decay channel N 1 → ην emerges. Since η 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 N 1 → γν decay channel, this has no gamma ray counterpart.
Unlike previous scenarios, η is extremely long-lived, and if sufficiently light, can contribute measurably to dark radiation at BBN or CMB [71,97,98]. A Goldstone that freezes JHEP02(2019)174 out above 100 MeV contributes ∼ 0.39 to N eff at CMB [99]; this is the case if the sterile neutrino annihilation to η is efficient or if sterile neutrinos decay dominantly to η. If η decays after neutrino decoupling, neutrinos from its decays provide additional radiation energy density in the CMB [75].
Finally, if η 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 [48,49,51,[58][59][60][61][62][63], with neutrino lines as an interesting signal [55].

Discussion
We studied the phenomenology of a pseudo-Goldstone boson η associated with a spontaneously broken global symmetry in a light (GeV scale) sterile neutrino sector. The presence of sterile neutrinos and η at similar mass scales gives rise to several novel possibilities for cosmology, DM, and direct searches. Primary among these are novel sterile neutrino DM production mechanisms from η-mediated scattering or decay, and new decay channels for heavy sterile neutrinos, which can alleviate BBN bounds and suppress standard search channels at direct search experiments, or provide distinct DM signals at neutrino detectors. Likewise, η 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 a few interesting phenomenological possibilities in this framework, and several directions, such as the effect of η on leptogenesis [7-9, 72, 100], or differences in the flavor structure and mixing angles from the hidden sector interpretation compared to the canonical seesaw mechanism, could be worthy of further detailed study.